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					Density functional theory (DFT) and the
concepts of the augmented-plane-wave plus
local orbitals (APW+lo) method



           Karlheinz Schwarz
          Institute of Materials Chemistry
                    TU Wien
Walter Kohn and DFT
   DFT         Density Functional Theory

Hohenberg-Kohn theorem
 The total energy of an interacting inhomogeneous electron gas in the
 presence of an external potential Vext(r ) is a functional of the density 

                                       
                  E   Vext ( r )  ( r )dr  F [  ]

In DFT the many body problem of interacting electrons and nuclei is mapped
to a one-electron reference system that leads to the same density as the real
system.


            DFT treats both, exchange and correlation effects,
            but approximately
         Kohn Sham equations
         Total energy
                                                            LDA, GGA
                                     1  ( r )  ( r )  
         E  To [  ]   Vext  ( r )d r     dr dr   E xc [  ]
                                           2 | r r |

           Ekinetic                 Ene                Ecoulomb Eee               Exc exchange-correlation
vary      non interacting


         1-electron equation (Kohn Sham)

              1                                                                   
            { 2  Vext ( r )  VC (  ( r ))  Vxc (  ( r ))} i ( r )   i  i ( r )
              2



                                
                              (r )       |  i |2
                                        i  EF
Walter Kohn, Nobel Prize 1998 Chemistry
A simple picture of LDA

Look at the “LDA” from a different angle
                                                       Slater,    X
                                                       Gunnarsson-Lundqvist
                                                       …………
               Exc = -∫ dx n(x)       e 2/   R(x)


      R(x) interpreted as the radius of the „exchange-
      correlation hole‟ surrounding an electron at the point x.



    R(x) is a length: What length could it be?
    Plausible assumption, the average distance between the
    electrons?                        R(x) ≈ γ-1 n-1/3(x)

                            Exc = - γ e2 ∫ dx n4/3(x)
Role of „Gradient corrected functionals“


Becke, Perdew, Wang, Lee,   Perdew ,Burke, Ernzerhof
Yang, Parr …… ‟87 – „92     PBE …… „96




Use n and ∂n/∂x to correct LDA in regions of low density



Substantial improvement in energy differences
DFT ground state of iron

                              LSDA
                                   NM
                                   fcc
                                   in contrast to
                                    experiment

                              GGA
        LSDA                        FM
  GGA                           

                                    bcc
                  GGA           

                                   Correct lattice
                                    constant
                              Experiment
                                   FM
                   LSDA            bcc
CoO AFM-II total energy, DOS
                   CoO
                        in NaCl structure
                        antiferromagnetic: AF II
                        insulator
                        t2g splits into a1g and eg‘
                        GGA almost splits the bands
 CoO why is GGA better than LSDA
                             Central Co atom distinguishes
                
Vxc  VxcGGA  VxcLSDA           between   Co 
                                  and       Co 
                             Angular correlation
DFT thanks to Claudia Ambrosch   (Graz)



                     GGA follows LDA
                           Overview of DFT concepts
                                        Form of     Full potential : FP
                                        potential   “Muffin-tin” MT
                                                    atomic sphere approximation (ASA)
                                                    pseudopotential (PP)
             Relativistic treatment
                of the electrons         exchange and correlation potential
fully-relativistic                                                  Local density approximation (LDA)
semi-relativistic                                                   Generalized gradient approximation (GGA)
non relativistic                                                    Beyond LDA: e.g. LDA+U



                  1 2           k
                      V ( r )i   ikik
                  2                                                Kohn-Sham equations
                               
                     Representation                   Basis functions
    non periodic
    (cluster)
                        of solid
                                                            plane waves : PW
    periodic
                                                            augmented plane waves : APW
    (unit cell)          Treatment of                       linearized “APWs”
                              spin                          analytic functions (e.g. Hankel)
    Spin polarized
                                                            atomic orbitals. e.g. Slater (STO), Gaussians (GTO)
    non spin polarized
                                                            numerical
         How to solve the Kohn Sham equations

          Total energy
                                                             LDA, GGA
                                      1  ( r )  ( r )  
          E  To [  ]   Vext  ( r )d r     dr dr   E xc [  ]
                                            2 | r r |

            Ekinetic                 Ene                Ecoulomb Eee               Exc exchange-correlation
vary       non interacting


         1-electron equation (Kohn Sham)

               1                                                                   
             { 2  Vext ( r )  VC (  ( r ))  Vxc (  ( r ))} i ( r )   i  i ( r )
               2



                                 
                               (r )       |  i |2
                                         i  EF
                         APW based schemes

   APW (J.C.Slater 1937)
      Non-linear eigenvalue problem
      Computationally very demanding


   LAPW (O.K.Anderssen 1975)
      Generalized eigenvalue problem
      Full-potential


   Local orbitals (D.J.Singh 1991)
        treatment of semi-core states (avoids ghostbands)
   APW+lo (E.Sjöstedt, L.Nordstörm, D.J.Singh 2000)
      Efficiency of APW + convenience of LAPW
      Basis for

                                       K.Schwarz, P.Blaha, G.K.H.Madsen,
                                     Comp.Phys.Commun.147, 71-76 (2002)
APW Augmented Plane Wave method

The unit cell is partitioned into:             Bloch wave function:
   atomic spheres                                  atomic partial waves
   Interstitial region                             Plane Waves (PWs)
                      unit cell



                                  Rmt
                                        rI



    Full potential

   VLM YLM (r )
             ˆ       r  R
                                                            
                                              PW: e                    Atomic partial wave
   LM
                                                      i ( k  K ). r
             
   VK e   iK . r
                     rI
   K
                                                 join                     aKmu ( r ,  )Ym ( r )
                                                                                                  ˆ
                                                                         m
  Slater‘s APW (1937)

                             Atomic partial waves


                               m
                               aK u (r ,  )Ym (r )
                               m
                                                    ˆ

                             Energy dependent
                             basis functions lead to



H Hamiltonian               Non-linear eigenvalue problem
S overlap matrix

                            Computationally very demanding
One had to numerically search for the energy, for which
the det(H-ES) vanishes.
        Linearization of energy dependence

LAPW suggested by

 O.K.Andersen,
 Phys.Rev. B 12, 3060
 (1975)


   kn    [ A
            m
                   m    ( k n )u ( E , r )  Bm ( k n )u ( E , r )]Ym ( r )
                                                                              ˆ




 join PWs in
 value and slope
                                                                   Atomic sphere
                                                                          LAPW
Plane Waves (PWs)
                                                                                     PW
                   
      i ( k  K n ). r
  e
Full-potential in LAPW

                              The potential (and charge density)
                               can be of general form
          SrTiO3               (no shape approximation)

                                               VLM (r )YLM (r )
                                                             ˆ          r  R




           Full
                               V (r )    {   LM



                                                    K
                                                        VK e
                                                                 
                                                               iK . r
                                                                        rI

           potential
                              Inside each atomic sphere a
                               local coordinate system is used
                               (defining LM)

           Muffin tin
           approximation


                                        Ti
           TiO2 rutile                                          O
   Core, semi-core and valence states

For example:   Ti            Valences states
                                  High in energy
                                  Delocalized wavefunctions
                             Semi-core states
                                  Medium energy
                                  Principal QN one less than valence
                                   (e.g. in Ti 3p and 4p)
                                  not completely confined inside
                                   sphere
                             Core states
                                  Low in energy
                                  Reside inside sphere
                 Problems of the LAPW method:
    EFG Calculation for Rutile TiO2 as a function of the
               Ti-p linearization energy Ep

                           exp. EFG                   Electronic Structure
                                                      E   Ti 3d / O 2p
                                                                          EF

                                                               O 2p
                                                          Hybridized w.
                               „ghostband“                 Ti 4p, Ti 3d


P. Blaha, D.J. Singh, P.I. Sorantin and K. Schwarz,
Phys. Rev. B 46, 1321 (1992).                                Ti- 3p
                       ONE SOLUTION

                       Treat all the states in a single energy
Electronic Structure   window:
                        • Automatically orthogonal.
E Ti 3d / O 2p
                        • Need to add variational freedom.
                  EF    • Could invent quadratic or cubic APW
                          methods.
       O 2p
  Hybridized w.
                                 -1/2    cG ei(G+k)r
                       (r) =   {  (A   G
   Ti 4p, Ti 3d
                                         lmul(r)+Blmůl(r)+Clmül(r))   Ylm(r)
                                 lm


                       Problem: This requires an extra matching
                       condition, e.g. second derivatives
      Ti- 3p           continuous method will be impractical
                       due to the high planewave cut-off needed.
Local orbitals (LO)

                         LOs are
                              confined to an atomic sphere
                              have zero value and slope at R
                              Can treat two principal QN n for
                               each azimuthal QN 
                               ( e.g. 3p and 4p)
                              Corresponding states are strictly
                               orthogonal
                                   (e.g.semi-core and valence)
                              Tail of semi-core states can be
                               represented by plane waves
                              Only slightly increases the basis set
                               (matrix size)


                          D.J.Singh,
                          Phys.Rev. B 43 6388 (1991)
                  THE LAPW+LO METHOD
Key Points:
 1.The local orbitals should only be
   used for those atoms and            Shape of H and S
   angular momenta where they
   are needed.
 2.The local orbitals are just
   another way to handle the             <G|G>
   augmentation. They look very
   different from atomic functions.
 3.We are trading a large number
   of extra planewave coefficients
   for some clm.
              New ideas from Uppsala and Washington

E.Sjöststedt, L.Nordström, D.J.Singh, SSC 114, 15 (2000)
•Use APW, but at fixed El (superior PW convergence)
•Linearize with additional lo (add a few basis functions)

  kn   Am ( kn )u ( E , r )Ym ( r )
                                       ˆ
         m




  lo  [ AmuE1  BmuE1 ]Ym ( r )
                                  ˆ         LAPW           PW
                                               APW

optimal solution: mixed basis
•use APW+lo for states which are difficult to converge:
(f or d- states, atoms with small spheres)
•use LAPW+LO for all other atoms and angular momenta
Improved convergence of APW+lo

                                         force (Fy) on oxygen in SES
                                          vs. # plane waves
                                         in LAPW changes sign
                                          and converges slowly
                                         in APW+lo better
                                          convergence
                                         to same value as in LAPW




 SES (sodium electro solodalite)

  K.Schwarz, P.Blaha, G.K.H.Madsen,
Comp.Phys.Commun.147, 71-76 (2002)
   Relativistic effects

For example:   Ti            Valences states
                                  Scalar relativistc
                                        mass-velocity
                                        Darwin s-shift
                                  Spin orbit coupling on demand by
                                   second variational treatment
                             Semi-core states
                                  Scalar relativistic
                                  No spin orbit coupling
                                  on demand
                                        spin orbit coupling by second
                                         variational treatment
                                        Additional local orbital (see Th-6p1/2)
                             Core states
                                  Full relativistic
                                        Dirac equation
Relativistic semi-core states in fcc Th

                         additional local orbitals for
                          6p1/2 orbital in Th
                         Spin-orbit (2nd variational method)




                     J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz,
                     Phys.Rev.B. 64, 153102 (2001)
   (L)APW methods

                                           spin polarization
  APW + local orbital method               shift of d-bands
  (linearized) augmented plane wave methodband
                                 Lower Hubbard
                                             


         k =  C k n  k n
                                                  (spin up)
               Kn
                                                Upper Hubbard band
Total wave function           k =  C k n  k(spin down)
                                               n     n…50-100 PWs /atom
                                   Kn

                                            <|H | >            < E >
Variational method:             < E >=
                                              <| >              Ck
                                                                            =0
                                                                        n




Generalized eigenvalue problem                       H C= ESC
 Flow Chart of WIEN2k (SCF)
                            Input n-1(r)

                    lapw0: calculates V(r)

          lapw1: sets up H and S and solves
          the generalized eigenvalue problem

                     lapw2: computes the
                     valence charge density

                                 lcore

                                mixer
                                                 yes
               no
                            converged?                  done!
WIEN2k: P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz
     Structure: a,b,c,,,, R , ...

    Structure optimization
                                        k ε IBZ (irred.Brillouin zone)
         iteration i
                                         [  2  V (  )] k  E k k
S
C
        DFT Kohn-Sham                                       Kohn Sham
F                                         k   C kn  kn
     V() = VC+Vxc Poisson, DFT     k            kn



       no     Ei+1-Ei <                 Variational      E
                                                         Ckn    0
                                         method
                       yes               Generalized
                                         eigenvalue      HC  ESC
            Etot,   force
                                          problem
        Minimize E, force0
                                               k * k
            properties                        Ek  E F
                             Brillouin zone (BZ)

   Irreducibel BZ (IBZ)
        The irreducible wedge
        Region, from which the
         whole BZ can be obtained
         by applying all symmetry
         operations

   Bilbao Crystallographic
    Server:
        www.cryst.ehu.es/cryst/
        The IBZ of all space groups
         can be obtained from this
         server
        using the option KVEC and
         specifying the space group
         (e.g. No.225 for the fcc
         structure leading to bcc in
         reciprocal space, No.229 )
WIEN2k software package

                  An Augmented Plane Wave Plus Local
                                 Orbital
                Program for Calculating Crystal Properties


                            Peter Blaha
                         Karlheinz Schwarz
                          Georg Madsen
                         Dieter Kvasnicka
                          Joachim Luitz

                               November 2001
                             Vienna, AUSTRIA
                       Vienna University of Technology
The WIEN2k authors
                         Development of WIEN2k

   Authors of WIEN2k
    P. Blaha, K. Schwarz, D. Kvasnicka, G. Madsen and J. Luitz
   Other contributions to WIEN2k
        C. Ambrosch-Draxl (Univ. Graz, Austria), optics
        U. Birkenheuer (Dresden), wave function plotting
        R. Dohmen und J. Pichlmeier (RZG, Garching), parallelization
        R. Laskowski (Vienna), non-collinear magnetism
        P. Novák and J. Kunes (Prague), LDA+U, SO
        B. Olejnik (Vienna), non-linear optics
        C. Persson (Uppsala), irreducible representations
        M. Scheffler (Fritz Haber Inst., Berlin), forces, optimization
        D.J.Singh (NRL, Washington D.C.), local orbitals (LO), APW+lo
        E. Sjöstedt and L Nordström (Uppsala, Sweden), APW+lo
        J. Sofo (Penn State, USA), Bader analysis
        B. Yanchitsky and A. Timoshevskii (Kiev), space group
   and many others ….
                  International co-operations

   More than 500 user groups worldwide
      25 industries (Canon, Eastman, Exxon, Fuji, A.D.Little, Mitsubishi,
       Motorola, NEC, Norsk Hydro, Osram, Panasonic, Samsung, Sony,
       Sumitomo).
      Europe: (ETH Zürich, MPI Stuttgart, Dresden, FHI Berlin, DESY, TH

       Aachen, ESRF, Prague, Paris, Chalmers, Cambridge, Oxford)
      America: ARG, BZ, CDN, MX, USA (MIT, NIST, Berkeley, Princeton,

       Harvard, Argonne NL, Los Alamos Nat.Lab., Penn State, Georgia Tech,
       Lehigh, Chicago, SUNY, UC St.Barbara, Toronto)
      far east: AUS, China, India, JPN, Korea, Pakistan, Singapore,Taiwan

       (Beijing, Tokyo, Osaka, Sendai, Tsukuba, Hong Kong)
   Registration at www.wien2k.at
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      code download via www (with password), updates, bug fixes, news

      User’s Guide, faq-page, mailing-list with help-requests

				
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