Semi-infinite metal: perturbative treatment based on semi-infinite jellium
Energy of electronic subsystem of semi-infinite metal is presented in the form of an expansion in powers of pseudo-potential. It is shown that generally electron many-particle density matrices are necessary for the energy calculation, whereas in case of a local pseudo-potential only diagonal eleme...
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Kostrobij, P.P. Markovych, B.M. 2017-06-07T12:52:15Z 2017-06-07T12:52:15Z 2008 Semi-infinite metal: perturbative treatment based on semi-infinite jellium / P.P. Kostrobij, B.M. Markovych // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 641-651. — Бібліогр.: 25 назв. — англ. 1607-324X PACS: 71.45.Gm, 71.10.-w, 73.20.-r DOI:10.5488/CMP.11.4.641 https://nasplib.isofts.kiev.ua/handle/123456789/119576 Energy of electronic subsystem of semi-infinite metal is presented in the form of an expansion in powers of pseudo-potential. It is shown that generally electron many-particle density matrices are necessary for the energy calculation, whereas in case of a local pseudo-potential only diagonal elements (electron distribution functions) are necessary. In a specific case of a local pseudo-potential within the first order of perturbation theory, our results for energy coincide with those widely applicable in the density functional theory. Представлено енергiю електронної пiдсистеми напiвобмеженого металу у виглядi ряду за степенями псевдопотенцiалу. Показано, що в загальному випадку для розрахунку цiєї енергiї необхiднi електроннi багаточастинковi матрицi густини, тодi як у частковому випадку локального псевдопотенцiалу лише її дiагональнi елементи (функцiї розподiлу електронiв). У першому порядку теорiї збурень у випадку локального псевдопотенцiалу наш результат для енергiї спiвпадає iз широко застосовуваними у теорiї функцiоналу густини. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Semi-infinite metal: perturbative treatment based on semi-infinite jellium Напiвобмежений метал: пiдхiд на основi моделi напiвобмеженого “желе” Article published earlier |
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Semi-infinite metal: perturbative treatment based on semi-infinite jellium |
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Semi-infinite metal: perturbative treatment based on semi-infinite jellium Kostrobij, P.P. Markovych, B.M. |
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Semi-infinite metal: perturbative treatment based on semi-infinite jellium |
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Semi-infinite metal: perturbative treatment based on semi-infinite jellium |
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Semi-infinite metal: perturbative treatment based on semi-infinite jellium |
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Semi-infinite metal: perturbative treatment based on semi-infinite jellium |
| title_sort |
semi-infinite metal: perturbative treatment based on semi-infinite jellium |
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Kostrobij, P.P. Markovych, B.M. |
| author_facet |
Kostrobij, P.P. Markovych, B.M. |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
| title_alt |
Напiвобмежений метал: пiдхiд на основi моделi напiвобмеженого “желе” |
| description |
Energy of electronic subsystem of semi-infinite metal is presented in the form of an expansion in powers
of pseudo-potential. It is shown that generally electron many-particle density matrices are necessary for the
energy calculation, whereas in case of a local pseudo-potential only diagonal elements (electron distribution
functions) are necessary. In a specific case of a local pseudo-potential within the first order of perturbation
theory, our results for energy coincide with those widely applicable in the density functional theory.
Представлено енергiю електронної пiдсистеми напiвобмеженого металу у виглядi ряду за степенями псевдопотенцiалу. Показано, що в загальному випадку для розрахунку цiєї енергiї необхiднi електроннi багаточастинковi матрицi густини, тодi як у частковому випадку локального псевдопотенцiалу лише її дiагональнi елементи (функцiї розподiлу електронiв). У першому порядку теорiї збурень у випадку локального псевдопотенцiалу наш результат для енергiї спiвпадає iз широко застосовуваними у теорiї функцiоналу густини.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119576 |
| citation_txt |
Semi-infinite metal: perturbative treatment based on semi-infinite jellium / P.P. Kostrobij, B.M. Markovych // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 641-651. — Бібліогр.: 25 назв. — англ. |
| work_keys_str_mv |
AT kostrobijpp semiinfinitemetalperturbativetreatmentbasedonsemiinfinitejellium AT markovychbm semiinfinitemetalperturbativetreatmentbasedonsemiinfinitejellium AT kostrobijpp napivobmeženiimetalpidhidnaosnovimodelinapivobmeženogožele AT markovychbm napivobmeženiimetalpidhidnaosnovimodelinapivobmeženogožele |
| first_indexed |
2025-11-24T23:41:04Z |
| last_indexed |
2025-11-24T23:41:04Z |
| _version_ |
1850498112092635136 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 4(56), pp. 641–651
Semi-infinite metal: perturbative treatment based on
semi-infinite jellium
P.P.Kostrobij, B.M.Markovych
Lviv Polytechnic National University, 12, S. Bandera Str., Lviv 79013, Ukraine
Received February 27, 2008
Energy of electronic subsystem of semi-infinite metal is presented in the form of an expansion in powers
of pseudo-potential. It is shown that generally electron many-particle density matrices are necessary for the
energy calculation, whereas in case of a local pseudo-potential only diagonal elements (electron distribution
functions) are necessary. In a specific case of a local pseudo-potential within the first order of perturbation
theory, our results for energy coincide with those widely applicable in the density functional theory.
Key words: partition function, pseudo-potential, many-particle density matrix
PACS: 71.45.Gm, 71.10.-w, 73.20.-r
1. Introduction
Theoretical studies of equilibrium properties of a metal surface turn out to be very difficult,
because an electronic subsystem of a bounded metal is spatially very nonuniform. This greatly com-
plicates the consecutive account of many-particle effects in an electronic subsystem. The greatest
successes in studying the electronic properties of a metal surface have been attained within den-
sity functional theory (DFT) [1]. However, the study of properties of a metal surface which are
caused by discreteness of the ionic subsystem is quite problematic, since there is no technique for
constructing necessary energy functionals.
The first attempt in this direction was made by Lang and Kohn [2]. At first they used a time
jellium model for a metal surface with the correction that takes into consideration discreteness of
the ionic subsystem. This correction was simple enough: self-consistent density was still that of
jellium while the perturbation (the difference between the lattice potential and that of the uniform
positive background) was averaged over the surface plane. Thus, they managed to consider the ion
cores via the first order perturbation theory and via the classical cleavage energy. In subsequent
works [3–6] surface energy calculations have been fulfilled taking into account the discreteness of the
ionic subsystem in the first order of the perturbation and with the use of variational methods. These
calculations showed that in case of metals with ionic charge Z = 1, 2 the surface energy is in good
agreement with experimental data. Whereas for chargesZ > 2 this model is not adequate. The first-
order perturbative results based on jellium for the surface energy of slabs of simple metals, using
various local pseudopotentials (Ashcroft, Heine-Abarenkov and evanescent core) were examined in
work [7].
Later, discreteness of the ionic subsystem was taken into account by Rose and Dobson [8,9] via
the second-order perturbation theory, which includes the linear response function of semi-infinite
jellium. They calculated the second-order surface energy terms. At the same time they used the
linear response of bulk jellium, in a kind of local approximation. Second-order perturbation theory
using the linear response of a jellium slab in the random phase approximation has been worked
out by Barnett and coworkes [10] as well as by Eguiluz [11]. These calculations show a noticeable
effect of the second-order term in the face-dependent surface energies. A systematic method for
constructing a perturbation theory for energy of an electronic subsystem in the field of static ions
in non-transitive metals having a surface was offered by Kaim [12].
c© P.P.Kostrobij, B.M.Markovych 641
P.P.Kostrobij, B.M.Markovych
The structureless pseudo-potential model which had been applied to the calculations of a metal
surface was developed in works [13–15]. This model is nothing else but a well-known zero model
of metal [16–19]. In this model the energy of nonuniform metal consists of the energy of electronic
gas in a positive jellium, the Madelung energy of pointwise ions in homogeneous electron density,
the energy caused by non-Coulomb character of an electron-ion interaction at a zero transfer
momentum and the value of the first-order pseudo-potential correction averaged by volume of
semi-infinite metal. This average is included in the self-consistent procedure of Lang and Kohn [2].
The derived values of a surface energy of simple metals are close to those in the paper [2].
Thus, a certain progress in the theory of metal surface has been reached, but mainly for the
surfaces of simple metals that can be adequately described by local pseudo-potentials. Therefore,
in the presented paper, systematic perturbation theory for bounded metals, described by nonlocal
pseudo-potentials, is developed. The semi-infinite jellium [20–23] is used as a reference system and
the perturbation theory with respect to the “difference potential”1 is constructed. In a specific case
of a local pseudopotential, this perturbation theory coincides with the results by Kaim [12].
In section 2, the model of semi-infinite metal is described, the definition of surface potential is
entered, a Hamiltonian of this system is written in the second-quantization form. In section 3, the
partition function is presented in the form of an expansion in powers of pseudopotential. Section 4
presents a specific case of local pseudo-potential and a comparison with the results of other papers.
Conclusions are presented in section 5. Appendix presents a useful proof of the statement.
2. Model
We consider a semi-infinite metal with ions having charges Ze and Cartesian coordinates Rj
(−∞ < Xj , Yj < +∞, Zj 6 Z0, Z0 = const, z = Z0 is the division plane (surface)), j = 1, . . . , Nion.
Electrons of a semi-infinite metal have coordinates ri, i = 1, . . . , N . A Hamiltonian of this model
has the following form
H = − ~
2
2m
N∑
i=1
∆i +
1
2
N∑
i6=j=1
e2
|ri − rj |
+
Nion∑
j=1
P2
j
2M
+
1
2
Nion∑
i6=j=1
(Ze)2
|Ri −Rj |
+
N∑
i=1
Nion∑
j=1
w(ri,Rj), (1)
where the first summand is the kinetic energy of electrons, the second summand represents the
potential energy of the interelectron interaction, the third summand is the kinetic energy of ions
(P is the operator of ion momentum), the fourth summand represents the potential energy of the
ion-ion interaction and the last one represents the energy of electron-ion interaction. We assume
that the system is electroneutral, that is
ZNion = N. (2)
We shall present the potential of the electron-ionic interaction as
w(ri,Rj) = w(|ri −Rj |) + ∆w(ri,Rj), (3)
where w(|ri − Rj |) is a periodic potential of the electron-ion interaction in the case of infinite
metal (pseudo-potential), ∆w(ri,Rj) is a deviation of the electron-ionic potential of the semi-
infinite metal from the periodic one.
From the Hamiltonian (1) we extract a Hamiltonian of the semi-infinite jellium model Hjell
which was explored in papers [20–22]. We shall use the model of semi-infinite metal as the reference
system for studying thermodynamic and structural properties of semi-infinite metal. Finally we
get
H = Hjell + δHii +
N∑
i=1
Nion∑
j=1
δw(|ri −Rj |), (4)
1That is the difference between the pseudo-potential of ions and the electrostatic potential of the semi-infinite
jellium positive background.
642
Semi-infinite metal: perturbative treatment based on semi-infinite jellium
where
δHii =
Nion∑
j=1
P2
j
2M
+
1
2
Nion∑
i6=j=1
(Ze)2
|Ri −Rj |
− 1
2
∫
dR
∫
dR′ ρjell(R)ρjell(R
′)
|R−R′| , (5)
Hjell = Hunif
jell +
N∑
i=1
V (ri), (6)
V (ri) = Vjell(ri) + Vion(ri) (7)
is the surface potential acting on electrons.
δw(|ri −Rj |) = w(|ri −Rj |) +
1
Nion
∫
dR
eρjell(R)
|ri −R| (8)
is the “difference potential” (see footnote on page 642),
Hunif
jell =−~
2
2m
N∑
i=1
∆i +
1
2
N∑
i6=j=1
e2
|ri − rj |
− e2N
V
N∑
i=1
∫
dR
1
|ri −R|+
(eN)2
2V 2
∫
dR
∫
dR′ 1
|R−R′| (9)
is the Hamiltonian of homogeneous jellium. V = SL is the volume of the system, S is the surface
area of semi-infinite metal, L determines the area of the change of the electron coordinate normal
to the surface: z ∈ (−L/2,+L/2), S → ∞, L→ ∞.
Vjell(ri) = e
∫
dR
eN/V − ρjell(R)
|ri −R| +
1
2N
∫
dR
∫
dR′ ρjell(R)ρjell(R
′) − (eN/V )
2
|R−R′| (10)
is the part of the surface potential formed by semi-infinite jellium.
Vion(ri) =
Nion∑
j=1
∆w(ri,Rj) (11)
is the part of the surface potential created by deviation ∆w(ri,Rj) of the true electron-ionic
potential of semi-infinite metal w(ri,Rj) from the space-periodical electron-ionic potential w(|ri −
Rj |).
ρjell(R) ≡ ρjell(Z) = ρ0θ(−d− Z), ρ0 =
eN
V/2
(12)
is a distribution of the ionic density in semi-infinite jellium, parameter d is defined by the elec-
troneutrality condition.
In what follows we assume that the electron-ionic potential w(|ri − Rj |) is modelled by a
nonlocal model pseudo-potential [24]
w(|ri −Rj |) = − Ze2
|ri −Rj |
+
l∑
l′=0
fl′(|ri −Rj |)Pl′ , (13)
where
Pl =
l∑
m=−l
|Yl,m〉〈Yl,m| ≡
l∑
m=−l
|l,m〉〈l,m| (14)
is the projection operator, and ∑
l
Pl = 1, (15)
l and m are the electronic orbital and magnetic quantum numbers, respectively. The concrete
expression for function fl(|ri −Rj |) depends on the selected model of pseudo-potential.
643
P.P.Kostrobij, B.M.Markovych
For a further discussion it is convenient to rewrite the Hamiltonian (10) in the second quantiza-
tion form. For this purpose, we introduce wave functions ψf (r) of an electron being in the surface
potential V (r) [
− ~
2
2m
∆ + V (r)
]
ψf (r) = Efψf (r). (16)
Then the field operators are introduced by definition
Ψ(r) =
∑
f
ψf (r)af ,
Ψ†(r) =
∑
f
ψ∗
f (r)a†f ,
where af and a†f are the annihilation and creation operators of electrons in the state f with an
energy Ef respectively,
{af , a
†
f ′} = δf,f ′ .
Since it is very difficult to solve the equation (16), we assume that the surface potential V (r)
is a function of the electron coordinate normal to the surface
V (r) ≡ V (z).
Then, the electron wave function is the product of plane wave and function depending only on
coordinate z:
ψf (r) =
1√
S
eipr||ϕα(z), r = (r||, z), f = (p, α), (17)
where function ϕα(z) is a solution of the following equation
[
− ~
2
2m
d2
dz2
+ V (z)
]
ϕα(z) = εαϕα(z).
Then, Ef ≡ Eα(p) = ~
2p2
2m
+ εα is the energy of electron in a state (p, α), ~p is a two-dimensional
momentum of electron in the plane parallel to the surface.
Then, the Hamiltonian (4) can be written as
H = Hjell + δHii +
Nion
SL
∑
q
∑
k
Sk(q)
∑
l
δwl
k(q)ρ̃l
k(q), (18)
where
Sk(q) =
1
Nion
Nion∑
j=1
e−iqR||j−ikZj (19)
is a geometrical structure factor of the ionic subsystem of semi-infinite metal,
δwl
k(q) = −Zνk(q) (1 − δq,0) + f l
k(q),
νk(q) = 4πe2/(q2 + k2) and f l
k(q) are three-dimensional Fourier-image of Coulomb potential and
nonlocal part of the pseudo-potential (13):
e2
|ri − rj |
=
1
SL
∑
q,k
νk(q) eiq(r||i−r||j)+ik(zi−zj),
fl(|ri −Rj |) =
1
SL
∑
q,k
f l
k(q) eiq(r||i−R||j)+ik(zi−Zj), R||j = (Xj , Yj),
ρ̃l
k(q) =
∑
m
∑
p1,α1
∑
p2,α2
〈p1, α1| eiqr||+ikz |l,m〉〈l,m|p2, α2〉a†α1
(p1)aα2
(p2), (20)
644
Semi-infinite metal: perturbative treatment based on semi-infinite jellium
〈p1, α1| . . . |l,m〉 =
1√
S
∫
dr||
∫
dz e−ip1r||ϕ∗
α1
(z) . . .Yl,m(θ, φ),
〈l,m|p2, α2〉 =
1√
S
∫
dr||
∫
dzY∗
l,m(θ, φ) eip2r||ϕα2
(z).
Let us note that in the case of a local pseudo-potential, the Hamiltonian (18) acquires the
following form:
H = Hjell + δHii +
Nion
SL
∑
q
∑
k
Sk(q)δwk(q)ρk(q), (21)
where
δwk(q) = −Zνk(q) (1 − δq,0) + fk(q),
ρk(q) ≡
∑
l
ρ̃l
k(q) =
∑
p,α1,α2
〈α1| eikz|α2〉a†α1
(p)aα2
(p − q), (22)
〈α1| . . . |α2〉 =
∫
dz ϕ∗
α1
(z) . . . ϕα2
(z).
Such representation of the Hamiltonian (see (18) or (21)) is convenient for calculation of the
partition function, which indicates thermodynamic characteristics of the system.
Let us note that since the electron has two possible orientations of a spin, a result of summation
on p in the formula (20) (or in (22)) should be doubled.
3. Partition function
We consider the partition function of the semi-infinite metal
Ξ = Sp e−β(H−µN ), (23)
where µ is a chemical potential of the electronic subsystem, N is an electron number operator.
Taking into account (18), in adiabatic approximation we get
Ξ = e−βδHii Sp e−β(Hjell−δVei), (24)
where
Hjell = Hjell − µN ,
δVei =
Nion
SL
∑
q
∑
k
Sk(q)
∑
l
δwl
k(q)ρ̃l
k(q). (25)
In the interaction representation, the partition function is presented as
Ξ = e−βδHiiΞjell〈S(β)〉jell, (26)
where
Ξjell = Sp e−βHjell (27)
is the partition function of semi-infinite jellium [20] (T is the time-ordering operator),
〈. . .〉jell =
1
Ξjell
Sp
(
e−βHjell . . .
)
, (28)
S(β) = T exp
−Nion
SL
β∫
0
dβ′
∑
q
∑
k
Sk(q)
∑
l
δwl
k(q)ρ̃l
k(q|β′)
, (29)
ρ̃l
k(q|β′) = eβ′Hjell ρ̃l
k(q) e−β′Hjell .
645
P.P.Kostrobij, B.M.Markovych
Transferring from the temperature representation to the frequency representation, according to
the rule:
ρ̃l
k(q|ν) =
1
β
β∫
0
dβ′ eiνβ′
ρ̃l
k(q|β′),
ρ̃l
k(q|β′) =
∑
ν
e−iνβ′
ρ̃l
k(q|ν),
where ν is the Bose frequency, we get
S(β) = T exp
[
−βNion
SL
∑
q
∑
k
Sk(q)
∑
l
δwl
k(q)ρ̃l
k(q|ν = 0)
]
(30)
and
〈S(β)〉jell = exp
∞∑
n=1
(
βNion
SL
)n
in
n!
∑
q1,...,qn
∑
k1,...,kn
Sk1
(q1) . . . Skn
(qn)
×
∑
l1,...,ln
δwl1
k1
(q1) . . . δw
ln
kn
(qn)Ml1,...,ln
k1,...,kn
(q1, . . . ,qn|ν = 0)
, (31)
where
M
l1,...,ln
k1,...,kn
(q1, . . . ,qn|ν = 0) = in
〈
Tρ̃l1
k1
(q1|ν = 0) . . . ρ̃ln
kn
(qn|ν = 0)
〉
jell,c
(32)
is the nth order irreducible correlation function of electrons.
Since the average value of the quantities ρ̃ln
kn
(qn|ν = 0) is equal to the average value of the
quantities ρ̃ln
kn
(qn) (see Appendix) it is possible to write
〈S(β)〉jell = exp
∞∑
n=1
(
βNion
SL
)n
in
n!
∑
q1,...,qn
∑
k1,...,kn
Sk1
(q1) . . . Skn
(qn)
×
∑
l1,...,ln
δwl1
k1
(q1) . . . δw
ln
kn
(qn)Ml1,...,ln
k1,...,kn
(q1, . . . ,qn)
, (33)
where
M
l1,...,ln
k1,...,kn
(q1, . . . ,qn) = in
〈
ρ̃l1
k1
(q1) . . . ρ̃
ln
kn
(qn)
〉
jell,c
. (34)
Calculation of M
l1,...,ln
k1,...,kn
(q1, . . . ,qn) can be made according to the definition (see (28)) using a
perturbation theory, but for the sake of comparison of our theories with others it is more convenient
to present M
l1,...,ln
k1,...,kn
(q1, . . . ,qn) through many-particle density matrices. According to Bogoljubov
[25] between M
l1,...,ln
k1,...,kn
(q1, . . . ,qn) and s-particle density matrix
N(N − 1) . . . (N − s+ 1)
V s
Fs(r1, . . . , rs|r′1, . . . , r′s)
there exists the following relation:
Fs(r1, . . . , rs|r′1, . . . , r′s) =
V s
N(N − 1) . . . (N − s+ 1)
∑
f1,...,fs
f ′
1,...,f ′
s
ψ∗
f1
(r1) . . . ψ
∗
fs
(rs)
×ψf ′
1
(r′1) . . . ψf ′
s
(r′s)
〈
a†f1
. . . a†fs
af ′
s
. . . a
f ′
1
〉
jell
. (35)
646
Semi-infinite metal: perturbative treatment based on semi-infinite jellium
The use of orthogonality of the wave functions (17),
∫
drψ∗
f1
(r)ψf2
(r) = δf1,f2
,
permits to write the expression (35) in the form
〈
a†f1
. . . a†fs
af ′
s
. . . af ′
1
〉
jell
=
N(N − 1) . . . (N − s+ 1)
V s
∫
dr1. . .
∫
drs
∫
dr′1. . .
∫
dr′s
×ψ∗
f1
(r1) . . . ψ
∗
fs
(rs)Fs(r1, . . . , rs|r′1, . . . , r′s)ψf ′
1
(r′1) . . . ψf ′
s
(r′s). (36)
Let us consider the first order correlation functions
M
l1
k1
(q1) = i
〈
ρ̃l1
k1
(q1)
〉
jell
= i
∑
m
∑
p1,α1
∑
p2,α2
〈
p1, α1| eiq1r||+ik1z |l1,m
〉
〈l1,m|p2, α2〉
〈
a†α1
(p1)aα2
(p2)
〉
jell
= i
N
V
∑
m
∫
dr1
∫
dr′1 eiq1r||1+ik1z1Yl1,m(θ1, φ1)F1(r1|r′1)Y ∗
l1,m(θ′1, φ
′
1) (37)
and the second order irreducible correlation functions
M
l1,l2
k1,k2
(q1,q2) = i2
〈
ρ̃l1
k1
(q1)ρ̃
l2
k2
(q2)
〉
jell,c
= i2
〈
ρ̃l1
k1
(q1)ρ̃
l2
k2
(q2)
〉
jell
− i
〈
ρ̃l1
k1
(q1)
〉
jell
i
〈
ρ̃l2
k2
(q2)
〉
jell
= i2
∑
m1,m2
∑
p1,α1
∑
p2,α2
∑
p3,α3
∑
p4,α4
〈
p1, α1| eiq1r||+ik1z|l1,m1
〉
〈l1,m1|p2, α2〉
×
〈
p3, α3| eiq2r||+ik2z|l2,m2
〉
〈l2,m2|p4, α4〉
×
[〈
a†α1
(p1)a
†
α3
(p3)aα4
(p4)aα2
(p2)
〉
jell
−
〈
a†α1
(p1)aα4
(p4)
〉
jell
δp2,p3
δα2,α3
]
−M
l1
k1
(q1)M
l2
k2
(q2)
= i2
N(N − 1)
V 2
∑
m1,m2
∫
dr1
∫
dr′1
∫
dr2
∫
dr′2 eiq1r||1+ik1z1+iq2r||2+ik2z2
×Yl1,m1
(θ1, φ1)Yl2,m2
(θ2, φ2)F2(r1, r2|r′1, r′2)Y ∗
l2,m2
(θ′1, φ
′
1)Y
∗
l1,m1
(θ′2, φ
′
2)
−i2
N
V
∑
m1,m2
∫
dr1
∫
dr′1 eiq1r||1+ik1z1Yl1,m1
(θ1, φ1)
×
〈
l1,m1| eiq2r||2+ik2z2 |l2,m2
〉
F1(r1|r′1)Y ∗
l2,m2
(θ′1, φ
′
1)
−M
l1
k1
(q1)M
l2
k2
(q2). (38)
In a similar way, it is possible to present the nth order irreducible correlation functions of
electrons through the one-, two-, . . . , n-particle density matrices.
Thus, the calculation of the partial function is reduced to the calculation of the many-particle
density matrices. Further it will be shown that in case of a local pseudo-potential it is necessary
to know only diagonal elements of the many-particle density matrices. They are many-particle
distribution functions of electrons which were already examined by us in [22]. The calculation of
the many-particle density matrices will be made in future.
647
P.P.Kostrobij, B.M.Markovych
4. Partial case: local pseudo-potential
In partial case of the local pseudo-potential w(|ri − Rj |) the function fl(|ri − Rj |) does not
depend on an orbital quantum number l:
fl(|ri −Rj |) ≡ f(|ri −Rj |)
and
w(|ri −Rj |) = − Ze2
|ri −Rj |
+ f(|ri −Rj |). (39)
Then we get
〈S(β)〉jell = exp
∞∑
n=1
(
βNion
SL
)n
in
n!
∑
q1,...,qn
∑
k1,...,kn
Sk1
(q1) . . . Skn
(qn)
× δwk1
(q1) . . . δwkn
(qn)Mk1,...,kn
(q1, . . . ,qn)
, (40)
The result (40) can be written down as
〈S(β)〉jell = exp
[
−β(δE(1) + δE(2) + . . .)
]
,
where δE(i) is the energy of the electronic subsystem in the field of the ions minus that in the
field of semi-infinite uniform charge background in the ith order approximation with respect to the
“difference potential” δw.
Let us consider this energy in the first order approximation with respect to δw.
δE(1) = −i
Nion
SL
∑
q1
∑
k1
Sk1
(q1)δwk1
(q1)Mk1
(q1). (41)
In this case
Mk1
(q1) =
∑
l1
M
l1
k1
(q1) = i
N
V
∫
dr1 eiq1r||1+ik1z1F1(r1|r1). (42)
F1(r1|r1) is the one-particle distribution function F1(r1) of electrons in semi-infinite jellium [22].
Since semi-infinite jellium is uniform in a plane parallel to the surface, then F1(r1) ≡ F1(z1) and
we can rewrite (42) as
Mk1
(q1) = i
N
V
∫
dr1 eiq1r||1+ik1z1F1(r1) = i
N
V
Sδq1,0
∫
dz1 eik1z1F1(z1). (43)
Substituting (43) in (41), we get
δE(1) =
N
V
Nion∑
j=1
∫
dz1 δw(q = 0|z1 − Zj)F1(z1). (44)
As
δw(q = 0|z1 − Zj) =
∫
dr|| δw
(√
r2|| + (z1 − Zj)2
)
= S〈δw〉plane(z1 − Zj),
is held where
〈δw〉plane(Zj − z) =
1
S
∫
dr|| δw
(√
r2|| + (Zj − z)2
)
(45)
648
Semi-infinite metal: perturbative treatment based on semi-infinite jellium
is δw averaged along the plane parallel to the surface, we can get
δE(1) = S
Nion∑
j=1
∫
dz1 〈δw〉plane(z1 − Zj)n(z1), (46)
where
n(z1) =
N
V
F1(z1)
is the electron density function of interacting electron gas. The expression for the energy in the
first order (46) coincides with that used in works [2–7].
Let us consider this energy in the second order approximation with respect to δw
δE(2) = β
N2
ion
(SL)2
∑
q1,q2
∑
k1,k2
Sk1
(q1)δwk1
(q1)Sk2
(q2)δwk2
(q2)Mk1,k2
(q1,q2). (47)
In this case
Mk1,k2
(q1,q2) =
∑
l1,l2
M
l1,l2
k1,k2
(q1,q2)
= i2
N(N − 1)
V 2
∫
dr1
∫
dr2 eiq1r||1+ik1z1+iq2r||2+ik2z2F2(r1, r2)
−i2
N
V
Sδq1+q2,0
∫
dz1 ei(k1+k2)z1F1(z1)
−i2
N2
V 2
S2δq1,0δq2,0
∫
dz1 eik1z1F1(z1)
∫
dz2 eik2z2F1(z2), (48)
where F2(r1, r2) ≡ F1(r1, r2|r1, r2) is the two-particle distribution function of electrons in semi-
infinite jellium [22]. Substituting (48) in (47), we get
δE(2) = −βN(N − 1)
2V 2
Nion∑
j=1
Nion∑
i=1
∫
dr1
∫
dr2δw(|r1 −Rj |)δw(|r2 − Ri|)F2(r1, r2)
+β
N
2V
Nion∑
j=1
Nion∑
i=1
∫
dr δw(|r −Rj |)δw(|r −Ri|)F1(z)
+β
N2
2V 2
S2
Nion∑
j=1
∫
dz1〈δw〉plane(z1 − Zj)F1(z1)
Nion∑
i=1
∫
dz2〈δw〉plane(z2 − Zi)F1(z2).(49)
As a rule, all calculations of energy are carried out in the first order in the pseudopotential [7].
In contrast, our approach has no basic difficulties in taking into consideration the second order
and higher.
5. Conclusion
The theory of semi-infinite metal is presented which takes into consideration the discreteness of
an ionic subsystem of metal. The semi-infinite model is used as the reference system for building the
perturbation theory powers of the “difference potential” (see footnote on page 642). Consideration
of non-local pseudo-potential is the main novelty in this theory. In the specific case of a local pseudo-
potential, this theory coincides with Kaim’s theory. It makes possible to consider structurally-
depending contributions to the energy of an electronic subsystem owing to indirect interaction
of ions through non-uniform electronic subsystem, interactions of ion groups with non-uniform
electronic subsystem. This is very important in order to understand the formation of the metal
649
P.P.Kostrobij, B.M.Markovych
static structure near the surface, to make research into interionic interactions and dynamics of ions
close to the surface, as well as to understand the effect of surface structure on surface energy, etc.
Generally, the structurally-depending contributions to energy of an electronic subsystem are
expressed through s-particle electron density matrices. However, in case of a local pseudo-potential,
it is necessary to know only their diagonal elements, that is electron distribution functions. Hence
non-locality of a pseudo-potential leads to the necessity of considering non-diagonal elements of
density matrices.
A. The proof of an equality〈
ρ̃
l1
k1
(q1) . . . ρ̃
ln
kn
(qn)
〉
jell
=
〈
Tρ̃
l1
k1
(q1|ν = 0) . . . ρ̃
ln
kn
(qn|ν = 0)
〉
jell
According to the definition (28)
〈
ρ̃l1
k1
(q1) . . . ρ̃
ln
kn
(qn)
〉
jell
=
1
Ξjell
Sp
(
e−βHjell ρ̃l1
k1
(q1) . . . ρ̃
ln
kn
(qn)
)
=
1
Ξjell
(−1)n
βn
∂
∂γl1
k1
(q1)
. . .
∂
∂γln
kn
(qn)
× Sp exp
(
− βHjell − β
∑
q,k,l
γl
k(q)ρ̃l
k(q)
)∣∣∣∣∣∣
γ
l1
k1
(q1)=0
...
γ
ln
kn
(qn)=0
.
Transferring to the interaction representation we get
〈
ρ̃l1
k1
(q1) . . . ρ̃
ln
kn
(qn)
〉
jell
=
1
Ξjell
(−1)n
βn
∂
∂γl1
k1
(q1)
. . .
∂
∂γln
kn
(qn)
× Sp
(
e−βHjellT exp
[
−
β∫
0
dβ′
∑
q,k,l
γl
k(q)ρ̃l
k(q|β′)
])∣∣∣∣∣∣
γ
l1
k1
(q1)=0
...
γ
ln
kn
(qn)=0
=
1
Ξjell
(−1)n
βn
∂
∂γl1
k1
(q1)
. . .
∂
∂γln
kn
(qn)
× Sp
(
e−βHjellT exp
[
− β
∑
q,k,l
γl
k(q)ρ̃l
k(q|ν = 0)
])∣∣∣∣∣∣
γ
l1
k1
(q1)=0
...
γ
ln
kn
(qn)=0
=
〈
Tρ̃l1
k1
(q1|ν = 0) . . . ρ̃ln
kn
(qn|ν = 0)
〉
jell
.
Thus, the statement
〈
ρ̃l1
k1
(q1) . . . ρ̃
ln
kn
(qn)
〉
jell
=
〈
Tρ̃l1
k1
(q1|ν = 0) . . . ρ̃ln
kn
(qn|ν = 0)
〉
jell
is proved.
650
Semi-infinite metal: perturbative treatment based on semi-infinite jellium
References
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Напiвобмежений метал: пiдхiд на основi моделi
напiвобмеженого “желе”
П.П.Костробiй, Б.М.Маркович
Нацiональний унiверситет “Львiвська полiтехнiка”, вул. С. Бандери, 12, Львiв 79013
Отримано 27 лютого 2008 р.
Представлено енергiю електронної пiдсистеми напiвобмеженого металу у виглядi ряду за степеня-
ми псевдопотенцiалу. Показано, що в загальному випадку для розрахунку цiєї енергiї необхiднi еле-
ктроннi багаточастинковi матрицi густини, тодi як у частковому випадку локального псевдопотенцi-
алу лише її дiагональнi елементи (функцiї розподiлу електронiв). У першому порядку теорiї збурень
у випадку локального псевдопотенцiалу наш результат для енергiї спiвпадає iз широко застосову-
ваними у теорiї функцiоналу густини.
Ключовi слова: велика статистична сума, псевдопотенцiал, багаточастинкова матриця густини
PACS: 71.45.Gm, 71.10.-w, 73.20.-r
651
652
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