1. Karlheinz Schwarz Institute of Materials Chemistry TU Wien
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

2. Walter Kohn and DFT

3. 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 
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

4. Kohn Sham equations Total energy LDA, GGA Ekinetic Ene Ecoulomb Eee
non interacting
Ene
Ecoulomb Eee
Exc exchange-correlation
vary 
1-electron equation (Kohn Sham)

5. Walter Kohn, Nobel Prize 1998 Chemistry

6. A simple picture of LDA Look at the “LDA” from a different angle
Slater,
Gunnarsson-Lundqvist
…………
Look at the “LDA” from a different angle
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)

7. 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

8. DFT ground state of iron
LSDA NM
fcc
in contrast to
experiment
GGA FM
bcc
Correct lattice constant
Experiment
LSDA
GGA
GGA
LSDA

9. CoO AFM-II total energy, DOS
in NaCl structure
antiferromagnetic: AF II
insulator
t2g splits into a1g and eg‘
GGA almost splits the bands

10. CoO why is GGA better than LSDA
Central Co atom distinguishes
between
and
Angular correlation

11. DFT thanks to Claudia Ambrosch (Graz)
GGA follows LDA

12. Overview of DFT concepts
Form of
potential
Full potential : FP
“Muffin-tin” MT
atomic sphere approximation (ASA)
pseudopotential (PP)
Relativistic treatment
of the electrons
exchange and correlation potential
fully-relativistic
semi-relativistic
non relativistic
Local density approximation (LDA)
Generalized gradient approximation (GGA)
Beyond LDA: e.g. LDA+U
Kohn-Sham equations
Representation
of solid
Basis functions
non periodic
(cluster)
periodic
(unit cell)
plane waves : PW
augmented plane waves : APW
linearized “APWs”
analytic functions (e.g. Hankel)
atomic orbitals. e.g. Slater (STO), Gaussians (GTO)
numerical
Treatment of
spin
Spin polarized
non spin polarized

13. How to solve the Kohn Sham equations
Total energy
LDA, GGA
Ekinetic
non interacting
Ene
Ecoulomb Eee
Exc exchange-correlation
vary 
1-electron equation (Kohn Sham)

14. K.Schwarz, P.Blaha, G.K.H.Madsen,
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, (2002)

15. 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
Full potential
PW:
Atomic partial wave
join

16. Non-linear eigenvalue problem
Slater‘s APW (1937)
Atomic partial waves
Energy dependent
basis functions lead to
Non-linear eigenvalue problem
H Hamiltonian
S overlap matrix
Computationally very demanding
One had to numerically search for the energy, for which
the det(H-ES) vanishes.

17. Linearization of energy dependence
LAPW suggested by
O.K.Andersen,
Phys.Rev. B 12, 3060
(1975)
join PWs in
value and slope
Atomic sphere
LAPW PW
Plane Waves (PWs)

18. Full-potential in LAPW
The potential (and charge density) can be of general form
(no shape approximation)
SrTiO3
Full
potential
Inside each atomic sphere a local coordinate system is used (defining LM)
Muffin tin
approximation Ti
TiO2 rutile O

19. 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

20. 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- 3p
O 2p
Hybridized w.
Ti 4p, Ti 3d
Ti 3d / O 2p EF
„ghostband“
P. Blaha, D.J. Singh, P.I. Sorantin and K. Schwarz, Phys. Rev. B 46, 1321 (1992).

21. {  (Almul(r)+Blmůl(r)+Clmül(r)) Ylm(r) ONE SOLUTION
Treat all the states in a single energy window:
Automatically orthogonal.
Need to add variational freedom.
Could invent quadratic or cubic APW methods.
Electronic Structure E
Ti 3d / O 2p EF
O 2p
Hybridized w.
Ti 4p, Ti 3d
-1/2  cG ei(G+k)r G {
(r) =
 (Almul(r)+Blmůl(r)+Clmül(r)) Ylm(r) lm
Problem: This requires an extra matching condition, e.g. second derivatives continuous method will be impractical due to the high planewave cut-off needed.
Ti- 3p

22. Local orbitals (LO) D.J.Singh, Phys.Rev. B 43 6388 (1991) 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 (1991)

23. THE LAPW+LO METHOD Key Points:
The local orbitals should only be used for those atoms and angular momenta where they are needed.
The local orbitals are just another way to handle the augmentation. They look very different from atomic functions.
We are trading a large number of extra planewave coefficients for some clm.
Shape of H and S
<G|G>

24. 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)
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

25. 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, (2002)

26. Relativistic effects For example: Ti Valences states Semi-core 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

27. 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, (2001)

28. (L)APW methods APW + local orbital method
spin polarization
shift of d-bands
Lower Hubbard band
(spin up)
Upper Hubbard band
(spin down)
APW + local orbital method
(linearized) augmented plane wave method
Total wave function
n… PWs /atom
Variational method:
Generalized eigenvalue problem

29. Flow Chart of WIEN2k (SCF)
Input rn-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
Um zu veranschaulichen, wie so eine Rechnung abläuft habe ich hier ein Flussdiagramm skizziert
Geeignete Startladungsdichte: Superposition/Überlagerung aus atomaren Dichten
Aus der Ladungsdichte wird das Potential konstruiert und zwar Vhartree Lsg der Poisson Gl.
Im Zwischenbereich dank Pseudoladungsmethode im reziproken Raum
In den MTs Randwertproblem aus der Bedingung der Stetigkeit des Potentials am Sphärenrand
Danach werden H und S in der Basis aufgestellt und das verallg. EW Problem gelöst. Dies ist meist der zeitaufwendigste Schritt. Danach wird aus den WF die neue Ladungsdichte berechnet und mit der Information aus vorigen Iterationen die neue Ladungsdichte erzeugt.
mixer
yes no
converged?
done!
WIEN2k: P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz

30. k ε IBZ (irred.Brillouin zone)
Structure: a,b,c,,,, R , ...
Structure optimization
k ε IBZ (irred.Brillouin zone)
iteration i
SCF
DFT Kohn-Sham
Kohn Sham
V() = VC+Vxc Poisson, DFT k
Ei+1-Ei < 
Variational
method no
yes
Generalized
eigenvalue
problem
Etot, force
Minimize E, force0
properties

31. 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:
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 )

32. 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

33. The WIEN2k authors

34. 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 ….

35. 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).
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Registration 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