- Email: [email protected]
A discussion of integral transform radial basis functions
Volume
5,
number
2
CHEMICAL
A DISCUSSION
OF
INTEGRAL
PHYSICS LETTERS
TRANSFORM
L hfarch 1970
RADIAL
BASIS
FUNCTIONS
F. P. BILLINGSLEY II’ and D. D. SHILLADY Uniuersiiy
of Virginia,
ClrarZoltesviZZe,
Virginia
22901,
USA
Received 9 January 1970
An improved transform function is given for the helium atom ground state. This function satisfies the nuclear cusp condition and yields an energy very close to the Hartrce-Fock value. Applications of this transform are discussed.
Several notes have appeared recently [l-4] extolling the virtues of integral transforms as improved radial basis functions for atomic and molecular electronic wave-functions. We wish to report some interesting aspects of a transform function which contributes to the understanding of integral transform bases but does not necessarily improve tractability. We believe this brief study demonstrates that a single transform function can give excellent energy values and still satisfy the cusp constraint at the nucleus. The particular transform we have chosen is more nearly optimum in the Hartree-Fock sense than those previously reported but is functionally related to the radial dependence tested’by Pruess on the hydrogen atom [5]. In keeping with previous terminology we will call his type of function a SIaterTransform-Pruess function (STP) and the extended transform STPX. We have previously shown for integer n [3], and Somorjai has extended the argument to noninteger r41, that
I*.(4 4
M, % a)> =
(2x-l) (2~2-2) (2,~-3)a(~~-~)
Gw!
gives a very good approximation carried out, we obtain
f (8,4)Y@-~) 1 YLM -(11-l)! s”0
to the Hartree-Fock
s,”exp[-b-+a)x]
xmdx
exp[-XY]
x(~‘-~) exp[-mx]
dx . (1)
radial function of helium. When the integration is
=
(2)
Using the function obtained from eqs. (1) and (2), Somorjai obtained a minimum energy of -2.860 555 91 au [4] when n and cr were both varied freely. However, when the cusp condition
is invoked, the energy was made worse, i.e., -2.859 155 44 au [4]. Zung and Parr [6] also noticed a similar worsening of the energy expectation value when the cusp condition was applied to a linear combination of OS and IS exponential functions. With a two-parameter function, they obtained an energy of -2.859207 au with the cusp condition and a value of -2.861673 au when four parameters were used and the cusp constraint was ignored In this note we wish to indicate one way in which STP functions can be improved to give good energies and still satisfy the cusp condition of eq. (3). Although the functional form is not especially tractable, it does illustrate the relationship between Slaier-Type-Orbital (STO) and STP functions, We began with one of Hastings’ [‘7] approximations for the exponential function emx = 1/[a+bx+cx2+d$]4 a=l, C+NASA
Predoctorai
E = 0.2507213,
f 0,00023,
c = 0.029 273 2,
d = 0.003 827 8.
(4)
Fellow.
97
Volume 5. number 2
CHEMICAL
:
.Using a modified form of Simpson’s
PHYSICS LETTERS
rule [8] to perform
1 M&b
the integrations,
this function,
1970
when normalized,
,yielded an energy of -0.499 99662 au for the ground state of the hydrogen atom. This served as a check on our small computer prOgram. In eq. (5) we see that the Fiastings’ approximation is a special case of a more general integral transform
Thus we.have used a function (STPX) of the form AN-11
I-L&e,
b(N. L. Al, a, 6, c, d)> = 9 La+br+C9+d934
9)
= bTPX).
(W
We restricted (nz+l) to 4 in order to lower the number of param eters and also to permit identification with the Hastings’ approxim ation more directly; q is a normalization constant determined by numerical integration After applying eq. (3) to eq. (6). with N = 1, one obtains the cusp constraint
1
$I~ST~)
b Z -=-SE a 4
-4b
= -2 =--;
.
We then wrote a small program to compute the energy of the ground state of the helium atom in atomic units. For the two-electron repulsion integral, we used the B-point algorithm for two-dtmensional integration, CsTPX (R(STPX) = - (STPX (v2 (STPX) - ZZ(STPX
(l/r
PTPX)
+ (STPX(l)STF’X(B)
11,+12 [STPX(l)STPX(2)) (8
(ref. [9]. p. 892, eq. 25.4.62) over a grid of 225 points which we found yielded nine significant figures when tested against finer grid systems. When we varied (1, b. c, and d freely, using Ransil’s [lo] algorithm, we obtained an energy of -2.86166452 au whi,ch differs from the Hartree-Fock value of Wilson and Lindsay [ll J (-2.861660) only in the fifth place. However, we were mainly interested in the cuspconstrained energy. so we used eq. (7) to eliminate one parameter and varied only b, c, and d, requiring that n = 2b. This procedure yielded an energy of -2.861554 74 au after eight cycles of Ftansil’s minimization technique_ This energy deviates from the Hartree-Fock value in the fourth place and is better than the fully optimized STP [4]. While these energies are not quite as good as the fully optimized fnncKon of Zung and Parr [6], the constrained value Is somewhat better than that of tho fully cons-ed function (2 = 2 and 5’ = 2t) of Zung and Parr. Since our constrained function has one more parameter (three) th&n the constrained function of Zung and Parr (two), the net effect is au expected improvement in the energy; but the noteworthy feature is that we have obtained a single analytical function which fully satisfies the cusp constraint and still yields an acellent value of the single-determinental energy. The constraint effect also seems less adverse for the STPX than for other fxnctions. Although eq. (5) is interesKng, one soon finds that even one-electror. operators are not easily handled due to unwieldy integrmds in molecular integrals. Furthermore, a basis set requiring four parameters per function has little computational advantage if many electron systems are to be treated_ Thus we have no illusions about the easy application of the STPX functions as general basis sets. .However, we can use this STPX calculation of the helium ground state energy to learn something about the more tractable STP [3,4] functions. We are presently working on applying the STP functions to molecular ,calculations and if we fail to develop analytical formulas for all the integrals, we can fall back to the brute force approach used by Wahl [12]. Thus there is hope of reducing basis set size with a concomitant decrease in the number of necessary molecular integrals without sacrificing accuracy. However, whatever analytical dsficlencies the STP functions may have shudld be characterized, and that is what we are trying to do here. In table 1 we compare tie radial characteristics of several helium wavefunctions. These functions. are all single-determinental functions and therefore should converge to the Hartree-Fock values as the energy is-minimized [13]. We first notice that the single scaled exponential (STO) function is poor because the electrons are concentrated too near the nucleus. This. is indicated-by the fact that (r), ti2), @3), aud g41 are all smaller than the corresponding Hartree-Fock values. Next we note that, while both [email protected] give reasonable agreement for (r) , the higher moments get prijgressively Torae. The STpX fuucUon is conside~rably l&roved o&r the 8TP functLons,- dthough the G-4 agyement is -98..
-_-.:_’
:
-:-
.. : .:
,,
..
.. ‘,
:
‘,
.,
..‘;
. .
.. ..
.:
.:‘. ..-’
-*.
. ..
‘.
_.
Volume 5. number 2
L March 1970
CEEMICAL PHYSICSLETTERS Table 1
Hartree-Focka)
bunk and Parrb) Unrastricted
(r>
0.927 2
0.927 0
(12)
1.1848
1.183 0
(4)
1.940 6
1.9317
k4>
3.8079
-@>
2.861690
n)
b) c) a) e) f)
STPX C) Conatralnad
3TP d) Unreetrictad
3TP e) Collstra.ined
Scaled 9 ExponentiaI
0.927 762 39
0.930202 17
0.935 913 04
o.eess
1.189 177 29
1.209 646 08
1.245 069 15
1.053 5
1.965 904 246
2.06650573
2.21689374
1.560-i
3.0476
4.013486993
4.46388365
5.08729273
2.7746
2.861673
2.m155474
2.060555Ql
2.65Q15544
e.64765625
Roothaa. Sachs and Weiss ST0 expansion [ltl]. Zung and Parr linear combination of 03 and 19 IS]. a = 3.793 430 5512. 6 = 1.890 715 177 56,’ c = 0.190 541644 57. d = 0.052 252 494 44. n = 12.811098 0. (Y = 6.704 3379 [4]. n = 10.068 288 20. d! = 5.034 144 1 [a]. Zung and Parr single scaled ST0 IS].
not quite as good as the Zung and Parr function. ComputaKonally, the Zung and Parr funcKon is at a
disadvantage however, since it is a linear combination of two terms which would generate times as many integrals as would the STPX single function This estimate is based on the LCAO scheme where the number of required integrals is proportional to the fourth power of basis functlans. The main point to be learned from this study is that, whilst exponential functions @TO) Hartree-Fock function from a situation where the electron density is too near the nucleus, STPX
functions
approach
from
a situation
where
Lhe electron
density
is too far
away from
about 16 Roothaau
of the number approach the the STP and the nucleus.
Thus the STPX function represents an improvement over the STP funcKons because it has the necessary flexibilitp to reduce the value of the function for large radii and still meet the cusp condition, while cusp-forcing the STP function worsens that function at large radii. During this study, we have learned some numerical techniques which may aid us in obtaining values of two-center repulsion and exchange integrals in either STP or STPX basis sets. Either would accomplish
our goal
of obtaining
accurate
basis
sets
which
are
truly
“minimum”
in that only one functioa
per
orbital is required. While the STPX functions are more accurate, the large number of parameters they require may prohibit their use unless a direct numerical integration approach is used simU= to that described by Wahl [12]. We have already found that two-center one-electron integrals can eastly be computed by numerical integration of an auxilliary function similar to the C-function of Roothaan [14]. Since the integration of the W-function is performed numerically (about 1.1 seconds on a Burroughs B-5500),
(9) the STPX functions could be treated similarly. In parKcular, it should be noted that only the two-center repulsion, kinek energy and potential energy Lntegrals need be calculated besides all one-center Megrals. This is due to the fact that tbe method of Billingsley and Bloor [15] can be used to compute the difficult two-center exchange and all 3- and 4-center two-electron integrals by an expansion technique which, although an approximation, is very accurate and has reproduced ab inltio values to within less tbsn 0.0003 atomic units. Thus, if an efficient numerical technique can be found for the two-center repulsion iutegral, one could perform molecular calculations in a minimum basis set which seems to promise near Hartree-Fock accuracy. Those already in possession of an SCF program similar to that of Wahl[12], which depends quite heavily on numerical integration, should be able to convert their programs with relative ease to this more efficient baais set (STPX or STP) and thereby greatly reduce the number of integrals required for given accuracy. The obvious way to get around the difficulty of the many parameters involved in the STPX fkcKo* is to fit them direct1y.k the Hartree-Fock orbit.& in a least squares sense or by maximizing the overlap. .At tbe very least, the 3TPX funcKons could be used to store numerical Harbee-Fock orbitala in compact form; per&pa by udng even more terms in the exponent of the weighting function-in eq, (5). -99
Volume
5. number
2
CHEMICAL
PHYSICS
LETTERS
1 March
We will continue to work toward a many-center capability here and we hope that the efficiency accurate minimum basis set is an advantage worthy of continued interest in the future.
1970
of an
We wish to thank Dr. Fred Richardson, at the University of Virginia, and Dr. John E. Bloor, at the University of Tennessee, for encouragement during this study. We also wish to thank the National Science Foundation for financial support administered through the Center for Advanced Studies, University of Virginia. under subgrant 3760-2195. REFERENCES [l] [Z] [3] ]-I] [5] !6] [7] [a] [9] [lo] [ll] [12] (131 [14] [15] [16]
100
R. L. Somorjai. Chcm. Phys. Letters 2 (1968) 399. D. D. Shillady. Chem. Phys. Letters 3 (1969) 17. D. D. Shillady. Chem. Phys. Letters 3 (1969) 104. R. L. Somorjai. Chem. Phys. Letters 3 (1969) 395. H.Pruess. Z.Naturforsch. 19 (1958) 439. J.T. Zung and R.G. Parr. J. Chem. Phys. 41 (1964) 2888. C. Hastings. Approsimations for digital computers (Princeton University Press, Princeton, 1955). W. Id. McKeemann. Commun. A.C.M. 5 (19G2) 604. 31. Abramowitz and I. A. Steyn. Handbook of mathematical functions (Dover Publications, New York, B J. Ransil. Rev. Mod. Phys. 32 (1960) 239. W. S. Wilson and R. B. Lindsay, Phys. Rev. 47 (1935) 661. A. C. Wahl and R. H. Land, J. Chem. Phys. 50 (1969) 8725. J. C. Slater. Phys. Rev. 35 (1930) 210. K. Ruedenberg. C. C. J. Roothaan and W. Jnunzemis. J. Chem. Phys. 24 (1956) 201. F. P. Billingsfey and J. E. Bloor. Chem. Phvs. Letters 4 (1969) 48. C. C. J_ Roothann, L M. Sachs and A. W. Weiss. Rev. Mod. Phys. 32 (1960) 186.
1965).
5,
number
2
CHEMICAL
A DISCUSSION
OF
INTEGRAL
PHYSICS LETTERS
TRANSFORM
L hfarch 1970
RADIAL
BASIS
FUNCTIONS
F. P. BILLINGSLEY II’ and D. D. SHILLADY Uniuersiiy
of Virginia,
ClrarZoltesviZZe,
Virginia
22901,
USA
Received 9 January 1970
An improved transform function is given for the helium atom ground state. This function satisfies the nuclear cusp condition and yields an energy very close to the Hartrce-Fock value. Applications of this transform are discussed.
Several notes have appeared recently [l-4] extolling the virtues of integral transforms as improved radial basis functions for atomic and molecular electronic wave-functions. We wish to report some interesting aspects of a transform function which contributes to the understanding of integral transform bases but does not necessarily improve tractability. We believe this brief study demonstrates that a single transform function can give excellent energy values and still satisfy the cusp constraint at the nucleus. The particular transform we have chosen is more nearly optimum in the Hartree-Fock sense than those previously reported but is functionally related to the radial dependence tested’by Pruess on the hydrogen atom [5]. In keeping with previous terminology we will call his type of function a SIaterTransform-Pruess function (STP) and the extended transform STPX. We have previously shown for integer n [3], and Somorjai has extended the argument to noninteger r41, that
I*.(4 4
M, % a)> =
(2x-l) (2~2-2) (2,~-3)a(~~-~)
Gw!
gives a very good approximation carried out, we obtain
f (8,4)Y@-~) 1 YLM -(11-l)! s”0
to the Hartree-Fock
s,”exp[-b-+a)x]
xmdx
exp[-XY]
x(~‘-~) exp[-mx]
dx . (1)
radial function of helium. When the integration is
=
(2)
Using the function obtained from eqs. (1) and (2), Somorjai obtained a minimum energy of -2.860 555 91 au [4] when n and cr were both varied freely. However, when the cusp condition
is invoked, the energy was made worse, i.e., -2.859 155 44 au [4]. Zung and Parr [6] also noticed a similar worsening of the energy expectation value when the cusp condition was applied to a linear combination of OS and IS exponential functions. With a two-parameter function, they obtained an energy of -2.859207 au with the cusp condition and a value of -2.861673 au when four parameters were used and the cusp constraint was ignored In this note we wish to indicate one way in which STP functions can be improved to give good energies and still satisfy the cusp condition of eq. (3). Although the functional form is not especially tractable, it does illustrate the relationship between Slaier-Type-Orbital (STO) and STP functions, We began with one of Hastings’ [‘7] approximations for the exponential function emx = 1/[a+bx+cx2+d$]4 a=l, C+NASA
Predoctorai
E = 0.2507213,
f 0,00023,
c = 0.029 273 2,
d = 0.003 827 8.
(4)
Fellow.
97
Volume 5. number 2
CHEMICAL
:
.Using a modified form of Simpson’s
PHYSICS LETTERS
rule [8] to perform
1 M&b
the integrations,
this function,
1970
when normalized,
,yielded an energy of -0.499 99662 au for the ground state of the hydrogen atom. This served as a check on our small computer prOgram. In eq. (5) we see that the Fiastings’ approximation is a special case of a more general integral transform
Thus we.have used a function (STPX) of the form AN-11
I-L&e,
b(N. L. Al, a, 6, c, d)> = 9 La+br+C9+d934
9)
= bTPX).
(W
We restricted (nz+l) to 4 in order to lower the number of param eters and also to permit identification with the Hastings’ approxim ation more directly; q is a normalization constant determined by numerical integration After applying eq. (3) to eq. (6). with N = 1, one obtains the cusp constraint
1
$I~ST~)
b Z -=-SE a 4
-4b
= -2 =--;
.
We then wrote a small program to compute the energy of the ground state of the helium atom in atomic units. For the two-electron repulsion integral, we used the B-point algorithm for two-dtmensional integration, CsTPX (R(STPX) = - (STPX (v2 (STPX) - ZZ(STPX
(l/r
PTPX)
+ (STPX(l)STF’X(B)
11,+12 [STPX(l)STPX(2)) (8
(ref. [9]. p. 892, eq. 25.4.62) over a grid of 225 points which we found yielded nine significant figures when tested against finer grid systems. When we varied (1, b. c, and d freely, using Ransil’s [lo] algorithm, we obtained an energy of -2.86166452 au whi,ch differs from the Hartree-Fock value of Wilson and Lindsay [ll J (-2.861660) only in the fifth place. However, we were mainly interested in the cuspconstrained energy. so we used eq. (7) to eliminate one parameter and varied only b, c, and d, requiring that n = 2b. This procedure yielded an energy of -2.861554 74 au after eight cycles of Ftansil’s minimization technique_ This energy deviates from the Hartree-Fock value in the fourth place and is better than the fully optimized STP [4]. While these energies are not quite as good as the fully optimized fnncKon of Zung and Parr [6], the constrained value Is somewhat better than that of tho fully cons-ed function (2 = 2 and 5’ = 2t) of Zung and Parr. Since our constrained function has one more parameter (three) th&n the constrained function of Zung and Parr (two), the net effect is au expected improvement in the energy; but the noteworthy feature is that we have obtained a single analytical function which fully satisfies the cusp constraint and still yields an acellent value of the single-determinental energy. The constraint effect also seems less adverse for the STPX than for other fxnctions. Although eq. (5) is interesKng, one soon finds that even one-electror. operators are not easily handled due to unwieldy integrmds in molecular integrals. Furthermore, a basis set requiring four parameters per function has little computational advantage if many electron systems are to be treated_ Thus we have no illusions about the easy application of the STPX functions as general basis sets. .However, we can use this STPX calculation of the helium ground state energy to learn something about the more tractable STP [3,4] functions. We are presently working on applying the STP functions to molecular ,calculations and if we fail to develop analytical formulas for all the integrals, we can fall back to the brute force approach used by Wahl [12]. Thus there is hope of reducing basis set size with a concomitant decrease in the number of necessary molecular integrals without sacrificing accuracy. However, whatever analytical dsficlencies the STP functions may have shudld be characterized, and that is what we are trying to do here. In table 1 we compare tie radial characteristics of several helium wavefunctions. These functions. are all single-determinental functions and therefore should converge to the Hartree-Fock values as the energy is-minimized [13]. We first notice that the single scaled exponential (STO) function is poor because the electrons are concentrated too near the nucleus. This. is indicated-by the fact that (r), ti2), @3), aud g41 are all smaller than the corresponding Hartree-Fock values. Next we note that, while both [email protected] give reasonable agreement for (r) , the higher moments get prijgressively Torae. The STpX fuucUon is conside~rably l&roved o&r the 8TP functLons,- dthough the G-4 agyement is -98..
-_-.:_’
:
-:-
.. : .:
,,
..
.. ‘,
:
‘,
.,
..‘;
. .
.. ..
.:
.:‘. ..-’
-*.
. ..
‘.
_.
Volume 5. number 2
L March 1970
CEEMICAL PHYSICSLETTERS Table 1
Hartree-Focka)
bunk and Parrb) Unrastricted
(r>
0.927 2
0.927 0
(12)
1.1848
1.183 0
(4)
1.940 6
1.9317
k4>
3.8079
-@>
2.861690
n)
b) c) a) e) f)
STPX C) Conatralnad
3TP d) Unreetrictad
3TP e) Collstra.ined
Scaled 9 ExponentiaI
0.927 762 39
0.930202 17
0.935 913 04
o.eess
1.189 177 29
1.209 646 08
1.245 069 15
1.053 5
1.965 904 246
2.06650573
2.21689374
1.560-i
3.0476
4.013486993
4.46388365
5.08729273
2.7746
2.861673
2.m155474
2.060555Ql
2.65Q15544
e.64765625
Roothaa. Sachs and Weiss ST0 expansion [ltl]. Zung and Parr linear combination of 03 and 19 IS]. a = 3.793 430 5512. 6 = 1.890 715 177 56,’ c = 0.190 541644 57. d = 0.052 252 494 44. n = 12.811098 0. (Y = 6.704 3379 [4]. n = 10.068 288 20. d! = 5.034 144 1 [a]. Zung and Parr single scaled ST0 IS].
not quite as good as the Zung and Parr function. ComputaKonally, the Zung and Parr funcKon is at a
disadvantage however, since it is a linear combination of two terms which would generate times as many integrals as would the STPX single function This estimate is based on the LCAO scheme where the number of required integrals is proportional to the fourth power of basis functlans. The main point to be learned from this study is that, whilst exponential functions @TO) Hartree-Fock function from a situation where the electron density is too near the nucleus, STPX
functions
approach
from
a situation
where
Lhe electron
density
is too far
away from
about 16 Roothaau
of the number approach the the STP and the nucleus.
Thus the STPX function represents an improvement over the STP funcKons because it has the necessary flexibilitp to reduce the value of the function for large radii and still meet the cusp condition, while cusp-forcing the STP function worsens that function at large radii. During this study, we have learned some numerical techniques which may aid us in obtaining values of two-center repulsion and exchange integrals in either STP or STPX basis sets. Either would accomplish
our goal
of obtaining
accurate
basis
sets
which
are
truly
“minimum”
in that only one functioa
per
orbital is required. While the STPX functions are more accurate, the large number of parameters they require may prohibit their use unless a direct numerical integration approach is used simU= to that described by Wahl [12]. We have already found that two-center one-electron integrals can eastly be computed by numerical integration of an auxilliary function similar to the C-function of Roothaan [14]. Since the integration of the W-function is performed numerically (about 1.1 seconds on a Burroughs B-5500),
(9) the STPX functions could be treated similarly. In parKcular, it should be noted that only the two-center repulsion, kinek energy and potential energy Lntegrals need be calculated besides all one-center Megrals. This is due to the fact that tbe method of Billingsley and Bloor [15] can be used to compute the difficult two-center exchange and all 3- and 4-center two-electron integrals by an expansion technique which, although an approximation, is very accurate and has reproduced ab inltio values to within less tbsn 0.0003 atomic units. Thus, if an efficient numerical technique can be found for the two-center repulsion iutegral, one could perform molecular calculations in a minimum basis set which seems to promise near Hartree-Fock accuracy. Those already in possession of an SCF program similar to that of Wahl[12], which depends quite heavily on numerical integration, should be able to convert their programs with relative ease to this more efficient baais set (STPX or STP) and thereby greatly reduce the number of integrals required for given accuracy. The obvious way to get around the difficulty of the many parameters involved in the STPX fkcKo* is to fit them direct1y.k the Hartree-Fock orbit.& in a least squares sense or by maximizing the overlap. .At tbe very least, the 3TPX funcKons could be used to store numerical Harbee-Fock orbitala in compact form; per&pa by udng even more terms in the exponent of the weighting function-in eq, (5). -99
Volume
5. number
2
CHEMICAL
PHYSICS
LETTERS
1 March
We will continue to work toward a many-center capability here and we hope that the efficiency accurate minimum basis set is an advantage worthy of continued interest in the future.
1970
of an
We wish to thank Dr. Fred Richardson, at the University of Virginia, and Dr. John E. Bloor, at the University of Tennessee, for encouragement during this study. We also wish to thank the National Science Foundation for financial support administered through the Center for Advanced Studies, University of Virginia. under subgrant 3760-2195. REFERENCES [l] [Z] [3] ]-I] [5] !6] [7] [a] [9] [lo] [ll] [12] (131 [14] [15] [16]
100
R. L. Somorjai. Chcm. Phys. Letters 2 (1968) 399. D. D. Shillady. Chem. Phys. Letters 3 (1969) 17. D. D. Shillady. Chem. Phys. Letters 3 (1969) 104. R. L. Somorjai. Chem. Phys. Letters 3 (1969) 395. H.Pruess. Z.Naturforsch. 19 (1958) 439. J.T. Zung and R.G. Parr. J. Chem. Phys. 41 (1964) 2888. C. Hastings. Approsimations for digital computers (Princeton University Press, Princeton, 1955). W. Id. McKeemann. Commun. A.C.M. 5 (19G2) 604. 31. Abramowitz and I. A. Steyn. Handbook of mathematical functions (Dover Publications, New York, B J. Ransil. Rev. Mod. Phys. 32 (1960) 239. W. S. Wilson and R. B. Lindsay, Phys. Rev. 47 (1935) 661. A. C. Wahl and R. H. Land, J. Chem. Phys. 50 (1969) 8725. J. C. Slater. Phys. Rev. 35 (1930) 210. K. Ruedenberg. C. C. J. Roothaan and W. Jnunzemis. J. Chem. Phys. 24 (1956) 201. F. P. Billingsfey and J. E. Bloor. Chem. Phvs. Letters 4 (1969) 48. C. C. J_ Roothann, L M. Sachs and A. W. Weiss. Rev. Mod. Phys. 32 (1960) 186.
1965).