How does one extract many-body interatomic potentials from ab-initio band structure calculations
An original approach to the ab-initio deriving many-body interatomic potentials in metals is discussed. It is based on calculating the Kohn-Sham total energy functional using the pseudopotential method. The local density approximation (LDA) is shown to be applied, within the pseudopotential concept...
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Gurskii, Z. Krawczyk, J. 2017-06-12T08:30:38Z 2017-06-12T08:30:38Z 1999 How does one extract many-body interatomic potentials from ab-initio band structure calculations / Z. Gurskii, J. Krawczyk // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 383-392. — Бібліогр.: 18 назв. — англ. 1607-324X DOI:10.5488/CMP.2.3.383 PACS: 34.20.Cf, 71.15.Nc, 71.15.Hx https://nasplib.isofts.kiev.ua/handle/123456789/120502 An original approach to the ab-initio deriving many-body interatomic potentials in metals is discussed. It is based on calculating the Kohn-Sham total energy functional using the pseudopotential method. The local density approximation (LDA) is shown to be applied, within the pseudopotential concept, to the analysis of the valence electron kinetic energy. Utilizing the LDA and linear superposition assumption for total electron density enables us to treat the exchange-correlation and kinetic energies of valence electrons in terms of contributions to indirect many-body interactions. Equations for the pair and for the triplet potentials in metals are given. Relationship between the method developed and other approaches is analyzed. Викладено підхід, в основі якого лежить використання методу псевдопотенціалів та формалізму Кона-Шема. Показано, як у наближенні локальної густини можна записати обмінно-кореляційну та кінетичну енергії валентних електронів у вигляді внесків у непрямі (опосередковані) багаточастинкові взаємодії. Аналізується зв’язок розвиненого підходу з відомим способом отримання міжатомних потенціалів у рамках теорії збурень за псевдопотенціалом. Подано явні вирази для парних і тричастинкових потенціалів. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics How does one extract many-body interatomic potentials from ab-initio band structure calculations Як можна отримати багаточастинкові взаємодії із розрахунків зонної структури із перших принципів ? Article published earlier |
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How does one extract many-body interatomic potentials from ab-initio band structure calculations |
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How does one extract many-body interatomic potentials from ab-initio band structure calculations Gurskii, Z. Krawczyk, J. |
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How does one extract many-body interatomic potentials from ab-initio band structure calculations |
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How does one extract many-body interatomic potentials from ab-initio band structure calculations |
| title_fullStr |
How does one extract many-body interatomic potentials from ab-initio band structure calculations |
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How does one extract many-body interatomic potentials from ab-initio band structure calculations |
| title_sort |
how does one extract many-body interatomic potentials from ab-initio band structure calculations |
| author |
Gurskii, Z. Krawczyk, J. |
| author_facet |
Gurskii, Z. Krawczyk, J. |
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1999 |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
| title_alt |
Як можна отримати багаточастинкові взаємодії із розрахунків зонної структури із перших принципів ? |
| description |
An original approach to the ab-initio deriving many-body interatomic potentials in metals is discussed. It is based on calculating the Kohn-Sham total
energy functional using the pseudopotential method. The local density approximation (LDA) is shown to be applied, within the pseudopotential concept, to the analysis of the valence electron kinetic energy. Utilizing the LDA
and linear superposition assumption for total electron density enables us to
treat the exchange-correlation and kinetic energies of valence electrons in
terms of contributions to indirect many-body interactions. Equations for the
pair and for the triplet potentials in metals are given. Relationship between
the method developed and other approaches is analyzed.
Викладено підхід, в основі якого лежить використання методу псевдопотенціалів та формалізму Кона-Шема. Показано, як у наближенні
локальної густини можна записати обмінно-кореляційну та кінетичну
енергії валентних електронів у вигляді внесків у непрямі (опосередковані) багаточастинкові взаємодії. Аналізується зв’язок розвиненого підходу з відомим способом отримання міжатомних потенціалів у
рамках теорії збурень за псевдопотенціалом. Подано явні вирази для
парних і тричастинкових потенціалів.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120502 |
| citation_txt |
How does one extract many-body interatomic potentials from ab-initio band structure calculations / Z. Gurskii, J. Krawczyk // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 383-392. — Бібліогр.: 18 назв. — англ. |
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2025-11-24T04:21:21Z |
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| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 3(19), pp. 383–392
How does one extract many-body
interatomic potentials from ab-initio
band structure calculations
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, ul.Oleska 48, 45–052 Opole, Poland
Received October 29, 1998
An original approach to the ab-initio deriving many-body interatomic poten-
tials in metals is discussed. It is based on calculating the Kohn-Sham total
energy functional using the pseudopotential method. The local density ap-
proximation (LDA) is shown to be applied, within the pseudopotential con-
cept, to the analysis of the valence electron kinetic energy. Utilizing the LDA
and linear superposition assumption for total electron density enables us to
treat the exchange-correlation and kinetic energies of valence electrons in
terms of contributions to indirect many-body interactions. Equations for the
pair and for the triplet potentials in metals are given. Relationship between
the method developed and other approaches is analyzed.
Key words: electron density, indirect interactions, kinetic energy
functional
PACS: 34.20.Cf, 71.15.Nc, 71.15.Hx
1. Introduction
The paper concerns a problem of derivation from the first principles multi-ion
interparticle potentials in metals. The approach suggested is based on the density-
functional and pseudopotential versions of the Kohn-Sham theory [1,2]. The focal
points of the method developed are as follows:
a) an accurate real-space representation of the metal total energy in terms of
widely transferable, structure-independent interatomic potentials [3];
b) treatment of the valence electron kinetic energy, determined through the ab-
initio band structure calculations in terms of appropriate contributions to the many-
ion indirect interactions [4].
There are two motivating factors behind this work. First, calculating the inter-
c© Z.Gurskii, J.Krawczyk] 383
Z.Gurskii, J.Krawczyk
atomic potentials is one of the central problems in the modern condensed matter
physics having a long history, see review of literature in [3–6]. A role of angular forces,
caused by the multi-ion interactions, is the subject of special interest in describing
metal properties microscopically [7].
A second motivating factor in this work is the prospect of doing molecular dy-
namics (MD) simulation of various metal properties having derived potentials [5,7].
Comparing the results of MD simulation with the corresponding experimental data
we get a convincing verification of a theoretical background used at the interatomic
potential evaluation.
2. History of the problem
One can distinguish several trends in deriving interatomic potentials in metals.
2.1. The rigorous development of interatomic potentials from quantum mechanics
is based on the pseudopotential method [8,5,6]. Working within the framework of
the perturbation theory in pseudopotential one can represent metal total energy in
the form of expansion in the effective multi-ion interactions [5–8]:
Etot = E0(Ω0) +
1
2!
∑
i,j
′
V2(Ri,Rj) +
1
3!
∑
i,j,l
′
V3(Ri,Rj,Rl) + . . . . (1)
The first term E0 depends on atomic volume Ω0 only and includes the energy of
the interacting electron gas as well as all one particle intraatomic contributions
to the Etot. The interatomic potentials V2, V3 etc. are implicitly volume depen-
dent but explicitly structure independent and thus rigorously transferable at a
given volume Ω0 to the all bulk structures, either ordered or disordered. To put
it another way the potentials themselves are independent of the absolute ion po-
sitions Ri and depend only on relative separations Rij = |Ri − Rj| [6,7]. For
example, the angular-force triplet potential V3 is the three-dimensional function:
V3(R1,R2,R3; Ω) ≡ V3(R12, R13, R23; Ω) [7]. The prime on each summation in (1)
denotes exclusion of all the self-interaction terms where two indices are equal. The
interatomic potentials V2, V3 etc. present on the right hand side of equation (1) are
completely transferable at a fixed volume because of their independence on metal
structure. The structure dependence of the total energy (1) appears through the
summations in (1) over all N ion positions. One should stress the important feature
of the equation (1) obtained within the pseudopotential theory. Expansion (1) is
rapidly convergent in the sense that the four-ion quadruplet potential V4 is smaller
than V2, V3 and higher potentials (V5, V6...) appear to be negligible [6,7].
2.2. The embedded atom method (EAM), see [9] and literature cited there, is a
semi-empirical approach. It is based on the assumption that the total energy of a
system can be represented as the sum of two parts: one of them having its origin
from the band structure is a function of the electron density and the other is a
short-ranged pairwise potential describing interactions between the atomic cores.
384
How does one extract many-body interatomic potentials . . .
Thus,
Etot =
∑
i
F (ρi) +
1
2
∑
i 6=j
ϕ(Rij). (2)
The embedding function F (ρ), the pairwise potential ϕ(Rij) and the charge density
ρi at the site Ri are modeled in the EAM. For example, the authors of [9] proposed
the following forms for ϕ(r) and ρi:
ϕ(r) =
6
∑
n=1
Kn[(r/R1 − 1)e−α(r/R1−1)2 ]n r 6 R5, (3)
ρi = ρ0
[
1 +
∑
j
f(Rij)
]
, (4)
where R1 and R5 are the nearest and fifth neighbour distances, respectively, Kn, α
and ρ0 are parameters to be determined [9].
The effects of the electron density Friedel oscillations are introduced in ρ i through
the function f(r) [9]
f(r) = ∆
cos (2KFr + δ)
(r/R1)3
. (5)
Here KF is the Fermi vector, ∆ and δ are free parameters. One can notice that there
are a lot of fitting parameters in the EAM. Besides, the introduced forms for F (ρi)
and ϕ(Rij) modelling functions demand theoretical justification.
2.3. The “pure phenomenological” approaches were exploited formerly [10–12]
with representation of the total energy entirely in terms of empirical pair potentials.
One should put E0 = V3 = V4 = ... = 0 in equation (1) to compare the ab-initio and
phenomenological approaches. Arbitrary forms for V2(Rij) containing a number of
adjustable parameters are used typically [10]. Such potentials are implicitly structure
dependent and hence their transferability is always strongly doubted. The authors of
[11] tried to determine a pair potential more systematically by inverting the energy-
band calculation of Etot as a function of volume Ω0. To put another way round
they obtained V2 ≡ V2(Rij ,Ω) [11]. Some attempts were also made to extract a pair
potential from experimental phonon spectrum [12].
In some cases an adequate theoretical description of metal and alloy properties
could not be achieved within the pair interatomic interaction model. It means the
many-body interactions (three-, four-particles) play essential role [13]. Their being
taken into account is especially important for transition metal investigations [7,14].
This brief review of literature shows clearly that the ab-initio approach is the
most consistent and promising on the problem of interatomic potential evaluation.
A somewhat different way of deriving the effective interatomic potentials in metals
within the ab-initio approach has been advanced in [3,4]. It is based on solution
of the Kohn-Sham equations as well as utilizing the pseudopotential concept and
the local density approximation [3,4]. The approach developed has the following
features:
a) the Kohn-Sham total energy functional is treated in terms of the multi-ion
interactions;
385
Z.Gurskii, J.Krawczyk
b) the many-body potentials are evaluated without using the perturbation theory
in pseudopotential.
3. Many-body potentials in metal
Let us examine an electron-ion system within the pseudopotential concept. It
means the core electrons are excluded from the explicit consideration in the all
electron problem and the true electron-ion potential is replaced by a pseudopotential
[15]. Thus, the pseudoions are placed in lattice sitesR and their interactions with the
valence electrons are described by a pseudopotential w(r−R) which is an operator
in a general case.
We start from the Kohn-Sham total energy functional of such a system [2]
E[ρ] = T [ρ] +
∫
W (r)ρ(r)dr+
1
2
∫
ρ(r)ρ(r′)
|r− r′|
drdr′ + Exc[ρ] + Ei−i . (6)
Here T [ρ] and Exc[ρ] are functionals of the kinetic and exchange-correlation energies,
respectively. The second and the third terms in (6) describe the energy of the electron
subsystem in the external field of the pseudoions
W (r) =
∑
R
w(r−R) (7)
and the Hartree energy, correspondingly. The last term Ei−i in (6) is an energy of
the ion-ion direct interactions. The total energy E[ρ] (6) is the universal functional
of an electron density
ρ(r) =
∑
k
Ψ∗
k
(r)Ψk(r) (8)
with Ψk(r) the valence electron pseudowave functions and k the wave vectors char-
acterising electron states. Summation in (8) is performed over all occupied states.
One can find the pseudowave functions Ψk(r) solving a system of the Kohn-
Sham equations with the pseudopotential [4]. As a result the pseudodensity ρ(r) (8)
of valence electrons that minimizes the total energy functionalE[ρ] (6) is known. The
pseudodensity differs from the true density of valence electrons ρv(r) in the regions
of ion cores and coincides with it outside cores. Unlike ρv(r) the pseudodensity is a
smooth function in the ion core region. The importance of this circumstance will be
demonstrated below.
In [3] the known quantity of ρ(r) (8) has been presented in the form
ρ(r) = ρ0 +
∑
R
ρi(r−R), (9)
where ρ0 = z/Ω0 (Ω0 = Ω/N) is a density of the uniform electron distribution, z an
ion valency, Ω0 the atomic volume and ρi(r − R) is an electron density related to
the pseudoion at the site R.
386
How does one extract many-body interatomic potentials . . .
The focal point of the linear superposition assumption (9) is an accurate real-
space treatment of the metal total energy in terms of well-defined interatomic po-
tentials [4]. The local density approximation (LDA) [2] and equation (9) makes it
possible to represent the T [ρ] and Exc[ρ] functionals in terms of appropriate contri-
butions to the many-body interatomic interactions and get the total energy in the
form of equation (1). Let us prove this statement.
The LDA is almost universally used for the Exc[ρ] functional in total energy
pseudopotential calculations [6,7,16]
Exc[ρ] =
∫
drρ(r)εxc (ρ(r)) (10)
with εxc (ρ(r)) the exchange-correlation energy per electron at point r which is equal
to the exchange-correlation energy of the homogeneous electron gas possessing the
same density ρ(r) as the electron subsystem under investigation.
Consider the kinetic energy functional T [ρ]. Within the pseudopotential concept
the unknown Ψk(r) functions of the valence electrons are searched on the plane-wave
basis [5,6,8], that is
Ψk(r) =
∑
G
aG(k) exp[i(k+G)r]. (11)
Here aG(k) are the expansion coefficients and G, the reciprocal lattice vectors.
Solution of the Kohn-Sham equations provides information on coefficients aG(k) as
eigenvectors of this problem. The valence electron kinetic energy with allowance for
(11) is equal to
T =
∑
k
∑
G6GM
1
2
|aG(k)|
2 · |k+G|2. (12)
The GM vector is determined by the cutoff energy |k + GM |2/2. Let us represent
the known quantity T (12) as follows
∑
k
∑
G
1
2
|aG(k)|
2 · |G+ k|2 =
∫
drρ(r)t (ρ(r)) (13)
that is in the form of LDA like Exc[ρ] (10). One can regard (13) as a constraint on
the unknown function t (ρ(r)).
In the case of simple metals the following form could be used for the t (ρ(r))
function [17]
t (ρ(r)) = C
3
10
[3π2ρ(r)]2/3. (14)
The value C = 1 corresponds to the homogeneous electron gas LDA expression. We
propose to treat C as a free coefficient fitting the right hand side of equation (13)
to the valence electron kinetic energy value (12).
The formula (14) has a physical justification. The valence electrons behave like
free electrons in nontransitive metals [5,8,13,15,16] and the Fermi-surface is close
to the spherical one. That is why, in simple metals the kinetic energy of valence
387
Z.Gurskii, J.Krawczyk
electrons can be approximated by functionals like (14), see [18] and literature cited
there. The results of [17] confirm this statement. Certainly, the form (14) fails in
the all electron problem when both the core and valence electrons are considered,
see review of literature in [18]. There are convincing reasons for explaining this
failure: behaviour (distribution) of the core electron density ρc(r) does not satisfy
the conditions of the Hohenberg-Kohn formalism applicability [1] (ρ c(r)-function is
not smooth).
The pseudopotential method permits to avoid these difficulties. The core elec-
trons are excluded from the explicit consideration but the true electron-ion potential
is replaced by a nonlocal operator that describes an effective electron-ion interaction
[5,8,15]. The pseudoelectron density ρ(r) (8), see (11) is used in the total electron
density calculations within the pseudopotential method [16]. It satisfies the con-
ditions of the Kohn-Hohenberg theory applicability better than the true electron
density because the potentials generated by ρ(r) are much smoother than those
caused by the true density [1]. We should emphasize that the approximation (13)
with (14) is used just for interpreting the valence electron kinetic energy in terms
of appropriate contributions to the many-ion indirect potentials in (1). Such an
interpretation is the most natural within the pseudopotential concept.
Let us expound briefly the main ideas of the approach advanced in [3,4]. The
regular metal of the valency z with one atom per primitive cell is considered. The
functions t(r) and εxc(r) are periodical ones in the regular crystal because of the
electron density ρ(r) periodicity. Therefore, one should consider them in the region
of the Wigner-Seitz cell only. Let us define
ρps(r−R1) = ρ0 + ρi(r−R1) (15)
as the electron density of a pseudoatom at the site R1 and expand t (ρ(r)) and
εxc (ρ(r)) in ρps(r−R1) for r ∈ Ω0 (1). Symbol r ∈ Ω0 (1) denotes that the running
variable r takes values in the unit cell (the Wigner-Seitz cell) centred at the site R 1.
For function f (ρ(r)) = t (ρ(r)) and f (ρ(r)) = ε (ρ(r)) such an expansion reads
f (ρ(r)) = f (ρps(r−R1)) +
∂f(r −R1)
∂ρ
∑
R 6=R1
ρi(r−R)
+
1
2!
∂2f(r−R1)
∂ρ2
[
∑
R 6=R1
ρi(r−R)
]2
+ . . . (16)
with r ∈ Ω0 (1) and
∂nf(r−R1)
∂ρn
≡
∂nf (ρ(r))
∂ρn
∣
∣
ρ(r)=ρps(r−R1) . (17)
It is seen from (15) and (9) that the total pseudoelectron density ρ(r) in the vicinity
of each atom is expressed as a sum of the density ρps contributed by the pseudoatom
in question plus contributions from the surrounding ions that is
ρ(r) = ρps(r−Rj) +
∑
R 6=Rj
ρi(r−R) for r ∈ Ω0(j). (18)
388
How does one extract many-body interatomic potentials . . .
The condition
∣
∣
∣
∣
∣
∣
∑
R 6=Rj
ρi(r−R)/ρps(r−Rj)
∣
∣
∣
∣
∣
∣
≪ 1 for r ∈ Ω0(j) (19)
provides convergence of the series for ρ(r)t (ρ(r)) and ρ(r)ε (ρ(r)) [4]. Thus, it is the
main trick to represent the known total electron density ρ(r) in each unit cell Ω0(j)
in such a manner (see equation (18)) that the condition (19) would take place in the
whole Ω0(j) region [3].
Owing to the (16) and (18) one can treat the Exc[ρ] and the T [ρ] in terms of
contributions of the exchange-correlation and the kinetic energy effects to the many-
body interatomic indirect interactions [4] and represent the sum T [ρ]+Exc[ρ] in the
form
N−1(T [ρ] + Exc[ρ]) = Ekin(Ω0) + Exc(Ω0) +
1
2!
∑
R2
V ind
2 (R1,R2)
(20)
+
1
3!
∑
R2,R3
V ind
3 (R1,R2,R3) + . . .+
1
n!
∑
R2,...,Rn
V ind
n (R1 . . .Rn) + . . . .
Here
Ekin(Ω0) =
∫
Ω0(1)
drρps(r−R1)t[ρps(r−R1)] (21)
and
Exc(Ω0) =
∫
Ω0(1)
drρps(r−R1)εxc[ρps(r−R1)] (22)
do not depend on ion configuration. Functions V ind
2 (R1,R2), V
ind
3 (R1 . . .R3) . . . de-
scribe indirect interactions between two, three. . . ions caused by the band structure
and exchange-correlation effects:
V ind
n (R1 . . .Rn) = Vkin(R1 . . .Rn) + Vxc(R1 . . .Rn) , (23)
where Vkin(R1 . . .Rn) and Vxc(R1 . . .Rn) are these terms of series for T [ρ]/N and
Exc[ρ]/N which depend on the coordinates of n-ions.
One can derive explicit expressions for the indirect many-ion interactions in metal
using equations (13), (16), (18) and (20)
1
2!
V ind
2 (R1,R2) =
N
∑
n=0
1
(n+ 1)!
∫
Ω0(1)
dr
∂nVind(r−R1)
∂ρn
ρn+1
i (r−R2), (24)
1
3!
V ind
3 (R1,R2,R3) =
∫
∂Vind(r−R1)
∂ρ
ρi(r−R2)ρi(r−R3)dr
389
Z.Gurskii, J.Krawczyk
(25)
+
N
∑
n=3
n−1
∑
m=1
Cm
n
n!
∫
Ω0(1)
∂n−1Vind(r−R1)
∂ρn−1
ρn−m
i (r−R2)ρ
m
i (r−R3)dr
with
∂nVind(r−R1)
∂ρn
≡
∂nVkin(r−R1)
∂ρn
+
∂nVxc(r−R1)
∂ρn
(26)
and Cm
n the binomial coefficients. The following notations are introduced in (24) to
(26):
∂nVind(r−R1)
∂ρn
≡
∂nVind(ρ(R))
∂ρn
∣
∣
∣
∣
ρ(r)=ρps(r−R1)
, (27)
Vkin[ρ(r)] =
∂
∂ρ
[ρ(r)t(ρ(r))] , (28)
Vxc[ρ] =
∂
∂ρ
[ρ(r)εxc(ρ)] . (29)
The first summand in (24) and the first term in (25) represent irreducible indirect
pair and triplet interactions, respectively, in metal. The next summands in (24) and
(25) are contributions to the V ind
2 (R1,R2) and V ind
3 (R1,R2,R3) arising from the
n-particle potentials (23) when (n− 2) or (n− 3) indices of ion coordinates coincide
accordingly.
They should emphasize that the present stage of the metal microscopic theory
permits one to estimate only the reducible terms of the third and forth order in
pseudopotential contributing to the V ind
2 (R1,R2) [13]. The problem of correct cal-
culation of irreducible contributions to the V ind
3 (R1 . . .R3), V
ind
4 (R1 . . .R4) has not
been solved completely yet. Unlike the perturbation theory in pseudopotential the
approach developed enables us to work out a general procedure of renormalizing the
n-ion potential by the (n + 1)-, (n + 2)-, etc. reducible interionic interactions. An
elegant analytical form for such a renormalization is obtained in the cases of the
pair and triplet potentials, see equations (24) and (25). That is why the proposed
method looks like a promising one.
4. Conclusions
A scheme of deriving the effective interatomic potentials is presented. Informa-
tion on the total valence electron density ρ(r) is needed only. The ρ(r) expansion
in contributions of the individual atoms is the essential point of the method sug-
gested. The linear superposition assumption (9), see also (18), for the total electron
density enables us to get the analytical expressions for the indirect interactions.
The n-particle potentials (n > 2) are formed by the kinetic (band structure) and
exchange-correlation effects only. Equation for the total pair potential is given in
[3]. Evaluation of the derived potentials as well as utilizing them in metal prop-
erty investigations are highly desirable for providing a convincing verification of the
approach proposed.
390
How does one extract many-body interatomic potentials . . .
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391
Z.Gurskii, J.Krawczyk
Як можна отримати багаточастинкові взаємодії із
розрахунків зонної структури із перших принципів ?
З.Гурський 1 , Є.Кравчик 2
1 Інститут фізики конденсованих систем НАН Укpаїни,
79011 Львів, вул. Свєнціцького, 1
2 Університет Ополє, вул. Олеска 48, 45–052 Ополє, Польща
Отримано 29 жовтня 1998 р.
Викладено підхід, в основі якого лежить використання методу псев-
допотенціалів та формалізму Кона-Шема. Показано, як у наближенні
локальної густини можна записати обмінно-кореляційну та кінетичну
енергії валентних електронів у вигляді внесків у непрямі (опосеред-
ковані) багаточастинкові взаємодії. Аналізується зв’язок розвинено-
го підходу з відомим способом отримання міжатомних потенціалів у
рамках теорії збурень за псевдопотенціалом. Подано явні вирази для
парних і тричастинкових потенціалів.
Ключові слова: електронна густина, непрямі взаємодії,
функціонал кінетичної енергії
PACS: 34.20.Cf, 71.15Nc, 71.15.Hx
392
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