Ab initio derivation of interatomic interactions in transition metals
An approach to the ab-initio calculation of many-body interatomic potentials in transition metals (TM) is developed. It is based on utilizing the local spin density approximation and linear superposition assumption for the density of the valence electrons. The analytic expressions for the spin de...
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irk-123456789-1197572017-06-09T03:05:06Z Ab initio derivation of interatomic interactions in transition metals Gurskii, Z. Krawczyk, J. An approach to the ab-initio calculation of many-body interatomic potentials in transition metals (TM) is developed. It is based on utilizing the local spin density approximation and linear superposition assumption for the density of the valence electrons. The analytic expressions for the spin dependent pair and triplet interatomic interactions are given. The many-body potentials for three particles and four particles are shown to be indirect and to describe interactions of the same spin orientation. Розвинуто оригінальний підхід обчислення багаточастинкових міжатомних потенціалів у перехідних металах. Він грунтується на використанні наближення локальної густини спінів та припущенні про те, що густину валентних електронів можна представити у вигляді лінійної суперпозиції. Подані явні вирази для парних та потрійних взаємодій, що залежать від орієнтації спіна. Показано, що багаточастинкові потенціали (три-, чотиричастинкові) описують опосередковані міжатомні взаємодії однакової орієнтації спінів. 2001 Article Визначення із перших принципів міжатомних взаємодій у перехідних металах / Z. Gurskii, J. Krawczyk // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 37-44. — Бібліогр.: 11 назв. — англ. 1607-324X PACS: 34.20.Cf, 71.15.Mb, 71.15.Nc, 71.20.Be DOI:10.5488/CMP.4.1.37 http://dspace.nbuv.gov.ua/handle/123456789/119757 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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An approach to the ab-initio calculation of many-body interatomic potentials
in transition metals (TM) is developed. It is based on utilizing the local spin
density approximation and linear superposition assumption for the density
of the valence electrons. The analytic expressions for the spin dependent
pair and triplet interatomic interactions are given. The many-body potentials
for three particles and four particles are shown to be indirect and to
describe interactions of the same spin orientation. |
| format |
Article |
| author |
Gurskii, Z. Krawczyk, J. |
| spellingShingle |
Gurskii, Z. Krawczyk, J. Ab initio derivation of interatomic interactions in transition metals Condensed Matter Physics |
| author_facet |
Gurskii, Z. Krawczyk, J. |
| author_sort |
Gurskii, Z. |
| title |
Ab initio derivation of interatomic interactions in transition metals |
| title_short |
Ab initio derivation of interatomic interactions in transition metals |
| title_full |
Ab initio derivation of interatomic interactions in transition metals |
| title_fullStr |
Ab initio derivation of interatomic interactions in transition metals |
| title_full_unstemmed |
Ab initio derivation of interatomic interactions in transition metals |
| title_sort |
ab initio derivation of interatomic interactions in transition metals |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119757 |
| citation_txt |
Визначення із перших принципів міжатомних взаємодій у перехідних металах / Z. Gurskii, J. Krawczyk // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 37-44. — Бібліогр.: 11 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT gurskiiz abinitioderivationofinteratomicinteractionsintransitionmetals AT krawczykj abinitioderivationofinteratomicinteractionsintransitionmetals |
| first_indexed |
2025-07-08T16:32:46Z |
| last_indexed |
2025-07-08T16:32:46Z |
| _version_ |
1837097149612949504 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 37–44
Ab initio derivation of interatomic
interactions in transition metals
Z.Gurskii 1 , J.Krawczyk 2
1 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 Opole University, 48 Oleska Str., 45-052 Opole, Poland
Received August 14, 2000
An approach to the ab-initio calculation of many-body interatomic potentials
in transition metals (TM) is developed. It is based on utilizing the local spin
density approximation and linear superposition assumption for the density
of the valence electrons. The analytic expressions for the spin dependent
pair and triplet interatomic interactions are given. The many-body poten-
tials for three particles and four particles are shown to be indirect and to
describe interactions of the same spin orientation.
Key words: local spin density, total energy functional, many-body
potentials
PACS: 34.20.Cf, 71.15.Mb, 71.15.Nc, 71.20.Be
1. Introduction
Constructing appropriate interatomic potentials is one of the central problems
in modern condensed matter physics. It is caused by utilizing these potentials in
many physical and material problems. The first-principles and the (semi)empirical
methods are the two approaches which dominate this field at present (see review of
literature in [1]). In some cases, an adequate theoretical description of metal and
alloy properties could not be achieved within the pair interatomic interaction model.
That means the many-body interactions play an important role. Transition metals
(TM) with partially filled d-bands belong to such objects.
Some many-body empirical models have been advanced to overcome the limita-
tions of the pair potential approximation (see introduction in [2] and papers cited
there). Recently, J.Moriarty developed a new theoretical scheme to obtain rigorously
triplet and quadruplet angular force interactions within the generalized pseudopo-
tential theory [3]. Another theoretical approach getting the angular forces in TM
based on the tight-binding method was proposed in [4].
c© Z.Gurskii, J.Krawczyk 37
Z.Gurskii, J.Krawczyk
Analyzing the results of [2–4], one can conclude that the three- and four-body
interactions play an important role in TM, especially in metals with partially filled
d-bands.
Effective interatomic potentials for TM introduced in [2–4] are spin independent.
The subject of the present paper is to get these potentials for the case when the
spin orientation is included into consideration explicitly. The method proposed in
[1,5] for simple metals is generalized here for TM.
2. The total energy functional within the local spin density
approximation
Let us examine an electron-ion-system of TM taking into account the spin ori-
entation of valence electron density
ρ(r) =
∑
σ
ρσ(r), (2.1)
where the symbol σ = ↑, ↓ distinguishes the up and the down spins. The outer d-
and s-electron shells of TM atoms are treated as valence electrons.
The Kohn-Sham total energy functional of the TM electron-ion system in the
spin polarized case has the form [6]
E[ρ] =
∑
σ
T [ρσ] +
∑
σ,σ′
∫
V σ′
ext(r)ρσ(r)dr
+
1
2
∑
σ,σ′
∫
ρσ(r)ρσ′(r′)
|r− r′|
drdr′ +
∑
σ
Exc[ρσ] + Ei−i (2.2)
(all equations are written in atomic units in the present paper). Here, T [ρσ] and
Exc[ρσ] are the kinetic and exchange-correlation energy functionals, respectively.
The second and the third term in (2.2) describe the energy of the valence electron
subsystem in the external field of ions V σ
ext(r) and the Hartree energy, correspond-
ingly. The last term in (2.2) is the energy of the ion-ion direct interactions.
The total energy E[ρ] is a universal functional of the electron density, and is
obtained by solving the set of the Kohn-Sham equations. The all-electron eigenvalue
problem for TM is usually solved by the full-potential-linearized-augmented-plane-
wave method [7].
Suppose the valence electron density ρ(r) is determined within the unrestricted
Hartree-Fock method, that is, in the form of (2.1). Let us represent ρ(r) as the
linear superposition of contributions from the individual “pseudoions” embedded in
the uniform background of density ρ0,
ρσ(r) = ρ0σ +
∑
R
ρi,σ(r−R). (2.3)
Here,
∑
σ
ρ0σ =
Z∗
s
Ω0
(2.4)
38
Interatomic interactions in transition metals
is the density of the uniform electron distribution, Z ∗
s is an effective ion valency
in TM, Ω0 is the atomic volume and ρi,σ(r −R) is the valence electron density of
σ-spin orientation related to the pseudoion on the site R. Electrons of s-symmetry
contribute to ρ0σ while d-electrons contribute to ρi,σ(r−R). The effective ion valency
takes noninteger values in TM owing to the s− d hybridization [8].
The approach proposed is based on two approximations:
i) linear superposition for the valence electron density, see equation (2.3);
ii) local spin density approximation (LSDA) for the exchange-correlation and
kinetic energy functionals.
The LSDA for the Exc[ρσ] functional in (2.2) reads [6]
Exc[ρσ] =
∫
drρσ(r)ε
0
xc
(
ρσ(r)
)
, (2.5)
where ε0xc(ρσ(r)) is the exchange-correlation energy per electron of the interacting
electron gas possessing the same density ρσ(r) at point r as the electron subsystem
under investigation.
We represent the known quantity Tσ for the valence electron kinetic energy ex-
tracted from the ab initio band structure calculations in the form of the LSDA, that
is,
T [ρσ] =
∫
drρσ(r)t
(
ρσ(r)
)
. (2.6)
One can regard equation (2.6) as a constraint on the unknown function t(ρσ(r)).
For explicit expressions for t(ρσ(r)) see [1,9]. As will be shown in the next section,
utilizing approximations (i) and (ii) allows one to develop the total energy of TM
as a multi-ion interaction expansion
E = E0(Ω0)+
1
2!
∑
σ,σ′
∑′
R1,R2
V
σ,σ′
2 (R1,R2)+
1
3!
∑
σ
∑′
R1...R3
V σ
3 (R1,R2,R3)+ . . . , (2.7)
where E0 represents a volume dependent term including one-ion contributions to E,
and V
σ,σ′
2 , V σ
3 , etc. are the two-particle, three-particle, etc. interatomic potentials
which are implicitly volume dependent and explicitly spin dependent. The prime on
each R-summation in (2.7) denotes the exclusion of all the self-interaction terms
where two indices are equal. It will be shown below that the many-body potentials
V σ
n (R1 . . .Rn), n > 3 describe indirect interatomic interactions via electrons of the
same spin orientation, that is they are diagonal in the σ-index.
3. Spin dependent interatomic potentials
Let us consider the regular crystal. The electron density ρ(r), (2.1), as well as
the functions ε0xc(ρσ(r)) and t(ρσ(r)) in (2.5) and (2.6) are periodical because of the
crystal periodicity. Therefore, one should consider (2.5) and (2.6) in the region of the
39
Z.Gurskii, J.Krawczyk
unit cell only [1,5]. The valence electron density ρσ(r), extracted from the ab initio
band structure calculations, can be represented in the unit cell region as follows
ρσ(r) = ρps,σ(r−R1) +
∑
R 6=R1
ρi,σ(r−R) for r ∈ Ω0(1), (3.1)
where
ρps,σ(r−R1) = ρ0σ + ρi,σ(r−R1) (3.2)
is the electron density of the σ-spin orientation related to the pseudoatom on the
site R1. The symbol r ∈ Ω0(1) in (3.1) denotes that the running variable r takes
values in the unit cell (the Wigner-Seitz cell) centred on the siteR1. The regular TM
with one atom per primitive cell is considered in the present paper. The condition of
periodicity does not restrict our approach because the potentials V σσ′
2 (R1,R2) and
V σ
3 (R1,R2,R3), that will be obtained, are structure independent. The structure
dependence of the total energy (2.7) appears through the summation in (2.7) over
all N ion positions [3,5].
Let us expand t(ρσ(r)) and ε0xc(ρσ(r)) in powers of ρps,σ(r −R1) for r ∈ Ω0(1).
For the function f(ρσ(r)) = t(ρσ(r)) or f(ρσ(r)) = ε0xc(ρσ(r)) such an expansion
reads
f
(
ρσ(r)
)
= f
(
ρps,σ(r−R1)
)
+
∂f(r−R1)
∂ρσ
∑
R 6=R1
ρi,σ(r−R)
+
1
2!
∂2f(r−R1)
∂ρ2σ
[
∑
R 6=R1
ρi,σ(r−R)
]2
+ . . . (3.3)
with r ∈ Ω0(1) and
∂nf(r−R1)
∂ρnσ
≡
∂nf
(
ρσ(r)
)
∂ρnσ(r)
∣
∣
ρσ(r)=ρps,σ(r−R1) . (3.4)
To demonstrate a way of treating the exchange-correlation and kinetic energy
functionals in terms of the many-body indirect interatomic interactions we consider
just equation (2.5) more in detail.
Calculation of the exchange-correlation energy is reduced to the following form
∫
Ω
drρσ(r)ε
0
xc
(
ρσ(r)
)
= N
∫
Ω0
drρσ(r)ε
0
xc
(
ρσ(r)
)
(3.5)
owing to the periodicity of the electron density ρσ(r). Here Ω0 is a volume of a unit
cell. In view of (3.1) to (3.3) the right hand side of equation (3.5) is rewritten as
follows
∫
Ω0(1)
ρσ(r)ε
0
xc
(
ρσ(r)
)
=
∫
Ω0(1)
drρps,σ(r−R1)ε
0
xc[ρps(r−R1)]
40
Interatomic interactions in transition metals
+
∑
R2
′
∫
Ω0(1)
drρi,σ(r−R1)V
σ
xc
(
ρps,σ(r−R1)
)
+
1
2
∑
R2
′
∫
Ω0(1)
dr
∂V σ
xc
(
ρps,σ(r−R1)
)
∂ρσ(r)
ρ2i,σ(r−R2)
+
∑′
R2,R3
∫
Ω0(1)
dr
∂V σ
xc
(
ρps,σ(r−R1)
)
∂ρσ
ρi,σ(r−R2)ρi,σ(r−R3) + . . . . (3.6)
Here
V σ
xc(r) =
∂
∂ρσ(r)
[
ρσ(r)ε
0
xc
(
ρσ(r)
)]
(3.7)
and the symbol
∫
Ω0(1)
. . . denotes integration over the unit cell on the site R1, the
prime on each summation excludes contributions with R i = Rj (i = 1, 2, 3). The
procedure to collect in (3.6) the terms of the same order with respect to ρi,σ(r−R) is
a simple one within the given approach. The items up to the order of ρ2
i,σ(r−R) only
are presented explicitly in (3.6) to avoid saturation of the present paper with too long
formulae. They add, see the right hand side of equation (3.6), to the E0(Ω0) and to
the energy of two-ion and three-ion indirect interactions. One can note from (3.5)–
(3.7) that the exchange-correlation effects contribute to the indirect interatomic
interactions via electrons of the same spin orientation.
The kinetic energy functional is treated in terms of contributions to the indirect
interatomic potentials in a similar way [1,5]. Then, using (3.1) to (3.7) and (2.5),
(2.6), one can represent the sum T [ρσ] + Exc[ρσ] as follows
N−1(T [ρσ] + Exc[ρσ]) = Eσ
kin(Ω0) + Eσ
xc(Ω0)
+
1
2!
∑
R2
V σ
2,ind(R1,R2) +
1
3!
∑
R2,R3
V σ
3,ind(R1,R2,R3) + . . . . (3.8)
Here,
Eσ
kin(Ω0) =
∫
(Ω0)(1)
drρps,σ(r−R1)t
(
ρps,σ(r−R1)
)
(3.9)
and
Eσ
xc(Ω0) =
∫
Ω0(1)
drρps,σ(r−R1)ε
0
xc
(
ρps,σ(r−R1)
)
(3.10)
contribute to E0(Ω0) in (2.7). Functions V σ
2,ind(R1,R2), V
σ
3,ind(R1,R2,R3), . . ., de-
scribe indirect spin dependent interactions between two, three,..., ions caused by the
band structure and exchange-correlation effects.
V σ
n,ind(R1 . . .Rn) = V σ
kin(R1 . . .Rn) + V σ
xc(R1 . . .Rn), n = 2, 3, 4 . . . , (3.11)
41
Z.Gurskii, J.Krawczyk
where V σ
kin(R1 . . .Rn) and V σ
xc(R1 . . .Rn) are these terms of series for N−1T (ρσ) and
N−1Exc(ρσ) which depend on the coordinates of n ions.
Expanding in (2.5) and (2.6) the integrands ρσ(r)ε
0
xc(ρσ(r)) and ρσ(r)t(ρσ(r))
with respect to ρi,σ(r−R) up to the n-th order, one can derive analytic expressions
for the pair and triplet indirect interatomic interactions via electrons in TM, see the
right hand side of (3.7) as an example. (Details of calculations for the simple metal
case are presented in [5,10]).
1
2!
V σ
2,ind(R1,R2) =
N−2
∑
n=0
1
(n+ 1)!
∫
Ω0(1)
dr
∂nV σ
ind(r−R1)
∂ρnσ
ρn+1
i,σ (r−R2), (3.12)
1
3!
V σ
3,ind(R1,R2,R3) =
∫
Ω0(1)
dr
∂V σ
ind(r−R1)
∂ρσ
ρi,σ(r−R2)ρi,σ(r−R3)
+
N
∑
n=3
n−1
∑
m=1
Cm
n
n!
∫
Ω0(1)
dr
∂n−1V σ
ind(r−R1)
∂ρn−1
σ
ρn−m
i,σ (r−R2)ρ
m
i,σ(r−R3), (3.13)
where Cm
n are the binomial coefficients. The following notations are introduced in
(3.11) to (3.13),
∂nV σ
ind(r−R1)
∂ρnσ
≡
∂nV σ
ind
(
ρσ(r)
)
∂ρnσ(r)
∣
∣
ρσ(r)=ρps,σ(r−R1) , (3.14)
V σ
kin(r) =
∂
∂ρσ(r)
[
ρσ(r)t
(
ρσ(r)
)]
. (3.15)
The first addend in (3.12) and the first term in (3.13) represent irreducible indirect
pair and triplet interactions, respectively, in TM. The next addends in (3.12) and
(3.13) are contributions to V σ
2,ind(R1,R2) and V σ
3,ind(R1,R2,R3) arising from the n-
particle potentials (3.11) when (n−2) or (n−3) indices of ion coordinates coincide,
accordingly.
Generalization of equation for the pair total potential [11] on the spin-polarized
case of TM does not face any difficulties. Omitting details we present the final result
(compare with equation (23) in [1] for V2(R1,R2) in simple metal)
V σσ′
2 (R1,R2) = V σσ′
i−i,dir(R1,R2) + V σ
2,ind(R1,R2)
+
1
2
Ω2
0
(2π)3
∫
dq[V σ′
ext(q) + V σ′
H (q)]ρi,σ(−q) exp[iq(R1 −R2)]
+
1
2
Ω2
0
(2π)3
∫
dq[V σ
ext(q) + V σ
H (q)]ρi,σ′(−q) exp[iq(R1 −R2)]. (3.16)
Here V σσ′
i−i,dir(R1,R2) is the potential of the direct ion-ion interaction, and
V σ
H (q) =
4πρps,σ(q)
q2
.
42
Interatomic interactions in transition metals
V σ
ext(q), ρi,σ(q) and ρps,σ(q) are the Fourier transforms of V σ
ext(r), ρi,σ(r) and ρps,σ(r),
respectively.
It is seen from (3.16) that just the Coulomb potentials (V dir
ii , Vext, VH) cause
interactions between atoms of different spin orientation. The band structure and
exchange-correlation effects (see V σ
2,ind (3.12)) contribute to the effective interatomic
potential V σσ
2 (R1,R2) that describes interactions of the same spin orientation.
4. Discussion
The focal point of the linear superposition assumption (2.3) and LSDA is an
accurate real-space treatment of the TM total energy in terms of well-defined inter-
atomic potentials, see (2.7). The V σσ′
2 , V σ
3 etc. potentials entering expansion (2.7)
are volume and spin dependent, but explicitly structure independent. Thus, they are
rigorously transferable at a given volume to all bulk structures, either ordered or
disordered. They can be used for example in computer simulations of TM properties
in solid, liquid and amorphous phases.
References
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p. 12431.
4. Carlsson A.E. // Phys. Rev. B., 1991, vol. 44, p. 6590.
5. Gurskii Z., Krawczyk J. // Cond. Matt. Phys., 1999, vol. 2, p. 383.
6. Callaway J., March N.H. // Solid State Physics, 1984, vol. 38, p. 136.
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p. 864.
8. Yukhnovskii I.R., Gurskii Z.A. Quantum-Statistical Theory of Disordered Systems.
Kyiv, Naukova Dumka, 1991 (in Russian).
9. Ghosh S.K., Deb B.M. // J. Phys. B.: At. Mol. Opt. Phys., 1994, vol. 27, p. 381.
10. Gurskii Z., Krawczyk J. // Metallofiz. Noveishie Tekhnol., 1999, vol. 21, No. 3, p. 10.
11. Gurskii Z., Krawczyk J. // Met. Phys. Adv. Tech., 1998, vol. 16, p. 1335.
43
Z.Gurskii, J.Krawczyk
Визначення із перших принципів міжатомних
взаємодій у перехідних металах
З.Гурський 1,2 , Є.Кравчик, 2
1 Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
2 Університет Ополє, Польща, 45-052 Ополє, вул. Олеска, 48
Отримано 14 серпня 2000 р.
Розвинуто оригінальний підхід обчислення багаточастинкових міжа-
томних потенціалів у перехідних металах. Він грунтується на вико-
ристанні наближення локальної густини спінів та припущенні про те,
що густину валентних електронів можна представити у вигляді ліній-
ної суперпозиції. Подані явні вирази для парних та потрійних взаємо-
дій, що залежать від орієнтації спіна. Показано, що багаточастинкові
потенціали (три-, чотиричастинкові) описують опосередковані міжа-
томні взаємодії однакової орієнтації спінів.
Ключові слова: локальна густина спінів, функціонал повної енергії,
багаточастинкові потенціали
PACS: 34.20.Cf, 71.15.Mb, 71.15.Nc, 71.20.Be
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