Transcript Document 7662574

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
vary 
Ene
non interacting
Ecoulomb Eee
Exc exchange-correlation
1-electron equation (Kohn Sham)
1





{  2  Vext ( r )  VC (  ( r ))  Vxc (  ( r ))} i ( r )   i  i ( r )
2

 (r ) 
2
|

|
 i
i  EF
Walter Kohn, Nobel Prize 1998 Chemistry
A simple picture of LDA
Look at the “LDA” from a different angle
Exc = -∫ dx n(x)
e 2/
R(x)
Slater,
X
Gunnarsson-Lundqvist
…………
R(x) interpreted as the radius of the ‘exchangecorrelation 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,
Yang, Parr …… ’87 – ‘92
Perdew ,Burke, Ernzerhof
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



GGA

LSDA
GGA

GGA



LSDA
NM
fcc
in contrast to
experiment
FM
bcc
Correct lattice
constant
Experiment


FM
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

Vxc  Vxc GGA  Vxc LSDA

Central Co atom distinguishes

between
Co 

and
Co 
Angular correlation
DFT thanks to Claudia Ambrosch
(Graz)
GGA follows LDA
Overview of DFT concepts
Form of
potential
Relativistic treatment
of the electrons
Full potential : FP
“Muffin-tin” MT
atomic sphere approximation (ASA)
pseudopotential (PP)
exchange and correlation potential
fully-relativistic
semi-relativistic
non relativistic
Local density approximation (LDA)
Generalized gradient approximation (GGA)
Beyond LDA: e.g. LDA+U
 1 2
 k
k k



V
(
r
)



i i
 2
 i
non periodic
(cluster)
periodic
(unit cell)
Representation
of solid
Spin polarized
non spin polarized
Treatment of
spin
Kohn-Sham equations
Basis functions
plane waves : PW
augmented plane waves : APW
linearized “APWs”
analytic functions (e.g. Hankel)
atomic orbitals. e.g. Slater (STO), Gaussians (GTO)
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
vary 
Ene
non interacting
Ecoulomb Eee
Exc exchange-correlation
1-electron equation (Kohn Sham)
1





{  2  Vext ( r )  VC (  ( r ))  Vxc (  ( r ))} i ( r )   i  i ( r )
2

 (r ) 
2
|

|
 i
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:
atomic spheres
Interstitial region
Bloch wave function:
atomic partial waves
Plane Waves (PWs)
unit cell
Rmt
rI
Full potential
VLM YLM ( rˆ)
LM
VK e
K
 
iK . r
r  R
rI
PW: e
  
i ( k  K ). r
join
Atomic partial wave
K
a
 mu (r,  )Ym (rˆ)
m
Slater‘s APW (1937)
Atomic partial waves
K
a
 mu (r,  )Ym (rˆ)
m
Energy dependent
basis functions lead to
H Hamiltonian
S overlap matrix
Non-linear eigenvalue problem
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
 ( E , r )]Ym (rˆ)
(kn )u ( E , r )  Bm (kn )u
m
join PWs in
value and slope
Atomic sphere
LAPW
Plane Waves (PWs)
e
 

i ( k  K n ). r
PW
Full-potential in LAPW

SrTiO3
The potential (and charge density)
can be of general form
(no shape approximation)
{
V (r ) 
Full
potential
VLM (r )YLM (rˆ)
LM
VK e
 
iK . r
rI
K

Inside each atomic sphere a
local coordinate system is used
(defining LM)
Muffin tin
approximation
TiO2 rutile
r  R
Ti
O
Core, semi-core and valence states
For example:
Ti

Valences states



Semi-core states




High in energy
Delocalized wavefunctions
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
„ghostband“
P. Blaha, D.J. Singh, P.I. Sorantin and K. Schwarz,
Phys. Rev. B 46, 1321 (1992).
O 2p
Hybridized w.
Ti 4p, Ti 3d
Ti- 3p
ONE SOLUTION
Electronic Structure
E Ti 3d / O 2p
EF
O 2p
Hybridized w.
Ti 4p, Ti 3d
Treat all the states in a single energy
window:
• Automatically orthogonal.
• Need to add variational freedom.
• Could invent quadratic or cubic APW
methods.
(r) =
-1/2
 cG ei(G+k)r
{  (A
G
lmul(r)+Blmůl(r)+Clmül(r))
Ylm(r)
lm
Ti- 3p
Problem: This requires an extra matching
condition, e.g. second derivatives
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
angular momenta where they
are needed.
2.The local orbitals are just
another way to handle the
augmentation. They look very
different from atomic functions.
3.We are trading a large number
of extra planewave coefficients
for some clm.
Shape of H and S
<G|G>
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 ( rˆ)
m
 lo  [ AmuE1  BmuE1 ]Ym ( rˆ)
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




SES (sodium electro solodalite)
K.Schwarz, P.Blaha, G.K.H.Madsen,
Comp.Phys.Commun.147, 71-76 (2002)
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
Relativistic effects
For example:
Ti

Valences states

Scalar relativistc




Spin orbit coupling on demand by
second variational treatment
Semi-core states



Scalar relativistic
No spin orbit coupling
on demand



mass-velocity
Darwin s-shift
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
shift of d-bands
APW + local orbital method 
Lower
Hubbard band
(linearized) augmented plane wave
method

(spin up)
k =  C k n  k n
Kn
Total wave function

Upper Hubbard band
down)
k =  C k n  k(spin
n…50-100 PWs /atom
n
Kn
Variational method:
< E >=
<|H | >
 < E >
<| >
Ck
Generalized eigenvalue problem
n
H C= ESC
=0
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
no
yes
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
S
C
F
[2  V (  )] k  Ek k
DFT Kohn-Sham
V() = VC+Vxc Poisson, DFT
no
Ei+1-Ei < 
yes
Etot,
force
Minimize E, force0
properties
Kohn Sham
k
 k   C kn  kn
kn
0
Variational
method
 E
Ck n
Generalized
eigenvalue
problem
HC  ESC

*

 k k
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 ….
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
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