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Approximations of exchange-correlation energy by functionals with kernels depends on difference of coordinates
0038-1098/84 $3.00 + .00 Pergamon Press Ltd
Solid State Communications, Vol 50, No 2, pp 121-123, 1984 Printed in Great Britain
APPROXIMATIONS OF EXCHANGE-CORRELATION ENERGY BY FUNCTIONALS WITH KERNELS DEPENDS ON DIFFERENCE OF COORDINATES V Iu. Kolmanovich and I M Reznlk Theoretical Physics Department, Donetsk Physics-Technical Institute Academy of Sciences of UkSSR, 340114, D.Ph-T.I, Roza Luxemburg str, 72, Donetsk, U S S R
(Recetved 7 November
1983
by V M Agranovtch)
An area has been found containing for any given electron density all possible values of exchange-correlation energy approxnnated by nonlocal expressions with kernels depends on the difference of coordinates It is assumed that such functlonals satisfy but a few evident physical restrictions Subsystem of valent electrons of germanium has been considered for sake of Illustration The details of kernels behavlour have been shown to weakly affect the total exchange-correlation energy 1 INTRODUCTION THEORETICAL EXPLORATIONS of the ground state properties and energy spectra of crystals may be carried out only within an approxlmatlve approach to the description of nonclassical part of the electronelectron interaction (exchange and correlation energies),
Exc[p]
1 =
-
-~
f
.
(p(r)p(r')g(r, r')/Ir -- r'l dr dr'
(1)
Here g(r, r') is in a simple way connected with the two-particle correlation function of electron system and describes the decrease of probability of another electron occurring in the vicinity of given one due to the Pau]l principle and singularity of the Coulomb potential, p(r) denotes electron density in the point r Local approximations
Exc[P]
f f[p(r)] dr,
(2)
rigorously justified only for the limit case of weakly nonhomogeneous electron gas and within a framework of the principle of local homogenlty [ 1] are the most widely used Respective condmons cannot be satisfied In the theory of the ground state of atoms, molecules and solids at normal conditions To regard equation (2) as zeroth order term of the gradient expansion [2] is not quite correct either The series of the gradient expansion starts diverging sooner or later [3] and in the case of exchange interaction even the first gradient amendment does not exist [4], although under empirical considerations it appears small [5] However, carefully carried out calculations based on functl6nals (2) fit the experimental data quite well Such calculattons give better spectra than the consistent
Hartree-Fock method gives (see review [6]). This IS caused by overestimation of the orbital dependency of effective potential by the single particle method. As it is known from the theory of homogeneous systems correlation effects compensate significantly this dependency [7] The effective potential corresponding to energy functional (2) is fully state-independent. Nevertheless local approximations are not satisfactory for a number of reasons So the values of energy gaps in semiconductors appear to be underestimated. The description of linear response of a homogeneous system to a small perturbation is rather crude [8] At last application of equation (2) causes much numerical trouble due to nonhnearlty of the corresponding potential in p This drawback may be eliminated by a transfer to nonlocal approximations [9] where the potential is connected to the electron density in an integral form but still does not depend on the state it is apphed to The first step in this direction appears to be representation of exchange-correlation energy as 1
E=[pl ~ - 7
f fp(r)p(r')Q(Ir -
r'l) dr dr'
(3)
The kernel Q may be chosen in different ways One may take some simple expression, choose its parameters to fit the sum relations for the exchangecorrelation hole [9] Correct polarizability of a homogeneous system x(q) =
xo(q) 1 -- U(q)[1
-- a(q)]
xo(q)'
where Xo(q) IS the polarizability of a system of noninteractive particles according to Llndhard, G(q) is a 121
(4)
Vol 50, No 2
APPROXIMATIONS OF EXCHANGE-CORRELATION ENERGY
122
local field correction [8], U(q) the Fourier-Image of the kernel of Coulomb lnteractmn may fit if we take
7
-- 2rr J S ( r ) Q ( r ) r 2 dr
-
k~e[O]
(9)
0
Q(q) = U ( q ) G ( q )
(5)
s(,) =
Various approaches to evaluatmn o f Q even within H a r t r e e - F o c k results for polanzabihty yield qmte different formulas [8] With other estimations of the local field correction the difference may be still greater Therefore the questmn arises of how differences in kernels affect the value of Exe accordingly to equation (3)~ The answer to it presents the aim of this paper Kernel Q is subject to quite few general conditions formulated in the next section
It IS natural to demand that under the condition of homogeneous distribution of charge in the considered system (3) should give results which we suppose known for this case
exe[Po ],
(6)
where ~2 IS the volume of the system, P0 is the constant density We also demand that the exchange-correlation potential should not change its sign and should be less or equal to that of Coulomb In all points of space and for all densities It is provided by a restriction
1
0 ~< Q(r) <~ -
(7)
/.
We seek the area of all possible values of Exe (3) under conditions (6) and (7). For the given distribution of charge this is a linear variational problem with respect to Q Its extremal solutions occur at the boundaries and appear to be discontinuous Therefore the classic vanatlonal methods cannot be applied here The solution may be obtained by using the theory of optimal regulation [ 10] The forces involved in the exchange-correlation interaction are assumed to be smaller than the Coulomb ones and to decrease monotonously with the distance Then the c o n d m o n (7) should be substituted by a more rigid condition
1
--r- ~ <~ Q(r) <~ O,
ffO(r +
dr' de ,
(10)
g2r are angle variables of r In order to formulate the problem in temrs of the theory of optimal regulation we transfer from the in integral representation {9), equation (6) to the dlffetentml one, introducing
2 PROBLEM FORMULATION
l p g J" Q ( r ) d r - Exc[P°~] -
where
(8)
where the point above denotes the derwative 3. SOLUTION Let us transform equation (3) having Integrated on all variables except the argument of kernel Then
qo = -- 2rrS(r)r2Q(r),
qo(0) = 0,
~ll
ql(O) = 0
2rrpZrZQ(r),
(1 I)
Let us consider restrictions (7) Then the problem is to determine boundaries of qo(°~) when q 1(~) = e~c [P0] equation (6) when the regulation Q changes in the region, given by equation (7) The respective Hamilton function [ 10] is
H(q, p, r) = - 2rrQ(r)r2( S (r )Po + Pl)
(12)
Conjugated to system (11)equations for " m o m e n t s " P provide for the conservation law, therefore PI and P0 are constants (by analogy with the Lagrange multlphers) The Pontryagln principle of maxunum reqmres that H be extremal over Q Taking into account equation (7) we obtain
Q(r) = l o ( s ( r ) P o + r
P1),
(13)
w h e r e 0 ( x ) = 1 when v ~ > 0 a n d 0 ( v ) - 0 m t h e opposite case To determine Po and PI we use the mteglal condition (6)
-- 2fro 2 ; O(S(r)Po + Pl)rl dr
exe[Po]
(14)
o
Evidently Po4- 0 and therefore only the sign ol P0 m the last equatmn IS significant Without loss of generahty P0 = +- 1 The upper sign as tt will be shown later corresponds to the maximum value of exchange-correlatmn energy, the lower corresponds to mmlnmm one (absolute values are assumed) It may be easdy shown that the solution in each case ~s smgle if it exists The d i s m b u t l o n of electron charge m a crystal ~s naturally determined by its Fourier components 0 6 , then p(r) =
~p6exp0Gr),
(15)
G
where G IS a vector of reciprocal lattice According to equation ( 1 0 ) w e get
Vol 50, No 2
APPROXIMATIONS OF EXCHANGE-CORRELATION ENERGY
3° hl
20
15 I0
B
J cVD
K~
Fig 1 Graphical solution of equation (14) sin Gr
S(r) = Y. IPa 12- G
(16)
Gr
This function (the term with G = 0 Is substlacted) is represented m Fig 1 For electron density we use the results of the empirical potential method apphed to subsystem of valence electrons of germanium [ 11 ] For exe [Po] pure exchange contribution was taken
Exe [rio] -
4~ 3 (3.rrZpo)4/3 _
l_ k 4 3 4rr3 r
mum value m gernlamum 1s near 27 (m the centre of bond), the average value Is 8 electrons per one elementary cell In metals varlanons of p(r) are much lower and so ex~[,o] are When restrictions of the type (8) are mvolved the solution may be carried out in a sunllar way assuming Q to be the regulanon, defining Q as an integral of the regulation and performing evtdent transformatmns of equation (9) In fins case the area of possible values of tile functional In question tends out to be 1 68 0 99 in units of exelPo] It should be pointed out that the areas that were found are wider than they may arise m real calculatums, because a number of resnlctlons on the kernel may be folmulated and only sonle of them were used m this papm The above discussed results show that different approxmlatmns of type (3) may be used for calculatmn the ground state propemes of sohds at normal condmons without fear o f a consldmable error due to ~ough approxmIatlons o[ the kernel m the case when It sansfles the conditions considered above Functionals (3) may be expected to slmphfy practmal calculations, lead to shorter computez programs and may ymld better preclsmn than local approxmlanons REFERENCES
(17) 1
Correlation energy is any mterpolatmn form may be included here without comphcatmg the problem However thxs is not necessary because the correlation contribution is quite insignificant for the densities in quesnon and does not affect the results quahtlvely
2 3 4 5
4 RESULTS AND DISCUSSION A graphm solution of equation (14) msshown m Fig I The kernel Q corresponding to the maximum is not equal to zero only at the AB interval, that corresponding to the minimum at the CD interval Tile value of exe [O] occurs m tire Interval 0 9 2 - 1 26 in units of exe [Po] It is clear that the area of possible values is quite narrow However ~t Is known that density oscillations in semiconductors are comparatively large the minimum value of density tends to zero, the maxl-
123
6 7 8 9 10
11
D A Klrzhmts, Yu E Lozovlk & G V Shpatakovskaya, Usp Phys Nauk 117, 3 (1975) P H o h e n b e r g & W Kohn,Phl,s Rev 136,864 (1964) D R Murphy,Phvs Rev A24, 1682(1981) D J W Geldart & M Rasolt Phvs Rev B13, 1477 (1976) C C Shin, D R Murphy & W Wang, J Chem Phys 73, 1340 (1980) D D Koelhng, Rep Progr Phys 44, 139(1981) D Pines, Elementary Excitations in Sohds, New York, Amsterdam (1963) V D Gorobchenkl & E G Maxmlov, Usp Ph)'s Nauk 130, 65 (1980) G P Kerker, Phys Rev B 2 4 , 3 4 6 8 ( 1 9 8 1 ) V G Boltyanskyl, Mathemattcal Methods oJ the Theory of Opttmal Regulanon, Nauka, Moskow (1969) J R Chehkowsky & M L Cohen, Phys Rev BI4, 556 (1976)
Solid State Communications, Vol 50, No 2, pp 121-123, 1984 Printed in Great Britain
APPROXIMATIONS OF EXCHANGE-CORRELATION ENERGY BY FUNCTIONALS WITH KERNELS DEPENDS ON DIFFERENCE OF COORDINATES V Iu. Kolmanovich and I M Reznlk Theoretical Physics Department, Donetsk Physics-Technical Institute Academy of Sciences of UkSSR, 340114, D.Ph-T.I, Roza Luxemburg str, 72, Donetsk, U S S R
(Recetved 7 November
1983
by V M Agranovtch)
An area has been found containing for any given electron density all possible values of exchange-correlation energy approxnnated by nonlocal expressions with kernels depends on the difference of coordinates It is assumed that such functlonals satisfy but a few evident physical restrictions Subsystem of valent electrons of germanium has been considered for sake of Illustration The details of kernels behavlour have been shown to weakly affect the total exchange-correlation energy 1 INTRODUCTION THEORETICAL EXPLORATIONS of the ground state properties and energy spectra of crystals may be carried out only within an approxlmatlve approach to the description of nonclassical part of the electronelectron interaction (exchange and correlation energies),
Exc[p]
1 =
-
-~
f
.
(p(r)p(r')g(r, r')/Ir -- r'l dr dr'
(1)
Here g(r, r') is in a simple way connected with the two-particle correlation function of electron system and describes the decrease of probability of another electron occurring in the vicinity of given one due to the Pau]l principle and singularity of the Coulomb potential, p(r) denotes electron density in the point r Local approximations
Exc[P]
f f[p(r)] dr,
(2)
rigorously justified only for the limit case of weakly nonhomogeneous electron gas and within a framework of the principle of local homogenlty [ 1] are the most widely used Respective condmons cannot be satisfied In the theory of the ground state of atoms, molecules and solids at normal conditions To regard equation (2) as zeroth order term of the gradient expansion [2] is not quite correct either The series of the gradient expansion starts diverging sooner or later [3] and in the case of exchange interaction even the first gradient amendment does not exist [4], although under empirical considerations it appears small [5] However, carefully carried out calculations based on functl6nals (2) fit the experimental data quite well Such calculattons give better spectra than the consistent
Hartree-Fock method gives (see review [6]). This IS caused by overestimation of the orbital dependency of effective potential by the single particle method. As it is known from the theory of homogeneous systems correlation effects compensate significantly this dependency [7] The effective potential corresponding to energy functional (2) is fully state-independent. Nevertheless local approximations are not satisfactory for a number of reasons So the values of energy gaps in semiconductors appear to be underestimated. The description of linear response of a homogeneous system to a small perturbation is rather crude [8] At last application of equation (2) causes much numerical trouble due to nonhnearlty of the corresponding potential in p This drawback may be eliminated by a transfer to nonlocal approximations [9] where the potential is connected to the electron density in an integral form but still does not depend on the state it is apphed to The first step in this direction appears to be representation of exchange-correlation energy as 1
E=[pl ~ - 7
f fp(r)p(r')Q(Ir -
r'l) dr dr'
(3)
The kernel Q may be chosen in different ways One may take some simple expression, choose its parameters to fit the sum relations for the exchangecorrelation hole [9] Correct polarizability of a homogeneous system x(q) =
xo(q) 1 -- U(q)[1
-- a(q)]
xo(q)'
where Xo(q) IS the polarizability of a system of noninteractive particles according to Llndhard, G(q) is a 121
(4)
Vol 50, No 2
APPROXIMATIONS OF EXCHANGE-CORRELATION ENERGY
122
local field correction [8], U(q) the Fourier-Image of the kernel of Coulomb lnteractmn may fit if we take
7
-- 2rr J S ( r ) Q ( r ) r 2 dr
-
k~e[O]
(9)
0
Q(q) = U ( q ) G ( q )
(5)
s(,) =
Various approaches to evaluatmn o f Q even within H a r t r e e - F o c k results for polanzabihty yield qmte different formulas [8] With other estimations of the local field correction the difference may be still greater Therefore the questmn arises of how differences in kernels affect the value of Exe accordingly to equation (3)~ The answer to it presents the aim of this paper Kernel Q is subject to quite few general conditions formulated in the next section
It IS natural to demand that under the condition of homogeneous distribution of charge in the considered system (3) should give results which we suppose known for this case
exe[Po ],
(6)
where ~2 IS the volume of the system, P0 is the constant density We also demand that the exchange-correlation potential should not change its sign and should be less or equal to that of Coulomb In all points of space and for all densities It is provided by a restriction
1
0 ~< Q(r) <~ -
(7)
/.
We seek the area of all possible values of Exe (3) under conditions (6) and (7). For the given distribution of charge this is a linear variational problem with respect to Q Its extremal solutions occur at the boundaries and appear to be discontinuous Therefore the classic vanatlonal methods cannot be applied here The solution may be obtained by using the theory of optimal regulation [ 10] The forces involved in the exchange-correlation interaction are assumed to be smaller than the Coulomb ones and to decrease monotonously with the distance Then the c o n d m o n (7) should be substituted by a more rigid condition
1
--r- ~ <~ Q(r) <~ O,
ffO(r +
dr' de ,
(10)
g2r are angle variables of r In order to formulate the problem in temrs of the theory of optimal regulation we transfer from the in integral representation {9), equation (6) to the dlffetentml one, introducing
2 PROBLEM FORMULATION
l p g J" Q ( r ) d r - Exc[P°~] -
where
(8)
where the point above denotes the derwative 3. SOLUTION Let us transform equation (3) having Integrated on all variables except the argument of kernel Then
qo = -- 2rrS(r)r2Q(r),
qo(0) = 0,
~ll
ql(O) = 0
2rrpZrZQ(r),
(1 I)
Let us consider restrictions (7) Then the problem is to determine boundaries of qo(°~) when q 1(~) = e~c [P0] equation (6) when the regulation Q changes in the region, given by equation (7) The respective Hamilton function [ 10] is
H(q, p, r) = - 2rrQ(r)r2( S (r )Po + Pl)
(12)
Conjugated to system (11)equations for " m o m e n t s " P provide for the conservation law, therefore PI and P0 are constants (by analogy with the Lagrange multlphers) The Pontryagln principle of maxunum reqmres that H be extremal over Q Taking into account equation (7) we obtain
Q(r) = l o ( s ( r ) P o + r
P1),
(13)
w h e r e 0 ( x ) = 1 when v ~ > 0 a n d 0 ( v ) - 0 m t h e opposite case To determine Po and PI we use the mteglal condition (6)
-- 2fro 2 ; O(S(r)Po + Pl)rl dr
exe[Po]
(14)
o
Evidently Po4- 0 and therefore only the sign ol P0 m the last equatmn IS significant Without loss of generahty P0 = +- 1 The upper sign as tt will be shown later corresponds to the maximum value of exchange-correlatmn energy, the lower corresponds to mmlnmm one (absolute values are assumed) It may be easdy shown that the solution in each case ~s smgle if it exists The d i s m b u t l o n of electron charge m a crystal ~s naturally determined by its Fourier components 0 6 , then p(r) =
~p6exp0Gr),
(15)
G
where G IS a vector of reciprocal lattice According to equation ( 1 0 ) w e get
Vol 50, No 2
APPROXIMATIONS OF EXCHANGE-CORRELATION ENERGY
3° hl
20
15 I0
B
J cVD
K~
Fig 1 Graphical solution of equation (14) sin Gr
S(r) = Y. IPa 12- G
(16)
Gr
This function (the term with G = 0 Is substlacted) is represented m Fig 1 For electron density we use the results of the empirical potential method apphed to subsystem of valence electrons of germanium [ 11 ] For exe [Po] pure exchange contribution was taken
Exe [rio] -
4~ 3 (3.rrZpo)4/3 _
l_ k 4 3 4rr3 r
mum value m gernlamum 1s near 27 (m the centre of bond), the average value Is 8 electrons per one elementary cell In metals varlanons of p(r) are much lower and so ex~[,o] are When restrictions of the type (8) are mvolved the solution may be carried out in a sunllar way assuming Q to be the regulanon, defining Q as an integral of the regulation and performing evtdent transformatmns of equation (9) In fins case the area of possible values of tile functional In question tends out to be 1 68 0 99 in units of exelPo] It should be pointed out that the areas that were found are wider than they may arise m real calculatums, because a number of resnlctlons on the kernel may be folmulated and only sonle of them were used m this papm The above discussed results show that different approxmlatmns of type (3) may be used for calculatmn the ground state propemes of sohds at normal condmons without fear o f a consldmable error due to ~ough approxmIatlons o[ the kernel m the case when It sansfles the conditions considered above Functionals (3) may be expected to slmphfy practmal calculations, lead to shorter computez programs and may ymld better preclsmn than local approxmlanons REFERENCES
(17) 1
Correlation energy is any mterpolatmn form may be included here without comphcatmg the problem However thxs is not necessary because the correlation contribution is quite insignificant for the densities in quesnon and does not affect the results quahtlvely
2 3 4 5
4 RESULTS AND DISCUSSION A graphm solution of equation (14) msshown m Fig I The kernel Q corresponding to the maximum is not equal to zero only at the AB interval, that corresponding to the minimum at the CD interval Tile value of exe [O] occurs m tire Interval 0 9 2 - 1 26 in units of exe [Po] It is clear that the area of possible values is quite narrow However ~t Is known that density oscillations in semiconductors are comparatively large the minimum value of density tends to zero, the maxl-
123
6 7 8 9 10
11
D A Klrzhmts, Yu E Lozovlk & G V Shpatakovskaya, Usp Phys Nauk 117, 3 (1975) P H o h e n b e r g & W Kohn,Phl,s Rev 136,864 (1964) D R Murphy,Phvs Rev A24, 1682(1981) D J W Geldart & M Rasolt Phvs Rev B13, 1477 (1976) C C Shin, D R Murphy & W Wang, J Chem Phys 73, 1340 (1980) D D Koelhng, Rep Progr Phys 44, 139(1981) D Pines, Elementary Excitations in Sohds, New York, Amsterdam (1963) V D Gorobchenkl & E G Maxmlov, Usp Ph)'s Nauk 130, 65 (1980) G P Kerker, Phys Rev B 2 4 , 3 4 6 8 ( 1 9 8 1 ) V G Boltyanskyl, Mathemattcal Methods oJ the Theory of Opttmal Regulanon, Nauka, Moskow (1969) J R Chehkowsky & M L Cohen, Phys Rev BI4, 556 (1976)