Полная энергия в приближении локальной плотности

COMPUTATIONAL PHYSICS
Computer Modeling & New Technologies, 1999, Volume 3, 19-24
Transport and Telecommunication Institute, Lomonosov Str.1, Riga, LV-1019, Latvia
TOTAL ENERGY OF SOLIDS
IN LOCAL DENSITY APPROXIMATION
YU. N. SHUNIN
Transport and Telecommunication Institute, Department of Fundamental and Social Sciences,
Divisiom of General and Applied Physics, Lomonosov Street 1, LV-1019, Riga, Latvia;
e-mail: shunin@tsi.lv
The method of the total energy estimation for atoms molecules and solids on the basement of developed analytical
electronic potentials is proposed. Numerical estimations of total energies are also presented.
1. Introduction
The most natural way of an estimation of a total energy of solids, apparently, is based on a
formalism of a functional of a density of electronic charge [6-8]. A total energy as the functional of a
H
H
density ρ (r ) for a ground state of a multi-electronic system if there is an external potential VEX ( r ) can
be noted as:
H H
H
H H 1 H H ρ ( r )ρ ( r ′)
(1)
E tot [VEX , ρ ] = T[ρ ] + d r VEX ( r )ρ ( r ) +
d r d r ′ H H + E EX [ρ ] ,
2
r − r′
ò
ò ò
where, accordingly, a kinetic energy, electrostatic energy of electronic gas with an external potential,
energy of interelectronic interaction and exchange-correlation energy are summarized. Usually in using of
H
this general formula there are following problems: 1) determinations ρ ( r ) , minimizing a functional
E tot [VEX , ρ ] ; 2) estimations of a functional of a kinetic energy T[ρ ] ; 3) estimations of a functional are
exchage-correlation energy E XC [ρ ] .
2. On the Kohn-Sham approach
Consequent, but the very laborious way of calculation was offered Kohn & Sham [7, 8] on the
basis of a solution of a system of the one-electron equations for varied wave functions of dummy noninteracting electrons:
{− ∇
2
}
H
H
H
+ VKS [ρ ( r )] ϕ j ( r ) = ε jϕ j ( r ) ,
(2)
H
where ϕ j and ε j are one-particle wave self-function and eigenvalue, VXS [ρ (r )] is the integro-differential
operator of a potential:
H
H
H
H
H ρ ( r ′) ∂E XC [ρ ( r )]
H
H
H
VXS [ρ ( r )] = VEX ( r ) + d r ′ H H +
= VEX ( r ) + Vc [ρ ( r )] + VXC [ρ ( r )] .
H
r−r
∂ρ ( r )
ò
(3)
H
Here VC , VXC are the Coulomb and exchange-correlation potentials, accordingly. ρ ( r ) then is defined by
a self-consistent image similarly to a procedure in the Hartree-Fock-Slater method. Then the kinetic
energy of a multielectronic system is found as:
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COMPUTATIONAL PHYSICS
T[ρ ] =
åϕ
ε j − VRS [ρ ] ϕ j .
j
(4)
j
H
Further, usually they use the local density approximation, which means, that ρ ( r ) is changed enough
slowly and it is possible to suppose:
H H
H
E XC [ρ ] ≅ d r ρ ( r )ε XC [ρ ( r )] ,
ò
(5)
H
where ε XC [ρ ( r )] is the exchange-correlation energy per one electron of homogeneous electronic gas of a
H
density ρ ( r ) . The appropriate exchage-correlation potential then is represented as:
H
H
∂E XC [ρ ( r )]
.
(6)
VXC [ρ ( r )] =
H
∂ρ ( r )
There are different submissions for E XC in the literature. In this case we shall mark only the
method of a determination ε XC in the form
ε XC = ε X + ε C ,
(7)
taking into account
ε X = −0,9164 / rS
(11)
and
(
)
ì
ï− 0,2846 / 1 + 1,9529 rs 2 + 0,0040 rS , (rS ≥ 1)
εC = í
ï
î− 0,0960 + 0,0622 ln rS − 0,0232 rS + 0,0040 rS ln rS ,
1
(12)
where rS the met earlier Wigner's radius (see, [1]. Then the exchange-correlation potential is received by
the way:
VXC = ε XC −
rs dε XC
.
3 drS
(13)
E.g., (10), (11), (12) allow to select «pure» exchange and correlation. However, using the Slater statistical
approximation, it is necessary to write
ε XC =
H
3 H
ρ ( r )VXC ( r ) ,
4
(14)
where
1
VXC = αVXS = VXα
H ö3 æ 2
ö
æ 3
= −6α ç
ρ ( r ) ÷ , ç ≤ α ≤ 1÷ .
è 8π
ø è3
ø
(15)
And, therefore, in (14), (15) the exchange and correlation are indiscernible. It is a feature and, in a way, a
lack of a statistical approximation.
Gaspar, Kohn and Sham [1, 10], as it was specified earlier in [*], used in a variational procedure
the Slater statistical approximation [VXiHF ]AV , that has given
H 2
H
VXGKS ( r ) = VXS ( r ) .
3
(16)
The exchange potential (16) is equal to the exchange potential for electronic gas (5) on the Fermi surface
for k = k F . It follows from here, that the exchange potential is:
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COMPUTATIONAL PHYSICS
1
VXGKS
H
H
(r ) = −4æç 3 ρ (r )ö÷ 3 .
8
π
è
ø
In connection with (5) choose select also exchange potential of Liberman-Slater [10]:
æ
~
H
H ö ç
kS
æ 3
VXLS (r , ε ) = −8ç
ρ (r )÷ Fç
8
π
è
ø ç
H 1
2
(
3
π
ρ
r) 3
è
(
)
ö
÷
÷,
÷
ø
~
where one of the quantum numbers k i is exchanged by the energy-dependent parameter k ε . It is
determined by the relationship:
æ~
k ö
ε =ç ε ÷
ç kF ÷
è
ø
2
2 ö
2
æ
ç ε F − VC + k F ÷ + VC − 2k F
ç
π ÷ø
π
è
æ~
k
Fç ε
ç kF
è
ö
÷.
÷
ø
The function F(η ) in this case plays a role trial parameter α . However, in practical calculations, as a
rule, the exchange-correlation potential (15) is used with tabulated values α [1, 10].
3. Total energy of a ground state of atom
The essence of the model of atom in the statistical approximation of distribution of electronic
charge is, that the electronic subsystem of atom is considered in a field of it's core as an electronic liquid
(or an electronic gas with the«turned on» exchange-correlation interaction. The approximation, however,
lies also in the fact that the electronic subsystem is considered locally free. It means, that in each point of
H
space r an electron of a subsystem is considered as a plane electronic wave. Then the energy of an atom
H
can be noted as a functional of a of electronic charge density ρ ( r ) :
H
E[ρ ( r )] =
ò ( )
3
3π 2
5
2
3
5
H H
H
H H 1
H
H H
H
H H
ρ 3 (r )d r + ρ ( r )v n ( r )d r +
ρ ( r )v e ( r )d r + ρ ( r )v X [ρ ( r )]d r ,
2
ò
ò
ò
(17)
where the first contribution is the total kinetic energy of electronic charge, the second - the electrostatic
energy of electronic gas with a core, the third - the energy of interelectronic interaction; v n , v e , v X are
corresponding interaction potentials. Considering the average electronic charge of atom being distributed
with the sphere symmetry around a core, it is possible to write down (17) as:
( )ò
3
E[ρ (r )] = 3π 2
5
2
3
1
é 5
ù
4
ê ρ 3 (r ) + 1 ρ (r )V (r ) − ρ (r ) 2 Z − 9 α æç 3 ö÷ 3 ρ 3 (r )ú 4πr 2 dr ,
e
ê
ú
2
r
2 è 8π ø
ëê
ûú
(18)
where Ve (r ) - electronic part of a potential V G (r ) (see [2,12]).
Total energy of an isolated atom then
E tot = Te + U en + U ee + U XC ,
(19)
where
Te =
2∞
3
( ) òρ
3
3π 2
5
5
3 ( r ) 4πr 2 dr
(20)
0
is the kinetic energy of electronic gas,
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COMPUTATIONAL PHYSICS
∞
U en = −
ò
0
2Z
ρ (r )4πr 2 dr ,
r
(21)
is the potential energy of interaction of an electron shell with a core,
∞
1
=
ρ ( r ) Ve (r )4πr 2 dr ,
2
ò
U ee
(22)
0
is the potential energy of an interelectronic interaction,
1
U XC
∞
4
9 æ 3 ö3
= − α ç ÷ ρ 3 (r )4πr 2 dr
2 è 8π ø
0
ò
(23)
is the energy of exchange-correlation interaction.
Using the results [2,12], it is possible to make an estimation of potential energy U en . Then
∞
U en =
ò
0
∞
7
2Z
ρ (r )4πr 2 dr = 2 Z 2 f ′′(r )dr = 2 Z 2 [f ′(∞ ) − f ′(0 )] = −2,787 Z 3 , Ry ,
r
ò
(24)
0
where
ρ (r ) =
(
)
Z
f ′′(r ) , f ′(r ) = − λ µ + A µ f (r ) ,
4πr
f (r ) = exp(− λr / µ ) / (1 + Ar / µ ), λ = 0,1837, A = 1,05, µ = 0,8853Z
−
1
3
.
The energy of an interelectronic interaction is also evaluated as:
U ee =
∞
∞
0
0
1
ρ (r )Ve (r )4πr 2 dr = Z 2
2
ò
ò (1 − f (r ))f ′′(r )dr .
(25)
−1
If now to take into account, that the potential of interelectronic interaction is changed ~ r , then, under
the virial theorem [9], it is possible to evaluate a kinetic energy of electronic gas
Te′ = − U ee / 2 ,
(26)
which , apparently, should be close to Te . Evidently, (20) and (26) are different methods of determination
of the same magnitude.
Energies of different isolated atoms [11] in comparison with results of self-consistent
calculations [1, 9, 10] are indicated in the Table.
TABLE. Energies of isolated atoms
Z
2
10
14
Te , R y
U en , R y
U ee , R y
U XC , R y
U XC , R y
E tot , R y
E tot , R y
(20),(26)
4.743
5.284
202.811
226.632
444.484
497.204
(21)
-14.003
-14.045
-597.858
-600.439
-1313.519
-1316.542
(22)
2.642
(23)
-1.542
[10]
-2.052
[1,9,10]
113.316
-21.346
-24.217
248.602
-37.288
(19)
-8.1611
-7.2446
-303.077
-264.907
653.103
571.258
22
-257.094
-577.638
COMPUTATIONAL PHYSICS
15
521.938
582.318
606.820
617.156
798.875
891.878
3059.055
3420.424
3286.751
3615.202
3523.840
3940.470
4026.540
4502.986
9069.757
10132.014
9490.281
10602.484
16
18
32
33
34
36
51
52
-1540.026
-1546.496
-1790.473
-1797.834
2357.109
-2366.493
-9014.161
-9060.503
-9685.038
-9734.984
-10384.777
-10437.273
-11867.235
-11926.386
-26740.928
-26882.308
-27980.574
-28128.316
291.159
-41.742
338.578
-45.396
445.939
-56.239
1710.212
-143.696
1837.601
-151.303
1970.235
-158.963
2251.493
-174.698
5066.007
-309.185
5301.242
-319.364
-60.370
-188.308
-768.681
-669.411
-891.472
-776.024
-1168.552
1016.516
-4388.585
-3818.840
-4712.566
-4099.992
-5049.660
-4392.476
5763.094
5012.140
-12914.347
-11217.332
-13508.414
-11732.898
681.297
-794.927
-1053.635
-5504.111
The difference of kinetic energies Te and Te′ for each Z included in in the odd and even table
rows, correspondingly, is stipulated by a method of their calculation. The values U en were calculated also
by two methods, namely:
∞
é 2Z 1
ù
+ Ve (r )ú 4πr 2 dr − U ee
U en = ρ (r )ê−
r
2
ë
û
0
ò
,
and as U ′en by the formula (21).
As a matter of fact these values should coincide. The divergence arises during numerical integrations,
that is explained by a weak convergence of integrals. By the way, the numerical integration under the
formulas (20), (21), (22), (23) requires a large caution and can under certain conditions lead to large
errors. This should be taken into account in usage of the given approach for more complicated systems.
The special attention should be paid on comparison of the exchange-correlation energies obtained on
the formula (23) with the data from [10], and also total energies (19) with the data [, 9, 10]. Thus, the
formula (19) has been applied twice for of different values of kinetic energies Te And Te′ . The value
E tot = Te + U en + U ee + U XC as a whole is more close on magnitude to values of self-consistent
calculations.
In conclusion it is necessary to note, that the divergence of estimations E tot And U XC On analytical
parities (ratio) (19-26) with self-consistent accounts does not exceed 10 %, that for absolute values it is
possible to consider(count) quite satisfied.
The given procedure can be utilised also for calculation of a total energy of more composite systems
(напр., molecules, clusters, rigid bodies). An indispensable condition in this case is the construction of
H
H
realistic frequency functions of electronic charge ρ ( r ) And one-electron potentials V( r ) , mentioned
above systems on the basis of a principle of superposition [2].
References
[1] Слэтер Дж. (1978) Методы самосогласованного поля для молекул и твердых тел. Мир.664 с.
[2] Shunin Yu. N., Shvarts K. K. (1986) Calculation of the Electronic Structure in Disordered
Semiconductors. Phys. Stat. Sol. (b), 135, 15-36.
[3] Достижения электронной теории металлов. (1984) Под ред. П. Цише, Г. Лемана. Т. 1. Мир,
Москва. 284 с.
[4] Кондратьев А. С., Кучма А. Е. (1980) Электронная жидкость нормальных металлов. Ленингр.
ГУ, Ленинград. 200 с.
[5] Райс Т., Хенселл Дж., Филлипс Т., Томас Г. (1980) Электронно-дырочная жидкость в
полупроводниках. Мир, Москва. 349 с.
[6] Hohenberg P. Kohn W. (1964) Inhomogeneous Electron Gas. Phys. Rev.136, 864-871.
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COMPUTATIONAL PHYSICS
[7] Kohn W., Sham L. J. (1965) Self-Consistent Equations Including Exchange and Correlation Effects. –
Phys. Rev.140, N 4A, 1113-1138.
[8] Srivastava G. P., Weaire D. (1987) The Theory of the Cohesive energies of Solids. Adv. Phys. 36,
463-517.
[9] Флайгер У. 1982. Строение и динамика молекул. Т. 1 – М., Мир, 408 с.
[10] Немошкаленко В. В., Кучеренко Ю. Н. (1986) Методы вычислительной физики в теории
твердого тела. Электронные состояния в неидеальных кристаллах.. Наукова Думка,
Киев 296 с.
[11] Шунин Ю. Н., Шварц К. К. (1988) Влияние состава на электронную структуру соединений
типа AXB1-X. Тез. 27 Всес. Семинара по моделированию радиационных и других дефектов в
твердых телах. ЛФТИ, Ленинград
[12] Shunin Yu.N.. (1998) Potentials and pseudopotentials. RAU Scientific Reports. Computer Modeling
& New Technologies 2, 34-49
Received on the 21st of May 1999
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