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Screened Coulomb interactions in metallic alloys. II. Screening beyond the single-site
and atomic-sphere approximations
Ruban, Andrei; Simak, S.I.; Korzhavyi, P.A.; Skriver, Hans Lomholt
Published in:
Physical Review B Condensed Matter
Link to article, DOI:
10.1103/PhysRevB.66.024202
Publication date:
2002
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Ruban, A., Simak, S. I., Korzhavyi, P. A., & Skriver, H. L. (2002). Screened Coulomb interactions in metallic
alloys. II. Screening beyond the single-site and atomic-sphere approximations. Physical Review B Condensed
Matter, 66(2), 024202. https://doi.org/10.1103/PhysRevB.66.024202
Screened Coulomb interactions in metallic alloys.
II. Screening beyond the single-site and atomic-sphere approximations
A. V. Ruban
Center for Atomic-scale Materials Physics and Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denmark
S. I. Simak
Department of Applied Physics, Chalmers University of Technology and Go
¨
teborg University, S-41296 Go
¨
teborg, Sweden
P. A. Korzhavyi
Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, 100 44 Stockholm, Sweden
H. L. Skriver
Center for Atomic-scale Materials Physics and Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denmark
共Received 9 August 2001; revised manuscript received 23 April 2002; published 26 June 2002兲
A quantitative description of the configurational part of the total energy of metallic alloys with substantial
atomic size difference cannot be achieved in the atomic-sphere approximation: It needs to be corrected at least
for the multipole-moment interactions in the Madelung part of the one-electron potential and energy. In the
case of a random alloy such interactions can be accounted for only by lifting the atomic-sphere and single-site
approximations, in order to include the polarization due to local environment effects. Nevertheless, a simple
parametrization of the screened Coulomb interactions for the ordinary single-site methods, including the
generalized perturbation method, is still possible. We obtained such a parametrization for bulk and surface NiPt
alloys, which allows one to obtain quantitatively accurate effective interactions in this system.
DOI: 10.1103/PhysRevB.66.024202 PACS number共s兲: 64.90.⫹b, 71.23.⫺k
I. INTRODUCTION
One of the main problems of modern alloy theory is to
establish a quantitatively accurate description of the
configuration-dependent part of the free energy, i.e., the dif-
ference in the total energies of alloys with different atomic
arrangements on the underlying lattice, in terms of effective
cluster interactions which may subsequently be used in sta-
tistical thermodynamics simulations.
1–3
Even without lattice
relaxation effects 共which are not considered here, although
they play an important role in the phase equilibria of many
alloy systems兲 a solution to the problem is still a challenge
especially in the case of inhomogeneous systems such as
surfaces in the presence of long-range and multisite interac-
tions which cannot be neglected.
The challenge originates from the fact that quantitatively
accurate and reliable 关within the accuracy of the approxima-
tion for the exchange-correlation part of the total energy of
the electronic subsystem in density-functional theory 共DFT兲
共Refs. 4 and 5兲兴 effective cluster interactions can be obtained
only by the Connolly-Williams 共CW兲 or structure inversion
method
3,6
on the basis of the total energies of a set of spe-
cifically chosen ordered structures calculated by the so-called
full potential 共FP兲 methods, which have no restrictions on the
form of the one-electron potential and density. However, the
structure inversion methods become practically unusable in
the case of an inhomogeneous system, not only because of
the large number of basic structures which must be calcu-
lated to extract position- 共layer-, for instance兲 dependent in-
teractions, but mainly because of the large size of those basic
structures 共supercells兲 which are needed to factorize a spe-
cific interaction inside homogeneous parts of the systems,
e.g., inside the layers parallel to a surface.
In this situation there appears to be only one alternative to
the structure inversion methods: The so-called generalized
perturbation method 共GPM兲, proposed by Ducastelle et al.
2,7
on the basis of the coherent potential approximation
共CPA兲,
8–10
and formulated within tight-binding 共TB兲 theory.
Later the GPM was generalized in a straightforward
manner
11–14
for use in ab initio calculations based either on
the Korringa-Kohn-Rostoker 共KKR兲 method or the linear-
ized muffin-tin orbitals 共LMTO兲 method in the atomic-
sphere approximation 共ASA兲. The main idea behind the
GPM is to calculate perturbatively the total-energy difference
between the alloy in the initial state, which is completely
random, and in a final state in which only one specifically
chosen atomic distribution correlation function or short-
range order parameter is different from that in the random
state. This makes the GPM very efficient and convenient to
use as it directly yields the needed effective cluster interac-
tions.
However, it is known, although rarely mentioned in the
literature, that the interactions obtained by the GPM yield a
quantitatively poor description of the ordering in real alloys
共see, for instance, Ref. 15兲 in those cases where there is a
substantial size mismatch between the alloy components.
This failure may only partly be attributed to lattice relaxation
effects. Rather, it originates not from the GPM method itself
but is a consequence of inappropriate approximations in the
basic methods underlying the GPM calculations. This is so,
because, as has been demonstrated by Bieber et al.
16
in pa-
rametrized tight-binding calculations and by Singh et al.
17
in
ab initio KKR-CPA calculations, the GPM interactions may
PHYSICAL REVIEW B 66, 024202 共2002兲
0163-1829/2002/66共2兲/024202共12兲/$20.00 ©2002 The American Physical Society66 024202-1
provide 共under certain conditions兲 a consistent description of
the ordering or configurational energy. That is, the ordering
energy obtained from the GPM interactions, calculated in the
framework of a particular technique, agrees reasonably well
with the ordering energy obtained directly from the total-
energy calculations by the same technique.
The ab initio techniques underlying GPM calculations are
usually the KKR-ASA, KKR-ASA-CPA, and LMTO-CPA
methods
18
which are based on a number of approximations,
such as the CPA, the single-site 共SS兲 approximation for the
electrostatic part of the DFT problem, and the spherical ap-
proximation for the form of the potential which, depending
on the geometry, is called either the muffin-tin 共MT兲 or
atomic-sphere approximation 共ASA兲. The question is which
of these approximations is the most severe in the cases where
the alloy components have a substantial size difference? To
answer the question, we note that a size difference leads to
so-called ‘‘charge-transfer effects’’ or, to be more precise, to
a nonzero net charge for each alloy component inside their
atomic spheres chosen to be of equal size.
Although there are systems where the CPA may lead to
substantial errors, it is clear from a general point of view that
the CPA cannot be responsible for the errors in the case of
pronounced charge-transfer effects, because the error of the
CPA is mainly related to specific features of the electronic
structure of the individual alloy components such as the dif-
ference in the position and overlap of the energy bands.
2,19
Moreover, there is a number of different calculations which
show that in such systems as, for instance, CuPd and CuAu,
where the alloy components have similar electronic struc-
tures but different atomic size, the CPA works fairly
well
20–22
as a method for obtaining the average electronic
structure of random alloys.
As far as the underlying KKR-CPA or LMTO-CPA meth-
ods are concerned, much larger errors may in fact come from
the use of the single-site approximation in the self-consistent
DFT part of the calculations as this yields no information
about the distribution of the charge outside the individual
atomic spheres of the alloy components. In fact, the effective
medium outside the individual atomic-spheres of the alloy
components is electroneutral, and therefore, if the net charge
of an atomic sphere is nonzero, Poisson’s equation cannot be
solved properly. A number of different models have been
proposed to include the missing screening charge in the so-
lution to Poisson’s equation
23–26
and, most recently, a gen-
eral formalism of screened Coulomb interactions 共SCI’s兲
based on the knowledge that the spatial distribution of the
screening charge around an impurity has been developed in
Ref. 27 together with a formalism for the SCI contribution to
the GPM interactions.
2
Although the SCI may now be in-
cluded in SS-DFT-CPA calculations, this does not solve all
problems connected with the description of the energetics of
alloys.
It is not surprising that the main source of inaccuracy in
the KKR-CPA and LMTO-CPA methods is the spherical ap-
proximation, MT or ASA, for the form of the electron den-
sity and potential 共in the following we will consider only the
ASA, since the difference between the ASA and MT is un-
important for the later discussion and results兲. For instance,
in the extreme case, where one of the alloy components is a
vacancy, the error due to the use of the ASA is about 100%
共or several eV in absolute values兲 for the vacancy formation
energy.
28
As has been shown by Korzhavyi et al.,
29
this kind
of error originates from the oversimplified description of the
nonspherical electrostatic contribution to the one-electron
potential and energy mainly from the charge density on the
atoms next to the vacancy.
This is similar to the case of surfaces where the quite
large ASA error may be substantially reduced by the inclu-
sion of the multipole moments of the electron charges inside
the atomic spheres.
30
The so-called ASA⫹M approach sig-
nificantly improves vacancy and defect formation ener-
gies,
29,31
surface energies,
32
and alloy energetics.
33
Recently,
Finnis et al.
34
have included the multipole moments in their
self-consistent tight-binding model which allowed them
to obtain a quite accurate description of the energetics of
zirconia.
In this paper we show that the use of the ASA⫹M ap-
proach leads to a representation of the configurational part of
the total energy, which is very close to the full-potential re-
sults. Since the polarization of the atoms in an alloy is almost
entirely determined by their closest local environment, it is
obvious that the effect of polarization cannot be described
properly in the single-site approximation. Nevertheless, the
SS-DFT-CPA methods may be still used for the electronic
structure and total-energy calculations of random alloys if
the definition of the SCI is modified. It is the main purpose
of the present paper to demonstrate how this may be done in
the cases of ordinary bulk homogeneous random alloys and
inhomogeneous systems such as surfaces.
The paper is organized as follows. In Sec. II we introduce
the ASA⫹M approximation and outline some details of our
calculations. In Sec. III we compare the ordering energies of
NiPt alloys, calculated by the KKR method in different ap-
proximations and by the Vienna ab initio simulation package
共
VASP兲.
35,36
In Sec. IV we define the on-site screening Made-
lung potential, which should be added to the one-electron
potential in the SS-DFT calculations. The SCI and the Made-
lung energy of a random alloy are defined in Sec. V. In Sec.
VI we calculate the intersite SCI in NiPt in different approxi-
mations and by different methods. In Sec. VII the screened
generalized perturbation method interactions are calculated
and compared with the Connolly-Williams interactions. In
Sec. VII we show how the formalism for the SCI should be
modified in the case of inhomogeneous systems where there
are several nonequivalent sublattices, like partially ordered
alloys or surfaces.
II. BEYOND THE ASA
In a companion paper,
27
in the following referred to as
paper I, we presented a consistent and variational, within
DFT, approach to the electrostatic screening effects in ran-
dom alloys, and within the ASA we found that these screen-
ing effects were almost independent of alloy composition,
lattice spacing, and crystal structure. However, in those cases
where the alloy components have a substantial size differ-
ence one cannot obtain a quantitatively correct description of
RUBAN, SIMAK, KORZHAVYI, AND SKRIVER PHYSICAL REVIEW B 66, 024202 共2002兲
024202-2
the configurational part of the total energy of metallic alloys
within the ASA. One must therefore go beyond the spherical
approximation at least for the density. In the following we
will show how this may be done. All other details of our
approach may be found in paper I.
A. Multipole correction to the atomic-sphere approximation:
ASA¿M
The idea behind the multipole correction is simply to in-
clude those contributions to the electrostatic multipole-
moment expansion of the intercell or Madelung part of the
one-electron potential and energy, which are neglected in the
ASA. If the multipole moments of the electron charge q
R
L
inside an atomic-sphere centered at R are defined as
q
R
L
⫽
冑
4
2l⫹ 1
冕
S
R
冉
r
S
R
冊
l
n
R
共
r
R
兲
Y
L
共
r
ˆ
R
兲
dr
R
⫺ Z
R
␦
0l
, 共1兲
where L is shorthand for the (l,m) quantum numbers, S
R
the
radius of the atomic sphere, n
R
the nonspherical charge den-
sity, and Y
L
a real harmonic, the Madelung contribution to
the one electron potential is given by
v
MR
L
⫽
1
S
兺
R
⬘
,L
⬘
M
R,R
⬘
L,L
⬘
q
R
⬘
L
⬘
, 共2兲
while the Madelung energy which now includes the
multipole-multipole electrostatic interactions between differ-
ent lattice sites may be written
E
M
⫽
1
2S
兺
R,L
q
R
L
兺
R
⬘
,L
⬘
M
R,R
⬘
L,L
⬘
q
R
⬘
L
⬘
. 共3兲
In these expressions, M
R,R
⬘
L,L
⬘
is the multipole Madelung ma-
trix which is equivalent to the conventional LMTO structure
constants and the number of multipoles included in the L,L
⬘
summations is determined by the angular momentum cutoff
l
max
in the basis set used in the Green’s-functions calcula-
tions. Owing to the properties of the Gaunt coefficients non-
zero multipole moments of the charge density may be gen-
erated for l values up to l
max
M
⫽ 2l
max
.
We note that, since in the ASA⫹M the one-electron po-
tential is still kept spherically symmetric inside each atomic
sphere, the only term which contributes to the one-electron
potential is the L⫽(0,0) or monopole term. This simple re-
striction on the form of the one-electron potential violates the
variational connection between the Madelung potential and
energy and, in turn, between the one-electron potential and
the total energy, i.e.,
v
MR
⬅
v
MR
00
⫽
␦
E
M
␦
n
R
. 共4兲
However, since this is just a consequence of the model, but
not of theory in general, it does not create any problems. On
the other hand, the reinstatement of the variational connec-
tion between the one-electron potential and the total energy
by keeping only the monopole-multipole term in Eq. 共3兲 may
lead to a substantial error in the total energy, as the
multipole-multipole interactions will not be accounted for.
B. Details of calculations
The Green’s-function technique has been used in both the
KKR-ASA and the locally self-consistent Green’s-function
共LSGF兲 calculations in the scalar relativistic and atomic-
sphere approximations. This part of the techniques is de-
scribed in Refs. 27, 32 and 37. The basis functions have been
expanded up to l
max
⫽ 3(spdf basis兲 inside the atomic
spheres, while the multipole moments have been calculated
up to l
max
M
⫽ 6. We have also performed a number of calcu-
lations in the spd basis, i.e., l
max
⫽ 2 and l
max
M
⫽ 4. The inte-
gration of the Green’s-function over energy was performed
in the complex plane over 16 energy points on a semicircular
contour using a Gaussian technique. We have used the gen-
eralized gradient approximation 共GGA兲 of Perdew and
co-workers.
38
For each structure the integration over the
Brillouin zone 共BZ兲 has been done by using equally spaced k
points in the irreducible part of the appropriate BZ and the
number of k points has been chosen to be equivalent to 500–
1000 uniformly distributed k points in the irreducible part of
the BZ of the fcc structure. Core states were recalculated at
each DFT iteration.
For benchmark calculations we applied the Vienna ab ini-
tio simulation package 共VASP兲 described in detail in Refs. 35
and 36. These calculations were performed in a plane-wave
basis, utilizing fully nonlocal Vanderbilt-type ultrasoft
pseudopotentials 共US-PP兲,
39
which allow the use of a mod-
erate cutoff in the construction of the plane-wave basis for
the transition metals. In the actual calculations the energy
cutoff was set to 302 eV, exchange and correlation were
treated in the framework of the GGA,
38
and the integration
over the Brillouin zone was performed on a Monkhorst-Pack
k mesh.
40
Test calculations showed that, depending on struc-
ture, the required convergence was reached for 35–275 k
points in the irreducible wedge of the BZ.
III. ORDERING ENERGIES IN Ni-Pt
We start by demonstrating the accuracy of the various
approximations which are usually used in KKR共LMTO兲-
ASA-like calculations. For this purpose we have calculated a
set of ordered fcc NiPt alloys by the KKR-ASA method and
by the US-PP for a fixed lattice constant without any local or
anisotropic relaxations. The lattice constant has been chosen
to be a⬇3.791 Å, which corresponds to an atomic Wigner-
Seitz radius of 2.8 a.u. The ordered structures include: L1
2
(Cu
3
Au-type兲,DO
22
(TiAl
3
-type兲, Z3,

,
41
␥
(Pt
2
Mo-type兲,
L1
0
共CuAu-type兲, CH or ‘‘40’’ 共NbP-type兲, Z2,
41
L1
1
共CuPt-
type兲, and the so-called SQS-16.
42
To simplify the comparison we present in Table I the val-
ues of the calculated ‘‘mixing’’ energies of the above-
mentioned ordered structures,
E
mix
Ni
m
Pt
n
⫽ E
tot
Ni
m
Pt
n
⫺
mE
tot
Ni
⫹ nE
tot
Pt
m⫹ n
, 共5兲
SCREENED COULOMB INTERACTIONS... II. ... PHYSICAL REVIEW B 66, 024202 共2002兲
024202-3
where E
tot
Ni
and E
tot
Pt
are the total energies of the pure com-
ponents calculated at the same lattice constant. All total en-
ergies are per atom. The mixing energies of the random al-
loys have been obtained on the basis of all the energies
included in Table I, except SQS-16, plus the energies of
Ni
7
Pt and Pt
7
Ni (CuPt
7
-type兲 ordered alloys 共not presented
in the table兲 by the Connolly-Williams method in which the
total-energy expansion included pair interactions at the first
four, seventh and tenth coordination shells 共these are the
largest pair interactions in this as well as in many other fcc
transition-metal alloys兲, the first four triangle interactions
and the two tetrahedron interactions corresponding to the
tetrahedra of nearest neighbors, and the straight line along
关111兴 direction 共the last being quite substantial in many
systems兲.
The SQS-16 is a so-called special quasirandom structure
44
which consists of eight atoms of one type and eight atoms of
another type distributed in the unit cell in such a way that the
first seven pairs, the nearest-neighbor triangle, and the tetra-
hedron atomic distribution correlation functions are the same
as in the random alloy. Hence the fact that the values of
E
mix
SQS⫺ 16
and E
mix
rand
are nearly equal indicates that: 共i兲 the
SQS-16 provides a good model for the random NiPt alloy,
and 共ii兲 the convergence of the CW method is reasonably
good. Part of the convergence of the CW method is provided
by the use of total energies on a fixed lattice, whereby the
volume dependent contribution to the total energy is not ex-
panded in terms of cluster interactions, which is an ill-
defined procedure in metallic systems and usually leads to
very bad convergence of the CW method 共see, for instance,
Ref. 43兲.
In Table I we present KKR-ASA results in the spdf as
well as the spd basis and in both cases we show results in
the pure ASA, i.e., without multipole moment contributions,
and in the ASA⫹M. We also include the results of neutral
sphere calculations, in which the atomic-sphere of Pt is cho-
sen to be larger than that of Ni in order to provide zero net
charges of the atomic spheres. Although the comparison of
the mixing energies should be done with some caution, be-
cause the ground-state properties of the alloys are different in
different approximations and because all the calculations
have been performed at the same fixed lattice constant, it is
clear that the ASA⫹M approach in the spdf basis leads to
values of E
mix
which are in considerably better agreement
with the US-PP results than any of the other approaches.
Using the results of the Connolly-Williams method for the
total energy of random alloys one can calculate the ordering
energies, defined as the difference between the total energies
of an ordered and a random alloy at the same composition. In
Table II we compare the ordering energies of different struc-
tures calculated in different approximations for Ni
3
Pt, NiPt,
and Pt
3
Ni. Such a comparison makes sense since the order-
ing energies are much less volume dependent than the total
energies themselves. Again, it is seen that, relative to the
US-PP results, the ASA⫹M approach in the spdf basis gives
not only the best, but also a quite accurate description of the
ordering effects in NiPt. It is also seen that the ordering
energies in the ASA⫹M but without multipole-multipole
contribution to the Madelung energy appear to be halfway
beteeen ASA and ASA⫹M results. As we will see below, this
is in fact the limit of accuracy which can be reached in con-
sistent single-site mean-field calculations.
It is also obvious from the table that the KKR-ASA does
not yield reasonable values for the ordering energies in the
case of transition-metal alloys unless f states are included in
the basis. These f states are needed to supply a better aug-
mentation of the basis functions at the atomic sphere and a
better interstitial charge density.
45
The neglect of f states can
only be partly compensated by the use of the so-called
combine-correction term in the LMTO method.
46
Another important conclusion, which can be drawn from
the results in Table II, is the fact that the use of neutral
spheres leads to a substantial underestimate of the ordering
effects in KKR-ASA共⫹M兲 calculations. In other words, al-
though the neutral-sphere approach formally solves the prob-
lem of the electrostatic interaction in an alloy, the electro-
static contribution to the one-electron potential and energy
being zero by definition, it introduces errors which are unac-
ceptable in a quantitative description of the configurational
energetics.
The reason for this failure is the following. If we com-
pare the values of E
mix
from Table I obtained with neutral
spheres with those obtained with equal spheres, we find that
the neutral sphere approach leads to substantial lowering of
the total energy of the ordered alloys. However, the amount
TABLE I. ‘‘Mixing’’ energies ⫺ E
mix
共in mRy/atom兲 of ordered
and random NiPt alloys obtained by different methods at a fixed
Wigner-Seitz radius of S⫽2.8 a.u.
KKR-ASA
Alloy spdf spd US-PP
ASA⫹M ASA Neutral ASA⫹MASA
Ni
25
Pt
75
L1
2
21.7 19.5 28.4 31.1 31.7 22.3
DO
22
20.6 18.3 26.4 30.3 30.1 21.0
Z3 13.2 8.6 23.4 18.1 17.0 14.5
Random 15.6 11.7 23.7 22.0 21.1 16.4
Ni
33
Pt
66
Pt
2
Mo 21.5 18.2 28.3 31.9 31.2 22.5

16.4 10.7 27.7 23.7 22.5 19.4
Ni
50
Pt
50
L1
0
27.5 23.8 33.1 39.0 38.4 28.0
Z2 13.4 5.1 25.1 17.0 13.4 15.0
CH 26.3 22.8 31.0 38.0 37.4 26.9
L1
1
22.7 17.5 32.8 30.8 30.1 23.0
SQS-16 20.7 21.7
Random 20.6 14.9 29.3 28.7 26.8 21.6
Ni
66
Pt
33

17.6 10.0 23.9 23.9 20.6 19.3
Pt
2
Mo 21.9 18.1 26.2 31.6 30.2 22.9
Ni
25
Pt
75
Z3 14.0 8.4 18.4 18.5 15.8 14.6
DO
22
19.7 16.7 21.8 28.3 27.0 20.2
L1
2
20.0 16.8 22.0 28.2 27.2 20.6
Random 15.4 10.7 20.4 21.1 19.1 16.4
RUBAN, SIMAK, KORZHAVYI, AND SKRIVER PHYSICAL REVIEW B 66, 024202 共2002兲
024202-4
