Algebraic modifications to
second quantization for non-
Hermitian complex scaled
Hamiltonians with application to a
quadratically convergent
multiconfigurational self-
consistent field method
Danny L. Yeager
Chemistry Department
Texas A&M University
College Station, TX
July 18, 2007 Casper Summer School 1
Acknowledgements:
Kousik Samanta
Dongxia Ma
Robert A. Welch Foundation (Grant A-770)
July 18, 2007 Casper Summer School 2
The algebraic structure for creation and annihilation operators
defined on orthogonal orbitals is generalized to permit easy
development of bound state techniques involving the use of non-
Hermitian Hamiltonians arising from the use of complex scaling in
the treatment of electron scattering and Auger resonances.
The ease of application is demonstrated by deriving the modified
equations for implementation of a quadratically convergent
multiconfigurational self-consistent field (MCSCF) method for
complex scaled Hamiltonians
This extends the domain of applicability of MCSCF, CC, MBPT and
methods based on MCSCF states to an accurate treatment of
resonances while still using L2 real basis sets.
Modification of all other bound state methods and codes should be
similarly straightforward.
July 18, 2007 Casper Summer School 3
Electron-Molecule Resonances
Electron-atom and electron-molecule resonances are
metastable electronic continuum states.
They play an important role in vibrational excitation of
molecules or molecular ions by electron impact and in
dissociative attachment and recombination which are
processes of considerable importance in outer space, in
the higher atmosphere, in plasmas and discharges, and
as a mechanism for DNA damage by low-energy
electrons.
Resonances are observed in scattering and Auger
experiments
July 18, 2007 Casper Summer School 4
Electron Scattering
Resonances
e + A → (A-)* → A + e
Eres = (K.E.)e = EA -E(A-)*
N N+1
= E0 - Es
July 18, 2007 Casper Summer School 5
Auger Resonances
hv + A → (A+)* + e-
↓
A++ + e-
Eres = (K.E.)e = E(A+)* -EA
N-1 N
= En - E0
July 18, 2007 Casper Summer School 6
Complex Scaling
The method of complex scaling makes
their investigation accessible to bound
state theoretical techniques as complex
eigenvalues of a Hamiltonian where all
electronic coordinates have been scaled
by a complex scaling factor = ei.
With this scaling H† () = H* () H ().
July 18, 2007 Casper Summer School 7
Modified Spectrum of H()
The bound eigenvalues are real and remain unchanged.
Ionization and excitation thresholds are unmodified.
The continua for each threshold are rotated by an angle
2.
Due to their finite lifetimes, the resonances need to be
described by complex eigenvalues and remain hidden
for Hermitian Hamiltonians.
The discrete complex eigenvalues of the complex scaled
Hamiltonian describe resonances and, once uncovered,
are invariant to changes in theta.
July 18, 2007 Casper Summer School 8
Modified Spectrum of H()
July 18, 2007 Casper Summer School 9
k E / k 0 k 1,2,3,...
For reasonably sized basis sets:
E
0
For ei
E / 0
E / 0
July 18, 2007 Casper Summer School 10
Square integrability
The wavefunctions res are square integrable for
> some system specific c and the
corresponding eigenvalues are quasi-stable with
respect to variations in near c.
Square integrability has led to bound state
methods like linearly convergent SCF, CI and
the usual perturbational electron propagator
theory to be employed for investigating these
scattering resonances.
July 18, 2007 Casper Summer School 11
We have developed a modification of the second
quantization algebra conventionally based on
creation and annihilation operators defined on
orthogonal orbitals to those defined on a
biorthogonal set of orbitals obtained from a bi-
variational procedure appropriate to a complex
scaled non-Hermitian Hamiltonian.
The ease of application is demonstrated by
deriving modified equations for a quadratically
convergent procedure developed previously for
multiconfigurational self-consistent field methods
(MCSCF) for real Hermitian Hamiltonians.
July 18, 2007 Casper Summer School 12
Multiconfigurational
wavefunctions
Single reference state based methods are often
inadequate for the accurate descriptions of resonances.
Complex bi-variational MCSCF equations will facilitate
their extension to other L2 techniques and computational
packages using MCSCF-based bound state techniques
allowing for a simple, economic and accurate treatment
of shape, Feshbach and Auger resonances.
Examples include the multiconfigurational spin-tensor
electron propagator method (MCSTEP).
July 18, 2007 Casper Summer School 13
In quantum mechanics we are often primarily interested in the
eigenvalue problem
ˆ
Hi Eii (1)
Electron-atom/molecule resonance positions and widths can be
obtained by scaling all the electron co-ordinates in by a
complex scale factor, . For this complex scaled Hamiltonian
H (),
Hˆ † H* ˆ (2)
where † is the Hermitian conjugate and * represents complex
conjugation.This in turn implies that for the wavefunction
j ij
*
i
(3)
i j ij
rather than
(4)
July 18, 2007 Casper Summer School 14
These biorthogonal states { i } can be expanded in
terms of linear combinations of Slater determinants
{ i } of spin-orbitals { i} where
i j ij
* (5)
and
(6)
i j ij
*
We have modified the usual second quantization
algebra of operators that add or remove electrons to
spin orbitals to account for this biorthogonality and to
develop an occupation number formalism similar to
that based on orthogonal spin orbitals of usual
second quantization.
July 18, 2007 Casper Summer School 15
Second quantization
In our occupation number formalism, a T adds an electron in
i
spin-orbital if it is unoccupied with the usual second
quantization sign conventions for fermions and gives
zero otherwise. Similarly, a i removes an electron from
spin-orbital with the usual second quantization sign
conventions for fermions if it is occupied and gives zero
otherwise. This gives the usual anticommutation
relations for fermions:
{a ,a j} a a a a ij
T
i
T
i j
T
j i
(7)
{a i ,a j} a i a j a ja i 0 (8)
{aT ,a T} a Ta T a Ta T 0
i j i j j i (9)
July 18, 2007 Casper Summer School 16
We define a general T operation in the following manner
ˆ ˆ
f A g AT*f g (10)
for arbitrary well-behaved states and with the complex
conjugate of a linear operator here defined:
ˆ ˆ
A* g* (A g )* (11)
The order of operations in Eq. (10) is that the right most
operation is performed first
Hence, from Eq. (10)
ˆ
ˆ † AT*
A (12)
July 18, 2007 Casper Summer School 17
Effect of operators on the bra
T
With this definition it can be shown that a i
removes an electron from orbital i in 0* if it is
occupied and gives zero if it is not and likewise
that a i adds an electron to orbital i in 0* if it is
not occupied and gives zero if it is.
July 18, 2007 Casper Summer School 18
Form of the Hamiltonian
By examining one-body and two-body operators with arbitrary * in the
bra and in the ket and comparing occupation number (second
quantization) results with wavefunction (first quantization), the form of
the Hamiltonian is easily shown to be
ˆ i* h j a T a 1 i* j* kl a T a Ta a
H ˆ (13)
i j i j l k
i, j 2 i, j,k,l
where is the usual one-body part of the Hamiltonian in the Born-
Oppenheimer approximation. Note that for convenience we have
used a simplified notation where
ˆ ˆ
i* h j * h j (14)
i
and
1
i j kl (1) (2)
* * *
i
*
j k (1)l (2) (15)
r12
July 18, 2007 Casper Summer School 19
The multiconfigurational self-consistent field
method (MCSCF)
In multiconfigurational self-consistent field calculations
an approximate wavefunction 0 is expanded as a
linear combination of Slater determinants of spin-
orbitals.
Both the orbitals and the determinant expansion
coefficients are then optimized by use of the variational
principle to obtain the final optimized state 0 .
Since the number of included Slater determinants is
obviously not infinite, this wave function is only an
approximation to the exact wave function.
July 18, 2007 Casper Summer School 20
A QUADRATICALLY CONVERGENT BI-
VARIATIONAL COMPLEX SCALED MCSCF
METHOD
ˆ
Let be an operator that takes 0 into 0
ˆ
0 0 (16)
where
0 0 0 0 1
* *
(17)
and
0* H 0 E 0* 0 E (18)
ˆ ˆ
0* T H 0 (19)
July 18, 2007 Casper Summer School 21
From Eq. (17)
ˆ T 0 1
0 0 0 ˆ
* * (20)
ˆ
ˆ T 1
(21)
ˆ
In other words, is an orthogonal operator.
July 18, 2007 Casper Summer School 22
To allow for an order-by-order expansion of the orbitals and the state expansion
coefficients
ˆ e eS
ˆ ˆ (22)
From Eq. (21),
rs (a Ta s a s a r )
(23)
ˆ T
r
and r s
ˆ S ( n 0* 0 n* )
S (24)
n0
n
where and S are complex coefficients that are solved for, the sum
rs n0
in Eq. (23) is only over non-redundant spin-orbital pairs, and the sum in Eq.
(24) is over the MCSCF orthogonal states complement to 0 .
The MCSCF orthogonal complement states are the remaining linearly independent states
that can be formed from the choice of MCSCF determinants.
July 18, 2007 Casper Summer School 23
Sn0
Second order expansion
An expansion of
ˆ eS 0
ˆ ˆ
ˆ ˆ
S
E 0 e e He
* (25)
through second order in rs and Sn0 followed by use of the
variational principle where the first derivatives with
respect to rs and Sn0 are set equal to zero yields
quadratically convergent complex scaled MCSCF
equations
1
1 G G S F rs
S 2 G G F (26)
S SS S
where and S are vectors containing the coefficients
and Sn0 respectively.
July 18, 2007 Casper Summer School 24
July 18, 2007 Casper Summer School 25
aT
Simple Result
These equations are the same as in the usual Newton-Raphson quadratically
convergent MCSCF in real space
but with
ˆ
a complex scaled H (η)
0* instead of 0 in the bra,
a T instead of a †
transfer operators 0 n* and n 0* instead of 0 n and n 0
However, because of previous equations (5)-(15), the resulting formulae for
matrix elements are exactly the same as in usual Newton-Raphson
quadratically convergent MCSCF in real space with {i } orbitals replacing {* }
i
orbitals in the bra and, of course, with all electronic coordinates complex
scaled.
July 18, 2007 Casper Summer School 26
Trust Region Schemes
Furthermore, trust region schemes can be likewise
straightforwardly implemented with the formulas
being exactly the same (with modifications
specified in the previous paragraph and some
other considerations for working in complex
space as well) as have been developed for
MCSCF with real Hermitian Hamiltonians. These
trust region methods are used to reliably bring
an MCSCF calculation into the local region
where quadratic convergence can be extremely
beneficial. These are also effective for complex
scaled MCSCF.
July 18, 2007 Casper Summer School 27
Application to other bound
state techniques
Of course, this powerful quadratically convergent
methodology can be used for complex scaled
SCF as well by specifying that the approximate
wavefunction has only a single determinant of
spin-orbitals.
Extensions to other single/multi-reference
complex scaled CC, MBPT, and other bound
state techniques and codes should be similarly
straightforward.
July 18, 2007 Casper Summer School 28
Preliminary
Results for the Be
Auger Resonance
July 18, 2007 Casper Summer School 29
Auger Resonances
hv + A → (A+)* + e-
↓
A++ + e-
Eres = (K.E.)e = E(A+)* -EA
N-1 N
= En - E0
July 18, 2007 Casper Summer School 30
Be Auger Resonance
hv + Be (1s22s2) → (Be+(1s2s2))* + e-
↓
Be++(1s2) + e-
Eres = (K.E.)e = E(Be+)*- EBe
3 4
= E1s2s - E1s2 2
2s2
July 18, 2007 Casper Summer School 31
k E / k 0 k 1,2,3,...
For reasonably sized basis sets:
E
0
For ei
E / 0
E / 0
July 18, 2007 Casper Summer School 32
ALPHA TRAJECTORIES FOR Be+ (1s^-1) Total Energy(E)
(Alpha increment = 0.02 au, starting w ith a value of 1.00 au at the top)
Basis: 14s11p; CAS: 1s2s2p3s3p
05/18/2007
0.00020
0.00010
0.00000
Im(E)/ au
-0.00010
-0.00020
-0.00030
-0.00040
-10.11900 -10.11890 -10.11880 -10.11870 -10.11860 -10.11850
Re(E)/ au
THETA=0.10RAD THETA=0.05 RAD
July 18, 2007 Casper Summer School 33
THETA TRAJECTORIES FOR Be+(1s^-1) Total Energy (E)
(Theta starts with a value of 0 in the right hand side;
theta decrement = 0.02 rad)
Basis: 14s11p; CAS: 1s2s2p3s3p
5/19/07
0.000030
0.000020
0.000010
Im(E)/ au
0.000000
-0.000010
-0.000020
-0.000030
-10.11904 -10.11900 -10.11896 -10.11892 -10.11888
Re(E)/ au
July 18, 2007 ALPHA = 0.84 AUSummer ALPHA = 0.86 AU
Casper School 34
ALPHA TRAJECTORY FOR Be+ (1s^-1) TOTAL ENERGY (E)
with optimized theta (0.11rad)
(Alpha starts w ith a value of 1.0au on top)
Basis: 14s11p; CAS: 1s2s2p3s3p
5/19/07
0.00015
0.00010
0.00005
0.00000
Im(E)/ au
-0.00005
-0.00010
-0.00015
-0.00020
-0.00025
-0.00030
-0.00035
-10.11895 -10.11890 -10.11885 -10.11880 -10.11875 -10.11870 -10.11865 -10.11860
Re(E)/ au
July 18, 2007 Casper Summer School
THETA = 0.11rad 35
Results For 2S Be+ (1s-1) Auger
Resonance
αopt = 0.84 au
θopt = 0.11 rad
E0 = -14.6522499 au
Er = (-10.11892837 - 0.00002104i) au
Position = Re(Er – E0) = 123.3589 eV
Width = |Im(Er)| × 2 = 0.001145 eV
o E0 and Er are total energies of neutral Be target and 2S Be+ (1s -1) shape resonance, respectively
o 1.0 au = 27.2116 eV
July 18, 2007 Casper Summer School 36
RESONANCE ENERGY OF 2S Be+ (1s-1)
AUGER RESONANCE (LITERATURE)
Y. Sajeev, M. K. Mishra, and S. Pal J. Chem. Phys. 120, 67 (2004)
July 18, 2007 Casper Summer School 37
Summary
We have provided a detailed generalization of the algebraic structure for creation and
annililation operators to permit easy development of bound state techniques involving
the use of non-Hermitian Hamiltonians arising from the use of complex scaling in the
treatment of electron scattering and auger resonances.
The resulting simplification is demonstrated by deriving the modified equations for
implementation of a quadratically convergent multiconfigurational self-consistent field
(MCSCF) method applied to non-Hermitian complex scaled Hamiltonians based on
an initial guess of complex biorthogonal orbitals resulting from a complex bi-
variational SCF scheme.
The resulting formulae make possible easy modification of currently available
standard MCSCF programs to perform quadratically convergent MCSCF calculations
for complex scaled non-Hermitian Hamiltonians.
The formal framework requires the use of orthogonal transformations of complex
biorthogonal orbitals as opposed to the customary unitary transformations of
orthogonal orbitals but preserves the full structural compatibility with standard
implementation of real MCSCF procedures.
July 18, 2007 Casper Summer School 38
Conclusions 1
Though the demonstrative application is for complex scaled bi-
variational MCSCF, the generalizations to second quantization
offered here are equally applicable to both single and multireference
CC, MBPT and other bound state techniques formulated with
second quantization algebra appropriate for non-Hermitian
Hamiltonians.
The generalizations discussed here allow for complex scaled
electron coordinates merely by letting the MCSCF expansion
coefficients and orbitals be complex instead of real numbers and the
integral transformation programs to use complex molecular orbital
expansion coefficients permiting the use of multiconfigurational
reference states for simple, economic and accurate treatment of
resonances.
July 18, 2007 Casper Summer School 39
Conclusions 2
Once the necessary coding modifications are done for
complex scaled MCSCF, these new second quantized
operators and complex scaled MCSCF can be used also
in multiconfiguration-based Green’s function/propagator
approaches such as the multiconfigurational spin tensor
electron propagator method (MCSTEP) and
multiconfigurational linear response (also known as
multiconfigurational time-dependent Hartree-Fock).
Based on the theoretical work reported here, the
resulting equations should be simple modifications of the
equations for multiconfigurational-based real Green’s
function/propagator approaches.
July 18, 2007 Casper Summer School 40
Conclusions 3
Hopefully, this will also motivate similar
attempts towards the developing of
single/multi-reference complex scaled
coupled cluster and many body
perturbation theory methods for the
treatment of resonances.
July 18, 2007 Casper Summer School 41