Potential regularization as the method accounting for short-range correlations in the electron liquid theory
In our previous paper [1] we proposed a Coulomb potential regularization as one of the methods for short-range correlation accounting in the electron liquid model. The present paper formulates the criterions of optimal choice of regularization as well as calculates the energetic, structural and diel...
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irk-123456789-1189632017-06-02T03:03:06Z Potential regularization as the method accounting for short-range correlations in the electron liquid theory Vavrukh, M.V. Tyshko, N.L. In our previous paper [1] we proposed a Coulomb potential regularization as one of the methods for short-range correlation accounting in the electron liquid model. The present paper formulates the criterions of optimal choice of regularization as well as calculates the energetic, structural and dielectric characteristics of the model. В попередній роботі авторів [1] запропоновано ідею регуляризації потенціалу Кулона як один із способів врахування короткосяжних кореляцій в моделі електронноїрідини. В даній роботі сформульовано критерії оптимального вибору регуляризаціїі розраховано енергетичні, структурні та діелектричні характеристики моделі. 2004 Article Potential regularization as the method accounting for short-range correlations in the electron liquid theory / M.V. Vavrukh, N.L. Tyshko // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 383–400. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 05.30.Fk DOI:10.5488/CMP.7.2.383 http://dspace.nbuv.gov.ua/handle/123456789/118963 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In our previous paper [1] we proposed a Coulomb potential regularization as one of the methods for short-range correlation accounting in the electron liquid model. The present paper formulates the criterions of optimal choice of regularization as well as calculates the energetic, structural and dielectric characteristics of the model. |
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Vavrukh, M.V. Tyshko, N.L. Potential regularization as the method accounting for short-range correlations in the electron liquid theory Condensed Matter Physics |
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Potential regularization as the method accounting for short-range correlations in the electron liquid theory |
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Potential regularization as the method accounting for short-range correlations in the electron liquid theory |
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Potential regularization as the method accounting for short-range correlations in the electron liquid theory |
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Potential regularization as the method accounting for short-range correlations in the electron liquid theory |
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Potential regularization as the method accounting for short-range correlations in the electron liquid theory |
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potential regularization as the method accounting for short-range correlations in the electron liquid theory |
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Potential regularization as the method accounting for short-range correlations in the electron liquid theory / M.V. Vavrukh, N.L. Tyshko // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 383–400. — Бібліогр.: 13 назв. — англ. |
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Condensed Matter Physics |
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AT vavrukhmv potentialregularizationasthemethodaccountingforshortrangecorrelationsintheelectronliquidtheory AT tyshkonl potentialregularizationasthemethodaccountingforshortrangecorrelationsintheelectronliquidtheory |
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Condensed Matter Physics, 2004, Vol. 7, No. 2(38), pp. 383–400
Potential regularization as the method
accounting for short-range correlations
in the electron liquid theory
M.V.Vavrukh, N.L.Tyshko
Ivan Franko National University of Lviv, Department of Astrophysics,
8 Kyrylo i Mefodii Str., Lviv, UA-79005, Ukraine
Received April 1, 2004
In our previous paper [1] we proposed a Coulomb potential regularization
as one of the methods for short-range correlation accounting in the elec-
tron liquid model. The present paper formulates the criterions of optimal
choice of regularization as well as calculates the energetic, structural and
dielectric characteristics of the model.
Key words: short-range correlation, local-field correction function,
correlation energy, binary distribution function, compressibility
PACS: 05.30.Fk
1. Introduction
As known, the random-phase approximation (RPA) is a universal method for
long-range correlation accounting in the fermi-particles systems (for example, in
degenerated electron systems with Coulomb interaction). The generally accepted
method for short-range correlation accounting in the modern theory of fermi-systems
is based on the local-field concept. Since, the calculation of the local-field correction
function (LFCF) for systems with intermediate and strong non-ideality is compli-
cated, only approximate solutions are available [2]. LFCF was investigated in the
common patterns but its microscopic theory is unfinished so far. In this situation
looking for alternative methods accounting for the short-range correlation between
particles remains urgent. It is very important to develop a simple and effective
method of calculating the characteristics of the metallic system models.
A simple and promising approach to the evaluation of this problem can be based
on the idea of correlation modelling for the degenerated fermi-systems particles with
Coulomb interaction. This idea has emerged due to the model approach to the de-
scription of electron-ion interactions in metals. The modelling of interaction poten-
tial between particles is equivalent to LFCF modelling. Similarly to the electron-ion
interaction theory, the effective interaction potential between electrons in an elec-
c© M.V.Vavrukh, N.L.Tyshko 383
M.V.Vavrukh, N.L.Tyshko
tron liquid model (ELM) should be chosen weak at short distances and tends to the
Coulomb potential e2/r at long distances. This approach permits to restrict our-
selves to the variants of perturbation theory that were developed for a description
of weak non-ideal systems in calculating the ELM characteristics in a wide range
of the coupling parameter. The idea of the model effective potential of interaction
between electrons can be derived from the Heisenberg indeterminacy principle. Inde-
terminacy of the electron position is close to the de Broglie wavelength λ. Therefore,
electrons can be described as spatially distributed charged particles of a linear size
about λ. The effective potential of interaction of two “smeared” electrons that de-
fines their cross-correlation, should have the properties of quantum wave packets
interaction potential
Vef(r) =
e2
r
f
( r
λ
)
, (1)
where f(r/λ) is the dimensionless linear function at r � λ and tends to unity in
case of r � λ (Vef(r) � e2/λ for r � λ and Vef(r) � e2/r for r � λ). Asymp-
totic properties of a weak long-range potential Vef(r) follow from general physical
principles. However, the selection of the function f(r/λ) is ambiguous.
If ρ(r) is charge distribution function of the “smeared” electron that satisfies the
normalization condition
e =
∫
drρ(r), (2)
then the interaction potential of two electrons can be written by the following ex-
pression:
Vef(r1 − r2) =
∫∫
dr′1dr
′
2
ρ(r′1 − r1)ρ(r′2 − r2)
|r′1 − r′2|
. (3)
Let us find explicit expressions for ρ(r) selecting various functions f(r/λ). Table 1
gives some variants of ρ(r) selection in the form of function |r|, Fourier-transform
of these functions
ρ(q) =
∫
drρ(r)eiqr, (4)
as well as a dimensionless function f(r/λ).
In table 1 we used the following notations: ρ̃(r) ≡ e−1ρ(r), ρ̃(q) ≡ e−1ρ(q), r∗ ≡
r/λ, q∗ ≡ qλ; K1(z) is the modified first order Bessel function [3] with an asymptotic
z−1 at z � 1 and π1/2(2z)−1/2 exp(−zλ) at z � 1. The parameters γ, α, β, δ are of
the order of unity. The criterions for the selection of these parameters and variants
can be formulated as the result of calculation of model physical characteristics.
Hereinafter we will use Fourier-transform for the potential:
Vef(r) =
1
V
∑
q
Vef(q) exp(iqr)
Vef(q) = Vqρ̃
2(q), Vq =
4πe2
q2
. (5)
384
Potential regularization
Table 1. Some variants of selecting the ρ(r), ρ(q) and f(r∗)
Variant ρ̃(r); ρ̃(q); f(r∗)
1
ρ̃1(r) = γ2λ−3[2π2r∗]−1K1(γr∗)
ρ̃1(q) = γ{γ2 + (q∗)2}−1/2
f1(r
∗) = 1 − exp{−γr∗}
2
ρ̃2(r) = α3/2λ−3π−3/2 exp{−α(r∗)2}
ρ̃2(q) = exp{−(q∗)2(4α)−1}
f2(r
∗) =
√
2/π
√
αr∗∫
0
dx exp(−x2/2)
3
ρ̃3(r) = β2λ−3{4πr∗}−1 exp{−βr∗}
ρ̃3(q) = β2{β2 + (q∗)2}−1
f3(r
∗) = 1 − {1 + (r∗β)/2} exp(−βr∗)
4
ρ̃4(r) = δ3λ−3{8π}−1 exp{−δr∗}
ρ̃4(q) = δ4{δ2 + (q∗)2}−2
f4(r
∗) = 1 − {1 + 11
16
x + 3
16
x2 + 1
48
x3} exp(−x)
(x = δr∗)
The function ρ̃(r) or ρ̃(q) can be modelled. In the latter case we can use the
following expression
f(r∗) = 1 − 1
π
∞∫
r∗
dr∗
∞∫
−∞
dq∗ρ̃2(q∗) exp iq∗r∗). (6)
2. Description of the ground state of the electron liquid model
in a random-phase approximation with effective potential
In the case of the ground state of the model, the de Broglie wavelength of electron
is defined by Fermi wave number. Therefore, we will use the potentials from table 1
at λ = k−1
F . To calculate the two-particle correlation function we restrict ourselves
at the beginning by RPA, because Vef(r) is a long-range but weak potential:
µRPA
2 (r) = N−2β−1
∑
q;ν
µRPA
2 (x,−x)eiqr, µRPA
2 (x,−x) = µ0
2(x,−x){1 + Lef(x)}−1,
(7)
Lef(x) = Vef(q)V
−1µ0
2(x,−x). Here µ0
2(x,−x) is spectral representation of two-
particle correlation function of the electron ideal system at T = 0 K, x = (q, ν), q is
wave vector, ν is the Bose-Matsubara frequency and r is the distance between two
particles [8]. Having compared the formula (7) with the expression for µ2(x,−x) in
the local-field representation [4]
µ2(x,−x) = µ0
2(x,−x){1 + V −1Vqµ
0
2(x,−x)[1 − G(x)]}−1, (8)
385
M.V.Vavrukh, N.L.Tyshko
we can see that Vef(q) is the modelling of a static variant of LFCF:
G0(q) = 1 − ρ̃2(q). (9)
The average value of the system energy will be calculated using the expression
E = E0 +
1
2βV
∑
q,ν
Vq
1∫
0
dλµλ
2(x,−x), (10)
where E0 is the ideal system energy and µλ
2(x,−x) is spectral representation of
two-particle correlation function of the model system with Fourier-representation
λVef(q) of interaction potential. Thus, Vef(q) plays the role of self-consistent effective
potential accounting for short-range correlation (local-field effects).
By extracting the ideal correlation contribution to the approximation µλ
2(x,−x) →
µ0
2(x,−x) we can represent the total energy in the usual dimensionless form
E = NRy{ε0(rs) + εHF(rs) + εc(rs)}, (11)
where ε0(rs) is the ideal system energy at T = 0 K in Ry per electron, εHF(rs) is the
Hartree-Fock energy contribution and εc(rs) is the so-called correlation energy,
ε0(rs) =
3
5
(
η
rs
)2
, εHF(rs) = − 3
2π
η
rs
, (12)
η = (9π/4)1/3, rs is non-ideality parameter (Wigner-Brueckner parameter).
According to expressions (7), (9)
εc(rs) =
3
2π
(
η
rs
)2
∞∫
0
du
∞∫
0
dqq3[ρ̃(q)]−2{ln[1 + Lef(q, u)] − Lef(q, u)}, (13)
where q ≡ |q|k−1
F , u ≡ ν(2εfq)
−1. One of the criterions of selecting distributions
ρ(r) or ρ(q) parameters can be based on the comparison of calculation results of
the energy correlation term equation (13) with the previous ones from Monte-Carlo
(MC) method [5]. In paper [6] there was proposed an approximative expression for
numeric values of energy, obtained in [5] according to which
εc(rs) = −2b0
∞∫
a
dx(b1 + x−1)[1 + b1x + b2x
2 + b3x
3]−1 (14)
at a = r
1/2
s , b0 = 0.0621814, b1 = 9.81379, b2 = 2.82214, b3 = 0.69699. However, the
energy being an integral characteristic is not very sensitive to the approximations
used in the calculation. Therefore, the mentioned criterion cannot be a single one
and other criterions should be added to it that follow from other characteristics as
functions of non-ideal parameter.
386
Potential regularization
As known, the binary distribution function F2(r) = 1+µ2(r) in a traditional ap-
proximation RPA (on Coulomb potential) has non-physical negative values at short
distances ((rkF � 1)) for rs � 0.82 [7]. According to the definition F2(r) must be
positive at any rs values. Also, from general physical principles there follows a mono-
tonicity dependence F2(r) at short distances where there are no Fridel oscillations
[2]. Thus, we have a second criterion for the selection of parameters
F2(0) = 1 + µ2(0) � 0; |µ2(r1)| > |µ2(r2)| for r1 < r2; r
∗
1,2 � 1. (15)
The next criterion can be obtained from a long-wave asymptote of the polariza-
tion operator in the static limit. The spectral representation of two-particle corre-
lation function µ2(x,−x) is determined by the polarization operator M2(x,−x) as
follows (see, for example [8])
µ2(x,−x) = M2(x,−x)
{
1 +
Vq
V
M2(x,−x)
}−1
. (16)
Using formula (7) from the last expression M2(x,−x) in RPA with an effective
potential can be written as follows:
MRPA
2 (x,−x) = µ0
2(x,−x)
{
1 − Vq
V
µ0
2(x,−x)[1 − ρ̃2(q)]
}−1
. (17)
Taking into account that M2(0, 0) = κN2/V (where κ = −V −1∂V /∂p is com-
pressibility of the system) we derive a relationship
κ0
κ
= 1 − 4rs
πη
{
1 − ρ̃2(q)
q2
}∣∣∣∣
q=0
, (18)
where κ0 = 3V (2εFN)−1 is the compressibility of the ideal system. On the other
hand, κ can be found based on the thermodynamic relation, using ground state
energy: as it is known p = −∂E/∂V , therefore, κ−1 = V (∂2E/∂V 2) or
κ0
κ
=
∂2E
∂V 2
{
∂2E0
∂V 2
}−1
. (19)
Using expression (11) and by transition from variable V to variable
rs = a−1
0
[
3V
4πN
]1/3
,
we will get the relationship
κ0
κ
= 1 +
rs[ε
′′
HF + ε′′c ] − 2[ε′HF + ε′c]
rsε
′′
0 − 2ε′0
, (20)
where ε′ = (d/drs)ε(rs), ε
′′ = (d2/dr2
s)ε(rs). According to the formulae (14), (20),
we obtain
387
M.V.Vavrukh, N.L.Tyshko
κ0
κ
= 1 − x2
πη
− b0x
4
6η2
[
1 + b1x + b2x
2 + b3x
3
]−1 ×
×
{
3 +
5
2
b1x + (1 + b1x)
[
b1 + 2b2x + 3b3x
2
] [
1 + b1x + b2x
2 + b3x
3
]−1
}
,
(21)
where x ≡ r
1/2
s . If the calculations of relationships (18) and (20) are correct, then
the results must be convergent, or the relationships of (18) and (21) must be close.
Namely, the values of non-ideality parameter must be close, that correspond to
κ0/κ = 0. As it follows from the formula (21) the change of κ0/κ sign will be at
r0
s = 5.2633. In the range of rs > r0
s the electron liquid is non-stable.
0 1 2 3 4 5 6 7 8 9 10
rs
-0.25
-0.20
-0.15
-0.10
-0.05
-0.00 εc(rs), Ry
γ = 1.00
γ = 1.25
γ = 1.50
ord. RPA
MC
Figure 1. Correlation energy εc(rs) in different approaches. Model potential in
variant 1 at γ = 1.25 (solid curve), at γ = 1.0 (top dashed curve) and at γ = 1.5
(low dashed curve); Coulomb potential: MC method (black filled circles), LFCF[9]
(the line with triangles).
Let us represent the calculation results of model characteristics of the electron
liquid in RPA with a model potential. Figure 1 shows the dependence of correlation
energy εc(rs) on non-ideality parameter rs. It was calculated using the expression
(19) for the first selection variant of function ρ̃(q) (see table 1). Here black filled
circles correspond to the MC results [5], triangles represent the best LFCF [9], low
curve corresponds to the traditional RPA with Coulomb potential, top dashed curve
matches up the parameter γ = 1.0 and low dashed curve corresponds to γ = 1.5.
388
Potential regularization
Bold solid curve is very close in the metals range to MC and corresponds to the case
of γ = 1.25. In the range of intermediate values of rs this curve is more close to MC
than the calculation results with LFCF [9].
0 1 2 3 4 5
r kF
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2 F2(r)
rs = 1
rs = 5
Figure 2. Binary distribution function for the model potential (variant 1) at
γ = 1.25.
Binary distribution function in this approach FRPA
2 (r) = 1+µRPA
2 (r) is shown in
figure 2. These results correspond to the value of γ = 1.25. As we can see, FRPA
2 (r)
has small negative values at short distances, and positive values are only in the range
of 0 � rs � 3.3.
0 1 2 3 4 5 6
rs
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 κ0/κ
MC
2
1
γ = 1.25
Figure 3. Inverse compressibility κ0/κ in different approaches. Calculated via
formulae (18) (curve 1), (20) – (curve 2) and the result of MC method (dashed
curve).
Figure 3 shows the dependence of inverse compressibility κ0/κ on rs in different
approaches. The dashed curve corresponds to the MC method [5], other curves
389
M.V.Vavrukh, N.L.Tyshko
represent RPA with the model potential at γ = 1.25 calculated via formulae (18)
and (20).
In the represented variant the LFCF G0(q) (9) has excessive values in the range
of small wave vectors, if 1.0 � γ � 1.5 (as we know, G(q) ∼ gq2 + · · · at q � 1,
where 1/4 � g � 1/3). This causes invalid behavior of compressibility calculated by
expression (18): at γ = 1.5 the compressibility will be equivalent to zero at r0
s ≈ 3.3.
0 2 4 6 8
rs
-0.3
-0.2
-0.1
εc(rs ), Ry
α = 2.13776
ORPA
MC
Figure 4. Correlation energy for the model potential (variant 2) at α = 2.13776
(solid curve), ordinary RPA (dashed curve), results of MC method (black filled
circles).
Let us consider the same model characteristics using the other selection variant of
the model potential. Namely, variant 2, where ρ̃(q) = exp{−q2/4α}. The dependence
of the correlation energy on rs is shown in figure 4. Solid bold curve corresponds to
the parameter α = 2.13776. To make a comparison, the dependence of correlation
energy in traditional RPA and the calculation results of MC method [5] are shown.
In the range of small rs in this variant, εc(rs) is better described than in the variant
1. But in the range of large rs values we have the opposite situation: better results
are received from variant 1. The same situation is observed for a binary function:
the results improve for a small rs (for an example, at rs = 1 we have F2(0) = 0.276
which corresponds to the better results obtained by other authors [10]), and become
worse for large rs. But in both selection variants of ρ̃(q) all the results are certainly
better than in ordinary RPA (ORPA) with Coulomb potential.
The fact that ρ̃1(q) better describes the model at a large value of rs and that ρ̃2(q)
is better for small rs, may be used for building a model potential as a superposition
of the first and the second variants. To amend the compressibility we will use variant
1 at γ = 1.5. To describe a model in a wide range of rs we shall build an effective
potential in which
ρ̃(q) = A(rs)ρ̃2(q) + [1 − A(rs)]ρ̃1(q), (22)
390
Potential regularization
or
ρ̃2(q) = A(rs)ρ̃
2
2(q) + [1 − A(rs)]ρ̃
2
1(q), (23)
where coefficient A(rs) is a function of rs such as A(rs) = δ ∗ (1 − rs/10), ρ1(q) =
γ[γ2 + q2]−1/2, ρ2(q) = exp(− 1
4α
q2), γ = 5/4, and α = 2.13776.
The compressibility which is calculated by the polarization operator, is defined
identically in both variants:
κ0
κ
= 1 − 4rs
πη
{
A(rs)
2α
+
1 − A(rs)
γ2
}
. (24)
If δ = 0.8, then compressibility vanishes near r0
s = 4.32.
0 2 4 6 8
rs
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
ε2(rs), Ry
ORPA
Figure 5. Correlation energy for the model potential corresponding to formula
(23) (top solid curve), model potential in variant 1 at γ = 1.25 (top dashed curve),
model potential in variant 2 at α = 2.13776 (low dashed curve), Coulomb poten-
tial via MC method (black filled circles), ordinary RPA for Coulomb potential
(low solid curve).
For all rs, the correlation energy, calculated in RPA according to the model
potential (23), better conforms to the MC method results [5] than variant 2, as
shown in figure 5. The dash curves represent the correlation energy top for variant 1
γ = 1.25 and the lower curves for variant 2 α = 2.13776. The binary function for this
approximation has a better behavior than in variant 2 but it has a small negative
value at r = 0 for intermediate and large rs. To amend the binary distribution
function for a wide region of rs we must use the next approximation over RPA
because this approach turns out to be deficient. The compressibility, calculated by
equation (24), is closer to the compressibility obtained via MC method [5].
Let us consider another important but poorly investigated one-particle charac-
teristic of the ELM, i.e., momentum distribution function [11,12]. We can use only
the case of absolute zero temperature to investigate the effect of the short-range
correlation rather than the temperature smearing of the Fermi surface.
391
M.V.Vavrukh, N.L.Tyshko
According to the definition
nk,s = 〈a+
k,sak,s〉H = Z−1(µ)Sp
{
a+
k,sak,se
−β(Ĥ−µN̂)
}
, (25)
where µ is chemical potential correspondence to the given particle density and tem-
perature, Z(µ) is the partition function. As it follows from the formula (25), nk,s
can be represented in the form of variation derivative of free energy F with respect
to the variable εk = �
2k2/2m,
nk,s =
1
2
δ
δεk
Ω(µ)
∣∣∣∣
µ
=
1
2
{
δΩ(µ)
δεk
− δµ
δεk
dΩ(µ)
dµ
}
=
1
2
δF
δεk
, (26)
because δΩ/δµ = −µN . In the T = 0 case, F coincides with the average energy
which is determined by formula (10). Thus, for the model potential in RPA case we
obtain the representation
nk,s = n0
k,s −
1
4βV
∑
q,ν
Vq
{
1 +
Vef
V
µ0
2(x,−x)
}−1
δµ0
2(x,−x)
δεk
, (27)
in which n0
k,s is electron momentum distribution function of an ideal system,
δ
δεk
µ0
2(x,−x) = 2
∑
σ=±1
n0
k,s − n0
k−qσ,s
{iνσ + εk − εk−qσ}2
. (28)
At the first glance the deviation |nk,s − n0
k,s|, calculated by formula (27) will
exceed the similar deviation for the Coulomb potential, due to a slighter screening
in (27). But the integral for ∂µ0
2(x,−x)/∂εk over the frequency is equal to zero.
Therefore, formula (27) can be presented in an equivalent form
nk,s = n0
k,s −
1
4βV 2
∑
q,ν
V 2
q ρ̃2(q)µ̂0
2(x,−x)
δµ̂0
2(x,−x)
δεk
×
×
{
1 +
Vq
V
ρ̃2(q)µ̂0
2(x,−x)
}−1
. (29)
From formula (29) it follows that deviation |nk,s − n0
k,s| will be smaller than for
Coulomb potential. Going over to undimentional variables (k = |k|/kF, q, u) and
integrating over angular variables of vector q we obtain a representation in the form
of two-dimensional integrals. At k < 1
nk = 1 − 4πα2
k
∞∫
0
1+k∫
1−k
dq
q
I
(0)
2 (q, u)ρ̃2(q)
q2 + 4παI
(0)
2 (q, u)ρ̃2(q)
×
− k + q/2
(k + q/2)2 + u2
+
1−k2
2q(
1−k2
2q
)2
+ u2
+
392
Potential regularization
+
∞∫
1+k
dq
q
I
(0)
2 (q, u)ρ̃2(q)
q2 + 4παI
(0)
2 (q, u)ρ̃2(q)
×
[
− k + q/2
(k + q/2)2 + u2
+
q/2 − k
(q/2 − k)2 + u2
]}
, (30)
at k > 1
nk =
4πα2
k
k+1∫
k−1
dq
q
∞∫
0
duI
(0)
2 (q, u)ρ̃2(q)
q2 + 4παI
(0)
2 (q, u)ρ̃2(q)
− k − q/2
(k − q/2)2 + u2
+
k2−1
2q(
k2−1
2q
)2
+ u2
,
where α = rsπ
−2η−1.
The results of numerical calculations are given in figure 6, where dashed curves
correspond to the Coulomb potential [11]. Therefore, the correct model considera-
tion of short-range correlation decreases the deviation |nk,s − n0
k,s|. Thus the model
approach, represented in this paper can be used as a simple and reliable method for
the calculation of electron momentum distribution in a metallic region. The relative
deviation of nk at k → 1 − 0 for rs = 9 takes the value of 20%.
0.0 0.5 1.0 1.5 2.0
k kF
-1
0.0
0.2
0.4
0.6
0.8
1.0
nk
rs = 1
rs = 1
rs = 9
rs = 9
3
5
7
3
5
Figure 6. Momentum distribution function for the model potential (solid curves)
and Coulomb potential (dashed curves) at rs = 1, 3, 5, 7, 9.
3. Free energy of the electron liquid model
Let us calculate the free energy of the ELM with temperatures minor in the
comparison to the temperature of degeneration. To this end we will use the model
potential with temperature independent parameters. As it is well known [7], the
contribution of interaction to F is defined by the charging energy
393
M.V.Vavrukh, N.L.Tyshko
F − F0 = (2βV )−1
∑
q,ν
Vq
1∫
0
dλµλ
2(x,−x). (31)
According to our model approach, function µλ
2(x,−x) should be calculated according
to the potential λVef(r) with finite temperature.
In this paper we show the results of calculations in the form of a series by dimen-
sionless parameter (πT ∗)2, where T ∗ ≡ kBT/εF is dimensionless temperature. Using
a power series of function µλ
2(x,−x) by this parameter obtained using Zommerfeld
method in the linear approximation we have
F = N{ε(rs) − (πT ∗)2ε2(rs)}Ry, (32)
where ε(rs) is the average energy in Rydberg per electron at T = 0 K and the
correction ε2(rs) in the RPA is equal to:
ε2(rs) =
η2
r2
s
+
η
2π2rs
∞∫
0
du
∞∫
0
dqqI
(2)
2 (q, u)
[
1 +
4rs
πη
I
(0)
2 (q, u)
q2
ρ̃2(q)
]−1
,
I
(0)
2 (q, u) =
3N
2εF
1
2
{
1 +
1
2q
(
1 + u2 − q2
4
)
ln
u2 + (1 + q/2)2
u2 + (1 − q/2)2
− u
∑
σ=±1
arctg
[
u−1
(
1 +
σ
2
q
)]}
,
I
(2)
2 (q, u) =
1
2
{
1
2q
ln
u2 + (1 + q/2)2
u2 + (1 − q/2)2
− 1
q
∑
σ=±1
σ(1 + σ
q
2
)[u2 + (1 + σq/2)2]−1
}
.
(33)
The first term in formula (33) caused by the temperature dependence of free
energy of ideal system. In the approximation (32) we obtain an expression for heat
capacity of the model:
Cv = −T
d2F
dT 2
= 2Nπ2T ∗
(
rs
η
)2
kBε2(rs) + · · · . (34)
Utilizing the well known relation between heat capacity and density of states on the
Fermi surface NF [7]
Cv
V
=
π2
3
kBTNF. (35)
From formula (34) and (35) we obtain
NF = 4N id
F
(
rs
η
)2
ε2(rs), (36)
394
Potential regularization
where N id
F = k3
F(2π2εF)−1 is the density of states of an ideal system. Figure 7 shows
the dependence of relative density of states NF/N id
F on the coupling parameter in
RPA with the model potential (23) (curve 1) and with Coulomb potential (curve 2).
0 2 4 6 8
rs
4.0
4.2
4.4
4.6
4.8
5.0
NF / NF
id
ORPA
γ = 1.25
Figure 7. Dependence of relative density of states NF /N id
F on coupling parameter
in RPA with model potential (23) (solid curve) and with Coulomb potential
(dashed curve).
4. Local-field approximation with an effective potential
We want to achieve the correctness of all characteristics of ELM in the region
of weak and intermediate non-ideality. Thus, we will overstep the RPA with model
potentials. As in the case of formula (8) of the traditional perturbation theory, by
using the summing of diagrams in the reference system approach [8] for the function
µ2(x,−x) we obtain a representation
µ2(x,−x) = µ0
2(x,−x)
{
1 +
Vef(q)
V
µ0
2(x,−x)[1 − GM(x)]
}−1
. (37)
Here GM(x) is a LFCF for the system of particles with a model potential Vef(q).
Equating expressions (8) and (37) we obtain a relation
G(x) = 1 − ρ̃2(q){1 − GM(x)}, (38)
which may be considered as one of the simple ways of calculating the LFCF. We use
the Geldart-Taylor approximation (these authors for the first time have calculated
the static correction for polarization operator for Coulomb potential in the linear
approximation [13]) because Vef(q) is a weak potential. In this approximation we
obtain
G
(1)
M (x) = −{2βVef(q)}−1{µ0
2(x,−x)}−2
∑
x1
Vef(q1)µ
0
4(x,−x, x1,−x1), (39)
395
M.V.Vavrukh, N.L.Tyshko
where µ0
4(x,−x, x1,−x1) is a four-particle semiinvariant correlation function of the
reference system [8]. Performing the summing over frequencies we get the following
representation:
G
(1)
M (x) = {Vef(q)}−1{µ0
2(x,−x)}−2Re
∑
k1,k2;s
Vef(k1 − k2)
×{n0
k1,s − n0
k1−q,s}{n0
k2,s − n0
k2−q,s}[iν + εk1 − εk−q1
]−1
×{[iν + εk1 − εk1−q]
−1 − [iν + εk2 − εk2−q]
−1}. (40)
Let us investigate the asymptote of (40) for small and large values of wave
vectors. In a long-wave limit we substitute ρ̃(q) = 1 and use the series n0
k−q,s =
n0
k,s + dn0
k,s/dεk(εk−q − εk) + · · · Going over from the sums over vectors k1,k2 to
integrals in a spherical system of coordinates we reduce the calculation to a one-
dimensional integral
G
(1)
M (x) = g1(ν)
(
q
kF
)2
+ · · · , q � kF. (41)
In a static limit we have
g1(0) =
1
8
1∫
−1
dtρ̃2([2(1 − t)]1/2) =
1
8
2∫
0
dq qρ̃2(q), (42)
and in the large frequencies limit we have
g1(∞) =
3
40
2∫
0
dq q(q2 − 1)ρ̃2(q). (43)
Here t = cos θ cosine of the angle between vectors k1 and k2. At ρ̃(q) = 1 we
obtain the corresponding asymptote for LFCF in Geldart-Taylor approximation for
the system with Coulomb potential (g(0) = 1/4, g(∞) = 3/20). In the large wave
vector limit we obtain the following asymptote:
G
(1)
M (x) =
1
2
− N−2q−4V −1
ef (q)
∑
k1,k2;s
Vef(k1 − k2)
× n0
k1,sn
0
k2,s{(k1q)2 + (k2q)2 − 2(k1q)(k2q)} + · · ·
=
1
2
− 3
4
ρ̃−2(q)
1∫
0
dk1 k2
1
1∫
0
dk2 k2
2
1∫
−1
dt ρ̃2([k2
1 + k2
2 − 2k1k2t]
1/2) + · · · .(44)
According to the last formula for Coulomb potential ρ̃(q) = 1 the LFCF asymptote
is equal to 1/3 [4]. For the potentials of quantum packet type, G
(1)
M (x) acquires
396
Potential regularization
negative values for the large wave vectors and essentially differs from LFCF in the
weakly coupled electron liquid case. However, according to formula (39) in the LFCF
for ELM G(x) in such an approximation remains positive and tends an asymptote
G(1)(x) ⇒ 1 − 1
2
ρ̃2(q) − 3
4
1∫
0
dk1 k2
1
1∫
0
dk2 k2
2
1∫
−1
dt ρ̃2([k2
1 + k2
2 − 2k1k2t]
1/2) (45)
at q � kF (at ρ̃(q) = 1 this limit is equal to 1/3). The numerical calculation of G
(1)
M (x)
using (40) can be done for an arbitrary effective potential. In the T = 0 K case and at
any given ρ̃(q), the calculation reduces to a five-dimensional integral. Let us proceed
from sums to integrals and use the dimensionless variables. We use the cylindric
coordinate system (axis OZ is parallel to vector q). Since ki = (kFρi, kFzi, ϕi), then
k−2
F (ki,q) = ziq, k−2
F (k1 − k2)
2 = ρ2
1 + ρ2
2 + (z1 − z2)
2 − 2ρ1ρ2 cos(ϕ1 − ϕ2). With
substitution ϕ = ϕ1 −ϕ2 after integration over ϕ2 we transform the expression into
the form
G
(1)
M (x) = ρ̃−2(q)I−2
2,0 (q, u)(16π)−1
×
∫
−1
1∫
dz1dz2
(1−z2
1 )1/2∫
0
dρ1 ρ1
(1−z2
2 )1/2∫
0
dρ2 ρ2
2π∫
0
dϕ
∑
σ1,σ2=±1
σ1σ2
× ρ̃2
(
P 1/2
σ1σ2
)
P−1
σ1σ2
{
zσ
1 − zσ
2
}{
(zσ
1 )2 zσ
2 − u2 [2zσ
1 + zσ
2 ]
}
× {u2 + (zσ
1 )2
}−2 {
u2 + (zσ
2 )2
}−1
, (46)
where zσ
1 ≡ z1 +1/2σ1q, zσ
2 ≡ z2 +1/2σ2q, Pσ1σ2 ≡ ρ2
1 + ρ2
2 + [zσ
1 − zσ
2 ]2 − 2ρ1ρ2 cos ϕ.
In the case of variants 1, 3 and 4 (see table 1) the calculations can be reduced
to a two-dimensional integral, since the integration over variables ϕi and ρi can be
performed in an analytical form. To illustrate we show an expression for G
(1)
M (x) for
variant 1 in which ρ̃2(q) = γ2(γ2 + q2)−1
G
(1)
M (q, u) = (γ2 + q2)[2γI2,0(q, u)]−2
∫
−1
1∫
dz1dz2
∑
σ1,σ2=±1
σ1σ2
× Ψ(z1 +
σ1
2
q; z2 +
σ2
2
q; 1 − z2
1 ; 1 − z2
2). (47)
Here the following notations were used:
Ψ(a; b; w; v) = Φ0(a; b; w; v) − Φγ(a; b; w; v),
Φγ(a; b; w; v) =
1
8
(a − b)J(s2
γ , w, v)(b2 + u2)−1(a2 + u2)−2{a2b − u2(2a + b)},
J(s2
γ, w, v) =
1
2
[W (s2
γ, w, v)− w − v − s2
γ]
397
M.V.Vavrukh, N.L.Tyshko
+ w ln{1
2
s−2
γ [W (s2
γ, w, v) + s2
γ + v − w]}
+ v ln{1
2
s−2
γ [W (s2
γ, w, v) + s2
γ + w − v]},
W (s2
γ, w, v) = {s4
γ + 2s2
γ(w + v) + (w − v)2}1/2, (48)
where s2
γ = γ2 + (a − b)2 and Φ0(a; b; w; v) coincide with Φγ(a; b; w; v) at γ = 0.
1 2 3 4 5 6 7
q
-0.5
0.0
0.5
1.0
1.5
2.0
GM
(1)
G (1)
G0
γ = 2.5
Figure 8. Local-field correction functions G
(1)
M (x) and G(1)(x) in the static case
for the system with model potential.
1 2 3 4 5 6 7 8 9 10
rs
-0.18
-0.16
-0.14
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02 εc(rs), Ry
Figure 9. Correlation energy calculated via LFCF for the model potential (top
solid curve), Coulomb potential: ordinary RPA (low curve), MC method (the line
with filled circles).
Figure 8 shows the properties of functions G
(1)
M (x) and G(1)(x) in the static
case. For γ = 2.5 function, G(1)(x) have a long-wave asymptote g(q/kF)2, where
398
Potential regularization
0 1 2 3 4 5
r kF
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2 F2(r)
rs = 1
rs = 7
Figure 10. Binary distribution function calculated via LFCF for the model po-
tential.
g ≈ 1/3, which gives r0
s ≈ 5.0 and provides a correct behaviour of compressibility.
The dependence of the correlation energy in this approximation is shown in figure 9:
the curve with filled circles corresponds to the results obtained with MC method [5],
dashed line corresponds to RPA with the model potential, solid curve corresponds
to LFCF G
(1)
M (x) and to the model potential (23). The average deviation from the
results obtained with MC method in the range of 0 � rs � 10 is 8%, while the
average deviation in RPA with the model potential is equal to 26%. The binary
distribution function in this approximation has a correct behaviour in the range of
0 � rs � 5.5 (figure 10).
5. Conclusions
The modern theory of strong degenerated systems is based on the local-field
concept. However, looking for alternative methods remains urgent because the mi-
croscopic local-field correction function theory is still unfinalized. The model ap-
proach, based on regularization of Coulomb potential was proposed by the authors
earlier. In this paper we suggest the criterions of selecting the form of the model
potential, which correct describes the set of characteristics of the electron liquid
model (i.e., the correlation energy, the binary distribution function, the compress-
ibility) in a wide region Wigner’s parameter rs. We have analyzed several weak
long-range interaction potentials between particles, which have an asymptote e2/r
at large distances and is regular when r → 0. The application of these potentials
gives good results for a small, intermediate and strong non-ideality of the system
by using the methods typical of weakly non-ideal systems. As it follows from our
calculations, one of the best model potentials are possessed by the Fourier-transform
Vef(q) = 4πe2q−2 exp{−(2α)−1(q/kF)2} at α = 2.1377 . . .. When utilizing the model
potentials we reduce to the calculation of the characteristics like the momentum
distribution of electron and local-field correction function. On the whole, the regu-
399
M.V.Vavrukh, N.L.Tyshko
larization of the Coulomb potential is a simple and promising method of calculating
the characteristics of metallic systems.
References
1. Vavrukh M., Koval’ S., Tyshko N. // Journal of Physical Studies (Lviv), 2000, vol. 4,
p. 403 (in Ukrainian).
2. Ziesche P., Lehmann G. Ergebnisse in der Elektronentheorie der Metalle. Berlin,
Akademie-Verlag, 1983.
3. Abramovwitz M., Stegun I. (Ed.). Handbook of Mathematical Functions with Formulas,
Graphs and Mathematical Tables. USA, National bureau of standards, 1964.
4. Vavrukh M. // Fizika Nizkikh Temperatur (Kharkiv), 1966, vol. 22, p. 1005 (in Rus-
sian); Vavrukh M. // Low. Temp. Phys., 1996, vol. 22, p. 767.
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8. Vavrukh M., Krokhmalskii T. // Phys. stat. sol. (b), 1991, vol. 168, p. 519;
Vavrukh M., Krokhmalskii T. // Phys. stat. sol. (b), 1992, vol. 169, p. 451.
9. Ichimaru S., Utsumi K. // Phys. Rev. B, 1981, vol. 24, p. 7385.
10. Lannto L.J. // Phys. Rev. B, 1980, vol. 22, p. 1380.
11. Daniel E., Vosko S.M. // Phys. Rev., 1960, vol. 120, p. 2041.
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Регуляризація потенціалу як спосіб врахування
короткосяжних кореляцій у теорії електронної
рідини
М.В.Ваврух, Н.Л.Тишко
Львівський національний університет імені Iвана Франка, кафедра
астрофізики, вул. Кирила і Мефодія, 8, Львів, 79005, Україна
Отримано 1 квітня 2004 р.
В попередній роботі авторів [1] запропоновано ідею регуляризації
потенціалу Кулона як один із способів врахування короткосяжних ко-
реляцій в моделі електронної рідини. В даній роботі сформульовано
критерії оптимального вибору регуляризації і розраховано енерге-
тичні, структурні та діелектричні характеристики моделі.
Ключові слова: модель електронної рідини, короткосяжні
кореляції, поправка на локальне поле, кореляційна енергія, бінарна
функція розподілу, стисливість
PACS: 05.30.Fk
400
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