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Silicon clusters Problems challenges and perspectives

VIEWS: 51 PAGES: 149

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									Journal of Computational Methods in Sciences and Engineering 7 (2007) 183                                          183
IOS Press



Editorial

Silicon clusters: Problems, challenges and
perspectives

George Maroulisa and Aristides Zdetsisb
a Department of Chemistry, University of Patras, Greece
E-mail: maroulis@upatras.gr
b Department of Physics, University of Patras, Greece

E-mail: zdetsis@upatras.gr


  The chemistry of silicon has shown dramatic expansion in recent years [1]. This is easily explained by
the importance of this chemical element in fields with high-level potential technological applications [2].
Atomic clusters form a link between molecules and solids with novel properties found neither in molecules
nor in solids. As silicon is the most important semiconducting element for the microelectronics industry,
device fabrication, and atomic scale engineering, the research on the subject grew dramatically over
the last five-ten years. There is currently a strong interest in the prospect of producing new materials
consisting of small atomic clusters, and in particular silicon clusters. Such cluster-assembled materials
may vary significantly from their crystalline counterparts, with different (and more useful) mechanical,
electronic, optical and other properties. In this volume we have collected a number of important studies
on structural, electronic, optical, and other fundamental properties of silicon clusters. The present
collection includes pure and doped silicon clusters and nanoclusters, as well as metal embedded (or
adsorbed) silicon clusters, including also the inverse process of doping metal clusters by silicon, in
view of the current interest for studying the metal-semiconductor interface at atomic or nano level. All
contributors are eminent specialists in their respective fields. The material of this Special Issue will be
of interest not only to silicon clusters specialists, but also to more general scientific audiences active
in physical chemistry, chemical physics, materials science, nanoscience and nanotechnology related
research fields.


References

[1]   P. Jutzi and U. Schubert, Silicon Chemistry: from the Atom to Extended Systems, Wiley-VCH, Weinheim, 2003.
[2]   V. Kumar, Nanosilicon, Elsevier, London, 2007.




1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
Journal of Computational Methods in Sciences and Engineering 7 (2007) 185–193                                              185
IOS Press




Exploring the lowest-energy structures of
group IV tetra-aurides: XAu4 (X = C, Si,
Ge, Sn)
Rhitankar Pal, Satya Bulusu and X.C. Zeng∗
Department of Chemistry and Center for Materials and Nanoscience, University of Nebraska-Lincoln,
Lincoln, NE 68588, USA

Received 3 August 2007
Accepted 1 September 2007

Abstract. We search for the lowest-energy structure of XAu4 (X = C, Si, Ge, Sn) clusters and calculate the photoelectron
spectrum for the lowest-energy isomer of each species. The gradual destabilization of the tetrahedral geometry from C → Si →
Ge → Sn, as well as the structural transition from the tetrahedral to the square planar geometry from SiAu4 to GeAu4 , indicate
that the change of electronic properties in the group IV elements also plays an important role in the structures of Group IV
tetra-aurides in addition to the known H/Au analogy in these species.


1. Introduction
   The structure and stability of SiAun and their close resemblance to SiH n (n = 1–4) were first
established through ab initio calculation and photoelectron spectroscopy measurement by Kiran et al. [1].
                                 −1/0        −1/0             −1/0/+1
Subsequent work on the Si 2 Au4 , Si2 Au2         and Si3 Au3          clusters [2,3] further confirmed the
structural similarity with the corresponding silanes as well as the Au/H analogy in these clusters. The
Au/H analogy has also been recognized from structural similarity and stability of certain gold/boron
clusters [4], observing one-to-one correspondence to boranes whose structures are well known. The
analogy of an Au atom to an H atom was attributed to strong relativistic effect of Au [5–9].
   The aim of this work is to explore whether the Au/H analogy can be extended to tetra-aurides with
other group IV elements. To this end, we performed a global-minimum search for the lowest-energy
structure of the XAu4 species (X = C, Si, Ge, Sn) using density-functional theory (DFT) methods.
It is known that the group IV hydrides XH 4 (X = C, Si, Ge, Sn) all exhibit the tetrahedral structure
although their ionized counterparts can deviate from the tetrahedral structure due to a symmetry breaking
mechanism [10]. We also investigated structures of anion XAu 4 species and how their structures
are related to the corresponding tetrahydride analogs. For SiAu 4 , our results of the simulated anion
photoelectron spectrum are in excellent agreement with those obtained by Kiran et al. [2]. For XH 4 (X
= Ge, Sn), we found their lowest-energy structures are very different from SiAu 4 . We found a structural
transition from the tetrahedral to the square pyramidal (or to a C 2v structure for the anion) from SiAu4
to GeAu4 . The tetrahedral isomers of GeAu 4 and SnAu4 have very high energy. Another isomer with
C3v symmetry for all the species was identified as the third low-energy structure.

  ∗
      Corresponding author. E-mail: xczeng@phase2.unl.edu.


1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
186     R. Pal et al. / Exploring the lowest-energy structures of group IV tetra-aurides: XAu4 (X = C, Si, Ge, Sn)

2. Theoretical method

   We employed the basin-hopping (BH) global optimization technique combined with DFT calculation
to search for the lowest-energy clusters [11]. The BH method essentially converts the potential energy
           ˜                                                                     ˜
surface (E ) of a cluster to a multidimensional “staircase” via the mapping E (X) = min{E (X)}, where
X denotes the nuclear coordinates of the cluster and “min” refers to the energy minimization performed
starting from X [12]. Specifically, the canonical Monte Carlo (MC) sampling method was used to sample
                                              ˜
the transformed potential energy surface E at a fixed temperature. With each MC move, coordinates
of all atoms are randomly displaced, followed by a geometry optimization based on a DFT method
within the generalized gradient approximation (GGA) in the Perdue-Burke-Ezerhof (PBE) form [13].
The DMol3 program [14] was utilized for the geometry optimization coupled with the BH search.
   The unbiased global search was performed for all the clusters XAu 4 (X = C, Si, Ge, Sn). Despite
marked differences in initial structures (randomly generated) for a given cluster, the BH-DFT search
consistently yields the same lowest-energy structure, typically, within 200 MC trial moves. Among the
low-lying isomers, those having energy values within 1.0 eV from the lowest-lying isomer were collected.
Electronic energies and relative energies of these low-energy isomers (with respect to the lowest-energy
isomer) were further evaluated using the PBE-DFT method with the cc-pVTZ [for C, Si and Ge] [15] and
SDD [for Au and Sn] basis sets [16] implemented in the GAUSSIAN 03 software package (Table 1) [17].
Vibrational frequency calculation was also carried out for all the optimized clusters to assure the absence
of imaginary frequencies. Finally, simulated anion photoelectron spectrum for the lowest energy isomer
was obtained, where the first vertical detachment energy (VDE) was calculated as the energy difference
of the neutral cluster (at the geometry of the anion) with respective to the anion counterpart. The binding
energies of the deeper energy levels were then added to the first VDE to give the electronic density of
states of the cluster. Each line of the energy levels was fitted with a Gaussian width of 0.04 eV to give the
simulated photoelectron spectrum. Finally the charge partitioning was calculated by using the Hirshfeld
method implemented in DMol3 program.


3. Results

3.1. Structures and stabilities of XAu 4 (X = C, Si, Ge, Sn)

3.1.1. CAu4
   The BH search gives rise to three low-energy isomers for CAu 4 , among which the isomer 1 has the
lowest energy while isomer 2 and 3 are a few tenth eV higher in energy (Table 1). Isomer 1 is highly
symmetric (Td ) with all bond lengths and bond angles being equal (Fig. 1), and it is resemblance to the
structure of a methane molecule. The structure of isomer 2 also has high symmetry (C 3v ) and the C-Au
bond lengths are very close to those of isomer 1. Since all the Au-Au bond lengths are equal, isomer
3 has square pyramid geometry, except two unequal C-Au bond lengths which make the geometry less
symmetric (C2v ) than the other two isomers. The relative energies among the three isomers show that
isomer 1 has much higher stability than isomer 2 and 3, and is expected to be the ground-state structure
for CAu4 .

3.1.2. SiAu4
  As in the case of CAu 4 , the BH search also yielded three low-energy structures for SiAu 4 . Isomer
1 has the lowest energy whereas isomers 2 and 3 are about two tenth eV higher in energy (Table 1).
          R. Pal et al. / Exploring the lowest-energy structures of group IV tetra-aurides: XAu4 (X = C, Si, Ge, Sn)      187




                                                        2.81A
                                                                                                  2.10A 2.73A
                                                     1.97A
                                                                                      2.13A

                                     1.99A


                             109.5




                          ∆E=0.00 eV                     ∆E=0.43 eV                      ∆E=0.76 eV
                              Td                                C3v                           C2v

Fig. 1. Optimized structures of three low-energy isomers of CAu4 . The relative energies and point-group symmetries are given
underneath each isomer structure.



                                                                                          2.79A
                                   2.33A               2.77A
                                                                                      2.34A
                                                        2.44A

                                     109.5




                          ∆E=0.00 eV                         ∆E=0.21 eV                  ∆E=0.24 eV
                              Td                                C4v                           C3v

Fig. 2. Optimized structures of three low-energy isomers of SiAu4 . The relative energies and point-group symmetries are given
underneath each isomer structure.

As such, isomers 2 and 3 are expected to be quite stable, if not the most stable one. Similar to
CAu4 , the lowest-energy structure of SiAu 4 (i.e. isomer 1) has a perfect tetrahedral geometry with
all the bond angles and bond lengths being equal (Fig. 2). This has already been shown through a
combined experimental/theoretical study by Kiran et al. [1] These authors have also noticed that the
square pyramidal (C4v ) isomer is higher in energy, which is analogous to the isomer 2 in this study. A
new low-energy isomer obtained from our BH search is the isomer 3 with energy close to that of the
isomer 2. Isomer 3 has a unique propeller like structure (C 3v symmetry) with all the Si-Au and Au-Au
bonds being nearly the same in length.

3.1.3. GeAu4
   In the case of GeAu 4, unlike its silicon and carbon counterparts, we found that the tetrahedral isomer
is no longer the most stable. The lowest-energy structure of GeAu 4 is a square pyramid (isomer 1) with
the Ge atom at the apex and four Au atoms forming the base (Fig. 3). It appears that from CAu 4 →
SiAu4 → GeAu4, there is a structural transition from the tetrahedral (in the case of CAu 4 and SiAu4 ) to
the square pyramidal geometry (in the case of GeAu 4). The square pyramidal structure has equal Au-Au
and Ge-Au bond lengths, respectively. The anion of isomer 1 has C 2v symmetry with all the Au-Au
bonds being equal and two types of diagonal Ge-Au bonds. Two other low-energy structures of this
188      R. Pal et al. / Exploring the lowest-energy structures of group IV tetra-aurides: XAu4 (X = C, Si, Ge, Sn)



                     2.78A                               2.76A

                                                     2.46A                                        2.37A
                      2.54A

                                                                                                      109.5




                       ∆E=0.00 eV                       ∆E=0.16 eV                      ∆E=0.52 eV
                              C4v                            C3v                             Td

Fig. 3. Optimized structures of three low-energy isomers of GeAu4 . The relative energies and point-group symmetries are
given underneath each isomer structure.

species have C 3v and Td symmetry, respectively, similar to the case of CAu 4 and SiAu4 . The difference
is in the relative energies from the corresponding lowest-energy structure. Here the C 3v isomer is closer
in energy to the square pyramid isomer while the tetrahedral isomer has a higher energy (Table 1).

3.1.4. SnAu4
   The low-energy isomer structures obtained in the case of SnAu 4 are nearly the same as in the case of
GeAu4. Again, we found three low-energy isomers with the square pyramid one being the lowest-energy
structure and the other two isomers having the C 3v and Td symmetry, respectively (Fig. 4). The square
pyramid isomer 1 has the Sn atom at the apex and four Au atoms as the base with equal Sn-Au and
Au-Au bond lengths, respectively. The C 3v isomer has a propeller like geometry and is closer in energy
to the isomer 1 while the Td isomer has a higher energy.

3.2. HOMO of XAu4 (X = C, Si, Ge, Sn)

  The highest occupied molecular orbital (HOMO) of the lowest-energy structure of each species can
provide valuable information about the electron distribution and charge transfer from the metal Au
atoms to the group-IV atom. Figure 5(a) shows the HOMO diagrams of XAu 4 (X = C, Si, Ge, Sn).
Interestingly, although the structure of GeAu 4 and SnAu4 are identical, their molecular orbital (MO)
diagrams are quite different [Fig. 5(b)]. However, the MO diagrams of CAu 4 and SiAu4 are similar.
Both CAu4 and SiAu4 have a t2 (triply degenerate) HOMO, while the HOMO of GeAu4 and SnAu4 is
doubly degenerate (e). Although the degeneracy of the HOMO for CAu 4 and SiAu4 is same, the extent
of charge transfer is high in the case of CAu 4 since all the four Au atoms contribute to the HOMO. In the
case of SiAu4 there is a node passing through the two Au and Si atoms, making less contribution from
the two Au atoms to the HOMO. In the cases of GeAu 4 and SnAu4 their HOMO is largely contributed
by two Au atoms. There is a node passing through the two Au and Ge atoms for GeAu 4, whereas the
same node passes between the two Au atoms for SnAu 4. From C to Sn in the group IV, the metallic
character increases, and the valence p orbital has a higher and higher energy level [Fig. 5(b)]. As a result,
the group IV atom contributes less and less to the HOMO of XAu 4 from X = C to Sn.

3.3. Hirsfeld charge analysis of XAu 4 (X = C, Si, Ge, Sn)

  Hirsfeld charge analysis was carried out for all the lowest-energy isomer of XAu 4 (X = C, Si, Ge,
Sn) to assess the magnitude of charge transfer from the Au atoms to the group IV atom. The calculation
          R. Pal et al. / Exploring the lowest-energy structures of group IV tetra-aurides: XAu4 (X = C, Si, Ge, Sn)      189

                                                            Table 1
                        Calculated electronic energies, relative energies with respective to the lowest-
                        energy isomer (relative energy in bold), and HOMO-LUMO gaps for three
                        low-energy isomers of XAu4 (X = C, Si, Ge, Sn). Bold faced energies mark
                        the lowest-energy isomer for each species
                         Cluster      Point      Energy (a.u.)      Relative energy   HOMO-LUMO
                         (isomer #)   group                              (eV)           gap (eV)
                         Au4 C (1)     Td       −581.1819381              0.00            2.53
                         Au4 C (2)     C3v      −581.1660237              0.43            1.83
                         Au4 C (3)     C4v      −581.1539868              0.76            1.78
                         Au4 Si (1)    Td       −832.6114997              0.00            2.09
                         Au4 Si (2)    C4v      −832.6028256              0.21            1.87
                         Au4 Si (3)    C3v      −832.6036545              0.24            2.12
                         Au4 Ge (1)    C4v      −2619.999025              0.00            2.12
                         Au4 Ge (2)    C3v      −2619.993000              0.16            1.95
                         Au4 Ge (3)    Td       −2619.980043              0.52            1.83
                         Au4 Sn (1)    C4v      −546.6744891              0.00            2.06
                         Au4 Sn (2)    C3v      −546.6690896              0.15            1.96
                         Au4 Sn (3)    Td       −546.6449101              0.80            1.85




                      2.81A                               2.77A

                       2.78A                          2.72A
                                                                                                           109.5

                                                                                           2.60A




                       ∆E=0.00 eV                        ∆E=0.15 eV                        ∆E=0.80 eV
                               C4v                            C3v                                  Td

Fig. 4. Optimized structures of three low-energy isomers of SnAu4 . The relative energies and point-group symmetries are given
underneath eath isomer structure.

shows a gradual decrease of negative charge on the group IV atom from C to Sn (Table 2), indicating that
the charge transfer from Au atoms to the group IV atom becomes less and less from C to Sn. As pointed
above, the MO diagrams [Fig. 5(b)] show that as the energy level of the valence p orbital increases from
C to Sn, an increasing contribution from the Au 6s orbitals to the HOMO occurs, resulting in a larger
charge partitioning for the Au atoms. This MO pictures are also consistent with the trend of decreasing
electronegativity for the group IV element from C to Sn.

3.4. Anion Photoelectron Spectrum of XAu 4 (X = C, Si, Ge, Sn)

  The simulated anion photoelectron spectroscopy (PES) spectrum for the lowest-energy structure of
XAu4 (X = C, Si, Ge, Sn) are shown in Fig. 6. The PES spectrum provides a fingerprint for each cluster
species and can be used to compare with simulated PES spectrum to identify the structure of the cluster.
Our simulated spectrum for SiAu− gives a very good match with the measured one, consistent with that
                                  4
reported by Kiran et al. [1]. The calculated vertical detachment energy (VDE) for this species is shown
in Table 3. The VDEs for all anion species are very close in value, as well as the gaps between the X
190     R. Pal et al. / Exploring the lowest-energy structures of group IV tetra-aurides: XAu4 (X = C, Si, Ge, Sn)

                                                        Table 2
                                        Hirshfeld charge analysis for XAu4 (X =
                                        C, Si, Ge, Sn)
                                        Cluster (isomer #)    Atom    Charge (e)
                                            CAu4 (1)           C       −0.2840
                                                               Au       0.0712
                                            SiAu4 (1)          Si      −0.170
                                                               Au       0.0426
                                           GeAu4 (1)           Ge      −0.0972
                                                               Au       0.0244
                                            SnAu4 (1)          Sn      −0.0256
                                                               Au       0.0064

                                                        Table 3
                                          Calculated VDE for anion XAu− (X
                                                                      4
                                          = C, Si, Ge, Sn)
                                           Anion cluster     Peak    VDE (eV)
                                              CAu−4           X        2.28
                                                              A        3.49
                                              SiAu−
                                                  4           X        2.19
                                                              A        3.90
                                              GeAu−
                                                  4           X        2.12
                                                              A        3.42
                                              SnAu−
                                                  4           X        2.03
                                                              A        3.44

and A peak, even though some of these species have very different structures. The large HOMO-LUMO
gaps for all the lowest-energy structures indicate that removal of the second electron takes large energy
(> 2eV) from the anion species. So the neutral structures of these species are expected to be very stable.
CAu4 may be chemically the most stable since it has the largest HOMO-LUMO gap among the XAu 4
species (Table 3).


4. Discussion and conclusion

   We have studied Au/H analogy in group IV tetra-aurides XAu 4 (X = C, Si, Ge, Sn). Toward this end,
we have searched the lowest-energy geometries and studied the charge partition between Au and Group
IV atom of X (X = C, Si, Ge, Sn). We found that the lowest-energy structures of CAu 4 and SiAu4
are the same as their corresponding hydrides. In contrast, GeAu 4 and SnAu4 adopt a very different
geometry (square pyramidal) rather than the tetrahedral geometry of the GeH 4 and SnH4 hydrides.
Further evidence for the lowest-energy structures was obtained from the simulated anion photoelectron
spectroscopy (PES) spectrum. A close match between the experimental and simulated PES spectrum was
confirmed for SiAu4 . Although the lowest-energy geometry of CAu 4 and SiAu4 has the Td symmetry,
a square pyramidal isomer as well as a propeller like isomer with C 3v symmetry has been found to be
another two low-energy isomers. From Si to Ge, our study suggests a structural transition from T d to C4v
(square pyramidal) geometry. The T d structure becomes less and less stable than the square pyramidal
structure from C to Sn. The physical/chemical reasons behind such a structural transition require future
studies. We note that the removal of degeneracy in the HOMO orbital (t 2 ) of the Td isomer gives rise
to doubly degenerate e g and a2g orbitals of the square pyramid isomer. This removal of degeneracy of
the t2 orbital is likely due to the higher energy level of the valence p orbitals of Ge and Sn, resulting
        R. Pal et al. / Exploring the lowest-energy structures of group IV tetra-aurides: XAu4 (X = C, Si, Ge, Sn)   191


                  (a)




                                                 CAu4                         SiAu4




                                                 GeAu4                        SnAu4

            (b)
                                 t2                                             t2

                                           a1                                        a1
                   4 X 6s                                            4 X 6s                               3p
                                 t2                                             t2
                                                              2p


                            Au                  CAu4     C               Au                 SiAu4   Si



                                      t2                                              t2

                                           b2                                              b2
                                                                                                          5p
                   4 X 6s                                     4p     4 X 6s

                                      e                                               e
                                           a2                                             a2


                        Au                      GeAu4    Ge            Au                 SnAu4     Sn

     Fig. 5. (a) HOMO diagrams and (b) energy diagrams for the lowest-energy isomer of XAu4 (X = C, Si, Ge, Sn).

in a less overlap with the 6s orbitals of Au atoms. Finally, we found that the lowest-energy geometry
of GeAu4 and SnAu4 differs from their corresponding tetrahedral hydrides. It can be concluded that
although a single Au atom can behave as H atom in SiAu 4 and CAu4 , other electronic factors can also
play an important role in the case of GeAu 4 and SnAu4 , addition to the Au/H analogy.
192       R. Pal et al. / Exploring the lowest-energy structures of group IV tetra-aurides: XAu4 (X = C, Si, Ge, Sn)




                                                                                                        A
                                        A


                              X                                                           X

            0      1      2         3       4   5     6                 0      1      2         3           4   5   6



                                  CAu4-                                                       SiAu4-




                          X             A                                             X             A


            0      1      2         3       4   5     6                 0      1      2         3           4   5   6




                              GeAu4-                                                          SnAu4-
         Fig. 6. Simulated anion photoelectron spectrum for the lowest-energy isomer of XAu− (X = C, Si, Ge, Sn).
                                                                                           4


Acknowledgments

  This research was supported in part by grants from DOE (DE-FG02-04ER46164), NSF (CHE-0427746
and CHE-0701540), and the Nebraska Research Initiative, and by the Research Computing Facility at
University of Nebraska-Lincoln.


References

 [1]   B. Kiran, X. Li, H.J. Zhai, L.F. Cui and L.S. Wang, Angew Chem Int Ed 43 (2004), 2125.
 [2]   X. Li, B. Kiran and L.S. Wang, J Phys Chem A 109 (2005), 4366.
 [3]   B. Kiran, X. Li, H.J. Zhai and L.S. Wang, J Chem Phys 125 (2006), 133204.
 [4]   H.J. Zhai, L.S. Wang, D. Yu Zubarev and A.I. Boldyrev, J Phys Chem A 110 (2006), 1689.
 [5]            o
       P. Pyykk¨ and J.P. Desclaux, Acc Chem Res 12 (1979), 276.
 [6]            o
       P. Pyykk¨ , Chem Rev 88 (1988), 563.
 [7]   P. Schwerdtfeger, M. Dolg and W.H.E. Schwartz, J Chem Phys 91 (1989), 1762.
 [8]   P. Schwerdtfeger, Chem Phys Lett 183 (1991), 457.
 [9]            o
       P. Pyykk¨ , Angew Chem Int Ed 41 (2002), 3573.
[10]   D. Balamurugan, M.K. Harbola and R. Prasad, Phys Rev A 69 (2004), 033201.
[11]   S. Yoo and X.C. Zeng, Angew Chem Int Ed 44 (2005), 1491.
[12]   D.J. Wales, and H.A. Scheraga, Science 285 (1999), 1368; J.P.K. Doye and D.J. Wales, New J Chem 773 (1998).
[13]   J.P. Perdew, K. Burke and M. Ernzerhof, Phys Rev Lett 77 (1996), 3865.
[14]   DMol3 is a density functional theory program distributed by Accelrys, Inc., B Delley, J Chem Phys 92 (1990), 508.
[15]   T.H. Dunning, Jr., J Chem Phys 90 (1989), 1007.
          R. Pal et al. / Exploring the lowest-energy structures of group IV tetra-aurides: XAu4 (X = C, Si, Ge, Sn)      193

[16]   P.J. Hay and W.R. Wart, J Chem Phys 82 (1985), 299.
[17]   Gaussian 03 (Revision C.02), M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, V.G.
       Zakrzewski, J.A. Montgomery, R.E. Stratmann, J.C. Burant, S. Dapprich, J.M. Millam, A.D. Daniels, K.N. Kudin, M.C.
       Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, S. Clifford, J. Ochterski, G.A.
       Petersson, P.Y. Ayala, Q. Cui, K. Morokuma, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J. Cioslowski,
       J.V. Ortiz, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R.L. Martin, D.J. Fox, T. Keith,
       M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P.M.W. Gill, B.G. Johnson, W. Chen, M.W.
       Wong, J.L. Andres, M. Head-Gordon, E.S. Replogle and J.A. Pople, Gaussian, Inc., Wallingford CT (2004).
Journal of Computational Methods in Sciences and Engineering 7 (2007) 195–217                                                   195
IOS Press




Prediction of structures and related properties
of silicon clusters
Marta B. Ferraro
                  ı              e
Departamento de F´sica “Juan Jos´ Giambiagi”, Facultad de Ciencias Exactas y Naturales,
                                                          ´
Universidad de Buenos Aires, Ciudad Universitaria, Pabell on I, Buenos Aires, Argentina
E-mail: ferraro@df.uba.ar

Received 5 January 2007
Accepted 6 April 2007

Abstract. Silicon is one of the most important semi conducting materials employed in microelectronic technologies. This
chapter is dedicated to review the enormous effort made during the last years to develop confident methods to determinate the
geometries and related properties of silicon clusters. Searching employing full global optimization of the energy surface, using
techniques like Molecular Dynamics (MD), algorithms, and local optimization of structures built onto generic structural motifs
are reported. Hybrid methodologies combining tight-binding (TB) with global optimization methods also contribute to the state
of the art in the prediction of the structures, and calculation of properties of silicon clusters, based on all-electron DFT methods
are also included.



1. Introduction

   The study of the structure and physical properties of atomic and molecular clusters is an extremely
active area of research due to their importance, both in fundamental science and in applied technology [1].
See as instance the discovery of luminescence in nanostructured silicon clusters at room temperature [2],
the presence of quantum size effects and the appearance of silicon photonic crystals with applications
in nanotechnology [3]. The properties of clusters differ markedly from those of molecules (with well
quantized states) and bulk, micro-crystalline materials, and depend also strongly on their size and
structure.
   Since nearly 20 to 26 years ago structural characterization of silicon and other semiconductor clusters
has been an area of enormous experimental and theoretical effort [4–6].
                    a
   Bachels and Sch¨ efer [7] produced neutral Si clusters in a laser vaporization beam and measured their
binding energy in samples from 65 to 890 atoms. They also produced metastable prolate structures at
sizes up to about 170 atoms.
   Existing experimental methods for structural determination seldom can obtain the structure of atomic
clusters directly. Therefore the calculation, using theoretical structures and comparison with experimental
values of their physical and optical properties is the most common way to obtain structural information
of atomic clusters. While the prediction of the structures of clusters with a small number of atoms is
well understood, the prediction of the structure and properties of medium size (10–100 atoms) clusters is
much less developed in spite to their critical importance in understanding the transition from microscopic
to macroscopic behavior of nano-materials and their possible technological applications.

1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
196                M.B. Ferraro / Prediction of structures and related properties of silicon clusters

   The numerous and interesting structures of carbon fullerenes promoted studies on silicon clusters, not
only because carbon and silicon belong to the same group in the periodic table, but because of the known
applicability of silicon in computer chips, microelectronic devices, catalysts, and new superconducting
compounds.
   Differences and similarities between both series of atomic clusters have been pointed out in numerous
publications [8]. For instance no fullerene like structures have been identified for Si n units, this is
attributable to the sp2 characteristic hybridization in fullerenes, which is more favorable for C n than for
Sin units [9], i.e., silicon clusters of five atoms form three-dimensional compact structures while pure
carbon clusters with ten or less atoms show linear and planar structures. For instance, properties related
to aromaticity of ring carbon clusters are not present in silicon clusters.


2. Prediction of structures

   Clusters with up to approximately ten atoms can be modeled using standard geometry optimization
techniques in conjunction with quantum chemical methods such as density-functional theory (DFT),
second-order Møller Plesset theory, coupled clusters, etc, and one of the objectives is to find, for a given
size, the geometry corresponding to the lowest potential energy.
   Global optimization is a non-trivial problem. For large clusters, ab initio calculations are still,
at present, not possible, because the systematic, global geometry optimization is complex, and time
consuming. The number of possible isomers of a cluster grows up with the number of atoms and
the global optimization of a system of ≈ 20 atoms is almost an intractable problem. For this and
larger clusters ab-initio calculation of energies are limited to a number of structures predetermined by
other methodology. At larger sizes, ion mobility spectrometry shown a shape transition [5,10,11]: S +     n
form aggregates up to n ≈ 25, adopt near spherical compact forms over n ≈ 26, and then grow three
dimensionally towards mesoscopic Si particles.
   Detailed structures for lowest energy prolate and compact silicon clusters, have been validated by
experimental results [12]. Also the calculation of excited state properties is fundamental for the right
description of optical properties of materials [13]. The right understanding of size and shape transition
dependence is a problem of technological interest.
   The electric polarizability, bonding energy and HOMO-LUMO gap are related with the size of the
cluster. The electric dipole polarizability is a basic property of electronic structures, related in bulk
Si to dielectric constant via the Claussius-Mosotti relation, and can be measured without touching the
clusters [14,15]. The importance of polarizability data to rationalize experimental observations is very
important [15] and the effort of looking for confidence theoretical models to calculate it is big. The
basic ideas of the research on atomic clusters has been reviewed from both, the experimental [16] and
the theoretical aspect [17]. When the number of atoms in the cluster increases there are many clusters
in a narrow range of energies, so the calculation of some property, like dipole polarizability, may be
compared with the experimental data to elucidate the isomer of lowest energy, or which is the most
favorable one. The dipole polarizability of clusters between 9 and 120 atoms have been measured [18].
The HOMO-LUMO gap correlates well with the polarizability of a system, being easier to polarize those
systems with a smaller HOMO-LUMO gap [19]. However, this correlation is not verified in medium
size clusters, unless that the shape of the clusters is not too different.
   Pouchan et al. [20] found a correlation between polarizability of Si n (n = 3–10) and the size of
the energy gap between symmetry-compatible bonding and antibonding molecular orbitals, that is the
“allowed gap”, instead the HOMO-LUMO gap.
                    M.B. Ferraro / Prediction of structures and related properties of silicon clusters   197

   Jackson et al. [21] computed polarizabilities for compact and prolate structures of Si n clusters (n =
20–28, and n = 50) and found that the charge density show a metalliclike response of the clusters to an
external field, and the calculated polarizabilities, are reproduced by the prediction of jellium-models for
spheres and cylinders, suggesting a metallic-like character for these medium-size clusters.
                                                          ¨
   The experimental polarizabilities reported by Sch afer et al. [18] vary irregularly around the bulk
limit (αbulk = 3.71 Å3 /atom) for n         9; i.e. 2.9, 5.5, 2.8 and 1.8 Å 3 /atom for n = 9, 10, 11 and
12, respectively. On the other hand, theoretical results reported in the literature are greater than 4.0
Å3 /atom [22–25] Becker et al. [14] have also reported experimental evidence that Si n clusters with 60
   n 120 are characterized by mean polarizabilities below the bulk limit.
   Photoelectron spectroscopy is another important tool for investigating the geometry of atomic clus-
ters [26,27]. The comparison of theoretical and experimental photoelectron spectra is much more
sensitive for structure elucidation, than the comparison of other observables, i.e., ionization potentials,
or electron affinities. Because of that, those spectra are very useful to confirm theoretical prediction of
atomic clusters structures.
   This chapter is dedicated to offer an update of the state of the art in the available methodologies to
determine silicon clusters structures of a few to sixty silicon atoms. There are two general strategies to
attack the problem that will be reviewed in the following sections: i) combination of several common
structural motifs observed in nanostuctures like small well determined silicon clusters (with at most ten
atoms), cages, wires, fullerenes; ii) global search of the geometric parameters space using searching tech-
niques like Molecular Dynamics (MD); algorithms, combined with a selected potential, semiempirical
or tight-binding (TB) method to analyze the potential energy surface (PES).

2.1. Silicon clusters structures predicted by combination of motifs and subunits

   Small silicon clusters has been the focus of theoretical and experimental investigations since nearly
three decades ago [28].
   Li et al. [29] obtained the experimental infrared spectra of Si 3 –Si7 in neon, argon and krypton matrices
at T = 4◦ K, in excellent agreement with ab initio calculations of their vibrational frequencies and relative
intensities.
   Zdetsis [30] studied the structure of the “magic” Si 6 cluster, and found that the ground state is not
well understood, since the distorted octahedron of D 4h symmetry is a transition state connecting two
nearly isoenergetic structures of lower symmetry. He compared results from Møller-Plesset perturbation
theory (MP2) [30,31] with Hartree-Fock gradient calculations [32]. His results, based in higher-order
perturbation theory, accurate coupled- cluster CCSD(T) calculations and density functional theory (DFT)
with gradients and hybrid-schemes like B3LYP, with the Becke [33] and the Lee-Yang and Parr gradient-
corrected corrections [34], shown that there are three lowest energy structures with symmetries D 4h ,
CS , and C2v . These structures are nearly isoenergetic and compete for the minimum in the energy
hypersurface. The two last structures show real frequencies at the B3LYP, MP3 and MP4 levels,
compatible with D4h active modes and comparable with experimental results. These findings suggest
that the “magic” properties of some clusters like Si 6 and Si10 , may be connected with the presence of
several isoenergetic isomers competing for a minimum as result of the flatness of the hypersurface near
the minimum.
   The structures of small (n < 8) silicon clusters have been studied by theoretical methods and confirmed
by anion photoelectron spectroscopy [27], or by Raman [31] and infrared [29] spectra on matrix-isolated
clusters. Ho et al. [4] reported geometries calculated for medium-size silicon clusters, for n = 12–26,
198                 M.B. Ferraro / Prediction of structures and related properties of silicon clusters

using an unbiased global search with a genetic algorithm [35]. They determined ion mobilities for
these geometries by trajectory calculations, in excellent agreement with the values that they measured
experimentally. For n = 12–18 the clusters are built on a structural motif consisting of a stack of Si 9
tricapped trigonal prisms (TTP). For n 19, they concluded that near-spherical cage structures become
the most stable. The transition to these more spherical geometries occurred in the measured mobilities
for slightly larger clusters than in the calculations, possibly because of entropic effects. They constructed
prolate isomers for n 19, by stacking Si 9 TTP prisms, following the patterns obtained for n < 19. In
spite of that their isomers are lower in energy than those previously reported in the literature, they do not
come from a global search in the parameters energy surface, and then there are not reasons to consider
that they are the best isomers for 19 n 26.
   Yoo and Zeng [36] found that the medium-size silicon clusters Si 15 -Si22 are, most of them, built
onto two generic structural motifs: the tricapped-trigonal-prism (TTP) Si9 motif, or the six/six Si6 /Si6
(sixfold-puckered hexagonal ring Si 6 plus six-atom tetragonal bypiramid Si6 ) motif. The location of
the transition from TTP to six/six depends on the DFT functional employed to perform the calculations.
This transition occurs at the surroundings of Si 16 . In order to examine this functional dependence, they
made full electron DFT calculations for two functionals: B3LYP [33,34,37,38] and the hybrid PBE
(PBE1PBE) [39,40]. They also computed all electron energies for Si 16 –Si22 employing a coupled-
cluster scheme, with single and double substitutions and triple excitations (CCSD(T)), and found that
the six/six-motif is the most favorable for B3LYP, while PBE1PBE1 calculations slightly favor the TTP-
motif-subunits. They concluded also that there is a transition at n ≈ 23, from six/six to six/ten motif, and
that the transition from prolate to compact (fullerene like structures) occurs at n ≈ 27. Ho et al. [4–6]
had also showed that the TTP motif prevails in the small-size low lying clusters Si 16 –Si19 , as it was
mentioned earlier in this section.
   Jackson et al. [41] studied the structures of Si n and Si+ with n = 20–27. They applied the “big-ban”
                                                              n
optimization method, an unbiased search algorithm, and validated the structures through independent
comparisons to measured ion mobilities and dissociation energies. The “big-ban” consists in creating
a big number of trials structures of n atoms, in a compressed space, typically ≈ 1/25 of the normal
volume. The structures are allowed to “explode” relaxing to local minima via a standard gradient-based
algorithm [42], using the density functional tight-binding (DFTB) method [43], which runs ≈ 10 3 to 104
times faster than DFT. The best isomers were relaxed again by DFT, using the gradient-corrected Perdew-
Burke-Ernzerhof (PBE) functional [44]. The Si+ systems were obtained by removing one electron from
                                                    n
Sin, and performing the DFT relaxation. They found that Si + structure are compact for n 23, and
                                                                    n
compete with prolate clusters for n = 24, and 25, that are the preferred shape for n 27. For neutrals
silicon clusters the transition occurs for n 26. They compared mobilities and dissociation energies
with experimental data [10,11] Si9 TTP units, Si6 octahedron and sixfold ring units appear in this range
of atoms, and the TTP units are present not only in elongated but also in some compact structures (Si + ). 24
They also proved [41] the statistical performance of the “big ban” method, using Lennard-Jones clusters
(LJn ) as test cases, and applied it to the series Si 20 –Si28 . They found three families of structures in the
shape transitions region (n ≈ 24), and relaxed nearly 300–500 structures by means of density functional
theory-tight binding (DFTB) in an implementation highly parallelized, and very efficient.
   In a recent paper, Jackson et al. [45], examined the utility of photoelectron spectroscopy as a structural
probe to examine Si− in the n = 20–26 size range. Across the entire size range, they consistently
                       n
reported a good agreement between the theory and experiment [46]. They employed structures previously
reported [41,47], and computed the theoretical spectra within the constant matrix approximation. They
constructed the DOS curves by solving the Khon-Sham equations for the fully relaxed clusters, using the
                         M.B. Ferraro / Prediction of structures and related properties of silicon clusters   199

higher order finite-difference pseudopotential method [48]. They found a shape transition from prolate to
compact (quasi spherical) isomers at n ≈ 25, that correlates with the measurements of ion mobilities [11]
For clusters with n 25, no signal corresponding to compact isomers is observed, and this feature is
reversed from n = 26 to 30, and at larger sizes all clusters are compact, not necessary special or perfect
cages, but they correspond to compact structures.
   Idrobo et al. [49] calculated the absorption spectra of Si 20 –Si28 , within the time-dependent local density
approximation (TDLDA) employing those structures of Refs [21,50,51] and compared with experimental
data of Ref [52]. They found that the general features of the absorption spectra in this range are size
independent, but they present distinct shape signatures, and proved that for the TDLDA approximation
the dynamical response of silicon clusters to time-varying electric fields provide further support for their
metallic behavior, as was suggested in Refs [21,22,41].
   Jarrold and Bower provided a wire isomer of Si 28 composed by TTP [53]. They measured the
mobilities of size selected silicon cluster ions, Si + (n = 10–60), using injected ion drift tube techniques,
                                                      n
and resolved two families of isomers by their different mobilities. From comparison of the measured
mobilities with the predictions of a simple model, they reported that clusters larger than Si + followed
                                                                                                    10
a prolate growth sequence to give sausage-shaped geometries; a more spherical isomer appeared for
clusters with n > 23, and isomers with this shape completely dominated for annealed clusters with
n > 35. According to their observations annealing converts the sausage-shaped isomer into the more
spherical form for n > 30.
   Sun et al. [8] studied the structure and energetic of Si 36 cluster. They proposed 17 structural isomers
built by different strategies: i) a set of 10 cages, i.e., a fullerene cage with D 6h symmetry, corresponding
to the ground state geometry of C 36; [54] ii) others composed by a ring of Si 12 where each silicon atom
has coordination number four; a dimmer of two Si 18 rings, and so on. iii) Four wire structures built by
TTP units of Si9 ; the Boerdijk-Coxeter helix structure, which is a linear stacking of regular tetrahedrons,
and iv) several stuffed structures composed by two silicon clusters: Si 28+8 , Si30+6 . All these structures
were optimized using gradient-corrected functionals [55] and the method of plane waves (PAW) adapted
by Kresse and Joubert [56]. They found that the most stable structure is a stuffed-fullerene cage, with a
6-atom unit encapsulated. The high cost of the ab-initio calculations precluded a more comprehensive
study of the parameters space, leaving open the possibility that other geometries not provided by the set
of selected motifs may be candidates for stable isomers of Si 36 , as it will be reported in next section.
   Wang et al. [57] studied the stuffed fullerene cages of Si 40 clusters using the density functional
theory. They considered the Si 40 as a prototype, of stuffed fullerene cages with different “stuffing/cages”
ratio: Si4 @Si36 , Si6 @Si34 , and Si8 @Si32 . They optimized these “handmade” structures. The study
is very interesting because they considered all the possible isomers for the fullerene structures and
then they stuffed inside the cage the necessary number of silicon atoms to reach n = 40. They also
generated a number of structures for bulk fragments of Si solids with diamond, and generated 30 initial
geometries of Si40 . These structures were relaxed applying a simulated annealing based on tight-binding
(TBMD),1 [58] and they optimized the resulting structures employing the DFT-PBE [40] exchange-
correlation functional. They only considered “classical” fullerenes cages with six- and five-membered
rings, and finally reported three isomers with outer cages symmetries C 3v , D2d , and D3h , competing for
the minima in the energy surface. The two lowest energy structures match those reported by Bazterra et
al. [59] and discussed in next section. Wang et al. [57] underlined that the inclusion of “non classical”
fullerene cages, with subsequent stuffing of atoms inside them, would dramatically increase the number
of isomers in the initial population.

  1
      Tight-binding (TB) method is briefly discussed in Section 2.3.
200                        M.B. Ferraro / Prediction of structures and related properties of silicon clusters

                                                        Table 1
      All electron and pseudopotential values for the dipole moment (in Debye) and the dipole polarizability per
      atom (in Å3 /atom) calculated within DFT-B3PW91a
      Cluster      Sym.          EB /atom           α           αb           αc exp        |µ|               Gap
                                 (Ev/atom)     (Å3 /atom)    (Å3 /atom)    (Å3 /atom)    (Debye)             (eV)
                       3
       Si1, p              P                     5.85          6.12d                       0.0            1.710 (α)
                                                               5.85e                       0.0            7.820 (β)
                       3
          Si1, s           P                      5.83         5.88f                       0.0            1.830 (α)
                                                                                           0.0            7.588 (β)
                       3
          Si1, s           P                      5.44                                     0.0            7.908 (α)
                                                                                           0.0           12.787 (β)
                       3
          Si1              P                      2.53                                     0.0            7.681 (α)
                                                                                           0.0           13.086 (β)
                   3
       Si2, s        D∞h           −1.509         4.78                                     0.0       3.416 (α) 1.458 (β)
       Sih2,
                   3
                     D∞h           −1.330         6.21          6.29d                      0.0       7.632 (α) 6.649 (β)
      Si2, sg      3
                     D∞h           −1.352         6.97                                     0.0       8.413 (α) 7.685 (β)
       Si3, p        C2v           −2.002         5.63          5.22d                     0.294             2.659
       Si3, s        C2v           −2.329         5.24          5.21e                     0.246             2.286
       Si4,p         Td            −1.922         5.41           4.7i                      0.0              1.666
       Si4,s         Td            −2.292         4.97          5.07d                      0.0              1.672
       Si4,p         D2h           −2.489         5.32           5.0i                      0.0              2.369
       Si4,s         D2h           −2.862         5.09                                     0.0              2.387
       Si5, p        C2            −2.679         5.18          4.81d                     ≈ 0.0             3.323
       Si5, s        C2            −3.057         4.92          4.82e                     ≈ 0.0             3.146
       Si6,p         C2v           −2.863         4.84          4.46d                     0.091             3.040
       Si6,s         C2v           −3.270         4.55           4.4i                     0.073             3.216
                   1
       Si6,p         D4h           −0.731         4.03          4.51e                      0.0              0.910
                   3
       Si6,s         D4h           −2.768         4.84           4.3i                      0.0       3.449 (α) 2.106 (β)
       Si7, p        C2v           −2.991         4.73          4.41f                     ≈ 0.0             3.297
       Si7, s        C2v           −3.375         4.45          4.36e                     ≈ 0.0             3.182
       Si8, p        C2v           −2.917         4.92          4.52d                     ≈ 0.0             2.553
       Si8, s        C2v           −3.299         4.64          4.54e                     ≈ 0.0             2.541
       Si9, p        C2v           −2.740         4.32          4.38d       2.9 ± 0.8     0.660             1.333
       Si9, s        C2v           −3.224         4.38          4.38e                     0.739             1.361
      Si10a, p       C3v           −4.370         4.25          4.31d                     0.940             3.132
      Si10b, p       Td            −4.305         4.31          4.34e       5.5 ± 0.8      0.0              3.552
      Si11a, p       CS            −2.945         4.27                      2.8 ± 0.8     0.770             2.195
      Si11b, p        E            −2.895         4.28                                    1.675             2.191
      Si12a, p       CS            −4.267         4.39          4.50e       2.3 ± 0.8     1.156             1.429
      Si12b, p        E            −4.293         4.43                                    0.530             1.914
      Si13, p        CS            −4.373         4.49          4.40f                     1.489             1.838
      Si13, p       C2V            −4.393         4.56          4.51e                     0.310             1.872
      Si13, p       C2V            −4.422         4.48                      1.8 ± 1.1     0.114             1.816
      a
        Sin,s : all electron calculations with Sadlej’s basis set; Sin,p : pseudopotential calculation. Binding energies
      respect to spin polarized atom EB /atom in (eV/atom), and HOMO/LUMO gap in (eV) are reported [65].
      b
        Results from other authors.
      c
        Exp. Data from Ref [68].
      d
        Ref [22].
      e
        Ref [87].
      f
        Ref [111].
      g
        CCSD(T) results with Sadlej basis set.
      h
        For some clusters CCSD(T) results performed with 6-311G* basis set are also displayed.
      i
        Ref [112].
      Reprinted with permission from V.E. Bazterra, M.C. Caputo, M.B. Ferraro, P. Fuentealba, J. Chem. Phys.
      117 (2002), 11158. Copyright (2002). American Institute of Physics.
                    M.B. Ferraro / Prediction of structures and related properties of silicon clusters   201

2.2. Global search of the potential energy surface of silicon clusters, by global optimization
     methodology

   Deaven and Ho performed an unbiased global search method to determine the ground state structures
of silicon clusters with 11–20 atoms, employing a genetic algorithm [35]. They employed a tight-binding
potential to calculate the energies and then relaxed the best isomers employing the LDA approximation.
   Yoo et al. [60] combined molecular mechanics/quantum mechanics procedure to predict isomers of
Si21 and Si25 . They combined a basin-hopping global optimization technique [61] with three empirical
model potentials to locate the structures of Si n (n = 21–30) [62]. The best 20 isomers of Si 21 and
Si25 were relaxed using the B3LYP functional, including the calculation of the vibrational spectra to
test the stability of the isomers. Finally they applied a coupled-cluster single and double substitution
(CCSD/6-31G(d) level), and found new isomers, not previously reported of both species, Si 21 and Si25 .
They reported, for both series, some spherical-like structures, which are lower in energy than the prolate
structures, suggesting that the prolate-to-spherical-like transitions are likely to appear for 21 n 25,
because the ionization potential (IP) measures show IP levels off for n 22 [63], suggesting that the
more “spherical like” clusters become energetically favorable for n 22.
   Bazterra et al. employed the modified genetic algorithm for crystals and clusters (MGAC) [64],
combined with the hybrid B3PW91 to produce intermediate size silicon clusters and found fourteen
stable structures of Si9 [65], some of them match those reported by Li et al. [66]. In the same article [65],
they evaluated the atomic dipole polarizability of silicon for the relaxed B3PW91 structures of Si n
(n = 4–13), at the same DFT level, employing pseudopotential and all electron calculation. The
corresponding values are compared with values reported by other authors [22,23,67] and experimental
data [68] in Table 1. An important observation is that the all electron and pseudopotential calculations
are very similar and they are in good agreement with the other reported values [22,23,67].
   More recently, they reported the use of a parallel genetic algorithm (PGA) [69] to predict the structure
of medium-size silicon clusters, using the MSINDO [70] semiempirical code to evaluate the energy of
the clusters, and made an application to the search of Si n (n = 4–16) as test cases, performing a full
search of the ground state structure of Si 36 .
   They defined a genome with enough information to calculate the associated fitness function. The
genome is quite simple because there are no symmetry or periodicity relationships that constrain the
parameters in the genome. It is given as an array containing the coordinates of the atoms. This array
has dimension 3N, were N is the number of atoms in the cluster. The first population, of size N pop ,
was constructed by generating a set of atomic coordinates at random. The GA operations of mating,
mutation and selection were used to evolve one generation into the next. The population replacement
was done through the steady-state genetic algorithm (SSGA), which typically replaces only a portion of
the individuals in each generation [71].
   This technique is also known as elitism, because the best individuals among the population, 50% in
our case, are copied directly into the next generation. The criteria for fitness probability, selection of the
individuals and mutation are discussed in detail in Ref [59]. Like any stochastic minimization procedure
the GA should be run several times to guarantee that the resulting structures are independent of the initial
population and statistically significant.
   The MGAC package has been implemented in C++ language using parallel techniques (MPI), making
it very portable as well as easy to maintain and upgrade. The parallel MGAC implementation of the GA
(PGA) is particularly efficient [72].
   The most stable structures were subject to further refinement by performing a local optimization
using the B3PW91 exchange-correlation functional and the LANL2DZ basis set with Los Alamos
202                    M.B. Ferraro / Prediction of structures and related properties of silicon clusters

                                                                  Si4




                                                            (a) 2.48 eV


                                                                  Si5




                                         (a) 2.67 eV                                  (b) 2.54 eV


                                                                  Si6




                               (a) 2.85 eV                  (b) 2.86 eV                   (c) 2.74 eV


Fig. 1. Optimized structures and binding energies per atom for Sin , n = 4–14 and 16 clusters. All energies in eV based on a Si
atomic energies of −102.449 eV. (Taken from ref. [69]). Reprinted from V.E. Bazterra, O. O˜ a, M.C. Caputo, M.B. Ferraro, P.
                                                                                            n
Fuentealba and J.C. Facelli, Phys. Rev. A 69 (2004), 53202. Copyright (2004) by the American Physical Society.

pseudopotential [73]. They found that for the small clusters, the structures were in very good agreement
with those previously reported and discussed in the preceding section, but for larger ones different
structures appeared. Figure 1 shows the structures obtained for the isomers up to 16 silicon atoms and
their corresponding binding energy per atom, based on a calculated atomic energy of −102.449 eV.
The isomers enclosed into frames are those previously reported by other authors, and those enclosed
by dashed lines, have been also reported, but they have at least one imaginary vibrational frequency
calculated by DFT [67].
   They reported three structures of Si 36 that have significant lower energy than those of Ref [8] the
most stable one has a binding energy of nearly 0.5 eV higher than the best isomer of Sun et al. [8].
The comparison is reported in Table 2 for B3PW91 [74] and LSDA calculations, and the geometries are
displayed in Fig. 2. This finding clearly highlights the importance of exploring the complete configuration
space when searching for atomic cluster.
   Bazterra et al. [75] employed their update parallel MGAC implementation of the parallel
GA(PGA) [72], which is particularly efficient to produce stable structures of Si 18 –Si40 , Si46 , and
Si60 . They reported the results of HOMO-LUMO gap, binding energies and polarizabilities for this
series. For the four best isomers of Si 60 produced by their approach they made a comparison with the
stuffed structures SF1 and SF2 reported in Ref [76]. Their lowest isomer (Si 60−a ) is very similar to the
SF1 structure. The four lowest Si60 isomers of Ref [75] are more stable than the stuffed SF1 structure
                   M.B. Ferraro / Prediction of structures and related properties of silicon clusters       203

                                                               Si6(Continued)




                                            (d) 2.58 eV                         (e) 2.58 eV


                                                                   Si7




                            (a) 2.98 eV                      (b) 2.85 eV                      (c) 2.83 eV




                              (d) 2.68 eV
                                                                                (e) 2.62 eV


                                                          Fig. 1, continued.

                                                      Table 2
                        Calculated binding energies per atom for Si36 isomers. All values in
                        eV based on Si atomic energies of −102.155 eV and -101.428 eV for
                        the LSDA and B3PW91 approaches, respectivelya
                        Structure                                               LSDA            B3PW91
                        Cage 2b                                                  4.79             3.93
                        Wire 2b                                                  4.95             4.18
                        Stuff30-Ab                                               5.00             4.16
                        MGAC/MSINDO                                              5.13             4.41
                        MGAC/MSINDO plus local DFT optimization                  5.19             4.46
                        a
                          All calculations done using the LanL2DZ basis set with its respective
                        Los Alamos pseudo potential.
                        b
                          Geometries from Reference [8].
                        Reprinted from Ref [69]. Copyright (2004) by the American Physical
                        Society.

of Ref [76], which presents five imaginary frequencies for the same level of approximation. The final
optimised structures for the four best isomers, Si 60 -(a) to Si60 -(d) are displayed in Fig. 3.
  In this article they also evaluated the static dipole polarizabities for each final B3PW91-LanL2DZ and
-SDDALL stable structure of the series. The polarizabilities evaluated using the B3PW91/SDDALL are
204                M.B. Ferraro / Prediction of structures and related properties of silicon clusters

                                                            Si8




                    (a) 2.91 eV                                        (b)     2.89 eV




                            (c) 2.88 eV                                        (d) 2.88V


                                                            Si9




                              (a) 3.00 eV                                    (b) 2.98 eV


                                                  Fig. 1, continued.

in the range ≈ 4.87 → 5.78 Å 3 /atom, while those calculated with the B3PW91/LanL2DZ, are in the
range ≈ 4.61 → 5.42 Å 3 /atom, and there is no difference on their tendency with respect to the cluster
size, as it is depicted in Fig. 4. The calculated values are in the same range of those calculated by other
authors. For instance, Jackson et al. [77] reported LDA polarizabilities for Si 20 and Si21 of 4.83 and 4.58
Å3 /atom. It is apparent that the experimental values [68] are always smaller than the calculated ones.
The same type of behaviour has been found for the polarizabilities taken from theoretical predictions of
                                   ˜
other authors [78]. Bazterra, Ona et al. [65,79] showed that the cited discrepancy cannot be attributed
to accuracy error introduced by the hybrid B3PW91 method, or bad efficiency of the basis sets, and
concluded that since all the known effects not considered in their calculations could only increase the
                   M.B. Ferraro / Prediction of structures and related properties of silicon clusters   205

                                                  Si9 (Continued)




                           (c) 2.94 eV                                            (d) 2.93 eV


                                                           Si10




                            (a) 3.02 eV                                   (b) 3.02 eV




                                             (c) 2.87 eV

                                                  Fig. 1, continued.

calculated values, and there is no evidence of noticeable errors in the geometries employed to make the
calculations, then the experimental values must likely be revised.
  The binding energies, polarizabilities, HOMO-LUMO gap, and dipole moments of Si 60 -(a) to Si60 -(d),
and SF1 and SF2 structures of Ref [76] are reported in Table 3 [79].
  The principal advantage of this hybrid technique to predict clusters structures is that it does not need
to make assumptions of any type on symmetry of combination of motif subunits, allowing for a full
exploration of the complete configuration space available for the cluster geometry.
  Yoo et al. [80] studied the medium-size silicon clusters (n = 27–39) employing an unbiased search for
the lowest-energy structures using also a genetic algorithm [35,81] and the non-orthogonal tight-binding
method [58], and relaxed the best candidates using basin hopping (BH) method [61] coupled with the
206                 M.B. Ferraro / Prediction of structures and related properties of silicon clusters

                                                                          Si11




                            (a) 3.04 eV                            (b) 3.03 eV                        (c) 3.02 eV




                                                (d) 3.00 eV                               (e) 2.99 eV

                                                                   Si12




                                          (a) 3.02 eV                                   (b) 3.00 eV


                                                              Fig. 1, continued.

                                                   Table 3
       B3PW91/SDDALL and B3PW91/LanL2DZ binding energies per atom (in eV/atom), dipole polarizability
       per atom (Å3 /atom), HOMO-LUMO gap (in eV) and dipole moment (in Debye) for the Si60
       Isomers                        SDDALL                                                        LANL2DZ
                    BE(eV)/at      α/at Gap(eV)                µ (Debye)           BE(eV)/at     α/at Gap (eV)      µ (Debye)
       Si60 -a       4.313         5.26   0.851                  2.508               4.592       4.95   0.846          2.567
       Si60 -b       4.310         5.55   1.091                  4.407               4.571       5.19   0.784          4.554
       Si60 -c       4.297         5.32   1.203                  3.444               4.557       5.12   0.885          3.893
       Si60 -d       4.275         5.45   1.194                  7.179               4.571       5.19   0.783          4.574
       Si60 -SF1I    4.239†         −       −                    3.196               4.520†       −       −           3.196
       Si60 -SF2I    4.232†         −       −                    1.869               4.504†       −       −           1.989
       †
        Imaginary Frequencies.
       I
        Ref [76].
                          n
       Reprinted from O. O˜ a, V.E. Bazterra, M.C. Caputo, J.C. Facelli, P. Fuentealba and M.B. Ferraro, Phys. Rev.
       A 73 (2006), 53203. Copyright (2006) by the American Physical Society.

general-gradient approximation(GGA) implemented within the CPMD source code [82]. They showed
that carbon fullerene cages may be considered as “magic cages” to form compact silicon structures in the
range n = 27–39. They found also no perfect fullerene cages for some global minima, which contain also
                   M.B. Ferraro / Prediction of structures and related properties of silicon clusters   207

                                                             Si12 (Continued)




                                                                        (c) 2.99 eV




                                                           Si13




                            (a) 3.06 eV                                           (b) 3.03 eV




                              (c) 3.02 eV                                   (d) 2.98 eV

                                                  Fig. 1, continued.

four- and seven-membered rings. In order to test the stability of the clusters, they relaxed the structures
employing the B3LYP/6-31G (d) level of theory. They found that the “stuffing” Si 3+m (m = 1, 2, . . .)
appear to be upper and lower limits, for the core-filling atoms in the Si 26+2m fullerene silicon cage.
  In a recent paper Ma et al. [83] constructed different endohedral fullerene cages for Si n (n = 30–39),
employing different possible combinations of outer endohedral cages and a number of filling atoms. The
hand-made structures were relaxed by DFT-MD methodology implemented in the CASTEP code [84].
After the MD relaxation these structures were further optimized using the all electron DFT program
DMOL [85]. They found that their structures had lower total energies or higher binding energies than
those from previous calculations for Si 33 and Si36 . For Si37 and Si38 , they found that the optimal
combinations are Si5@Si32 and Si6@Si32. This article indicates that as the number of atoms grows up,
one has to examine absolutely all the possible isomers to find the lowest-energy configuration for the
208                M.B. Ferraro / Prediction of structures and related properties of silicon clusters

                                                         Si14




                              (a) 3.11 eV                               (b) 3.10 eV




                         (c) 3.08 eV                                         (d) 3.05 eV

                                                  Fig. 1, continued.

clusters.

2.3. Tight binding methods to predict structures of small to medium size silicon clusters

  Hybrid methodologies combining tight-binding (TB) with global optimization methods have been
mentioned in previous sections. The tight-binding scheme consists in determining the eigenvalues of the
equation

      (H − En S)Cn = 0

where H and S are the Hamiltonian and overlap matrices, respectively. The elements of the Hamiltonian
are obtained employing a set of parameters depending on the atomic species and different techniques
that characterize each tight-binding scheme. Mennon and Subbaswamy [58] designed a tight-binding
                    M.B. Ferraro / Prediction of structures and related properties of silicon clusters    209

                                                           Si16




                                (a) 3.02 eV                             (b) 3.01 eV




                                   (c) 3.00 eV                                        d) 2.91 eV

                                                   Fig. 1, continued.

method which incorporates explicitly the non-orthogonally in the basis set to describe silicon clusters.
They found good results for band structures, phase diagrams, and bulk phonon, for cluster with 5–10
atoms, and then improved the method to get a better transferability of characteristic parameters to silicon
clusters of arbitrary sizes [86].
   Sieck et al. [87] identified low-energy structures of silicon clusters with 9 to 14 atoms using a
nonorthogonal tight-binding method (NTB) based on density-functional theory (DFT). They found
equilibrium structures for Si9 and Si14 , and isomers near to the ground states for Si 11 , Si12 , and
Si13 . The most stable structures, characterized by low energies and large HOMO-LUMO gaps, have
similar common subunits. They computed the full vibrational spectra of the structures, along with the
Raman activities, IR intensities, and static polarizabilities, using SCF-DF theory within the local-density
approximation (LDA). This method has already been successfully applied to the determination of Raman
and IR spectra of silicon clusters with 3–8, 10, 13, 20, and 21 atoms [23,88].
   Deng et al. [89] performed an hybrid DFT study on Si 13 clusters, employing three geometries(C 3V ,
C2V , CS ) taken from the literature. The C3V structure is the ground state of R othlisberger et al. [90]; the
                                                                                 ¨
C2V structure was presented by Sieck et al. [25] and the C S one was presented with energy even lower
than that of the C3V one [91]. They employed pure DFT exchange functionals plus the exact Hartree-
Fock (HF) exchange. In particular they choosed B3LYP and B3PW91, with the Becke corrections and
the Perdew-Wang, gradient corrected corrections [38,39]. They confirmed that the ground state is C S as
it was proposed by a previous local-density-approximation (LDA) result [91]. They performed also the
calculation of the vertical ionization potential (VIP) and vertical electron affinity (VEA) for the three
isomers, in good agreement with the experimental data [63].
210                  M.B. Ferraro / Prediction of structures and related properties of silicon clusters




Fig. 2. Comparison of optimized structure of Si36 found by MGAC/MSINDO plus local DFT optimization (B3PW91/
Lanl2DZ) and those from Reference [8]. Reprinted from Ref [69]. Copyright (2004) by the American Physical Society.

   Khakimov et al. [92] proposed a non conventional tight-binding method for the calculation of the total
energy and spectroscopic energies of atomic clusters, and applied it to obtain those parameters with
accuracy comparable to the state-of-the -art of ab-initio methods, for small and medium-size silicon
clusters.
   In the preceding section we cited an article of Yoo et al. [80] that combines TB with genetic algorithms
to predict structures of silicon clusters (n = 27–39). The literature shows that hybrid methods are usually
the preferred ones to explore the potential energy surface for medium-size silicon clusters. In a further
contribution, they [93] performed an unconstrained search for low-lying structures of Si 31 –Si40 , and Si45 ,
by means of the minimum-hopping global optimization method coupled with a density-functional theory
with PBE, BLYP and B3LYP functionals to determine the relative stability of the candidates produced
by the global optimization search. The initial configuration were taken from previous studies [57,80,94],
and after the DFT optimization, the top ten candidates for each species, were identified with the lowest
DFTB isomers. They found that most of the new isomers belong to the same fullerene- cage family
previously reported.
   They also found another family called the “Y-shaped-a three-arm” in the size range 31 n               40,
which are characterized by three arms built by various arrangements of the three magic-number clusters
Si6 , Si7 , Si10 and the TTP Si9 . It is very interesting to note that these Y-shaped- three-arm neutral
clusters provide an explanation to the photodissociation results for the medium size clusters beyond n =
30. Smalley and co-workers [95] studied the photodissociation process in neutral cluster containing up
to 60 atoms, and found that the medium-sized clusters larger than 30 atoms dissociate mainly by loss of
                       M.B. Ferraro / Prediction of structures and related properties of silicon clusters                  211

                                                            Si60




                                    Ly


                              Lz             Lx




                              (a) 4.313 eV                                          (b) 4.310 eV




                              (c) 4.297 eV                                          (d) 4.275 eV

Fig. 3. Optimized structures and binding energies per atom for stable Si60 cluster, at the B3LYP-sddall level of theory. The
binding energies are based on calculated silicon atomic energy of −101.536 eV. Lx Ly        Lz , are drawn for the most stable
isomer of Si60 . They represent the maximum extension (absolute values measured in Å) along the direction of its principal axis
of inertia. (Taken. From Ref. [75])

the magic-number clusters Si10 .
   In Ref [94] the authors show very interesting results of a combined photoelectron spectroscopy and
first principle studies of Si− clusters in the range 20 n 45. They compared the experimental [96]
                             n
and calculated photoelectron spectra, and found evidence for a prolate-to-near-spherical shape transition
at n = 27. For the range n = 30–45, the low-lying isomers are most likely near spherical in shape
and exhibit “stuffed-cage” – like structures. On the basis of this study, they identified new structural
motifs for silicon clusters in this size-range. These new motifs contain a diamond-like fragment, either
a six-atom or nine-atom subunit coupled with one or two magic clusters or the TTP Si 9 .
   Zhou and Pan [97] made and extensive search of the structure of Si 45 employing the tight-binding
potential proposed by Wang et al. [98] employed also with success, by Ho et al. [4] They picked up
the best 200 isomers and relaxed them by using the SIESTA code [99]. The best 20 isomers of this
optimization were relaxed again with a better methodology: Perdew-Wang 91 exchange correlation
functional within GGA. The three lowest final isomers exhibit structures of two cages, one into the other.
The outer cages show fourfold-coordinated atoms, and fivefold-coordinated atoms, which are all active
212                       M.B. Ferraro / Prediction of structures and related properties of silicon clusters
 α (¯ /atom)




                6
3




               5.5




                5




               4.5




                4




               3.5




                3
                                                                                                        B3Pw91-LanL2dz

               2.5                                                                                      B3Pw91-sddall

                                                                                                        Experimental data
                2




               1.5


                     20             25           30           35           40           45            50       55         60

                                                                                                               Cluster size n

Fig. 4. Comparison between B3PW91 atomic dipole polarizability and experimental data: ( ) SDDALL basis set; ( )
                                                             n
                                                                                                                         •
LanL2DZ basis set; (*) experimental data from Ref. [18]. O. O˜ a, V.E. Bazterra, M.C. Caputo, J.C. Facelli, P. Fuentealba and
M.B. Ferraro, Phys. Rev. A 73 (2006), 53203. Copyright (2006) by the American Physical Society.

sites. These structures are new ones, not previously reported at all. The authors demonstrated that these
isomers might be associated with the lowest reactivity of the Si 45 cluster.

2.4. Molecular dynamics applications

  The discrepancies between theoretical predictions and experimental measurements to determine silicon
clusters geometries occur very often when the number of atoms increases, i.e., n 20 [4,100,101].
  Common techniques to take into account the impact of dynamical and temperature effects in the
                    M.B. Ferraro / Prediction of structures and related properties of silicon clusters     213

energetic ordering of the isomers are the employment of methods of molecular dynamics (MD). One
of the most popular is the quantum Monte Carlo (QMC). The advantage of QMC is that many-body
correlation effects might be described employing an explicit correlation of the trial wave function and
                                         ¨
through a stochastic solution of Schr odinger equation. The trial wave function is a product of Slater
determinants of spin-up and spin-down orbitals, and a correlation factor that describes electron-electron,
electron-electron-ion, and many body effects in function of the distances between those particles. The
method is applied for T0 = 0 and T > 0, in the later case, combined with accurate calculations of
the energies, i.e., employment of DFT functionals. This method is described in Ref [102] and applied
to the study of isomers of Si20 and Si25 . The dynamical simulations further elucidated the structural
behavior and stability of the competing structures. The authors employed three geometries taken from
the literature for Si20 [4,103], and built another one by matching Si 9 and Si11 clusters, with a subsequent
optimization, and found by their MD-QMC search, an additional structure, 0.65 eV lower than the best
one reported previously. This new ground state belongs to the same “Si 10 + Si10 ” class of elongated
structures [4,101]. Using a combination of MD at T=0 they also constructed new elongated and compact
isomers of Si25 , with difference energies of 0.15 eV and 0.63 eV, for PW91 and LDA, functionals,
respectively. This finding reveals the coexistence of elongated and spherical-like isomers, as it was
suggested by the experiment [100]. The elongated structure contains “Si 10 + Si5 + Si10 ” subunits, and
the compact structure shows internal atoms encapsulated in the cage. The MD simulations lets understand
the mechanism of the transition from elongated to compact structures observed in the range 20 n
25.
   Li et al. [66,104] used a full-potential linear-muffin-tin orbital molecular dynamics (FP-LMTO-MD)
to study the structure of Si9 . They found fourteen stable structures, including structures reported before
by other authors.
   Li and Cao [105] found 15 stable structures for Si 20, most of them built by: i) stacking Si9 TTP subunits,
and relaxed applying the full-potential linear-muffin-tin-orbital molecular dynamics method (FP-LMTO-
MD) [106]. The resulting ground state is composed of Si 10 subunits. ii) Stacking of pentagonal bypiramid
Si7 subunits, that produced not stable structures, and iii) stacking of triangle structure and ring structures,
which produced sub stable clusters, and finally three structures corresponding to cage like and planar
structure which are the less stables ones of their production.
   Li [107] employed also the method FP-LMTO-MD to find the ground state structures of Si n (n = 26,
30), and for n = 20, 24, 26, 28, 30 and 32 [104]. They found that the compact structures compete with
the stacked prolate structure for n = 24, and that the prolate structures transit into the compact ones at
n = 27 for neutral silicon clusters. The same transition occurs at n = 28 for anionic and cationic silicon
clusters. For n 29, the stable ground state structures found are most of them compact structures. The
fullerene cages are not stable, and relax into structures which are distorted.
   Li et al. [108] also applied this FP-LMTO-MD for Si50 fullerene like cages, built with hexagons and
pentagons, as initial structures. The final structures are distorted cages, and appeared like puckered balls.
These distorted cages of Si 50 are stable. They repeated the procedure taking as initial structures two
stacked configurations built from tricapped trigonal prism (TTP) subunit. After relaxation the structures
resulted prolated, stable (but not the most stable ones) and had the“wire” characteristic.
   For the Si60 cage, they also found that a distortion to lower symmetry than that of the fullerene like
one, produce the stable structure. Their conclusion is that the distortion of the fullerene cages for silicon
clusters, makes some atoms four- and six-fold coordinated, with average coordination number larger
than 4. The Si20 fulllerene like cage is stable but it is not the ground state.
   Goedecker et al. [109] presented a method referred to as the dual minima hopping (MH) within
density functional theory. They explored the PES performing a global search by the minima hopping
214                 M.B. Ferraro / Prediction of structures and related properties of silicon clusters

method [110]. The first step of the search corresponds to a combination of MH and a certain number
of molecular dynamics (MD) moves until at least one barrier is overcome. In the second step the best
candidates are relaxed applying a standard optimization with a more accurate method of DFT. They
calculated accurate final energies and zero-point energies with the CPMD program [82], using the PBE
functional [44] and made a successful application for silicon clusters. Even though these systems had
already been extensively studied by other methods, they found new global minimum candidates for Si 16
and Si19 , as well as new low-lying isomers for Si16 , Si17 , and Si18 .


3. Conclusions

   In this review chapter the effort is focus in presenting an overview of the available methodologies to
predict the structure of silicon clusters, with the aim of taking into account the importance of the energetic
factors in building up the structures that are produced in the experiments or observed in the simulations.
The physics of silicon clusters might be fully interpreted taking into account also thermodynamics, and
kinetic factors. The potential range is always the strongest factor and builds up a guide for understanding
the qualitative features of structural properties and transformations.
   The joint experimental and theoretical study confirms, as example, that neutral and anionic silicon
clusters undergo a major structural change at n ≈ 26–28, and the well resolved photoelectron spectra of
clusters with more than 20 silicon atoms, is a formidable tool to identify generic structural features for
the low-lying silicon clusters.
   The development of reliable methods for modeling the energetics of nanoclusters, is a field of great
researching, both from the point of view of ab initio calculations, advisable for small systems, and from
the point of view of semi-empirical methods, and modeling, which are needed for large systems. These
approaches are complementary and are both necessary. The advantage of modeling and employment of
semi-empirical methods resides in the computational costs. Then, the recipe is using as starting point,
that kind of methodology that produces good candidates for subsequent accurate local optimization. This
last step is made usually by means of Density Functional Theory methods, but in many cases it is also
possible to use ab initio methods at MP2, CCSD, CCSD(T) level to find the lowest energy isomers in the
energy hypersurface. The intrinsic problem is that the most efficient algorithms transform the original
PES to a multidimensional system, and then it is necessary to apply, and also design, very efficient
methods to treat systems which are complicated and/or show large sizes.
   The effort in developing these methodologies and their applications has exploded in the last ten
years, with a great progress in understanding the difficulties involved for building up efficient global
optimization methods. When the total energy difference between the low-lying isomers is less than the
typical accuracy of total-energy calculations of the selected all electron method, the energy ordering of
the low-lying isomers depends strongly on the method selected to relax the best candidates for the global
minima structure.
   Genetic Algorithms have generally been shown to be more efficient than Simulated Annealing tech-
niques. The hybrid technique of combining the GA with full-electron calculations on the best candidate
structures, allows for a full exploration of the complete space available for the cluster geometry, including
those regions that represent the desirable low-energy configurations.
   The presence of structural motifs as TTP or six/six, and six/ten units offers information about the
structural evolution of silicon clusters, because each structural motif is more or less favorable for
determined clusters sizes. The hybrid technique of employing a “soup of handmade” structures built up
                     M.B. Ferraro / Prediction of structures and related properties of silicon clusters            215

with motif subunits to apply MD methodologies and final relaxation of the best candidates by an accurate
all electron method, is a very useful technique for large systems.
   In brief, we hope that this review article has given at least some overview of the state of the art on the
prediction of atomic cluster structures.


Acknowledgment

  Financial support from UBACYT(X-035), and CONICET is gratefully acknowledged.


References
  [1] U. Landman, R.N. Barnett, A.G. Scherbakov and P. Avouris, Phys Rev Lett 85 (2000), 1958.
  [2] K.D. Hirschman, L. Tsybeskov, S.P. Duttagupta and P.M. Fauchet, Nature (London) 384 (1996), 338.
  [3] S.Y. Lin, J.G. Fleming, D.L. Hetherington, B.K. Smith, R. Biswas, K.M. Ho, M.M. Sigalas, W. Zubrzycki, S.R. Kurtz
      and J. Bur, Nature (London) 394 (1998), 251.
  [4] K.M. Ho, A.A. Shavartsburg, B. Pan, Z.-Y. Lu, C.Z. Wang, J.G. Wacker, J.L. Fye and M.F. Jarrold, Nature (London)
      392 (1998), 582.
  [5] A.A. Shavrstburg et al., Chem Soc Rev 30 (2001), 26.
  [6] B. Liu, Z.Y. Lu, B. Pan, C.Z.W. Ang, K.M. Ho, A.A. Shavartsburg and M.F.J. arrold, J Chem Phys 109 (1998), 9401.
                              a
  [7] T. Bachels and R. Sch¨ fer, Chem Phys Lett 324 (2000), 365.
  [8] Q. Sun, Q. Wang, P. Jena, S. Warterman and Y. Kawazoe, Phys Rev A 67 (2003), 063201.
  [9] M. Menon and K.R. Subbaswamy, Chem Phys Lett 219 (1994), 219.
 [10] E.C.H.M.F. Jarrold, J Phys Chem 95 (1991), 9181.
 [11] M.I.R. Hudgins and M.F. Jarrold, J Chem Phys 111 (1999), 7865.
 [12] Cambridge database, http://www-wales.ch.cam.ac.uk/CCD.html.
 [13] J. Keilson, Markov Chain Models-rariry and Exponentiality, Springer Berlin (1979).
                                       a
 [14] J.A. Becker S. Schlecht, R. Sch¨ fer, J. Woenckhaus and F. Hensel, Mater Sci Eng A 217/218 (1996), 1.
 [15] J.A. Becker, Angew Chem Int 1390 (1997).
 [16] W.A. de Heer, Rev Mod Phys 65 (1993), 611.
 [17] M. Brack, Rev Mod Phys 65 (1993), 677.
             a
 [18] R. Sch¨ fer, J. Schletcht, J. Woenckhaus and J.A. Becker, Phys Rev Lett 76 (1996), 471.
 [19] J. Wang, M. Yang, G. Wang and J. Zhan, Chem Phys Lett 367 (2003), 448.
 [20] D.B.C. Pouchan and D. Zhang, J Chem Phys 121 (2004), 4628.
 [21] K.A. Jackson, M. Yang, I. Chaudhuri and Th. Frauenheim, Phys Rev A 71 (2005), 033205.
                          u
 [22] I. Vasiliev, S. Oug¨ ut and J. Chelikowsky, Phys Rev Lett 78 (1997), 4805.
 [23] K. Jackson and M.R. Pederson, Phys Rev B 55 (1997), 2549.
 [24] K. Deng, J. Yang and T. Chan, Phys Rev A 61 (2000), 25201.
 [25] A. Sieck, D. Porezag, Th. Frauenheim, M.R. Pederson and K. Jackson, Phys Rev A 56 (1997), 4890.
 [26] T.N. Kitsopoulos, C.J. Chick, A. Weaver and D.M. Neumark, J Chem Phys 93 (1990), 6108.
 [27] C.C. Arnold and D.D. Newmark, J Chem Phys 99 (1993), 3353.
 [28] K. Balasubramanian, Chem Phys Letters 125 (1986), 400; K. Raghavachari and L.A. Curtiss, in: Kluwer, Dordrecht,
      S.R. Langoff, ed., 1995, pp. 173–207; S. Li, R.J. Van Zee and W. Weltner, Jr., J Chem Phys 100 (1994), 7079.
 [29] S. Li, R.J. Van Zee, W. Weltner, Jr. and K. Raghavachari, Chem Phys Letters 243 (1995), 275.
 [30] A. Zdetsis, Phys Rev A 64 (2001), 023202.
 [31] E. Honea, A. Ogura, C. Murray, K. Raghavachari, W. Sprenger, M. Jarrold and W. Brown, Nature 366 (1993), 42.
 [32] K. Raghavachari, J Chem Phys 84 (1986), 5672.
 [33] A.D. Becke, J Chem Phys 98 (1993), 5648.
 [34] C. Lee, W. Yang and R.G. Parr, Physical Review B 37 (1988), 785.
 [35] M. Deaven and K.M. Ho, Phys Rev Lett 75 (1995), 288.
 [36] S. Yoo, X.C. Zeng, J Chem Phys 123 (2005), 164303; S. Yoo and X.C. Zeng, Agnew Chem Inst 44 (2005), 1491.
 [37] J.P. Perdew, Phys Rev B 33 (1986), 8822.
 [38] A.D. Becke, Phys.Rev A 38 (1988), 3098.
 [39] J.P. Perdew, K. Burke and Y. Wang, Phys Rev B 54 (1996), 16533.
 [40] J.P. Perdew, K. Burke and M. Ernzerhof, Phys Rev Lett 78 (1997), 1396.
216                   M.B. Ferraro / Prediction of structures and related properties of silicon clusters

[41]   K.A. Jackson, M. Horioi, I. Chaudhuri, T. Frauenheim and A.A. Shvartsburg, Phys Rev Lett 93 (2004), 13401.
[42]   A.C. Liu and J. Nocedal, Math Prog 45 (1989), 403.
[43]                                        o
       D. Porezag, Th. Frauenheim, Th. K¨ hler, G. Seifert and R. Kaschner, Phys Rev B 12 (1995), 947.
[44]   J.P. Perdew, K. Burke and M. Ernzerhof, Phys Rev Lett 77 (1996), 3865.
[45]   O. Guliamov, L. Kronik and K.A. Jackson, J Chem Phys 123 (2005), 204312.
[46]                                                      u               o
       M.A. Hoffmann, G. Wrigge, B.v. Issendorff, J. M¨ ller, G. Gantef¨ r and H. Haberland, Eur Pys J D 16 (2001), 9.
[47]   K.A. Jackson, M. Horoi, I. Chaudhuri, T. Frauenheim and A.A. Shvartsburg, Comp Mater Sci 35 (2006), 232.
[48]   J.R. Chelikowsky, L. Kronik, I. Vasiliev, M. Jain and Y. Saad, in: Handbook of Numerical Analysis, (Vol. X), C.L. Bris,
       ed., Elsevier, Amsterdam, 2003, pp. 613–637.
[49]                                      ¨
       J.C. Idrobo, K.A. Jackson and S. Ogut, Phys Rev B 74 (2006), 153410.
[50]   K.A. Jackson, M. Horoi, I. Chaudhuri, T. Frauenheim and A.A. Shvartsburg, Phys Rev Lett 93 (2004), 013401.
[51]   K.A. Jackson, M. Horoi,I. Chaudhuri, T. Frauenheim and A.A. Shvartsburg, Comp Mater Sci 35 (2006), 232; Phys Rev
       Lett 93 (2004), 013401.
[52]   K.D. Rinnen and M.L. Mandich, Phys Rev Lett 69 (1992), 1823.
[53]   M.F. Jarrold and J.E. Bower, J Chem Phys 96 (1992), 9180.
[54]   P.G. Collins, J.C. Grossman, M. Ct, M. Ishigami, C. Piskoti, S.G. Louie, M.L. Cohen and A. Zettl, Phys Rev Lett 82
       (1999), 165.
[55]   J.P. Perdew, J.A. Chevary, K. A. Jackson, S.H. Voksko, M.R. Pedersen, D.J. Singh and C. Fiolhais, Phys Rev B 46
       (1992), 6671.
[56]   G. Kresse and D. Joubert, Phys Rev B 59 (1999), 1758.
[57]   J. Wang, X. Zhou, G. Wang and J. Zhao, Phys Rev B 71 (2005), 113412.
[58]   M. Menon and K.R. Subbaswamy, Phys Rev B 50 (1994), 11577.
[59]   V.E. Bazterra, M.B. Ferraro and J.C. Facelli, J Chem Phys 116 (14) (2002), 5984.
[60]   S. Yoo, X. Zhou, X. Zhu and J. Bai, J Am Chem Phys 125 (2003), 13318.
[61]   D.J. Wales and H.A. Scheraga, Science 285 (1999), 1368.
[62]   S. Yoo and X. Zhou, J Chem Phys 119 (2003), 1442.
[63]   K.T.K. Fuke, F. Misaizu and M. Sanekata, J Chem Phys 99 (1993), 7807.
[64]   V. Bazterra, M.B. Ferraro and J.C. Facelli, J Chem Phys 116 (2002), 5984; V. Bazterra, M.B. Ferraro and J.C. Facelli, J
       Chem Phys 116 (2002), 5992.
[65]   V.E. Bazterra, M.C. Caputo, M.B. Ferraro and P. Fuentealba, J Chem Phys 117(24) (2002), 11158.
[66]   B. Li and P. Cao, Phys Rev B 256 (2005), 235311.
[67]   A.G. Cullis and L.T. Canham, Nature 353 (1991), 335.
[68]           a             a
       R. Sch¨ fer and S. Sch¨ fer Schletcht, J. Woenckhaus and J. Becke, Phys Rev Lett 76 (1996), 471.
[69]                       n
       V E. Bazterra, O. O˜ a, M.C. Caputo, M.B. Ferraro, P. Fuentealba and J.C. Facelli, Phys Rev A 69 (2004), 53202.
[70]   B. Ahlswede and K. Jug, J Comp Chem 20 (1999), 563; B. Ahlswede and K. Jug, J Comp Chem 20 (1999), 572; T.
       Bredow, G. Geudtner and K. Jug, J Comp Chem 22 (2001), 861.
[71]   D. Whitley, Presented at the Rocky Mountain Conference on Artificial Intelligence, Denver, Colorado, 1988 (unpub-
       lished); D. Whitley, presented at the Third International Conference on Genetic Algorithms, San Mateo, California,
       1989 (unpublished); G. Syswerda, presented at the Third International Conference on Genetic Algorithms, San Mateo,
       California, 1989 (unpublished).
[72]   V.E. Bazterra, M. Cuma, M.B. Ferraro and J.C. Facelli, J of Parallel and Distrib Comput 65 (2005), 48.
[73]   P.J. Hay and W.R. Wadt, J Chem Phys 82 (1985), 270.
[74]   L.W.S.H. Vosko and M. Nusair, Can J Phys 58 (1980), 1200.
[75]         n
       O. O˜ a, V.E. Bazterra, M. Caputo, P. Fuentealba, J. Facelli and M.B. Ferraro, Phys. Rev A 73 (2006), 053203.
[76]   Q. Sun, Q. Wang, P. Jena, S. Warterman and Y. Kawazoe, Phys Rev Lett 90 (2003), 135503.
[77]   K. Jackson, M.R. Pederson, D. Porezag, Z. Hajnal and T. Frauenheim, Phys Rev B 55 (1997), 2549.
[78]   K.A. Jackson, M. Yang, I. Chaudhuri and F.T. Phys, Rev A 71 (2005), 033205.
[79]        n
       O.O˜ a, V.E.B. Azterra, M.C. Caputo, J.C. Facelli, P. Fuentealba and M.B. Ferraro, Phys Rev A 73 (2006), 53203.
[80]   S. Yoo, J. Zhao, J. Wang and X.C. Zeng, J Am Chem Phys 126 (2004), 13845.
[81]   M. Deaven and K.M. Ho, Phys Rev Lett 75 (1995), 288; J.J. Zhao and R.H. Xie, J Compt Theoret Nanosci 1 (2004),
       117.
[82]   Michele Parrinello, Jurg Hutter, D. Marx, P. Focher, M. Tuckerman, W. Andreoni, A. Curioni, E. Fois, U. Roetlisberger,
       P. Giannozzi, T. Deutsch, A. Alavi, D. Sebastiani, A. Laio, J. VandeVondele, A. Seitsonen, S. Billeter and others,
       Car-Parinello Molecular Dynamics CPMD is copyrighted by IBM Corp and MPI Stuttgart, 1993–2006.
[83]   J.Z. Li Ma, Jianguang Wang, Baolin Wang, and Guanghou Wang, Phys Rev A 73 (2006), 063203.
[84]   M.D. Segall, P.J.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark and M.C. Payne, J Phys Condensed
       Matter 14 (2002), 2717.
[85]   B. Delley, J Chem Phys 92 (1990), 508.
                       M.B. Ferraro / Prediction of structures and related properties of silicon clusters                 217

 [86]   M. Menon and K.R. Subbaswamy, Phys Rev B 55 (1997), 9231.
 [87]   A. Sieck, D. Porezag, T. Frauenheim, M.R. Pederson and K. Jackson, Phys Rev A 56(6) (1997), 4890.
 [88]   M.R. Pederson, K. Jackson, D. Porezag, D. Hajnal and Th. Frauenheim, Phys Rev B 54 (1996), 2863.
 [89]   K. Deng, J. Yang, L. Yuan and Q. Zhu, Phys Rev A 62 (2000), 045201.
 [90]        o
        U. R¨ thlisberger, W. Andreoni and P. Giannozzi, J Chem Phys 96 (1992), 1248.
 [91]   B. Liu, Z.Y. Lu, B. Pan, C.Z. Wang, K.M. Ho, A.A. Shvartsburg and M.F. Jarrold, J Chem Phys 109 (1998), 9401.
 [92]   Z.M. Khakimov, P.L. Tereshchuk, N.T. Sulaymanov, F.T. Umarova and M.T. Swihart, Phys Rev B 72 (2005), 115335.
 [93]   S. Yoo, N. Shao, C. Koehler, T. Frauunham and X.C. Zeng, J Chem Phys 124 (2006), 164311.
 [94]   J. Bai, L. Cui, J. Wang, S. Yoo, X. Li, J. Jellimek, C. Koehler, T. Frauenheim, L. Wang and X.C. Zeng, J Phys Chem A
        110 (2006), 908.
 [95]   J.L. Elkind, J.M. Alford, F.D. Weiss, R.T. Laaksonen and R.E. Smalley, J Chem Phys 87 (1987), 2397; Q.L. Zhang, Y.
        Liu, R.F. Curl, F.K. Tittel and R.E. Smalley, J Chem Phys 88 (1988), 1670.
 [96]   C.C. Arnold and D.D. Newmark J Chem Phys 99 (1993), 3353.
 [97]   A.B.C.P.R.L. Zhou, Phys Rev B 73 (2006), 045417.
 [98]   C.Z. Wang, B.C. Pan and K.M. Ho, J Phys Condens Matter 11 (1999), 2043.
 [99]        a                        o
        D. S´ nchez-Portal, P. Ordej´ n, E. Artacho and J.M. Soler, Int J Quantum Chem 65 (1997), 453; N. Troullier and J.L.
        Martins, Phys Rev B 43 (1991), 1993.
[100]   M.F. Jarrold, J Phys Chem 99 (1995), 11.
[101]   A. Shvartsburg, M.F. Jarrold, B. Liu, Z.Y. Lu, C.Z. Wang and K.M. Ho, Phys Rev Lett 81 (1998), 4616.
[102]   G.L. Mitas, J.C. Grossman, I. Stich and J. Tobik, Phys Rev Lett 84 (2000), 1479.
[103]   J. Song, S. E.Ulloa and D.A. Drabold, Phys Rev B 53 (1996), 8042.
[104]   B. Li, and P. Cao, J Phys: Condens Matter 13 (2001), 10865.
[105]   B. Li and P. Cao, Phys Rev A 62 (2000), 023201.
[106]   M. Methfessel and M.V. Schilfgaarde, Phys Rev B 48 (1993), 4937; M. Methfessel, Phys Rev B 38 (1988), 1537; M.
        Methfessel, C.O. Rodriguez and O.K. Andersen, 40 (1989), 2009.
[107]   B. Li and P. Cao, Phys Rev B 71 (2005), 235311.
[108]   B. Li, P. Cao, B. Song and Z. Ye, J Mol Structu (Theochem) 620 (2003), 189.
[109]   S. Goedecker, W. Hellmann and T. Lenosky, Phys Rev Lett 95 (2005), 55501.
[110]   S. Goedecker, J Chem Phys 120 (2004), 9911.
[111]   K. Jackson, M. Pederson, C.-Z. Wang and K.-M. Ho, Phys Rev A 59(5) (1999), 3685.
[112]                                                a
        A. Rubio, J.A. Alonso, X. Blase, L.C. Balb´ s and S.G. Louie, Phys Rev Lett 77 (1996), 5442E; Phys Rev Lett 77 (1996),
        247.
Journal of Computational Methods in Sciences and Engineering 7 (2007) 219–232                                                 219
IOS Press




Role of surface passivation and doping in
silicon nanocrystals

R. Magria,∗ , E. Degolib , F. Ioria , E. Luppia , O. Pulcic , S. Ossicinib, G. Canteled , F. Tranid and
D. Ninnod
a CNR-INFM-S3-CNISM                                            `
                         and Dipartimento di Fisica, Universit a di Modena e Reggi Emilia, Via Campi
213/A, I-41100 Modena, Italy
b CNR-INFM-S3 and Dipartimento di Scienze e Metodi dell’Ingegneria, Universit a di Modena e Reggio
                                                                                `
Emilia, via G. Amendola 2, I-42100 Reggio Emilia, Italy
c European Theoretical Spectroscopy Facility (ETSF) and Dipartimento di Fisica – Universit a di Roma
                                                                                           `
“Tor Vergata” Via della Ricerca Scientifica 1, I-00133 Roma, Italy
d CNR-INFM-Coherentia and Dipartimento di Fisica, Universit a di Napoli “Federico II”, Complesso
                                                                `
Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy

Received 14 March 2007
Accepted 8 May 2007

Abstract. The absorption and the emission spectra of undoped and doped silicon nanocrystals of different size and surface
terminations have been calculated within a first-principles framework. The effects induced by the creation of an electron-hole pair
on the atomic structure and on the optical spectra of hydrogenated silicon nanoclusters as a function of dimension are discussed
showing the strong interplay between the structural and optical properties of the system. Starting from the hydrogenated clusters,
(i) different Si/O bonding at the cluster surface and (ii) different doping configurations have been considered. We have found
that the presence of a Si-O-Si bridge bond at the nanocrystal surface gives rise to significant excitonic luminescence features in
the near-visible range that are in fair agreement with photoluminescence (PL) measurements on oxidized and SiO2 embedded
nanocrystals. The study of the structural, electronic and optical properties of simultaneously n- and p-type doped hydrogenated
silicon nanocrystals with boron and phosphorous impurities have shown that B-P co-doping is energetically favorable with
respect to single B- or P-doping and that the two impurities tend to occupy nearest neighbors sites. The co-doped nanocrystals
present band edge states localized on the impurities that are responsible of a red-shifted absorption threshold with respect to
that of pure un-doped nanocrystals in agreement with the experiment.



1. Introduction

  The extreme integration levels reached nowadays by Si microelectronics industry have permitted high
speed performance and unprecedented interconnection levels. However, the present interconnection
degree is sufficient to cause interconnect propagations delays, overheating and information latency.
To overcome these problems, photonic materials, in which light can be generated, guided, modulated,
amplified and detected, need to be integrated with standard electronics circuits to combine the information
processing capabilities of electronic data transfer and the speed of light. In particular, chip to chip or

  ∗
      Corresponding author. E-mail: magri.rita@unimo.it.


1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
220                 R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals

even intra-chip optical communications all require the development of efficient optical functions and
their integration with state-of-the-art electronic functions [1]. Silicon is the desired material, because
Si-based optoelectronics would open the door to faster data transfer and higher integration densities at
low cost.
   The main limitation of a silicon based photonics remains the lack of any practical Si-based light
sources. Several attempts have been employed to engineer luminescent transitions in an otherwise
indirect material [1]. Following the discovery of photoluminescence (PL) from porous silicon [2],
extensive experimental and theoretical work has been devoted to nanostructured silicon [3], with the
aim to get relevant optoelectronic properties from it. Optical gain was then found in Si-nc embedded in
SiO2 [4]. It is generally accepted that the quantum confinement (QC), caused by the nanometric size, is
essential for obtaining PL emission in Si-nc, but some PL features, such as: (i) the red-shift of the PL
peak with respect to the absorption onset (Stokes Shift); (ii) the substantial redshift (RS) of the PL energy
with respect to the theoretical predictions based merely on the QC model and its independence from the
size for small (< 3 nm) crystallites, need still to be explained. Baierle et al. [5] and G. Allan et al. [6]
stressed the importance of bond distortion at the Si-nc surface in the excited state (EXC) in creating an
intrinsic localized state responsible of the PL emission. Wolkin et al. [7] observed that also oxidation
introduces states in the gap, which pin the transition energies. They and others [8,9] suggested that the
formation of a Si=O double bond is responsible of the RS of the optical absorption edge upon oxidation.
On the contrary, Vasiliev et al. [10] showed that similar absorption gaps can be obtained also in the
case of the O atom connecting two Si atoms (bridge bond) at the Si-nc surface. Recently TD-ALDA
calculations of Gatti and Onida [11] on six small different prototypical oxidized Si clusters found that
the RS of the absorption edge is much more pronounced in the case of the double Si = O bond than
for bridge bonds. Same results were reached also in Ref. [12] where many oxidized configurations for
silicon nanocrystals were considered using a TDDFT-B3LYP approach. Although all these calculations
address the absorption, yet the large majority of the experimental results and the most interesting ones
are relative to PL measurements, thus are strictly related to the excited state.
   Another way to circumvent the indirect gap behavior of bulk Si is given by the introduction in the
Si-nc of an isoelectronic impurity or by a simultaneous n- and p-type impurity doping [13]. It has
been shown that the PL peak can be tuned also below the bulk Si band gap by properly controlling the
impurities, for example by B and P co-doping [13–15]. Besides, under resonant excitation condition,
the co-doped samples did not exhibit structures related to momentum-conserving phonons, suggesting
that in this case the quasidirect optical transitions are predominant. Only few first principles studies of
impurities in silicon quantum dots are present in the literature, and they are mainly devoted to quantum
confinement effects in singly doped Si-nc [16–18]. The results point out that the ionization energy of
the singly doped Si-nc is virtually size independent and that the donor and acceptor binding energies are
substantially enhanced.
   In this paper we address the issue of the role played by the excited state in the determination of the
Stokes Shifts and the PL peak position in undoped and doped small hydrogenated Silicon clusters of
different size and different surface terminations. In Section 2 we will describe the Constrained Density
Functional Method applied to hydrogenated silicon clusters and report our results for the Stokes Shifts.
In Section 3 we will show the absorption and emission spectra calculated for a small hydrogenated
silicon cluster with an oxygen atom adsorbed on its surface. The optical spectra have been calculated
also including excitonic effects. Finally, in Section 4 we discuss the effects of boron and phosphorus
doping and co-doping on the impurity formation energies and on the electronic and optical properties of
Si-nc. Some conclusions will be drawn at the end.
                    R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals     221

2. Ground and excited state of hydrogenated Si nanocrystals

   The study of the silicon nanocrystals (Si-nc) has been done within the DFT, using a pseudopotential,
planewave approach [19]. The calculations of this section have been performed with the ABINIT
code [20]. Norm-conserving, non-local Hamann-type pseudopotentials have been used. The Kohn-
Sham wave functions have been expanded in a plane-wave basis set, with an energy cutoff of 32 Ry.
Each Si-nc has been embedded within a large cubic supercell, containing sufficient vacuum in order to
make the interactions between nanocrystals negligible. Convergence with respect to both the energy
cutoff and supercell side has been carefully checked. A gradient corrected Perdew-Burke-Ernzerhof
(GGA-PBE) exchange-correlation functional has been used for both structural and electronic properties
calculations.
   In the constrained DFT method, also referred to as the ∆SCF approach, the excited state corresponds
to the electronic configuration in which the highest occupied single-particle state (HOMO) contains a
hole (h), while the lowest unoccupied single-particle state (LUMO) contains the corresponding electron
(e). The nanocluster excitations are thought to occur when the atomic positions are fixed in their
ground-state equilibrium geometry. E(N ) is the N -electron ground-state energy and E(N ; e − h)
the total energy of the nanocluster calculated with the electron-hole pair constraint. The difference
εA = E(N ; e − h) − E(N ) gives the energy needed for the creation of the pair, and defines the
absorption edge. After the excitation, due to the change in the charge density, atomic relaxation occurs
until the atoms reach a new minimum energy configuration, in the presence of the electron-hole pair.
The emission energy is defined as ε E = E (N ; e − h) − E (N ), where E (N ; e − h) and E (N ) are the
total energies evaluated in the presence and in the absence of the electron-hole pair, respectively, when
the atoms occupy the equilibrium positions appropriate to the excited (e − h) state. The absorption and
emission processes are represented schematically in Fig. 1.
   The difference ∆EStokes = εA − εE defines the Stokes shift. Thus, the Stokes shift arises from
atomic relaxation following the excitation process. This model relies on the assumption that the atomic
relaxation under excitation is faster than the radiative electron-hole recombination. Our calculations are
not spin-polarised, however similar computations performed by Franceschetti and Pantelides [21] within
local spin-density approximation, showed that the singlet-triplet splitting is significantly smaller than the
Stokes shift.
   Because of the large surface to volume ratio, the presence of an electron-hole pair in the clusters causes
a strong structural deformation with respect to the ground-state atomic geometry. The atomic deformation
is larger for the smaller systems. As a result, the difference between absorption and HOMO-LUMO
ground state (GS) gap and between emission and HOMO-LUMO excited state (EXC) gap increases
with the diminishing of the nanocluster dimension. In particular, the GS HOMO-LUMO gap tends to
be smaller than the absorption energy while the EXC HOMO-LUMO gap tends to be larger than the
emission energy, so that trying to deduce the Stokes Shift simply from the HOMO-LUMO gaps leads to
errors especially large for small clusters.
   Our results are collected in Table 1. We see that the Stokes shifts diminish with the increase of the
H-Si-nc size and are substantially in agreement with those of Puzder et al. [22] and Franceschetti et
al. [21] although our values lie between theirs. The discrepancies could be due to the different codes
used to perform the calculations.
   We have also calculated the optical spectra for some small clusters using: (i) LDA Kohn and Sham
eigenvalues and eigenvectors in the Random Phase Approximation (RPA-LDA, neglecting Local Fields),
(ii) quasi-particle self-energy corrected eigenvalues within the GW method (RPA-GW), (iii) a time de-
pendent adiabatic local density approximation (TD-ALDA), and (iv) a many-body perturbation approach
222                   R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals

                                                         Table 1
                   Stokes Shift values for hydrogenated Si clusters: present work versus theoretical data
                   present in literature
                   H-Si       Diameter                          Theory
                   clusters     (nm)       This work        Ref. 22       Ref. 21    Ref. 25    Ref. 26
                   Si1 H4        0.0         8.38
                   Si5 H12       0.45        5.67
                   Si10 H16      0.55        4.40       LDA      QMC
                   Si29 H36      0.9         1.35       0.69      1.0       2.92       0.22       0.70
                   Si35 H36      1.1         0.92       0.57      0.8                             1.67
                   Si66 H64      1.3                    0.50
                   Si87 H76      1.5                    0.22                0.32
                   Si29 H24      0.8          0.84      0.34      0.4                             1.17




Fig. 1. Schematic representation of the absorption and emission processes assumed in the Constrained Density Functional
Approach.


through the solution of the Bethe-Salpeter equation (BSE). Both LDA-RPA and GW-RPA are within
the scheme of the independent particle RPA (IP-RPA). For Si 5 H12 (see Fig. 2) the RPA-LDA scheme
underestimates the gap with respect to the experimental value (6.5 eV) [23]. The RPA-GW method
(which includes the self-energy corrections to the Kohn-Sham eigenvalues) opens the gap. The effects of
the electron-hole interaction on the optical properties (through BSE) are also quite large, the calculated
excitonic binding energy being of the order of 3 eV. The optical gap is reduced partially compensating the
GW opening. Interestingly, the BSE and TDLDA results are similar for the absorption onset with the first
excitonic peak around 6.5 eV in agreement with the experimental result. A very accurate calculation of
the optical gap of Si5 H12 using both a TDDFT- B3LYP and a multi-reference second-order perturbation
theory MR-MP2 approach has given the values 6.66 eV and 6.76 eV, respectively, in good agreement
with our theoretical value [24].
                       R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals                 223




Fig. 2. (Color online) Absorption spectra of the Si5 H12 cluster calculated using different DFT based methods: LDA-RPA black
(black), GW-RPA red (dark gray), BSE violet (light gray), TDLDA blue (light black).

3. Absorption and emission spectra: The role of nanocrystal surface oxidation

   The aim of the present section is to investigate the mechanisms involved in the modification of the
electronic and optical properties of Si-nc when they are oxidized. We start from an hydrogenated cluster
e.g. the Si10 H16 nanocrystal, and add an oxygen atom to its ground state atomic structure. This can be
made either without removing H atoms, hence leading to Si 10 H16 O, with an O atom placed on one of the
12 equivalent Si-Si bonds, or by replacing a pair of hydrogen atoms with one oxygen (in this case, no
Si-Si bond is broken). In the latter case, one is led to one of four different isomers of Si 10 H14 O. In these
isomers, oxygen is covalently bonded to silicon, with either a double bond (“double” isomer, containing
a Si = O bond like in the silanone H 2 SiO structure), or with different bridge-bonds (i.e., Si-O-Si bonds).
In particular, oxygen can make a bridge between two “first neighbors” or “second neighbors” Si atoms
(called “asym” and “sym” isomers) or it can lay in an “interstitial” position inside the Si 10 cage (“interst”
isomer). The cluster structures we have considered are reported in Fig. 3.
   We have found that the isomer with double Si = O bond (i.e., Si 10 H14 O-double) undergoes smaller
atomic relaxations than isomers with bridge bonds: as a consequence the Si 10 H14 O-sym is 1.7 eV more
stable than Si10 H14 O-double. Oxidation induces significant changes in the electronic properties. First,
one expects a splitting of the degenerate Kohn-Sham levels of Si 10 H16 , due to the symmetry reduction;
second, adding an oxygen atom will introduce electronic states with respect to the non-oxidized cluster,
third, the oxygen-related states appear inside the energy region of the Si 10 H16 HOMO-LUMO gap. It is
then interesting to calculate the absorption and emission optical spectra. For both the ground and excited
state optimized geometries, the transition energies and the optical response, Im ε(ω) (the imaginary
part of the nanocrystal dielectric function), are evaluated through first-principles calculations beyond
the one-particle approach. We consider Si 10 H16 , Si10 H14 O-double (Si10 H14 = O), and Si10 H14 O-sym
(Si10 H14 > O). First, we have calculated the Stokes Shifts as the difference between the band edge states
in absorption and emission. The Stokes-Shifts are calculated using: (1) TDLDA, (2) GW-BSE and (3)
the constrained-LDA and are reported in Fig. 4. We can see that the Stokes Shifts calculated using the
three different approaches for the hydrogenated system and for the cluster with the double Si = O bond
agree, while for the system with the bridge bond there are relevant differences. In this last case the BSE
approach shows that more than one e − h pair has a substantial contribution to the lower energy exciton.
224                     R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals




Fig. 3. (Color online) Structure of Si10 H16 , Si10 H16 O and the four isomers of Si10 H14 O. In the ball and stick representation,
Si is light blue (gray), H is gray (light gray), and O is red (dark gray).

Obviously CDFT works best when the excited state is a linear combination of e − h pairs with only one
single dominant component (e.g. the transition between the HOMO and LUMO levels). However we
notice that, by performing a CDFT calculation, one allows all the orbitals to relax hence the resulting
excited state, even if represented by one electron-hole pair, is not the same as the unrelaxed state which
enter the linear combination of e − h pairs building up the excited state in the BSE scheme. Thus some
orbital mixing is present even at the CDFT level.
   For the absorption and emission spectra we consider the self-energy corrections [27] by means of the
GW method and the excitonic effects through the solution of the Bethe-Salpeter equation [28]. The
effect of local fields is included, to take into account the inhomogeneity of the systems. Actually, we
found that, in the case of finite systems, such as clusters, the effect of the local fields on the optical
spectra (both absorption and emission) is the stronger components, much more relevant than the excitonic
effects. The inclusion of the local fields, even in the simple RPA approach, leads to a blueshift of the
main peaks and substantially changes the transition oscillator strengths. In order to perform emission
spectra calculations, we use the excited state atomic geometry together with the ground state electronic
configuration. Thus, strictly speaking, Im ε(ω) corresponds to an absorption spectrum in the new
structural geometry. In other words, we consider the emission, in first approximation, simply as the time
reversal of the absorption [29]. We show the results in Fig. 5. Each panel reports the imaginary part
of the dielectric function for absorption (dashed line) and emission (solid line) for the three considered
                        R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals                    225




Fig. 4. (Color online) Calculated Stokes shift for the Si10 H16 , Si10 H14 > O and Si10 H14 = O double clusters. The calculation
have been performed using GW + BSE: left, red (dots), TDLDA: middle, blue (squares), constrained LDA: right, purple
(triangles) methods.

clusters. Self-energy, local-fields and excitonic effects (BSE-LF) are taken into account. The absorption
features of the three cases are similar showing an increase of the absorption with the energy. On the
contrary, the emission-related spectra are clearly different. Whereas the fully hydrogenated Si 10 H16
cluster and the Si10 H14 = O cluster show similar emission, in the case of the Si-O-Si bridge bond
(bottom panel) an important excitonic peak, separated from the rest of the spectrum, is evident at 1.5 eV.
Actually, bound excitons are present also in the fully hydrogenated (at 0.4 eV) and in the Si 10 H14 = O
(at 1.0 eV) clusters, with calculated binding energies even larger than in the case of the Si-O-Si bridge
bond (3.4 and 3.6 eV respectively, to be compared with a binding energy of 2.0 eV in the case of the
bridge bond cluster). Nevertheless, the related transitions are almost dark and the emission intensity is
very low. Only in the case of the Si-O-Si bridge bond the photoluminescence peak appears thanks to the
strong oscillator strength of the related transition.
   The comparison of our results with the experiment suggests that the presence of a Si-O-Si bridge bond
at the surface of a Si-nc can explain the nature of luminescence: only in this case the presence of an
excitonic peak in the emission related spectra, red shifted with respect to the absorption onset, provides
an explanation for both the observed Stokes Shift and the PL in the near-visible range. It is worth to
stress that the role of the interface has been experimentally proven to be important for the PL properties
of embedded Si-nc in SiO2 and suggested to play a role in the mechanism of population inversion at
the origin of the optical gain [30]; besides, Monte Carlo approaches have demonstrated that Si-O-Si
bridge bonds are the main building blocks in the formation of Si-SiO 2 flat interfaces [31] and form the
low energy structures at the interface of Si-nc embedded in silicon dioxide [32]. In conclusion, our
226                     R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals




Fig. 5. (Color online) Absorption (dashed line, black) and Emission (solid line, red) spectra in the ground state and excited state
geometries, respectively for Si10 H16 (top panel), Si10 H14 = O (central panel) and Si10 H14 > O (bottom panel).

theoretical results, obtained by ab-initio calculations including excitonic effects, suggest that the Si-O-Si
bridge bond may be responsible for the observed strong PL peak, and shed some light on the role of the
Si-nc-SiO2 interface.


4. Doping silicon nanocrystals

  There is experimental evidence that doping control at the nanoscale can add optical properties which
cannot be achieved in pure systems. In the case of silicon nanocrystals it has been shown that the PL
peak can be tuned even below the bulk Si band gap by properly controlling the impurities, for example by
B and P co-doping [15]. We have investigated the structural changes of Si-nc after inserting the dopant
impurity as a function of the: (i) nc size; (ii) the impurity position within the nanocluster and (iii) the
number of doping species. We performed our calculations using a plane-wave, pseudopotential density
functional approach. We consider B and P impurities in substitutional sites within spherical Si-nc, with
diameter ranging from 1.04 nm (Si29 H36 ) to 2.24 nm (Si29 3H172 ). Full relaxation with respect to the
atomic positions is performed for both doped and undoped systems using the ESPRESSO package [33],
within the GGA approximation using Vanderbilt ultrasoft [34] pseudopotentials. The Si-nc have been
embedded in large supercells in order to prevent interactions between the periodic replicas.
  First, we have studied the change of the structural properties due to the impurity presence. It comes
out that the amount of relaxation around the impurity is directly related to the impurity valence. A
more significant distortion is found for the trivalent impurity (boron). In fact, for the B-doped clusters,
while the Si-Si bond lengths remain almost unchanged, some relaxation occurs around the impurity. The
overall structure acquires a C 3v symmetry, with the impurity displaced along the <111> direction from
                    R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals      227

the nanocluster center. Such displacement leads to one longer and three shorter (and equal) Si-impurity
distances. While the longer bond length does not depend much on the nc size, the shorter ones decrease
when the nc size increases. It is interesting to note that the relaxation of bulk Si containing one B
impurity leads to an “almost” Td symmetry, in which the four B-Si bonds are practically the same. For
a pentavalent impurity, such as P, the relaxation leads to a nearly T d symmetry, in which the differences
between the four P-Si bonds are negligible, less than 0.7%.
  The formation energy (FE) of a neutral X impurity is defined as the energy needed to insert the X
atom with chemical potential µX within the cluster after removing a Si atom which is transferred to the
external chemical reservoir, assumed to be bulk Si:

    Ef = E (Sin−1 XHm ) − E (Sin Hm ) + µSi − µX

where E is the total energy of the system, µ Si the total energy per atom of bulk Si, µ X the total energy per
atom of the impurity (the total energy per atom in the tetragonal B 50 structure for B, and the orthorhombic
black phosphorus for P). Our calculations show that for smaller Si-nc a larger energy is required to bind
the impurity. For B-doped Si-nc we found a decreasing behavior of E f vs. 1/R, that can be described by
the linear formula:

    Ef = 0.80 + 4.64/R

where R is expressed in Å and E f in eV, and the value E f = 0.80 eV corresponds to the B impurity in
bulk Si. For P-doped Si-nc the same decreasing behavior of E f vs. 1/R has been found, with a linear
formula:

    Ef = 0.21 + 4.98/R

   The calculated formation energy is lower for larger Si-nc. This behaviour is in qualitative agreement
with the observed suppression of the PL in doped Si nanocrystals, since doping of the nanocrystal with
III or V group shallow dopants tends to suppress the photoluminescent emission. Fujii et al. [15] have
indeed shown that on increasing the annealing temperature both the Si-nc size increases and a stronger
PL suppression is observed. This effect is a signature of an higher impurity concentration, thus showing
that larger Si-nc can more easily sustain the doping.
   The formation energy changes also as a function of the impurity site within the Si-nc. It is much lower
when the B impurity is located in the sub-surface Si layer. In Fig. 6 we show the formation energy of
a B neutral impurity in the Si146 BH100 cluster. The impurity is located in many sites from the cluster
center toward the surface along two paths, as shown in Fig. 6(a). The calculated energies are shown in
Fig. 6(b). On the x-axis we put the distance from the center of the replaced Si atom in the Si 147 H100
cluster. The more stable sub-surface sites are explained by considering that atomic relaxation around
the impurity is easier in such positions. Thus, as the B atom is moved toward the surface the formation
energy decreases. The local structure around the impurity has a C 2v symmetry, with two shorter and two
longer Si-impurity distances (the two bonds with the Si surface atoms). Recently these findings have
received an experimental support [35].
   When the Si nc is doped with both a B and a P atoms (co-doping) the differences among the impurity-Si
bond lengths are smaller and an almost T d symmetry is recovered.
   This fact is reflected in the formation energy results, which are reported in Fig. 7, for singly B-,
P-doped and (B-P)-co-doped Si-nc. In all cases the impurities are located in subsurface positions. In the
figure at the top the neutral impurities are located at the largest possible distances, at the bottom they are
228                     R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals




Fig. 6. Formation energies for neutral impurities as a function of the impurity position within the cluster (b). The impurity is
moved along two different paths toward the surface, as shown in (a).

nearest neighbors. Figure 7 shows that the simultaneous (P-B) doping strongly reduces (by about 1 eV)
the formation energy with respect to both the two single-doped cases and, in the case the two dopants
are near-neighbours, the co-doped Si-nc is even stable (negative FE). The formation energy reduction
is almost independent from the Si-nc size. Thus, Si-nc can be more easily simultaneously doped than
singly doped; this is a consequence of a charge transfer which entails a minor structural deformation.
Moreover the formation energy is lower when the impurities are nearest neighbours, thus confirming the
important role played by the electrostatic attraction.
   The dopants insert donor or acceptor states within the Si-nc gap lowering the HOMO-LUMO energy
gap EG . For single-doped Si-nc the dopant level falling within the gap is strongly localized either on
B or P impurity. For example in the case of Si 86 BH76 nanocrystals the defect level is located above
the valence band and the energy gap is reduced from 2.59 to 2.31 eV, whereas for the Si 86 PH76 doped
nanocrystals the defect level is located below the conduction band and the energy gap is 2.34 eV. The
electronic properties of (B-P)-codoped Si-nc are qualitatively and quantitatively different from those of
either B- or P- doped Si-nc. For the Si85 BPH76 nanocrystals EG is lowered from 2.59 eV (pure Si-nc)
to 1.82 eV (co-doped Si-nc). The results for the Si 145 H100 case are similar: EG is reduced by codoping
from 2.30 eV (pure Si-nc) to 1.56 eV. In the case of Si-nc larger than those considered here, which have
a smaller EG , it would be possible by co-doping to obtain an E G even smaller than that of bulk Si in
agreement with the experiment [12,14]. In the co-doped case the HOMO is strongly localized on the B
impurity and the LUMO on the P impurity.
   We have calculated the absorption and emission spectra of co-doped Si-nc, comparing the IP-RPA
                        R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals                        229




Fig. 7. Formation energy of the neutral impurities located at subsurface positions as a function of doping: B (left), P (right) and
(B-P) (middle) doped nanoclusters. The lines are a guide for the eyes. Dashed lines (green and blue): the neutral impurities are
separated from each other by the largest possible distance within the Si-nc; solid lines (black and red): the neutral impurities are
located at nearest neighbor distances in the codoped clusters. Squares (green and black) are related to the Si87 H76 nanoclusters,
circles (red and blue) to the Si147 H100 ones.

spectra with those where the many body effects are included, within a GW-BSE approach. This approach
takes into account the self-energy correction (in the GW approximation), the local fields, and the electron-
hole interaction. We analyze separately the role on the optical spectra of: (i) the nanocrystal dimension,
(ii) the distance between the B and P impurities and (iii) the many body effects.
   We find that the dopant states in the Si-nc gap give rise to new optical transitions below the absorption
onset of the un-doped Si-nc.
   In Fig. 8 we report the IP-RPA absorption spectra of the Si 145 BPH100 cluster. The energy range at the
absorption onset where the optical transitions due to the dopant levels contribute most is shown in detail.
The figure shows how the features at the absorption onset change with the distance between the B and P
impurities. We can see that by increasing the B and P distance the onset shifts to lower energies while
the dipole oscillator strength of the first transition diminishes. This behavior is due to the reduction of
the interaction between the acceptor and donor levels, which opens the gap, and the diminishing of the
electron (localized on the P atom) and the hole (localized on the B atom) wave function overlap.
   To enlighten the effects of the local fields and of the electron-hole coupling on the optical absorption and
emission spectra of the co-doped nanocrystals we have applied the GW-BSE approach to the calculation
of the optical spectra of the Si33 BPH36 cluster. For the calculation of the emission spectra the cluster
atoms are located in the equilibrium positions appropriate to the approximated excited state which has
an electron in the HOMO and a hole in the LUMO. The comparison of the spectra with those obtained
using the simple IP-RPA approach reveals the presence of sharp excitonic features at the spectrum onset
when the many-body effects are taken into account. As in the case of the oxidize clusters, the emission
spectrum is red shifted to lower energies with respect to the absorption spectrum. This energy red-shift
allows us to estimate the Stokes Shift which we compare with that obtained using the simple DFT
approach in Table 2.
   As we can see from the table the difference in the estimation of the Stokes Shift provided by the two
theories is only 0.16 eV. In Fig. 9 we show the absorption and emission spectra of Si 33 BPH36 obtained
including the many body effects through the GW + BSE approach. We can see the strong red-shift of
the first excitonic peak and the increase in its intensity.
230                   R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals

                                                          Table 2
                                        Absorption gap, emission gap, and Stokes
                                        Shift ∆ calculated as the HOMO-LUMO
                                        difference in DFT and the GW + BSE ap-
                                        proaches
                                        Si33 BPH36 DFT GW + BSE              ∆
                                        Abs. (eV)      2.77       3.35      0.58
                                        Ems. (eV)      1.78       2.20      0.42
                                        ∆ (eV)         0.99       1.15      0.16




Fig. 8. Absorption spectra of Si145 BPH100 nanocrystal with the B and P impurities located at the largest (10th neighbors,
slashed filling) and the shortest (2nd neighbors, dark homogeneous filling) distance.

5. Conclusions

  In this paper we have presented the results of first-principles calculations of the structural, electronic
and optical properties of hydrogenated silicon-based nanoclusters. We have analysed the effects on
the atomic structure and the optical spectra of: (1) the surface bonding by an oxygen atom; (2) the
doping by B, P and both B and P atoms; (3) the electronic excitation. This last point is essential to
correctly describe photoemission experiments where the cluster initial state is indeed an excited state.
The Si-based nanocrystals have diameters ranging from few angstroms to 2.5 nm. In particular we have
found that:
    i) the presence of a single electron-hole pair causes a strong deformation of the structures with
       respect to their ground-state configuration, which is more relevant for the smaller systems;
   ii) a significant contribution to the Stokes shift arises from structural relaxation after excitation of
       the nanocluster. Thus considering the HOMO-LUMO gaps of the ground and excited state as the
       proper absorption and emission energies provides a prediction which is worse the smaller is the
       cluster;
                       R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals                   231




      Fig. 9. Absorption (filled region) and emission (empty region) spectra including many body effects of Si33 BPH36 .

 iii) the inclusion of the many-body effects through the BSE or the TDLDA gives results for the absorp-
      tion spectra that are quite similar regarding the absorption onset and substantially in agreement
      with the available experimental data;
 iv) the oxidation of a Si-nc induces structural modifications and significant changes in the electronic
      and optical properties that depend on the type of the Si-O bond;
  v) the inclusion of the excitonic effects in the calculation of the emission spectra suggests that the
      Si-O-Si bridge bond might be responsible for the strong PL peak experimentally observed in
      oxidized Si-nc;
 vi) Si-nc can be more easily simultaneously doped than singly doped;
 vii) finally, co-doping is a way to engineering the PL properties of Si-nc.


Acknowledgments

  The authors thank INFM/CNR/CNISM and CNISM “Progetto Innesco” and the MIUR COFIN-PRIN
2005.


References
[1]   S. Ossicini, L. Pavesi and F. Priolo, Light Emitting Silicon for Microphotonics, Springer Tracts on Modern Physics 194,
      Springer-Verlag Berlin, 2003.
[2]   L.T. Canham, Appl Phys Lett 57 (1990), 1046.
[3]   O. Bisi, S. Ossicini and L. Pavesi, Surf Sci Reports 38 (2000), 5.
[4]   L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franz¶o and F. Priolo, Nature 408 (2000), 440.
[5]   R.J. Baierle, M.J. Caldas, E. Molinari and S. Ossicini, Solid State Commun 102 (1997), 545.
[6]   G. Allan, C. Delerue and M. Lannoo, Phys Rev Lett 76 (1996), 2961.
[7]   M.V. Wolkin, J. Jorne, P.M. Fauchet, G. Allan and C. Delerue, Phys Rev Lett 82 (1999), 197.
232                   R. Magri et al. / Role of surface passivation and doping in silicon nanocrystals

 [8] A. Puzder, A.J. Williamson, J.C. Grossman and G. Galli, Phys Rev Lett 88 (2002), 097401.
 [9] M. Luppi and S. Ossicini, J Appl Phys 94 (2003), 213, Phys Rev B 71 (2005), 035340.
[10] I. Vasiliev, J.R. Chelikowski and R.M. Martin, Phys Rev B 65 (2002), 121302(R).
[11] M. Gatti and G. Onida, Phys Rev B 72 (2005), 045442.
[12] C.S. Garoufalis and A.D. Zdetsis, Phys Chem Chem Phys 8 (2006), 808.
[13] M. Fujii, Y. Yamaguchi, Y. Takase, K. Ninomiya and S. Hayashi, Appl Phys Lett 87 (2005), 211919.
[14] M. Fujii, K. Toshikiyo, Y. Takase, Y. Yamaguchi and S. Hayashi, J Appl Phys 94 (2003), 1990.
[15] M. Fujii, Y. Yamaguchi, Y. Takase, K. Ninomiya and S. Hayashi, Appl Phys Lett 85 (2004), 1158.
[16] D.V. Melnikov and J.R. Chelikowsky, Phys Rev Lett 92 (2004), 046802.
[17] G. Cantele, E. Degoli, E. Luppi, R. Magri, D. Ninno, G. Iadonisi and S. Ossicini, Phys Rev B 72 (2005), 113303.
[18] Z. Zhou, M.L. Steigerwald, R.A. Friesner, L. Brus and M.S. Hybertsen, Phys Rev B 71 (2005), 245308.
[19] E. Degoli, G. Cantele, E. Luppi, R. Magri, D. Ninno, O. Bisi and S. Ossicini, Phys Rev B 69 (2004), 115411.
[20] ABINIT software project (URL http://www.abinit.org), X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs,
     G.-M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.-Y. Raty
     and D.C. Allan, Comp Mat Science 25 (2002), 478.
[21] A. Franceschetti and S.T. Pantelides, Phys Rev B 68 (2003), 033313.
[22] A. Puzder, A.J. Williamson, J.C. Grossman and G. Galli, J Am Chem Soc 125 (2003), 2786.
[23] U. Itoh, Y. Toyoshima, H. Onuki, N. Washida and T. Ibuki, J Chem Phys 85 (1986), 4867.
[24] C.S. Garoufalis, A.D. Zdetsis and S. Grimme, Phys Rev Lett 87 (2001), 276402.
[25] M. Hirao, in: Microcrystalline and Nanocrystalline Semiconductors, L. Brus, M. Hirose, R.W. Collins, F. Koch and C.C.
     Tsai, eds, Mater Res Soc Symp Proc 358, Material Research Society, Pittsburgh, 1995.
[26] O. Lehtonen, and D. Sundholm, Phys Rev B 72 (2005), 085424.
[27] L. Hedin, Phys Rev 139 (1965), A796.
[28] G. Onida, L. Reining and A. Rubio, Rev of Mod Phys 74 (2002), 601 and references therein.
[29] F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids, Pergamon Press, New York
     1975.
[30] N. Daldosso, M. Luppi, S. Ossicini, E. Degoli, R. Magri, G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, L. Pavesi, S.
     Boninelli, F. Priolo, C. Spinella and F. Iacona, Phys Rev B 68 (2003), 085327.
[31] Y. Tu and J. Tersoff, Phys Rev Lett 89 (2002), 086102.
[32] G. Hadjisavvas and P. Kelires, Phys Rev Lett 93 (2004), 226104.
[33] ESPRESSO package: S. Baroni, A. Dal Corso, S. de Gironcoli, P. Giannozzi, C. Cavazzoni, G. Ballabio, S. Scandolo, G.
     Chiarotti, P. Focher, A. Pasquarello, K. Laasonen, A. Trave, R. Car, N. Marzari and A. Kokalj, http://www.pwscf.org/.
[34] D. Vanderbilt, Phys Rev B 41 (1990), R7892.
[35] E. Garrone, F. Geobaldo, P. Rivolo, G. Amato, L. Boarino, M. Chiesa, E. Giamello, R. Gobetto, P. Ugliengo and A.
     Vitale, Adv Mat 17 (2005), 528.
Journal of Computational Methods in Sciences and Engineering 7 (2007) 233–240                                            233
IOS Press




Monte Carlo geometry optimization of Sin
(n 71) clusters

Nazım Dugan and Sakir Erkoc∗
                ¸         ¸
Department of Physics, Middle East Technical University, 06531 Ankara, Turkey

Received 29 August 2006
Revised /Accepted 17 October 2006

Abstract. Optimum geometries of silicon clusters up to 71 atoms have been found by a recently developed Monte Carlo based
global optimization method. Structural properties of these clusters have been investigated and the results have been compared
with available results obtained by other methods. Radial distribution of atoms of Si71 have been compared with the silicon
crystal structure.

Keywords: Silicon clusters, Monte Carlo optimization, empirical potential energy functions

PACS: 02.60.Pn, 02.70.Uu, 36.40.-c



1. Introduction

   Geometry of an atomic cluster has a great importance since it determines the electronical and ther-
modynamical properties of the cluster, together with the cluster size. Because of this importance, there
are lots of theoretical work on the determination of stable structures of atomic clusters which is an
nondeterministic polynomial-time hard (NP-Hard) [1,2] global optimization problem. Using empirical
potential energy functions (PEF) to define the interaction between the atoms, reduces the exponential
scaling of the computation time with increasing cluster size [3], to a polynomial cubic scaling. This
approximation becomes very useful when relatively larger clusters are considered, with the price of
making the reliability of the results dependent on the accuracy of the PEF used. Global minimum on the
potential energy surface (PES) is usually found by simulated annealing [4] or by some kind of genetic
algorithm (GA) [2,3,5–8]. Results of the small clusters may be compared with the more accurate ab
initio calculations or with the quantum Monte Carlo (QMC) [9,10] results in order to test the accuracy
of the PEF. Covalently bonded semiconductor clusters, such as C, Si and Ge, have been given much
attention since they have different properties than the bulk materials and form unique structures which
may be useful in generating new materials with novel and unusual properties [11]. In these elements, Si
has a special importance since it enabled new fields such as electronics and information technologies.
   There are lots of ab initio calculations [12–27] and some QMC computation results [11,28,29] about
Si clusters in the range of up to 70 atoms. Most of them is about determination of the geometries in which

  ∗
      Corresponding author. E-mail: erkoc@erkoc.physics.metu.edu.tr.


1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
234              N. Dugan and S. Erkoc / Monte Carlo geometry optimization of Sin (n
                              ¸      ¸                                                 71) clusters

clusters posses stable structures. In this study, using an empirical potential energy function to define the
interactions between atoms, stable geometries of Si clusters have been found by a recently developed
Monte Carlo based global optimization method. Results of small clusters have been compared with ab
initio calculations and QMC results. Structure of a relatively larger cluster is compared with Si crystal
structure.


2. Method of calculation

   We have used a fast local optimization procedure together with a technique to escape from local minima
in order to find the minimum energy configurations of atomic clusters composed of Si atoms. Local
optimization method is very similar to the well known Metropolis algorithm [9,10,30,31]. Individual
atoms make a random walk in the three dimensional space, in which the step sizes are chosen from a
Gaussian distribution. After each step, total potential energy of the system is calculated and the steps
resulting decrease in the potential energy are accepted and the steps resulting increase in the potential
energy are rejected.
   Global minimum on the potential energy surface (PES) may be found by running the procedure
described above many times with different initial conditions or different seeds of the random number
generator. However this is not an efficient way and takes a long computation time. We have used a more
efficient technique to find the global minimum in a recent study on the Zinc-Cadmium and Aluminium-
Titanium-Nickel clusters [32]. We have inspired from the crossing-over technique of Deaven and Ho [33],
while developing this technique.
   In this technique, a plane that passes through the center of the cluster is chosen. Then, one half
of the cluster is rotated about an axis passing through the center of the cluster and perpendicular to
the chosen plane, by an arbitrary angle. This operation is applied when a minimum is found by
the local optimization method and it allows the method to escape from the local minimum without
altering the optimized geometry too much. Then, the local optimization method is applied again to find
another minimum energy configuration and the energy of this configuration is compared with that of the
previous one. The configuration with the lower energy is chosen and the rotation operation is applied
again. The rotation procedure continues several times in this way and the minimum energy value tends to
decrease in each rotation cycle. This rotation procedure speeds up optimization to reach global minimum
configuration.
   Rata et al. [34], also uses this rotation operation together with another operation called ‘piece reflection’
between conjugate gradient type local optimization and they also relate their method with genetic
algorithms.


3. Potential functions used in the calculations

  An empirical potential energy function (PEF), known as the Stillinger-Weber (SW) potential, which
was developed for silicon has been used in the computations [35,36]. The PEF is composed of two-body
and three-body interaction terms:

      φ = φ2 + φ3 =         Uij +           Wijk                                                            (1)
                      i<j           i<j<k
                  N. Dugan and S. Erkoc / Monte Carlo geometry optimization of Sin (n
                               ¸      ¸                                                          71) clusters   235

                                                            Table 1
                                    Comparison of current binding energy per atom re-
                                    sults (in eV) with the results obtained by GA [6] for
                                    Sin clusters. Stillinger-Weber PEF has been used in
                                    both cases
                                    n E(GA)              E       n     E(GA)       E
                                     3 −1.481 −1.480 10 −2.994 −2.991
                                     4 −2.037 −2.035 11 −3.001 −2.990
                                     5 −2.169 −2.167 12 −3.077 −3.074
                                     6 −2.367 −2.365 13 −3.089 −3.086
                                     7 −2.558 −2.554 14 −3.137 −3.134
                                     8 −2.869 −2.867 15 −3.127 −3.120
                                     9 −2.880 −2.877

where, the two-body interaction term Uij is defined as

    Uij = εf2 (rij /σ)                                                                                          (2)

and the three-body interaction term W ijk is defined as

    Wijk = εf3 (rij /σ, rik /σ, rjk /σ).                                                                        (3)

The symbol r , in the above equations denote the distance between two atoms. Functions f 2 and f3 are
defined as
                                              −1
                 A(Br −p − r −q )e(r−a)            for r < a,
    f2 (r) =
                 0                                 for r a.

    f3 (rij , rik , rjk ) = h(rij , rik , θjik ) + h(rji , rjk , θijk ) + h(rki , rkj , θikj )                  (4)

where
                                           −1 +γ(r         −1 ]
                              λe[γ(rij −a)           ik −a)       × ( 1 + cos θjik )2 for rij and rik < a,
    h(rij , rik , θjik ) =                                            3
                              0                                                       for rij or rik a.
Parameters in the above equations have the following values for Si [35]: A = 7.049556277, B =
0.6022245584, p = 4, q = 0, a = 1.80, λ = 21.0, γ = 1.20, σ = 2.0951 Å, ε = 50 kcal/mol.
Geometry optimization of Si clusters by a GA, using SW potential, can be found in the reference [6] and
investigation of nucleation process of Si, again using SW potential, can be seen in the reference [37].


4. Results and discussion

  Geometries of Si clusters up to 71 atoms have been optimized by using the optimization method
discussed in the Section 2, starting from totally random configurations. Method have been adjusted to
run until there was no progress in the minimum energy value for 20 successive rotation operations. SW
PEF has been used to define the interactions between Si atoms. Optimized geometries of Si n clusters (n =
3–29,35,45,47,71) are given in Fig. 1. Planar geometries have been obtained for up to 5 atoms. When
the combination of Lennard-Jones (LJ) and Axilrod-Teller (AT) functions has been used [38] instead of
SW function, obtained geometries were planar for up to 13 atoms. This is not the behavior observed in
the ab initio calculations, so we decided to use SW function instead of LJ-AT combination since it gives
236              N. Dugan and S. Erkoc / Monte Carlo geometry optimization of Sin (n
                              ¸      ¸                                                   71) clusters

                                                        Table 2
            Average bond lengths of surface atoms (< rv >) and average bond lengths of volume atoms
            (< rs >) are given in the units of Angstroms. Comparison of number of surface atoms (nv ) with
            number of volume atoms (ns ) for Sin clusters is also given
            n    nv    < rv >    ns    < rs >     nv /ns    n    nv     < rv >     ns    < rs >    nv /ns
             3    0      −        3     2.56      0.000     19    1      2.48      18     2.41     0.056
             4    0      −        4     2.39      0.000     20    1      2.48      19     2.43     0.053
             5    0      −        5     2.35      0.000     21    1      2.48      20     2.40     0.050
             6    0      −        6     2.53      0.000     22    2      2.45      20     2.41     0.100
             7    0      −        7     2.47      0.000     23    2      2.59      21     2.46     0.095
             8    0      −        8     2.41      0.000     24    2      2.42      22     2.43     0.091
             9    0      −        9     2.40      0.000     25    3      2.44      22     2.43     0.136
            10    0      −       10     2.40      0.000     26    2      2.42      24     2.41      0.083
            11    0      −       11     2.38      0.000     27    1      2.44      26     2.40      0.038
            12    0      −       12     2.38      0.000     28    3      2.46      25     2.43      0.120
            13    0      −       13     2.40      0.000     29    2      2.44      27     2.40      0.074
            14    0      −       14     2.37      0.000     35    5      2.44      30     2.43      0.167
            15    0      −       15     2.40      0.000     45    7      2.39      38     2.44      0.184
            16    0      −       16     2.36      0.000     47    6      2.46      41     2.40      0.146
            17    0      −       17     2.38      0.000     71   12      2.44      59     2.45      0.203
            18    1     2.42     17     2.43      0.059

                                                                                ¸
more reliable results for Si clusters. A recent GA study of author Erkoc et al. on small Si clusters [7]
also shows how different empirical PEFs may result in different geometries for clusters. Binding energy
per atom results of the current optimization method have been compared with the results obtained by a
GA [6], using the same PEF, for Si n (n = 3–15) clusters. Results of these two methods have come out to
be very close to each other. This comparison can be seen in Table 1. Figures of the optimum geometries
of the GA method are given in the reference mentioned above and it has been observed that only the
geometry of 11 atom cluster is significantly different from the current result but binding energy results
of two methods are very close to each other (GA method gives slightly lower energy). Current results
have also been compared with the ab initio and QMC results by looking at the figures of the optimum
geometries. While results of these more accurate methods are significantly different from current results
for very small Si clusters, similar results have been obtained for larger Si clusters. Ab initio calculations
suggest an isosceles triangle with one angle being approximately 80 degrees for Si 3 , rhombus geometry
for Si4 , trigonal bipyramid for Si5 , tetragonal bipyramid (may be tilted) for Si6 , pentagonal bipyramid
or tetragonal bipyramid with an additional atom forming a trigonal pyramid on one surface for Si 7 and
parallelpiped (tilted cube) or distorted bicapped octahedrod for Si 8 [12,14,16,17,19,24–27]. However
current result is an equilateral triangle for Si3 , square geometry for Si4 , pentagonal geometry for Si 5 ,
trigonal prism for Si6 and cubic geometry for Si 8 (see Fig. 1). Since all these geometries for small
clusters are different from the ab initio results it can be concluded that the SW PEF is not suitable for
studying small Si clusters. For larger clusters (n 20), similar structures have been obtained with ab
initio and QMC methods but in current results there is no elongated structures, as in the studies [28,
29]. Average bond length (< r >) values of the clusters have been obtained from the current results. In
these calculations, surface atoms and the volume atoms have been considered separately. This separation
has been done by looking at the coordination numbers (number of neighbors) and radial distributions of
atoms. An atom has been identified as a surface atom if its coordination number is less than four and if its
distance to the center of the cluster is not less than half of the distance of the outer most atom. Remaining
atoms have been identified as volume atoms. It has been observed that there was no volume atoms
until Si18 . This separation is not much meaningful for smaller clusters, but it gives some information
about the structure of larger clusters. Average bond length values of the surface atoms and the volume
N. Dugan and S. Erkoc / Monte Carlo geometry optimization of Sin (n
             ¸      ¸                                                 71) clusters   237




                   Fig. 1. Optimized geometries of Sin clusters.
238                N. Dugan and S. Erkoc / Monte Carlo geometry optimization of Sin (n
                                ¸      ¸                                                   71) clusters




Fig. 2. Comparison of radial distribution peaks of Si71 and Si crystal. Sharp peaks are for the crystal structure and the
Gaussian-broadened peaks are for Si71 .

atoms have been given in Table 2, together with the amounts of the surface atoms and the volume atoms.
Ratios of these numbers are also given in this table. For Si dimer, bond length, r 0 , has the value of
2.246 Å [39] and nearest neighbor distance, d nn , of the bulk has the value of 2.35 Å for Si [40]. Average
bond length values of the metal clusters are usually come out to be between these two values, namely
r0 < < r > < dnn . However, in semiconductor clusters, as observed in the present investigation, this
situation may not hold. Finally, structure of Si 71 has been compared with the Si crystal structure by
looking at the radial distributsion of atoms plots. These plots of the two cases are given in Fig. 2 in which,
sharp peaks are for the diamond type crystal structure and the Gaussian-broadened peaks are for Si 71 .
Places of peaks are in good aggreement with some small exceptions in these two cases. This situation
suggests a resemblance between the structures of larger Si clusters and the Si crystal. As a conclusion,
this study about the Si clusters suggests that the optimization method discussed in the Section 2 can be
used for global geometry optimization of atomic clusters and the SW PEF is not accurate enough for
studying very small Si clusters but it gives relatively better results for larger Si clusters. This is due to
the fact that this type of PEFs are usually parameterized using bulk properties.


Acknowledgements

   The authors would like to thank METU and TUBITAK for partial support through the projects METU-
BAP-2006-07-02-00-01 and TUBITAK-TBAG-107T142, respectively. One of the authors (N.D.) would
like to thank Turkish Petroleum Foundation for partial financial support.


References
 [1]   L.T. Wille and J. Vennik, Computational complexity of the ground-state determination of atomic clusters, Journal of
       Physics A: Mathematical and General 18 (1985), L419–L422.
                    N. Dugan and S. Erkoc / Monte Carlo geometry optimization of Sin (n
                                 ¸      ¸                                                       71) clusters                 239

 [2]   J. Zhao and R. Xie, Genetic algorithms for the geometry optimization of atomic and molecular clusters, Journal of
       Computational and Theoretical Nanoscience 1 (2004), 117–131.
 [3]   B. Hartke, Global Cluster Geometry Optimization by a Phenotype Algorithm with Niches: Location of Elusive Minima,
       and Low-Order Scaling with Cluster Size, Journal of Computational Chemistry 20 (1999), 1752–1759.
 [4]   S. Kirkpatrick, C.D. Gelatt and M.P. Vecchi, Optimization by Simulated Annealing, Science 220 (1983), 671–680.
 [5]   S.K. Gregurick, M.H. Alexander and B. Hartke, Global geometry optimization of (Ar)n and B(Ar)n clusters using a
       modified genetic algorithm, The Journal of Chemical Physics 104 (1996), 2684–2691.
 [6]   M. Iwamatsu, Global geometry optimization of silicon clusters using the space-fixed genetic algorithm, The Journal of
       Chemical Physics 112 (2000), 10976–10983.
 [7]   ¸        ¸
       S. Erkoc, K. Leblebicioglu and U. Halici, Application of Genetic Algorithms to Geometry Optimization of Microclusters:
       A Comparative Study of Empirical Potential Energy Functions for Silicon, Materials and Manufacturing Processes 18
       (2003), 329–339.
 [8]                                   ¸      ¸
       N. Chakraborti, P. Mishra and S. Erkoc, A study of the Cu clusters using gray-coded genetic algorithms and differential
       evolution, Journal of Phase Equilibria 25 (2004), 16–21.
 [9]   W.M.C. Foulkes, L. Mitas, R.J. Needs and G. Rajagopal, Quantum Monte Carlo simulations of solids, Reviews of Modern
       Physics 73 (2001), 33–83.
[10]                                                                                              o
       A. Aspuru and W.A. Lester, Quantum Monte Carlo methods for the solution of Schr¨ dinger equation for molecular
       systems, arXiv: cond-mat/0204486 v2 12 Jun 2002.
[11]                              a˘
       J.C. Grossman and L. Mit´ s, Quantum Monte Carlo Determination of Electronic and Structural Properties of Sin Clusters
       (n     20), Physical Review Letters 74 (1995), 1323–1326.
[12]   K. Raghavachari and V. Logovinsky, Structure and Bonding in Small Silicon Clusters, Physical Review Letters 55 (1985),
       2853–2856.
[13]              e                  u
       D. Toman´ k and M.A. Schl¨ ter, Calculation of Magic Numbers and the Stability of Small Si Clusters, Physical Review
       Letters 56 (1986), 1055–1058.
[14]              e
       D. Toman´ k and M.A. Schluter, Structure and bonding of small semiconductor Clusters, Physical Review B 36 (1987),
       1208–1217.
[15]   E. Kaxiras, Effect of Surface Reconstruction on Stability and Reactivity of Si Clusters, Physical Review Letters 64 (1990),
       551–554.
[16]   C.H. Patterson and R.P. Messmer, Bonding and structures in silicon clusters: A valence-bond interpretation, Physical
       Review B 42 (1990), 7530–7555.
[17]   O.F. Sankey, D.J. Niklewski, D.A. Drabold and J.D. Dow, Molecular-dynamics determination electronic and vibrational
       spectra, and equilibrium structures of small Si Clusters, Physical Review B 41 (1990), 12750–12759.
[18]   D.A. Jelski, B.L. Swift, T.T. Rantala, X. Xia and T.F. George, Structure of the Si 45 cluster, The Journal of Chemical
       Physics 95 (1991), 8552–8560.
[19]   N. Binggeli, J.L. Martins and J.R. Chelikowsky, Simulation of Si Clusters via Langevin Molecular Dynamics with
       Quantum Forcesm, Physical Review Letters 68 (1992), 2956–2959.
[20]   E. Kaxiras and K. Jackson, Shape of Small Silicon Clusters, Physical Review Letters 71 (1993), 727–730.
[21]   M.R. Pederson, K.J. Jackson, D.V. Porezag, Z. Hajnal and Th. Frauenheim, Vibrational signatures for low-energy
       intermediate-sized Si clusters, Physical Review B 54 (1996), 2863–2867.
[22]   J. Song, S.E. Ulloa and D.A. Drabold, Exciton-induced lattice relaxation and the electronic and vibrational spectra of
       silicon clusters, Physical Review B 53 (1996), 8042–8050.
[23]   J. Wang, G. Wang, F. Ding, H. Lee, W. Shen and J. Zhao, Structural transition of Si clusters and their thermodynamics,
       Chemical Physics Letters 341 (2001), 529–534.
[24]   C. Xiao and F. Hagelberg, Geometric, energetic, and bonding properties of neutral and charged copper-doped silicon
       clusters, Physical Review B 66 (2002), 075425.
[25]   S. Zhan, B. Li and J. Yang, Study of aluminum-doped silicon clusters, Physica B, in press.
[26]   A.D. Zdetsis, The story of the Si6 magic cluster, Computing Letters 1 (2005), 337–342.
[27]   Y. Gao, C. Killblane and X.C. Zeng, Structures and Stabilities of Small Silicon Cluster: High-Level Ab initio Calculations
       of Si6 , Computing Letters 1 (2005), 343–347.
[28]                              a˘
       J.C. Grossman and L. Mit´ s, Family of low-energy elongated Sin (n        50) clusters, Physical Review Letters 52 (1995),
       16735–16738.
[29]   L. Mitas, J.C. Grossman, I. Stich and J. Tobik, Silicon Clusters of Intermediate Size: Energetics, Dynamics, and Thermal
       Effects, Physical Review Letters 84 (2000), 1479–1482.
[30]   N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, Equation of State Calculations by Fast Computing
       Machines, The Journal of Chemical Physics 21 (1953), 1087–1092.
[31]   M. Lewerenz, Monte Carlo Methods: Overview and Basics, NIC Series 10 (2002), 1–24.
[32]                    ¸      ¸
       N. Dugan and S. Erkoc, Geometry optimization of Znn Cdm and (AlTiNi)n clusters by the modified diffusion Monte
       Carlo method, Computing Letters, submitted.
240                N. Dugan and S. Erkoc / Monte Carlo geometry optimization of Sin (n
                                ¸      ¸                                                    71) clusters

[33]   D.M. Deaven and K.M. Ho, Molecular Geometry Optimization with a Genetic Algorithm, Physical Review Letters 75
       (1995), 288–291.
[34]   I. Rata, A.A. Shvartsburg, M. Horoi, T. Frauenheim, K.W.M. Siu and K.A. Jackson, Single-Parent Evolution Algorithm
       and the Optimization of Si Clusters, Physical Review Letters 85 (2000), 546–549.
[35]   F.H. Stillinger and T.A. Weber, Computer simulation of local order in condensed phases of silicon, Physical Review B
       31 (1985), 5262–5271.
[36]   ¸        ¸
       S. Erkoc, Empirical potential energy functions used in the simulations of materials properties, in: Annual Reviews of
       Computational Physics IX, D. Stauffer, ed., World Scientific, Singapore, 2001, pp. 1–103.
[37]   P. Beaucage and N. Mousseau, Nucleation and crystallization process of silicon using the Stillinger-Weber potential,
       Physical Review B 71 (2005), 094102–094109.
[38]   E. Pearson, T. Takai, T. Halicioglu and W.A. Tiller, Computer modeling of Si and SiC surfaces and surface processes
       relevant to crystal growth from the vapor, Journal of Crystal Growth 70 (1984), 33–40.
[39]   K.P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules, Van
       Nostrand Reinhold, New York, 1979.
[40]   C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1996.
Journal of Computational Methods in Sciences and Engineering 7 (2007) 241–256                                                 241
IOS Press




Structure and relative stability of Sin, Si−,
                                           n
and doped SinM clusters (M = Sc−, Ti, V+)
in the range n = 14–18

M.B. Torresa,∗ , E.M. Fern´ ndezb and L.C. Balb´ sc
                          a                    a
a Departamento            ´                  ´              e
                de Matem aticas y Computacion, Escuela Polit´ cnica Superior, Universidad de Burgos,
Burgos, Spain
b Center for Atomic-Scale Materials Design, Department of Physics, Building 307, Technical University

of Denmark, DK-2800 Lyngby, Denmark
c Departamento de F´sica Teorica, Atomica y Optica, Universidad de Valladolid, Valladolid, Spain
                    ı       ´       ´        ´


Received 4 April 2007
Revised /Accepted 1 June 2007


Abstract. We report on first-principles quantum mechanical optimizations of the minimum energy equilibrium structure of
neutral, Sin , and anionic, Si− , pure silicon clusters, as well as the isoelectronic Sin M doped clusters (M = Sc− , Ti, V+ ) for
                              n
n = 14–18. We have published previously some of these results, but additional analysis is contributed here for the first time,
particularly for the pure anionic silicon clusters and doped Sin Ti compounds. The lowest energy isomer of the anionic Si−      n
cluster shows different geometry than the neutral cluster, except for n = 15, 17. The geometries of a few low-lying energy
isomers of doped Sin M does not relate to those of pure silicon clusters in the range of sizes considered in this work. For
both pure and doped Si clusters, we analyze the trend of several electronic properties with the cluster size, like the binding
energy, the addition energy of the impurity M to pure Si clusters, the second difference of total energy, the Homo-Lumo gap,
the average Si-Si and Si-M distance, and the electron affinity. For Si16 M doped clusters we found the largest binding energy,
the highest second difference of energy, and the highest Homo-Lumo gap. These facts are manifestations of the special stability
of Si16 M clusters found in recent experimental mass spectra, which was rationalized in previous works as a combination of
geometrical (near spherical cage-like structure) and electronic effects (l-selection rule of the spherical potential model). Here
we present additional arguments, by comparing the partial orbital density of states of the near-spherical Frank-Kasper isomer of
Si16 Ti, with that of a non-spherical isomer of Si16 Sc− anion. We have also computed the adiabatic electron affinity of pure and
doped Si clusters. For doped clusters, the computed electron affinities are in very good agreement with available estimations
from experimental photoelectrons spectra, but for pure neutral clusters the calculations underestimate by more than 18% the
experimental values.

Keywords: Electronic and geometrical properties, stability, silicon-doped clusters

Mathematics Subject Classification: PACS: 36.40.Cg, 36.40.Qv




  ∗
      Corresponding author. E-mail: begonia@ubu.es.


1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
242             M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                 n


1. Introduction

   The electronic and structural properties of pure and doped silicon nanoclusters are of interest in many
areas of technology where the nanoscale is being reached, as those of electronic [1] and optoelectronic [2]
devices. The shape of low-lying isomers of Si n are preferentially prolate for n < 27 and became near
spherical for n > 27, as it has been inferred from a variety of experimental measurements [3–6] and
computational studies [1,7–18]. Most clusters in the range n = 10–18 contain the tricapped-trigonal-
prism (TTP) motif [1,6] (Si9 subunit). Nevertheless, in a recent calculation [12] was obtained a new
global minimum of Si16 , as well as new low-lying isomers of Si 17 , Si18 and Si22 , which were built on a
different generic motif based on the Si 6 tetragonal bi-pyramid plus the Si6 six fold puckered ring structural
subunits (called six/six structural motif). Another recent result, using a refined structural optimization
method [14], regains the TTP motif as the ground state of Si 16 instead of the six/six structural motif.
On the experimental side, photoelectron spectroscopy (PES) spectra are used, whenever it is possible, to
elucidate the geometry of these clusters [6]. Moreover, the morphology of the ground state of Si n and
Si+ clusters in the range n = 14–20 can be different from that of Si − anions [5], as it was demonstrated
   n                                                                   n
       u
by M¨ ller et al. [6], Nigam et al. [18], Li and Xu [19], and Shvarstburg et al. [20].
   With regard to the growth behavior of transition metal-doped silicon clusters, Si n M, recent first-
principles calculations [21,22] have found that open basket like structures are the most favorable for
n = 8–12, while for n = 13–16 the metal atom becomes completely surrounded by Si atoms. A recent
theoretical study [23] address the question of the special stability of MSi 12 (M = Hf, Ta, W, Re, Os,
Ir, Pt, and Au) clusters from the point of view of geometrical and electronic relations. For n = 16 it is
obtained the optimal Si cage for metal-encapsulation [24]. Experiments confirmed later this prediction.
Thus, experiments on photo-dissociation of MSi n clusters [25] indicate that, for M = Cr, encapsulation
of Cr occurs at n = 15–16. A mass spectrometric stability study of binary MS n clusters [26], with
S = Si, Ge, Sn, Pb, and M = Cr, Mn, Cu, Zn, reveals interesting trends. For example, Cr doped
silicon cationic clusters are peculiarly abundant at sizes n = 15, 16, as was reported by Beck twenty
years ago [27,28]. These experiments provide support for those encapsulated structures calculated by
Kumar and Kawazoe [29]. Other experiments, using mass spectrometry, a chemical probe method, and
photoelectrons spectroscopy [30], revealed that one metal atom (M = Ti, Hf, Mo, W) can be encapsulated
inside a Sin cage at n 15.
   In recent mass spectrometry experiments, Nakajima and coworkers [31] have demonstrated the size-
selective formation of Si16 Sc− , Si16 Ti, and Si16 V+ cluster. More details about these experiments,
combining mass spectrometry, anion photoelectron spectroscopy, and adsorption reactivity towards
H2 O, has been published very recently [32]. There are several theoretical studies, which use a variety
of methods, of Sin M clusters for different cluster sizes and impurity atoms or ions [15,19,20,27,33,34].
We have found very few works [29,35–37] concerning to the special stability of Si 16 M for the type
of impurity involved in the experiments of Nakajima and coworkers [31]. Kumar and Kawazoe [29]
obtained for Si16 Ti a truncated tetrahedral structure, called the Frank-Kasper (FK) polyhedron, and
explained in further works [22] (see reference 2 for a review and earlier references) the special stability
of that cluster in terms of the spherical potential model [38–40], as a combination of geometrical and
electronic shell effects. In the work of Reveles and Khana [35], cationic, neutral, and anionic doped
clusters Sin M with n = 15–17, were optimized. These authors found that the ground state of Si 16 M
clusters with M = Sc− , Ti, V+ , adopt the FK-polyhedron structure, and have both, the atomization energy
and Homo-Lumo Gap larger than the same clusters within other charge states. These facts manifest the
special stability of these clusters against changes in their electronic charge. This was explained [35] on
                 M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                  n                              243

the basis of a 20 electron rule, assuming that only one electron is contributed by a Si atom to the valence
manifold when that Si atom is bonded to the metal atom, whereas the other three valence electrons of Si
belongs to the Si cage.
  We have recently [36,37] studied Si n M clusters (M = Sc− , Ti, V+ ) in the range n = 14–18, and
found at n = 16 a positive peak for the second difference in the total energy (see Eq. (4)), which is
related directly to the higher abundance of these clusters in the mass spectrometry experiment [31,32]. In
the present work, we report on systematic first-principles computational studies of the relative stability
of Si− anions, and we present an additional analysis of the partial density of states of spherical and
      n
non-spherical isomers of doped silicon clusters.
  In Section 2 is outlined the computational method. In Section 3 we discuss previous and new results.
We present the structures of Sin and Si− in Section 3.1, and those of Si n M in Section 3.2. The average
                                        n
distances between Si-Si and Si-M atoms are compared and discussed in Section 3.3. The electronic
properties are presented in Section 3.4, and comprise the study of the different bond energy trends
(Section 3.4.1), the discussion of the total, partial, and orbital density of states for spherical and non-
spherical clusters (3.4.2) and the comparison of calculated adiabatic electron affinity with available
experimental estimations (Section 3.4.3). Conclusions are given in Section 4.


2. Computational methods

  The first-principles code SIESTA [41] is used to solve fully self-consistently the standard Kohn-
Sham equations [42] of density functional theory (DFT) within the spin-dependent generalized gradient
approximation (GGA) for the exchange-correlation effects as parameterized by Perdew, Burke and
Ernzerhof [43]. Norm-conserving scalar relativistic pseudo-potentials [44] are used in their fully nonlocal
form [45]. They are generated from the atomic valence configuration 3s 2 3p2 for Si (with core radii 1.9
a.u. for s, p and d orbitals), and the semi-core valence configuration 4s 2 3p6 3dx for Sc (x = 1), Ti (x = 2),
and V (x =3), all of them with core radii, in a.u., 2.57, 1.08, and 1.37 for s, p, and d orbitals, respectively.
Flexible linear combinations of numerical (pseudo) atomic orbitals are used as the basis set, allowing
for multiple-ζ and polarization orbitals. In order to limit the range of the basis pseudo atomic orbitals
(PAO), they are slightly excited by a common energy shift (0.068 eV in this work), and truncated at the
resulting radial node. In the present calculations we used a double- basis s, p (for Si) and s, p, d for the
impurity M, with single polarization d (for Si) and p for M, having maximum cutoff radius 7.47 a.u. (for
p of Si), and 8.85 a.u., 8.45 a.u., and 8.08 a.u for s of Sc, Ti, and V, respectively. The basis set of the 3d
metal was tested in previous works [46,47]. The basis functions and the electron density are projected
onto a uniform real space grid in order to calculate the Hartree and exchange-correlation potentials and
matrix elements. The grid fineness is controlled by the energy cutoff of the plane waves that can be
represented in it without aliasing (120 Ry in this work).
  We found the equilibrium geometries from an unconstrained conjugate-gradient structural relaxation
using the DFT forces. We tried several initial structures for each cluster (typically more than twenty) until
the force on each atom was smaller than 0.005 eV/Å. For pure neutral and anionic Si clusters we have
optimized the low-lying energy isomeric geometries obtained previously from a genetic algorithms code,
as well as those suggested by previous calculations [1,12,14,19,48], and from many other reasonable
configurations. For Sin M we started with several geometries of doped Si n M isomers obtained in previous
works [21,22], and we use our optimized low-lying geometries of neutral Si n adding a M impurity at
different sites.
244             M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                 n


  As further tests of the pseudo potentials, basis sets, and cutoff energy, the relative stability, bond
distance, and dipole moment of different spin states of SiM ν monosilicides (M = Sc− , Ti, V+ ; ν = ±1,
0) have been calculated and the results compared with those calculated recently by Wu and Su [52] using
a standard all-electron density functional method, see Table 1. In all the cases, the spin multiplicity
of the lower energy state is the same than in the Wu and Su calculations. Specifically, a triplet state
for SiSc− and a quintuplet for SiTi and SiV+ were obtained, with bond distances (in Å) 2.50, 2.51,
and 2.51, respectively, to be compared with the values 2.43, 2.45, and 2.50 obtained by Wu and Su.
The calculated electric dipole moments for the neutral species are (in Debye) 3.17 (SiSc quadruplet),
3.36 (SiTi quintuplet), and 3.13 (SiV sextuplet), to be compared with the values 3.62, 3.58, and 3.20,
respectively, reported by Wu and Su [49].


3. Results and discussions

  We present results for the geometry of several low-lying energy isomers of pure neutral (Si n ) and
anionic (Si− ) clusters in Section 3.1, and for doped silicon clusters (Si n M) in Section 3.2. Several
            n
neutral Sin geometries, and all geometries of anionic species, were not considered in our previous
works [36,37]. In Section 3.3 we study the Si-Si average distance, d av (Si-Si), and their standard
deviations of the ground state geometry of pure Si n clusters compared with the average distances d av (Si-
Si) and dav (Si-M), and their standard deviations for doped Si n M clusters. In Section 3.4 are discussed
several electronic properties and comprise the study of several bond energy trends (Section 3.4.1), the
discussion of the total, partial, and orbital density of states for spherical and non-spherical clusters
(Section 3.4.2) and the comparison of calculated adiabatic electron affinity with available experimental
estimations (Section 3.4.3).

3.1. Structure of neutral (Si n ) and anionic (Si− ) clusters for n = 14–18
                                                 n

  The equilibrium structures of several low-lying energy isomers of neutral and anionic pure silicon
clusters are represented in Fig. 1. In order to identify the isomers of a particular size and specie (neutral
or anionic), we have ordered the isomers according their excess energy above the lowest energy cluster
and labeled clusters 1, 2, 3, . . ., starting with ground state. Note that the structure of the ground state
isomer is generally different for neutral and anionic species. For each isomer is given the excess energy
with respect to the ground state and the Homo-Lumo gap. In all these structures, the spin multiplicity is
singlet for the neutrals, and doublet for the anionic species.
  We discuss first the structure of neutral clusters. Different calculations using a variety of numerical
codes and/or xc-energy functional lead to different energy order of these structures, particularly for the
Si16 isomers [14,53]. The geometry of the lowest isomer of Si n is elongated and, except for Si 14 and
Si16 , is similar to that reported by K.M. Ho and coworkers [1], which is based on the TTP motif. The
first lowest energy isomer of Si14 contains a central trigonal prism with the edges of two rectangular
faces capped by eight Si atoms, and is similar to the ground state of Si 14 reported by Bazterra and
coworkers [54]. This geometry is different to the one that we found in a previous work [36]. In the
second, third and fourth isomers of Si 14 , is added to the TTP a group of four Si atoms plus a fifth Si atom
which is placed in a different site for each one of the mentioned isomers. The second (fourth) isomer is
near degenerate in energy with the first (third) isomer, although their respective Homo-Lumo gaps are
quite different. The lowest energy isomer of Si 15 contains a TTP unit plus a ring of six atoms, and it
was obtained before using different methods [1,11]. The second and third isomers of Si 15 are the union
               M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                n                              245

                                                         Table 1
                           Spin multiplicity, bond distance, in Å, excess energy, in eV, with
                           respect to the lowest energy configuration, and electric dipole mo-
                           ment, in Debye, for several monosilicides SiM molecules. For
                           the sake of comparison are given in parenthesis the values calcu-
                           lated by Wu and Su [49] using an all-electron density functional
                           method. Available experimental values for neutral species are
                           also given
                           Molecule     2S+1          d (Å)          ∆E (eV)        Dipole (D)
                            SiSc+         1       2.402 (2.566)     0.63 (0.68)        6.37
                                          3       2.349 (2.544)     0.00 (0.00)        5.71
                                          5       2.561 (2.495)     0.15 (0.22)        4.76
                             SiSc         2       2.470 (2.456)     0.25 (0.14)        3.96
                                          4       2.588 (2.519)     0.00 (0.00)     3.17 (3.62)
                             exp49        4           2.520            0.00
                             SiSc−         1      2.389 (2.326)     0.05 (0.17)        2.49
                                           3      2.502 (2.430)     0.00 (0.00)        1.44
                                           5      2.774 (2.726)     0.48 (0.17)        1.03
                             SiTi+         2      2.548 (2.229)     0.52 (0.53)        4.95
                                           4      2.561 (2.367)     0.26 (0.19)        4.75
                                           6      2.518 (2.447)     0.00 (0.00)        3.69
                              SiTi         1      2.300 (2.214)     0.76 (0.62)        3.38
                                           3      2.446 (2.364)     0.79 (0.63)        3.78
                                           5      2.515 (2.447)     0.00 (0.00)     3.36 (3.58)
                             exp50         5          2.410            0.00
                             SiTi−         2      2.328 (2.290)     0.08 (0.15)        2.03
                                           4      2.438 (2.372)     0.00 (0.00)        2.32
                                           6      2.660 (2.589)     0.58 (0.48)        1.77
                             SiV+          1      2.293 (2.221)     0.95 (1.78)        4.90
                                           3      2.498 (2.440)     0.81 (0.25)        4.53
                                           5      2.507 (2.504)     0.00 (0.00)        3.89
                                           7                                           2.85
                              SiV          2      2.394 (2.348)     0.64 (0.34)        3.43
                                           4      2.425 (2.379)     0.33 (0.00)        3.27
                                           6      2.460 (2.399)     0.00 (0.00)     3.13 (3.20)
                             exp51         2
                             SiV−          1      2.378 (2.089)     0.68 (1.94)        1.83
                                           3      2.368 (2.335)     0.00 (0.00)        3.09
                                           5      2.398 (2.375)     0.34 (0.36)        3.11

of two TTP sharing a triangular face, but with different relative orientations. To our knowledge, these
two-TTP structures of Si15 are reported here for the first time, although they have some resemblance to
the third and fourth isomers, respectively, found by Zhu and coworkers [11]. The fourth isomer of Si 15
resembles the structure of Si14 TTP type of isomers with an additional Si atom.
  The four isomers of neutral Si16 in Fig. 1 are very close in energy. Our first and second isomers are
similar those reported by Goedecker and coworkers [14]. The first isomer contains a TTP unit plus seven
atoms, and the second shows a symmetric and compact structure. The structure of the third isomer of
Si16 was reported as the lowest energy isomer by Bazterra et al. [54]. Our fourth isomer is similar to
the ground state of Si16 found by Yoo and coworkers [12]. The fifth isomer of Si 16 (not shown), with
0.109 eV excess energy, is similar to the first isomer reported by Ho and coworkers [1] and Zhu and
coworkers [11].
  The lowest energy isomer for Si 17 coincides with the first isomer found by Ho et al. [1] and by
Goedecker et al. [14] using different methods. The second isomer is similar to the lowest energy
246                M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                    n




Fig. 1. Several low-lying energy isomeric geometries of neutral and anionic silicon clusters for n = 14–18. For each isomer
the excess energy with respect to the ground state, and the Homo-Lumo gap is given in eV, as well as its ordinal number. The
symmetry is also given.
                M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                 n                              247

structure of Si17 found by Yoo et al. [13], and it does not contain the TTP unit. The third and fourth
isomers of Si17 are practically degenerate. Their geometries coincide, respectively, with the isomers
denoted by Si17a and Si17b in the work of Goedecker et al. [14]. Si 17a contains two separated TTP units,
but Si17b has no trace of TTP unit. Our geometries for the first, second, and third isomers of Si 18 in Fig. 1
are analogous to the first lowest energy isomer found by Ho et al. [1], Yoo et al. [12], and Goedecker et
al. [14], respectively. The geometry of the 4th isomer of Si 18 , which coincides with the ground state of
Si− , is particularly interesting because it suggest the possibility of formation of Si wires formed by TTP
   18
Si9 units. The 5th isomer of Si18 has a considerable excess energy and shows a structure more symmetric
than the others.
   The geometries of the anionic species are generally different from those of the neutral clusters. The
exceptions are Si15 and Si17 , which have the same ground state geometry as the corresponding anions.
The geometry of the first isomer of Si− coincides with that of Shvartsburg and coworkers [20], and
                                          14
corresponds to a Si atom on a TTP structure. The second isomer of Si − is also the second isomer found
                                                                           14
by Batzerra et al. [54], and the 3 rd and 4th isomers coincide with the 3 rd and 2nd of the neutral specie.
The 5th isomer of Si− coincides with the ground state given by Li et al. [19]. The geometries of the
                        14
first and second isomers of Si − , which are practically degenerate, resemble those of Ho el at. [1], and
                                 15
Li et al. [19], respectively. The second isomer of Si − is similar to that of the 15th isomer of the neutral
                                                        15
specie, and coincides with those given by Li el al. [19] and by Rata el al. [9]. Using the Mulliken charge
population analysis, we found that the charge of the extra electron in Si − with respect to Si15 ground
                                                                              15
state is located outside of the three central atoms which connect the TTP motif with the quasi-planar
6-atom triangle, mainly in this planar area of the anion.
   The ground state geometry of Si − is similar to that found by Li et al. [19], and Shvartsburg et
                                       16
al. [20], and to our 9th isomer for the neutral cluster. However, the second isomer of Si − is similar to
                                                                                               16
the 4th isomer of the neutral cluster, which coincides with the ground state of Si 16 found by Yoo and
coworkers [12]. The 3 th and 4th isomers of Si− correspond to the 11 th and 8th isomers of the neutral
                                                  16
cluster, respectively, that is, the addition of one electron changes radically the relative energy of the
equilibrium geometries. This effect seems to be less severe for Si − and Si− clusters. The ground state
                                                                       17       18
geometry of Si− is similar to the 4th isomer of the neutral species, and is composed by TTP units sharing
                 18
a triangular face. That geometry coincides with those given by Li et al. [19], and Shvartsburg et al. [20].
   We observe in Fig. 1 that the ground state of Si − anions contains the TTP structural subunit. Thus,
                                                       n
the addition of one electron to neutral Si clusters tends to stabilize structures that contain this TTP unit,
as can be seen when we compare the two first isomers of Si 16 with those of Si− . The energy gained
                                                                                      16
from tying up the dangling on Si9 TPP units benefits the stacked geometries. Therefore, we expect the
prolate-to-spherical transition for anionic Si clusters to occur at larger size compared with the neutral
clusters.

3.2. Structure of Si n M doped clusters (M = Sc − , Ti, V+ ; n = 14–18)

   The equilibrium geometry of the lowest energy isomer of iso-electronic Si n M doped clusters (M =
Sc− , Ti, V+ ) is represented in Fig. 2 for the sizes n = 14–18. In the structures of Fig. 2 Si atoms
surround the impurity. We have reported higher energy isomers in previous works [36,37]. For each
structure and type of impurity, are given the excess of total energy with respect to the lowest energy
isomer and the Homo-Lumo gap, which will be discussed in Section 3.4 together with other electronic
properties. The structure of the first isomer is the same for all doped clusters, except for Si n V+ with
n = 16, 17, 18. The first isomer of Si n V+ , for n = 16, 17, is the second isomer of Si n M (M = Sc− , Ti),
248                M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                    n




Fig. 2. Geometry of the lowest energy isomer of Sin M clusters, with M = Sc− , Ti, V+ , in the range n = 14–18. For n =
16–17 the lowest energy isomer for V impurity coincides with the second lowest energy isomer for Sc− and Ti doped clusters.
The first isomer of Si18 Sc− and Si18 Ti is the fifth isomer of Si18 V+ , and the first isomer of Si18 V+ is the 5th and 8th isomer
of Si18 Sc− and Si18 Ti, respectively. For each structure and impurity the total energy with respect to the lowest energy isomer
and the Homo-Lumo Gap are given (in eV). In all these structures, the spin is zero.

and the first isomer of Si18 V+ has a elongated structure which corresponds to the 5 th and 8th isomers
of Si18 Sc− and Si18 Ti, respectively [37]. In general [37], the structure of the third, and higher, isomers
of Sin V+ and Sin Ti depart from the sequence for Si n Sc− . Notice that there is no relation between the
equilibrium structures of Sin M and those of Sin or Sn+1 depicted in Fig. 1.
   A common structural motif for the ground state structure of Sin M with n = 14, 17, 18, and M = Sc − ,
Ti, is a distorted hexagonal prism (DHP) of Si atoms surrounding the M impurity, with additional Si
atoms and dimmers decorating the lateral prism faces [36]. That DHP motif resembles the structures C s
(ground state) and C2h , which have been reported recently [34] for Si 12 Ni. This Cs structure has been
found [34] as the new ground state of Si 12 Ni, instead of the C5v Frank-Kasper structure, which results
unstable.
   For the first isomer of Si14 M we obtain a DHP structure decorated with a Si 2 dimmer on a side.
That structure is similar to the first isomer of ZrSi14 found by Lu and Nagase [55], and to the second
isomer of ZrSi14 found by Wang and Han [33]. We have tested other structures of Si 14 M given in the
literature [21,22]. Thus, the first isomer of Si14 Ti reported by Kawamura et al. [22] corresponds to the
isomers [37] 12th , 7th , and 9th for Sc− , Ti, and V+ impurities, respectively, with excess energy 0.57 eV,
0.53 eV, and 0.39 eV, and spin zero. When we optimize that structure for the triplet spin state, results a
triplet-singlet energy difference of 0.62 eV, 0.74 eV, and 0.62 eV for M = Sc − , Ti, and V+ , respectively.
Similarly, the structure of the first isomer of Si14 Ti reported in other work by Kawamura et al. [21]
corresponds to our isomers 16 th , 10th , and 11th of Si14 M, with excess energy 0.63 eV, 0.76 eV, and
0.68 eV, and triplet-singlet energy difference 0.27 eV, 0.10 eV, and 0.09 eV, for M = Sc − , Ti, and V+ ,
                M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                 n                              249

respectively.
   The structure of the ground state of Si 15 M coincides with that previously obtained for the second
isomer of Si15 Ti [22], and for the third isomer of Si15 Cr [21]. That structure (and the one for the second
isomer [35] of Si15 M) reminds the cubic structure of the ground state of Si 14 Fe [29] with an additional
Si atom. That cubic structure was also obtained by Kawamura et al. [21] for the third isomer of Si 14 Cr.
   For the first and second isomers of Si 16 M we obtain two structures practically degenerate, especially
for M = V+ . These structures were obtained by Kumar and coworkers [22,29] as the second and first
isomers, respectively, of Si16 Ti. The structure of the second isomer is the Frank-Kasper (FK) polyhedron,
with a nearly spherical structure and T h symmetry [24]. It consists of a central M atom surrounded by
16 Si atoms within two closely spaced shells: one with 12 atoms (all equidistant from central atom), and
an internal shell with 4 Si atoms forming a perfect tetrahedron. The ground state structure can be seen as
a distortion of the FK polyhedron (16-II), with the triangle along the three-fold symmetric axis rotated
by 30◦ . Analogous to the case of pure Si 16 clusters discussed in Section 3.1, several isomers of doped
Si16 M clusters are found in a narrow energy interval [37], and then the determination of the ground state
geometry is a difficult task [53]. We have reported in a previous work four isomers of Si 16 M very close
in energy but with very different symmetry.
   We are not aware of previous calculations, except our previous work [36], reporting the structures of
the two isomers of Si17 M represented in Fig. 2. We note that the sequence of V + and Ti doped isomers
departs considerably from the sequence for Sc − doped clusters. For higher energy isomers [37], the
tendency to the spherical form observed for the isomers of Si 15 M and Si16 M disappears. The structure
of the lowest energy isomer of Si18 Sc− and Si18 Ti can be seen as three pairs of silicon atoms decorating
alternating lateral faces of a DHP motive. Instead, the ground state of Si 18 V+ adopts a more elongated
structure. Analogous to Si17 M clusters, the low-lying isomers of Si18 M departs from the spherical form
adopted by the Si 16 M and Si15 M clusters. Thus, in the range n = 14–18, the optimal coordination of the
M impurity to Si atoms of Sin M clusters is found for n = 15, 16, as we discus in the next section.
   The fact that the ground state geometry of Si 18 V+ differs from that of Sc− and Ti doped clusters is
due to the different electronegativity of the metal impurity relative to a Si atom; Sc − and Ti (V+ ) have
smaller (larger) electronegativity than Si. Using the Mulliken population analysis it is found that the
V+ cation gains 0.047 electrons per Si atom in the 18-V isomer, which is preferred to the 18-I isomer
because in that geometry V + donates 0.026 electrons per Si atom. In contrast, Si18Sc − prefers the 18-I
geometry instead of the 18-V because the Si cage gains 0.063 electrons per Si atom in the 18-I isomer,
and only 0.043 electrons per Si atom in the 18-V isomer.
   The preference of Si17 V+ for 17-II isomer instead of 17-I is due to a size effect. It cannot be due
to charge transfer optimization because the charge transfer between the M impurity and the Si cage is
nearly identical for both geometries. Specifically, the number of electrons per Si atom transferred to the
Si cage in (17-I, 17-II) geometries is (0.067, 0.065) for Sc − , (0.018, 0.018) for Ti, and (−0.037, −0.034)
for V+ doped clusters, respectively.
   On the other hand, the atomic radius of the impurity, to be commented in Section 3.3, correlates well
with the average Si-M distances given in Fig. 3, namely 3.12 (Sc-in 17-I), 3.09 (Ti in 17-I), and 2.98
(V+ in 17-II). This argument cannot be applied to isomers 18-I and 18-V because the 18-V geometry
departs from the spherical shape, and the Si-M average distances have a broad dispersion.

3.3. Average Si-Si and Si-M distances

  The detailed atomic coordinates of the clusters reported in Figs 1 and 2 are available upon request to
the corresponding author (begonia@ubu.es). In Fig. 3 are represented the average distance d av (Si-Si)
250                M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                    n




Fig. 3. Average distance dav (Si-Si) among Si atoms (dashed lines), and its standard deviation (continuous bars), for the ground
state geometry of pure Sin and doped Sin M clusters with n = 14–18. The average distance dav (M-Si) among the impurity M
and Si atoms of doped clusters, and its standard deviation, is also represented (dots line and bars, respectively).

between Si atoms, and its standard deviation, for the ground state geometry of pure Si n and doped
Sin M clusters (anionic Si− clusters are not analyzed here). The average distance d av (M-Si) between the
                          n
impurity M and Si atoms of doped clusters, and its standard deviation, is also represented in Fig. 3.
  For pure Si clusters we see that d av (Si-Si) increases sharply at n = 16, where a broad maximum appears
in the distribution of Si-Si distances. These facts indicate a less compact geometry, which leads to a
decrease in the binding energy, as will be discussed in Section 3.4. A detailed analysis of the d av (Si-Si)
distributions for n = 14–18 reveals two peaks, at distances d 1 and d2 , which are larger, respectively,
than the first and second neighbors distances of bulk Si, namely, 2.45 Å and 4.00 Å. For n = 16 these
two peaks are broader than for the other clusters, and for n = 17 only a broad peak is distinguished.
Specifically, the distances (d 1 , d2 ) for n = 14–18 are (2.60, 4.20), (2.62, 4.25), (3.12, 4.61), (3.88), and
(3.13, 4.25), respectively. The deviation of the clusters Si-Si lengths from the crystalline silicon first
and second neighbor lengths, illustrate the frustration of atomic clusters, where most atoms cannot adopt
their favorite fourfold coordination.
  In the case of doped clusters, the distance d av (Si-Si) is smaller for Si15 M and Si16 M clusters, which
are also the “more spherical” and the more bonded ones (see next section). Notice that d av (Si-Si)
decrease smoothly at a given size from Si n Sc− to Sin Ti to Sin V+ , following the same tendency than
the metallic bond radius of the atomic impurities, namely 1.63 Å, 1.4 Å, and 1.3 Å, for Sc, Ti, and V,
respectively [56]. The distribution of Si-Si distance in doped clusters shows two maximal for n 16, at
about (roughly) 2.5 Å and 4.5 Å, which are larger than the first and second neighbor Si-Si distances in
                M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                 n                              251

bulk Si, respectively. For n = 17, 18 isomers there are only one broad maximum in the Si-Si distance
distributions.
   Concerning the average value of the distance Si-M, d av (Si-M), in doped clusters we see in Fig. 3 that
it is ∼ 1 Å smaller than dav (Si-Si), and the standard deviation is considerably smaller. That means that
doped clusters are more spherical and much more bonded than the pure Si clusters. The state of charge has
also influence on the overall size of doped clusters, as noted by Koyasu et al. [31], being the anions larger
than the cations. There are two or more maximal in the distributions of Si-M distances, depending on the
isomeric geometry. For example, the maxima of Si-M distance distribution for the four first isomers of
Si16 M, correspond to well defined radial shells of Si atoms around the central M impurity [37]. Thus, for
the second isomer of Si 16 Sc− and Si16 Ti (16-II isomer with FK geometry), there are two atomic shells
composed of 4 and 12 Si atoms, respectively, the first shell forming a perfect tetrahedron. It is interesting
to note that the electronic charge distribution of a perfect tetrahedral molecule X 4 has zero dipole and
quadrupole electric moments, and then it behaves approximately as a spherical charge distribution.

3.4. Electronic properties of Si n and Sin M clusters (M = Sc− , Ti, V+ ; n = 14–18)

3.4.1. Relative energy and stability
  For Sin M doped clusters, we will compare the binding (atomization) energy per atom,
    Eb (Sin M ) = [E(M ) + nE(Si) − E(Sin M )]/(n + 1),                                                         (1)
the addition energy of a M impurity to a Sin cluster,
    Ead (Sin M ) = E(Sin ) + E(M ) − E(Sin M ),
     M
                                                                                                                (2)
the addition energy of a Si atom to a Si n−1 M cluster,
    Ead (Sin M ) = E(Sin−1 M ) + E(Si) − E(Sin M ),
     Si
                                                                                                                (3)
the second difference of the cluster energy,
    ∆2 En (Sin M ) = E(Sin+1 M ) + E(Sin−1 M ) − 2E(Sin M ),                                                    (4)
and the energy difference between the eigenvalues of the lowest unoccupied (LUMO) and the highest
occupied (HOMO) molecular orbital, ∆gap . In the expressions above E(X) is the total energy of system
                                           n
X. The second difference energy ∆ 2 En , is equivalent to ESi (n)-ESi (n + 1). This second difference
                                                               ad      ad
is proportional to log(In /In+1 ), where In is the intensity of the Sin M signal in the experimental mass
spectra [57]. Similar relations to Eqs (1)–(4) can be defined for pure Si clusters.
   Figure 4 shows, for the lowest energy isomer of pure and doped Si clusters, the evolution with the
cluster size of the several quantities defined in Eqs (1)–(4). In panel a) of Fig. 4 we see that, for doped
clusters, the second energy difference has a positive peak at n = 16 and a negative value at n = 17,
which indicate a high abundance of Si 16 M relative to the neighbouring clusters. This fact agrees with
the mass spectrometry experiments of Nakajima and coworkers [31,32]. Contrarily, in panel b) we see
that the second energy difference of both, neutral and anionic pure Si clusters with even number of
valence electrons is smaller than for these clusters with an odd number of valence electrons. Thus, the
odd-even effect in panels (a) and (b) of Fig. 4 corresponds to minimal-maximal for doped clusters and to
maximal-minimal for pure Si clusters. We are not aware of previous notice of such a different behavior.
   In panel c) of Fig. 4 is depicted the binding energy per atom of the lowest energy isomer of pure and
doped Si clusters. In panel d) are shown the addition energy of the impurity M to pure Si n clusters, as
252                 M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                     n




Fig. 4. Evolution with cluster size of several properties of pure (Sin and Si− ) and doped (Sin M; M = Sc− , Ti, V+ ) silicon
                                                                                 n
clusters. In panels a) and b) is given the second difference of the total energy for doped (a) and pure (b) Si clusters, respectively.
In panel c) is represented the binding energy per atom of these clusters, and in panel d) is compared the addition energy of the
impurity M to pure Sin clusters (filled symbols) with the addition energy of Si to doped Sin−1 M clusters (empty symbols). The
addition energies of a Si atom to neutral and anionic Sin−1 are also represented in panel (d). Circles, squares, and triangles
represent Sc− , Ti, and V+ doped clusters, crosses represent pure Sin , and diamonds represent Si− clusters. The lines stand
                                                                                                       n
only to guide the eye.

well as the addition energy of Si to doped Si n−1 M clusters. The addition energies of a Si atom to neutral
and anionic Sin−1 are also represented in panel (d). For doped clusters we see in panels c) and d) local
peaks at n = 16, whereas for pure neutral Si clusters there are local minima at n = 16, and for anionic
Si clusters no special features can be distinguished.
  We see in Fig. 2 that a peak at n = 16 is also obtained for the Homo-Lumo gap of Si n M, which
can be considered as an additional signature of the special stability of Si 16 M clusters. The measured
Homo-Lumo gap for Si16 Ti is 1.9 eV [31,32], a slightly smaller value than our calculated value for the
ground state isomer 16-I. The Homo-Lumo gap of Si 16 M FK (16-II) clusters calculated by Khana and
Reveles [35] is 2.26 eV, 2.34 eV, and 2.42 eV for Si 16 Sc− , Si16 Ti, and Si16 V+ , respectively. For these
FK isomers, we obtain 2.12 eV, 2.17 eV, and 2.25 eV, respectively. However, for the near degenerate
16-I structure of these clusters, we obtain 1.91 eV, 2.09 eV, and 2.25 eV, respectively. Thus, we hope
that the measurement of magnitudes that are strongly dependent on the Homo-Lumo gap, as the dipole
polarizability, will allow us discriminate the isomers 16-I and 16-II in the case of Si 16 Sc− .
  The binding energy of Si n clusters with n = 14–18 fits quit well the phenomenological expression [5]
Eb (n) = Eb (∞) − cn−1/3 , where Eb (∞) is the binding energy of the bulk Si solid (experiment: 4.63 eV;
DFT-Siesta-PBE-DZbasis [41]: 4.84 eV). By fitting our calculations to that expression, taking E b(∞) =
4.84 eV, we obtain the coefficient c = 2.34 eV, to be compared with the value c = 2.23 eV, which was
obtained from a fit to experimental data [1,58] for Si n clusters in the range n = 25–70.

3.4.2. Symmetry and bonding: partial density of states (PDOS)
   The enhanced stability of the Si 16 M species with respect to the other (n = 16) Si n M clusters is due
to a special relationship between the cage symmetry and the electronic structure. Kumar [24] discussed
recently this issue for Si16 Ti in the context of the spherical potential model [38–40]. A similar discussion
                   M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                    n                                       253




Fig. 5. The total density of states (upper panel), the Si cage and impurity M projected density of states (middle panel), and the
M orbital projected density of states (lower panel) is shown for a) the isomer 16-II of Si16 Ti, with symmetry Td (FK structure),
and b) the ground state Si14 Sc− , with Cs symmetry. The Fermi energy was set to zero. The splitting of the levels of the
spherical potential model in Si16 Ti is caused by the molecular symmetry (see text).

can be found in reference 37. In that model, the orbital angular moment is a good quantum number, that
is, the single particle states belong to the irreducible representations of the rotation group O + (3). For an
empty cage, these states have predominantly zero radial nodes [59], and are labeled as 1s, 1p, 1d, 1f ,
1g, 1h, . . . The predominant covalent bonding in Si 16 M is the result of hybridization of the empty-cage
states and the endohedral-atom valence states, both with the same orbital angular moment character. This
approximate l-selection rule, has been postulated for endohedral Zr@C 28 and Zr@Si20 by Jackson and
coworkers [40], and is because only those orbitals transforming in the same irreducible representation of
the point group of the endohedral complex can be mixed in a given bonding state.
   This can be clearly seen in the total density of states (DOS) of the nearly spherical isomer 16-II of
Si16 Ti, which is represented in the upper panel of Fig. 5a), where the electronic levels are grouped and
labeled according to the spherical model. In the middle and lower panels of Fig. 5 is shown the partial
density of states (PDOS) for the impurity atom and the Si cage, and the contribution of 3d, 4s, and 4p
orbitals of the impurity to the PDOS, respectively.
   The valence electrons of Si 16 Ti can be associated with the sequence of levels 1s, 1p, 1d, 1f , 2s, 1g, 2p,
2d, 1h, 3s, 3p, 1i. . . of the spherical potential model, leading to a shell closing with 68 electrons when
the orbital 2d is completed, which is the HOMO. This HOMO orbital is the bonding orbital resulting
from the hybridization of a 3d orbital of Ti with the 2d (e + t2 ) orbital of the Si FK-cage, whereas the
antibonding hybrid orbital forms a part of the t2 components of the LUMO orbital, which is grouped in
the 1h state of the spherical model. Similarly, the 4s orbital of Ti hybridizes with the 2s orbital of the
FK structure, giving the bonding orbital 2s and the antibonding 3s orbital of the compound Si 16 Ti. A
similar analysis can be performed for the isomers 16-I of Si 16 M, with D3 symmetry, as we have shown
in a previous work [37]. For non-spherical clusters, the bonding is not based on the l-selection rule. It
can be seen by inspection of Fig. 5b) for the ground state of Si 14 Sc− , that a definite symmetry cannot be
254             M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                 n


                                                           Table 2
                           Calculated adiabatic electron affinity (EAcal , in eV) of neutral
                           Sin Sc, Sin Ti, Sin V, and Sin . For Sin Ti and Sin the experimental
                           values (EAexp ) from Ohara et al. [30] are also given
                                   Sin Sc         Sin Ti          Sin V              Sin
                           Size    EAcal     EAcal EAexp          EAcal     EAcal       EAexp
                            14      3.17      2.72      2.56       3.06      2.12     3.2 ± 0.19
                            15      3.21      2.57      2.78       2.36      2.06     3.1 ± 0.19
                            16      3.31      1.88      1.81       3.03      2.77     3.2 ± 0.19
                            17      3.09      2.57      2.47       2.58      2.38     3.1 ± 0.19
                            18      2.91      2.79      2.82       2.85      2.62     3.1 ± 0.19

recognized, and it is clear that the 3d orbital of the impurity is fragmented among several levels near the
HOMO, and, mainly, at higher energies than the HOMO.

3.4.3. Electron affinity
   In Table 2 is given the adiabatic electron affinity (EA cal ) of neutral Sin M0 doped clusters (M0 = Sc,
Ti, V) and pure Sin clusters, calculated as the total energy difference between neutral Si n M0 and anionic
Sin M− species in their respective lowest energy states. We see in Table 2 that our results for the adiabatic
electron affinities of Sin Ti− , compare well with the experimental values for the threshold detachment
energies of Sin Ti− , which corresponds to the upper limit of the EA of Si n Ti. Another experimental value
for the adiabatic detachment energy of Si 16 Ti− , 2.03 ± 0.09, was given by Koyasu et al. [31,32]. Two
different theoretical estimations of the EA of Si16 Ti [22,35] yield the value 1.91 eV.
   The calculated vertical (adiabatic) electron affinity of Si 16 Sc is 3.56 (3.31) eV, which compare well
with the experimental detachment energy of Si 16 Sc− [31,32] 4.25 (3.41 ± 0.12) eV. The vertical affinity
is obtained as the difference of energy between the neutral and anion clusters but allowing relaxation
of the neutral cluster with the same geometry as the anion. The experimental adiabatic detachment
energy of Si16 V− determined recently by Koyasu et al. [32] is 3.08 ± 0.13 eV, which is close to our
calculation, 3.03 eV, in Table 2. On the other hand, the calculated electron affinities of pure Si n clusters
are considerably lower than the measured ones.


4. Conclusions

   The structural and electronic properties of the low lying energy isomers of pure Si n and doped Si n M
(M = Sc− , Ti, V+ ) clusters in the range n = 14–18 have been studied, and several new geometries of low
lying isomers have been obtained, extending previous theoretical results [35–37] and giving a detailed
explanation of recent experimental results [31,32]. For doped clusters, the metal impurity becomes
encapsulated in all cases, leading to geometries without structural relation to those of pure Si clusters.
Near spherical geometries are obtained for the lowest energy isomers of Si 16 M. Doped clusters with n =
16 exhibit smaller Si-M average distance than clusters with other sizes, which correlates with the higher
stability of Si16 M compared to neighbour clusters found in recent experiments [31,32].
   That enhanced stability is demonstrated here by means of first principles calculations leading to positive
peaks at n = 16 in the trend of second difference of total energy versus the cluster size. Similar trends
are reported and discussed for the binding energy per particle, the addition energy of the impurity M to
pure Si clusters, the adiabatic electron affinity, and the HOMO-LUMO gap. A detailed comparison of
the bonding properties of the low lying energy isomers Si 14 Sc− and Si16 Ti is given by analyzing their
partial density of states in the context of the spherical potential model. This study has identified the
                   M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                    n                                  255

interplay among geometrical and electronic factors which determine the stability and therefore the high
abundance of Si 16 M clusters detected in the experiments.
  The calculated electron affinities of pure Si n are systematically underestimated by more than 18%
compared to experimental estimations [30]. Nevertheless the calculated adiabatic electron affinity of
neutral species of Sin Ti clusters, as well as the HOMO-LUMO gap of Si n M, is in good agreement with
available experimental measurements [30,32]. These results give us confidence for the future research
of metal doped silicon materials formed by aggregation of very stable Si n M units.


Acknowledgments

  The authors wish to thank the support of the Spanish Ministry of Science (Grant MAT2005-03415),
                                                                 ´
of FEDER of the European Community, and Junta de Castilla y Le on (Grant No. VA068A06).


References

 [1]   K.-M. Ho, A.A. Shvartsburg, B. Pan, Z.-Y. Lu, C.-Z. Wang, J.G. Wacker, J.L. Fye and M.F. Jarrold, Nature 392 (1998),
       582.
 [2]   V. Kumar, Computational Material Science 36 (2006) 1.
 [3]   E.C. Honea, A. Ogura, C.A. Murray, K. Raghavachari, W.O. Sprenger, M.F. Jarrold and W.L. Brown, Nature 366 (1993),
       42.
 [4]   C.C. Arnold and D.M. Neumark, J Chem Phys 99 (1993), 3353.
 [5]   M.F. Jarrold and E.C. Honea, J Phys Chem 95 (1991), 9181.
 [6]        u
       J. M¨ ller, B. Liu, A.A. Shvartsburg, S. Ogut, J.R. Chelikowsky, K.W.M. Siu, K.-M. Ho and G. Gantefor, Phys Rev Lett
       85 (2000), 1666.
 [7]   N. Binggeli, J.L. Martins and J.R. Chelikowsky, Phys Rev Lett 68 (1992), 2956.
 [8]   I.-H. Lee, K.J. Chang and Y.H. Lee, J Phys: Condens Matter 6 (1994), 741.
 [9]   I. Rata, A.A. Shvartsburg, M. Horoi, T. Frauenheim, K.W.M. Siu and K. Jackson, Phys Rev Lett 85 (2000), 546.
[10]   B.-X. Li, P.-L. Cao and S.-Ch. Zhang, Phys Lett A 316 (2003), 252.
[11]   X.L. Zhu, X.C. Zeng and Y.A. Lei, J Chem Phys 120 (2004), 8985.
[12]   S. Yoo and X.C. Zeng, Angew Chem Int Ed 44 (2005), 1491.
[13]   S. Yoo, J.J. Zhao, J.L. Wang and X.C. Zeng, J Am Chem Soc 126 (2004), 13845.
[14]   S. Goedecker, W. Hellmann and T. Lenosky, Phys Rev Lett 95 (2005), 055501.
[15]   J. Bai, L. Cui, J. Wang, S. Yoo, X. Li, J. Jellinek, C. Koehler, T. Frauenheim, L. Wang and X.C. Zeng, J Phys Chem A
       110 (2006), 908.
[16]   M.A. Belkhir, S. Mahtout, I. Belabbas and M. Samah, Physica E: Low-dimensional Systems and Nanostructures 31
       (2006), 86.
[17]   S. Mahtout and M.A. Belkhir, Acta Physica Polonica A 109 (2006), 685.
[18]   S. Nigam, Ch. Majumder and S.K. Kulshreshtha, J Chem Phys 125 (2006), 074303.
[19]   B.-X. Li and Q. Xu, Phys Stat Sol (b) 241 (2004), 990.
[20]   A.A. Shvartsburg, B. Liu, M.F. Jarrold and K.M. Ho, J Chem Phys 112 (2000), 4517.
[21]   H. Kawamura, V. Kumar and Y. Kawazoe, Phys Rev B 70 (2004), 245433.
[22]   H. Kawamura, V. Kumar and Y. Kawazoe, Phys Rev B 71 (2005), 075423.
[23]   N. Uchida, T. Miyazaki and T. Kanayama, Phys Rev B 74 (2006), 205427.
[24]   See ref 2 and: V. Kumar, Computational Material Science 35 (2006), 275; ibid 30 (2004), 260.
[25]   J.B. Jaeger, T.D. Jaeger and M.A. Duncan, J Phys Chem A 110 (2006), 9310.
[26]   S. Neukermans, X. Wang, N. Veldeman, E. Janssens, R.E. Silverans and P. Lievens, International Journal of Mass
       Spectrometry 252 (2006), 145.
[27]   S.M. Beck, J Chem Phys 87 (1987), 4233.
[28]   S.M. Beck, J Chem Phys 90 (1989), 6306.
[29]   V. Kumar and Y. Kawazoe, Phys Rev Lett 87 (2001), 055503.
[30]   M. Ohara, K. Koyasu, A. Nakajima and K. Kaya, Chem Phys Lett 371 (2003), 490.
[31]   K. Koyasu, M. Akutsu, M. Mitsui and A. Nakajima, J Am Chem Soc 127 (2005), 4998.
256                M.B. Torres et al. / Structure and relative stability of Sin , Si− , and doped Sin M clusters
                                                                                    n


[32]   K. Koyasu, J. Atobe, M. Akutsu, M. Mitsui and A. Nakajima, J Phys Chem A 111 (2007), 42.
[33]   J. Wang and J.-G. Han, J Chem Phys 123 (2005), 064306.
[34]   E.N. Koukaras and C.S. Garoufalis and A.D. Zdetsis, Phys Rev B 73 (2006), 235417.
[35]   J. Ulises Reveles and S.N. Khanna, Phys Rev B 74 (2006), 035435.
[36]                               a
       M.B. Torres and L.C. Balb´ s, European Physics Journal D 43 (2007), 217.
[37]                           a                   a
       M.B. Torres, E.M. Fern´ ndez and L.C. Balb´ s, Phys Rev B 75 (2007), 205425.
[38]   H.-P. Cheng, R.S. Berry and R.L. Whetten, Phys Rev B 43 (1991), 10647.
[39]   J. Jackson, E. Kaxiras and M.R. Pederson, J Phys Chem 98 (1994), 7805.
[40]   J. Jackson and B. Nellermoe, Chem Phys Lett 254 (1996), 249.
[41]                                             ı                        o           a
       J.M. Soler, E. Artacho, J.D. Gale, A. Garc´a, J. Junquera, P. Ordej´ n and D. S´ nchez-Portal, J Phys Condens Matter 14
       (2002), 2745, http://www.uam.es/siesta.
[42]   W. Kohn and L.J. Sham, Phys Rev 145 (1965), 561.
[43]   J.P. Perdew, K. Burke and M. Ernzerhof, Phys Rev Lett 77 (1996), 3865.
[44]                             ı
       N. Troullier and J.L. Mart´ns, Phys Rev B 43 (1991), 1993.
[45]   L. Kleinman and D.M. Bylander, Phys Rev Lett 438 (1982), 1425.
[46]              a                                a
       E.M. Fern´ ndez, M.B. Torres and L.C. Balb´ s, Int J Quantum Chem 99 (2004), 39.
[47]                           a                   a
       M.B. Torres, E.M. Fern´ ndez and L.C. Balb´ s, Phys Rev B 71 (2005), 155412.
[48]   The Cambridge Cluster Database, http://www-wales.ch.cam.ac.uk/CCD.html.
[49]   Xiao, A. Abraham, R. Quinn, F. Hagelberg and W.A. Lester, Jr., J Phys Chem A 106 (2002), 11380.
[50]   Tomonari and K. Tanaka, Theor Chem Acc 106 (2001), 188.
[51]   M. Hamrick and W. Weltner, Jr., J Chem Phys 94 (1991), 3371.
[52]   Z.J. Wu and Z.M. Su, J Chem Phys 124 (2006), 184306.
[53]   W. Hellmann, R.G. Henning, S. Goedecker, C.J. Umrigar, B. Delley and T. Lenosky, Phys Rev B 75 (2007), 08541.
[54]                       n
       V.E. Bazterra, O. O˜ a, M.C. Caputo, M.B. Ferrero, P. Fuentaelba and J.C. Facelli, Phys Rev A 69, (2004), 0532202.
[55]   J. Lu and S. Nagase, Phys Rev Lett 90 (2003), 115506.
[56]   D.R. Lide, CRC Handbook of Chemistry and Physics, (79th Edition), CRC Press: Boca Raton, FL, 1999.
[57]   C.E. Klots, J Chem Phys 92 (1988), 5864.
[58]   T. Bachels and R. Schafer, Chem Phys Lett 324 (2000), 365.
[59]              a                              a
       E.M. Fern´ ndez, J.M. Soler and L.C. Balb´ s, Phys Rev B 74 (2006), 235433.
Journal of Computational Methods in Sciences and Engineering 7 (2007) 257–272                                                       257
IOS Press




The full story of the Si6 magic cluster

Aristides D. Zdetsis
Department of Physics, University of Patras, GR-26500 Patras, Greece
E-mail: zdetsis@upatras.gr

Received 20 July 2007
Revised /Accepted 15 September 2007

Abstract. The structural, electronic, dynamical and spectral properties of Si6 and its ions (Si1− , Si2− , and Si1+ ) have been
                                                                                                      6      6          6
examined using a variety of high level ab initio techniques, including quadratic configuration interaction, coupled cluster, and
density functional theory (DFT) with the hybrid B3LYP functional. Various high quality correlation-consistent basis sets,
ranging from 2Z up to 5Z quality, were employed for the DFT calculations. It is shown that not only the ground state structure,
but also the structure of the excited states, as well as the structure of the anions and cations of Si6 are controversial. Each
one of the three competing structures for the ground state, with Cs /C2v , D4h , and C2v symmetry, has been considered by
different investigators as the lowest energy structure either of the neutral cluster, or of its anion or cation (or both). In a spirit of
“structural democracy” it is demonstrated that the Cs /C2v , D4h , and C2v structures can be safely assigned as the ground states
of the neutral, anion, and cation clusters respectively. The present results, which support the structural plasticity (fluxionality)
of Si6 , are in excellent agreement with experiment, including Raman and IR spectra, ionization energies, electron affinities as
well as vibrationally resolved photoelectron spectra. The paradigm of Si6 could be very helpful for other silicon clusters as
well.



1. Introduction

   The field of atomic clusters, and in particular silicon clusters, is a well established important interdis-
ciplinary field of research which is increasingly active over the last thirty years due to its fundamental
scientific and technological importance [1–10]. Yet, as it was illustrated earlier [6,7], the structure of
small magic clusters as Si6 , which is considered as one of the best understood and extensively studied
small clusters, is controversial even now [5–10]. The reason for this paradox and the discrepancies in
the structure of Si6 seem to have been finally understood, at least according to the present author, only
recently [10]. As was recently demonstrated [10], Si 6 is fluxional without a well defined single geomet-
rical structure. The structure of a molecule or a cluster as Si 6 is usually visualized by the traditional
ball-and-stick diagrams, reflecting the common notion of rigid structures in which the atoms execute
small harmonic vibrations about their equilibrium positions [11]. This ideal “Platonic” picture is not
always valid [12]. Isomerization, and fluxional rearrangements, common in Organic and Organometallic
Chemistry, are processes which go beyond this simple picture [11,13]. Fluxional behavior, characterized
by rapid molecular rearrangements, is widespread in electron-deficient molecules such as boranes and
carboranes [13]. Such behavior is rather unexpected for Si 6 , the structure of which has been extensively
studied [4–10] -not always unambiguously- within the traditional picture of rigid spheres connected
with harmonic springs. The traditional “picture” is based on the adiabatic and harmonic approximations
which allow the separate treatment of electrons and nuclei. According to these approximations most of
the molecular systems have a single, well-defined equilibrium configuration, around which the atoms

1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
258                             A.D. Zdetsis / The full story of the Si6 magic cluster

execute approximate harmonic vibrations. Besides and/or together with the well known Jahn-Teller
(first and second order) effects [9] which go beyond but can be approximately described within the
traditional framework (even as “small” perturbations) are various types of nuclear rearrangements such
as isomerization and fluxional processes [11,13]. Isomerization or tautomerization characterizes nuclear
rearrangements in which two or more configurations are not chemically equivalent. Isomerization is
very important for organic chemistry and for the living cells (for instance the cis-trans isomerization
in the retina is very important for vision). In the fluxional rearrangements the nuclear configurations
(structures) are chemically equivalent, but the bonding pattern is changing. Fluxional rearrangements
are common in electron-deficient molecules, such as boranes and carboranes, because there is not a set
bonding pattern. In fluxional molecules there is a network of multi-center bonds, which can interconvert
with little or marginal energy cost. In electron-precise molecular systems such rearrangements involve
a relatively high energy cost. Yet, as has been demonstrated [10], Si 6 exhibits fluxional behavior. This
fluxional behavior is driven by the coupling of the electronic motion with soft nuclear vibrations (second
order Janh-Teller effect) under the influence of the “charge smoothing process”. The consequences of
the Janh-Teller effect for Si6 have been discussed recently by Karamanis et al. [9].
   The objective of the present analysis, which is largely based on older [6] and recent [7,10] work of the
present author, is to present a high level (as complete as possible) description of the Si 6 cluster and its
ions, including structural, electronic, and spectral properties. This will elucidate existing controversies
and ambiguities [8–11,14–22] about the exact structure and properties of these clusters.
   The presentation is organized as follows: In Section 2 some technical details of the calculations are
presented. In Section 3 the structural and electronic properties of the neutral and ionic clusters are given,
whereas the low-lying excited structures are described in Section 4. Finally, comparison with experiment
is given in Section 5, and some concluding remarks are summarized in Section 6.


2. Technical details

   The theoretical techniques for the geometry optimizations include the quadratic configuration in-
teraction (QCI) and coupled cluster (CC) methods with single and double excitations (QCISD, and
CCSD respectively) and the density functional theory (DFT) using the hybrid B3LYP functional. The
correlation-consistent cc-pvdz and cc-pvtz basis sets [23,24] were used for all calculations. The B3LYP
results have been also tested with higher order correlation-consistent basis sets, such as cc-pvqz and
cc-pv5z [23]. At the optimized QCISD, CCSD and B3LYP equilibrium geometries single point en-
ergy calculations have been performed at various levels of Moller-Plesset perturbation theory as well
as QCISD and CCSD including quasiperturbative corrections for the triple excitations, QCISD(T) and
CCSD(T) respectively. The bulk of the calculations were performed with the GAUSSIAN-03 [23] and
the TURBOMOL [24] program packages. Finally, for the calculation of the optical excitations and the
optical absorption spectrum, the time dependent DFT method (TDDFT) has been used with the B3LYP
hybrid functional.


3. Structural and electronic properties of the neutral and ionic clusters

3.1. The neutral cluster

   The three competing structures for the ground state are shown in Fig. 1(a), 1(b) and 1(c). Each of these
structures has been considered at different times [4–10] as the real equilibrium geometry of Si 6 . This
                                     A.D. Zdetsis / The full story of the Si6 magic cluster                               259

                                                         Table 1
                            Absolute energies of the three structures in atomic units (hartree),
                            evaluated at the QCISD optimized geometry using the cc-pvtz basis
                            sets
                                                                Structure
                                                 D4h            CS /Cc2v             C2V
                            Method            (Fig. 1a)         (Fig. 1b)         (Fig. 1c)
                            MP2             −1734.081679 −1734.080323 −1734.079998
                            MP4SDTQ −1734.153700 −1734.152899 −1734.152633
                            CCSD(T)        −1734.129129 −1734.129187 −1734.129171
                            QCISD(T)       −1734.130095 −1734.130207 −1734.130200
                            QCISD(T)       −1734.316252 −1734.316711 −1734.316080
                            B3LYP          −1737.064145 −1737.064789 −1737.064918
                                                1                  1                 1
                            State                 A1g                A1                A1

                                                         Table 2
                  Comparison of the DFT/B3LYP energies (in Hy) for various basis sets. The energy
                  differences from the corresponding lowest energy state (shown with bold characters) are
                  given in parenthesis (in eV)
                              B3LYP/ cc-pvdz       B3LYP/ cc-pvtz       B3LYP/ cc-pvqz        B3LYP/ cc-pv5z
                                    hy                   hy                    hy                   hy
                  D4h         −1737.000826          −1737.06415          −1737.082326          −1737.102644
                                (+0.02 eV)           (+0.02 eV)            (+0.02 eV)           (+0.02 eV)
                  C2v         −1737.001447          −1737.06492          −1737.083138          −1737.103401
                                    –                    –                     –                    –
                  CS/C2v      −1737.001324|         −1737.06479          −1737.083017          −1737.103295
                               (+0.003 eV)          (+0.003 eV)           (+0.003 eV)          (+0.003 eV)
                  CS          −1737.0013213        −1737.064803          −1737.083029          −1737.103306
                               (+0.003 eV)          (+0.003 eV)           (+0.003 eV)          (+0.003 eV)




           Fig. 1. The three energetically lowest structures of Si6 : (a) the D4h , (b) the Cs /C2v , and (c) the C2v .

can be easily understood in view of the extremely small energy differences between the three structures,
obtained with a variety of high level ab initio theoretical methods, shown in Tables 1 and 2. As can be
seen in Table 1, the various methods give fluctuating energy differences of the order of 0.005 eV, which
is marginal.
   The same is true for the DFT/B3LYP results in Table 2, obtained with a variety of large correlation-
consistent basis sets (up to 5Z level). As we can see, contrary to the absolute energies, the relative
energies between the three structures do not change as the size and the quality of the basis sets increases.
The ordering (and the magnitude) of the marginal energy differences between the structures does not
change (at least in detectable amounts) when the zero energy corrections to the total energy are taken
into account.
   By truncating the level or type of the correlation and/or the quality of the technical approximations
260                              A.D. Zdetsis / The full story of the Si6 magic cluster

                                                       Table 3
                         Calculated vibrational frequencies of the D4h Si6 cluster by various
                         methods, in units of cm−1
                         Exp                                 D4h
                                Sym.     MP2/       MP2       QCISD/       B3LYP/         B3LYP/
                                        6-31g*     cc-pcdz    cc-pvdz      cc-pvdz        cc-pvtz
                          –     Eu         52         61         i62         i57            i62
                          –     B2u       197        183        124          113            125
                         252    B2g       220        231        242          241            240
                         300    A1g       314        297        308          303            302
                          –     A2u       358        332        319          306            315
                         386    B1g       396        371        385          375            384
                         404    Eg        447        423        388          380            392
                         458    A1g       481        459        457          440            446
                          –     Eu        482        460        453          436            446

and details (e.g. basis sets) the ordering of the structures can be easily reversed (within the small margins
of the energy differences). Actually as the level of theoretical treatment increases, the magnitude of the
small energy differences decrease. Such small differences are barely in the limits of numerical accuracy
and, in any case, smaller than the average vibrational zero point energy. In addition, all three structures
have very soft vibrations (with frequencies around 60–70 cm −1 ), while the high symmetry D4h structure
in Fig. 1(a) in all levels of theory except second order perturbation theory (MP2) has soft modes with
imaginary frequencies, as shown in Table 3. As has been shown [10], not only the three structures have
marginal energy differences, but they also have marginal energy barriers during the transformation from
one to another, following the displacement patterns of the imaginary frequency or soft vibrational modes.
Thus, in reality all three structures can be considered as different instances of the same fluxional structure.
Actually, not only these three, but also all intermediate structures during fluxional rearrangements are
different instances of the same multi-structure.
   This type of structural plasticity, accompanied by mixing of different nuclear and electronic configura-
tions is largely responsible for the fluctuation of energy differences at different levels of theory. For the
same reason, earlier calculations [5], based on simple second order Moller-Plesset perturbation theory
(MP2) identified the D4h structure as the real structure of Si6 .
   In structure 1(b), which for bookkeeping purposes can be considered as the equilibrium structure (by
roughly 0.005 eV energy difference from the other two), the number of atoms with five-connections
(atoms 1 and 2) are balanced out by an equal number of threefold-connected atoms (atoms 3 and 6).
Since, as population analysis shows, the charge distributions on 3-connected and 5-connected atoms have
opposite signs, this is roughly equivalent with a manifestation of charge smoothing, which is responsible
for fluxional rearangements [10]. A similar balancing occurs in the T d structure [25] of Si10 which is the
second lowest by a small energy difference from the C 3v ground state). This balancing could possibly
be a general trend for larger silicon systems (e.g. amorphous silicon). In this case one could infer that
structures with a balanced number of dangling (3-fold coordinated) and floating (5-fold coordinated)
bonds would be particularly stable, and vice-versa.

3.2. Results and discussion for the anions and cations

  On the basis of the connection of fluxional behavior and electron deficiency it would be expected that
the Si1− anion would be more stable and more symmetric [10]. Apparently, this would favor from the
      6
three competing structures the D 4h structure for the Si1− anion. Indeed, all calculations for this anion
                                                        6
                                     A.D. Zdetsis / The full story of the Si6 magic cluster                                261




Fig. 2. Six stages of the geometry optimization of the Cs /C2v anion. The geometry finally converges to the D4h anion structure.

at various levels and orders of (perturbation) theory, as well as density functional theory, converge to
one and the same D 4h structure, unlike the neutral cluster. This illustrated in Fig. 2, which shows some
instances (stages) of the geometry optimization of the C s /C2v anion (in this particular case with the
B3LYP/cc-pvtz method).
   In Table 4 the QCISD(T) and B3LYP results for Si 6 , Si1− and Si2− and Si1+ are summarized in
                                                                6          6          6
a concise form. The energies in Table 4 are given at the QCISD(T) level, evaluated at the QCISD
optimized geometry, using correlation consistent basis sets of double and triple Z quality (cc-pvdz and
cc-pvtz respectively). Similar results obtained at the B3LYP/cc-pvtz (both geometries and energies) are
also given for comparison. The cohesive or atomization energies per atom are referred to the neutral
silicon atom in its normal triplet configuration, 3 P. The cohesive energy of Si 6 at the QCISD(T)/cc-pvTZ
level is calculated at 3.22 eV/atom (3.11 at the B3LYP level). Thus the atomization energy is predicted
to be 19.92 eV. Although there are not experimental measurements to compare, this value is in excellent
agreement with the G2 computed atomization results of Zhao et al. [19]. The lower B3LYP value
(18.66 eV) is consistent with similar results [19] which show smaller values of the Si 5 cluster at the
B3LYP level (14.29 eV) compared to MP2 (15.26 eV).
   The comparison with other theoretical results for the structural and electronic properties has the added
difficulty of the ambiguity with regard the real structure of the neutral cluster as well as ambiguities in the
structure of anions and cations [14,18–21]. As a matter of fact each the structures of Fig. 1(a), (b), and (c)
262                                    A.D. Zdetsis / The full story of the Si6 magic cluster

                                                          Table 4
                   QCISD(T) and B3LYP Total energies E (in Hy), cohesive energies εb (in eV/atom),
                   and relative (to neutral) energies ∆E (in eV) of charged Si6 clusters. The “cohesive
                   energies” of the charged clusters are relative to the neutral silicon atom
                    Symmetry       State        QCISD(T)     E       QCISD(T)     E        B3LYP       E
                                                cc-pvDZ     εb       cc-pvTZ     εb        cc-pvTZ    εb
                          Charge                           ∆E                   ∆E                   ∆E
                    D2
                     4h                A2u       −1734.1942453        −1734.3895609        −1737.1387357
                                                          3.20                 3.65                 3.45
                            −1                           −1.75                 −2.0                −2.01
                                       1
                    C2v (Cs )              A1    −1734.1302066        −1734.3161663         −1737.064917
                                                          2.90                 3.32                 3.11
                            0                               −                    −                    −
                                       2
                    C2v                    B1    −1733.8578971        −1734.0361338        −1736.7894234
                                                          1.66                 2.05                 1.86
                            +1                           +7.40                +7.60                +7.49
                                   4
                    Oh                 A1g       −1733.8414947        −1734.0232934        −1736.7773601
                                                          1.59                 1.99                 1.81
                           +1                           + 7.85               + 7.98               + 7.82

have been considered at various levels of theory as ground states of the anion or of the cation (in addition
to the neutral) cluster. This could be referred as the principle of structural democracy and it is certainly
related to the fluxionality of Si6 . In spite of this, the D4h distorted octahedron structure is a rather well
established [10,14,19,20] (for reasons which have been clearly explained in ref 10) ground state for the
anion. Yet its structural characteristics (bond lengths and angles) are different from the corresponding
D4h neutral “ground state”. For instance the long bond of 2.74 Å between the diagonal equatorial atoms
1 and 2 (with reference the Fig. 1a) is broken and the distance between these atoms becomes (12) =
3.184 Å. Similarly the bond lengths (31) = 2.760 Å and (24) = 2.385 Å become 2.600 Å and 2.43 Å,
whereas the bond-angle (415) in Fig. 1, or (215) in Fig. 2a, from 109.8 ◦ in the neutral cluster, in the
anion becomes 98.3 ◦. This is a rather large relaxation, producing large differences between the vertical
and adiabatic electron affinities which could show up as an effective breaking of time reversal symmetry
in photoemission and its inverse process [20]. The cation Si 6 obtained here by QCISD(T), CCSD(T)
and B3LYP calculations, using the correlation consistent cc-pvtz basis set, is the edge-capped trigonal
bipyramid of C2v symmetry (Fig. 1c). However, as was explained elsewhere [10], a very stable 4 A1g
state of full Oh symmetry exists about 0.4 eV higher than the 2 B1 (C2v ) ground state of the cation. In
Table 5, following the principle of structural democracy we have assigned (for book-keeping purposes)
the bicapped tetrahedron of Fig. 1b (with C s near C2v symmetry) as the ground state of the neutral cluster.
Thus, all three structures in Figs 1a, 1b, and 1c are respectively the ground states of Si 1− , Si6 , and Si1+ ,
                                                                                           6                6
whereas the Oh structure is unambiguously the ground state of the Si 2− cluster. On the basis of these
                                                                           6
assignments we can easily obtain from Table 4 the adiabatic electron affinities and ionization potential
of Si6 . This is left for Section 5.1 for reasons of homogeneity in the structure of the paper.


4. Low-lying excited structures

  In addition to the “multistructure” of Fig. 1 based on structures (a) (b) and (c) with symmetries D 4h ,
C2v and Cs /C2v respectively there are at least two more structures with energies up to around 1eV from
the ground state. These structures are shown schematically in Fig. 3, whereas their energetic, vibrational,
and structural characteristics are summarized in Tables 5 and 6.
                                       A.D. Zdetsis / The full story of the Si6 magic cluster                                263

                                                            Table 5
              Energy differences (in eV) and vibrational frequencies (in cm−1 ) of the higher energy states obtain
              with two different methods (QCISD and B3LYP). The frequencies with bold characters correspond
              to the higher intensity IR frequencies
               State (sym)     Method      Frequencies (symmetry)
               ∆E (in eV)
               3
                 A1g (D4h )     QCISD      84.6 (B2u ), 119.0 (Eu ), 316.1 (Eg ), 319.3 (B1g ), 359.8 (A1g ),
               +0.69 eV                    372.6 (Eu ), 377.8 (A2u ), 419.6 (B2g ), 454.9 (A1g )
               3
                 A1g (D4h )     B3LYP      78.0 (B2u ), 127.5 (Eu ), 312.5 (Eg ), 319.4 (B1g ), 354.9 (A1g ),
               +0.58 eV                    367.7 (A2u ), 375.9 (Eu ), 413.8 (B2g ), 443.3 (A1g )
               3
                 A1g (D3d )     QCISD      124.3 (A1u ), 136.0 (Eu ), 323.9 (Eg ), 349.9 (A1g ), 356.4 (Eu ),
               +1.08 eV                    363.3 (A2u ), 441.0(A1g ), 721.8 (Eg )
               3
                 A1g (D3d )     B3LYP      −654.2 (Eg )∗ , 126.1 (A1u ), 137.5 (Eu ), 346.0 (A1g ), 354.5 (Eu ),
               +0.92 eV                    363.8 (Eg ), 365.8 (A2u ), 430.5 (A1g )
               3
                 A1g (D3d )     CCSD       −711.0 (Eg )∗ , 126.5 (A1u ), 142.2 (Eu ), 350.9 (A1g ), 363.0 (Eu ),
               +1.08 eV                    367.8 (A2u ), 411.2 (Eg ), 441.8 (A1g )
               3
                A1g (D3d )       MP2       162.0 (A1u ), 214.2 (Eu ), 369.3 (A1g ), 382.8 (Eu ), 445.2 (A2u ),
               +1.14 eV                    451.7 (Eu ), 463.6 (A1g ), 796.9 (Eg )
               1
                A’ (Cs )        QCISD      74.7 (A’), 89.1 (A"), 96.2 (A’), 197.0 (A’), 241.0 (A’), 256.8 (A"),
               +1.83 eV                    322.5 (A’),327.1(A"), 388.6 (A’), 406.6 (A"), 477.3 (A’), 541.6 (A")
               1
                A’ (Cs )        B3LYP      81.5 (A’), 94.7 (A"), 99.0 (A’), 204.0 (A’), 238.8 (A’), 258.8 (A"),
               +1.42 eV                    320.7 (A’), 333.4 (A"), 383.6 (A’), 410.3 (A"), 473.6 (A’), 531.8 (A")
              ∗
                  imaginary frequencies.




                                                                                                        (b)
                                                             (a)




                                                             (c)                                        (d)

Fig. 3. Low-lying structures discussed in the text: D4h (a), D3d chair structure (b), and peacock-tail structure in front (c) and
side (d) view.

  The first is a cubic structure with full Oh symmetry, which has a triplet electronic configuration (as is
clear from the energy level diagram in Fig. 3 of ref. 10).
264                               A.D. Zdetsis / The full story of the Si6 magic cluster

                                                      Table 6
                      Bond-lengths of the higher energy states. The numbering of atoms is re-
                      flected in Fig. 3(a, c)
                      State          method      Bond-lengths of the higher energy states
                      3
                        A1g (D4h )   QCISD       (12) = (15) = (42) = (45) = 2.401 Å
                      Fig. 7(a)                  (31) = (32) = (34) = (35) = 2.556 Å
                      3
                       A1g (D4h )     B3LYP      (12) = (15) = (42) = (45) = 2.377 Å
                      Fig. 7(a)                  (31) = (32) = (34) = (35) = 2.532 Å
                      3
                       A1g (D3d )     QCISD      (12) = (45) = (32) = (31) = 2.472 Å
                      Fig. 7(a)                  (15) = (52) = ( 34) = (35) = 2.530 Å
                      3
                        A1g (D3d )    B3LYP      (12) = (45) = (32) = (31) = 2.448 Å
                      Fig. 7(a)                  (15) = (52) = ( 34) = (35) = 2.501 Å
                      3
                       A1g (D3d )     CCSD       (12) = (45) = (32) = (31) = 2.472 Å
                      Fig. 7(a)                  (15) = (52) = (34) = (35) = 2.530 Å
                      3
                       A1g (D3d )      MP2       (12) = (45) = (32) = (31) = 2.419 Å
                      Fig. 7(a)                  (15) = (52) = (34) = (35) = 2.464 Å
                      1
                       Å (Cs )        QCISD      (61) = (65) = 2.337 Å, (62) = (64) = 2.483 Å,
                      Fig. 7(c)                  (63) = 2.703 Å
                                                 (12) = (45) = 2.311 Å, (23) = (34) = 2.373 Å
                      1
                       Å (Cs )       B3LYP       (61) = (65) = 2.319 Å, (62) = (64) = 2.463 Å,
                      Fig. 7(c)                  (63) = 2.671 Å
                                                 (12) = (45) = 2.287 Å, (23) = (34) = 2.346 Å

   This structure which at the B3LYP/cc-pvtz level is 1.3 eV higher from the ground state, is dynamically
unstable (it has one doubly degenerate imaginary frequency).
   At this region of energy (∆E = 1.26 eV) Xu et al. [14] have found at the MP2/6-31G* level) a
3 E (D ) state, which has an imaginary frequency [14,19] and thus undergoes Jahn-Teller distortion
   g    4h
to a lower D2h state [19]. Distortion of our Oh structure according to the displacement patterns of the
imaginary frequency mode followed by re-optimization leads to a stable 3 A1g state of D4h symmetry,
shown in Fig. 3(a), with energy ∆E = 0.58 eV above the ground state. At the QCISD(T)/cc-pvtz level
the energy separation of the 3 A1g state is ∆E = 0.69 eV. The characteristics of this structure (energy
difference from the ground state(s), bond lengths and frequencies) are given in Tables 6, 7. As we can
see in Table 6, the frequencies of this D 4h structure at all levels of theory (B3LYP, MP2, and QCISD)
are real. In addition this structure has no long bonds, as the corresponding singlet in Fig. 1(a). All atoms
are 4-coordinated and the general geometry is different compared to the corresponding singlet of the
same symmetry. Thus, it is clear from the way it was constructed and from its geometrical properties,
that this structure should be considered as originating from the full cubic O h structure (which has the
same spin multiplicity) rather than from the D4h “ground state” structure through spin rearrangement.
Zhao et al. [19] have found at the CCSD(T)/6-31+G* level a similar 3 B2g (D4h ) state at ∆E = 0.77 eV.
Apparently this must be the same state.
   Another low-lying structure is a hexagonal chair, shown in Fig. 3(b). This structure of (initiallyD 3d and
after Jahn –Teller distortion and re-optimization, finally) C 2h symmetry (similar to the one obtained in
the original work of Raghavachari [4] and Zdetsis [6a]) which exists in both singlet and triplet electronic
configurations. The singlet configuration with energy separation ∆E = 1.1 eV (at the B3LYP/cc-pvtz
level) is dynamically unstable. Distortion according to the eigenvectors of the imaginary frequency
modes leads to C i symmetric structure. This structure after re-optimization finally approaches (and after
symmetrization coincides with) the D4h “ground state” (which has also imaginary frequencies). The
                                A.D. Zdetsis / The full story of the Si6 magic cluster                     265

fully optimized triplet chair in D3d symmetry, as we can see in Table 4, has real frequencies at the QCISD
and MP2 levels but, surprisingly enough has imaginary frequencies at the B3LYP and CCSD levels.
Following the displacement pattern of the B3LYP imaginary frequency we are lead to a 3 Bg state of C2h
symmetry and real frequencies with energy separation from the ground state(s) ∆E = 0.58 eV (at the
B3LYP/cc-pvtz level). This energy is practically the same as for the 3 A1g . Indeed, this chair structure is
practically of D4h symmetry (as in Fig. 3a). By simple resymmetrization with looser symmetry criteria
we are lead to the D4h triplet structure discussed before. Therefore the 3 A1g (D4h ) state and the 3 Bg (C2h )
states are practically the same. The 3 Bg (C2h ) state at D4h symmetry coincides with the 3 B2g (D4h ) state
of ref. [19]. However, following different routes (and/or methods) of reoptimizations, of the D 3d chair
structure in singlet and triplet configurations (see also Table 1 of ref. 6a) structures of C 2h symmetry
both triplet and singlet are obtained similar to the ones found by Zhao et al. [19]. However, these states
found at 1.12 and 1.14 eV respectively have imaginary frequencies at both B3LYP and QCISD levels. It
is possible that these states are different manifestations of an open shell singlet state (at about the same
energy) which cannot be properly treated with the present theoretical approach. Such possibility was also
speculated earlier by Xu et al. [14]. In the same energy region is the D 3d state described in Tables 5 and
6, discussed above, which at the QCISD (and MP2) level has real frequencies. Thus there is multitude
of states in a very narrow energy region. This is reminiscent of the structural plasticity of the ground
state(s) which we have examined in the previous section. As a matter of fact such a connection would
be also consistent with the high intensity of the transition to this state during photo-detachment [14] of
one electron from the D4h symmetric anion. This will be discussed in section D. Finally, by successive
reoptimazations of the initial Oh structure (from which the D3d structure was obtained) we obtain the
structure of Fig. 3(c,d) with a peacock tail shape, which viewed from the side it could also be called
butterfly shape. This structure is quite higher in energy (1.42 eV at the B3LYP/cc-pvtz level) but is
the only structure after the D4h triplet which has unambiguously real frequencies at all levels of theory
examined here (QCISD, CCSD, MP2, B3LYP).


5. Comparison with experiment

5.1. Electron affinity and ionization potential

   From Table 4 we can see that the calculated electron affinity of Si 6 is 2.0 eV (2.01 eV at the B3LYP/cc-
pvtz level). This value is in excellent agreement with the experimental values of 2.2 eV, by Maus et
al. [17], and 1.8 eV obtained by Cheshnovsky et al. [16]. Also, the values of 2.0 eV and 2.01 eV obtained
here at the QCISD(T) /cc-pvtz and B3LYP/cc-pvtz levels are in very good agreement with the values
of 1.92 eV and 2.08 eV obtained by Zhao et al. [?] at the MRSDCI+Q /6-31G* and B3LYP levels
respectively. The corresponding values obtained by Zhao et al. at the CCSD(T) /6-31G* MP2/6-31G*
levels are 1.93 and 1.78 eV respectively.
   From Table 4 we see that the calculated adiabatic ionization energy is 7.60 eV at the QCISD(T) /cc-pvtz
level (7.49 eV at the B3LYP/cc-pvtz level). This value is in excellent agreement with the experimental
value [15,20] of 7.7–7.9 eV found by Fuke et al. [15] from photoionization thresholds. This value is also
in good agreement with the theoretical value of 7.77 obtained by Ishii et al. [20] using the GW method.
The theoretical values computed by Zhao et al. [19] (8.44 eV at the MSRDCI level) are considerably
larger. This could be due to their D 4h symmetry assignment for the cation, instead of the real C 2v
symmetry found here. Apparently this state is higher in energy than the C 2v state. Furthermore as we
can see in the last row of Table 4, very close to the real C 2v ground state of the cation there is a high
266                            A.D. Zdetsis / The full story of the Si6 magic cluster

symmetry high spin 4 A1g (Oh ) state, with reference to which the adiabatic ionization energy becomes
7.98 eV. This state is considerably stable with all frequencies real.

5.2. Photoelectron spectra

   Vibrationally resolved spectra have been reported by Xu et al. [14] for Si 1− in the region of 355 nm.
                                                                                6
These authors have identified three vibrationaly resolved bands. The first band, labeled X, is a weak
unstructured band assigned to the transition from the Si 1− anion to the neutral ground state(s) of Si 6
                                                             6
on the basis of Frank-Condon simulations using the frequencies and normal coordinate displacements
calculated by ab initio method. Attempts by Xu et al. [14] to observe vibrational structure in this band
by measuring its photoelectron spectrum at 416 nm were unsuccessful. Also the simulation of this band
is considerably broader than the experimental band, indicating, according to Xu et al., that the calculated
normal coordinate displacements and vibrational frequencies (on the basis of the D 4h neutral ground
state) need to be adjusted. All these characteristics (broad unstructured band, etc.) are consistent with
and highly suggestive for the present interpretation of a fluxional ground state.
   The second band, band A, represents a transition to an excited state of Si 6 . This band consists of
seven resolved peaks exhibiting a single progression of 323 ± cm −1 between 0.25 and 0.50 eV starting
at electron kinetic energy (eKE) of 0.476 eV. The energy separation of the excited Si 6 from the ground
state, which exhibits a maximum at eKE = 1.13 eV, is 0.75 eV. Furthermore, the presence of a single
progression indicates, as Xu et al. [14] have argued, that the excited Si 6 state responsible for this band
should have equal or higher symmetry than the D 4h symmetry of the Si1− anion. Thus, the symmetry
                                                                             6
should be either D4h or Oh . It is clear the 3 A1g (D4h ) state at 0.69 eV above the ground state, described
in detail in section B, fits perfectly to this description, explaining fully the position and structure of
band A. In addition this also supports the D 4h symmetry of the Si1− anion. Zhao et al. [19] have also
                                                                        6
identified and assign to the A band such a state, located at 0.77 eV (at the CCSD(T)/6-31g* level) above
the ground state according to their calculations. The same authors [19] have also corrected an erroneous
energy scale in Fig. 2 of the original work of Xu et al. [14], thus making it consistent with the discussion
in the text of Xu et al. [14], and the present band assignments (as well as their own assignments).
   The final and most intense band in the photoelectron spectra is band B. However, Xu et al. could only
observe the onset of this band and could not explain its large intensity. From their spectra (after the
scale correction) Zhao et al. [19] have estimated the B state to be about 1 eV above the ground state.
In this energy region Zhao et al. [19] have identified at the CCSD(T)/6-31G* level two states, 3 B2g
(D2h ) and 1 B2g (D2h ), at 1.05 and 1.09 eV respectively. These states are similar to (or originate from)
the D3d singlet and triplet states found earlier by the author in the same energy region (see Table 1 in
Zdetsis [6a]). However as was pointed out in Section 4, the singlet state has imaginary frequencies in
all levels of theory examined here, whereas the triplet D 3d state located at 1.08 eV above the ground
state has real frequencies at the QCISD and MP2 levels but imaginary frequencies at the CCSD and
B3LYP levels (see Table 5). At this energy region there are also two C 2h structures almost identical to
the two D2h structures of Zhao et al. [19] which, however, have imaginary frequencies at both QCISD
and B3LYP levels. Thus, there exists a multitude of states (some with imaginary frequencies in all levels
of theory and some in only one or two levels of theory) in a very narrow energy region around 1 eV
above the ground state. This is reminiscent of the structural plasticity of the ground state(s) which we
have examined in the previous section. Apparently the B band must be associated with this multitude
of states, which are also responsible for the high intensity of this band. This assignment is in perfect
agreement with the present interpretation of the structure(s) of Si 6 .
                                    A.D. Zdetsis / The full story of the Si6 magic cluster                              267




Fig. 4. The experimental (a) and calculated Raman spectra of Si6 for each of the “ground states” of Fig. 1, together with an
averaged (over the three structures) calculated spectrum. The experimental data are read from the figures of ref. [5].

  Slightly higher in energy at the 1.42 and 1.83 eV, at the B3LYP and QCISD levels respectively, is
located the peacock-tail or butterfly structure of C s symmetry, described in the last rows of Tables 5 and
6. It is not easy to say whether or not this structure is part or a continuation of the B band, since only the
onset of the B band was observed.
  It is clear therefore that the present results and interpretations are fully compatible with the experi-
mentally measured vibrationaly resolved photoelectron spectrum.

5.3. Raman spectra

   In Fig. 4 we show the calculated Raman spectrum of all three structures of Si 6 in Fig. 1, together
with the average of this spectrum and the experimentally measured Raman spectrum. Since all three
structures coexist and transform into each other, according to the present results, the experimental values
in reality must reflect time-averages over the three structures, which are certainly dominated by the
highest symmetry structure. As we can see in Fig. 4, looking at the raw experimental data (read from
the figures of ref. [5]) and the corresponding calculated results at the B3LYP/cc-pvtz level, it is not so
easy to conclusively assign the observed spectrum solely to the D 4h structure. This is on top of several
possible experimental uncertainties, such as that the experiment cannot detect all active Raman modes
below some threshold value, or that it could not resolve frequencies which are “nearby”.
   These possibilities acquire more importance if one compares the last two columns of Table 4 in ref.
6a, which show the extracted experimental Raman frequencies together with some selective frequencies
268                                  A.D. Zdetsis / The full story of the Si6 magic cluster




Fig. 5. Calculated Raman spectra for the charged clusters in comparison with the corresponding spectrum of the neutral cluster
(averaged over the tree structures) and the D4h neutral cluster.

of the C2v and Cs /C2v structures, compatible with the D4h Raman active modes and with large intensities
(Raman activities). As we can see, these frequencies are comparable in magnitude to the experimental
results and to the values predicted by MP3, MP4, B3LYP (and MP2) for the corresponding D 4h structure.
  Furthermore, since ambiguities exist also for the charged species [4–10,18–21], in order to facilitate
possible future experimental characterization, the calculated Raman spectrum for these species has been
plotted in Fig. 5. No experimental data exist up today for such spectra.

5.4. Infrared spectrum

  The experimental IR spectrum of Si 6 obtained by Li et al. [23] is dominated by a single high intensity
band around 460 cm −1 . Li et al. [23] assigned this band to the E u vibrational mode of the D4h structure.
The calculated value of this mode at the MP2/6-31G* level is 482 cm −1 , which after scaling by 5%
reduces to 458 cm −1 , in apparently good agreement with experiment. At the B3LYP/cc-pvtz level the
same mode is calculated (without any kind of scaling) at 446 cm −1 .Two more modes, A 2u and Eu ,
calculated by Li et al. (at the MP2/6-31G* level) at 340 and 49 cm −1 (after 5% scaling) could not be
observed due to their low intensities (5% and 1% the intensity of the E u mode, respectively). Obviously
the experiment cannot detect low intensity modes and also cannot resolve different close-lying modes.
This last effect is reflected in the calculated IR spectra (after suitable Gaussian broadening) in Fig. 6, for
the D4h , the Cs /C2v assigned ground state structures, together with an averaged spectrum obtained in a
similar way as the Raman spectrum structures.
  As we can see in Fig. 6, the spectra are practically indistinguishable. Therefore, the experientially
measured IR spectrum is also compatible with the present assignment of the C s /C2v ground state.
  In Fig. 7 the calculated IR spectra of Si 1+ , Si1− , and Si2− are plotted to facilitate possible future
                                               6     6            6
characterization of these species.
                                                                     A.D. Zdetsis / The full story of the Si6 magic cluster                  269

                                                    40
                                                    30
                                                    20
                                                                                                                       Average
                                                    10
                     Infrared Intensity (KM/Mole)

                                                     0
                                                      -100               0           100          200                300         400   500
                                                    40
                                                    30
                                                    20                                                                D4h
                                                    10
                                                     0
                                                      -100               0           100          200                300         400   500
                                                    40
                                                    30
                                                    20                                                               CsC2v
                                                    10
                                                     0
                                                     -100                0           100          200                300         400   500
                                                                                                                -1
                                                                                            Frequency (cm )

Fig. 6. Infrared spectra for D4h and Cs /C2v assigned ground state structures (see text), together with an average spectrum (top).

                                                    16

                                                    12

                                                     8                                                  minus2
                                                     4
                     Infrared Intensity (KM/Mole)




                                                     0
                                                    16-100               0           100          200                300         400   500
                                                    12

                                                     8
                                                                                                        minus1
                                                     4

                                                     0
                                                    16-100               0           100          200                300         400   500
                                                    12

                                                     8
                                                                                                          plus1
                                                     4

                                                     0
                                                         -100            0           100          200                300         400   500
                                                                                                           -1
                                                                                           Frequency (cm )

                                                                Fig. 7. Calculated Infrared spectra for the charged Si6 clusters.

5.5. Optical absorption spectrum

  The TDDFT method in particular with B3LYP functional has been proven very successful for the
calculation of the optical excitations of silicon nanocrystals [6b,26]. The same method (TDDFT-B3LYP)
270                             A.D. Zdetsis / The full story of the Si6 magic cluster

                                                       Table 7
                        The lowest ten excitations for the three structures of Si6 clusters.
                        Excitation energies (in eV) and oscillator strengths in parenthesis
                        D4h            a2u                   eu
                                4.052 (0.41E-01)      3.416 (0.58E-02)
                                 4.490 (0.19E-01      3.892 (0.29E-02)
                                5.284 (0.85E-02)      4.215 (0.18E-02)
                                5.727 (0.11E+00)      4.288 (0.34E-02)
                                6.564 (0.19E-03)      5.177 (0.22E+00)
                                6.749 (0.28E-01)      5.695 (0.17E-02)
                                7.397 (0.89E-01)      5.802 (0.26E-03)
                                8.883 (0.97E-04)      5.986 (0.54E-03)
                                9.051 (0.32E+01)      6.131 (0.25E-01)
                               10.232 (0.53E+00)      6.496 (0.40E+00)
                         C2v            a1                   b1                    b2
                                2.540 (0.19E-19)      2.100 (0.16E-19)      1.976 (0.69E-21)
                                2.545 (0.17E-18)      2.128 (0.26E-20)      2.384 (0.17E-18)
                                2.754 (0.17E-03)      2.765 (0.11E-02)      2.437 (0.21E-03)
                                2.998 (0.14E-01)      2.860 (0.67E-03)      2.691 (0.88E-19)
                                3.013 (0.20E-20)      2.907 (0.24E-19)      2.897 (0.32E-04)
                                3.207 (0.18E-17)      3.178 (0.59E-19)      3.540 (0.98E-18)
                                3.614 (0.28E-02)      3.281 (0.17E-02)      3.658 (0.19E-18)
                                3.685 (0.15E-18)      3.345 (0.87E-19)      3.720 (0.92E-02)
                                3.813 (0.62E-20)      3.708 (0.20E-19)      3.821 (0.35E-03)
                                3.855 (0.50E-04)      3.793 (0.46E-04)      3.866 (0.18E-19)
                         Cs             a                    a"
                                2.574 (0.258-02)      2.329 (0.19E-04)
                                2.706 (0.35E-02)      2.415 (0.12E-03)
                                3.064 (0.13E-03)      2.738 (0.68E-05)
                                3.171 (0.67E-02)      2.833 (0.55E-04)
                                3.346 (0.18E-02)      3.326 (0.25E-03)
                                3.546 (0.41E-02)      3.573 (0.85E-03)
                                3.665 (0.28E-04)      3.654 (0.27E-03)
                                3.801 (0.17E-02)      3.764 (0.37E-02)
                                4.057 (0.11E-02)      3.807 (0.46E-03)
                                4.162 (0.95E-03)      4.015 (0.16E-01)


is used here to calculate the optical absorption spectrum for each lowest lying structure of Si 6 . The
results of the calculations are shown in Table 7 and Fig. 8.
   In Table 7 the first ten optical excitations compatible with the selections rules are shown separately
for each structure and symmetry (irreducible representation) together with the corresponding oscillator
strength (in the length representation). As we can see in Table 7, the optical gap defined either as the
energy of the first allowed excitation or the first allowed excitation with “appreciable” oscillator strength,
is larger for the D4h structure due to its higher symmetry which places more stringent restrictions.
   The full structure of the spectrum up to 10 eV is shown in Fig. 8. On the basis of the present
interpretation, the “observed” absorption spectrum (whenever becomes available) would be some kind
of average of the three structures as the one on the panel.
   For comparison, the corresponding absorption spectra of the anion and cation are plotted in Fig. 9.
The high spin Oh cation, lying higher by 0.06 eV from the C 2v cation is also included for comparison.
As in the case of the neutral C 2v structure, although due to its lower symmetry this cation has a smaller
optical gap, many of the lower excitations are characterized by extremely small oscillator strengths. As
a result, the two spectra do not look that much different in the low energy region.
                                    A.D. Zdetsis / The full story of the Si6 magic cluster                                  271




        Fig. 8. (Color on line). Absorption spectrum in the region of 2 to 10 eV for the D4h , Cs , and C2v structures.




    Fig. 9. Absorption spectrum in the region of 2 to 7 eV for the D4h anion (bottom) and C2v cation(s) (top) structures.

6. Concluding remarks

  In conclusion we see that the present results can reconcile the existing ambiguities in the structure of
Si6 and its ions. At the same time the emerging picture is fully compatible with the existing experimental
measurements of Raman and IR spectra, and in particular the vibrationally resolved photoelectron spectra,
as well as the experimental values of adiabatic ionization energies and electron affinities.
272                                  A.D. Zdetsis / The full story of the Si6 magic cluster

  Concerning the disputes about the real structure of Si 6 , we see that all claims could be characterized
as correct but “one-sided”. We can also see that the D 4h structure is dynamically stable either as a triplet
(but at a higher energy, about 0.7 eV) or as an anion in the lowest (anionic) state. This seems to be the
full story and apparently a story with happy ending; but again, since the story of Si 6 if full of surprises,
one never knows for sure. Nice stories usually have some (not so nice) sequels!


References

 [1]   R.P. Andres et al., J Mat Res 4 (1989), 704.
 [2]   K. Raghavachari, Phase Transitions, 24–26 (1990), 61–90.
 [3]   M.F. Jarrold, Science 252 (1991), 1085.
 [4]   K. Raghavachari, J Chem Phys 84 (1986), 5672; K. Raghavachari and V. Logovinsky, Phys Rev Letters 55 (1985), 2853.
 [5]   H.C. Honea et al., Nature 366 (1993), 42; E.C. Honea et al., J Chem Phys 110 (1999), 12161.
 [6]   A.D. Zdetsis, Phys Rev A 64 (2001), 023202; A.D. Zdetsis, Rev Adv Mater Sci 11 (2006), 56.
 [7]   A.D. Zdetsis, Computing Letters 1 (2005), 337; A.D. Zdetsis Structures and Properties of Clusters: From a few Atoms
       to Nanoparticles, in: Lecture Series on Computer and Computational Sciences, (Vol. 5), G. Maroulis, ed., 2006, p. 193.
 [8]   Y. Gao, C. Killblane and X.C. Zeng in Structures and Properties of Clusters: From a few Atoms to Nanoparticles, Lecture
       Series on Computer and Computational Sciences, (Vol. 5), G. Maroulis, ed., 2006, p. 199; Computing Letters 1 (2005).
 [9]   P. Karamanis, D. Zhang-Negrerie and C. Pouchan, Chem Phys 331 (2006), 417.
[10]   A.D. Zdetsis, J Chem Phys 127 (2007), 014314, in press.
[11]   F.A. Cotton and G. Wilkinson, Advanced Inorganic Chemistry, (5th edition), Wiley, New York, 1988, 1318–1334.
[12]   J.M. Bowman, Science 290 (2000), 724.
[13]   M.L. McKee, in: Encyclopedia of Computational Chemistry, (Vol. 2), John Wiley & sons, 1998, pp. 1002–1013.
[14]   C. Xu, T.R. Taylor, G.R. Burton and D.M. Neumark, J Chem Phys 108 (1998), 1395.
[15]   K. Fuke, K. Tsukamoto and F. Misaizu, Z Phys D: At, Mol Clusters 26 (1993), S204; K. Fuke, K. Tsukamoto, F. Misaizu
       and D.M. Sanekata, J Chem Phys 99 (1993), 7807.
[16]   O. Cheshnovsky, S.H. Yang, C.L. Pettiette, M.J. Craycraft, Y. Liu and R.E. Smalley, Chem Phys Lett 138 (1987), 119.
[17]   M. Maurs, G. Gantefor and W. Eberhardt, Appl Phys A: Mater Sci Process 70 (2000), 535.
[18]   K. Raghavachari and C.M. Rohlfing, J Chem Phys 94 (1991), 3670.
[19]   C. Zhao and K. Balasubramanian, J Chem Phys 116 (2002), 3690.
[20]   S. Ishii, K. Ohno, V. Kumar and Y. Kawazoe, Phys Rev B 68 (2003), 195412.
[21]   A. Shvartsburg, B. Liu, M.F. Jarrold and K.M. Ho, J Chem Phys 112 (2000), 4517.
[22]   S. Li, R.J. Van Zee, W. Weltner, Jr. and K. Raghavachari, Chem Phys Lett 243 (1995), 275.
[23]   M.J. Frisch et al., Gaussian 03-Revision C.02 Program Package, Gaussian, Inc., 2004.
[24]   TURBOMOL (Version 5.6) Program package for ab initio electronic structure calculations, Universitat Karlsruhe, 2000.
[25]   A.D. Zdetsis, in: Stability of Materials, (Vol. 355), A. Gonis, P.E.A. Turchi and J. Kudrnovsky, eds, NATO ASI Series
       B: Plenum Press N. York, 1996, p. 455; A.D. Zdetsis, Phys Rev B 75 (2007), 085409.
[26]   C.S. Garoufalis, A.D. Zdetsis and S. Grimme, Physical Review Letters 87 (2001), 276402.
Journal of Computational Methods in Sciences and Engineering 7 (2007) 273–286                                                  273
IOS Press




Ab initio investigation of structures and
properties of mixed silicon-potassium SinKp
and SinK+ (n 6 , p 2 ) clusters
          p

F. Rabilloud∗ and C. Sporea
                        ´                   e                             e
Laboratoire de Spectrom etrie Ionique et Mol´ culaire, UMR 5579 (Universit´ Claude Bernard Lyon 1 &
CNRS), 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France

Received 17 September 2006
Revised /Accepted 17 October 2007

                      (+)
Abstract. The Sin Kp (n           6, p     2) clusters with different spin configurations have been systematically investigated by
using the density functional theory with B3LYP. Equilibrium geometries, population analysis, binding energies, adiabatic and
vertical ionization potentials as well as electric dipole moments and static dipolar polarizabilities, have been calculated and are
discussed for each considered size. For the most stable isomers, the structure of the neutral Sin Kp and cationic Sin K+ clusters
                                                                                                                         p
are found to keep the frame of the corresponding Sin , potassium atoms being adsorbed at the surface. The localization of the
                                                                       +
potassium cation is not the same one as that of the neutral atom. K ion is preferentially located on a Si atom while the K atom
is preferentially attached at a bridge site. The population analysis show that the electronic structure of Sin Kp can be described
as Sip− + pK + for the small sizes considered here. Binding energies and ionization potentials are compared to those of sodium
     n
and lithium-doped silicon clusters.



1. Introduction

  The study of silicon is important due to its technological relevance towards the development of
nanoelectronics, and the comprehension of the properties of silicon with miniaturization is a true chal-
lenge. Hence, Si clusters have been studied most extensively using both theoretical and experimental
techniques [1–12]. The recent experimental evidence of the formation of stable transition-metal en-
capsulating silicon cage clusters ions for MSi + , with n ranging from 9 to 14, by Hiura et al. [13] has
                                               n
revived the interest to investigate the interactions of metal atoms with silicon clusters. The princi-
pal question in studying metal-doped silicon clusters is the comprehension of the modifications of the
properties compared to the case of bare silicon clusters. Although several reports are available on the
interaction of transition-metal atom with Si clusters, similar investigations with alkali atoms are very
few. To our knowledge, the only works published on neutral or positively charged alkali-silicon system
have concerned the interaction between small Si clusters with sodium or lithium atoms. Majunder and
Kulshreshtha [14] have investigated impurity-doped Si 10 clusters (Si10 M, M = Li, Be, B, C, Na, Mg,
Al ans Si) showing that the location of the impurity atom on the host cluster depends on the atomic size

  ∗
      Corresponding author. Tel.: +33 4 72 43 29 31; Fax: +33 4 72 43 15 07; E-mail: franck.rabilloud@lasim.univ-lyon1.fr.


1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
274     F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium

and the nature of interaction between the host cluster and the impurity atoms. The ionization potentials
of Sin Nap , 3 n 11, 1 p 4, have been experimentally determined from the threshold energies
of their ionization efficiency curves [15]. On the theoretical aspect, the structural and electronic prop-
erties of Sin Na (n 10) have been studied through calculations based on Moller Plesset (MP2-MP4)
method [15] or in the framework of the density functional theory (DFT) [16]. It has been found that
the ionization potential for Sin Na clusters presents local minima for n = 4, 7, and 10, correlating with
the measured low values [17] of the electron affinity of bare silicon clusters Si n for these sizes. Very
recently, Wang et al. [18] have investigated the adsorption of a lithium atom on small Si n clusters (n =
2–7), while Wu et al. [19] have studied the geometrical structures of Si 7−m Lim (m = 1–6). In previous
works [20,21], we have investigated the neutral SiNa, Si n Na2, Sin Li, and Sin Li2 clusters and their cations
with DFT/B3LYP type calculations. A clear parallelism between the structures of Si n Nap and those of
Sin Lip appeared. The geometrical structure of the most stable isomers of Si n M2 clusters was found to
be similar to that of Sin M in which a second M alkali atom is located on a site far from the first one.
No alkali-alkali bonding was found. Our ab initio calculations indicate that on doping with Li or Na
atoms, the electron charge migrates from alkali atoms to the silicon cluster without disturbing seriously
the original framework of the bare silicon cluster.
  In the present work, we have investigated the electronic and structural properties of neutral or cationic
                                         (+)
potassium-doped silicon clusters Si n Kp . We have considered sizes up to n = 6 and p = 2. To our
                                                                         (+)
knowledge, present work is the first theoretical investigation of Si n Kp . Details of the calculations are
introduced in Section 2. In Section 3, we present and discuss the calculated properties of the clusters
including structures, charge transfer, adsorption energies, adiabatic and vertical ionization potentials,
electric dipole moment and static dipolar polarizability.


2. Computational details

   Calculations were achieved with the same approach as that used in our previous theoretical works on
silicon-sodium [20] and silicon-lithium clusters [21]. This approach has been validated in the sense that
it provided theoretical results in good agreement with the experimental data concerning the ionization
potential for Sin Nap . Calculations were carried out with the Gaussian 98 program package [22] and
the graphical interface Gabedit [23]. The Gaussian basis set used was 6–31+G(d) which is (17s11p1d)
contracted into [5s4p1d] on silicon atoms and (23s17p1d) contracted into [6s5p1d] on potassium atom.
The electronic calculations performed in the framework of the density functional theory used the hybrid
B3LYP functional which involves Becke’s three-parameter exchange functional [24].
   In the optimization process of cluster geometries, a number of structures were tested for each size.
We have initiated the geometry optimization process of Si n Kp clusters from the known frame of the
corresponding Sin cluster on which potassium atoms were located either on Si atoms or bridged over
two atoms or capped on several atoms. We have also tested structures for which the frame of the
corresponding Sin cluster used as initial geometry was deformed. We have also taken advantage of the
known structures of Sin Nap [20] and Sin Lip [21]. Of course, the explicit treatment of all the electrons in a
cluster having a large number of atoms constitutes a demanding computational task and the search for the
lowest isomer can not include a global optimization procedure of the potential energy surface. So we can
not be sure that a more stable cluster than those found in our calculations does not exist. All optimizations
were carried on without symmetry constraints (C 1 symmetry group). Harmonic frequency analysis was
performed to guaranty that the optimized structures are local minima. In the discussion we only report on
        F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium   275

the lowest-energy stable isomers determined in our optimizations. For each stable structure, the charge
on K atoms have been estimated through a natural population analysis (NPA) [25,26]. Furthermore, the
dipole moment µ as well as the averaged static dipolar polarizability α = (α xx + αyy + αzz )/3 have
been calculated.


3. Results and discussion

3.1. Lowest-energy structures and isomers

   First, we have investigated SiK dimer for which no data is available in the literature to our knowledge.
The ground state for SiK dimer is found to be a 4 Σ− , a2 Π excited state being located 0.448 eV above.
The bond lengths are found to be 3.13 Å and 3.36 Å for quartet and doublet states respectively. The latter
are much longer than those found for SiNa and SiLi dimers, since the ground states 4 Σ− calculated with
B3LYP/6–31+G(d) for SiNa and SiLi have bond lengths of 2.72 and 2.39 Å respectively [20,21].
   The optimized geometries of the Si n K clusters, together with the energies of isomers relative to that of
the lowest-energy one as well as the spin multiplicities, are shown in Fig. 1. Spatial symmetries are given
in Table 1. As for Sin Na and Sin Li clusters, the ground state of all Sin K (n > 1) clusters examined is a
doublet. The structure of the most stable isomer keeps the frame of the corresponding Si n unchanged.
This means that the Si-Si bond predominates on the Si-K one. The adsorbing site of a potassium atom is
a bridge-site type in which the K atom is bridged over two Si atoms for n = 2, 3, 4 in a planar structure.
For Si2 K, the lowest-energy structure is found to be of C 2v symmetry. The structures (2b) and (2e) are
linear structures with the Si-K bonds of 3.15 Å and 3.24 Å, respectively. For Si 4 K, the structure (4b),
in which the K atom bridges over the four Si atoms, is only slightly less stable than the lowest energy
structure (4a) by 0.027 eV, this structure being the lowest-energy isomer for Si 4 Na [20]. The Si5 K and
Si6 K clusters present the same structure that for sodium and lithium doped-system. For Si 5 K, the K atom
bridges over three Si atoms in structure (5a), this latter can be described as K substituting one of the Si
atoms in the distorted Si6 clusters. The structures in which K is adsorbed on one Si atom were found not
to be stable. The absorbing site of K for Si 6 K is a bridge-site type in which the K is capped over four Si
atoms. This structure, which presents a C 2v symmetry, keeps unchanged the frame of the corresponding
Si6 cluster. The isomer (6a) is about 0.1–0.2 eV lower than the three similar (6b) (6c) (6d) structures in
which K is adsorbed on another site.
   To check the structures and energetics, we have performed single-point calculations with the coupled
cluster theory using the version involving single and double substitutions and taking into account the
effect of the triple substitutions (CCSD(T)). We used the 6–31+G(d) basis set. We have checked the
relative order in energy for the two first isomers for n = 2, 3 and 4. For Si 2 K, the isomer (2b) was found
to lie 0.364 eV above the isomer (2a). For Si 3 K, the isomer (3b) was found 0.818 eV above the isomer
(3a) while for Si4 K the isomer (4b) was found 0.016 eV above the isomer (4a). All these results are in
very good agreement with the B3LYP values of 0.255 eV, 0.723 eV and 0.027 eV respectively. Both the
relative order of isomers and the relative energies between two isomers are similar.
   The geometries of the Sin K2 clusters as well as the spin multiplicities and relative energies for all
isomers are shown in Fig. 2. Spatial symmetries are given in Table 2. The ground state of Si n K2 clusters
is found to be a singlet, except in the case of SiK 2 for which it is found to be a triplet. The geometrical
structure of the most stable isomers is similar to that of Sin K (except for n = 4) in which a second K atom
is located on a site far from the first K atom. The adsorbing site of the second K atom is a bridge-site type
in which the K atom is bridged over two Si atoms for n = 2, 3 and 6, or a bridge-site type in which the K
276     F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium

                                                            Table 1
                    Calculated relative energies, potassium binding energies (Eb ), dipole moment (µ),
                    static dipolar polarizabilities (α) and charge on potassium atoms from the natural
                    population analysis (qNP A ) for Sin K clusters
                          Cluster         Energy (eV)    Eb (eV)    µ (Debye)     α(Å3 )   qNP A (a.u.)
                        SiK (C∞v )           0.000        1.24         9.62       19.51       0.77
                    Si2 K 2a (C2v )          0.000        2.06         9.64       26.83       0.84
                            2b (C∞v )        0.255                    12.95       22.83
                            2d (C2v )        1.528                     8.96       26.75
                            2e (C∞v )        1.610                     9.46       32.17
                            2e (D∞h )        3.964                     0.00       38.32
                    Si3 K 3a (C2v )          0.000         1.20       11.03       22.00        0.91
                            3b (Cs )         0.723                     9.69       24.12
                            3c (C2v )        0.998                     9.48       24.89
                            3d (Cs )         1.474                     8.20       50.19
                    Si4 K 4a (Cs )           0.000         1.40       12.60       27.38        0.93
                            4b (Cs )         0.027                    10.05       26.05
                    Si5 K 5a (Cs )           0.000         1.85       10.97       29.93        0.94
                            5b (Cs )         0.231                    10.84       28.52
                            5c (C2v )        0.353                    10.47       29.66
                            5d (C4v )        1.164                     9.75       28.28
                    Si6 K 6a (C2v )          0.000         1.57       10.50       32.21        0.95
                            6b (Cs )         0.114                    13.37       34.99
                            6c (C1 )         0.136                    11.75       33.25
                            6d (C2v )        0.165                    13.32       34.24
                            6e (Cs )         0.296                    11.89       33.16
                            6f (Cs )         1.485                    10.50       32.21
                            6g (Cs )         1.488                    11.48       35.24
                            6h (Cs )         1.556                    13.22       35.06


atom is capped over three Si atoms for n = 5. Si 2 K2 and Si3 K2 present a C2v symmetry while Si4 K2 is
of Cs symmetry. For Si2 K2 , both singlet and triplet are found to be of C 2v symmetry, the D2h structure
being not stable. For Si3 K2 , the lowest-energy isomer is found to have a symmetric planar structure in
which the K atoms bridge a side of an isosceles triangle Si 3 . For Si4 K2 , the K atoms are located on both
sides of the rhombus Si 4 .
  The optimized geometries of the singly charged Si n K+ clusters, together with the energies of isomers
relative to that of the lowest-energy one as well as the spin multiplicities, are shown in Fig. 3. Spatial
symmetries of the lowest-energy isomers are given in Table 3. The ground state is found to be a singlet
for all size except for n = 2 for which it is found to be a triplet. The structure of the most stable isomer
keeps the frame of the corresponding Si n unchanged, but the localization of the potassium ion is not the
same one as that of the neutral atom in Si n K. For the lowest-energy isomer, the K + ion is located on
a Si atom for all size. In this configuration the K + charge generates an induced dipole in the globally
neutral system Sin (as shown with the atomic NPA charges in Fig. 3 for isomers (2a) (3a) and (4a)). On
the contrary, if the K+ bridges over two or more silicon atoms, it does not generate an induced dipole as
shown in the Fig. 3 in the case of Si 2 K+ (isomer (2b)). Thus the K+ ion tends to bind over one silicon
atom to minimize the interaction energy. The electronic structure of Si n K+ can be described as Si n + K+ .
For Si3 K+ we find two planar equilibrium geometries ((3a) and (3c)) and two tridimensional structures
((3b) and (3d)). The most stable isomer is formed from one irregular triangle structure Si 3 where the
Si-K distance is 3.45Å. The second lowest-energy structure is constructed on a isosceles triangle. For
the triplet configuration, the structure (3c) is of C2v symmetry. Si4 K+ have two quasi-degenerated planar
          F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium         277




                                                                                                 2e) 3.964 [2]
                      2a) 0.000 [2]      2b) 0.255 [2]     2c) 1.528 [4]      2d) 1.610 [4]




                     3a) 0.000 [2]       3b) 0.723 [2]     3c) 0.998 [2]      3d) 1.474 [4]




                      4a) 0.000 [2]      4b) 0.027 [2]




                      5a) 0.000 [2]      5b) 0.231 [2]     5c) 0.353 [2]      5d) 1.164 [4]




                      6a) 0.000 [2]      6b) 0.114 [2]     6c) 0.136 [2]      6d) 0.165 [2]      6e) 0.296 [2]




                      6f) 1.485 [4]      6g) 1.488 [4]     6h) 1.556 [4]

Fig. 1. Optimized geometries of neutral Sin K clusters. The relative energies (eV) and the spin multiplicities (in square brackets)
are shown under the structure of each isomer. Charges from the natural population analysis (qNP A ) are indicated for structures
(2a), (2b), (4a) and (4b), as well as the Si-K distances in Å for some clusters.

configurations with a K+ tail located on one Si at a distance of 3.44 Å and 3.41 Å respectively. Si 5 Li+
is of C2v symmetry.
   As for neutral species, to check the relative order in energy of isomers, we have performed single-point
CCSD(T)/6–31+G(d) calculations for n = 2, 3 and 4. The structure (2a) was found to be more stable
than structure (2b) by 0.146 eV. The structure (3a) was found to be more stable than the structure (3b)
by 0.193 eV, while the structure (4a) was found to be more stable than structure (4b) by 0.021 eV. These
results are in reasonable agreement with the 0.215 eV, 0.182 eV and 0.003 eV values found with B3LYP
for n = 2, 3 and 4 respectively.
278     F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium

                                                           Table 2
                  Calculated relative energies, potassium binding energies (Eb ), dipole moment (µ), static
                  dipolar polarizabilities (α) and charge on potassium atoms from the natural population
                  analysis (qNP A ) for Sin K2 clusters
                        Cluster          Energy (eV)      Eb (eV)   µ (Debye)     α (Å3 )     qNP A (a.u.)
                  SiK2     1a (D∞v )        0.000          2.16        0.00        27.33       0.64; 0.64
                           1b (C2v )        0.114                      5.71        59.73
                           1c (C∞v)         2.016                    10.95         67.79
                  Si2 K2 2a (C2v )          0.000          3.78        6.70        34.51       0.80; 0.80
                           2b (C2v )        1.398                      1.34        43.27
                  Si3 K2 3a (C2v )          0.000          3.54        9.34        37.83       0.88; 0.90
                           3b (C2v )        1.041                      3.75        31.94
                           3c (Cs)          1.333                      7.42       132.65
                           3d (C1 )         1.335                      4.30       148.38
                  Si4 K2 4a (Cs )           0.000          2.44        3.59        35.94       0.83; 0.75
                           4b (Cs )         0.184                      2.53        34.44
                           4c (C1 )         0.259                    11.01         70.21
                           4d (C1 )         0.447                      7.54        58.87
                  Si5 K2 5a (C1 )           0.000          3.79        8.55        36.73       0.94; 0.86
                           5b (Cs )         0.040                    11.36         36.99
                           5c (Cs )         1.020                      8.43        39.70
                           5d (C2v )        1.696                      6.44        68.72
                           5e (C1 )         1.702                      8.53        50.74
                  Si6 K2 6a (C2v )          0.000          2.97        2.38        40.33       0.94; 0.86
                           6b (Cs )         0.115                    12.91         43.08
                           6c (Cs )         0.115                      0.35        41.91
                           6d (D4h )        0.536                      0.20        65.84

  The geometries of the singly charged Si n K+ clusters, together with the relative energies of isomers
                                                   2
and spin multiplicities, are shown in Fig. 4. The ground state of all Si n K+ clusters examined is a doublet,
                                                                            2
except for n = 1 for which it is found to be a quartet. The electronic structure is describable as Si − +n
2K+ . The structures are approximatively similar for both Si n K+ and Sin K2 clusters for n = 1, 2 and
                                                                     2
6, but are different for n = 3, 4 and 5. For Si n K+ clusters, the K+ ions are located on a Si atom or
                                                       2
bridge over two Si atoms. As for neutral Sin K2 clusters, the second K atom is located on a site far from
the first one. The lowest-energy isomer of SiK + , Si3 K+ and Si4 K+ have a planar structure. For Si 2 K+
                                                    2      2           2                                   2
cluster, all Si-K distances are equals to 3.48 Å. For Si 6 K+ the isomers (6a) and (6b) are found to be
                                                                2
quasi degenerated.
  In conclusions, in all cases, the structure of the most stable isomer keeps the frame of the corresponding
Sin unchanged or slightly deformed, the K atoms being adsorbed at the surface. Most of isomers are
similar for the adsorption of Li or Na atoms [20,21] but the relative order in energy is sometimes changed.
Several structures are found stable for one alkali and not stable for the other one. The lowest-energy
isomers are found to be different for the adsorption of alkali atoms (M = Li, Na or K) in the cases of
Si4 M, Si5 M and Si4 M2 .

3.2. Charge transfer

  To further understand the interaction between the silicon clusters and the potassium atoms, natural
population analysis were performed. The Tables 1 and 2 give the calculated charges q N P A on the
potassium atoms for the most stable clusters listed in Figs 1–4. For Si n K clusters, the charge in near to
one. It increases monotonically with n from +0.77 for SiK to +0.95. For all the clusters investigated
         F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium   279




                    1a) 0.000 [3]     1b) 0.114 [3]      1c) 2.016 [1]




                    2a) 0.000 [1]     2b) 1.398 [3]




                    3a) 0.000 [1]     3b) 1.041 [3]      3c) 1.333 [3]     3d) 1.335 [3]




                    4a) 0.000 [1]     4b) 0.184 [3]      4c) 0.259 [1]     4d) 0.447 [3]




                    5a) 0.000 [1]     5b) 0.040 [1]      5c) 1.020 [3]     5d) 1.696 [3]     5e) 1.702 [3]




                    6a) 0.000 [1]     6b) 0.115 [1]      6c) 0.115 [1]     6d) 0.536 [1]

Fig. 2. Optimized geometries of neutral Sin K2 clusters. The relative energies (eV) and the spin multiplicities (in square
brackets) are shown. The Si-K distances are given in Å for some clusters.


here, the valence electron 4s of the K atom is transferred to the LUMO (lowest unoccupied molecular
orbital) of Sin and the electronic structure of Sin K clusters corresponds to that of Si − + K+ . This latter
                                                                                         n
conclusion is in agreement with the fact that the frame of Si n in Sin M is similar to that of bare Si−     n
cluster [15,18]. In the Fig. 1, the calculated q N P A charges for Si atoms are indicated for isomers (a) and
(b) of Si2 K and Si4 K. They show that the transferred charge from the alkali is located on the nearest Si
neighbors. In the case of Si 2 K, one can see that the structure is most stable when the transferred electron
can be shared by two silicon (isomer (2a) Fig. 1) rather than located on only one silicon (isomer (2b)
Fig. 1). For Sin K2 clusters, the charge on potassium atoms are in the 0.80–0.94 range, except for SiK 2
for which the qN P A charge are 0.64 for both potassium atoms (Table 2). The electronic structure of
280       F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium




                                              2b) 0.215 [3]
                       2a) 0.000 [3]                                 2c) 0.727 [1]           2d) 4.623 [1]




                                              3b) 0.182 [1]          3c) 0.189[3]            3d) 1.330 [3]
                       3a) 0.000 [1]




                       4a) 0.000 [1]          4b) 0.003 [1]




                       5a) 0.000 [1]          5b) 0.552 [3]          5c) 0.821 [1]
                                              6b) 0.090 [1]          6c) 0.188 [1]




                                                                                             6d) 1.385 [1]
                       6a) 0.000 [1]

Fig. 3. Optimized geometries of cationic Sin K+ clusters. The relative energies (eV) and the spin multiplicities (in square
brackets) are shown. Charges from the natural population analysis (qNP A ) are indicated for structures (2a), (2b), (3a) and (4a),
as well as the Si-K distances in Å for some clusters.

Sin K2 clusters corresponds approximatively to that of Si 2− + 2 K+ . The charge on alkali is larger in the
                                                          n
case of potassium compared to sodium and lithium cases [20,21] because the atomic ionization potential
of potassium is smaller.

3.3. Energetic properties

3.3.1. Potassium binding energy
  The binding energy of K to Si n cluster calculated as E b = − [E(Sin Kp ) – E( Sin ) – p E(K)] is listed
in Tables 1 and 2 for the most stable isomer of each size. In all cases, K atoms are stably adsorbed on
the Sin frame. The evolution of Eb against the number of silicon atoms is shown in Fig. 5 for Si n K
compared to that of Sin Li [21] and Sin Na [20] clusters. Eb oscillates as a function of n, showing local
         F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium   281




                    1a) 0.000 [4]     1b) 0.906 [2]      1c) 1.610 [2]
                                      1b) 0.906 [2]




                    2a) 0.000 [2]     2b) 0.130 [2]     2c) 1.634 [4]      2d) 1.760 [4]




                    3a) 0.000 [2]      3b) 0.024 [2]     3c) 0.081 [2]    3d) 1.381 [4]      3e) 1.412 [4]




                    4a) 0.000 [2]      4b) 0.024 [2]     4c) 1.341 [4]     4d) 1.458 [4]     4e) 1.903 [4]




                    5a) 0.000 [2]      5b) 1.293 [4]     5c) 1.297 [4]     5d)1.331 [4]      5e) 2.492 [4]




                    6a) 0.000 [2]      6b) 0.005 [2]     6c) 0.130 [2]     6d) 0.193 [2]     6e) 1.381 [4]

Fig. 4. Optimized geometries of cationic Sin K+ clusters. The relative energies (eV) and the spin multiplicities (in square
                                                2
brackets) are shown. The Si-K distances are given in Å for some clusters.

maxima for n = 2 and 5. Our calculations show that alkali binding energy is similar for both Si n K and
Sin Na and is higher for Sin Li by about 0.4 eV. This relative higher value for Si n Li can be understood
since the Sin -alkali distance is much shorter for lithium than for the other alkali (because lithium atom
is smaller) and so the electrostatic interaction between Si − and Li+ is larger. The binding energy of two
                                                            n
K atoms is approximatively equal to twice the binding energy of one K atom.
   Binding energies of potassium cation to Si n cluster calculated as E b (Sin K+ ) = − (E(Sin K+ )–E(Sin )–
E(K+ )) are listed in Table 3. As already discussed above, for the lowest-energy isomer, the K + ion binds
over one silicon atom, minimizing the interaction energy between the K + charge and the induced dipole
in the globally neutral system Sin . This leads to binding energies of about 0.5 eV. The binding energies of
282     F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium

                                                          Table 3
                                          Spatial symmetries of the lowest-
                                          energy isomers and binding energies
                                          of potassium ion and atom to Sin clus-
                                          ter for cationic species calculated as
                                          Eb (Sin K+ ) = −(E(Sin K+ )–E(Sin )–
                                          E(K+ )) and Eb (Sin K+ ) = −(E(Sin
                                                                2
                                          K+ )–E(Sin )–E(K+ )–E(K))
                                           Cluster      Symmetry        Eb (eV)
                                           SiK+         C∞v             0.274
                                           Si2 K+       C∞v             0.705
                                           Si3 K+       Cs              0.499
                                           Si4 K+       C2v             0.512
                                           Si5 K+       C2v             0.523
                                           Si6 K+       Cs              0.573
                                           SiK+2        D∞h             2.678
                                           Si2 K+2      C2v             3.496
                                           Si3 K+2      Cs              3.254
                                           Si4 K+2      D2h             2.765
                                           Si5 K+2      C1              3.070
                                           Si6 K+2      C2v             2.734




                     Fig. 5. Binding energy (Eb ) of lithium, sodium or potassium atom to Sin cluster.

potassium atom and cation to Si n in Sin K+ clusters, calculated as E b (Sin K+ ) = − (E(Sin K+ )–E(Sin )–
                                          2                                   2
E(K+ )–E(K)), are also given in Table 3.

3.3.2. Vertical and adiabatic ionization potentials
  We now discuss the ionization potentials for Si n Kp clusters . We have calculated both the vertical
ionization potential vIP (when the ion geometry is considered as identical to the geometry of the neutral)
and the adiabatic ionization potential aIP (including the relaxation of the ion geometry) for Si n K and
Sin K2 clusters.
  Results for Sin K and Sin K2 clusters are compared with those for lithium and sodium-doped silicon
        F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium   283




              Fig. 6. Calculated adiabatic ionization potential for Sin Li [21], Sin Na [20] and Sin K clusters.




               Fig. 7. Calculated vertical ionization potential for Sin Li [21], Sin Na [20] and Sin K clusters.

clusters in Figs 6, 7, 8 and 9. For Si n K clusters vIP values vary in the range of 5.55–6.78 eV and aIP in
the range of 5.40–6.08 eV (Table 4). The evolution of IPs against n presents a local minimum for n = 4
and two local maxima for n = 3 and 5, similarly to previous results for Si n Na [15,16,20] and Si n Li [21]
(Figs 6 and 7). The evolution is similar to that of the electron affinity of Si n , the latter displaying a
minimum for n = 4 and a maximum for n = 5 [17]. This parallelism had already been noticed by Kishi
et al. [15] and can be easily understood as the HOMO of Si n K is the LUMO of Sin . The IPs for the
potassium-doped cluster are significantly lower (2–3 eV) than those for the parent Si n clusters [16]. The
decrease reflects the change in the orbital being ionized, which is in Si n K of similar character as the
LUMO of the parent Sin . For Sin K2 , the evolution of IPs against n is similar to those already observed
for Sin Na2 and Sin Li2 (Figs 8 and 9). The IPs of potassium-doped clusters are lower than those of
284     F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium

                                                           Table 4
                                              Vertical and adiabatic ionization
                                              potentials for Sin K and Sin K2
                                              Cluster    vIP (eV)    aIP (eV)
                                              SiK          5.65        5.49
                                              Si2 K        6.25        6.08
                                              Si3 K        6.53        5.93
                                              Si4 K        5.55        5.40
                                              Si5 K        6.78        5.83
                                              Si6 K        6.11        5.50
                                              SiK2         3.99        3.99
                                              Si2 K2       4.92        4.79
                                              Si3 K2       4.95        4.79
                                              Si4 K2       4.83        4.20
                                              Si5 K2       5.69        5.23
                                              Si6 K2       5.01        4.74




            Fig. 8. Calculated adiabatic ionization potential for Sin Li2 [21], Sin Na2 [20] and Sin K2 clusters.

sodium-doped by about 1 eV and than those of lithium-doped by about 1.5–2 eV. This is consistent with
the respective values of IPs for alkali atoms which are 5.92 eV, 5.139 eV and 4.341 eV for Li, Na and K
respectively.

3.4. Dipole moments and polarizabilities

  Dipole moments and static dipolar polarizabilities are two properties which could in principle make
possible discrimination between different isomers. The dipole moments are interesting observables since
they probe the charge distribution. We have calculated the dipole moment for all stable isomers. The
results are exposed in Table 1 for Si n K and Table 2 for Sin K2 . For Sin K clusters, the transfer of one
electron from the K atom to the silicon cluster leads to a dipole moment µ oriented from the center of
mass of the silicon atoms toward the potassium atom. It has a quite large value within the 9.62–12.60 D
range for the most stable isomer of Sin K. It is much larger than those of Li and Na-doped systems [20,
        F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium   285




             Fig. 9. Calculated vertical ionization potential for Sin Li2 [21], Sin Na2 [20] and Sin K2 clusters.

21] because the transferred charge is higher and the distance between the barycenter of the positive and
negative charges is also larger. For Si n K2 clusters, the dipole moment depends on the relative position of
the two K atoms. For the lowest-energy isomer of each size, values of the dipole moment are in the range
of 2.38–9.34 D, significantly smaller than values for Si n K. Furthermore, for a given size the dispersion
of the dipole moment for the various isomers is larger for Si n K2 than for Sin K. The calculated averaged
static dipolar polarizabilities, α = (αxx + αyy + αzz )/3 are also given in Tables 2 and 3. The values
increase with the size of the cluster. They are significantly higher for Si n K2 than for Sin K.


4. Conclusions

   We have presented results of the first theoretical investigation on neutral and positively charged
      (+)
Sin Kp clusters. Calculations have been carried out in the framework of the density functional theory
with B3LYP/6–31+G(d). The structure of Sin Kp keeps the frame of the corresponding Si n cluster
unchanged, and the electronic structure of Si n Kp corresponds approximatively to that of Si p− + p K+ for
                                                                                                n
the small sizes considered here. For all clusters studied here, the spin multiplicity of the lowest-energy
isomer is found to be the lowest one for each size (doublet for Si n K and Sin K+ , singlet for Sin K2 and
                                                                                    2
Sin K+ ). The localization of the potassium cation is not the same one as that of the neutral atom. The
K+ ion is preferentially located on a Si atom while the K atom is preferentially attached at a bridge
site. The K+ ion tend to bind over one silicon atom to minimize the interaction energy in generating an
induced dipole in the globally neutral system Si n , while for the neutral system the structure is more stable
when the transferred charge can be shared by several silicon neighbors. In the geometrical structure
of the most stable isomers of Sin K2 clusters, the second potassium atom is located far from the first
one due to the electrostatic repulsion between their positive charges. Binding energies and ionization
potentials are compared to those of sodium or lithium-doped silicon clusters. Values of dipole moments
are tightly connected to geometrical structures. We hope that our theoretical predictions will provide
strong motivation for further experimental studies of these important silicon clusters and their cations.
286       F. Rabilloud and C. Sporea / Ab initio investigation of structures and properties of mixed silicon-potassium

References

 [1]   J. Yang, W. Xu and W. Xiao, The small silicon clusters Sin (n = 2–10) and their anions: sturctures, thermochmistry, and
       electro affinities, J Mol Struct: THEOCHEM 719 (2005), 89–102, and references therein.
 [2]   C.C. Arnold and D.M. Neumark, Study of Si4 and Si− using threshold photodetachment (ZEKE) spectroscopy, J Chem
                                                              4
       Phys 99 (1993), 3353–3362.
 [3]   E.C. Honea, A. Ogura, C.A. Murray, K. Raghavachari, W.O. Sprenger, M.F. Jarrold and W.L. Brown, Raman spectra of
       size-selected silicon clusters and comparison with calculated structures, Nature 366 (1993), 42–44.
 [4]   S. Li, R.J. Van Zee, W. Weltner, Jr. and K. Raghavachari, Si 3 –Si7 . Experimental and theoretical infrared spectra, Chem
       Phys Lett 243 (1995), 275–280.
 [5]   S. Yoo and X.C. Zeng, Global geometry optimization of silicon clusters described by three empirical potentials, J Chem
       Phys 119 (2003), 1442–1450.
 [6]   L.R. Marin, M.R. Lemes and A. Dal Pino, Jr., Ground-state of silicon clusters by neural network assisted genetic
       algorithm, J Mol Struct: THEOCHEM 663 (2003), 159–165.
 [7]   J.C. Grossman and L. Mitas, Quantum Monte Carlo determination of electronic and structural properties of Sin clusters
       (n 20), Phys Rev Lett 74 (1995), 1323–1326.
 [8]   S. Nigam, C. Majumder and S.K. Kulshreshtha, Structural and electronic properties of Sin , Si+ , and AlSin−1 (n = 2–13)
                                                                                                    n
       clusters: Theoretical investigation based on ab initio molecular orbital theory, J Chem Phys 121 (2004), 7756–7763.
 [9]   A. Tekin and B. Hartke, Global geometry optimization of small silicon clusters with empirical potentials and at the DFT
       level, Phys Chem Chem Phys 6 (2004), 503–509.
[10]   I. Rata, A.A. Shvartsburg, M. Horoi, T. Frauenheim, K.W. Siu and K.A. Jackson, Single-Parent Evolution Algorithm and
       the Optimization of Si Clusters, Phys Rev Lett 85 (2000), 546–549.
[11]   M.F. Jarrold and V.A. Constant, Silicon cluster ions: Evidence for a structural transition, Phys Rev Lett 67 (1991),
       2994–2997.
[12]   K. Fuke, K. Tsukamoto, F. Misaizu and M. Sanekata, Near threshold photoionization of silicon clusters in the 248–146
       nm region: Ionization potentials for Sin , J Chem Phys 99 (1993), 7807–7812.
[13]   H. Hiura, T. Miyazaki and T. Kanayama, Formation of metal-encapsulating Si cage clusters, Phys Rev Lett 86 (2001),
       1733–1736.
[14]   C. Majunder and S.K. Kulshreshtha, Impurity-doped Si10 cluster: Understanding the structural and electronic properties
       from first-principles calculations, Phys Rev B 70 (2004), 245426 (7 pages).
[15]   R. Kishi, S. Iwata, A. Nakajima and K. Kaya, Geometric and electronic structures of silicon – sodium binary clusters. I.
       Ionization energy of Sin Nam , J Chem Phys 107 (1997), 3056–3070.
[16]   S. Wei, R.N. Barnett and U. Landman, Energetics and structures of neutral and charged Sin (n 10) and sodium-doped
       Sin Na clusters, Phys Rev B 55 (1997), 7935–7944.
[17]   O. Cheshnovsky, S.H. Yang, C.L. Pettiette, M.J. Craycraft, Y. Liu and R.E. Smalley, Ultraviolet photoelectron spec-
       troscopy of semiconductor clusters: Silicon and germanium, Chem Phys Lett 138 (1987), 119–124.
[18]   H. Wang, W. Lu, Z. Li and C. Sun, Theoretical investigation on the adsorption of lithium atom on the Sin cluster (n =
       2–7), J Mol Struct (THEOCHEM) 730 (2005), 263–271.
[19]   J. Wu and F. Hagelberg, Equilibrium geometries and associated energetic properties of mixed metal-silicon clusters from
       global optimization, J Chem Phys A 110 (2006), 5901–5908.
                                                                                                              (+)
[20]   C. Sporea, F. Rabilloud, A.R. Allouche and M. Fr´ con, Ab initio study of neutral and charged Sin Nap (n 6, p 2)
                                                          e
       clusters, J Phys Chem A 110 (2006), 1046–1051.
[21]                                                                            e
       C. Sporea, F. Rabilloud, X. Cosson, A.R. Allouche and M. Aubert-Fr´ con, Theoretical study of mixed silicon-lithium
                      (+)
       clusters Sin Lip (n = 1–6, p = 1–2), J Phys Chem A 110 (2006), 6032–6038.
[22]   M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, V.G. Zakrzewski, J.A. Montgomery,
       Jr., R.E. Stratmann, J.C. Burant, S. Dapprich, J.M. Millam, A.D. Daniels, K.N. Kudin, M.C. Strain, O. Farkas, J. Tomasi,
       V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G.A. Petersson, P.Y.
       Ayala, Q. Cui, K. Morokuma, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J. Cioslowski, J.V. Ortiz,
       B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R.L. Martin, D.J. Fox, T. Keith, M.A.
       Al-Laham, C.Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P.M.W Gill, B. Johnson, W. Chen, M.W. Wong,
       J.L. Andres, C. Gonzalez, M. Head-Gordon, E.S. Replogle and J.A. Pople, Gaussian 98, Revision A.6; Gaussian, Inc.:
       Pittsburgh, PA, 1998.
[23]   Gabedit is a graphical user interface for Gaussian, Molpro, Molcas and MPQC ab initio programs available from
       http://lasim.univ-lyon1.fr/allouche/gabedit.
[24]   A.D. Becke, Density-functional thermochemistry. III. The role of exact exchange, J Chem Phys 98 (1993), 5648–5652.
[25]   A.E. Reed and F. Weinhold, Natural bond orbital analysis of near-Hartree – Fock water dimer, J Chem Phys 78 (1983),
       4066–4073.
[26]   A.E. Reed, R.B. Weinstock and F. Weinhold, Natural population analysis, J Chem Phys 83 (1985), 735–746.
Journal of Computational Methods in Sciences and Engineering 7 (2007) 287–296                                             287
IOS Press




Electric polarizabilities of the
CxSi4−x (0 x 4 ) clusters. A conventional
and time-dependent density functional theory
study

Demetrios Xenidesa,b,1 and Christos S. Garoufalisc
a Laboratory  of Computational Sciences, Department of Computer Science and Technology, Faculty of
Sciences and Technology, University of Peloponnese, GR-22 100 Tripolis, Greece
b Division of Theoretical Chemistry, Department of General, Inorganic, and Theoretical Chemistry,

University of Innsbruck, A-6020, Innrain 52a, Austria
E-mail: xenides@uop.gr
c Department of Physics, Faculty of Natural Sciences, University of Patras, GR-26 500 Patras, Campus

at Rion, Greece
E-mail: garoufal@physics.upatras.gr

Received 22 April 2007
Revised /Accepted 20 June 2007

Abstract. The dipole polarizabilities of the pure tetramers (C4 , Si4 ) and mixed carbon silicon hetero-clusters (Cx Si4−x , 0
    x     4) have been calculated within the framework of time independent and Time-Dependent (TD) Density Functional
Theory (DFT) methods. The convergence of the two approaches is remarkably good revealing the absence of any systematic
error. The Si- substitution leads to clusters with enhanced properties. The effect is more pronounced in the case of second
                                   −3
hyperpolarizability (x 103 e4 a4 Eh ): 8.51(C4 ) → 18.04(SiC3 ) → 32.45(Si2 C2 ) → 60.45(Si3 C) → 100.79(Si4 ). To further
                                0
extend our study, we have performed a spectral decomposition of dipole polarizabilities by employing the TD excitation
energies and their corresponding oscillator strengths. Such a decomposition allows for a pictorial insight into some of the
factors controlling the evolution of the property for these clusters.

Keywords: Dipole polarizability, hyperpolarizability, silicon-carbon clusters, DFT, TDDFT, spectral decomposition



1. Introduction

  The chemistry and physics of heteroatomic clusters is a tedious task for every quantum chemical
method, thus they provide fertile ground for the evaluation of the performance of different methods.
The potential of the Density Functional Theory (DFT) methods has been widely recognized, however
there is ongoing research towards their improvement as to bridge the gap between them and the highly

  1
      Present address: University of Peloponnese Greece.


1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
288           D. Xenides and C.S. Garoufalis / Electric polarizabilities of the Cx Si4−x (0   x   4) clusters

sophisticated ab initio methods. An important step towards the understanding of the chemistry and
physics of these species will be the calculation of their electric (hyper)polarizabilities [1–3], while at
the same time this will serve as an evaluation test of different conventional and TDDFT methods. The
present study is not focused on providing final estimates (this is the subject of a subsequent paper) but
onto shed some light on the behaviour of the employed methods.
  For our calculations we relied on several pure, hybrid, and gradient corrected DFT functionals (for
details see next section). A unique, large and flexible basis set has been constructed upon an initial
substrate by Ahlrichs et al. [4] for each one of the different moieties (except in the case of Si 4 when a
basis set by Maroulis and Pouchan [5] has been used). Geometries of the systems were either computed
or collected from the available literature.


2. Theory

2.1. Computational details

  DFT calculations have been performed via the Finite Field (FF) approach which, in brief, has as
follows: when a weak, static and homogeneous electric field is applied to a system (atom, molecule,
cluster) it causes its energy to be shifted. Let us denote the shifted energy as E p . Buckingham in 1978 [1]
showed that this energy shift can be related to the permanent and induced electric properties of the system
via the following (the compact form doesn’t lead to any loss of accuracy) equation:
                              1             1               1
      E p = E 0 − µα Fα −        ααβ Fα Fβ − βαβγ Fα Fβ Fγ − γαβγδ Fα Fβ Fγ Fδ + . . .                          (1)
                              2!            3!              4!
E 0 is the energy of the unperturbed molecule, F α . . . is the strength of the applied electric field, µ α is the
dipole moment, ααβ is the dipole polarizability, and βαβγ and γαβγδ are the first and second hyperpo-
larizability tensors, respectively. The number of independent components is regulated by the symmetry
of the system via the “the higher the symmetry the lesser the independent components” principle. Thus,
for the C4 , Si4 , and C2 Si2 clusters (D2h group of symmetry) they are: three for the dipole polarizability
(αxx , αyy , αzz ), and six for the second hyperpolarizability (γ xxxx ,γyyyy ,γzzzz ,γxxyy ,γxxzz ,γzzyy ) respec-
tively; for the C3 Si and CSi3 clusters (C2v group of symmetry) the additional properties – independent
components – are one for the dipole moment (µ z ) and three for the first hyperpolarizability (βzzz , βzxx ,
βzyy ) [1]. Mean values and anisotropy have been calculated via the following formulas:
         1
      α = (αxx + αyy + αzz )
      ¯                                                                                                         (2)
         3
      ¯ 3
      β = (βzzz + βzxx + βzyy )                                                                                 (3)
         5
         1
      γ = (γxxxx + γyyyy + γzzzz + 2(γxxyy + γyyzz + γzzxx ))
      ¯                                                                                                         (4)
         5
and
                                                                               1
           1                                                                   2
      ∆α =   (αxx − αyy )2 + (αyy − αzz )2 + (αzz − αxx )2                                                      (5)
           2
Finite differences have been calculated between the perturbed energies (fields scaled as F, 2F, 4F with F
= 0.0025 e−1 a−1 Eh ) and the energy of the unperturbed system. All calculations were performed with
                0
Gaussian 03 [6] program.
                    D. Xenides and C.S. Garoufalis / Electric polarizabilities of the Cx Si4−x (0   x   4) clusters   289

2.2. Basis set

  DFT methods do not depend on the basis set as much as the wavefunction methods do, however an
appropriate basis set is always a demand in quantum chemical calculations. Thus, a large and flexible
basis set (for every given system) has been constructed upon an initial substrate by Ahlrichs et al. [4]
added with s- and p- type Gaussian Type Functions (GTFs) as:
 C:       [1s (0.039421656411) and 1p(0.035011522946)] ≡ [6s4p],
 Si:      [1s (0.03098958575) and 1p(0.02384352131)] ≡ [6s5p], respectively.

  Further augmentation with tight and diffuse GTFs of d- and f- type has been performed as follows:
 C4 :        [C /C] ≡          [6s4p/6s4p] + 3d(0.5135, 0.1234, 0.0605/0.6446, 0.1719, 0.0880)
                               + 2f(0.5135, 0.1234/0.6446, 0.1719) ≡ [6s4p3d2f/6s4p3d2f] 188 GTFs.
 C3 Si:      [C/C/Si]≡         [6s4p/6s4p/6s5p] + 4d(0.7765, 0.3090, 0.1230, 0.0490/0.5039, 0.2266,
                               0.1019, 0.0458/0.4033, 0.1931, 0.0925, 0.0443)
                               + 1f(0.1230 / 0.1019 / 0.0925) ≡ [6s4p4d1f/6s4p4d1f/6s5p4d1f] 183 GTFs.
 C2 Si2 :    [C/Si]≡           [6s4p/6s5p] + 4d( 0.5175, 0.2019, 0.0788, 0.0307/0.4059, 0.1872, 0.0863,
                               0.0398)
                               + 1f( 0.0788/0.0863) ≡ [6s4p4d1f/6s5p4d1f] 188 GTFs.
 CSi3 :      [C/Si/Si] ≡       [6s4p/6s5p/6s5p] + 4d(0.4172, 0.1973, 0.0933, 0.0471/0.4004, 0.1762,
                               0.0775, 0.0341/0.2500, 0.1183, 0.0560, 0.0265)
                               + 1f(0.0933/ 0.0775/ 0.0560) ≡ [6s4p4d1f/6s5p4d1f/6s5p4d1f] 189 GTFs.
 Si4 :       [Si]              [5s4p3d1f] Basis set corresponds to SV2 from Ref. [5].



2.3. Geometry

  Geometrical features have been obtained either from the available literature (C 4 , SiC3 , Si3 C, Si4 )
or calculated in the context of the present study (Si 2 C2 ). The orientation of the molecules and the
geometrical parameters are:

 C4 :        RCC = 1.442 Å, ∠ CCC = 62.75◦ from Ref. [7].
 C3 Si:      RSiC = 1.8290 Å, RCC(b) = 1.4342 , RCC(d) = 1.4830 Å from Ref. [8].
 C2 Si2 :    RSiC = 1.824 Å, RCSi = 1.448 Å, present investigation MP2(full)/cc-pVQZ.
 CSi3 :      RSiC = 1.858 Å, RCSi = 1.876 Å, ∠ SiCSi = 62.75◦ from Ref. [9], and
 Si4 :       RSiSi(b) = 2.4449 Å, RSiSi(d) = 2.3382 Å, from Ref. [5];

and they are illustrated in Fig. 1.

2.4. Methods

  To provide reliable remarks on the DFT performance, a large number of DFT functionals have been
implemented in present the study. These are BLYP [10,11], BP86 [10,12], BPW91 [10,13], OLYP [11,
15], OP86 [12,15], OPW91 [13,15], B3LYP [11,14], B3P86 [12,14], B3PW91 [13,14], B98LYP [11,
17], B98P86 [12,17], B98PW91 [13,17], O3LYP [11,16], mPW1PW91 [13,19], PBE1PBE [18a],
PBEPBE [18], PBEPW91 [13,18], mPW1PBE [18,19]. 2

  2
    Part of obtained results has been presented in the International Conference on Computational Methods in Science and
Engineering, Chania, Crete, Greece, October 27 – November 1, 2006.
290            D. Xenides and C.S. Garoufalis / Electric polarizabilities of the Cx Si4−x (0   x    4) clusters




                     Fig. 1. Illustration of the geometrical parameters (large spheres denote Si- atoms).

  For some of the aforementioned functionals, the mean dipole polarizabilities have also been calculated
analytically by TDDFT (which, for the static case is equivalent to the CPHF equations) using either the
TURBOMOLE [20] or PCGAMESS [21] programs. The spectral decomposition is performed on the
basis of the sum over state (SOS) formulation, as:
           2        f n0
      ¯
      α=               2
                                                                                                                  (6)
           3   n
                   ∆En0
                 2
In this case, ∆En0 is the excitation energy from the ground state to the nth excited state and f n0 is the
                                                                      2
corresponding oscillator strength of the transition. Both f n0 and ∆En0 are calculated in the framework
of TDDFT. Each excitation contributes a value of:
            2 f n0
     αn =        2
                                                                                                       (7)
            3 ∆En0
In this way, a detailed decomposition of the calculated polarizabilities to contributions from different
distinct excitations can be obtained. However, it should be noted that TDDFT provides accurate excitation
energies only for low-energy transitions involving valence states. The evolution of polarizability as a
function of the number of excitations can be expressed as α(∆E k0 ) = k αi .
                                                                           i=1


3. Results

  Results and discussion will be focused on the common (meaning appearing in clusters of both sym-
metries) properties. Emphasis will be put on the anisotropy of polarizability, a property that is very
sensitive to changes of the independent components, thus clearly mirrors the behaviour of different
functionals, while at the same time mean values can blur this picture. A separate subsection with details
on calculated properties and relevant discussion will be devoted to two characteristic molecules (C 4 and
SiC3 , belonging to D2h and C2v groups of symmetry, respectively).
             D. Xenides and C.S. Garoufalis / Electric polarizabilities of the Cx Si4−x (0     x      4) clusters         291

                                                     Table 1
                                                                                                       −1
             Conventional and TD- (in parentheses) DFT values for the dipole polarizability (in e2 a2 Eh )
                                                                                                    0

 Method      αxx       αyy       αzz        ¯
                                            α        ∆α    Method             αxx       αyy         αzz        α¯      ∆α
 BLYP       39.73     30.83     45.37     38.64     12.70 O3LYP              39.00     29.79       44.47      37.75   12.85
           (39.73)   (30.82)   (45.37)   (38.64)   (12.70)
 BP86       39.44     30.70     45.12     38.42     12.59 B98LYP             46.27     36.78       53.37      45.47   14.42
           (39.32)   (30.66)   (45.09)   (38.36)   (12.58)
 BPW91      39.00     30.36     44.77     38.04     12.56 B98P86             45.79     36.56       52.90      45.08   14.19
 B3LYP      39.19     29.85     44.02     37.69     12.48 B98PW91            45.14     36.09       52.28      44.50   14.05
           (39.19)   (29.85)   (44.02)   (37.69)   (12.48)
 B3P86      38.68     29.57     43.60     37.28     12.33 PBE1PBE      38.82   29.56   43.66   37.35                   12.41
                                                                      (38.82) (29.56) (43.65) (37.4)                  (12.41)
 B3PW91     38.71     29.55     43.67     37.31     12.41    PBEPBE    39.36   30.71   45.14   38.40                   12.58
           (38.70)   (29.55)   (43.67)   (37.31)   (12.41)            (39.36) (30.70) (45.13) (38.4)                  (12.58)
 OLYP       39.25     30.28     45.31     38.28     13.09    PBEPW91   39.34   30.68   45.10   38.38                   12.57
                                                                      (39.34) (30.68) (45.10) (38.38)                 (12.57)
 OP86      38.83     30.06     45.12     38.00     13.10     mPW1PW91 38.68    29.41   43.50   37.19                   12.40
 OPW91     38.49     29.78     44.64     37.64     12.93     mPW1PBE   38.70   29.42   43.53   37.22                   12.42

                                                        Table 2
                                                                                                −3
                     Conventional DFT values for the second hyperpolarizability (in 103 xe4 a4 Eh )
                                                                                             0

             Method             γxxxx      γyyyy      γzzzz       γxxyy       γyyzz           γzzxx         ¯
                                                                                                            γ
             BLYP               12.24       4.91      20.55        2.56        5.94            4.25       12.64
             BP86              −28.65      16.79      41.08        2.64       21.22            4.31       17.11
             BPW91              10.69       4.28      18.22        2.24        5.48            3.68       11.20
             B3LYP               9.94       3.88      14.75        2.09        4.12            3.10        9.44
             B3P86               8.85       3.46      13.16        1.86        3.80            2.73        8.45
             B3PW91              9.10       3.56      13.73        1.91        3.91            2.83        8.74
             OLYP               12.46       5.04      22.18        2.68        6.16            4.58       13.31
             OP86               38.28      25.27     −39.16       16.66      −10.91            3.32        8.50
             OPW91              10.81       4.31      19.44        2.30        5.59            3.91       11.63
             O3LYP              10.88       4.33      17.68        2.33        4.90            3.68       10.94
             B98LYP             30.96      12.19      60.18        6.46       15.44           12.91       34.59
             B98P86            −24.00      32.19      18.85       −6.66       40.70          −43.05        1.80
             B98PW91            26.23      10.44      51.74        5.51       13.94           10.78       29.77
             PBE1PBE             9.09       3.54      13.49        1.92        3.81            2.83        8.65
             PBEPBE             11.40       4.61      19.57        2.44        5.92            4.07       12.09
             PBEPW91            11.37       4.59      19.43        2.43        5.88            4.04       12.02
             mPW1PW91            8.91       3.53      13.32        1.89        3.72            2.78        8.51
             mPW1PBE             8.93       3.54      13.40        1.90        3.74            2.79        8.55

3.1. Pure carbon cluster (C 4 )

  Results obtained for this cluster are presented in Tables 1 and 2. Anisotropic dipole polarizabilities
are nicely grouped in three distinct sets. In different cases ([22–24]) mPW1PW91 and PBE1PBE have
shown reasonable agreement to highly sophisticated ab initio methods (i.e., CCSD(T)) (let us name this
group as d), while the B98LYP, and B98PW91 have shown the worst (let us name this group as b).
Between these two extremes another group is formed having as members the BLYP, BPW91, OLYP, and
OPW91 functionals (let us name this group as c). The BP86, OP86 and B98P86 functionals are giving
dipole polarizability values closer to b group. Results from the B3P86 and B3PW91 methods are closer
to those of the d group, while B3LYP and O3LYP results to those of the c group. Similar trends have
been observed when we classified the methods by using results of the mean dipole polarizability.
  Second hyperpolarizability is much more sensitive to the method/basis set used in calculations, thus
292           D. Xenides and C.S. Garoufalis / Electric polarizabilities of the Cx Si4−x (0      x      4) clusters

                                                        Table 3
                                                                                                                       −1
 Conventional and TD- (in parentheses) DFT values for the dipole moment (in ea0 ) and dipole polarizability (in e2 a2 Eh )
                                                                                                                    0

Method   µz      αxx         αyy       αzz        ¯
                                                  α        ∆α      Method       µz    αxx        αyy         αzz        α¯      ∆α
BLYP   −1.4908 50.15        44.05     75.47     56.56     28.86    O3LYP      −1.4537 49.16      43.17       73.54     55.29    27.86
               (50.15)     (44.06)   (75.48)   (56.56)   (28.86)
BP86   −1.4511 49.83        43.91     75.07     56.27     28.67    B98LYP     −1.3483 59.02      52.19       87.13     66.11    32.08
               (49.67)     (43.79)   (74.96)   (56.14)   (28.68)
BPW91 −1.4429 49.21         43.35     74.42     55.66     28.60    B98P86     −1.3080 58.17      51.78       86.11     65.36    31.62
B3LYP −1.5250 49.00         43.07     72.71     54.93     27.17    B98PW91    −1.2978 57.43      51.04       85.28     64.58    31.53
               (49.00)     (43.08)   (72.72)   (54.93)   (27.17)
B3P86 −1.4988 48.38         42.65     71.98     54.34     26.93          −1.4854 48.56
                                                                   PBE1PBE                       42.81     71.97      54.45     26.75
                                                                                (48,55)         (42.81)   (71.98)    (54.45)   (26.75)
B3PW91 −1.4856 48.43 42.66 72.12 54.40                    27.04 PBEPBE   −1.4455 49.75           43.84     74.99      56.19     28.66
               (48.43) (42.67) (72.12) (54.41)           (27.04)                 (49.75)        (43.84)   (74.99)    (56.19)   (28.66)
OLYP   −1.4197 49.79 43.68 75.15 56.21                    28.90 PBEPW91 −1.4500 49.72            43.81     74.94      56.16     28.64
                                                                                (49.72)         (43.81)   (74.94)    (56.16)   (28.64)
OP86  −1.3787 49.23         43.38    74.65     55.75      28.80 mPW1PW91 −1.4885 48.33           42.57     71.71      54.20     26.73
OPW91 −1.3696 48.82         43.01    74.07     55.30      28.60 mPW1PBE −1.4841 48.36            42.60     71.75      54.24     26.74

                                                        Table 4
                                                                                              −2
                          Conventional DFT values for the first hyperpolarizability (in e3 a3 Eh )
                                                                                           0

              Method      βzxx      βzyy     βzzz          ¯
                                                           β       Method       βzxx    βzyy         βzzz       β¯
              BLYP        64.2      85.9     246.0       237.7     O3LYP         55.3    61.6        220.4     202.4
              BP86        54.9      77.7     252.9       231.3     B98LYP       142.3   194.0        332.1     401.0
              BPW91       58.1      78.0     237.6       224.3     B98P86       122.6   147.2        292.1     337.1
              B3LYP       53.4      59.8     204.5       190.6     B98PW91      119.9   172.2        309.8     361.1
              B3P86       50.3      56.1     202.1       185.1     PBE1PBE       50.6    54.2        199.4     182.5
              B3PW91      50.8      56.1     202.9       185.8     PBEPBE        61.4    83.3        244.3     233.4
              OLYP        61.9      75.6     247.1       230.8     PBEPW91       61.3    83.2        243.8     232.9
              OP86        77.5      83.6     247.5       245.2     mPW1PW91      49.7    52.8        196.7     179.5
              OPW91       55.4      67.3     238.6       216.8     mPW1PBE       49.8    52.8        197.1     179.8

it provides a more rigorous test. Results presented in Table 2 reveal the rather erratic behaviour of the
BP86, OP86 and B98P86 (along with the aforementioned B98LYP and B98PW91 ones) functionals.
Apart from this noticeable exception the grouping of the rest methods is not further affected.

3.2. Mono Si- substituted heterocluster (SiC 3 )

  In Tables 3, 4, 5 the results for the independent components, mean values and anisotropy have
been summarized. The behaviour of the functionals is almost unaltered, therefore the previous made
classification retains its validity.
  The first hyperpolarizability values (Table 4) can serve as a first, rough though, evaluation criterion.
The erratically behaved functionals are still the same, while, in the same time, the ones with the best
performance are the mPW1PW91, mPW1PBE, and PBE1PBE. The B3P86, B3PW91 and to a lesser
degree B3LYP, can be regarded as members of the latter set. This is not the case for all the Bx and Ox
(x = LYP, P86 and PW91) based functionals which provide results at least 30 % higher.
  To further refine the already made observations the second hyperpolarizability values (Table 5) will be
used. In this case the negative BP86 (group a) values and the extremely large OP86, B98LYP, B98P86
and B98PW91 ones (b) are at least awkward results. Overestimated results have been obtained from
BLYP, BPW91, OLYP, OPW91, B3LYP, O3LYP, PBEPBE and PBEPW91 methods, thus they are also
on the wrong side (group c). The remaining methods are forming the group (d) with the acceptable
performance.
             D. Xenides and C.S. Garoufalis / Electric polarizabilities of the Cx Si4−x (0    x     4) clusters   293

                                                       Table 5
                                                                                               −3
                    Conventional DFT values for the second hyperpolarizability (in 103 xe4 a4 Eh )
                                                                                            0

                    Method           γxxxx     γyyyy     γzzzz     γxxyy     γyyzz    γzzxx           ¯
                                                                                                      γ
                    BLYP              2.36      1.25      5.05      0.54      1.30     0.89        2.82
                    BP86             −4.14     −2.84      2.13     −2.60     −0.93    −0.74       −2.68
                    BPW91             1.98      1.08      4.37      0.45      1.13     0.75        2.42
                    B3LYP             1.77      0.97      3.66      0.41      0.85     0.65        2.04
                    B3P86             1.54      0.87      3.26      0.36      0.76     0.57        1.81
                    B3PW91            1.59      0.89      3.37      0.37      0.78     0.59        1.86
                    OLYP              2.40      1.22      5.06      0.55      1.25     0.91        2.82
                    OP86              7.72      4.44      6.00      4.65      3.10     3.13        7.98
                    OPW91             1.99      1.04      4.33      0.46      1.08     0.76        2.39
                    O3LYP             2.00      1.04      4.14      0.46      0.97     0.74        2.30
                    B98LYP            6.63      3.26     13.12      1.48      3.93     2.58        7.80
                    B98P86            8.84      1.71     16.77      3.57      5.00     5.67       11.16
                    B98PW91           5.30      2.68     10.81      1.19      3.30     2.06        6.38
                    PBE1PBE           1.58      0.88      3.28      0.37      0.75     0.58        1.83
                    PBEPBE            2.15      1.16      4.65      0.50      1.23     0.82        2.61
                    PBEPW91           2.15      1.16      4.64      0.50      1.22     0.82        2.60
                    mPW1PW91          1.55      0.88      3.25      0.36      0.74     0.57        1.80
                    mPW1PBE           1.55      0.88      3.26      0.36      0.74     0.58        1.81

                                                           Table 6
                                mPW1PW91 results for the mean value and the anisotropy
                                                                    −1
                                of dipole polarizability (in e2 a2 Eh )
                                                                 0

                                 Cluster    αxx         αyy       αzz        ¯
                                                                             α        ∆α
                                 C4         38.68      29.41      43.50     37.19    12.40
                                 SiC3       48.33      42.57      71.71     54.20    26.73
                                 Si2 C2    106.72      57.07      60.31     74.70    48.11
                                 Si3 C     152.65      73.49      97.61    107.91    70.28
                                 Si4       137.54      91.89     187.94    139.12    83.22

3.3. Evolution of the properties with the cluster size

  From the above discussion on the functionals’ performance, it becomes clear that the use of
mPW1PW91 for the analysis of the substitution effect is adequately justified. For this reason the
values for all the independent and dependent components (i.e., calculated from Eqs (2)–(5)) have been
collected in Tables 6 and 7.
  Mean dipole polarizabilities increase by ≈ 50 % after every Si- substitution step. The effect is almost
uniform for all the independent components. The effect to the anisotropy is rather large ≈ 100% for the
C4 to SiC3 to Si2 C2 clusters. Further Si- addition has a much smoother effect on this property, that is ≈
50 % from Si2 C2 to Si3 C and ≈ 20 % from Si3 C to Si4 .
  Second hyperpolarizability is much more sensitive to the heavy atom substitution. This can be seen in
the evolution of the respective mean values, that is > 100% from C 4 to SiC3 and ≈ 100% from Si2 C2 to
Si3 C; for each of the other substitution processes the respective enhancement is around 80 %.

3.4. Spectral decomposition of mean dipole polarizability

  In Fig. 2 we have plotted the distinct contributions to the polarizability according to Eq. (7). The
excitation energies and oscillator strengths correspond to TDDFT/B3LYP calculations, which include
the 120 lowest spin and symmetry allowed transitions. Since TDDFT provides accurate excitation
294   D. Xenides and C.S. Garoufalis / Electric polarizabilities of the Cx Si4−x (0    x   4) clusters




                       Fig. 2. Spectral decomposition of mean dipole polarizability.
             D. Xenides and C.S. Garoufalis / Electric polarizabilities of the Cx Si4−x (0     x    4) clusters   295

                                                   Table 7
                      mPW1PW91 results for the mean value of second hyperpolarizability (in
                                   −3
                      x 103 e4 a4 Eh )
                                0

                      Cluster    γxxxx    γyyyy      γzzzz      γxxyy     γyyzz    γzzxx        ¯
                                                                                                γ
                      C4          8.91     3.53      13.32       1.89      3.72     2.78       8.51
                      SiC3       15.49     8.76      32.47       3.62      7.39     5.73      18.04
                      Si2 C2     65.58    15.04      19.87      13.13      5.10    12.63      32.45
                      Si3 C      96.22    26.27      60.61      19.19     13.44    26.94      60.45
                      Si4        98.65    47.22     177.30      22.41     35.39    32.58     100.79




                   Fig. 3. Accumulated polarizability (left), percentage of total polarizability (right).

energies only for low-energy transitions involving valence states, such a decomposition may give some
reasoning on the ability of DFT calculations to produce accurate values for the different clusters. For
example C4 exhibits comparably large contributions from high energy excitations (for which the TDDFT
is expected to perform poorly due to incorrect asymptotic behavior) while the contributions for the case of
Si4 come from lower energy excitations. As a result, it may be speculated that C 4 is a more difficult system
for DFT, which might need larger and more flexible basis sets compared to Si 4 . For the intermediate cases
of SiC3 , Si3 C, the percentage of total polarizability retrieved from the 120 lowest allowed transitions (the
evolution of the property is shown in Figs 4a and 4b) is 69% and 70% respectively, while the energy range
of the excitations is approximately 2 eV–14 eV. As a result, their enhanced polarizability compared to C 4
is mainly originated from an accumulation of a larger number of contribution in a smaller energy range.
This can be quantified by evaluating the percentage of polarizability retrieved by the most pronounced
contributions (e.g. contributions larger than 1% of the total value), which gives 73%, 68%, 61%, 47%
and 44% for the Si4 , Si2 C2 , C4 , Si3 C, and SiC3 clusters, respectively.


4. Conclusions

  The electric response properties of the C x Si4−x (0 x  4) clusters have been calculated with a
vast number of conventional and TDDFT methods. The erratic behaviour of the BP86, OP86 and B98
296             D. Xenides and C.S. Garoufalis / Electric polarizabilities of the Cx Si4−x (0   x     4) clusters

{LYP, P86, PW91} methods is noted. On the other hand mPW1PW91, mPW1PBE and B3PW91 seem
to provide reliable results.


Acknowledgments

  DX wishes to express his gratitude for the warm and generous hospitality of the Institute of Theo-
retical Chemistry of the University of Innsbruck, and his indebtedness the personnel of the Zentraler
                             a
InformatikDienst der Universt¨ t Innsbruck for their expert assistance. CSG thanks the European So-
cial Fund (ESF), Operational Program for Educational and Vocational Training II (EPEAEK II), and
particularly the Program PYTHAGORAS, for funding the above work.


References
 [1]   A.D. Buckingham, Basic theory of intermolecular forces: Applications to small molecules in Intermolecular Interactions:
       From Diatomics to Biopolymers, B. Pullman, ed., Wiley, New York, 1978, pp. 1–67.
 [2]   U. Hohm, J Phys Chem A 104 (2000), 8418.
 [3]   U. Hohm, Vacuum 58 (200), 117.
 [4]           a
       A. Sch¨ fer, C. Huber and R. Ahlrichs, J Chem Phys 100 (1994), 5829.
 [5]   G. Maroulis and C. Pouchan, Phys Chem Chem Phys 5 (2003), 1992.
 [6]   M.J. Frisch et al., Gaussian 03 Revision C.03, Gaussian Inc. Wallingford CT, 2004.
 [7]   J.D. Watts, J. Gauss, J.F. Stanton and R.J. Bartlett, J Chem Phys 97 (1992), 8372.
 [8]   J.F. Stanton, J. Gauss and O. Christiansen, J Chem Phys 114 (2001), 2993.
 [9]                                    e
       J.F. Stanton, J. Dudek, P. Theul´ , H. Gupta, M.C. McCarthy and P. Thaddeus, J Chem Phys 122 (2005), 124314.
[10]   A.D. Becke, Phys Rev A 38 (1988), 3098.
[11]   a) C. Lee, W. Yang and R.G. Parr, Phys Rev B 37 (1988), 785, b) B. Miehlich, A. Savin, H. Stoll and H. Preuss, Chem
       Phys Lett 157 (1989), 200.
[12]   J.P. Perdew, Phys Rev B 33 (1986), 8822.
[13]   a) K. Burke, J.P. Perdew and Y. Wang, in: Electronic Density Functional Theory: Recent Progress and New Directions,
       J.F. Dobson, G. Vignale and M.P. Das, eds, Plenum, 1998, b) J.P. Perdew, in: Electronic Structure of Solids 91, P. Ziesche
       and H. Eschrig, eds, Akademie Verlag, Berlin, 1991, 11, c) J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R.
       Pederson, D.J. Singh and C. Fiolhais, Phys Rev B 46 (1992), 6671, d) J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A.
       Jackson, M.R. Pederson, D.J. Singh and C. Fiolhais, Phys Rev B 48 (1993), E4978, and e) J.P. Perdew, K. Burke and Y.
       Wang, Phys Rev B 54 (1996), 16533.
[14]   A.D. Becke, J Chem Phys 98 (1993), 5648.
[15]   N.C. Handy and A.J. Cohen, Mol Phys 99 (2001), 403.
[16]   A.J. Cohen and N.C. Handy, Mol Phys 99 (2001), 607.
[17]   H.L. Schmider and A.D. Becke, J Chem Phys 108 (1998), 9624.
[18]   a) J.P. Perdew, K. Burke and M. Ernzerhof, Phys Rev Lett 77 (1996), 3865, b) J.P. Perdew, K. Burke and M. Ernzerhof,
       Phys Rev Lett 78 (1997), 1396.
[19]   C. Adamo and V. Barone, J Chem Phys 108 (1998), 664.
[20]                         a        a             o
       a) R. Ahlrichs, M. B¨ r, M. H¨ ser and C. K¨ lmel, Chem Phys Lett 162 (1989), 165, b) R. Ahlrichs and M. von Arnim,
       Methods and Techniques in Computational Chemistry: METECC-95 STEF, Vol. , Cagliari, 1995, c) M. von Arnim and
       R. Ahlrichs, J Comput Chem 19 (1998), 1746.
[21]   A.A. Granovsky, http://classic.chem.msu.su/gran/gamess/index.html.
[22]   G. Maroulis, Comp Lett 1 (2005), 31.
[23]   G. Maroulis and D. Xenides, Comp Lett 1 (2005), 246.
[24]   D. Xenides, J Mol Struct (THEOCHEM) 804 (2007), 41.
Journal of Computational Methods in Sciences and Engineering 7 (2007) 297–304                                               297
IOS Press




Ab initio investigation on the nonlinear optical
properties of silicon clusters Sin (n = 3–8)

Benoˆt Champagnea,∗ , Maxime Guillaumea, Didier B´ gu´ b and Claude Pouchanb
    ı                                            e e
a Laboratoire              e                e        e
              de Chimie Th´ orique Appliqu´ e, Facult´ s Universitaires Notre-Dame de la Paix, rue de
Bruxelles, 61, B-5000 Namur, Belgium
b Laboratoire de Chimie Th´ orique et de Physico-Chimie Mol eculaire, IPREM – ECP – UMR 5254,
                           e                                  ´
         e
Universit´ de Pau et des Pays de l’Adour, IFR, Rue Jules Ferry, BP27540 64075 PAU Cedex, France

Received 6 February 2007
Revised /Accepted 20 March 2007

Abstract. First and second hyperpolarizabilities of small silicon clusters have been calculated using conventional ab initio
methods systematically increasing the amount of electron correlation. Besides Si5 , upon successive addition of electron
correlation in the MP2, MP3, MP4, CCSD, and CCSD(T) series, all clusters display the same behavior: i) the HF γ// values
are the smallest, ii) the MP2 γ// values the largest, and iii) the latter values are good approximate to the reference CCSD(T)
results because the overestimation is smaller than 10%. Contrary to the polarizability per Si atom, which decreases with the
cluster size until reaching the bulk limit, the average second hyperpolarizability per Si atom presents a sawtooth behavior with
maxima in γ// associated with even numbers of Si and minima with odd numbers of Si atoms.



1. Introduction

   Clusters are intriguing systems bridging the gap between atoms and solids, constituting therefore the
first species along atomic aggregation [1–5]. Predicting and understanding their properties, in particular
as a function of their size, has been the topic of a considerable number of experimental and theoretical
investigations, in particular because they present a potential for applications in nanotechnology [6–26].
These investigations can be classified into two main streams, although complementary. In the first,
the geometries of the clusters are optimized, which enables systematic comparison with experimental
data provided the level of theory is sufficient. These studies generally reveal that the small clusters
present distinct properties from the bulk, the properties of which can be determined by extrapolation.
On the other hand, the number of stable atomic arrangements increases rapidly with the size of the
clusters – for instance, Si clusters can be oblate or prolate –, which often turned out to be an usual
computational bottleneck. The approaches of the second class aim at assessing directly the bulk properties
by characterizing representative clusters. In this case the number of arrangements is limited to the bulk
structure whereas the specificities of the (small) cluster characteristics are not tackled. The latest
breakingphysics aspects in this field encompass i) the synthesis and characterization of nano-objects
made of specific combinations/aggregations of clusters as well as ii) the search for stable magic clusters,

  ∗
      Corresponding author. E-mail: benoit.champagne@fundp.ac.be.


1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
298   B. Champagne et al. / Ab initio investigation on the nonlinear optical properties of silicon clusters Sin (n = 3–8)

i.e. clusters with specific numbers of atoms, which display peculiar and outstanding properties, in view
of grafting them to molecules or surfaces.
   Besides the huge number of structural studies several works have addressed their polarizabilities (α)
but less have tackled their first and second hyperpolarizabilities (β and γ) [9,27–33]. α is the linear
response of the dipole moment to an external electric field whereas β and γ are the first and second
nonlinear responses:
                      1      1
      µ(E) = µ0 + αE + βE 2 + γE 3 + . . .                                                                                  (1)
                      2      6
In particular, numerous of these investigations aimed at determining bulk properties from increasingly
large clusters with specific shapes and boundaries as recently reviewed [34]. Here we adopt the approach
of Refs. 19 and 21 and determine the static second hyperpolarizability of small silicon clusters (Si n , n =
3–8). We concentrate on the closed-shell singlet structures, though for n = 3 and 4, the open-shell singlet
is more stable than the closed-shell singlet by a few kcal/mol. These are determined using a hierarchy
of ab initio approaches, which enables to assess the impact of electron correlation effects. For n = 3
and n = 5, the first hyperpolarizabilities are also analyzed. Among the other objectives of this work, i)
analyzing the size effects on γ in the smallest Si clusters and ii) comparing these with the evolution of
the polarizability.


2. Computational aspects

  The geometrical structures of the Si n clusters taken from Ref. 21 were determined using density
functional theory with the B3LYP exchange-correlation functional and the 6-311G* basis set and are
represented in Fig. 1. These clusters present high symmetry [6,8,14–16,21]. Si 6 recently investigated
using post HF and DFT methods with large polarized basis sets exhibits an interesting PJTE (pseudo-
Jahn-Teller effect) [26] leading to deformation around his D 4h symmetry, the structure that has been
chosen here for hyperpolarizability calculations.
  The Si3 cluster presents C2v symmetry and is the only one having a dipole moment (µ) and therefore
a β// value. β// is the first hyperpolarizability that can be extracted from electric-field induced second
harmonic generation (EFISHG) experiment and that corresponds to the projection of the vector part of
β on the dipole moment vector,
                                     3         µi                                    3       µi βi
      β// (−2ω; ω, ω) = β// =                             (βijj + βjij + βjji ) =                                           (2)
                                     5        ||µ||                                  5       ||µ||
                                          i           j                                  i

where ||µ|| is the norm of the dipole moment and µ i and βi the components of the µ and β vectors. For
symmetry reasons, the Si 4 (D2h ), Si6 (D4h ), Si7 (D5h ), and Si8 (C2h ) clusters present a permanent dipole
moment, a EFISHG β// response, as well as a hyper-Rayleigh scattering (HRS) β HRS response, which
are all equal to zero. On the other hand, the Si 5 (D3h ) cluster displays a βHRS responses, as well as
Si3 . In the case of plane-polarized incident light and observation made perpendicular to the propagation
plane:

      βHRS =           2      2
                      βZZZ + βXZZ                                                                                           (3)
     B. Champagne et al. / Ab initio investigation on the nonlinear optical properties of silicon clusters Sin (n = 3–8)   299




             n = 3 (C2v)                               n = 4 (D2h)                           n = 5 (D3h)




             n = 6 (D4h)                               n = 7 (D5h)                           n = 8 (C2h)

                       Fig. 1. Structures of the Sin clusters with n = 3–8 drawn with MacMolPlot [40].

while the associated depolarization ratio (DR) is given by:
             2
            βZZZ
    R=       2
                                                                                                                           (4)
            βXZZ
                        2            2
Full expressions for βZZZ and βXZZ can be found in Ref. 35. They correspond to orientational
averages of the β tensor components that contain in principle β ijk (i = j = k = i) contributions
whereas the latter contributions do not appear in Eq. (2). Nevertheless, contrary to helical systems and
Td compounds, βxyz = 0 for both Si3 and Si5 due to symmetry reasons. All compounds possess non-zero
second hyperpolarizability. Here we concentrate on the quantity that can be extracted from EFISHG
experiment – although we consider its static limit – and that correspond to an isotropic average:
            1
    γ// =             γiijj                                                                                                (5)
            5
                i,j

The different β and γ tensor components were evaluated by using the finite field approach, which consists
in evaluating the system energy for different amplitudes and directions of the applied external electric
field and, subsequently, in differentiating it numerically. To determine all tensor components needed
to evaluate the quantities in Eqs (1)–(4), the following electric fields were applied: (0, 0, 0), (F, 0, 0),
(−F, 0, 0), (F, F, 0), (−F, −F, 0), (F, −F, 0), (−F, F, 0), (F, F, F), and (−F, −F, −F), as well as the other
combinations obtained by permuting the electric field Cartesian components, where F = 2 k × 10−4 a.u.
with k = 2–6. To improve its accuracy, the finite difference expressions were combined with Romberg
procedure [36] iterations to eliminate high-order contaminants. The Taylor series expansion convention
(usually called T convention) is chosen for defining β and γ .
  In addition to finite field Hartree-Fock (HF) values, electron correlation was included using Møller-
Plesset nth-order perturbation theory [MPn (n = 2–4)] and the coupled-cluster method with single and
300   B. Champagne et al. / Ab initio investigation on the nonlinear optical properties of silicon clusters Sin (n = 3–8)

                                                           Table 1
                                         Static average second hyperpolarizability
                                         (in 100 au) of small silicon clusters calcu-
                                         lated using the 6-311+G* basis set and a
                                         hierarchy of ab initio approaches
                                                           Si3       Si4       Si5
                                          RHF              493       669       839
                                          MP2              606       898       834
                                          MP3              547       789       849
                                          MP4(DQ)          519       795       870
                                          MP4(SDQ)         540       800       725
                                          MP4(SDTQ)        626       878       672
                                          CCSD             543       815       831
                                          CCSD(T)          601       879       832
                                                           Si6       Si7       Si8
                                          RHF              822       902      1204
                                          MP2             1202      1255      1692
                                          MP3              950      1012      1313
                                          MP4(DQ)         1041      1123      1495
                                          MP4(SDQ)        1021      1135      1523
                                          CCSD            1011      1096      1440
                                          CCSD(T)         1094      1187      1560

double excitations (CCSD) as well as with a perturbative treatment of triple excitations, [CCSD(T)].
The triple ζ 6-311+G* basis set, which also includes a set of polarization and a set of diffuse functions,
was employed to determine β and γ . Although it does not contains d diffuse functions, the 6-311+G*
= [7s6p1d] basis set is close to the SV0 = [5s4p2d] and D1 = [8s6p3d] basis sets using in Ref. 19.
Moreover, the SV0 basis set was recommended for studying the polarizability in large Si clusters [19].
All calculations were performed using the GAUSSIAN 03 program [37].


3. Results and discussions

   The average static second hyperpolarizabilities are listed in Table 1 while their evolutions with the
number of Si atoms are displayed in Fig. 2. Considering the reference CCSD(T) data, the global behavior
of γ// between Si3 and Si8 is an increase although i) γ // (Si5 ) < γ// (Si4 ) and ii) the γ// increase between
Si6 and Si7 is smaller than between the other consecutive clusters. This can be explained in particular
by the fact that the distortion of Si6 due to the PJTE is not taken into account in our γ calculations and
that lower symmetry is generally associated with smaller γ // values. Similar behaviors are found at the
MP2, MP4(SDQ), and MP4 levels of approximation. At the MP3 and CCSD level, γ // continuously
increases whereas the jumps are smaller between Si 4 and Si5 as well as between Si 6 and Si7 . At the HF
level, a maximum in γ// is obtained for Si5 while a minimum for Si6 .
   Except for Si5 , the HF γ// value is the smallest whereas the MP2 value is the largest. Moreover, the
MP2 method overestimates γ // by less than 10% with respect to the CCSD(T) results – for Si 3 , Si5 , and
to a lesser extent Si4 , the agreement is even very good. The inverse agreement as a function of the system
size is found for the MP4-SDQ method. Indeed, for n = 3–5, it underestimates the CCSD(T) values
by as much as 13% whereas the agreement improves for the three largest clusters. The fourth-order
singles contribution is particularly large for Si5 , which explains the particularly small γ // MP4(SDQ)
and MP4(SDTQ) = MP4 values. The triples contributions are analyzed by comparing the CCSD and
CCSD(T) values as well as the MP4(SDQ) and MP4 results. With the exception of Si 5 where it is
      B. Champagne et al. / Ab initio investigation on the nonlinear optical properties of silicon clusters Sin (n = 3–8)     301

                                                             1800.0
                                                                              HF
                                                                              MP2




                           hyperpolarizability (in 100 au)
                                                             1600.0
                                                                              MP3

                              Average static second
                                                                              MP4(SDQ)
                                                             1400.0           CCSD
                                                                              CCSD(T)
                                                             1200.0


                                                             1000.0


                                                              800.0


                                                              600.0


                                                              400.0
                                                                      2   3       4      5    6     7   8   9
                                                                                   Number of Si atoms

Fig. 2. Evolution of the static average second hyperpolarizability as a function of the cluster size calculated at different levels
of approximation using the 6-311+G* basis set.

negligible, the triples contribution to CCSD(T) is positive and amounts to 7–11% of the CCSD γ // value.
It is also positive and ranges from 10 to 15% in the MP4 γ // values of Si3 and Si4 . On the other hand,
for Si5 it is negative, which further decreases the MP4 γ // value with respect to the other methods. In
general, for the six clusters, electron correlation effects play a minor role on γ // in comparison with
π -conjugated systems [38,39] where electron correlation effects could lead to increase of γ // larger than
100%. Indeed, for all clusters except Si 5 , γ// is underestimated by 18–25% when employing the HF
scheme. Nevertheless, electron correlation effects on γ // are larger than on the polarizability as discussed
in a recent publication by Maroulis and two of us [19]. As a matter of illustration, using the D2 basis
set, the α(HF)/α[CCSD(T)] ratios amount to 1.026 and 1.011 for Si 3 and Si4 , respectively whereas the
corresponding ratios for γ are 0.820 and 0.761.
   Figure 3 sketches the evolution of the average second hyperpolarizability per Si atom, γ // /nSi . Besides
the HF and MP3 methods that provide slightly different trends, γ // /nSi displays a sawtooth behavior,
with larger γ// /nSi values associated with even numbers of Si atoms and smaller γ // /nSi values with
odd numbers. Moreover, contrary to the polarizability divided by the number of Si atoms [15,19,21,22],
γ// /nSi does not exhibit a global decrease with n Si . Indeed, using the SV0 basis set, Ref. 19 reports
successive values of 35.6, 34.3, 32.9, 29.8, and 30.5 au for the polarizabilities per Si atom of Si 3 -Si7
calculated at the CCSD(T) level. Since the clusters present the same shape, this evidences that γ //
behaves differently with respect to the number of non-interacting atoms or to the number of dangling
bonds as well as with respect to the system size like in conjugated systems. The CCSD(T) γ // /nSi values
oscillate around 18 × 10 3 au, which is about one order of magnitude larger than the corresponding bulk
values estimated by Jansik et al. [30]. Indeed, using clusters containing up to 54 Si atoms where the
dangling bonds are saturated by H atoms, the HF/ECP γ // /nSi ranges between 2.6 and 3.1 × 10 3 au for
nSi = 18–54. Assuming that the differences between Ref. 30 and the present data are not dominated
by the methods of calculations – which would be very peculiar – this difference between small and
large clusters second hyperpolarizability per Si atom indicates that γ // /nSi will finally decrease in larger
clusters.
302   B. Champagne et al. / Ab initio investigation on the nonlinear optical properties of silicon clusters Sin (n = 3–8)

                                                                                               Table 2
                                                                            Static first hyperpolarizabilities (in au) of
                                                                            Si3 and Si5 clusters calculated using the 6-
                                                                            311+G* basis set and a hierarchy of ab initio
                                                                            approaches. The quantities in parentheses are
                                                                            the depolarization ratios. Due to symmetry,
                                                                            the DR of Si5 is 1.5 no matter which method
                                                                            is employed
                                                                                                    Si3            Si5
                                                                                             β//       βHRS       βHRS
                                                                            RHF             −139     310 (1.65)    245
                                                                            MP2             −139     212 (1.84)    266
                                                                            MP3             −70      230 (1.57)    279
                                                                            MP4(DQ)         −20      252 (1.50)    294
                                                                            MP4(SDQ)          37     261 (1.51)    275
                                                                            MP4(SDTQ)       −56      217 (1.54)    254
                                                                            CCSD              42     197 (1.50)    168
                                                                            CCSD(T)           4      238 (1.50)    222


                                                                                                             HF
                          hyperpolarizability per Si atom (in 100 au)




                                                                        240.0                                MP2
                                                                                                             MP3
                                                                                                             MP4(SDQ)
                                                                        220.0                                CCSD
                                                                                                             CCSD(T)
                                   Average static second




                                                                        200.0


                                                                        180.0


                                                                        160.0


                                                                        140.0


                                                                        120.0
                                                                                2    3       4       5       6       7      8   9
                                                                                             Number of Si atoms

Fig. 3. Evolution of the static average second hyperpolarizability per Si atom as a function of the cluster size calculated at
different levels of approximation using the 6-311+G* basis set.

  The β// and βHRS values of Si3 and Si5 are listed in Table 2. β// , which reduces here to 3/5 µ z βz =
3/5 µz [βxxz + βyyz + βzzz ], presents large electron correlation effects. This fact combined with the
small β// amplitude would probably deserve further investigations on the basis set effects. The large
variations in β// upon inclusion of electron correlation is mostly explained by the small β // value, which
results from compensations between the β iiz tensor components. On the other hand, the β HRS values
are larger and display less electron correlation effects. It is however interesting to point out that the
fourth-order triples contribution increases the CCSD values whereas it decreases the MP4(SDQ) results.
Moreover, the Si3 and Si5 clusters present very similar βHRS and DR values. It is interesting to point
out that the Si3 DR value coincides with the DR value of a system with C 3 symmetry axis whereas the
valence Si-Si-Si angle amounts to 83.2 ◦ , quite far from the C3 120◦ angle.
       B. Champagne et al. / Ab initio investigation on the nonlinear optical properties of silicon clusters Sin (n = 3–8)   303

4. Conclusions and outlook

  First and second hyperpolarizabilities of small silicon clusters have been calculated using conventional
ab initio methods systematically increasing predictive capability. Besides Si 5 , all clusters display the
same behavior as a function of including electron correlation, i.e. the HF γ // values are the smallest, the
MP2 γ// values the largest and less than 10% larger than the reference CCSD(T) values. Contrary to the
polarizability per Si atom, which decreases with the cluster size until reaching the bulk limit, the average
second hyperpolarizability per Si atom presents a sawtooth behavior with maxima in γ // associated with
even numbers of Si and minima with odd numbers of Si atoms.
  Future investigations will tackle the importance of the vibrational contribution and frequency-
dispersion effects while they will address the reliability of density functional theory schemes in view of
applications to larger systems.


Acknowledgements

  B.C. thanks the Belgian National Fund for Scientific Research for his Research Director position. Part
of the calculations have been performed on the Interuniversity Scientific Computing Facility (ISCF),
                       e
installed at the Facult´ s Universitaires Notre-Dame de la Paix (Namur, Belgium), for which the authors
gratefully acknowledge the financial support of the FNRS-FRFC and the “Loterie Nationale” for the
convention n◦ 2.4578.02, and of the FUNDP. D.B. also acknowledges the Centre Informatique National
                          e
de l’Enseignement Sup´ rieur (CINES) for support.


References
 [1]   E.L. Muetterties, T.N. Rhodin, E. Band, C.F. Brucker and W.R. Pretzer, Chem Rev 79 (1979), 91.
 [2]   J. Koutecky and P. Fantucci, Chem Rev 86 (1986), 539.
 [3]   J. Sauer, Chem Rev 89 (1989), 199.
 [4]   J.A. Alonso, Chem Rev 100 (2000), 637.
 [5]   Atomic Clusters and Nanoparticles, NATO Advanced Study Institute, les Houches session LXXIII, les Houches, (2000),
       C. Guet, P. Hobza, F. Spiegelman and F. David, eds, EDP Sciences and Springer Verlag, Berlin, 2001.
 [6]   K. Raghavachari and V. Logovinsky, Phys Rev Lett 55 (1985), 2853.
 [7]   P.S. Bagus, C.J. Nelin and C.W. Bauschlicher, J Surf Sci 156 (1985), 615.
 [8]   K. Raghavachari, J Chem Phys 84 (1986), 5672.
 [9]   T.T. Rantala, M.I. Stockman, D.A. Jelski and T.F. George, J Chem Phys 93 (1990), 7427.
[10]                           o
       R.O. Jones, G. Gantef¨ r, S. Hunsicker and P. Pieperhoff, J Chem Phys 103 (1995), 9549.
[11]   I.G. Kaplan, R. Santamaria and O. Novaro, Int J Quant Chem 55 (1995), 237.
[12]   V. Bonacic-Koutecky, J. Pittner, C. Fruchs, P. Fantucci, M.F. Guest and J. Koutecky, J Chem Phys 104 (1996), 1427.
[13]   J.A. Becker, Angew Chem Int Ed 36 (1997), 1390.
[14]       ¨ u
       S. Og¨ t and J.R. Chelikowsky, Phys Rev B 55 (1997), 4914.
[15]                   ¨ u
       I. Vasiliev, S. Og¨ t and J.R. Chelikowsky, Phys Rev Lett 78 (1997), 4805.
[16]                                              e
       E.C. Honea, A. Ogura, D.R. Peale, C. F´ lix, C.A. Murray, K. Raghavachari, W.O. Sprenger, M.F. Jarrold and W.L.
       Brown, J Chem Phys 110 (1999), 12161.
[17]   P. Fuentealba, Phys Rev A 58 (1998), 4232.
[18]        e e
       D. B´ gu´ and C. Pouchan, J Comput Chem 22 (2001), 230.
[19]                       e e
       G. Maroulis, D. B´ gu´ and C. Pouchan, J Chem Phys 119 (2003), 794.
[20]         a
       H. H¨ kkinen, M. Moseler, O. Kotsko, N. Morgner, M. Astruc Foffmann and B.V. Issendorff, Phys Rev Lett 93 (2004),
       093401.
[21]                      e e
       C. Pouchan, D. B´ gu´ and D.Y. Zhang, J Chem Phys 121 (2004), 4628.
[22]   K.A. Jackson, M. Yang, I. Chaudhuri and Th. Frauenheim, Phys Rev A 71 (2005), 033205.
[23]   M. Yang and K.A. Jackson, J Chem Phys 122 (2005), 184317.
304    B. Champagne et al. / Ab initio investigation on the nonlinear optical properties of silicon clusters Sin (n = 3–8)

[24] M.B. Knickelbein, Phys Rev B 71 (2005), 184442.
[25] K. Fink, Phys Chem Chem Phys 8 (2006), 1482.
[26] P. Karamanis, D. Zhang and C. Pouchan, Chem Phys Lett 331 (2006), 417.
[27] P.P. Korambath and S.P. Karna, J Phys Chem A 104 (2000), 4801.
[28] A. Banerjee and M.K. Harbola, J Chem Phys 113 (2000), 5614.
[29] D.M. Bishop and F.L. Gu, Chem Phys Lett 317 (2000), 322.
[30] B. Jansik, B. Schimmelpfennig, P. Norman, Y. Mochizuki, Y. Luo and H. Ågren, J Phys Chem A 106 (2002), 395.
[31] J.L. Wang, M.L. Yang, G.H. Wang and J.J. Zhao, Chem Phys Lett 367 (2003), 448.
[32] Y.Z. Lan, W.D. Cheng, D.S. Wu, J. Shen, S.P. Huang, H. Zhang, Y.J. Gong and F.F. Li, J Chem Phys 124 (2006), 094302.
[33] A.C. Pineda and S.P. Karna, Chem Phys Lett 429 (2006), 169.
[34] B. Champagne and D.M. Bishop, Adv Chem Phys 126 (2003), 41.
[35] R. Bersohn, Y.H. Pao and H.L. Frisch, J Chem Phys 45 (1966), 3184.
[36] P.J. Davis and P. Rabinowitz, in: Numerical Integration, Blaisdell Publishing Company, London, 1967, p. 166.
[37] Gaussian 03, Revision C.02, M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A.
     Montgomery, Jr., T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M.
     Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M.
     Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E. Knox, H.P. Hratchian, J.B. Cross, V. Bakken, C.
     Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y.
     Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain,
     O. Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford,
     J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A.
     Al-Laham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez
     and J.A. Pople, Gaussian, Inc., Wallingford CT, 2004.
[38] B. Champagne, F.A. Bulat, W. Yang, S. Bonness and B. Kirtman, J Chem Phys 125 (2006), 194114.
[39] M. Spassova, B. Champagne and B. Kirtman, Chem Phys Lett 412 (2005), 217.
[40] B.M. Bode and M.S. Gordon, J Mol Graphics and Modeling 16 (1999), 133.
Journal of Computational Methods in Sciences and Engineering 7 (2007) 305–317                                               305
IOS Press




Computational approach to the reaction
dynamics associated with the formation and
crystallization of hydrogenated silicon
clusters

Grygoriy Dolgonos, Ning Ning and Holger Vach ∗
LPICM, Ecole Polytechnique, CNRS, 91128 Palaiseau, France

Received 23 December 2006
Revised /Accepted 29 March 2007

Abstract. The present paper focuses on the application of semiempirical quantum molecular dynamics to explore: 1) the
reaction dynamics of the elementary reaction Si+ (2 P) + H2 , which is believed to play a significant role in the growth of thin
silicon films, and 2) the dynamics of hydrogenated silicon nanoparticle growth (up to a size of about 1.1 nm) making use of
predetermined parameters obtained from fluid model dynamics calculations to describe the experimentally employed silane
plasma conditions. It has been shown that PM3 gives an adequate description of the silicon-hydrogen interactions that allows us
to use it in conjunction with molecular dynamics simulation techniques to explore more in detail the dynamics of the Si+ (2 P) +
H2 reaction and even the growth and crystallization dynamics of small silicon nanoclusters. This success makes PM3 molecular
dynamics a promising candidate for future in-depth explorations of chemical reactions involving silicon and hydrogen atoms.

Keywords: Molecular dynamics, semiempirical methods, reaction mechanisms, silicon nanostructures, thin films, plasma
conditions



1. Introduction

   Semiempirical methods [1–3] are becoming increasingly important tools for the study of chemical
reactions since they are much less computationally intensive than conventional ab initio methods, giving
at the same time a sufficiently correct description of the potential energy surfaces of the interacting atoms
or ions. This fact allows one not only to determine the main thermodynamic characteristics of a given
chemical reaction, but also to explore more in detail the reaction dynamics by performing semiempirical
molecular dynamics simulations. Unlike molecular dynamics (MD) with empirical potentials, semiem-
pirical MD takes into account the quantum-chemical nature of chemical bond formation and dissociation,
ensuring its adequate description over a wide range of interatomic distances.
   There exist different integral approximations to semiempirical methods to solve the Schr odinger  ¨
equation on the basis of molecular orbital theory. Among the most popular NDDO-based [4] methods –

  ∗
      Corresponding author. E-mail: vach@leonardo.polytechnique.fr.


1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
306                G. Dolgonos et al. / Computational approach to the reaction dynamics associated

MNDO [5], AM1 [6] and PM3 [7–9] – the last one was shown to be the most accurate as concerns the
description of ground-state properties (heats of formation, bond lengths and angles, ionization potentials
and dipole moments) of organic and inorganic molecules [3,8,10]. Generally, the best result is obtained
for systems similar to those, for which a given semiempirical method has been parameterized, leading to
a remarkable accuracy in comparison to ab initio results with large basis sets [8,11]. In the following, we
will thus mainly concentrate on the results of quantum MD simulations using the PM3 Hamiltonian to
evaluate interatomic potential energies “on the fly” of each MD step. This methodology has been applied
for the systems involving silicon and hydrogen atoms to explore the reaction mechanisms associated
with the silicon nanocluster growth and will be presented more in detail below.
   The organization of this paper is as follows. In Section 1, the main features of our semiempirical
quantum MD method will be briefly introduced. Then, we will present its application to study the
reaction dynamics of H 2 with Si+ (2 P). In Section 3, the growth dynamics of small H-Si nanostructures
under realistic plasma conditions (deduced from fluid model calculations) together with the H-promoted
crystallization of amorphous silicon as revealed by quantum MD will be discussed.


2. PM3 molecular dynamics

  PM3 [7–9] is one of the most widely used semiempirical methods. It is based on the NDDO (Neglect
Diatomic Differential Overlap) approximation [4], i.e. assuming no overlap between the atomic orbitals
residing on different atoms. This approximation leads to avoiding the computationally expensive two-
center repulsion integrals that, together with the use of only valence electrons within molecular orbital
theory, makes PM3 calculations much faster than ab initio ones. The errors due to the neglecting of
those integrals are compensated by introducing empirical parameters and functions, which are fitted to
reproduce experimental or accurate theoretical reference data – such as heats of formation and geometries,
as well as ionization potentials and dipole moments. Being extensively parameterized against organic
and many inorganic molecules, PM3 performs very good for the compounds of first- and second-row
elements [12,13], which makes it the method of choice to study reactions involving silicon and hydrogen
atoms. Since we are interested in the dynamics of the elementary Si + + H2 reaction, the reliability of
the PM3 method to correctly describe reactants and products of this reaction needs to be proven by a
detailed comparison between PM3 potential energy surfaces with those of high-level ab initio methods.
  Figure 1 shows potential energy curves for the Si + –H and the H–H interactions obtained with PM3
(using MOPAC program [14]) and highly correlated ab initio coupled-cluster CCSD(T)/6-311G(2df,2p)
(as implemented in Gaussian 03 software [15]) model chemistry calculations. An inspection of this
figure reveals that PM3 performs nearly as well as CCSD(T) in terms of binding energy (well depth) and
equilibrium geometry of SiH+ also in agreement with the experimental values of 1.4990 Å and 3.20 ±
0.08 eV for the equilibrium bond length and binding energy, respectively [16]. However, the shape of
the potential energy curves is slightly different, especially, in its repulsion part, leading to errors, which
may exceed 1 eV at short interatomic distances. But, since there are not that many atoms, which can
approach each other closer than 1.3 Å under realistic plasma conditions, one can clearly neglect this
discrepancy.
  It should be noted that this difference in potential energy surface (PES) profiles obtained with PM3
and CCSD(T) originates from the PM3 parameterization procedure, which is performed mainly against
available experimental data for the standard neutral tetra coordinated silicon compounds. For the
unsaturated case of SiH + (see SiH+ profiles in Fig. 1), the PM3 method yields a more shallow PES than
CCSD(T) leading to a lower PM3 value for the Si + –H stretching frequency (1901 cm −1 vs. 2130 cm −1
                   G. Dolgonos et al. / Computational approach to the reaction dynamics associated                 307


                                  1.5
                                                                                          CCSD(T)/6-311G(2df,2p)
                                  1.0                                                     PM3
                                  0.5

                                  0.0

                                  -0.5
                      V(r) [eV]



                                  -1.0

                                  -1.5

                                  -2.0

                                  -2.5

                                  -3.0

                                  -3.5
                                               1.0         1.5     2.0          2.5        3.0     3.5     4.0
                                                                            +
                                                                         r(Si -H) [A]

                                  1


                                  0


                                  -1
                      V(r) [eV]




                                  -2

                                                                                        CCSD(T)/6-311G(2df,2p)
                                  -3                                                    PM3

                                  -4


                                  -5


                                         0.5         1.0     1.5    2.0          2.5       3.0    3.5    4.0

                                                                         r(H-H) [A]

Fig. 1. Comparison between PM3 (using spin-unrestricted Hartree-Fock method) and CCSD(T)/6-311G(2df,2p) potential
energy curves for the Si+ –H (top) and H–H (bottom) interactions.

for PM3 and CCSD(T)/6-311G(2df,2p) models, respectively, compared to the experimental value of
2157 cm−1 [16]). In general, force constants and vibrational frequencies obtained with semiempirical
methods unfortunately suffer from some quantitative errors which cannot simply be improved through
scaling factors in a systematic way [17]. As mentioned before, however, this problem does not play a
significant role in our present MD simulations due to a negligible number of atoms interacting at shorter
distances than 1.3 Å.
  In the case of the H–H interaction, PM3 overestimates the binding energy of the H 2 molecule by
about 0.4 eV and gives an equilibrium bond length that is 0.043 Å shorter than the corresponding
CCSD(T) value. On the other hand, at larger interatomic distances (from 1.1 Å to 2.5 Å), PM3 tends to
underestimate the H–H interaction with the largest error of about 0.5 eV between 1.3 and 1.5 Å. As a
rough approximation, one can assume that these two errors partially compensate each other during our
308               G. Dolgonos et al. / Computational approach to the reaction dynamics associated

MD simulations as the fraction of atoms near the equilibrium distance is comparable to that between 1.1
and 2.5 Å.
  To further validate the applicability of the PM3 method for our purposes, the relative energies and
geometries for the intermediates and products of the Si + + H2 reaction have also been determined and it
turned out that PM3 performs even better than sophisticated ab initio methods in some cases [18].
  Employing the PM3 interaction potential, one can further perform constant-energy molecular dynamics
simulations by solving Newton’s equations of motion using discrete integration over a large number of
finite time steps. For this purpose, a fifth-order Gear algorithm has been used [19]. The time step for
the numerical integration was chosen to be 0.001 fs, and, to ensure the appropriate precision of the
calculated PM3 energy and forces at each time step, the self-consistent-field convergence criterion of
10−9 kcal/mol has been used. Using these criteria, the total energy of each trajectory conserves within
1%, independently of the impact energy during the reaction. Finally, we would like to mention that
the whole PM3 molecular dynamics procedure has been implemented in the Venus-Mopac computer
program [20].


3. Reaction dynamics of H 2 with Si+ [18,21,22]

   The unraveling of the nature of interactions between silicon and hydrogen atoms is of considerable in-
terest since the latter have been shown to strongly influence the structure and properties of semiconductor
materials. For example, hydrogen atoms are known [23] to cause light-induced metastable changes in
the properties of hydrogenated amorphous silicon (Staebler-Wronski effect [24]), leading to significant
changes in the optical and electronic properties of silicon thin films, which are used in solar cells and
displays [25]. The simplest case of these interactions can be given by the elementary reaction of molec-
ular hydrogen H2 with the silicon cation Si+ (2 P), which can also serve as a model for the experimental
study of the ion-molecule reaction dynamics [26]. This reaction can also play a key role in the growth
of silicon thin films under Plasma Enhanced Chemical Vapor Deposition (PECVD) conditions [27] as
the final film structure depends on the detailed interaction between hydrogen atoms of the plasma and
the solid silicon matrix. In addition, this reaction is of astrophysical importance since SiH molecules
were identified in the solar atmosphere [28] and SiH + cations may also exist in the interstellar clouds or
circumstellar shells [29,30].

3.1. Direct and indirect reactive scattering processes

   A detailed knowledge of elementary reaction processes is essential to understand the mechanisms of
silicon cluster formation and growth. For the above-mentioned reaction, one can assume the following
mechanisms:
      Si+ + H2 → SiH+ + H                                                                              (1)

      Si+ + H2 → Si+ + H + H                                                                           (2)

      Si+ + H2 → [SiH2 ]+ → Si+ + H2 ,                                                                 (3)
i.e., this reaction can proceed through direct or through indirect complex-mediated reaction paths. An
analysis of the reactive cross sections for all of these paths helps us to gain a deeper insight onto the
                      G. Dolgonos et al. / Computational approach to the reaction dynamics associated                       309

                                                                         +               +
                                                               2        Si + H2         Si H + H        (a)

                                                              1.5




                                         cm ]
                                        2
                                                               1




                                        -16
                                                              0.5



                                         Cross Section [ 10
                                                                                        Experiment
                                                               0                        MD Simulation

                                                               2         +
                                                                        Si + H2
                                                                                         +
                                                                                        Si + H + H

                                                              1.5
                                                                             MD Simulation

                                                               1

                                                              0.5

                                                               0                                        (b)

                                                                    0        2      4        6     8          10
                                                                         Impact Energy [ eV, CM ]

Fig. 2. Calculated reaction cross-sections σ as a function of relative impact kinetic energy (a) for SiH+ production and (b) for
complete dissociation.

conditions, under which a given reaction pathway dominates. It is commonly accepted to define the
reactive cross section σ as a function of both the maximum impact parameter b max (which is equal to 1.5
Å for the present case) and the probability of product formation P [31]:

     σ = πb2 P
           max                                                                                                              (4)

P can be determined from our MD simulations simply as a fraction of the number of trajectories leading
to a given reaction over the total number of simulated trajectories.
   The cross sections of the reactions (1) and (2) for H 2 molecules in the vibrational v = 0 and rotational
J = 1 states (corresponding to room temperature conditions) as a function of relative impact kinetic
energy of the reactants are depicted in Fig. 2. It can easily be seen that the curve in Fig. 2, representing
SiH+ production, reaches its maximum around 4.0–4.5 eV. However, at higher impact energies, the
complete dissociation channel (reaction (2)) starts to dominate, leading to the shown decrease in SiH +
formation. These simulation results are in perfect agreement with the experimental observations of
Elkind and Armentrout [26], who have found a decrease of the cross-section for SiH + at the impact
energies above 4.5 eV. Moreover, their absolute σ values correspond very closely to ours even without
any renormalization.
   It should be mentioned that SiH + formation is very unfavorable – its binding energy is relatively small
                                    2
(around 0.7 eV [18]) and for the almost entire range of impact energies it cannot redistribute the excess
energy to its degrees of freedom without causing dissociation. In addition, SiH 2 by itself is an extremely
reactive species, which has a high propensity for a wide range of chemical reactions [32].
   The probability P of the formation of the reaction products due to direct processes is shown in
Fig. 3. It can be inferred from these results that the probability of finding products on account of direct
processes for the reactions (1) and (2) becomes important at impact energies above 5.5 eV, and reaches
310                  G. Dolgonos et al. / Computational approach to the reaction dynamics associated


                                                                1




                                   P = Ndir / (Ndir + Ncom )
                                                               0.8


                                                               0.6


                                                               0.4

                                                                                                 +
                                                                                                Si + H + H
                                                               0.2                                  +
                                                                                                SiH + H
                                                                                                  +
                                                                                                Si + H2



                                                                0
                                                                     0   2      4       6       8            10
                                                                         Impact Energy [ eV , CM ]

 Fig. 3. Probability P that a given product is formed in a direct process versus one that passes by an intermediate complex.

its maximum value at ∼ 7.5 eV. It can also be seen that for the complete dissociation (reaction (2)), a
small number of trajectories (about 5%) led to the formation of a three-atom complex even at impact
energies between 8.0 and 10.0 eV. These findings agree with a general idea that for the above-mentioned
reactions, the hydrogen molecule needs to dissociate first. Since the binding energy of the H 2 molecule
is 4.748 eV [33], the highly mobile hydrogen atoms can further form a transition complex with Si + at
energies, which exceed 4.748 eV. This effect is conspicuous in the case of SiH + formation (cf. Fig. 3) as
the corresponding curve changes its curvature at the impact energy value close to that of the H 2 binding
energy. At smaller values, the formation of SiH + is still possible and proceeds through the intermediate,
loosely bound complex without breaking the H–H bond of the H 2 molecule, but the probability of this
process is relatively low. There is also no evidence for the products of the complete dissociation reaction
(2) at impact energies below the H 2 binding energy value.

3.2. Role of the reactant excitation for the reaction dynamics

   It has been shown [26] that the experimentally observed cross section of SiH + at ambient conditions
reaches its maximum of 1.63 Å 2 at 4.0 eV. In order to improve the reactivity of the Si + + H2 system,
one can excite hydrogen molecules by thermal heating or upon laser irradiation. Under realistic plasma
conditions, the maximum temperature of H 2 gas in the plasma can rise up to 1000 K [34]. This
corresponds to the most probable vibrational and rotational quantum numbers of H 2 molecules of v = 0
and J = 2, respectively. The average rotational quantum number lies around J = 3 at this temperature.
Therefore, the (v = 0, J = 3) state has been chosen for the further exploration of the reaction dynamics
under such conditions.
   Figure 4 shows the reactive cross section dependence on the relative impact energy of the reactants for
the reactions (1) and (2). The maximum value of 2.2 Å 2 at 4.0 eV for the SiH+ cross section has been
found, which corresponds to an increase of only about 25% in comparison to the value reported for the
room temperature case. Therefore, thermal heating of the reactants from 300 K to 1000 K leads to only
marginal changes in the SiH + formation efficiency.
   Alternatively, one can also selectively excite the internal degrees of freedom of the reactants in order
to reach a given quantum state of the H 2 molecules to increase the reactive cross section of SiH + . The
difference between the minimum and maximum values of the SiH + cross section at room temperature
is at least 1.8 eV (cf. Fig. 2). This difference corresponds to various combinations of vibrational and
                      G. Dolgonos et al. / Computational approach to the reaction dynamics associated                      311

                                                                      2.5
                                                                            (a)
                                                                       2

                                                                      1.5




                                          Cross section σ [ 10 cm ]
                                          2
                                                                       1



                                          -16
                                                                      0.5

                                                                        0
                                                                      2.5
                                                                            (b)
                                                                       2

                                                                      1.5

                                                                       1

                                                                      0.5

                                                                       0
                                                                        0         2     4     6     8    10
                                                                                  Impact energy [ eV ]

Fig. 4. Calculated reaction cross sections σ as a function of relative impact kinetic energy for a thermal excitation of the H2
reactants to 1000 K corresponding to H2 (v = 0, J = 3): (a) for SiH+ production and (b) for complete dissociation. The
continuous lines only serve as a guide for the eye.

rotational quantum numbers. In the following, two distinct cases with (v = 0, J = 14) and (v = 3, J =
1) have been chosen for the hydrogen molecules. The corresponding cross sections for these extreme
cases are depicted in Fig. 5. It clearly shows for reaction (1) that the maximum value is now 4.8 Å 2
reached at 2.5 eV, i.e., it is almost three times higher than the experimental maximum value at room
temperature [26]. In addition, the impact energy necessary for the maximum product cross section for
reaction (1) is shifted toward lower energies by ∼ 1.5 eV in comparison to the room temperature case
with v = 0, J = 1 state of H2 [18]. This value is comparable to the energy difference upon excitation of
H2 molecules from (v = 0, J = 1) to (v = 3, J = 1). Consequently, the cross section for the complete
dissociation via reaction (2) starts to increase at about 3 eV (Fig. 5b), which is ∼ 1.5 eV lower than in
the case of dissociation at room temperature (Fig. 2b). This difference is slightly lower than the applied
laser excitation value of 1.8 eV due to the initial internal energy difference of the H 2 molecules at these
conditions. Hence, the behavior of reactants upon laser excitation of the internal degrees of freedom
strongly influences the reaction dynamics.
   An analysis of the [SiH+ ] complex lifetimes helps to identify regions, where direct and indirect reactive
                           2
scattering processes occur. The dependence of the averaged lifetime τ of [SiH + ], which leads to SiH+
                                                                                   2
formation versus kinetic impact energy is presented in Fig. 6. This figure clearly demonstrates that the
SiH+ formation becomes direct when the impact energy reaches at least 6 eV. This finding can also be
supported by a detailed analysis of the vibrational, rotational and translational energy dependencies on
the kinetic impact energies for all reaction paths [21]. Being plotted on a logarithmic scale, the slope
resulting from the data of Fig. 6 indicates a barrier height of 2.2 eV that needs to be overcome in order
to form the SiH+ product. This result is close to the situation at room temperature (cf. Fig. 3), where
the probability of SiH+ formation via direct processes becomes nonzero starting from impact energies
312                    G. Dolgonos et al. / Computational approach to the reaction dynamics associated

                                                                                    6
                                                                                            (a)
                                                                                    5                         (J=14,v=0)
                                                                                                              (J=1,v=3)
                                                                                    4
                                                                                    3




                                                       Cross section (10 -16cm 2)
                                                                                    2
                                                                                    1
                                                                                    0
                                                                                    5       (b)
                                                                                    4
                                                                                    3
                                                                                    2
                                                                                                               (J=14,v=0)
                                                                                    1                          (J=1,v=3)
                                                                                    0
                                                                                        0         2    4     6     8        10
                                                                                                  Impact energy (eV)

Fig. 5. Calculated reaction cross sections σ as a function of relative impact kinetic energy for an initial laser excitation of the
H2 reactant to (v = 3, J = 1) and (v = 0, J = 14), both corresponding to an internal energy Eint of 1.8 eV (see text): (a) for
SiH+ production, and (b) for complete dissociation.


                                             300




                                             200
                                  τ [ fs ]




                                             100




                                               0
                                                   0                                               2              4              6
                                                                                              Impact energy [ eV ]

Fig. 6. Calculated [SiH2 ]+ complex lifetime τ as a function of kinetic impact energy E0 for complexes leading to SiH+
formation. The continuous line corresponds to an exponential fit.

of 1.5–2.0 eV [18].
  It should be mentioned that both the shift to lower impact energies as well as the absolute maximum
value of a cross section depend mainly on the amount of initial internal excitation rather than its origin – as
rotational and vibrational excitations enhance the reactivity in a similar manner according to Fig. 5 [21].
  To summarize, semiempirical molecular dynamics was successfully applied to reproduce the reactive
cross sections for the Si + + H2 reaction at room temperature. In order to enhance the SiH + formation,
                   G. Dolgonos et al. / Computational approach to the reaction dynamics associated                  313




         Fig. 7. Typical geometrical structures obtained from a cluster growth under realistic plasma conditions.

the H2 molecules were proposed to be excited by laser irradiation rather than thermally heated up. The
scattering dynamics of the direct and indirect mechanisms has also been explored and the corresponding
regions, where a given mechanism dominates, were identified. Taking into account the relatively low
computational cost of PM3 molecular dynamics simulations, this method may also be extended to larger
systems in order to shed light onto the analogous chemical reaction mechanisms, which take place during
chemical vapor deposition of silicon.


4. Silicon nanocluster growth under realistic plasma reactor conditions [35,36]

   It is commonly admitted that hydrogenated silicon nanoparticles play a crucial role in the formation
of silicon thin films under PECVD conditions [27]. Plasma-generated nanocrystals deposited on an
amorphous matrix (called “polymorphous silicon”) have been demonstrated [37] to improve structural
and transport properties of thin films compared to those of hydrogenated amorphous silicon. Therefore,
polymorphous silicon becomes an attractive candidate for photovoltaic applications. The formation of
nanocrystals was shown [38] to take place in the gas phase rather than on the substrate itself. The
detailed mechanisms of nanocrystal formation under conditions of a plasma reactor, however, are not
well understood yet. To get a deeper insight into them, one needs a relatively fast and reliable tool to
describe the dynamics of reactants and products on a relatively long time scale. Since pure ab initio MD
simulations for those relatively large systems are possible only for very short simulation times and will
thus be too expensive in terms of computer power, semiempirical MD simulations present an efficient
alternative to explore the growth of silicon nanostructures.

4.1. Growth dynamics of silicon nanoparticles

  Since the growth of silicon nanoparticles occurs under experimental plasma conditions, one needs
to know the main silane/hydrogen plasma characteristics. Such information can be obtained from
fluid model calculations [39], which provide the relative densities of radicals in the plasma, their
temperatures and collision interval times, which are used as input data for our semiempirical MD
314                     G. Dolgonos et al. / Computational approach to the reaction dynamics associated




      Fig. 8. Typical geometrical structures obtained for the growth of Sin Hm clusters considering impact energies of 2 eV.

simulations. According to the time-averaged density profile [36], the most abundant radicals in the
plasma, created in a mixture of 2% SiH 4 in a hydrogen gas, are SiH 3 and H. Therefore, they mainly
contribute to the initial nucleation and growth of silicon nanostructures. The plasma can be characterized
by a room temperature distribution for all species [39].
   The initial stage of a silicon nanoparticle nucleation results from the interaction between a SiH 3
radical and a silane molecule. Thereafter, the cluster growth is ensured by the subsequent capture
of SiH4 molecules onto the growing cluster fragment leading to different Si n Hm structures. Typical
structures obtained during this process using PM3 molecular dynamics are shown in Fig. 7. During the
growth, some of the hydrogen atoms can desorb from the cluster. In addition, H atoms can migrate along
the surface of the growing cluster by saturating/unsaturating the dangling bonds.
   The dynamics of the cluster growth can be further investigated by using higher impact energies of
SiH4 molecules compared to the room temperature value. At an impact energy of 2.0 eV, it turns
out that the cluster starts ‘to lose’ hydrogen atoms leading to less saturated, but more compact Si n Hm
structures – as those shown in Fig. 8. These structures can be envisaged as building blocks of the growing
nanocrystalline phase. For the clusters with n = 6 − 8, the obtained structures are quite similar to the
global-minimum-energy structures of pure Sin ones calculated with ab initio methods [40–42].

4.2. Hydrogen-promoted crystallization of amorphous silicon structures

   It should be mentioned that during the MD simulations the growing Si n Hm cluster has enough time
to cool down to room temperature between subsequent SiH 4 impingements according to the current
experimental plasma conditions. This growth dynamics results in amorphous, hydrogen-rich silicon
structures as that shown in Fig. 9a. If however hydrogen atoms are present in a sufficient amount
during the cluster growth, structural relaxation and the formation of more compact and symmetric silicon
structures become feasible – similar to what has been found at the elevated impact energies of the SiH 4
molecules described above. Such structures, presented in Figs 9b and 9c, are formed as a result of low and
high atomic hydrogen flux, respectively. It has been found that in the case of relatively small hydrogenated
silicon cluster Sin Hm (n < 12), the hydrogen flux causes structural changes between local minimum
and metastable Sin Hm structures during MD simulations whereas for larger structures no more structural
changes are further observed once a local-minimum structure has been formed. In addition, silicon
atoms may self-organize around a given central silicon atom leading to silicon nanowire-like structures
(cf. Fig. 9d). Overall, the above-mentioned findings allow one to control the resulting structure of the
growing cluster by “programming” the impact flux of atomic hydrogen in the experimental conditions.
   To summarize, semiempirical MD simulations in conjunction with fluid model dynamics reveal the
reaction dynamics associated with the growth of hydrogenated silicon nanoclusters under realistic ex-
perimental plasma conditions. The significant role of hydrogen atoms has undoubtedly been shown
                      G. Dolgonos et al. / Computational approach to the reaction dynamics associated                         315




Fig. 9. Typical structures of hydrogenated silicon nanoparticles created in a plasma reactor. (a) An amorphous structure
resulting from a growth mechanism in a pure silane plasma at room temperature; (b) typical example for a low atomic hydrogen
flux giving rise to crystalline structures that are rich in hydrogen; (c) typical example for a high atomic hydrogen flux yielding
crystalline structures relatively poor in hydrogen that are similar to those predicted for pure silicon clusters; (d) side and top
view of a typical tubelike structure obtained with an intermediate atomic hydrogen flux.

to influence the formation of polymorphous silicon materials, which is of utmost importance in the
fabrication of photovoltaic devices based on thin film technology.


5. Conclusions

   Computational chemistry provides substantial tools for the in-depth understanding of chemical process-
es. Among the different quantum chemistry approaches, semiempirical methods are becoming popular
to determine the main chemical reaction characteristics mainly due to their relatively low computational
cost and acceptable accuracy. The hybrid semiempirical molecular dynamics gives the necessary step
forward to explore the reaction dynamics. We have demonstrated that PM3 molecular dynamics with
emphasis on the reactions involving silicon and hydrogen atoms gives useful insights into the reaction
dynamics of an elementary model reaction Si + + H2 : the main reaction paths have been identified and the
316                   G. Dolgonos et al. / Computational approach to the reaction dynamics associated

reactive cross section of SiH + formation agrees perfectly well with the experimental one. By examining
the different vibrational and rotational states of the interacting hydrogen molecules, it becomes possible
to predict that laser excitation of the reactants may lead to a significant enhancement in the SiH + for-
mation rather than their thermal heating. Moreover, the growth dynamics of silicon nanoparticles under
realistic experimental plasma conditions has been investigated using the same methodology, which is of
practical importance to control the structure of silicon thin films. It has been shown that hydrogen atoms
play a crucial role for the final structure of silicon clusters, which are responsible for the formation of
the nanocrystalline phase of polymorphous silicon. Therefore, we believe that PM3 molecular dynamics
simulations may serve as an efficient tool to further understand and control chemical reactions involving
silicon and hydrogen atoms, especially those, which take place in the gas phase and on the silicon
surfaces.


Acknowledgments

  G.D. thanks EADS for the financial support. N.N. is the recipient of a PhD studentship from the
        e                            e
“Minist` re de l’Enseignement Sup´ rieur et de la Recherche”. This work has been partially financed
                      e
by the “French Minist` re de Recherche” in the framework of the program “ACI Nanostructures” under
                                                       ´
contract No. N52-01. We also thank the “Institut du D eveloppement et des Ressources en Informatique
Scientifique” (IDRIS), who supported different aspects of the computational works discussed herein.
The authors are also grateful to Dr. Q. Timerghazin for his generous help and discussion.


References

 [1]   T. Clark, A Handbook of Computational Chemistry, Wiley, New York, 1985.
 [2]   J.N. Murrell and A.J. Harget, Semiempirical Self-Consistent Field Molecular Orbital Theory of Molecules, Wiley, New
       York, 1972.
 [3]   W. Thiel, Adv Chem Phys 93 (1996), 703.
 [4]   J.A. Pople, D.P. Santry and G.A. Segal, J Chem Phys 43 (1965), S129.
 [5]   M.J.S. Dewar and W. Thiel, J Am Chem Soc 99 (1977), 4899.
 [6]   M.J.S. Dewar, E.G. Zoebisch, E.F. Healy and J.J.P. Stewart, J Am Chem Soc 107 (1985), 3902.
 [7]   J.J.P. Stewart, J Comput Chem 10 (1989), 209.
 [8]   J.J.P. Stewart, J Comput Chem 10 (1989), 221.
 [9]   J.J.P. Stewart, J Comput-Aided Mol Des 4 (1990), 1.
[10]   W. Thiel, Thermochemistry from semiempirical molecular orbital theory, in" Computational Thermochemistry,
       (Vol. 677), K.K. Irikura and D.J. Frurip, eds, American Chemical Society, Washington, 1998, p. 142.
[11]   M.J.S. Dewar and D.M. Storch, J Am Chem Soc 107 (1985), 3898.
[12]   W. Thiel and A.A. Voityuk, J Phys Chem 100 (1996), 616.
[13]   W. Thiel and A.A. Voityuk, Theor Chim Acta 81 (1992), 391.
[14]   J.J.P. Stewart, MOPAC 7.0, Frank J. Seiler Research Laboratory, US Air Force Academy: Colorado Springs, 1993.
[15]   M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery, Jr., T. Vreven,
       K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N.
       Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima,
       Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E. Knox, H.P. Hratchian, J.B. Cross, C. Adamo, J. Jaramillo, R.
       Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y. Ayala, K. Morokuma,
       G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K.
       Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford, J. Cioslowski, B.B.
       Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A.
       Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez and J.A. Pople, Gaussian
       03, Revision B.05, Gaussian Inc.: Pittsburgh PA, 2003.
[16]   A.E. Douglas and B.L. Lutz, Can J Phys 48 (1970), 247.
                      G. Dolgonos et al. / Computational approach to the reaction dynamics associated                   317

[17]   A.P. Scott and L. Radom, J Phys Chem 100 (1996), 16502.
[18]                     a
       H. Vach, N. Chaˆ bane and G.H. Peslherbe, Chem Phys Lett 352 (2002), 127.
[19]   W.H. Press, W.T. Vetterlin, S.A. Teukolsky and B.P. Flannery, Numerical Recipes in Fortran: The Art of Scientific
       Computing, Cambridge University Press, Cambridge, UK, 1992.
[20]   W.L. Hase, R.J. Duchovic, X. Hu, A. Komornicki, K.F. Lim, D. Lu, G.H. Peslherbe, K.N. Swamy, S.R. Vande Linde,
       A. Varanda, W. Haobin and R.J. Wolf, VENUS96: General Chemical Dynamics Computer Program, Department of
       Chemistry, Wayne State University, Detroit, MI, 1996.
[21]           a
       N. Chaˆ bane, H. Vach and P. Roca i Cabarrocas, J Phys Chem A 108 (2004), 1818.
[22]           a
       N. Chaˆ bane, H. Vach and G.H. Peslherbe, J Non-Cryst Solids 299 (2002), 42.
[23]   T. Kruger and A.F. Sax, Physica B 353 (2004), 263.
[24]   D.L. Staebler and C.R. Wronski, Appl Phys Lett 31 (1977), 292.
[25]   A. Shah, P. Torres, R. Tscharner, N. Wyrsch and H. Keppner, Science 285 (1999), 692.
[26]   J.L. Elkind and P.B. Armentrout, J Phys Chem 88 (1984), 5454.
[27]   U. Kroll, J. Meier, A. Shah, S. Mikhailov and J. Weber, J Appl Phys 80 (1996), 4971.
[28]   H.D. Babcock, Astrophys J 102 (1945), 154.
[29]   T.A. Carlson, J. Copley, N. Duric, N. Elander, P. Erman, M. Larsson and M. Lyyra, Astron Astrophys 83 (1980), 238.
[30]   A. Kalemos, A. Mavridis and A. Metropoulos, J Chem Phys 116 (2002), 6529.
[31]   J.I. Steinfeld, J.S. Francisco and W.L. Hase, Chemical Kinetics and Dynamics, Prentice-Hall, Upper Saddle River, NJ,
       1998.
[32]   J.M. Jasinski, R. Becerra and R. Walsh, Chem Rev 95 (1995), 1203.
[33]   L. Wolniewicz, J Chem Phys 103 (1995), 1792.
[34]   V.M. Donnelly, D.L. Flamm and G. Collins, J Vac Sci Technol 21 (1982), 817.
[35]   H. Vach and Q. Brulin, Phys Rev Lett 95 (2005), 165502.
[36]   H. Vach, Q. Brulin, N. Chaabane, T. Novikova, P.R.I. Cabarrocas, B. Kalache, K. Hassouni, S. Botti and L. Reining,
       Comp Mater Sci 35 (2006), 216.
[37]   P. Roca i Cabarrocas, A. Fontcuberta i Morral, S. Lebib and Y. Poissant, Pure Appl Chem 74 (2002), 359.
[38]           a
       N. Chaˆ bane, A.V. Kharchenko, H. Vach and P. Roca i Cabarrocas, New J Phys 5 (2003), 1.
[39]   T. Novikova, B. Kalache, P. Bulkin, K. Hassouni, W. Morscheidt and P. Roca i Cabarrocas, J Appl Phys 93 (2003), 3198.
[40]   K. Raghavachari and V. Logovinsky, Phys Rev Lett 55 (1985), 2853.
[41]   K. Raghavachari, J Chem Phys 84 (1986), 5672.
[42]   X.L. Zhu and X.C. Zeng, J Chem Phys 118 (2003), 3558.
Journal of Computational Methods in Sciences and Engineering 7 (2007) 319–335                                                319
IOS Press




Theoretical study of pure (Sin) and doped
silicon (AlSin−1 and PSin−1) clusters (n =
2–13) using ab initio molecular orbital theory
Sandeep Nigam, S.K. Kulshreshtha and Chiranjib Majumder ∗
Chemistry Division, Bhabha Atomic Research Center, Trombay, Mumbai 400 085, India

Received 4 March 2007
Accepted 7 March 2007

Abstract. The geometric and electronic structures of Sin , Si+ , Si− , AlSin−1 and PSin−1 clusters (2
                                                                    n    n                                      n 13) has been
investigated using the ab initio molecular orbital theory under the density functional theory formalism. Relative stabilities
of these clusters have been analyzed based on their binding energies, second difference in energy (∆2 E) and fragmentation
behavior. The equilibrium geometry of the neutral and charged Sin clusters shows similar structural growth. The geometries
of the Si+ and AlSin−1 are similar to those of the Sin , but with small distortions. The ground state geometries of the AlSin−1
          n
clusters shows that the impurity Al atom prefers to substitute for the Si atom, that has the highest coordination number in the
host Sin cluster. However for Si− , n = 6, 8, 11, and 13, significant changes have been observed in the ground state geometries
                                   n
of the negatively charged clusters as compared to their neutral counterparts. In general the geometries of the P substituted
silicon clusters remain similar to that of negatively charged Sin clusters with small local distortions. In general, the average
binding energy of charged clusters is found to be higher than that of neutral Sin clusters. However, significant differences
have been observed in the electronic structure of neutral and charge cluster leading to their different stability pattern. While
for neutral clusters, the Si10 is magic, the extra stability of the Si+ cluster over the Si+ and Si+ bears evidence for the magic
                                                                      11                   10      12
behavior of the Si+ cluster, which is in excellent agreement with the recent experimental observations (ref. [29]). Similarly for
                    11
AlSin−1 clusters, which is iso-electronic with Si+ clusters show extra stability of the AlSi10 cluster suggesting the influence
                                                     n
of the electronic structures for different stabilities between neutral and charged clusters. For iso-electronic PSin−1 clusters,
it is found that although for small clusters (n < 4) substitution of P atom improves the binding energy of Sin clusters but
for larger clusters (n > 4), the effect is opposite. The fragmentation behavior of all these clusters shows that while small
clusters prefers to evaporate monomer, the larger ones dissociates into two stable clusters of smaller size. Finally, a good
agreement between experimental and our theoretical results suggests good prediction of the lowest energy isomeric structures
for all clusters calculated in the present study.



1. Introduction

  The study of atomic and molecular clusters has attracted the attention of a large number of investigators
because of their unique size dependent properties and non monotonic behaviour of their physico-chemical
characteristics. Heteroatomic clusters are composed of two different elements and it is of interest to
know the ground state geometry and location of the individual atoms in these clusters as the chemical
reactivity of metal clusters and many of their physicochemical properties such as evaporation, melting
etc. are governed by the surface atoms.

  ∗
      Corresponding author. E-mail: chimaju@barc.gov.in.


1472-7978/07/$17.00  2007 – IOS Press and the authors. All rights reserved
320        S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters

   In the recent past a large number of investigations have been performed on the impurity atom doped
Gr-IV elemental clusters. This is due to their technological relevance towards the development of
nanoelectronics, which gives an extra impetus to understand the properties of these materials with its
miniaturization. In particular, Si clusters have been studied most extensively using both theoretical and
experimental techniques [1–26]. To date, the global (potential-energy) minima of small silicon clusters
(Sin ) up to n = 11 have been well established through all-electron molecular-orbital calculations [15–
20]. Recently, Rata et al. [27] have reported an unbiased global-minimum search for Si n (n 23) using
a novel single-parent evolution algorithm. Zeng and co-worker [18] have performed a series of studies
on silicon cluster up to n = 30 using pseudo-potential as well as all electron methods. They have found
that cluster in range of n = 15 to 30 has two structural motifs, either the tricapped-trigonal-prism (TTP)
motif or Si6 /Si6 (six fold puckered hexagonal ring Si 6 plus six atom tetragonal bipyramid Si6 ) motif.
They have done the simulation studies [18] on medium size silicon cluster ranging from n = 31 to 45.
   Although significant work has been carried out for the neutral Si clusters, only a few studies are
available for charged clusters [19,23,25,28–44]. In fact, the ionic clusters are mainly detected in most
of the experimental study. Cheshnovsky et al. [11] have carried out systematic study to measure the
electron detachment energies of Sin anion using the ultraviolet photoelectron spectroscopy. The electron
affinities of Si− (n = 2–12) were reported based on these results. The electron affinities of Si 2 dimer
                n
was reported by Blondel et al. [32] and Nimlos et al. [33] using the electron spectrometry and photo-
electron spectroscopy, respectively. Arnold, Neumark and coworkers [34–40] measured photoelectron
spectroscopy for pure silicon cluster anions.
   Although photo-electron spectroscopy is a useful experimental tool to determine the electronic energy
levels of the negatively changed systems and predicting possible geometrical structures however, a
combination of theoretical calculation of electronic and geometrical structure would certainly provide
the fundamental insight to analyze the spectroscopic data more clearly. Ab initio molecular orbital
theory is commonly used to obtain direct information about the ground state geometries of atomic and
molecular clusters.
   Raghavachari and co-workers [19] have systematically studied small silicon clusters using molecular-
orbital theory taking all electrons into account In these calculations they have optimized several isomers
at the HF/6-31G(d) level, followed by a total energy calculation at MP2, MP3, and MP4 levels with the
polarized 6-31G(d) basis set. Due to large computational requirements these calculations were limited
to small size clusters. In another theoretical investigation [41–44], Curtiss et al. have reported accurate
binding energy and electron affinities based on G2 level of calculations [41]. Recently, Yang et al. have
reported the structure and electronic properties of silicon anion clusters upto n = 10 using the hybrid
and pure density functional methods [44].
   Apart from the studies on homo-atomic clusters, a large number of investigations have been performed
on the impurity atom doped Si elemental clusters recently [31,45–52] Nakajima and coworkers have
carried out the photoelectron spectroscopy primarily on binary clusters of Si nC− , Sin F− and Sin Na− [45–
47]. It is found that the incorporation of an impurity atom can lead to reorder the energy levels and
thereby alter the geometry and chemical stability of the system more precisely. One of the most important
examples of this feature is the transformation of non-compact Si clusters into fullerene like compact
structure by doping with transition metal atoms [50]. Although several reports are available on the
interaction of transition metal atom with Si clusters, similar investigations with simple metal atoms are
very few. Kishi et al. [47,49] carried out a combined experimental and theoretical study of NaSi n (n <
7), and found that the Na atom acts as an electron donor to the Si n framework and the most stable isomer
of NaSin retains the framework of corresponding Si n cluster nearly unchanged upon the adsorption of
Na.
             S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters   321

  Here we present the geometric and electronic structure of Si n , Si+ , Si− , AlSin−1 and PSin−1 clusters
                                                                     n     n
(2 n 13). The objective of the present study is two fold, the first one being to understand the influence
of a positive and negative charge on the structural and electronic properties of neutral Si clusters and
secondly, to compare its physicochemical properties to that of a neutral iso-electronic AlSi n−1 , PSin−1
cluster.


2. Computational details

   The geometry optimization and the electronic structure calculation have been carried out under the
framework of the LCAO-MO approach as implemented in the GAMESS software [53]. The geometry
optimization was carried out at the B3LYP/6-31+G(d) level. The B3LYP represents Becke’s three
parameter hybrid functional, which uses part of the Hartree-Fock exchange and Becke’s exchange
functional [54] in conjunction with the Lee – Yang – parr functional [55] for correlation. B3LYP has
evolved into a widely applicable computation technique, which require less computational effort than
convergent quantum mechanical technique such as MP2, coupled cluster theory and reasonably good
accuracy has been obtained. B3LYP has been employed earlier on silicon cluster to get accurate geometry
and energetics [18,28,44]. Since diffuse function are important for proper description of anionic cluster,
a standard split-valence with polarization and diffuse functions (6-31G+(d)) was employed as basis
set for this purpose. The 6-31G+(d) basis set has been shown to yield very adequate results for Si n
clusters [16,17]. All calculations were carried out using the spin polarization option. In order to verify
the reliability of the B3LYP/6-31G+(d) level of calculations, an initial test was carried out on Si 2 , Si+2
Si− , PSi, AlSi dimers. The inter-atomic distances for these dimers are 2.18, 2.31, 2.12, 2.00 Å and
  2
2.45 Å, respectively and the corresponding dissociation energies are 2.92, 3.30 3.70, 3.32 eV and 2.32
eV. It was found that the minimum energy structure of neutral Si 2 favors triplet spin multiplicity and other
two i.e. Si+ and AlSi favor quartet spin multiplicity as the lowest energy configuration. Si − and PSi favor
           2                                                                                 2
doublet spin multiplicity as the lowest energy configuration. The energy differences between different
possible multiplicities are listed in Table 1. Comparison of these values with the available experimental
results [56] showed a good agreement suggesting the reliability of the computational method applied in
this work.


3. Results and discussion

  In the following section we have described the geometry of homoatomic silicon clusters. Figure 1
displays the optimized geometries of lowest energy isomers of Si n , Si+ and Si− clusters. The relaxation
                                                                         n       n
energies of the cation and anion clusters are important to identify the amount of structural changes by the
addition or removal of an electron from Sin clusters. In Fig. 2 we have displayed the relaxation energy
as a function of size of the cluster. Relaxation energy has been calculated in the following manner:
  Relaxation Energy cation [∆R(+)] = VIP-AIP
  Relaxation Energy anion [∆R(−)] = VEA-AEA

3.1. N = 3

  The Si3 cluster forms open triangle with Si-Si bond length of 2.19 Å forming an angle of 83.51 ◦. The
distance between two extreme Si atoms is 2.87 Å. In contrast to this the lowest energy configuration of
other cluster Si+ , Si− trimers favor isosceles triangles. In case of Si + cation the three arms of isosceles
                3     3                                                  3
are 2.26,2.26 and 2.52 Å in length. In Si − the Si-Si-Si angle is 65.66◦.
                                           3
322          S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters

                                                       Table 1
                   Bond energy (eV) bond length (angstrom) and spin multiplicity of dimers calculated
                   at B3LYP/6-31+G(d) level of theory. The experimental (Exp.) Values has been taken
                   from Ref. 56
                          Bond energy Bond length Multiplicity Exp. bond                Exp. bond
                                                                          energy          length
                    Si2        1.90          2.07             1             −               −
                               2.90          2.17             3            3.33            2.25
                    Si+
                      2        1.90          2.16             2             −               −
                               3.30          2.31             4             −               −
                    Si−
                      2        3.70          2.12             2             −         2.21*(ref. 33)
                               2.21          2.33             4             −               −
                    AlSi       1.35          2.28             2             −               −
                               2.32          2.45             4            2.34             −
                    P-Si       3.33          2.00             2            3.73             −
                               1.38          2.23             4             −               −

3.2. N = 4

  Four atom clusters are important as it is the smallest size of a cluster which can achieve three
dimensional structures. The Si 4 forms planar rhombus with side arms of 2.33 Å and the distances
between Si atoms along the short diagonal is 2.44 Å. The Si 4 anion has D2h structure similar to neutral
where the side arms of planar rhombus are 2.34 Å and the short diagonal is 2.39 Å. The Si2-Si4-Si3
angle in neutral is 63.04 ◦ while in anion this angle is 61.23. For Si 4 cation also the symmetry (D2h ) is
similar to that of neutral cluster. The lengths of side arms increases upto 3.31 Å and the short diagonal
length increased upto 2.68 Å.

3.3. N = 5

   The lowest energy isomer of Si 5 cluster shows elongated trigonal bipyramid structure. This can other
way be viewed as four Si atoms forming a rhombus and the addition Si atom is connected to two Si atoms
placed along the diagonal leading a capped bend rhombus. The point group symmetry of this structure
is D3h . All Si-Si bonds are equal with a value of 2.33 Å same to that for Si 4 planar rhombus. After
addition of one electron also retain the capped bend rhombus structure where the length of the side arm
of rhombus has changed slightly from 2.33 to 2.36 Å. The bond between Si3-Si4 gets elongated to 3.46
Å. The removal of one electron from Si 5 cluster leads to geometrical distortions for Si+ cluster. While
                                                                                        5
the Si-Si distances in the rhombus is elongated upto 2.39 Å, the distance between the capping Si atom
and the rhombus is compressed upto 2.27 Å.

3.4. N = 6

   Several possible geometries of Si 6 clusters were considered to locate the lowest energy structure. The
edge capped trigonal bipyramid (C 2v ) and crossed rhombus (D 4h ) were found to be degenerate in energy
with a difference of 0.03 eV, former being the lowest energy isomer. Interestingly, it has been reported
earlier that B3LYP always finds C 2v structure as the global minimum while this (the edge capped trigonal
bipyramid) structure collapses to the crossed rhombus structure at the MP2 optimization [43]. Honea et
al. [12] and Zhao et al. [43] has found the crossed rhombus structure as the ground state in their MP2
calculation. For Si6 cation the relative stability order remains same as in case of neutral with an increase
in the energy difference of 0.13 eV but in case of Si 6 anion the stability order get reversed and the crossed
rhombus (compressed octahedron) structure was found to be lowest energy isomer.
             S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters   323




                       Fig. 1. Representative lowest energy structure of neutral and charged clusters.

3.5. N = 7

   The ground state structure of the Si 7 cluster is a pentagonal bipyramid (D 5h ). The addition or removal
of one electron distorts the symmetry for Si7 cluster. In neutral the two axial atoms are 2.63 Å apart while
in case of anion this separation increase upto 2.92 Å resulting in opening of geometry.

3.6. N = 8

  The lowest energy structure of Si 8 is bicapped distorted octahedron. Another structure which is lying
just above 0.23 eV higher in energy is two parallel bent rhombus. The capped pentagonal pyramidal
isomer is 0.64 ev higher in energy as compared to the bicapped octahedron. Addition of one electron
324          S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters




                                                     Fig. 1. continued.

to neutral Si8 cluster leads to changes the relative stability of the low lying isomers. Two parallel bent
rhombus structure was found to be the lowest energy isomer while the bicapped distorted octahedron
was 0.37 eV higher in energy. The ground state geometry of the Si 8 cation is similar to that of neutral
counterpart of bicapped octahedron isomer with small contractions in the Si-Si bond lengths.

3.7. N = 9

  The lowest energy structure of the Si 9 cluster shows capped tetragonal antiprism structure as shown
in Fig. 1. Another structure, tricapped trigonal prism (TTP), which has been predicted to be the lowest
energy structure using the DFT/LDA technique [21] is significantly 1.63 eV higher in energy. However,
            S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters   325

TTP is an important framework, which acts as building block for building larger size Sin clusters as
predicted by Zeng and co-workers also [18]. The Si − clusters also shows similar structure as that of the
                                                    9
neutral part with small changes in the inter-atomic distances between Si atoms. In neutral the capping
silicon atom is tetrahedrally co-ordinated while in anionic cluster the capping silicon atom is attached
to three silicon atom only as can be seen from Fig. 1, the bond Si1-Si5 of Si − stretches to 3.02 Å. Si+
                                                                              9                         9
shows geometry very similar as that of neutral.

3.8. N = 10

   The lowest energy structure of the Si 10 cluster shows tetracap trigonal prism structure. This is similar
to that what has been reported by several groups using the DFT or HF/MP2 level of theory. This structure
can otherwise be viewed as bi-capped tetragonal anti-prism. The Si − cluster also shows similar structure
                                                                      9
as that of the neutral part with small changes in the interatomic distances between Si atoms. The bonds
Si4-Si5, Si5-Si6, Si4-Si6 in anionic cluster get shrinked from 2.77 Å to 2.67 Å as compared to the
neutral counterpart. The removal of one electron i.e. for Si 10 cation cluster the relaxed geometry leads
to lowering of symmetry by distortion in the bond lengths.

3.9. N = 11

   This is an important cluster as it shows different stability behavior in the neutral and positively charged
states. Several initial geometries as obtained by capping the additional silicon atom at different site of
tricapped trigonal prism (TTP) have been taken to optimize the geometry of the Si 11 cluster. The lowest
energy structure has been shown in Fig. 1. But for Si 11 anion we have found different structure than that
of the neutral as may be seen from the Fig. 1. For the Si 11 cation, shows similar structure as that of the
neutral part with small relaxations in the inter-atomic distances among them.

3.10. N = 12

  The lowest energy structure of the Si 12 cluster is obtained by adding one more Si atom on the Si 11
cluster having the trigonal prism framework as shown in Fig. 1. Similar structure was also obtained after
addition or removal of one electron from it with small changes in the bond lengths.

3.11. N = 13

   For Si13 clusters, it has been found that Si 13 also follows the Si12 framework by capping the additional
Si at the four fold site to make the structure more symmetric (C2v ). For Si13 anion the ground state
structure of neutral totally changes to another isomer having C s symmetry. The C2v isomer was 0.34 ev
higher in energy. For Si13 cation, the geometry remains similar to that of neutral structure with small
relaxation in the interatomic distances.


4. Energetics

  The average binding energies per atom as a function of cluster size for Si n , Si+ and Si− are shown in
                                                                                   n       n
Fig. 3. The average binding energies of these clusters are calculated as
  BE(Sin ) = [E(Sin ) – n × E(Si)]/n
326         S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters


                                                                                      anion
                                              1.0                                     cation

                       Relaxation Energy-eV   0.8



                                              0.6



                                              0.4



                                              0.2



                                              0.0


                                                    2   4   6        8         10        12         14

                                                                 N

                   Fig. 2. Relaxation energy of cationic and anion silicon cluster as a function of size.

   BE(Si+ ) = [E(Si+ ) – (n − 1) × E(Si)–E(Si+ )]/n
          n          n
   BE(Si− ) = [E(Si− ) – (n − 1) × E(Si)–E(Si− )]/n
          n          n
   It is seen that the binding energy increases as the cluster size grows with small humps or dips for
specific size of clusters indicating their relative stabilities. The binding energies of these three types
of clusters are in the order of (Si− ) > (Si+ ) > (Sin ). The highest binding energy of Si − is due to the
                                   n          n                                                n
increase in the number of electrons i.e. more interactions. The higher binding energy of cation clusters is
due to higher bond strength of Si + or in other words the higher effective nuclear charge of the Si cation.
                                  2
Although in all three clusters the trend in the binding energy curve is similar, significant difference has
been observed at n = 11. Whereas in the case of Si + the binding energy of Si+ shows an upward
                                                           n                             11
trend, for both neutral Sin and anion Si− it shows a downward trend. This is in excellent agreement with
                                          n
the recent experimental observations of the mass abundance spectrum of the positively charged silicon
clusters made by Castleman and coworkers [29].
   From theoretical calculations one can search for magic clusters by calculating the second energy
difference in energy, which has been calculated as
   ∆2 E = [2E(Sin )–E(Sin+1 )–E(Sin−1 )].
   ∆2 E = [2E(Si+ )–E(Si+ )–E(Si+ )].
                  n        n+1        n−1
   ∆2 E = [2E(Si− )–E(Si− )–E(Si− )].
                  n       n+1        n−1
   From the above expressions it is clear that the clusters that have negative values of ∆ 2 E are more
stable than their nearest neighbors. We have plotted the ∆ 2 E for Sin , Si+ and Si− clusters as a function
                                                                            n         n
of cluster size as shown in Fig. 4. It is clear from this figure that in general, clusters with n = 4, 6 and 7
are relatively more stable than their nearest neighbors. This is in agreement with previous experimental
observations. However, significant differences have been observed for n = 10 and 11. Whereas for
neutrals, Si10 is more stable than Si9 and Si11 , for positively charged clusters Si + is more stable than
                                                                                        11
Si+ and Si+ clusters.
  10        12
            S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters    327

                               3.4

                               3.2

                               3.0

                               2.8
               B.E./atom(eV)



                               2.6
                                                                                                   Sin
                               2.4                                                                     -
                                                                                                   Sin
                               2.2                                                                     +
                                                                                              Sin
                               2.0
                                                                                          B3LYP-6-31+G(d)
                               1.8

                               1.6

                               1.4

                                        2           4          6           8         10          12           14

                                                                       n
                                     Fig. 3. Average binding energy of neutral and charged silicon cluster.

   Based on these results we have found two important differences in the stability of neutral and charged
clusters. For example, while in case of n = 6 the neutral are more stable, the anion does not show similar
behavior. On the other hand we note that while Si − shows higher stability than its neighbor clusters
                                                        12
which is different in cases of neutral clusters. In case of anion cluster with n = 4, 5, 7 10 and 13 were
found to be more stable than their neighbors. For n = 6 while neutral and cation are magic but anion is
not. It is interesting to note that Si− is a magic cluster while Si+ and Si13 do not show magic behavior.
                                      13                            13
   To further study the relative stability of the silicon cluster we have calculated the vertical ionization
potential (VIP), adiabatic ionization potential (AIP), adiabatic electron affinity (AEA) and vertical
detachment energies (VDE). These terms are evaluated as the difference of total energies in the following
manner:
   AEA = E (optimizedneutral) –E(optimizedanion)
   VDE = E (neutralatoptimizedaniongeometry) –E(optimizedanion)
   VIP = E (cationatoptimizedneutralgeometry) – E(optimizedneutral)
   AIP = E (optimizedcation) –E (optimizedneutral)
   In Table 2 we have compared the ionization potential and electron affinities with the available experi-
mental results.
   Understanding the fragmentation behavior is another important criteria to infer about the stability of
smaller clusters. In experiment generally clusters are produced in ionic form, and there fragmentation
behavior and stability governs the mass spectrum therefore it is of interest to see the fragmentation
behavior of charged cluster. Although it is known that the fragmentation process involves dissociation
barrier, entropy or free energy changes however, in the present work we have assumed the spontaneous
fragmentation with no barrier thereby leading to infer about the relative stability of these clusters in the
328         S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters

                    1.5



                    1.0



                    0.5



                    0.0
           ∆ E eV
          2




                    -0.5                                                                          (Sin)
                                                                                                          -
                                                                                                  (Sin )
                    -1.0
                                                                                                          +
                                                                                                  (Sin )
                    -1.5                                                                 B3LYP-6-31+G(d)

                              2           4           6           8          10          12
                                                            n

                            Fig. 4. Second difference in energy of neutral and charged silicon cluster.

                                                              Table 2
                    Adiabatic ionization potential (AIP) , adiabatic electron affinities (AEA), vertical detach-
                    ment energy (VDE) , relaxation energy for anion (∆R(−)) and cations (∆R(+))
                     n     AIP VIP Exp. I. P*. AEA Exp# . VDE Exp$ . ∆R(−) ∆R(+)
                     2    7.64 7.78        > 8.49      2.04      2.2     2.06      −       0.02       0.14
                     3    7.94 8.07        > 8.49      2.23      2.0     2.51      −       0.28       0.13
                     4    7.75 8.02      7.97–8.49     2.01      1.8     2.03    2.19      0.02       0.27
                     5    7.87 8.01      7.97–8.49     2.34     2.50     3.08    3.38      0.74       0.14
                     6    7.45 7.90        ∼ 7.90      2.02      1.8     2.75    2.40      0.73       0.45
                     7    7.63 7.92        ∼ 7.90      1.87      1.7     2.30    2.33      0.42       0.29
                     8    7.06 7.20      7.46–7.87      2.5      2.3     3.22    2.66      0.72       0.14
                     9    7.30 7.46      7.46–7.87     2.08      2.4     2.57    3.53      0.49       0.16
                    10 7.52 7.77           ∼ 7.90      2.25      2.2     2.56    2.66      0.31       0.25
                    11 6.43 6.67         7.46–7.87     2.33      2.5     2.78    2.93      0.45       0.24
                    12 6.55 6.82         7.17–7.46     2.84      2.6     3.13      −       0.29       0.27
                    13 6.51 6.76         7.17–7.46      2.4       −      3.39      −       0.99       0.25
                    *The experimental I.P. has been taken from Fuke et al. [30].
                    #The experimental adiabatic electron affinity (AEA) has been taken from cheshnovsky et
                    al. [11].
                    $ The experimental Vertical detachment energy (VDE) has been taken from Yang et al. [44].

ground state. For this purpose the fragmentation energies have been calculated for all possible channels,
which can be expressed as
  Ef (Si+ ) = E(Si+ )–E (Sin−x )–E(Si+ )
         n          n                  x
  Ef (Si− ) = E(Si− )–E (Sin−x )–E (Si− )
         n          n                   x
  For simplicity we have plotted the fragmentation energies of the lowest energy channels as a function
of the cluster size as shown in Fig. 5. From this figure it is clear that while smaller size clusters favor
monomer evaporation, medium size clusters dissociate into two stable clusters. This has been the typical
           S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters                   329


                                                          (The first number in the brackets indicates)
                                                          cation and second number indicate neutral

                        Fragmentation energy
                                               4.5              (3,1)
                                                                                (5,1)
                                               4.0

                                               3.5                                  (6,1)
                                                              (2,1)                             (4,5)
                                               3.0    (1,1)
                                                                        (4,1)               (7,1)        (7,4)
                                               2.5                                                   (6,4)
                                                                                                                        (6,7)
                                               2.0
                                                                                                                (6,6)
                                                      2          4          6               8           10       12         14
                                                                                    N
                                                             (The first number in the brackets indicates))
                                                             anion and second number indicate neutral
                                               4.5
                                                      [2,1]      [3,1]
                     Fragmentation energy




                                               4.0
                                                                         [4,1]
                                               3.5   [1,1]                         [6,1]
                                                                         [5,1]                          [6,4]
                                               3.0                                 [4,4]
                                                                                                [5,4]
                                               2.5

                                               2.0                                                      [4,7]
                                                                                                                  [5,7]
                                               1.5
                                                                                                                      [6,7]
                                                      2          4          6               8           10       12         14
                                                                                        N

                                                     Fig. 5. Fragmentation of cationic and anion clusters.

nature of the fragmentation of covalently bonded clusters as has already been reported for other Group-IV
elements [52]. For larger charged clusters it has been noticed that in general after fragmentation the
charge prefers to reside on the heavier fragments. This is attributed to more delocalization of charge on
the larger clusters.


5. Doped cluster: AlSin−1 and PSin−1

   As described earlier, in contrast to homoatomic silicon cluster, much fewer effort have been devoted
to doped silicon cluster. The problem of the heteroatom silicon bond like Al-Si, Na-Si, Si-F, Si-O
was a topic of number of studies, both experimentally [45–47,49] and theoretically [31,51,49]. The
motivation for the great and persisting interest in this topic consists of both its fundamental relevance
to the understanding of the silicon based composite material and its high degree of practical importance
in microelectronic technology. It appears that the basic process governing the interaction between the
impurity atom and the silicon host can be studied more easily in a finite cluster than in a extended
system, as a cluster is more easily accessible to accurate computational analysis than the surface or bulk
system. This consideration provides a strong motivation for the study of the doped silicon cluster like
330         S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters

AlSin−1 and PSin−1 . Because impurity atom addition can lead to change in electronic structure thereby
influencing the chemical stability, we have compare neutral doped cluster with there isoelectronic charge
cluster.
   The ground state geometries of the AlSi n−1 clusters also show similar atomic configurations where
the Al atom has replaced the Si-1 atom from the host neutral Si n clusters. In general the geometries of
the Al substituted silicon clusters shows small local distortions due to asymmetric charge distributions.
Significant differences were found for small clusters like n = 3, 6 etc. Unlike the case of neutral Si 3
cluster, AlSi2 favor closed equilateral triangles. It is worth mentioning that although Al-Si bond is larger
(2.45 Å) than Si-Si (2.17 Å), but in AlSi2 cluster the Al-Si bond shrinks significantly and it becomes
almost equal to that of Si-Si bonds (2.36 for Al-Si and 2.38 for Si-Si). For AlSi7 cluster, the bicapped
octahedron structure is deformed significantly as the Si1-Si5 bond becomes very long and new bonds like
Si2-Si3 and Si4-Si5 are formed. For n = 9, 10, 11, 12, and 13, no significant changes have been observed
in the ground state geometries of the Al substituted clusters as compared to their neutral counterparts.
   The optimized geometries of P substituted Si n clusters (P atom replaced the Si-1 in Si − clusters ) adopts
                                                                                           n
similar atomic configuration to that of negatively charged Si n clusters with small local distortion. Like
Si− , the iso-electronic PSi2 cluster favor isosceles triangle with <Si-P-Si angle of 70.19 ◦ in comparison
  3
to that of 65.66◦ observed for Si− . The PSi3 cluster favors rhombus configuration with small local
                                      3
distortion as shown in Fig. 1. The P-Si bond forming edges of the deformed rhombus is 2.20 Å. Other
isomer of PSi3 cluster, where P atom occupies the center position of the triangle formed by three Si
atoms (D3h ) is found to be 1.70 eV higher in energy than that of the rhombus configuration. For PSi 4
cluster, the structure is similar to that of the Si− , where the heteroatom replaces one of the Si atom from
                                                   5
the base triangle, which is distorted by two different bond lengths of P-Si and Si2-Si5 as 2.54 and 2.86 Å,
respectively. This structure can be viewed as a capped bent rhombus formed by four Si atoms with Si-Si
bond lengths of 2.34 Å. In case of PSi 6 cluster, the P atom prefers to replace one of the Si atoms (Si-1) in
the base plane. Another isomer of PSi 6 cluster, where the P atom occupies the apex position shows 0.30
eV higher in energy with respect to that of lowest energy isomer. It may be noted the P atom prefers to
occupy lower coordination (base atom) over apex position indicates strong directional bonding between
Si-P bonds. For n = 7, 8, 9, 10, 11, and 12, no significant changes have been observed in the ground
state geometries of the P substituted clusters as compared to their negatively charged Si n counterparts.


6. Energetics of doped cluster

  To check the influence of Al and P atom on the host Si n clusters we have compare the binding energies
of Sin , AlSin−1 and PSin−1 . The binding energies for heteroatomic system were calculated as following:
  BE(AlSin−1 ) = [E(AlSin−1 )–(n − 1) × E(Si)–E(Al)]/n
  BE(PSin−1 ) = [E(PSin−1 )–(n − 1) × E(Si)–E(P)]/n
  The lower binding energy of AlSi n−1 is due to the substitution of one Si atom by Al atom, which has
one electron less. It is important to notice that although Si-P bond is stronger than Si-Si, for P substituted
Si clusters, the binding energy show difference in the small and larger size range. It is clear from this
figure that although for small clusters (n = 2, 3) substitution of P atom increases the average binding
energy but for n 4, it becomes smaller than Si n clusters. The reduction in the binding energy by the
inclusion of P atom is attributed to the smaller cohesion of bulk phosphorous (bulk cohesive energy of
P = 3.43 eV/atom) in comparison to that of Si (bulk cohesive energy of Si = 4.63 eV/atom).
  To understand the electronic structure of isoelectronic systems comparison was done between the
binding energies of Si− , PSin−1 and Si+ , AlSin−1 . From Fig. 6 it is clear that, although incorporation of
                        n                 n
            S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters   331

                                  3.4
                                  3.2
                                  3.0
                                  2.8
                  B.E./atom(eV)   2.6
                                  2.4                                               (Sin)
                                  2.2
                                  2.0                                               (AlSin-1)
                                  1.8                                               (PSin-1)
                                  1.6
                                  1.4
                                                                               B3LYP-6-31+G(d)
                                  1.2
                                  1.0
                                        2          4           6           8            10        12   14
                                  3.4
                                                                       n
                                  3.2

                                  3.0

                                  2.8
                  B.E./atom




                                  2.6                                                   -
                                                                                    (Sin )
                                  2.4
                                                                                    (PSin-1)
                                  2.2

                                  2.0
                                                                               B3LYP-6-31+G(d)
                                  1.8

                                  1.6

                                        2          4           6           8            10        12   14

                                  3.4                                  n
                                  3.2
                                  3.0
                                  2.8
                                  2.6
                  B.E./atom




                                  2.4
                                  2.2                                                        +
                                  2.0
                                                                                      (Sin )
                                  1.8                                                 (AlSin-1)
                                  1.6
                                  1.4
                                                                                B3LYP-6-31+G(d)
                                  1.2
                                  1.0
                                        2          4           6           8            10        12   14

                                                                       n

                                            Fig. 6. Binding energy per atom of doped clusters.


P atom in Sin changes the stability order, but they show similar trend as that of iso-electronic Si − clusters.
                                                                                                    n
Similar correlation between the iso-elcetronic clusters of Si + and AlSin−1 was found. Therefore, it can
                                                              n
be inferred that the stability of such clusters is strongly influenced by the removal or addition of an
electron. Here also for AlSin−1 clusters, which is iso-electronic with Si+ clusters show extra stability
                                                                             n
of the AlSi10 cluster suggesting the influence of the electronic structures for different stabilities between
neutral and charged clusters.
  From theoretical calculations one can search for magic clusters by calculating the second energy
difference in energy, which has been calculated as. The second order difference in energy (∆ 2 E) for
impurity cluster has been elucidated in the following manner:
  ∆2 E = [2E(AlSin−1 )–E(AlSin )–E(AlSin−2 )]
  ∆2 E = [2E(PSin−1 )–E(PSin )–E(PSin−2 )]
  It is seen that substitution of P atom with Sin changes the relative stability order. For example, while
332        S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters

                                                              (Sin)
                                                                  +
                                                              (Sin )
                                                              (AlSin-1)
                                                   B3LYP-6-31+G(d)
                                   1.5

                                   1.0

                                   0.5
                          ∆ E eV

                                   0.0
                          2




                                   -0.5

                                   -1.0

                                   -1.5

                                            2         4                   6       8   10     12
                                   1.5              (Sin)                     N
                                                          -
                                                     (Sin )
                                   1.0               (PSin-1)
                                                B3LYP-6-31+G(d)
                                   0.5
                         ∆ E eV




                                   0.0

                                   -0.5
                         2




                                   -1.0

                                   -1.5

                                            2         4                   6       8   10     12
                                                                              n

                                     Fig. 7. Second difference in energy of doped cluster.

for Sin clusters, n = 4, 6, 7 and 10 shows higher stability in comparison to its neighbors, for PSi n−1 ,
n = 3, 5, 7, 10 and 12 shows higher stability. The second order difference in energy of Si n , AlSin−1 and
PSin−1 has been compared in Fig. 7.
   To further check the stability of doped cluster we have calculated the fragmentation energy of these
clusters. The fragmentation energies have been calculated for all possible channels, which can be
expressed as
   Ef (AlSin−1 ) = E(AlSin−1 )–E(AlSin−1−p )– E(Sip )
   Ef (PSin−1 ) = E(PSin−1 )–E(PSin−1−x )–E (Six )
   The fragmentation energies of the lowest energy channels as a function of the cluster size has been
shown in Fig. 8.
   The fragmentation behavior of AlSi11 and AlSi12 is different than others. The lowest energy frag-
mentation of the AlSi11 and AlSi12 clusters favors fission type fragmentation into two stable clusters
(AlSi11 → AlSi5 + Si6 ; AlSi12 → AlSi6 + Si6 ) than even Al atom dissociation. It is found from the
fragmentation energy that while the dissociation of Al atom requires 2.08 and 2.84 eV which are much
larger than the dissociation energies required for the above mentioned fragmentation channels. On the
contrary to this behavior, from the bond energy (Table 1) of dimers it is known that the Si-Si bond is
much stronger than the Al-Si bond. Therefore, the fragmentation behavior of the AlSi 11 and AlSi12
            S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters         333

                                                         (The First number in the bracket is AlSin-p-1 )
                                                         and second number is Sip

                       Fragmentation Energy -ev   4.5
                                                                    (3,1)
                                                  4.0
                                                  3.5                       (2,4)
                                                                                    (6,1)
                                                  3.0                    (4,1)
                                                                 (2,1)                  (4,4)
                                                  2.5                                           (4,6)
                                                  2.0    (1,1)                          (2,7)           (4,7)
                                                  1.5                                                           (6,7)
                                                                                                     (6,6)
                                                         2         4        6       N   8       10        12       14

                                                             (The First number in the bracket is PSin-x-1 )
                                                             and second number is Six
                                                  4.5
                           Fragmentation Energy




                                                  4.0        [2,1]
                                                  3.5                 [4,1]
                                                  3.0   [1,1]     [3,1] [5,1]
                                                  2.5
                                                                            [3,4]       [3,5]
                                                  2.0                                            [3,7]
                                                  1.5
                                                                                        [3,6]                  [1,1]
                                                  1.0                                           [4,7]
                                                                                                           [5,7]
                                                         2         4        6    N      8       10        12       14


                     Fig. 8. Fragmentation energy of doped silicon cluster as function of cluster size.

clusters implies the weakening of Si-Si bonds in the larger AlSi n−1 clusters as compared to that of Al-Si
bonds. Further, it has been noticed that the fragmentation behavior of most PSi n−1 (except PSi7 and
PSi4 ) clusters favors fission type fragmentation into two stable clusters rather than P atom dissociation.
It is found from the fragmentation energy that while the dissociation of P atom requires much larger
energy than the dissociation energies required for the above mentioned fragmentation channels. This
phenomenon can be explained from the bond energy (Table 1) of dimers, it is known that the P-Si bond
is much stronger than the Si-Si bond. Therefore, the fragmentation of P atom from impurity cluster is
more difficult than the Si atom dissociation.


7. Conclusion

   The geometric and electronic structure calculations have been carried out for Si n , Si− , PSin−1 , Si+ and
                                                                                          n              n
AlSin−1 clusters under the LCAO-MO approach at the B3LYP/6-31+G(d) level. It has been observed
that the geometries of the positively charged clusters remain almost similar to those of the neutral ones
with small changes in the inter-atomic distances, which is due to asymmetric charged distribution. While
in case of anionic cluster generally, good correspondence between the ground state structure of the neutral
species and the negative ion was observed. However for n = 6, 8, 11, and 13, significant changes have
334           S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters

been observed in the ground state geometries of the negatively charged clusters as compared to their
neutral counterparts. The changes are more evident for small clusters as compared to the larger clusters.
The effect of impurity substation on the geometry and electronic structures was observed by doping the
Sin clusters with Al and P atom. The geometries of the AlSi n−1 clusters also favor similar structural
growth as that prevails for neutral clusters. The impurity Al atom substitutes one of the Si atoms from
the host Sin clusters, causing a small local distortion. In most of the cases the choice of the aluminum
substitutional site is the highest coordination site. It is found that P substituted silicon clusters follow
similar structural pattern as that of negatively charged Si n clusters with small local distortions.
   Although a similar structural pattern was observed for Si n , Si− PSin−1 , Si+ and AlSin−1 clusters,
                                                                        n            n
significant differences have been observed in their electronic structures, which have been illustrated based
on their relative stability orders. Differences in the relative stability pattern have been observed between
the neutral and charged clusters. Doped cluster follow same stability order as that of corresponding
isoelectronic one. The experimentally observed magic behavior of the Si + clusters has been established
                                                                               11
theoretically from the extra stability of it as compared to that of Si+ and Si+ clusters. Based on the
                                                                           10       12
energetics it was also found that the substitution of P atom with Si n clusters changes the relative stability
order. For example, while for Sin clusters, n = 4, 6, 7 and 10 shows higher stability in comparison to
its neighbors, for PSin−1 , n = 3, 5, 7, 10 and 12 shows higher stability. The fragmentation behavior of
the iso-electronic Si+ , AlSin−1 , Si− and PSin−1 clusters suggest that while small size clusters tend to
                      n               n
dissociate by monomer evaporation, larger clusters prefer to dissociate into two stable clusters.


Acknowledgements

  We are thankful to the members of the Computer Division, BARC, for their kind cooperation during
this work.


References
 [1]   M.F. Jarrold, Science 25(2) (1991), 1085.
 [2]   W.L. Brown, R.R. Freeman, K. Raghavachari and M. Schluter, Science 23(5) (1987), 860.
 [3]   S. Hayashi, Y. Kanzawa, M. Kataoka, T. Nagarede and K. Yamamoto, Z Phys D: At Mol Clusters 2(6) (1993), 144.
 [4]   L.A. Bloomfield, R.R. Freeman and W.L. Brown, Phys Rev Lett 5(4) (1985), 2246.
 [5]   L.A. Bloomfield, M.E. Guesic, R.R. Freeman and W.L. Brown, Chem Phys Lett 12(1) (1985), 33.
 [6]   K.D. Rinnen and M.L. Mandich, Phys Rev Lett 6(9) (1992), 1823.
 [7]   W. Begemann, K.H. Meiwes-Broer and H.O. Lutz, Phys Rev Lett 7(3) (1986), 2248.
 [8]   M.F. Jarrold and E.C. Honea, J Phys Chem 9(5) (1991), 9181.
 [9]   J.M. Hunter, J.L. Fye, M.F. Jarrold and J.E. Bower, Phys Rev Lett 7(3) (1994), 2063.
[10]   T.P. Martin and H. Schaber, J Chem Phys 8(3) (1985), 855.
[11]   O. Cheshnovsky, S.H. Yang, C.L. Pettiette, M.J. Craycraft, Y. Liu and R.E. Smalley, Chem Phys Lett 13(8) (1987), 119.
[12]   E.C. Honea, A. Ogura, D.R. Peale, C. Felix, C.A. Murray, K. Raghavachari, W.O. Sprenger, M.F. Jarrold and W.L.
       Brown, J Chem Phys 110 (1999), 12161.
[13]   M.F. Jarrold and W.L. Brown, Nature (London) 36(6) (1993), 42.
[14]   S. Li, R.J. Van Zee, W. Weltner and K. Raghavachari, Jr., Chem Phys Lett 24(3) (1995), 275.
[15]   K. Raghavachari and V. Logovinsky, Phys Rev Lett 5(5) (1985), 2853.
[16]   K. Raghavachari, J Chem Phys 8(4) (1986), 5672.
[17]   K. Raghavachari and C.M. Rohlfing, J Chem Phys 8(9) (1988), 2219; C.M. Rohlfing and K. Raghavachari, Chem Phys
       Lett 16(7) (1990), 559; J Chem Phys 9(6) (1992), 2114.
[18]   X.L. Zhu, X.C. Zeng and Y.A. Lei, J Chem Phys 118 (2003), 3558; X.L. Zhu, X.C. Zeng and Y.A. Lei, J Chem Phys 120
       (2004), 8985; S. Yoo and X.C. Zeng, J Chem Phys 123 (2005), 164303; S. Yoo and X.C. Zeng, J Chem Phys 124 (2006),
       54304; S. Yoo, N. Shao, C. Koehler, T. Fraunhaum and X.C. Zeng, J Chem Phys 124 (2006), 164311.
              S. Nigam et al. / Theoretical study of pure (Sin ) and doped silicon (AlSin−1 and PSin−1 ) clusters        335

[19]   K. Raghavachari and C.M. Rohlfing, Chem Phys Lett 9(4) (1991), 3670.
[20]   K. Raghavachari and C.M. Rohlfing, Chem Phys Lett 19(8) (1992), 521.
[21]   K.M. Ho, A.A. Shvartsburg, B. Pan, Z.Y. Lu, C.Z. Wang, J.G. Wacker, J.L. Fye and M.F. Jarrold, Nature (London) 39(2)
       (1998), 582. M.F. Jarrold and V.A. Constant, Phys Rev Lett 6(7) (1991), 2994; M.F. Jarrold and J.E. Bower, J Phys Chem
       9(6) (1992), 9180;
[22]   H. Haberland, Clusters of Atoms and Molecules: Theory, Experiment, and Clusters of Atoms, Springer Verlag, New
       York, 1994.
[23]   B. Liu, Z.Y. Lu, B. Pan, C.-Z. Wang, K.-M. Ho, A.A. Shvartsburg and M.F. Jarrold, J Chem Phys 10(9) (1998), 9401;
       Z.Y. Lu, C.Z. Wang and K.M. Ho, Phys Rev B 6(1) (2000), 2329.
[24]   B.X. Li, P.L. Cao and M. Jiang, Phys Status Solidi B 21(8) (2000), 399; B.X. Li and P.L. Cao, Phys Rev A 6(2) (2000),
       023201; B.X. Li and P.L. Cao, J Phys: Condens Matter 1(3) (2001), 1.
[25]   S. Wei, B.N. Barnett and U. Landman, Phys Rev B 5(5) (1997), 7935.
[26]   I. Vasiliev, S. Ogut and J.R. Chelikowsky, Phys Rev Lett 7(8) (1997), 4805.
[27]   Ionel Rata, Alexandre A. Shvartsburg, Mihai Horoi, Thomas Frauenheim, K.W. Michael Siu and Koblar A. Jackson,
       Phys Rev Lett 85 (2000), 546.
[28]   C. Xiao, F. Hagelberg and W.A. Lester, Jr., Phys Rev B 66 (2002), 075425.
[29]   D.E. Bergeron and A.W. Castleman, Jr., J Chem Phys 117 (2002), 3219.
[30]   K. Fuke, K. Tsukamoto, F. Misaizu and M. Sanekata, J Chem Phys 99 (1993), 7907.
[31]   S. Nigam, C. Majumder and S.K. Kulshrestha, J Chem Phys 121 (2004), 7756; S. Nigam, C. Majumder and S.K.
       Kulshrestha, J Chem Phys 125 (2006), 74303.
[32]   C. Blondel, C. Delsart and F. Goldfarb, J Phys B 34 (2001), L281.
[33]   M.R. Nimlos, B.L. Harding and G.B. Ellison, J Chem Phys 87 (1987), 5116.
[34]   C.C. Arnold, T.N. Kitsopoulos and D.M. Neumark, J Chem Phys 99 (1993), 766.
[35]   C.C. Arnold and D.M. Neumark, J Chem Phys 100 (1994), 1797.
[36]   C.C. Arnold and D.M. Neumark, J Chem Phys 99 (1993), 3353.
[37]   T.N. Kitsopoulos, C.J. Chick, A. Weaver and D.M. Neumark, J Chem Phys 93 (1990), 6108.
[38]   C. Xu, T.R. Taylor, G.R. Burton and D.M. Neumark, J Chem Phys 108 (1998), 1395.
[39]   T.N. Kitsopoulos, C.J. Chick, Y. Zhao and D.M. Neumark, J Chem Phys 95 (1991), 1441.
[40]   C.C. Arnold, T.N. Kitsopoulos and D.M. Neumark, J Chem Phys 99 (1993), 766.
[41]   L.A. Curtiss, P.W. Deutsch and K. Raghavachari, J Chem Phys 96 (1992), 6868.
[42]   A.A. Shavartsburg, B. Liu, M.F. Jarrold and K.M. Ho, J Chem Phys 112 (2000), 4517.
[43]   C. Zhao and K. Balasubramanian, J Chem Phys 116 (2000), 3690.
[44]   J.Yang, W. Xu and W. Xiao, J Mol Struct (Theochem) 719 (2005), 89.
[45]   A. Nakajima, T. Taguwa, K. Nakao, M. Gomei, R. Kishi, S. Iwata and K. Kaya, J Chem Phys 103 (1995), 2050.
[46]   H. Kawamata, Y. Negishi, R. Kishi, S. Iwata, A. Nakajima and K. Kaya, J Chem Phys 105 (1996), 5369.
[47]   R. Kishi, H. Kawamata, Y. Negishi, S. Iwata, A. Nakajima and K. Kaya, J Chem Phys 107 (1997), 10029.
[48]   W. Zheng, J.M. Nilles, D. Radisic and K.H. Bowen, Jr., J Chem Phys 122 (2005), 071101.
[49]   R. Kishi, S. Iwata, A. Nakajima and K. Kaya, J Chem Phys 107 (1997), 3056.
[50]   V. Kumar and Y. Kawazoe, Phy Rev Lett 87 (2001), 045503.
[51]   C. Majumder and S.K. Kulshreshtha, Phys Rev B 69 (2004), 115432.
[52]   C. Majumder, V. Kumar, H. Mizuseki and Y. Kawazoe, Phys Rev B 64 (2001), 233405.
[53]   M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S. Gordon, J.H. Jensen, S. Koseki, N. Matsunaga, K.A.
       Nguyen, S.J. Su, T.L. Windus, M. Dupuis and J.A. Montgomery, J Comput Chem 14 (1993), 1347.
[54]   A.D. Becke, Phys Rev A 38 (1988), 3098.
[55]   C. Lee, W. Yang and R.G. Parr, Phys Rev B 37 (1988), 785.
[56]   CRC Handbook of Chemistry and Physics, (49th ed.), R.C. Weast, ed., CRC press, Cleveland, Ohio, 1969.

								
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