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The coulomb hole and the correlation energy for the electron gas at intermediate densities
Volume 81, number I
CHEMICAL
PHYSICS
THE COULOMB HOLE AND THE CORRELATION FOR THE ELECTRON
GAS AT INTERMEDIATE
LETTERS
I July 1981
ENEiRGY DENSITIES
M. BERRONDO* 1nsri.Ute for Advanced Studies, The Hebrew University of Jermtdem, Jertmdetn, Israel and 0. GOSCINSKI ** Quantum Chemistry Group, Universzty of UppsaLz, S-751 20 Uppsala. Sweden Received 19 January 1981; in fmal form 24 March 1981
An approximation for the correIation energy of the electron gas is derived (employing a direct modelling of the Coulomb hole) for the pair distribution function: ec(rs) = -0.303&, + 1_108) rydberg. It is of the same form as the expressions of Wigner and of McWeeny.
The aim of this work is to fmd an approximate expression for the correlation energy of the homogeneous electron gas derived from the study of the Coulomb hole. Although the model we adopt is very simple, the result compares ftily well with interpolation formulae based on a frequency-dependent dielectric fur&ion formulation [ 1,Z] _ A most important feature of this work however, is the possibility of extending the method to the case of non-homogeneous systems, such as the electrons in an atom or a molecule where the nuclear singula&ies play an essentialrole. This point of view has recently been taken with respect to the exchange energy [3] and the exchange potential [4], and the results thereof seem promising. The resulting expression for the correlation energy in the present work is a simple function of the electron density (or rather of the ubiquitous z-r).It has the same functional form as the venerabIe interpolation formulaintroduced by Wigner. [S] , as welJ 9s the one proposed by McWeeny [6]. TUG Coulomb hole surroundhrg the electron ap* On Xtike from Instit+o
de F&a,
University of Mexico,
Mexico20 D-F., Mexico. *f Support&lby the Swedish Natural Sciences Resekrch coun&L
pears as a consequence of the iuterelectronic repulsion. It is thus a measure of the correlation present in the motion of the electrons. It is a (dynamic) many-electron effect, Its explicit form is obtained 17J comparing the pair distribution function obtained from a correlated wave&n&ion, wi*h that of Hartree-Fock_ For the latter, the wavefunction incfudes the (static) part of the correlation due to the antisymmetry principle, and provides-the standard point from which to measure the correlation [S] _ Since we are interested in the correlation energy, we can look directly at the pair distribution function g(r). This gives the probability density for finding an electron at a distance r from a fixed electron. We obtain the form of the correlated distribution function using a HylleraasLJastrow correlation factor for each pair of electrons_A cluster expansion is then introduced [9], keeping only-the twoelectron clusters.This should yield a good approximation, except in the limit of very h&h de&ties (rs < l), where higher-order clustershave to be considered. In the Hartree approximation, the pair distribution function is equal to unity for a hoinogeneous electron gas. When we take into account the antisymmetry principle, a Fermi hole is formed for electrons with like spin. The distribution function is hence 3 at the origin.
0 009-2614/81/0000-0000/$02.50 0 North-Holland Publishing Company
75
CHEMICAL
Volume 8 1, number 1
The Hartree-Fock q&-)
= 1 -
PHYSICS
approximation yields [lo,1 1]
%VIWFrYI(-‘F4
2
,
zero c (1)
where jl (z) is a spherical Bessel function, and z = KE r, with KE the Fermi momentum. The usual relation between K,, the electronic density, and the parameter rs is given by density = N/V = Kz/3n2 = (3/47r)rsd3_
(2)
The further correlation induced by the Coulomb repulsion should lower g(r) close to the origin. In fact, the simple random phase approximation overestimates this effect [12] , so that g becomes negative at the o&in. We shall include this correlation in the pair distribution function starting from the N-electron wavefunction written as the Hartree-Fock determinant of plane waves times a product of correlation factors. Each factor involves one electron pair at a time and is assumed to depend on the interelectronic distance only:
The two-electron density matrix can then be expressed as a linked cluster expansion [9] . The fust correction to Hartree-Fock involves only fi 2 _ This can easily be understood since fij + 1 for large ‘ii and hence only the pair considered should give a correction to this order. Computing the pair function from this density matrix immediately yieids &)
=gHF(r)lf,,(r)12-
(4)
The next stage consists in using a Hylleraas type of function [13] for the correlation factor fi3. For small r we expand fin powers of r. Asymptot&lly however, this function should go to I_ The correlation energy for intermediate densities is not very sensitive to the specific way it tends to 1. Hence we shall use the following expression for the approximate pair distribution function, ~r)=g,,(r)(a~+a,r+a2~) =gHF
(r b).
The value of b determines the size of the Coulomb hole and depends on the electronic density. It is hence chosen such that it corresponds to the first 76
LETTERS
(5)
1 July 1981
of the spherical Bessel functionjl(z)
b = 4.493409/K,
= 2.341344rs.
[14] : (6)
Beyond this point, the Bessel function oscillates with a very small (decreasing) amplitude, so that gHF is always very close to unity. According to eq. (S), we have taken the pair function equal to the HartreeFock value for r > b. We now determine the coefficients a0 , al, and a2 in eq. (5) in terms of the cut-off b or, equivalently, of rs_ Let us fust consider the cusp condition for electrons 1 and 2. For r12 = 0, the correlation factor f(r12) must behave as d hf
(‘>lW,=, = $ -
(7)
Hence the pair function is proportional to 1 f r, and eq. (5) yields a0 = al _Secondly, we want the fimction g(r) to be continuous at the cut-off r = b, so that a2 =
(1 - ao)/b2 - so/b.
(8:s)
Lastly, the relation between a0 and the density follows from physical considerations. The Coulomb hole is formed by displacing part of the electronic charge away from the origin [7] when we include the effect of the electronic repulsion_ However, the total charge enclosed within the sphere r < b must remain constant_ This conservation equation thus reads
8)
~gOPdr=~gHF(r)rZclr.
0
0
Substituting eqs. (1) and (5) into eq. (9), we obtain (aa0- 1)D2 + (aoKF)D3
+ (a21@Dq
= 0,
(IO)
where the integrals Dn are simply defined as Dn=j
z” Cl - 4 [0‘&9Y4
21*,
(11)
0
4.493409 being the first zero ofjl(z) 1131. Displayed in table 1 are the required values of Dn, computed numerically_ The shape of the pair distribution function g(r) is given in fig_ 1 for rs equal to 1 and 4. The HartreeFock function is included for comparison. We notice that g(r) remains positive at the origin and has a unit slope. Close to the boundary, however, g(r) becomes larger than 1, an undesirable feature due to the limited
with c =
Volume 81, number 1
CHEMICAL
PHYSICS
Finally, using the values of table 1, together with eqs. (61, (8) and (lo),
Table 1 Values of Dn as defmed in eq. (11)
n
4
e&)
= -0_303/(r,
1 2 3 4
9.02345 28.38122 97.94850 356.72837
At this point it is important to compare with the work by Colle and Salvetti [15] and McWeeny [6]. Colle and Salvetti have actually suggested the choice ffr, J2)
mt
=Eint/V=
Hence,
(N/v)Jg(r) r--l
the correlation
energy
dr.
is &en
(12) by
@(r1,t2)
e,(r,> = (4J$/379 X [(a0 - KID1 + (Qo/$)D2
+
(Q&PJ
-
(14)
a
-SF
Fig. 1. Fair distriiution
functiong(rs)
for rs = 1 md rs = 4.
(15)
(16)
L =e-~29
[l --@(I?)(1
iV/2Qo)],
07)
where R = $(rl + r2), and r12 = r2 - r1 _ The form (17) ensures that the Coulomb hole dies off. The function
which in turn can be expressed in terms of the integrals defmed in eq. (1 l),
+ 1.108) rydberg.
= 1 - @@I ‘51’
with number of terms considered in the expansion in eq. (5). We fmaIiy proceed to compute the correlation energy, using this pair distribution function. The interaction energy (in rydberg) per particle can be expressed dire&y in terms of this function. For the electron gas, we have [I I] E-
1 suly 1981
LETIERS
+ 2.032)
rydberg.
(18)
Even though the coefficients differ somewhat, it is interesting to see that (5) leads perhaps to an understanding of the under&&g reasons for such a form. Our result, valid for intermediate densities, and McWeeny’s coincide for r, = I .2.5. In fig. 2 we have plotted the results obtained with eq. (18). We have included also, for the sake of comparison: (a) Gel&Mann and Brueckner’s result for the high-den+y limit fl& J , (b) Wigner’s interpolation formula [S J , (c) NoziBres and Pines’ interpolation between high- and lay-momentum transfer [2,1 l] , (d) Clementi’s empirical formula for the correlation energy of atoms, taken for a constant density f17, 181 *, and(e) MgWEny*s formula [6]_ The interpolation formula
Volume 81, number 1
CHEMICAL
PHYSICS
LETTERS
1 July 1981
studied in a few cases [19-211 and information about their nature is emerging so that estimates of the correlation energy along the lines presented here will become feasible. The main problem is to ascertam to what extent the correlation hole is affected by the non-homogeneous character of the density. The results of CoUe and SaIvetti [15] are promising in this respect. One of us (MB) would like to thank Professor P.-O. Lijwdin and the members of the Quantum Chemistry Group for their hospitality. References
[II 3. Hubbard, Proc. Roy. Sot. A243 (1957) 336.
121 P. Nozi&es and D. Pmes, Phys. Rev. 111 (1958) 442;
Fig. 2. Correlation energy as a function of rr as predicted by various models. (a) GeIl-Mann and Brueckner high-density limit [ 16 ] ; (6) Wiiner’s interpolation form&r [S ] ; (c) Nozi&e and Pines’ interpolation formula [2,1 l] ; (d) Clementi’s empirical formula [ 17,181; (e) McWeeny’s formula, and our results from eq. (18): “Ours”_
ems, the empirical formula (d) of Clementi et al. [17] gives excellent results. The approximate form which we have found, eq. (15), is appropriate for intermediate densities. In the high-density limit, the pair distribution function (4) should include three and more electron clusters. For the low-density regime, the expansion in powers of rs for the correlation factor (5) should be improved. The region where g(r) is larger than one becomes relevant in this limit. Again, for intermediate and high densities, we do not expect that the precise form of the cut-off should Play a role. In fact, the interaction energy is given by a weighted average over the normalized distribution function (a, first moment), and beyond b this function is almost unity. Rather than implementing the improvements suggested above, we want to look at the case of a homogeneous electron gas as a test case for our method which models the Coulomb hole. The study of non-homogeneous systems, as the electron cl&d in atoms, is in progress, by combining these ideas with those used to compute the exchange energy in atoms IS] _ The results will be published elsewhere. Correlation holes in molecular systems have been
78
]31 [41
[51 161
L. Hedin and S. Lundqvist, Solid State Phys 23 (1969) 1. M. Berrondo and A. Flores, to be published. M. Berrondo and 0. Goscinski, Chem. Phys. Letters 62 (1979) 31. E.P. Wiiner, Phys. Rev. 46 (1934) 1002; Trans. Faraday Sot. 34 (1938) 678. R. McWeeny, in: The new world of quantum chem-
istry, eds. B. Pullman and R. Parr (Reidel, Dordrecht, 1976) p. 3. 171 CA. Coulson and A.H. Neilson, Proc. Phys. Sot. (London) 78 (1961) 831; R.J. Boyd, Chem. Phys. Letters 44 (1976) 363. PI P--O. Lijwdin, Advan Chem. Phys. 2 (1959) 207. [91 M. Gaudin, J. Gillespie and G. Ripka, NucI. Phys. Al76 (1971) 237. 1101 F. Seitz. Theory of solids (McGraw-Hill, New York, 1940). IllI D. Pmes, The many body problem (Benjamin, New York, 1962). P*: K.S. Singwi, M.P. Tosi, R-H_ Land and A. Sjiilander, Phys. Rev. 176 (1968) 589. 1131 E.A. Hylleraas, Z. Physik 54 (1929) 347_ 1141 M. Abramowitz and LA. Stegun, eds., Handbook of mathematical functions (National Bureau of Standards, Washington, 1967). [I51 R. Colle and 0. Salvetti, Theoret. Chim. Acta 37 (1975) 329; R. CoIle, F. Moscard6, P-L. Riani and 0. Salvetti, Theoret. Chim. Acta 44 (1977) 1. WI M. GeIl-Mann and K_ Brueckner, Phys. Rev. 106 (1957) 364. [17] G.C. Lie and E. Clementi, J. Chem. Phys. 60 (1974) 1275. 1181 P. Gomb&, PseudopotentiaIe (Springer, Berlin, 1967) [19] 1-L. Cooper and C.N.M. Pounder, Theoret. Chim. Acta 47 (1978) 51. [20] R. Resta, Intern. J. Quantum Chem. 14 (1978) 171. [21] G. Doggett. Mol. Phys. 34 (1977) 1739.
CHEMICAL
PHYSICS
THE COULOMB HOLE AND THE CORRELATION FOR THE ELECTRON
GAS AT INTERMEDIATE
LETTERS
I July 1981
ENEiRGY DENSITIES
M. BERRONDO* 1nsri.Ute for Advanced Studies, The Hebrew University of Jermtdem, Jertmdetn, Israel and 0. GOSCINSKI ** Quantum Chemistry Group, Universzty of UppsaLz, S-751 20 Uppsala. Sweden Received 19 January 1981; in fmal form 24 March 1981
An approximation for the correIation energy of the electron gas is derived (employing a direct modelling of the Coulomb hole) for the pair distribution function: ec(rs) = -0.303&, + 1_108) rydberg. It is of the same form as the expressions of Wigner and of McWeeny.
The aim of this work is to fmd an approximate expression for the correlation energy of the homogeneous electron gas derived from the study of the Coulomb hole. Although the model we adopt is very simple, the result compares ftily well with interpolation formulae based on a frequency-dependent dielectric fur&ion formulation [ 1,Z] _ A most important feature of this work however, is the possibility of extending the method to the case of non-homogeneous systems, such as the electrons in an atom or a molecule where the nuclear singula&ies play an essentialrole. This point of view has recently been taken with respect to the exchange energy [3] and the exchange potential [4], and the results thereof seem promising. The resulting expression for the correlation energy in the present work is a simple function of the electron density (or rather of the ubiquitous z-r).It has the same functional form as the venerabIe interpolation formulaintroduced by Wigner. [S] , as welJ 9s the one proposed by McWeeny [6]. TUG Coulomb hole surroundhrg the electron ap* On Xtike from Instit+o
de F&a,
University of Mexico,
Mexico20 D-F., Mexico. *f Support&lby the Swedish Natural Sciences Resekrch coun&L
pears as a consequence of the iuterelectronic repulsion. It is thus a measure of the correlation present in the motion of the electrons. It is a (dynamic) many-electron effect, Its explicit form is obtained 17J comparing the pair distribution function obtained from a correlated wave&n&ion, wi*h that of Hartree-Fock_ For the latter, the wavefunction incfudes the (static) part of the correlation due to the antisymmetry principle, and provides-the standard point from which to measure the correlation [S] _ Since we are interested in the correlation energy, we can look directly at the pair distribution function g(r). This gives the probability density for finding an electron at a distance r from a fixed electron. We obtain the form of the correlated distribution function using a HylleraasLJastrow correlation factor for each pair of electrons_A cluster expansion is then introduced [9], keeping only-the twoelectron clusters.This should yield a good approximation, except in the limit of very h&h de&ties (rs < l), where higher-order clustershave to be considered. In the Hartree approximation, the pair distribution function is equal to unity for a hoinogeneous electron gas. When we take into account the antisymmetry principle, a Fermi hole is formed for electrons with like spin. The distribution function is hence 3 at the origin.
0 009-2614/81/0000-0000/$02.50 0 North-Holland Publishing Company
75
CHEMICAL
Volume 8 1, number 1
The Hartree-Fock q&-)
= 1 -
PHYSICS
approximation yields [lo,1 1]
%VIWFrYI(-‘F4
2
,
zero c (1)
where jl (z) is a spherical Bessel function, and z = KE r, with KE the Fermi momentum. The usual relation between K,, the electronic density, and the parameter rs is given by density = N/V = Kz/3n2 = (3/47r)rsd3_
(2)
The further correlation induced by the Coulomb repulsion should lower g(r) close to the origin. In fact, the simple random phase approximation overestimates this effect [12] , so that g becomes negative at the o&in. We shall include this correlation in the pair distribution function starting from the N-electron wavefunction written as the Hartree-Fock determinant of plane waves times a product of correlation factors. Each factor involves one electron pair at a time and is assumed to depend on the interelectronic distance only:
The two-electron density matrix can then be expressed as a linked cluster expansion [9] . The fust correction to Hartree-Fock involves only fi 2 _ This can easily be understood since fij + 1 for large ‘ii and hence only the pair considered should give a correction to this order. Computing the pair function from this density matrix immediately yieids &)
=gHF(r)lf,,(r)12-
(4)
The next stage consists in using a Hylleraas type of function [13] for the correlation factor fi3. For small r we expand fin powers of r. Asymptot&lly however, this function should go to I_ The correlation energy for intermediate densities is not very sensitive to the specific way it tends to 1. Hence we shall use the following expression for the approximate pair distribution function, ~r)=g,,(r)(a~+a,r+a2~) =gHF
(r
The value of b determines the size of the Coulomb hole and depends on the electronic density. It is hence chosen such that it corresponds to the first 76
LETTERS
(5)
1 July 1981
of the spherical Bessel functionjl(z)
b = 4.493409/K,
= 2.341344rs.
[14] : (6)
Beyond this point, the Bessel function oscillates with a very small (decreasing) amplitude, so that gHF is always very close to unity. According to eq. (S), we have taken the pair function equal to the HartreeFock value for r > b. We now determine the coefficients a0 , al, and a2 in eq. (5) in terms of the cut-off b or, equivalently, of rs_ Let us fust consider the cusp condition for electrons 1 and 2. For r12 = 0, the correlation factor f(r12) must behave as d hf
(‘>lW,=, = $ -
(7)
Hence the pair function is proportional to 1 f r, and eq. (5) yields a0 = al _Secondly, we want the fimction g(r) to be continuous at the cut-off r = b, so that a2 =
(1 - ao)/b2 - so/b.
(8:s)
Lastly, the relation between a0 and the density follows from physical considerations. The Coulomb hole is formed by displacing part of the electronic charge away from the origin [7] when we include the effect of the electronic repulsion_ However, the total charge enclosed within the sphere r < b must remain constant_ This conservation equation thus reads
8)
~gOPdr=~gHF(r)rZclr.
0
0
Substituting eqs. (1) and (5) into eq. (9), we obtain (aa0- 1)D2 + (aoKF)D3
+ (a21@Dq
= 0,
(IO)
where the integrals Dn are simply defined as Dn=j
z” Cl - 4 [0‘&9Y4
21*,
(11)
0
4.493409 being the first zero ofjl(z) 1131. Displayed in table 1 are the required values of Dn, computed numerically_ The shape of the pair distribution function g(r) is given in fig_ 1 for rs equal to 1 and 4. The HartreeFock function is included for comparison. We notice that g(r) remains positive at the origin and has a unit slope. Close to the boundary, however, g(r) becomes larger than 1, an undesirable feature due to the limited
with c =
Volume 81, number 1
CHEMICAL
PHYSICS
Finally, using the values of table 1, together with eqs. (61, (8) and (lo),
Table 1 Values of Dn as defmed in eq. (11)
n
4
e&)
= -0_303/(r,
1 2 3 4
9.02345 28.38122 97.94850 356.72837
At this point it is important to compare with the work by Colle and Salvetti [15] and McWeeny [6]. Colle and Salvetti have actually suggested the choice ffr, J2)
mt
=Eint/V=
Hence,
(N/v)Jg(r) r--l
the correlation
energy
dr.
is &en
(12) by
@(r1,t2)
e,(r,> = (4J$/379 X [(a0 - KID1 + (Qo/$)D2
+
(Q&PJ
-
(14)
a
-SF
Fig. 1. Fair distriiution
functiong(rs)
for rs = 1 md rs = 4.
(15)
(16)
L =e-~29
[l --@(I?)(1
iV/2Qo)],
07)
where R = $(rl + r2), and r12 = r2 - r1 _ The form (17) ensures that the Coulomb hole dies off. The function
which in turn can be expressed in terms of the integrals defmed in eq. (1 l),
+ 1.108) rydberg.
= 1 - @@I ‘51’
with number of terms considered in the expansion in eq. (5). We fmaIiy proceed to compute the correlation energy, using this pair distribution function. The interaction energy (in rydberg) per particle can be expressed dire&y in terms of this function. For the electron gas, we have [I I] E-
1 suly 1981
LETIERS
+ 2.032)
rydberg.
(18)
Even though the coefficients differ somewhat, it is interesting to see that (5) leads perhaps to an understanding of the under&&g reasons for such a form. Our result, valid for intermediate densities, and McWeeny’s coincide for r, = I .2.5. In fig. 2 we have plotted the results obtained with eq. (18). We have included also, for the sake of comparison: (a) Gel&Mann and Brueckner’s result for the high-den+y limit fl& J , (b) Wigner’s interpolation formula [S J , (c) NoziBres and Pines’ interpolation between high- and lay-momentum transfer [2,1 l] , (d) Clementi’s empirical formula for the correlation energy of atoms, taken for a constant density f17, 181 *, and(e) MgWEny*s formula [6]_ The interpolation formula
Volume 81, number 1
CHEMICAL
PHYSICS
LETTERS
1 July 1981
studied in a few cases [19-211 and information about their nature is emerging so that estimates of the correlation energy along the lines presented here will become feasible. The main problem is to ascertam to what extent the correlation hole is affected by the non-homogeneous character of the density. The results of CoUe and SaIvetti [15] are promising in this respect. One of us (MB) would like to thank Professor P.-O. Lijwdin and the members of the Quantum Chemistry Group for their hospitality. References
[II 3. Hubbard, Proc. Roy. Sot. A243 (1957) 336.
121 P. Nozi&es and D. Pmes, Phys. Rev. 111 (1958) 442;
Fig. 2. Correlation energy as a function of rr as predicted by various models. (a) GeIl-Mann and Brueckner high-density limit [ 16 ] ; (6) Wiiner’s interpolation form&r [S ] ; (c) Nozi&e and Pines’ interpolation formula [2,1 l] ; (d) Clementi’s empirical formula [ 17,181; (e) McWeeny’s formula, and our results from eq. (18): “Ours”_
ems, the empirical formula (d) of Clementi et al. [17] gives excellent results. The approximate form which we have found, eq. (15), is appropriate for intermediate densities. In the high-density limit, the pair distribution function (4) should include three and more electron clusters. For the low-density regime, the expansion in powers of rs for the correlation factor (5) should be improved. The region where g(r) is larger than one becomes relevant in this limit. Again, for intermediate and high densities, we do not expect that the precise form of the cut-off should Play a role. In fact, the interaction energy is given by a weighted average over the normalized distribution function (a, first moment), and beyond b this function is almost unity. Rather than implementing the improvements suggested above, we want to look at the case of a homogeneous electron gas as a test case for our method which models the Coulomb hole. The study of non-homogeneous systems, as the electron cl&d in atoms, is in progress, by combining these ideas with those used to compute the exchange energy in atoms IS] _ The results will be published elsewhere. Correlation holes in molecular systems have been
78
]31 [41
[51 161
L. Hedin and S. Lundqvist, Solid State Phys 23 (1969) 1. M. Berrondo and A. Flores, to be published. M. Berrondo and 0. Goscinski, Chem. Phys. Letters 62 (1979) 31. E.P. Wiiner, Phys. Rev. 46 (1934) 1002; Trans. Faraday Sot. 34 (1938) 678. R. McWeeny, in: The new world of quantum chem-
istry, eds. B. Pullman and R. Parr (Reidel, Dordrecht, 1976) p. 3. 171 CA. Coulson and A.H. Neilson, Proc. Phys. Sot. (London) 78 (1961) 831; R.J. Boyd, Chem. Phys. Letters 44 (1976) 363. PI P--O. Lijwdin, Advan Chem. Phys. 2 (1959) 207. [91 M. Gaudin, J. Gillespie and G. Ripka, NucI. Phys. Al76 (1971) 237. 1101 F. Seitz. Theory of solids (McGraw-Hill, New York, 1940). IllI D. Pmes, The many body problem (Benjamin, New York, 1962). P*: K.S. Singwi, M.P. Tosi, R-H_ Land and A. Sjiilander, Phys. Rev. 176 (1968) 589. 1131 E.A. Hylleraas, Z. Physik 54 (1929) 347_ 1141 M. Abramowitz and LA. Stegun, eds., Handbook of mathematical functions (National Bureau of Standards, Washington, 1967). [I51 R. Colle and 0. Salvetti, Theoret. Chim. Acta 37 (1975) 329; R. CoIle, F. Moscard6, P-L. Riani and 0. Salvetti, Theoret. Chim. Acta 44 (1977) 1. WI M. GeIl-Mann and K_ Brueckner, Phys. Rev. 106 (1957) 364. [17] G.C. Lie and E. Clementi, J. Chem. Phys. 60 (1974) 1275. 1181 P. Gomb&, PseudopotentiaIe (Springer, Berlin, 1967) [19] 1-L. Cooper and C.N.M. Pounder, Theoret. Chim. Acta 47 (1978) 51. [20] R. Resta, Intern. J. Quantum Chem. 14 (1978) 171. [21] G. Doggett. Mol. Phys. 34 (1977) 1739.