# ws2003_schwarz

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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
…………
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)

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)
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

 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

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

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)

 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)

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

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
reciprocal space, No.229 )
WIEN2k software package

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

Peter Blaha
Karlheinz Schwarz
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,
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