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Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbitals (APW+lo) method
Karlheinz SchwarzInstitute of Materials Chemistry
TU Wien
Walter Kohn and DFT
DFT Density Functional Theory
Hohenberg-Kohn theoremThe total energy of an interacting inhomogeneous electron gas in the presence of an external potential Vext(r ) is a functional of the density
][)()( FrdrrVE ext
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
][
||
)()(
2
1)(][ xcexto Erdrd
rr
rrrdrVTE
Total energy
Ekinetic non interacting
Ene Ecoulomb Eee Exc exchange-
correlation
1-electron equation (Kohn Sham)
)()())}(())(()(2
1{ 2 rrrVrVrV iiixcCext
FEi
ir
2||)(
LDA, GGA
vary
Walter Kohn, Nobel Prize 1998 Chemistry
A simple picture of LDA
Look at the “LDA” from a different angleSlater,
Gunnarsson-Lundqvist
…………Exc = -∫ dx n(x) e2/ 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)
X
Role of „Gradient corrected functionals“
Becke, Perdew, Wang, Lee, Becke, Perdew, Wang, Lee, Yang, Parr …… ’87 – ‘92Yang, Parr …… ’87 – ‘92
Perdew ,Burke, Ernzerhof Perdew ,Burke, Ernzerhof PBE …… ‘96PBE …… ‘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 FM bcc Correct lattice
constant Experiment
FM bcc
GGAGGA
LSDA
LSDA
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
between
and
Angular correlation
LSDAxc
GGAxcxc VVV Co
Co
DFT thanks to Claudia Ambrosch (Graz)
GGA follows LDA
Overview of DFT concepts
fully-relativisticsemi-relativisticnon relativistic
Full potential : FP“Muffin-tin” MTatomic sphere approximation (ASA)pseudopotential (PP)
Local density approximation (LDA)Generalized gradient approximation (GGA)Beyond LDA: e.g. LDA+U
Spin polarizednon spin polarized
non periodic (cluster)periodic (unit cell)
plane waves : PWaugmented plane waves : APWlinearized “APWs” analytic functions (e.g. Hankel)atomic orbitals. e.g. Slater (STO), Gaussians (GTO)numerical
Basis functions
Treatment of spin
Representationof solid
Form ofpotential
exchange and correlation potential
Relativistic treatment of the electrons
Kohn-Sham equationski
ki
kirV
)(
2
1 2
How to solve the Kohn Sham equations
][
||
)()(
2
1)(][ xcexto Erdrd
rr
rrrdrVTE
Total energy
Ekinetic non interacting
Ene Ecoulomb Eee Exc exchange-
correlation
1-electron equation (Kohn Sham)
)()())}(())(()(2
1{ 2 rrrVrVrV iiixcCext
FEi
ir
2||)(
LDA, GGA
vary
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)
PW:
APW Augmented Plane Wave method
The unit cell is partitioned into:
atomic spheresInterstitial region
Bloch wave function:atomic partial wavesPlane Waves (PWs)
rKkie
).( Atomic partial wave
mm
Km rYrua
)ˆ(),( join
Rmt
unit cell
Full potential
LM
LMLM rYV )ˆ(
K
rKiK eV
.
Rr
Ir
Ir
Slater‘s APW (1937)
Atomic partial waves
Energy dependent
basis functions lead to
m
mKm rYrua
)ˆ(),(
Non-linear eigenvalue problem
One had to numerically search for the energy, for which the det(H-ES) vanishes.
Computationally very demanding
H HamiltonianS overlap matrix
Linearization of energy dependence
LAPW suggested by
)ˆ()],()(),()([ rYrEukBrEukA mnm
mnmkn
rnKkie
).(
Atomic sphere
Plane Waves (PWs)
PW
O.K.Andersen,Phys.Rev. B 12, 3060 (1975)
join PWs in value and slope
LAPW
Full-potential in LAPW
The potential (and charge density) can be of general form (no shape approximation)SrTiO3
Fullpotential
Muffin tinapproximation
Inside each atomic sphere a local coordinate system is used (defining LM)
LM
LMLM rYrV )ˆ()( Rr
K
rKiK eV
.
Ir
TiO2 rutile
TiO
)(rV {
Core, semi-core and valence states
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
For example: Ti
Problems of the LAPW method:
EFG Calculation for Rutile TiO2 as a function of the Ti-p linearization energy Ep
P. Blaha, D.J. Singh, P.I. Sorantin and K. Schwarz, Phys. Rev. B 46, 1321 (1992).
exp. EFG
„ghostband“
Electronic Structure
E
Ti- 3p
O 2pHybridized w.Ti 4p, Ti 3d
Ti 3d / O 2p
EF
ONE SOLUTION
Electronic Structure
E
Ti- 3p
O 2pHybridized w.Ti 4p, Ti 3d
Ti 3d / O 2p
EF
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
G
(Almul(r)+Blmůl(r)+Clmül(r)) Ylm(r)lm
ProblemProblem: This requires an extra matching : This requires an extra matching condition, e.g. second derivatives condition, e.g. second derivatives continuous continuous method will be impractical method will be impractical due to the high planewave cut-off needed.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
)ˆ(),()( rYrEukA mm
nmkn
)ˆ(][ 11 rYuBuA mE
mE
mlo
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)
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
LAPW
APW
PW
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
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
For example: Ti
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
kkK
k nn
n
C =
spin polarization shift of d-bands
Lower Hubbard band (spin up)
Upper Hubbard band (spin down)
0 = C
>E<
>|<
>|H|< = >E<
k n
kkK
k nn
n
C =
C S E= CH
APW + local orbital method (linearized) augmented plane wave method
Total wave function n…50-100 PWs /atom
Variational method:
Generalized eigenvalue problem
Flow Chart of WIEN2k (SCF)
converged?
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
no yesdone!
lcore
mixer
WIEN2k: P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz
Structure: a,b,c,,,, R , ...
Ei+1-Ei <
Etot, force
Minimize E, force0
properties
yes
V() = VC+Vxc Poisson, DFT
DFT Kohn-Sham
Structure optimization
iteration i
no
SCF
k ε IBZ (irred.Brillouin zone)
kkk EV )]([ 2
Kohn Sham
nknknkk C
Variationalmethod
0nkC
E
Generalized eigenvalue problem
ESCHC
FEkE
kk *
k
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 SchwarzGeorg Madsen
Dieter KvasnickaJoachim Luitz
November 2001Vienna, AUSTRIA
Vienna University of Technology
The WIEN2k authors
Development of WIEN2k
Authors of WIEN2kP. 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 400/4000 Euro for Universites/Industries code download via www (with password), updates, bug fixes, news User’s Guide, faq-page, mailing-list with help-requests