Solid State Calculations Using WIEN2k

Computational Materials Science 28 (2003) 259–273

www.elsevier.com/locate/commatsci

Solid state calculations using WIEN2k

Karlheinz Schwarz *, Peter Blaha

Institute for Materials Chemistry, Vienna University of Technology, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria

Abstract

To study solid materials on the atomic scale one often starts with an ideal crystal at zero temperature and calculates

its electronic structure by means of density functional theory (DFT). This allows a quantum mechanical treatment of

the physics that underlines properties such as relative stability, chemical bonding, relaxation of the atoms, phase

transitions, electrical, mechanical, optical or magnetic behavior, etc. For the solution of the DFT equations several

methods have been developed. The linearized-augmented-plane-wave method is one of the most accurate methods. It is

embodied in the computer code––WIEN2k––which is now used worldwide by more than 500 groups to solve crystal

properties on the atomic scale (see www.wien2k.at). Nowadays calculations of this type can be done––on sufficiently

powerful computers––for systems containing about 100 atoms per unit cell. Chromium dioxide CrO2 is selected as a

representative example using both, bulk and surface structures. References to other applications are given.

� 2003 Elsevier B.V. All rights reserved.

PACS: 71.10; 71.15; 71.20; 71.22

Keywords: WIEN2k; Density functional theory (DFT); Electronic structure

1. Introduction

In advanced solid materials, which are of great

technological interest, several parameters such as

temperature, pressure, structure, composition, and

disorder determine their properties. They are

governed by very different length and time scales

which may vary by many orders of magnitudedepending on their applications. Let us focus on

the length scale, where from meters (m) down to

micrometers (lm) classical mechanics and contin-

uum models are the dominating concepts to in-

* Corresponding author.

E-mail address: [email protected] (K. Sch-

warz).

URL: http://info.tuwien.ac.at/theochem/.

0927-0256/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/S0927-0256(03)00112-5

vestigate the properties of the corresponding

materials. However, when one comes to the nano-

meter (nm) scale or atomic dimensions measured

in �AA, the properties are determined by the elec-

tronic structure of the solid.

In the development of modern materials an

understanding on the atomic scale is frequently

essential in order to replace trial and error proce-dures by a systematic materials design. Modern

devices in the electronic industry provide such an

example, where the increased miniaturization is

one of the key advances. Other applications are

found in the area of magnetic recording or other

storage media. In chemistry heterogeneous catal-

ysis is a typical example, in which one likes to

understand the details of catalytic processes be-tween molecules interacting with a solid surface as

for example in a zeolite.

ed.

http://www.wien2k.at

mail to: [email protected]

http://info.tuwien.ac.at/theochem/

260 K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273

One possibility to study complex systems is to

perform computer simulations. Calculation of

solids in general (metals, insulators, semiconduc-

tors, minerals, etc.) can be performed with a va-

riety of methods from classical to quantummechanical approaches. The former are force field

schemes, in which the forces that determine the

interactions between the atoms are parameterized

in order to reproduce a series of experimental data

such as equilibrium geometries, bulk moduli or

special vibrational frequencies (phonons). These

schemes have reached a high level of sophistication

and are often used within a given class of materialsprovided good parameters are already known

from closely related systems. If, however, such

parameters are not available, or if a system shows

unusual phenomena that are not yet understood,

one often must rely on ab initio calculations. These

are more demanding in terms of computer re-

quirements and thus allow only the treatment of

smaller unit cells than force-field calculations. Theadvantage of first-principle (ab initio) methods lies

in the fact that they can be carried out without

any experimental knowledge of the system. In the

following we will restrict ourselves to ab initio

methods and sketch their main characteristics.

The fact that electrons are indistinguishable and

are Fermions requires that their wave functions

must be anti-symmetric when two electrons areinterchanged. This situation leads to the phe-

nomenon of exchange. There are two types of

approaches for a full quantum mechanical treat-

ment, HF and DFT. The traditional scheme is (or

was) the Hartree Fock (HF) method which is

based on a wave function description (with one

Slater determinant). Exchange is treated exactly

but correlation effects are ignored by definition.The latter can be included by more sophisticated

approaches such as the configuration interaction

scheme that progressively requires more computer

time with a scaling as bad as N 7 when the system

size (N ) grows. As a consequence it is only feasible

to study small systems, which contain a few atoms.

An alternative scheme is density functional

theory (DFT) that is commonly used to calculatethe electronic structure of complex systems con-

taining many atoms such as large molecules or

solids. It is based on the electron density rather

than on wave functions and treats exchange and

correlation, but both approximately. It will be

discussed in the next section.

The ideal crystal is defined by the unit cell

which may contain several (up to about 100)atoms. Periodic boundary conditions are used to

describe the infinite crystal by knowing the prop-

erties in one unit cell. Translational and point

group symmetry operations (inversion, rotation,

mirror planes, etc.) that leave the ideal crystal in-

variant allow to simplify the calculations, which all

correspond to the absolute temperature zero

(T ¼ 0).

2. Density functional theory

2.1. The Kohn–Sham equation

The well-established scheme to calculate elec-

tronic properties of solids is based on DFT, for

which Walter Kohn has received the Nobel Prize

in chemistry in 1998. DFT is a universal approach

to the quantum mechanical many-body problem,

where the system of interacting electrons is map-ped in a unique manner onto an effective non-

interacting system with the same total density.

Hohenberg and Kohn [1] have shown that the

electron density . uniquely defines the total energy

Etot of a system and is a functional EtotðqÞ of the

density:

EtotðqÞ ¼ TsðqÞ þ EeeðqÞ þ ENeðqÞ þ ExcðqÞ þ ENN:

ð1ÞThe different electronic contributions are conven-

tionally labeled as, respectively, the kinetic energy

(of the non-interacting particles), the electron–

electron repulsion, nuclear–electron attraction,and exchange-correlation energies. The last term

corresponds to the repulsive Coulomb energy of

the fixed nuclei ENN.

The most efficient way to minimize Etot by

means of the variational principle is to introduce

orbitals vik constrained to construct the densities

as a sum over occupied orbitals according to the

aufbau principle (with increasing energy)

qðrÞ ¼Xi;k

qikjvikðrÞj2: ð2Þ

K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273 261

Here, the qik are occupation numbers such that

06 qik 6 1=wk, where wk is the symmetry-required

weight of point k. Then variation of Etot gives a set

of effective one-particle Schr€oodinger equations(written for an atom in Rydberg atomic units), the

so-called Kohn–Sham equations [2],

½�r2 þ VNe þ Vee þ Vxc�vikðrÞ ¼ �ikvikðrÞ; ð3Þ

which must be solved. The non-interacting parti-

cles of this auxiliary system move in an effective

local one-particle potential, which consists of a

classical mean-field (Hartree) part and an ex-change-correlation part Vxc (due to quantum me-

chanics) that, in principle, incorporates all

correlation effects exactly. The KS equations must

be solved in an iterative process till self-consistency

is reached, since finding the KS orbitals requires

the knowledge of the potentials which themselves

depend on the density and thus on the orbitals

again.The exact functional form of Exc or the poten-

tial Vxc is not known and thus one needs to make

approximations. Early applications were done by

using results from quantum Monte Carlo calcula-

tions for the homogeneous electron gas, for which

the problem of exchange and correlation can be

solved numerically with high accuracy, leading to

the original local density approximation (LDA).LDA works reasonably well but has some short-

comings mostly due to the tendency of overbind-

ing, which cause e.g. too small lattice constants.

Modern versions of DFT, especially those using

the generalized gradient approximation (GGA),

improve the LDA by adding gradient terms of the

electron density and reach (almost) chemical ac-

curacy. At present the version by Perdew–Burke–Ernzerhof is the recommended option for solids

[3].

In the study of large systems the strategy of

schemes based on HF or DFT differ. In HF based

methods the Hamiltonian is well defined but can

be solved only approximately. In DFT, however,

one must first choose (approximate) the functional

that is used to represent the exchange and corre-lation effects but then this effective Hamiltonian

can be solved almost exactly, i.e. with very high

numerical accuracy. Thus the approximation nec-

essarily enters in both cases but the sequence is

reversed. This perspective illustrates the impor-

tance of improving the functional in DFT calcu-

lations, since this defines the quality of the

calculation.

2.2. Spin polarization and magnetism

Magnetism can come from localized electrons

(e.g. from the 4f electron in rare earth elements) or

from delocalized electrons. In the latter case the

itinerant electrons can be well described by band

theory. Without spin–orbit coupling the magne-

tism originates from the spin polarization, whereasthe orbital moment is often quenched. For exam-

ple, Fe, Co, Ni are ferromagnets (FM) which fall

in this category. In this case DFT is generalized to

allow for a spin polarization, in which the spin-up

and spin-down electrons move in spin-dependent

potentials, which can be deduced from the corre-

sponding spin densities, for example using the

local spin density approximation. Another case areantiferromagnets (AFM) (e.g. bcc Cr) in which

neighboring atoms have the opposite spin align-

ment but the magnitude of the moment at each site

is the same. The net bulk magnetization vanishes

for AFM. When the magnitude differs we have

ferrimagnetism and a net magnetization. All these

cases belong to collinear magnets. The orientation

of the magnetic moment with respect to the crystalaxis is only defined when spin–orbit coupling is

included. There are more complicated magnetic

structures such as canted magnetic moments or

spin spirals. In such cases the direction of the

magnetization varies with position and this is

called non-collinear magnetism.

2.3. Choice of basis set, wave function, and potential

Many methods are available to solve the DFT

equations. Essentially all use a linear combination

of basis functions. Some use a linear combination

of atomic orbitals (LCAO) method with Gauss-

ian or Slater type orbitals (GTOs or STOs) others

use muffin tin orbitals (MTOs) as in linear com-

bination of MTOs or augmented spherical wave.In the former cases the basis functions are given in

analytic form, but in the latter the radial wave

functions are obtained by numerically integrating

II

II

Fig. 1. Partitioning of the unit cell into atomic spheres (I) and

an interstitial region (II).


the radial Schr€oodinger equation. Alternatively

plane waves (PWs) form in principle a complete

basis set. However, convergence is slow and thus

one must use either a pseudo-potential or augment

them with some other functions. Usually thisaugmentation is done with some atomic-like basis

set.

In the muffin tin or the atomic sphere approxi-

mation (MTA or ASA) an atomic sphere, in which

the potential (and charge density) is assumed to be

spherically symmetric, surrounds each atom in the

crystal. While these approximations work reason-

ably well in highly coordinated, closely packedsystems (as for example face centered cubic metals)

they become very approximate in all non-isotropic

cases (e.g. layered compounds, semiconductors, or

other open structures). Schemes that make no

shape approximation in the form of the potential

are termed full-potential schemes (see Section 3.2).

Closely related to the basis sets is the explicit form

of the wave functions, which should be well rep-resentable. These can be node-less pseudo-wave-

functions or all-electron wave-functions including

the complete radial nodal structure and a proper

description close to the nucleus.

With a proper choice of pseudo-potential one

can focus on the valence electrons, which are rel-

evant for chemical bonding, and replace the inner

part of their wave functions by node-less pseudo-functions that can be expanded in PWs with good

convergence.

2.4. Choice of method

As a consequence of the aspects described

above different methods have their advantages or

disadvantages when it comes to compute variousquantities. For example, properties that rely on the

knowledge of the density close to the nucleus

(hyperfine fields, electric field gradients, etc.), re-

quire an all-electron description rather than a

pseudo-potential approach with un-physical wave

functions near the nucleus. On the other hand for

studies, in which the shape (and symmetry) of the

unit cell changes, the knowledge of the corre-sponding stress tensor is needed for an efficient

structural optimization. These tensors are much

easier to obtain in pseudo-potential schemes and

thus are available there. In augmentation schemes

such algorithms become more tedious and conse-

quently are often not implemented. On the other

hand all-electron methods do not depend on

choices of pseudo-potentials and contain the fullwave function information. Thus the choice of

method for a particular application depends on

the property of interest and may affect the ac-

curacy, ease or difficulty to calculate a given

property.

3. The linearized-augmented-plane-wave method

One among the most accurate schemes for

solving the Kohn–Sham equations (see Section 2)

is the full-potential linearized-augmented-plane-

wave (FP-LAPW) method suggested by Andersen

[4] on which our WIEN code is based. For further

details see e.g. the book by Singh [5] with many

references and details. During the last two decadesthis FP-LAPW code has been developed in our

group and is used worldwide by more than 500

groups at universities and industrial laboratories.

The original version was the first LAPW code that

was published and thus was made available for

other users (WIEN) [6].

In the LAPW method [4] a basis set is intro-

duced that is especially adapted to the problem bydividing the unit cell into (I) non-overlapping

atomic spheres (centered at the atomic sites) and

(II) an interstitial region (Fig. 1). For the con-

struction of basis functions––but only for that

purpose––the MTA is used according to which the

potential is assumed to be spherically symmetric

within the atomic spheres but constant outside. In


the two types of regions different basis sets are

used.

(I) inside atomic sphere t, of radius Rt, a linear

combination of radial functions times spherical

harmonics YlmðrÞ is used (we omit the index t whenit is clear from the context)

/kn¼

Xlm

½Alm;knulðr;ElÞ þ Blm;kn _uulðr;ElÞ�Ylmðr̂rÞ;

ð4Þwhere ulðr;ElÞ is the (at the origin) regular solutionof the radial Schr€oodinger equation for energy El

(usually chosen at the center of bands with the

corresponding ‘-like character) and _uulðr;ElÞ is theenergy derivative of ul evaluated at the same en-

ergy El. A linear combination of these two func-

tions linearize the energy dependence of the radial

function; the coefficients Alm and Blm are functions

of kn (see below) and are determined by requiringthat this basis function matches (in value and

slope) the PW labeled with kn, the corresponding

basis function of the interstitial region. The func-

tions ul and _uul are obtained by numerical inte-

gration of the radial Schr€oodinger equation on a

radial mesh inside sphere t within the spherical

part of the potential.

(II) in the interstitial region a PW expansion isused

/kn¼ 1ffiffiffiffi

xp eikn�r; ð5Þ

where kn ¼ kþ Kn; Kn are the reciprocal lattice

vectors and k is the wave vector inside the first

Brillouin zone. Each PW is augmented by an

atomic-like function in every atomic sphere asdescribed above.

The solutions to the Kohn–Sham equations are

expanded in this combined basis set of LAPW�saccording to the linear variation method

wk ¼Xn

cn/knð6Þ

and the coefficients cn are determined by the

Rayleigh–Ritz variational principle. The conver-

gence of this basis set is controlled by a cutoff

parameter RmtKmax ¼ 6–9, where Rmt is the smallest

atomic sphere radius in the unit cell and Kmax is the

magnitude of the largest Kn vector in Eq. (6).

In order to improve upon the linearization (i.e.

to increase the flexibility of the basis) and to make

possible a consistent treatment of semicore and

valence states in one energy window (to ensure

orthogonality) additional (kn independent) basisfunctions can be added. They are called ‘‘local

orbitals (LO)’’ according to Singh [7] and consist

of a linear combination of two radial functions at

two different energies (e.g. at the 3s and 4s ener-

gies) and one energy derivative (at one of these

energies):

/LOlm ¼ ½Almulðr;E1;lÞ þ Blm _uulðr;E1;lÞ

þ Clmulðr;E2;lÞ�Ylmðr̂rÞ: ð7Þ

The coefficients Alm, Blm and Clm are determined by

the three requirements that /LO should be nor-malized and has zero value and slope at the sphere

boundary.

3.1. Different implementations of APW

The energy dependence of the atomic radial

functions described above can be treated in dif-

ferent ways. In APW this is done by finding theenergy that corresponds to the eigenenergy of each

state. This leads to a non-linear eigenvalue prob-

lem, since the basis functions become energy de-

pendent. In LAPW a linearization of the energy

dependence is introduced by solving the radial

Schr€oodinger equation for a fixed linearization en-

ergy but adding the energy derivative of this

function [4,5]. The corresponding two coefficientscan be chosen such as to match (at the atomic

sphere boundary) the atomic solution to each PW

in value and slope, which determine the two co-

efficients of the function and it�s derivative (see Eq.(4)). LAPW leads to a standard general eigenvalue

problem but the PW basis is less efficient than in

APW. A new scheme, APW plus local orbitals, [8]

combines the advantages of both methods. As inLAPW a fixed linearization energy is used but the

energy dependence is covered by using a kn-inde-pendent local orbital. The matching is only done in

value, but this new scheme leads to a significant

speed-up of the method (up to an order of mag-

nitude) while keeping the high accuracy of LAPW

[9]. A description of these three types of schemes


(APW, LAPW, APW+ lo) mentioned above is

given in [10], which contains many additional ref-

erences. A combination of these schemes (LAPW

and APW+ lo) is the basis for the new WIEN2k

program [11].

3.2. The full-potential scheme

The MTA was frequently used in the 1970s and

works reasonable well in highly coordinated (me-

tallic) systems such as face centered cubic (fcc)

metals. However, for covalently bonded solids,

open or layered structures, MTA is a poor ap-proximation and leads to serious discrepancies

with experiment. In all these cases a treatment

without any shape approximation is essential.

Both, the potential and charge density, are ex-

panded into lattice harmonics (inside each atomic

sphere) and as a Fourier series (in the interstitial

region).

V ðrÞ ¼P

LM VLMðrÞYLMðr̂rÞ inside sphere;PK VKe

iK�r outside sphere:

�

ð8Þ

Thus their form is completely general so that such

a scheme is termed full-potential calculation. The

choice of sphere radii is not very critical in full-

potential calculations in contrast to MTA, in

which one would obtain different radii as optimumchoice depending on whether one looks at the

potential (maximum between two adjacent atoms)

or the charge density (minimum between two ad-

jacent atoms). Therefore in MTA one must make a

compromise but in full-potential calculations one

can efficiently handle this problem.

3.3. Relativistic effects

In a solid that contains only light elements, a

non-relativistic calculation is justified. As soon as

heavier elements are involved, however, relativistic

effects can no longer be neglected. Core states are

treated fully relativistically. For valence states(and high lying semi-core states one shell below the

valence states) relativistic effects of atoms in the

medium range of atomic numbers (up to about 54)

can be included in a scalar relativistic treatment. It

describes the main contraction or expansion of

various orbitals (due to the Darwin s-shift or the

mass–velocity term) [12] but this scheme omits the

spin–orbit splitting. The latter can be included in a

second-variational treatment [13]. An additionalbasis function in form of a local orbital can be

added for p1=2 which differs from the scalar-rela-

tivistic p orbital [14]. For very heavy elements it

may be necessary to solve Dirac�s equation, whichhas all these terms included.

3.4. Computational aspects

In the newest version WIEN2k [11] the alter-

native basis set (APW+ lo) is used inside the

atomic spheres for those important ‘-orbitals(partial waves) that are difficult to converge (out-

ermost valence p-, d-, or f-states) or for atoms,

where small atomic spheres must be used [9,10,15].

For all the other partial waves the LAPW scheme

is used. In addition new algorithms for solving thecomputer intensive general eigenvalue problem

were implemented. The combination of algorith-

mic developments and increased computer power

has led to a significant improvement in the possi-

bilities to simulate relatively large systems on

moderate computer hardware. Now PCs or a

cluster of PCs can be efficiently used instead of the

powerful workstations or supercomputers thatwere needed about a decade ago. Several consid-

erations are essential for a modern computer code

and were made in the development of the new

WIEN2k package [11]:

• Accuracy is extremely important in the present

case. It is achieved by the well-balanced basis

set, which contains numerical radial functionsthat are recalculated in each iteration cycle.

Thus these functions adapt to effects due to

charge transfer or hybridization, are accurate

near the nucleus and satisfy the cusp condition.

• The PW convergence can essentially be con-

trolled by one parameter, namely the cutoff

energy corresponding to the highest PW com-

ponent. There is no dependence on selectingatomic orbitals or pseudo-potentials. It is a full-

potential and all electron method. Relativistic

effects (including spin orbit coupling) can be


treated with a quality comparable to solving Di-

rac�s equation.• Efficiency and good performance should be as

high a possible. The smaller matrix size of thenew mixed basis APW+ lo/LAPW helps to save

computer time or allows studying larger sys-

tems.

• User friendliness is achieved by a web based

graphical user interface (w2web) and is comple-

mented by an automatic choice of default op-

tions. In addition an extensive User�s Guide is

provided.

A modern software package needs a good graph-

ical user interface (GUI). In WIEN2k this is im-

Fig. 2. Structure generator illustrated for CrO2

plemented as a web-based GUI, called w2web. As

one example the structure generator is illustrated

below and is discussed with Fig. 2 for the case of

chromium dioxide CrO2. One can give a title forthe calculation, chromium dioxide, and select (the

default) a relativistic treatment of the core states.

In this example the known space group P42/mnm,

number 136, was selected from the list of 230

space groups. In this structure we have two in-

equivalent atoms. Next one specifies the lattice

parameters, either in �AA or in atomic units (a.u.).

The latter are used in all the calculations but forthe input �AA is often more convenient. For CrO2

we have a tetragonal structure with a¼ b but

different from c. The three angles (a, b, c) are all

using the graphical user interface w2web.


90�. Now the two inequivalent atoms must be

specified. The first is Cr, defined by the chemical

symbol, from which the atomic number Z is gen-

erated automatically. The NPT (number of pointsin the logarithmic radial mesh) and r0 (the first

radial mesh-point) are set to default values, but

the crucial parameter is the muffin tin radius. It

should be set as large as possible to save computer

time but overlapping spheres must not occur.

Then one must specify the atomic position, where

only one of the equivalent atoms must be speci-

fied, whereas the others are generated using thesymmetry operation of the specified space group.

The procedure must be repeated for the second

atom. This generates the most important file that

characterizes the structure. In cases where the

space group is not known, for example in an ar-

tificially constructed super-cell, auxiliary pro-

grams can find the space group. All other input

files are generated during initialization with somedefault values, which need to be changed only in

few special cases.

On the left side of Fig. 2 one sees the various

options, for example to initialize a calculation or

to run a SCF cycle. There are several utility pro-

grams and for the most common tasks like band

structure, electron density, or density of states, a

guided menu simplifies their generation.

3.5. WIEN2k hardware and software considerations

The hardware and software are changing rap-

idly and require a continuous adaptation of an

application software. We attempt to use high level

languages for the program development to sim-

plify such adaptations and try to hide the detailsfrom the regular user. In addition we provide in-

stallation support for most platforms which

should make it easier for a users to optimally in-

stall the program package on the computer system

that is available for a research group. For the in-

terested reader we summarize some details and

mention the most important aspects for running

the WIEN2k code efficiently.

• WIEN2k runs on any Unix/Linux platform

from PCs, cluster, workstations to supercom-

puters.

• FORTRAN90 is used mainly to allow dynami-

cal storage allocation. The program size is auto-

matically adapted to the problem size at

runtime.• The program consists of several separate pro-

grams (modules) that are linked by means of

C-shell or perl-scripts.

• Architecture. The hardware in terms of proces-

sor speed, memory access, and communication

is crucial. Depending on the given architecture,

optimized algorithms and libraries are used dur-

ing installation of the program package withsupport for most platforms.

• Portability requires the use of standards as much

as possible, such as FORTRAN90, BLAS (level

3), MPI, SCALAPACK, etc., standard software

that is essentially public domain. Besides that

perl5, ghostview, gnuplot, a pdf-reader, Tcl/

Tk, and a web-browser should be available.

• Inversion symmetry determines whether the ma-trix elements are real or complex numbers. The

latter case requires about twice as much memory

and about four times more computer time for

the calculations. The proper version of the rele-

vant programs is automatically selected.

• A system (with inversion symmetry) requires

about 128 MB (1–2 GB) for about 10 (100)

atoms per unit cell. With modern processorsnot only the speed of the CPU is crucial but also

the memory bus that determines how fast data

can be accessed from memory.

• Parallelization. The program can run in paral-

lel, either in a coarse grain version where each

k-point is computed on a single processor,

or––if the memory requirement is larger than

that available on one CPU––in a fine grainscheme that requires special attention for the

eigensolver, the most time consuming part.

The k-point parallelization works very well on

a cluster of PCs connected with a common

NFS filesystem and needs only a slow network

(100 MB/s). The fine-grain parallelization with

MPI/SCALAPACK requires either a shared

memory machine or a very fast communication.Depending on the system size both options,

full (LAPACK) and iterative (Block Davidson

method) diagonalization, are implemented to

allow the selection of the most efficient routines.


4. Chromium dioxide as a representative application

with WIEN2k

The WIEN2k code can solve the Kohn–ShamDFT equations with high accuracy. It provides

many results like energy bands, electron density,

total energy, forces acting on the atoms, spectra,

and related quantities. Such results are often very

useful to study various materials science problems.

In this paper we select one system, for which we

illustrate a few representative results, namely

chromium dioxide, CrO2, which has been inten-sively studied over the years. Originally the main

interest was due to its use as a magnetic recording

material. Then came the discovery that CrO2 was

predicted to be a half-metallic ferromagnet [16],

i.e. it has a metallic behavior in one spin direction,

but a gap for the other spin. This property makes

it very interesting for example in connection with

spin-electronic devices, since there is a 100% spinpolarization at the Fermi energy.

CrO2 crystallizes in the rutile structure whose

electronic structure was described in detail in [17].

The structure definition was explained in connec-

tion with the graphical user interface w2web in

Section 3.4.

(a)

X ∆ Γ Z R

Ene

rgy

(eV

) 0.0

1.0

2.0

3.0

4.0

5.0

-1.0

-2.0

-3.0

-4.0

-5.0

-6.0

-7.0

-8.0

(b)

X ∆ Γ Z R

Fig. 3. Energy bands of ferromagnetic CrO2 for spin-up (a, b) and sp

by the circle size and shows the Cr-d admixture (a, c) and the O-p ad

4.1. Energy band structure and density of states

As a first example we show the energy band

structure of CrO2, a half-metallic ferromagnet. InFig. 3 the left two panels show the spin-up and the

right the spin-down bands. The bands are only

shown along a few high symmetry lines in the

Brillouin zone. It can be clearly seen that spin-up

electrons correspond to a metallic state, whereas

the spin-down bands show a gap. This constitutes

the half-metallic nature of CrO2 in agreement with

previous work [16]. To each eigenvalue Enk wehave the corresponding wave functions, from

which we can calculate how much each atomic

orbital contributes. This can be done by calculat-

ing the fraction of charge that is found in each

atomic sphere or in the interstitial region. Inside

each sphere the charge can be further decomposed

into contributions from s-, p-, d-, f-, etc. partial

waves. This is illustrated (Fig. 3) by showing theCr-d and O-p contributions in the adjacent panels

(a, b) for spin up and (c, d) for spin-down elec-

trons. The low energy region ()2 to )7 eV) is

dominated by O-p states with little admixture from

Cr and only a small spin polarization. Thus these

bands look very similar in both spin channels. For

(c)

X ∆ Γ Z R

(d)

X ∆ Γ Z R

EF

in-down (c, d) electrons. The character of the bands is indicated

mixture (b, d), respectively.


the Cr-d bands, however, a large exchange split-

ting can be seen (compare the bands in panel a

starting around )1 eV with the bands above +1 eV

for the spin-down case in panel c). It should be

noted that a classification according to irreduciblerepresentations is available and can be color-

coded.

The corresponding density of states (DOS) is

shown in Fig. 4. The charge decomposition of each

state as discussed above for the character of the

energy bands can also be used to define the cor-

responding partial DOS showing the contributions

from the chromium (dashed line) and oxygen(dotted line) sphere. In the range between )1.5 and)7.5 eV the O-DOS dominates with an admixture

of Cr-DOS. The Cr-DOS dominates above )1.5eV and shows a large exchange splitting that leads

to the half-metallic character of CrO2. For the

spin-up electrons the Cr bands are significantly

Fig. 4. Density of states (DOS) of ferromagnetic CrO2 for spin-

up (left) and spin-down (right) electrons. The partial DOS show

the Cr and O contributions to the DOS.

lower in energy and thus the Cr admixture in the

oxygen bands is higher than in the spin-down case.

It should be mentioned that the decomposition

into partial DOS is model dependent and is af-

fected by the choice of the sphere radii. If onewould make the chromium sphere larger and the

oxygen sphere smaller the details of the decom-

position will slightly change. Therefore one should

not put too much emphasis on quantitative details

but focus on the trends. There is another aspect

that should be understood, namely that the APW

scheme uses a spatial decomposition of wave

function and consequently its character changesfrom sphere to sphere in contrast to an LCAO

scheme, where the atomic orbitals are atom-cen-

tered and extend way beyond the atomic region.

For example, when an oxygen 2p orbital enters the

neighboring chromium sphere this tail must be

represented as Cr partial waves of a certain ‘-character inside the Cr sphere. We call this an off-

site component in contrast to on-site componentslike Cr-d or O-p states which reside mostly inside

the Cr or O sphere, respectively.

4.2. Electron and spin density

The electron density is a useful quantity for

analysis and interpretation. It can be calculated for

all occupied states, or a certain energy region ofinterest, or even for a single state Enk. This allows

to visualize the character of bands and helps to

interpret chemical bonding. In a magnetic case it is

useful to visualize the spin magnetization density

that is defined as the difference between the spin-

up and spin-down electrons, respectively. Their

integral gives the spin magnetic moment, which

can be decomposed into atomic contributionscorresponding to the atomic spheres. The sum of

these spin densities q" and q# yields the total

density. We illustrate the spin magnetization den-

sity for CrO2 in the plane through chromium and

the two types of ligands, the equatorial and axial

oxygen atoms (Fig. 5). A discussion about the

comparison with experimental data and the inter-

pretation is given in Section 5.1.The magnetization density around the chro-

mium site shows a d-character with t2g symmetry

forming bonds with the oxygen ligands. The

Fig. 5. Spin magnetization density, q"–q# of ferromagnetic

CrO2 in a plane through chromium and four of the six oxygen

ligands. This figure is produced with XCrysDen [18] which

provides a color coding for the positive and negative regions.

Fig. 6. X-ray absorption (XAS) spectrum for the O–K edge of

CrO2 with a Gaussian broadening of 1 eV.


magnetization is localized near chromium and

shows that even in an itinerant ferromagnet the

magnetic moment comes from a local region in

space.

4.3. Spectra

Core-level spectra are particularly useful becausetheir interpretation is simplified by the localized

nature of the core hole, that leads to atom-selective

information. A typical core wave function is com-

pletely confined inside the corresponding atomic

sphere. Consequently the intensity of core-level

spectra require only information from the atomic

sphere, where the core hole occurs. These are for

example X-ray emission and absorption spectra,or electron energy loss (XES, XAS, EELS). In

XES a core hole is created and a transition occurs

from the occupied valence states to the core hole

governed by dipole selection rules. In XAS a

transition from an occupied core state to an un-

occupied valence state determines the intensity.

Besides all broadening effects, such as lifetime, or

spectrometric function the intensity of such spectracan be obtained from a ground state calculation.

We illustrate that for O-K XAS in Fig. 6 where a

transition from the oxygen 1s level to unoccupied

valence states that have O-p symmetry inside theoxygen sphere determine the spectrum. It consists

mainly of the O-p DOS modified by proper tran-

sition matrix elements. In our case we can have

transitions for spin-up or spin-down electrons that

would lead to different spectra which will be av-

eraged in the experiment.

4.4. Structure optimization

In this example we illustrate two aspects, the

calculation of a surface using a super-cell ap-

proach and the structural relaxations at the sur-

face. A surface is modeled by a slab with a certain

number of layers and a vacuum region separating

these slabs. Such a unit cell is repeated with peri-

odic boundary conditions in all three dimensions.It should be noted that such a slab has two sur-

faces which must be thick enough to reduce the

artificial interaction between the top and bottom

surface. Furthermore, it is desirable that the cen-

tral layer represents the bulk and the vacuum re-

gion is at least about 10 �AA.

For CrO2 we model a (1 0 0) surface with Cr-

termination (see Fig. 7 (left panel)). In this case wehave seven layers of chromium. One can see the

distorted oxygen octahedra surrounding a Cr

center. The initial structure is obtained by cutting

Fig. 7. Initial (unrelaxed) structure of a CrO2 slab with a Cr-terminated (1 0 0) surface (left panel). Structure of the CrO2 slab after

relaxation (right panel). Chromium (large grey sphere) and oxygen (small black sphere).


such layers from a bulk material but keeping allstructural parameters as in the bulk. Now we allow

the structure to relax using the forces acting on the

atoms to move them to the equilibrium positions,

where the forces vanish and the corresponding

total energy has a minimum. The result is shown in

Fig. 7 (right panel) which shows that the structure

remains approximately unchanged in the central

layers, whereas a significant relaxation takes placeat the surface. The outermost Cr layer moves

inward and the neighboring oxygens outward

forming a buckled structure. Such a surface re-

construction has important consequences on the

electronic structure and can be compared to ex-

perimental data.

5. Other applications with WIEN2k

In addition to the results explained in connec-

tion with CrO2 we provide a few examples that are

important in our opinion, namely the electron

density and its relation between theory and ex-

periments, the electric field gradients including

interpretations, and a summary of features andtopics for which references to the original litera-

ture is provided.

5.1. Electron density

The electron density is the key quantity in DFT

and is determined in the self-consistent-field (SCF)

cycle. It is a ground state property that can be

related to experiments. A Fourier transform of the

electron density q leads to the static structure

factor. In the experiment (e.g. X-ray diffraction)

additional factors must be considered, e.g. finitetemperature, extinction, and absorption. A de-

tailed comparison between theory and experiment

was presented by Zuo et al. [19] for silicon, one of

the best studied materials for which excellent

samples exist.

In contrast to the simple diamond structure of

silicon with only two atoms per unit cell, the In-

ternational Union for Crystallography (IUCr) haschosen corundum, Al2O3 as the test compound for

the multipole refinement project [20]. In this work

the refinement of experimental data was investi-

gated by using noise-free theoretical data for

comparison. A detailed discussion of the compar-

ison was given.

As a third example we mention the polymorphs

of Al2SiO5 with the minerals andalusite, sillimaniteand kyanite [21]. This structure contains four

formula units (Z ¼ 4) and thus 32 atoms per unit


cell. In all three cases the structure consists of

chains of octahedrally coordinated aluminum

atoms parallel to the c-axis sharing edges with

alternating silica tetrahedra and polyhedra of

aluminum in 4-, 5-, and 6-fold coordination. Thesedifferences had been analyzed on the basis of DFT

calculations.

The use of high-energy synchrotron radiation

for studying the electron-density distribution is

discussed for stishovite, SiO2 [22]. In addition to

the comparison between theory and experiment a

topological analysis is presented using the concept

of atoms in molecules. Here the bond criticalpoints, where rqðrÞ vanishes, were discussed.

Another example is cuprite, for which the charge

density and chemical bonding is discussed [23], a

controversial issue also with respect to the high

temperature superconductors. Since LDA has

shortcomings for this system, different versions of

LDA+U are tried to go beyond the conventional

LDA or GGA.

5.2. Electric field gradients

Nuclear quadrupole interactions (NQI) can be

measured by several experimental techniques such

as M€oossbauer (MB), nuclear magnetic resonance

(NMR), nuclear quadrupole resonance (NQR) or

perturbed angular correlations (PAC). The iso-topes of certain atoms have a nuclear quadrupole

moment Q provided the nuclear quantum number

IP 1. Representative examples are 57Fe and 17O.

The electric field gradient (EFG) is mainly deter-

mined by the anisotropy of the charge density

surrounding the corresponding nucleus. It has

been shown in 1985 that the EFG can be calcu-

lated from first principles from the knowledge ofthe complete charge distribution [24].

An interpretation of EFG�s was given for ex-

ample for the four oxygen positions in the high

temperature superconductor YBa2Cu3O7 [25],

where it was shown that the different occupation

numbers of the px, py, and pz orbitals inside the

corresponding oxygen atomic spheres are directly

related to the principle components of the EFGtensor. The difference in occupation numbers can

easily be understood on the basis of the electronic

structure. In this structure two of these orbitals

point in a direction where no nearby neighbor is

present and thus they are essentially non-bonding

(i.e. fully occupied), whereas in the third direction

(copper) neighbors are present leading to a large

splitting between (Op–Cud) bonding and antib-onding states, where the latter are only partially

occupied due to the large splitting. This difference

in occupation numbers causes a non-spherical

charge distribution that is responsible for the re-

lated EFG.

It should be mentioned that the situation for the

copper positions is different in two respects. On the

one hand, at a Cu-position not only the p- but alsothe d-orbitals may contribute to the EFG often

with opposite sign, which complicates the inter-

pretation. On the other hand, copper oxides can be

strongly correlated systems for which standard

DFT may not be sufficiently accurate. This seems

to be the case for Cu(2) in the buckled Cu–O layer,

which shows large deviations between theory and

experiment. The Cu(1) in the Cu–O linear chainhas an EFG, however, that agrees well.

Experiments always measure the NQI coupling,

that is proportional to the product of Q, the nu-

clear quadrupole moment, and the EFG tensor.

Often Q is known and can be used to convert the

calculated EFG to the experimental coupling

constant. In the case of Fe, the most important

M€oossbauer isotope, the value of Q was re-exam-ined by calculating EFGs for a series of Fe com-

pounds [26]. By relating these EFG values to the

experimental data we could estimate the propor-

tionality Q that was found twice as big as the

previous value in literature.

In the meantime the EFGs of many systems

have been studied from metals to minerals [27,28].

In most cases the agreement between theory andexperiment is often within 10–20%. For strongly

correlated systems larger deviations can be found

[23,25].

5.3. Further examples and new program features

The WIEN code was used in many research

topics to carry out electronic structure calcula-tion to solve material science problems. A list

of references, although by no means complete,

can be found on the web-page (www.wien2k.at).



A few representative examples are summarized

below.

• Test of new density functionals [29]

• LDA+U for highly correlated systems [30]• Atomic core hole spectra [31]

• Optical properties [32]

• Detailed comparison with high resolution UPS

spectroscopy for surface states, spin orbit split-

ting and spin splitting [33–35]

• Unconventional heavy Fermion systems [36]

• Mixed valence systems [37]

• Intercalation compounds [38]• Alkali-electro-sodalites [15]

• Thermoelectric materials [39]

At present several new features are developed and

tested before they will be included in the WIEN2k

package. These are the treatment of non-collinear

magnetism (by R. Laskowski), non-linear optics

(B. Olejnik), the calculation of phonons usinga direct method (linking to the PHONON pro-

gram by Parlinski [40]), and transport properties

(G.K.H. Madsen).

6. Conclusion

DFT calculations can provide useful informa-tion concerning the electronic structure of ordered

crystal structures. The results depend on the choice

of the exchange and correlation potential that is

used in the calculations. In comparison with real

systems it should be stressed that all kinds of im-

perfections are ignored. Some of them can be

simulated. For example, it is possible to represent

defects, e.g. an impurity, by studying a super-cellthat consists (in a simple case) of 2 · 2 · 2 times the

original unit cell. When one atoms in this large

unit cell is replaced by an impurity atom, the

neighboring atoms can relax to new equilibrium

positions. Such a super-cell is repeated periodically

and approximately represents an isolated impurity.

This example illustrates how theory can carry out

computer experiments. Hypothetical or artificialstructures can be considered and their properties,

irrespective of whether or not they can be made,

can be calculated. Complete information con-

cerning the electronic, optic or magnetic properties

is available. Therefore calculations allow us to

predict a system to be an insulator, a metal or

magnetic. The chemical bonding can be analyzed

and this allows to make predictions, of how the

system may change with deformations, underpressure or substitutions. Pressure is an easy para-

meter for DFT calculations in contrast to experi-

ments. The opposite is true for estimating

temperature effects which is easier for experiment

than for theory. Pressure and finite temperature

effects can be included using lattice vibrations,

which may be included from phonon data or cal-

culations. The examples given here are just a smallselection to illustrate some possibilities that DFT

and the WIEN2k code can provide.

Acknowledgements

The Austrian Science Fund (FWF Projects

P8176, P9213, M134, and SFB 011/8) supported

this work at the TU Wien. It is a pleasure to thank

all coworkers who contributed to the present

paper: G.K.H. Madsen, who implemented the new

APW+ lo scheme, J. Luitz (graphical user inter-face) and D. Kvasnicka (matrix diagonalization).

The large community of WIEN users has con-

tributed many improvements of the code (see

www.wien2k.at). Their contributions are ac-

knowledged with great thanks. In particular we

want to thank the following persons and their

group who substantially improved the features of

the WIEN code.

• C. Ambrosch-Draxl (Graz) optics

• L. Nordstr€oom (Uppsala) APW+ local orbitals

• D. J. Singh (Naval Research Lab, Washington,

DC) APW+ local orbitals

• M. Scheffler (FHI, Berlin) parallelization, opti-

mization, surfaces

• P. Novak (Prague) relativistic effects (spin–orbit), LDA+U

• V. Petricek (Prague) space groups

• C. Persson (Uppsala) irreducible representa-

tions

• B. Yanchitsky, A. Timoshevskii (Kiev) symme-

try



• J. Sofo (Penn State) topological analysis, Ba-

der�s analysis• R. Laskowski (Vienna) non-collinear magne-

tism• B. Olejnik (Vienna) non-linear optics

References

[1] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864.

[2] W. Kohn, L.S. Sham, Phys. Rev. A 140 (1965) 1133.

[3] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77

(1996) 3865.

[4] O.K. Andersen, Phys. Rev. B 12 (1975) 3060.

[5] D.J. Singh, Plane Waves, Peudopotential and the LAPW

Method, Kluwer Academic Publishers, Boston, Dortrecht,

London, 1994.

[6] P. Blaha, K. Schwarz, P. Sorantin, S.B. Trickey, Comput.

Phys. Commun. 59 (1990) 399–415.

[7] D.J. Singh, Phys. Rev. B 43 (1991) 6388.

[8] E. Sj€oostedt, L. Nordstr€oom, D.J. Singh, Solid State Com-

mun. 114 (2000) 15.

[9] G.H.K. Madsen, P. Blaha, K. Schwarz, E. Sj€oostedt, L.

Nordstr€oom, Phys. Rev. B 64 (2001) 195134-1.

[10] K. Schwarz, P. Blaha, G.K.H. Madsen, Comput. Phys.

Commun. 147 (2002) 71.

[11] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J.

Luitz, An augmented plane wave plus local orbitals

program for calculating crystal properties, Vienna Univer-

sity of Technology, Austria, 2001, ISBN 3-9501031-1-2.

[12] D.D. Koelling, B.N. Harmon, Solid State Phys. 10 (1977)

3107.

[13] A.H. MacDonnald, W.E. Picket, D.D. Koelling, J. Phys.

C: Solid State Phys. 13 (1980) 2675.

[14] J. Kunes, P. Nov�aak, R. Schmid, P. Blaha, K. Schwarz,

Phys. Rev. B 64 (2001) 153102.

[15] G.H.K. Madsen, B.B. Iversen, P. Blaha, K. Schwarz, Phys.

Rev. B 64 (2001) 195102-1-6.

[16] K. Schwarz, J. Phys. F: Met. Phys. 16 (1986) L211.

[17] P.I. Sorantin, K. Schwarz, Inorg. Chem. 31 (1992) 567.

[18] A. Kokalj, J. Mol. Graph. Model. 17 (1999) 176.

[19] J.M. Zuo, P. Blaha, K. Schwarz, J. Condens. Matter 9

(1997) 7541.

[20] S. Pillet, M. Souhassou, C. Lecomte, K. Schwarz, P. Blaha,

M. Rerat, L. Lichanot, P. Roversi, Acta Cryst. A 57 (2001)

290.

[21] M. Iglesias, K. Schwarz, P. Blaha, D. Baldomir, Phys.

Chem. Minerals 28 (2001) 67.

[22] A. Kirfel, H.-G. Krane, P. Blaha, K. Schwarz, T.

Lippmann, Acta Cryst. A 57 (2001) 663.

[23] R. Laskowski, P. Blaha, K. Schwarz, Phys. Rev. B 67

(2003) 075102.

[24] P. Blaha, K. Schwarz, P. Herzig, Phys. Rev. Lett. 54 (1985)

1192.

[25] K. Schwarz, C. Ambrosch-Draxl, P. Blaha, Phys. Rev. B

42 (1990) 2051.

[26] P. Dufek, P. Blaha, K. Schwarz, Phys. Rev. Lett. 75 (1995)

3545.

[27] P. Blaha, K. Schwarz, W. Faber, J. Luitz, Hyperfine Int.

126 (2000) 389.

[28] B. Winkler, P. Blaha, K. Schwarz, Am. Minerolog. 81

(1996) 545.

[29] J.P. Perdew, S. Kurth, J. Zupan, P. Blaha, Phys. Rev. Lett.

82 (1999) 2544.

[30] P. Mohn, C. Persson, P. Blaha, K. Schwarz, P. Nov�aak, H.

Eschrig, Phys. Rev. Lett. 87 (2001) 196401-1.

[31] J. Luitz, M. Maier, C. Hebert, P. Schattschneider, P.

Blaha, K. Schwarz, B. Jouffrey, Eur. Phys. J. B 21 (2001)

363.

[32] C. Ambrosch-Draxl, J. Majewski, P. Vogl, G. Leising,

Phys. Rev. B 51 (1995) 9668.

[33] J. Sch€aafer, E. Rotenberg, G. Meigs, S.D. Kevan, P. Blaha,

S. H€uufner, Phys. Rev. Lett. 83 (1999) 2069.

[34] J. Sch€aafer, E. Rotenberg, S.D. Kevan, P. Blaha, R.

Claessen, R.E. Thorne, Phys. Rev. Lett. 87 (2001)

196403-1.

[35] G. Nicolay, F. Reinert, S. H€uufner, P. Blaha, Phys. Rev. B

65 (2001) 033407-1.

[36] D.J. Singh, P. Blaha, K. Schwarz, I.I. Martin, Phys. Rev. B

60 (1999) 16359–16363.

[37] P. Nov�aak, F. Boucher, P. Gressier, P. Blaha, K. Schwarz,

Phys. Rev. B 63 (2001) 235114-1.

[38] X. Rocquefelte, F. Boucher, P. Gressier, G. Ouvrard, P.

Blaha, K. Schwarz, Phys. Rev. B 62 (2000) 2397.

[39] G.K.H. Madsen, K. Schwarz, P. Blaha, D.J. Singh, Phys.

Rev. B, in press (2003).

[40] K. Parlinski, Software PHONON, 2002.

Documents

Solid State Calculations Using WIEN2k