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