solid state calculations using wien2k
DESCRIPTION
WIEN2kTRANSCRIPT
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.
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).
262 K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273
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
K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273 263
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
264 K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273
(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
K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273 265
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.
266 K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273
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.
K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273 267
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.
268 K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273
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.
K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273 269
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).
270 K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273
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
K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273 271
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).
272 K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273
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
K. Schwarz, P. Blaha / Computational Materials Science 28 (2003) 259–273 273
• J. Sofo (Penn State) topological analysis, Ba-
der�s analysis• R. Laskowski (Vienna) non-collinear magne-
tism• B. Olejnik (Vienna) non-linear optics
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