Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbital (L)APW+lo method

Density functional theory (DFT)
and the concepts of the
augmented-plane-wave plus local orbital
(L)APW+lo method
Karlheinz Schwarz
Institute for Material Chemistry
TU Wien
Vienna University of Technology

K.Schwarz, P.Blaha, S.B.Trickey,
Molecular physics, 108, 3147 (2010)
DFT
APW LAPW
J.C.Slater O.K.Andersen
Wien2k is used worldwide
by about 2600 groups
Electronic structure of
solids and surfaces
hexagonal boron nitride on Rh(111)
2x2 supercell (1108 atoms per cell)
Phys.Rev.Lett. 98, 106802 (2007)

The WIEN2k code: comments
 Walter Kohn: density functional theory (DFT), 1965
 J.C.Slater: augmented plane wave (APW) method, 1937
 O.K.Andersen: Linearized APW (LAPW), 1975
 Wien2k code: developed during the last 35 years
 In the year 2000 (2k) the WIEN code (from Vienna) was called wien2k
 One of the most accurate DFT codes for solids
 All electron, relativistic, full-potential method
 Widely used in academia and industry
 Applications:
 solids: insulators , covalently bonded systems, metals
 Surfaces: catalysis
 Electronic, magnetic, elastic , optical ,…properties
 Many application in literature
 See www.wien2k.at

Review articles
K.Schwarz, P.Blaha, S.B.Trickey, Electronic structure of solids with WIEN2k
K.Schwarz, Computation of Materials Properties at the Atomic Scale
InTech, (2015) Chapter 10, 275-310, open access book
ISBN 978-953-51-2126-8 (2015) dx.doi.org/10.5772/59108
K.Schwarz, P.Blaha, G.K.H.Madsen,
Electronic structure calculations of solids using the WIEN2k package for material Sciences
Comp.Phys.Commun.147, 71-76 (2002)
K.Schwarz, DFT calculations of solids with LAPW and WIEN2k
Solid State Chem.176, 319-328 (2003)
K.Schwarz, P.Blaha, Electronic structure of solids and surfaces with WIEN2k
in Practical Aspects of Computational Chemistry I: An Overview of the Last Two Decades and Current Trends,
J.Leszczyncski, M.K.Shukla (Eds),
Springer Science+Business Media B.V. (2012) Chapter 7, 191-207,
ISBN 978-94-007-0918-8
S.Cottenier,Density Functionl Theory and the famliy of (L)APW methods :
A step by step introduction
2002-2012 /(2nd edition); ISBN 978-90-807215-1-7
Freely available at: http://www.wien2k.at/reg-user/textbooks I

Aspects at this workshop
 Atomic structure
 Periodic boundary condition (approximation)
 Quantum mechanical treatment
 DFT (functionals) and beyond (GW, DMFT, RPA, BSE, …)
 How to solve the QM (basis set)
 LAPW method and local orbitals as implemented in WIEN2k
 Applications
 Structure, surfaces, core-level spectra, NMR, hyperfine, Wannier,…
 Software development
 Accuracy, efficiency, system size, user-friendliness, commercial
 Insight and understanding
 Analysis to find trends, computer experiments (artificial cases)
 Combination of expertise
 Chemistry, physics, mathematics, computer science, application

Four big challenges for theorists :
 large scale applications to simulate “real” materials
 at the atomic scale
 A proper quantum mechanical treatment (accurate)
 develop more accurate DFT functionals
 beyond DFT (GW, DMFT, BSE, RPA…)
 Efficiency (make calculations faster):
 improve numerics, parallelization, algorithms (iterative
diagonalization)
 calculate “new” properties
 for direct comparison with experiment

The atomic structure
 A crystal is represented by a unit cell
 We assume periodic boundary condition (approximation)
 The unit cell is repeated to infinity (makes calculations feasible)
 A real crystal is finite (with surfaces, impurities, defects …)
 Nano materials differ from bulk
 Symmetry helps (space group, Bloch theorem, …)
 In theory
 The atomic structure is an input and thus well defined.
 Artificial structures can be studied too
 In experiment
 The atomic structure is not perfectly known
 Single crystals, micro crystals, powder samples, nano
 e.g. by X-ray: averaged with uncertainties (defects, disorder)

A few solid state concepts
 Crystal structure
 Unit cell (defined by 3 lattice vectors) leading to 7 crystal systems
 Bravais lattice (14)
 Atomic basis (Wyckoff position)
 Symmetries (rotations, inversion, mirror planes, glide plane, screw axis)
 Space group (230)
 Wigner-Seitz cell
 Reciprocal lattice (Brillouin zone)
 Electronic structure
 Periodic boundary conditions
 Bloch theorem (k-vector), Bloch function
 Schrödinger equation (HF, DFT)

b
c
a
b
g
a
Unit cell: a volume in space that
fills space entirely when translated
by all lattice vectors.
The obvious choice:
a parallelepiped defined by a, b, c,
three basis vectors with
the best a, b, c are as orthogonal
as possible
the cell is as symmetric as
possible (14 types)
A unit cell containing one lattice point is called primitive cell.
Unit cell
Assuming an ideal infinite crystal we define a unit cell by

Axis system
primitive
a = b = c
a = b = g = 90°
body centered face centered
P (cP) I (bcc) F (fcc)
Crystal system: e.g. cubic

3D lattice types:
Triclinic 1 “no” symmetry
Monoclinic (P, C) 2 Two right angles
Orthorhombic (P, C, I, F) 4 Three right angles
Tetragonal (P, I) 2 Three right angles + 4 fold rotation
Cubic (P, I, F) 3 Three right angles + 4 fold + 3 fold
Trigonal (Rhombohedral) 1 Three equal angles (≠ 90o)+ 3 fold
Hexagonal 1 Two right and one 120o angle + 6 fold
7 Crystal systems and 14 Bravais lattices

Wigner-Seitz Cell
Form connection to all neighbors and span a plane normal
to the connecting line at half distance

n=3 3x3x3 particle
surface
edge
Fraction of atoms on surface (black) or edge (red) as function of particle size n
Finite particle with a length in nm

The quantum mechanical treatment
 The electronic structure requires a QM treatment
 The main scheme is density functional theory (DFT)
 It is a mean field approach and requires approximations
 According to Hohenberg Kohn, it is sufficient to know the electron
density of a system to determine its total energy. The many electron
wave function (which depends on many variables) is not needed.
In principle this is an enormous simplification, but in practice
approximations must be made.
 The direction of improving the QM treatment is summarized pictorially
in Jabob’s ladder:
 There are schemes which go beyond DFT:
 GW method (for excitations or band gaps)
 The Bethe Salpeter equation (BSE) for excitons (core hole - electron)
 Dynamical mean field theory (DMFT) based on DFT (wien2wannier)

Bloch-Theorem:
  )()()(
2
1 2
rErrV 
1-dimensioanl case:
V(x) has lattice periodicity (“translational invariance”):
V(x)=V(x+a)
The electron density r(x) has also lattice periodicity, however,
the wave function does NOT:
1)()(
:)()()()(
*
*



rr
xax
butxxaxx
Application of the translation t g-times:
)()()( xgaxx gg
 t

periodic boundary conditions:
 The wave function must be uniquely defined: after G
translations it must be identical (G a: periodicity volume):
k
ika
ika
G
g
i
G
GG
xeaxconditionBloch
e
G
g
a
kDef
ge
xxGaxx





)()(:
2
:.
,....21,0
1
)()()()(
2




t

a
G a

Bloch functions:
 Wave functions with Bloch form:
)()(:)()( axuxuwherexuex ikx
k 
Phase factor lattice periodic function
Re [y(x)]
x
Replacing k by k+K, where K is a reciprocal lattice vector,
fulfills again the Bloch-condition.
 k can be restricted to the first Brillouin zone .
a
k
a
e
K
a
i 
1
2

Concepts when solving Schrödingers-equation in solids
non relativistic
semi-relativistic
fully-relativistic
(non-)selfconsistent
“Muffin-tin” MT
atomic sphere approximation (ASA)
Full potential : FP
pseudopotential (PP)
Hartree-Fock (+correlations)
Density functional theory (DFT)
Local density approximation (LDA)
Generalized gradient approximation (GGA)
Beyond LDA: e.g. LDA+U
Non-spinpolarized
Spin polarized
(with certain magnetic order)
non periodic
(cluster)
periodic
(unit cell)
plane waves : PW
augmented plane waves : APW
atomic oribtals. e.g. Slater (STO), Gaussians (GTO),
LMTO, numerical basis
Basis functions
Treatment of
spin
Representation
of solid
Form of
potential
exchange and correlation potential
Relativistic treatment
of the electrons
Schrödinger – equation
(Kohn-Sham equation)
k
i
k
i
k
irV  



 )(
2
1 2

DFT vs. MBT (many body theory)

Coulomb potential:
 nuclei
 all electrons
 including
self-interaction
Quantum mechanics:
 exchange
 correlation
 (partly) cancel
self-interaction

Jacob‘s ladder: The grand challenge:
 Find a functional which leads to “DFT heaven”:
predictive power with chemical accuracy
 J. Perdew‘s „Jacob‘s ladder“ into DFT heaven:
• DFT heaven
• unoccupied orbitals (ACFDT-RPA)
• occupied orbitals (hybrid-DFT)
• meta-GGAs (kinetic energy density t)
• GGA (r)
• LDA (r)
• Hartree

Hohenberg-Kohn theorem: (exact)
The total energy of an interacting inhomogeneous electron gas in the
presence of an external potential Vext(r ) is a functional of the density r
][)()( rr FrdrrVE ext  

In KS 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.
][
||
)()(
2
1
)(][ r
rr
rr xcexto Erdrd
rr
rr
rdrVTE 


 




Ekinetic
non interacting
Ene Ecoulomb Eee Exc exchange-correlation
Kohn-Sham: (still exact!)
DFT Density Functional Theory

Exchange and correlation
 We divide the density of the N-1 electron system
into the total density n(r) and an exchange-
correlation hole:
Properties of the exchange-correlation hole:
 Locality
 Pauli principle
 the hole contains ONE electron
 The hole must be negative
 The exchange hole affects electrons with the
same spin and accounts for the Pauli principle
 In contrast, the correlation-hole accounts for the
Coulomb repulsion of electrons with the opposite
spin. It is short range and leads to a small
redistribution of charge. The correlation hole
contains NO charge:

Kohn-Sham equations
rd
rr
r 


  ||
)(r
r
r

 )(xcE
][
||
)()(
2
1
)(][ r
rr
rr xcexto Erdrd
rr
rr
rdrVTE 


 




1-electron equations (Kohn Sham)
)()())}(())(()(
2
1
{ 2
rrrVrVrV iiixcCext

 rr



FEi
ir

r 2
||)(

LDA, GGA
vary r
LDA treats both,
exchange and correlation effects,
GGA but approximately
-Z/r
drrrFrE
drrrE
GGA
xc
xc
LDA
xc
)](),([)(
)]([)(
.hom
rrr
rr




New (better ?) functionals are still an active field of research

ESSENCE OF DENSITY-FUNTIONAL THEORY
• Every observable quantity of a quantum system can
be calculated from the density of the system ALONE
(Hohenberg, Kohn, 1964).
• The density of particles interacting with each other
can be calculated as the density of an auxiliary
system of non-interacting particles (Kohn, Sham,
1965).
Walter Kohn’s 80
Ecole Normale Supérieur

Walter Kohn
• 1923 born in Vienna
• 1938 had to leave Vienna
• 1946 Univ. Toronto (master, Math)
• 1948 Harvard (PhD, Schwinger)
• Carnegie Mellon, Pittsburgh
(Luttinger), Bell Labs, Copenhagen,
Washigton (Seattle), Paris, Imperial
College (London), ETH Zürich,
Unv. California, San Diego
• 1979 Founding director Santa
Barbara, California
• 1964 Hohenberg Kohn
• 1965 Kohn Sham
• 1998 Nobel prize: Chemistry
• 2016, he died on April 19
With 80 years

Walter Kohn, Nobel Prize 1998 Chemistry
“Self-consistent Equations including Exchange and Correlation Effects”
W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)
Literal quote from Kohn and Sham’s paper:“… We do not expect
an accurate description of chemical binding.”
50 years ago

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

DFT thanks to Claudia Ambrosch (previously in Graz)
GGA follows LDA

Kopmanns‘ and Janak‘s theorem
Slater‘s transion state:
Cl atom
K.Schwarz
Chem. Physics 7, 100 (1975)
Cl
Cl+
Single-electron picture
i
i
n
E


Janak‘s theorem:
Koopmans
theorem:

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

CoO why is GGA better than LSDA
 Central Co atom distinguishes
 between
 and
 Angular correlation
LSDA
xc
GGA
xcxc VVV 
 
Co

Co

Co

Co

Co
O
Co

FeF2: GGA works surprisingly well
FeF2: GGA splits
t2g into a1g and eg’
Fe-EFG in FeF2:
LSDA: 6.2
GGA: 16.8
exp: 16.5
LSDA
GGA
agree

Accuracy of DFT for transition metals
 3d elements:
 PBE superior, LDA
much too small
 4d elements:
LDA too small, PBE too large
 New functional
Wu-Cohen (WC)
 5d elements:
 LDA superior, PBE
too large
Exp. LDA PBE WC
Co 2.51 2.42 2.49 2.45
Ni 3.52 3.42 3.52 3.47
Cu 3.61 3.52 3.63 3.57
Ru 2.71 2.69 2.71 2.73
Rh 3.80 3.76 3.83 3.80
Pd 3.88 3.85 3.95 3.89
Ag 4.07 4.01 4.15 4.07
Ir 3.84 3.84 3.90 3.86
Pt 3.92 3.92 4.00 3.96
Au 4.08 4.07 4.18 4.11
Lattice parameters (Å)
Z.Wu, R.E.Cohen,
PRB 73, 235116 (2006)

accuracy: “DFT limit”
 Testing of DFT functionals:
 error of theoretical lattice
parameters for a large
variety of solids (Li-Th)
me
(Å)
mae
(Å)
mre
(%)
mare
(%)
LDA -0.058 0.058 -1.32 1.32
SO-GGA -0.014 0.029 -0.37 0.68
PBEsol -0.005 0.029 -0.17 0.67
WC 0.000 0.031 -0.03 0.68
AM05 0.005 0.035 0.01 0.77
PBE 0.051 0.055 1.05 1.18
LDA
PBE
PBEsol
AM05
WC

Can LDA be improved ?
 better GGAs and meta-GGAs (r, r, t):
 usually improvement, but often too small.
 LDA+U: for correlated 3d/4f electrons, treat strong Coulomb
repulsion via Hubbard U parameter (cheap, “empirical U” ?)
 Exact exchange: imbalance between exact X and approximate C
 hybrid-DFT (mixing of HF + GGA; “mixing factor” ?)
 exact exchange + RPA correlation (extremely expensive)
 GW: gaps in semiconductors, expensive!
 Quantum Monte-Carlo: very expensive
 DMFT: for strongly correlated (metallic) d (f) -systems (expensive)

Treatment of exchange and correlation

Hybrid functional: only for (correlated) electrons
 Only for certain atoms
and electrons of a given
angular momentum ℓ
The Slater integrals Fk are calculated according to
P.Novák et al., phys.stat.sol (b) 245, 563 (2006)

Structure: a,b,c,a,b,g, Ra , ...
Ei+1-Ei < 
Etot, force
Minimize E, force0
properties
yes
V(r) = VC+Vxc Poisson, DFT
DFT Kohn-Sham
Structure optimization
iteration i
no
S
C
F
k  IBZ (irred.Brillouin zone)
kkk EV r  )]([ 2
Kohn Sham
 
nk
nknkk C
Variational method 0
nkC
E


Generalized eigenvalue problem
ESCHC 


FEkE
kk r *
k
unit cell atomic positions
k-mesh in reciprocal space

Solving Schrödingers equation:
  cannot be found analytically
 complete “numerical” solution is possible but inefficient
 Ansatz:
 linear combination of some “basis functions”
 different methods use different basis sets !
 finding the “best” wave function using the variational principle:
 this leads to the famous “Secular equations”, i.e. a set of linear
equations which in matrix representation is called “generalized
eigenvalue problem”
H C = E S C
H, S : hamilton and overlap matrix; C: eigenvectors, E: eigenvalues
k
i
k
i
k
irV 



 )(
2
1 2
 
n
nn
K
kkk c
0*
*






nk
k
kk
kk
k
c
EH
E

Basis Sets for Solids
 plane waves
 pseudo potentials
 PAW (projector augmented wave) by P.E.Blöchl
 space partitioning (augmentation) methods
 LMTO (linear muffin tin orbitals)
 ASA approx., linearized numerical radial function
+ Hankel- and Bessel function expansions
 full-potential LMTO
 ASW (augmented spherical wave)
 similar to LMTO
 KKR (Korringa, Kohn, Rostocker method)
 solution of multiple scattering problem, Greens function formalism
 equivalent to APW
 (L)APW (linearized augmented plane waves)
 LCAO methods
 Gaussians, Slater, or numerical orbitals, often with PP option)

pseudopotential plane wave methods
 plane waves form a “complete” basis
set, however, they “never” converge
due to the rapid oscillations of the
atomic wave functions  close to the
nuclei
 let´s get rid of all core electrons and
these oscillations by replacing the
strong ion–electron potential by a
much weaker (and physically dubious)
pseudopotential
 Hellmann´s 1935 combined
approximation method


r
Veff
Pseudo-
Pseudo-r
Pseudo-potential
x
x
x
r
“real” potentials vs. pseudopotentials
• “real” potentials contain the Coulomb singularity -Z/r
• the wave function has a cusp and many wiggles,
• chemical bonding depends mainly on the overlap of the
wave functions between neighboring atoms (in the region
between the nuclei) 
exact form of V only needed beyond rcore rcore
exact 
exact V
exact r

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,
DFT calculations of solids with LAPW and WIEN2k
Solid State Chem.176, 319-328 (2003)
K.Schwarz, P.Blaha, S.B.Trickey,

PW:
APW Augmented Plane Wave method
The unit cell is partitioned into:
atomic spheres
Interstitial region
rKki
e

).( 
Atomic partial waves
 
m
m
K
m rYruA

 )ˆ(),( 
join
Rmt
unit cell
Basis set:
ul(r,) are the numerical solutions
of the radial Schrödinger equation
in a given spherical potential
for a particular energy 
Alm
K coefficients for matching the PW
Ir 
Plane Waves
(PWs)
PWs atomic

Slater‘s APW (1937)
Atomic partial waves
Energy dependent basis functions
lead to a
Non-linear eigenvalue problem
 
m
m
K
m rYrua

 )ˆ(),( 
Numerical search for those energies, for which
the det|H-ES| vanishes. Computationally very demanding.
“Exact” solution for given MT potential!
H Hamiltonian
S overlap matrix

Linearization of energy dependence
LAPW suggested by
)ˆ()],()(),()([ rYrEukBrEukA mn
m
mnmkn 


 
Atomic sphere
PW
O.K.Andersen,
Phys.Rev. B 12, 3060
(1975)
expand ul at fixed energy El and
add
Alm
k, Blm
k: join PWs in
value and slope
 General eigenvalue problem
(diagonalization)
 additional constraint requires
more PWs than APW
LAPW
 /ll uu
APW
bonding
antibonding
center

shape approximations to “real” potentials
 Atomic sphere approximation (ASA)
 overlapping spheres “fill” all volume
 potential spherically symmetric
 “muffin-tin” approximation (MTA)
 non-overlapping spheres with spherically
symmetric potential +
 interstitial region with V=const.
 “full”-potential
 no shape approximations to V

Full-potential in LAPW (A.Freeman et al)
 The potential (and charge density)
can be of general form
(no shape approximation)SrTiO3
Full
potential
Muffin tin
approximation
 Inside each atomic sphere a
local coordinate system is used
(defining LM)

LM
LMLM rYrV )ˆ()( aRr 

K
rKi
K eV

.
Ir 
TiO2 rutile
Ti
O
)(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 (charge leakage)
 Core states
 Low in energy
 Reside inside sphere
For example: Ti
-356.6
-31.7 Ry
-38.3 1 Ry =13.605 eV

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)
Ti atomic sphere
)ˆ(][ 211
rYuCuBuA m
E
m
E
m
E
mLO 
 

An alternative combination of schemes
)ˆ(),()( rYrEukA m
m
nmkn 


E.Sjöstedt, L.Nordström, D.J.Singh,
An alternative way of linearizing the augmented plane wave method,
Solid State Commun. 114, 15 (2000)
• Use APW, but at fixed El (superior PW convergence)
• Linearize with additional local orbitals (lo)
(add a few extra basis functions)
)ˆ(][ 11
rYuBuA m
E
m
E
mlo 

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
e.g. 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)
Representative Convergence:
SES

Summary: Linearization LAPW vs. APW
 Atomic partial waves
 LAPW
 APW+lo
 Plane Waves (PWs)
 match at sphere boundary
 LAPW
value and slope
 APW
value
)ˆ()],()(),()([ rYrEukBrEukA mn
m
mnmnk 

  
rnKki
e

).( 
)ˆ(),()( rYrEukA m
m
nmnk 


)( nm kA
)(),( nmnm kBkA 
Atomic sphere
PW
plus another type of local orbital (lo)
LAPW
APW
Fe

Method implemented in WIEN2k
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
A summary is given in

The WIEN2k authors
G.Madsen
P.Blaha
D.Kvasnicka
K.Schwarz
J.Luitz
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
http://www.wien2k.at

International users
about 2600 licenses worldwide
100 industries (Canon, Eastman, Exxon,
Fuji, Hitachi, IBM, Idemitsu Petrochem.,
Kansai, Komatsu, Konica-Minolta, A.D.Little,
Mitsubishi, Mitsui Mining, Motorola, NEC,
Nippon Steel, Norsk Hydro, Osram,
Panasonic, Samsung, Seiko Epson,
Siemens, Sony, Sumitomo,TDK,Toyota).
Europe: A, B, CH, CZ, D, DK, ES, F,
FIN, GR, H, I, IL, IRE, N, NL, PL, RO,
S, SK, SL, SI, UK (ETH Zürich, MPI
Stuttgart, FHI Berlin, DESY, RWTH
Aachen, ESRF, Prague, IJS Ljubjlana,
Paris, Chalmers, Cambridge, Oxford)
America: ARG, BZ, CDN, MX, USA
(MIT, NIST, Berkeley, Princeton,
Harvard, Argonne NL, Los Alamos
NL, Oak Ridge NL, Penn State,
Purdue, Georgia Tech, Lehigh, John
Hopkins, Chicago, Stony Brook,
SUNY, UC St.Barbara, UCLA)
far east: AUS, China, India, JPN,
Korea, Pakistan, Singapore,Taiwan
(Beijing, Tokyo, Osaka, Kyoto,
Sendai, Tsukuba, Hong Kong)
mailinglist: 10.000 emails/6 years

The first publication of the WIEN code

Europa Austria Vienna  WIEN
In the Heart of EUROPE
Austria
Vienna

In Japan
 Book published by
Shinya Wakoh (2006)

Development of WIEN2k
 Authors of WIEN2k
P. Blaha, K. Schwarz, D. Kvasnicka, G. Madsen and J. Luitz
 Other contributions to WIEN2k
 C. Ambrosch-Draxl (Free Univ. Berlin) optics
 E. Assmann (Vienna) Wannier functions
 F. Karsai (Vienna) parallelization
 R. Laskowski (Singapore), non-collinear magnetism, NMR chemical shifts, BSE
 L. Marks (Northwestern, US) , various optimizations, new mixer
 P. Novák and J. Kunes (Prague), LDA+U, SO
 B. Olejnik (Vienna), non-linear optics,
 C. Persson (Uppsala), irreducible representations
 V. Petricek (Prague) 230 space groups
 O. Rubel (McMaster Univ. Hamiton, ON) Berry phases
 M. Scheffler (Fritz Haber Inst., Berlin), forces
 D.J.Singh (NRL, Washington D.C., Oak Ridge), local oribtals (LO), APW+lo
 E. Sjöstedt and L Nordström (Uppsala, Sweden), APW+lo
 J. Sofo (Penn State, USA) and J. Fuhr (Barriloche), Bader analysis
 F. Tran (Vienna) Hartree Fock, DFT functionals
 B. Yanchitsky and A. Timoshevskii (Kiev), space group
 and many others ….

2650 users
 1st Vienna April 1995 Wien95
 2nd Vienna April 1996
 3rd Vienna April 1997 Wien97
 4st Trieste, Italy June 1998
 5st Vienna April 1999
 6th Vienna April 2000
 7th Vienna Sept. 2001 Wien2k
 8th Esfahan, Iran April 2002
 Penn State, USA July 2002
 10th Penn State, USA July 2004
 11th Kyoto, Japan May 2005
 IPAM, Los Angeles, USA Nov. 2005
 13th Penn State, USA June 2007
 14th Singapore July 2007
 15th Vienna March 2008
 17th Nantes, France July 2010
 19th Tokyo, Japan Sept 2012
 20th Penn State, USA Aug. 2013
 21th Nantes, France July 2014
 Warsaw, Poland Oct. 2014
 22nd Singapore Aug. 2015
 23rd McMaster, Canada June 2016
A series of WIEN workshops were held

(L)APW methods
kk
K
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




kk
K
k nn
n
C= 
APW + local orbital method
(linearized) augmented plane wave method
Total wave function n…50-100 PWs /atom
Variational method:
Generalized eigenvalue problem: H C=E S C
Diagonalization of (real or complex) matrices of
size 10.000 to 50.000 (up to 50 Gb memory)
upper bound minimum

The Brillouin zone (BZ)
 Irreducible 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 )

Self-consistent field (SCF) calculations
 In order to solve H=E we need to know the potential V(r)
 for V(r) we need the electron density r(r)
 the density r(r) can be obtained from (r)*(r)
 ?? (r) is unknown before H=E is solved ??
SCF cycles
Start with rin(r)
Calculate Veff (r) =f[r(r)]
)()()}(
2
1
{Solve 2
rrr iiieffV  


Fi E
i

r 2
|)(|)(Compute rr
Do the mixing of r(r)
SCF cycles
Start with rin(r)
Calculate Veff (r) =f[r(r)]
)()()}(
2
1
{Solve 2
rrr iiieffV  


Fi E
i

r 2
|)(|)(Compute rr
Do the mixing of r(r)

Effects of SCF
Band structure of fcc Cu

Program structure of WIEN2k
 init_lapw
 initialization
 symmetry detection (F, I, C-
centering, inversion)
 input generation with
recommended defaults
 quality (and computing time)
depends on k-mesh and R.Kmax
(determines #PW)
 run_lapw
 scf-cycle
 optional with SO and/or LDA+U
 different convergence criteria
(energy, charge, forces)
 save_lapw tic_gga_100k_rk7_vol0
 cp case.struct and clmsum files,
 mv case.scf file
 rm case.broyd* files

Flow Chart of WIEN2k (SCF)
converged?
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
no
yes
done!
lcore
mixer
WIEN2k: P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz

Workflow of a WIEN2k calculation
• individual FORTRAN programs linked by shell-scripts
• the output of one program is input for the next
• lapw1/2 can run in parallel on many processors
LAPW0
LAPW1
LAPW2
LCORE
LAPW0
LCORE
MIXER
SUMPARA
LAPW1
LAPW2
SCF cycle
single mode parallel mode
Iteration
Iteration
75 %
20 %
3 %*
1%
1%
self
consistent? END
yesno
MIXER
self
consistent?END
yes no
k-point parallelization
* fraction of total computation time

Advantage/disadvantage of WIEN2k
+ robust all-electron full-potential method (new effective mixer)
+ unbiased basisset, one convergence parameter (LDA-limit)
+ all elements of periodic table (comparable in CPU time), metals
+ LDA, GGA, meta-GGA, LDA+U, spin-orbit
+ many properties and tools (supercells, symmetry)
+ w2web (for novice users)
? speed + memory requirements
+ very efficient basis for large spheres (2 bohr) (Fe: 12Ry, O: 9Ry)
- less efficient for small spheres (1 bohr) (O: 25 Ry)
- large cells, many atoms (n3, but new iterative diagonalization)
- full H, S matrix stored  large memory required
+ effective dual parallelization (k-points, mpi-fine-grain)
+ many k-points do not require more memory
- no stress tensor
- no linear response

w2web GUI (graphical user interface)
 Structure generator
 spacegroup selection
 import cif file
 step by step initialization
 symmetry detection
 automatic input generation
 SCF calculations
 Magnetism (spin-polarization)
 Spin-orbit coupling
 Forces (automatic geometry
optimization)
 Guided Tasks
 Energy band structure
 DOS
 Electron density
 X-ray spectra
 Optics

Structure given by:
spacegroup
lattice parameter
positions of atoms
(basis)
Rutile TiO2:
P42/mnm (136)
a=8.68, c=5.59 bohr
Ti: (0,0,0)
O: (0.304,0.304,0)
Wyckoff position: x, x, 0
Spacegroup P42/mnm
2a
4f
Ti
O

Quantum mechanics at work
thanks to Erich Wimmer

TiC electron density
 NaCl structure (100) plane
 Valence electrons only
 plot in 2 dimensions
 Shows
 charge distribution
 covalent bonding
 between the Ti-3d and C-2p
electrons
 eg/t2g symmetry
C
Ti

TiC, three valence states at Δ
Energy bands
C-2p
C-2s
Ti-3d
Ti-4s
Cp-Tid σ Tid-Tid σ
Cp-Tid 
P.Blaha, K.Schwarz,
Int.J.Quantum Chem. 23, 1535 (1983)
(100) plane

TiC, energy bands
P.Blaha, K.Schwarz,
spaghetti irred.rep. character bands
unit cell interstitial atom t ℓ=s, p, d, …


t
t
out qq1

TiC, bonding and antibonding states
P.Blaha, K.Schwarz,
C-2p
O-2p
Ti-3d
bonding
antibonding
weight: C Ti
O Ti
C-2s
O-2s

Bonding and antibondig state at Δ1
antibonding
Cp-Tid σ
bonding
Cp-Tid σ

TiC, TiN, TiO
P.Blaha, K.Schwarz,
Rigid band model: limitations
TiC TiN TiO
Electron density r: decomposition
unit cell interstitial atom t ℓ=s, p, d, …


t
t
out qq1

TiC, TiN, TiO Atomic form factors for Ti and C
Paired reflections
Experimental difference electron density

Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbital (L)APW+lo method

The most accurate code
AE: all electron

Precision of DFT calculations illustrated for Si

Crucial aspects for a simulation
These aspects need to be considered when comparing theory with experiment.
stoichiometry electron core-hole vacuum average
disorder satellites supercell vibrations
impurities, defects all electron ℓ quantum n.
relativistic effects
Theory vs. experiment: Agreement or disagreement:
What can cause it?

Vienna, city of music
and the Wien2k code

Walter Kohn
Walter Kohn’s 80
Ecole Normale Supérieur
 1997 DFT Conferene in Vienna
 Walter Kohn Main speaker
 1998 Nobel Prize
for Chemmistry
 2001 invited lecturer
 My 60th birthday

WIEN2k
W
I
E
N
2
k
orld of
nteracting
lectrons and
uclei
ohn
due
walter

Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbital (L)APW+lo method

More Related Content

What's hot (20)

Similar to Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbital (L)APW+lo method (20)

More from ABDERRAHMANE REGGAD (20)

Recently uploaded (20)

Density functional theory (DFT) and the concepts of the augmented-plane-wave plus local orbital (L)APW+lo method