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Slater transform functions. Another class of approximate wave functions
Volume
3. number
CHEMICAL
1
SLATER ANOTHER
PHYSICS
LETTERS
TRANSFORM
CLASS
OF
January
1969
FUNCTIONS.
APPROXIMATE
WAVE
D. D. SHILLADY
FUNCTLONS
*
Debartment of Chemistry, University of Virginia, Charlottesville, Virginia 22903, USA Received
16 December
1968
The construction form of a weighting
of a new class of approximate wave functions is described using the Laplace transfunction similar to a Slater Function. The proposed Slater Transform Functions (STF) are then tested variationally on the hydrogen atom ground state.
Recently in this journal [l] a discussion of a new type of approximate wave function pointed out that the two-parameter Hulthen functions used by Parr and coworkers [Z] can be considered as an infinite number of equally weighted screened 1s functions with orbital exponents ranging continuously in a variationally optimized interval
Somorjai suggested a more general form amenable to well known Laplace transform suggested the addition of an orbital exponent weighting function. Thus
F(r) = s,” e-xyftcy, where (Y and /3 are again variationally optimized constants. filnction could well be a sliding gaussian such as
operations 131; he
P, x) dx, The most conceptionally
desirable
envelope
f(a, P, x) = exp I-(x - P)2/4~2]f
(3)
but this leads, unfortunately, to an expression with a parameter-dependent might be avoided with some other envelope function, thus F(Y) = Lw eexy e -[!x -
,+/4a2J
&u
=
2a
e[&2 - P-1
limit of integration which
SW_e-x2ti_
ray-wwl
(4)
It is the purpose of this note to suggest another envelope function, so as to provide an infinite numType Orbitals (STO) with a parametric weighting of the orbital exponents_ It has Long been lrnown that a fewer number of ST0 functions are required as basis functions than for Gaussian Type Orbitals (GTO), but many calculations are more easily performed with a large number of GTO functions. It would obviously be desirable to use functions related to the central field eigenfunctions and stiIL have fewer functions required for an adequate basis. The GTO functions offer many computational advantages, but an undesirable fe8ture of GTO functions is their behavior near r = 0. As Schwartz [4] has shown very clearly, even a very good GTO function behaves poorly near the nucleus. Clearly a function should have a better representation of the cusp at Y = C! than do GTO functions. The Laplace transform functions suggested by Somorjai can represent the cusp, but we suggest here such a transform with a different weighting function, thus ber of Slater
s(n,a,x) * N. I. H. Predoctoral
Fellow
=zP
e-p%,
n 2.1.
(5)
1968/69. 17
Volume 3, number 1
CHEMICAL PHYSICS LETTERS
This weighting function is suggested by the Slater terest as follows:
Type
Orbitals
January 1969
themselves
and have properties
of in-
s(n, (Y, 0) = 0,
v 72,o!,
(64
s(72, a!, 4
V&a,,
(W
= 0,
Iim s(72,aopt, x) = 6(x- (Zme2/ti2)) , n-+-J
(64
where aopt is defined in eq. (14). Note that if iz is increased and./or (Y is decreased, the maximum of the envelope is shifted to larger values of X, which corresponds to more heavily weighting the large orbital exponent values in the radial function. We shall also show that 7t and ci --dlt: not independent of each other when the-minimization of energy is carried out. Thus we form the radial function.
R(n.
(Y,
7) =
irnem%'*s(n,
CY,
x) dr
=
Jrn e -(‘Y+ cY)Y %7Zdx
=
(7)
After normalization of the radial function we form the new Slater Transform cluding the angular momentum eigenfunctions,
Function (STF) by in-
As yet the multicenter integrals involving these functions have not been evaluated, but this note shows the results of this type of function when applied to the standard test case, namely the hydrogen atom, so that this work may be extended even as it extends that of Somorjai. Using the following Hamiltonian:
we will investigate
/+(I, 0, O,)z,
(Y)) by means of the variation
(+(1,0,0,?2,cY)I9&&(1,0,0,?2,cu))
= (4) Z>lo!
theorem.
[(2”-t()2(;;;)3)(pz)]
We find - (g)
[qq
(10)
Note that I @(l, 0. 0, ?Z,(Y)) = J-2 (212 1)(21Z-2j(212-3)(u(2”‘3) whence
-- 1 (r+(Y)
1 ?LGG9
(11)
(12) and (13) Setting
yields an optimum value for (Y, thus (14)
18
January 1969
CHEMICAL PHYSICS LETTERS
Volume 3, number 1
Table 1 Variationally determined energies of the hydrogen atom with STF orbitals Smin (a.u.)
n
2
-0.312 500
3
-0.437 500
6
-0.487500
10
-0.495833
20
-0.499013
100
-0.499962 _-
Using aopt in the energy expression Emin
we find
=(~I~I~)I,,,,=-[~)-6,(,3_,,
] (9).
(15)
Thus as n - 00, Emin --t -$ atomic units, We list some selected values of Emin for various values of n in table 1. We should remember that the best single gaussian function yields an energy of about -0.424 a-u. [5] whereas a single STF with ?Z= 3 yields an energy of -0.4375 a.u. and only one STF with 12= LOO yields an energy of -0.499 962 12 a.u. The main point c;f interest is the accuracy obtained by a single analytical expression which may be more easily treated in the multicenter case than are STff functions. This remains to be shown and work will continue here as time permits. However, the prize to be gained in improved behavior at the origin for these STF orbitals may be sufficient incentive to find the multicenter expressions. The value of IL could then be determined for arbitrary accuracy of each orbital and (Y evaluated from eq. (14). We should note that by using an infinite number of exponential functions, we have tried to find the ground state of the hydrogen atom which is actually a single exponential function. Hence we must use the limiting form of the envelope function which is a Dirac delta function [S]. However, in the case of Larger atoms with many electrons, Hartree-Fock functions can require two or more zeta values using exponential functions [7] or many more gaussian functions for proper representation of radial orbitam [a]. Using STF orbitals it may not be necessary to use as many basis functions, since for finite n and optimum CY values, the envelope function can select a continuous range of zeta values so as to require perhaps onLy one analytical expression per one-electron orbital. This is especially important for future work in accurate electronic calculations for large metal complexes, where one faces not only evaluation of multicenter integrals, but also a tremendous increase in the number of such multicenter integraIs for very large basis sets. Even so-called “minimum basis sets” can be in the hundreds quite easiIy for some orgar.ometallic compounds, and for this reason it is highly desirable to have efficient one-electron functions so as to avoid linear combinations as much as possible. It is hoped that the STF orbi+als can meet this need in the future.
Acknowledgement should be given to Dr. J. E, Bloor for allowing the aclthor some time for his project and to the author’s wife for typing the manuscript.
BEFEBENCES R. L. Somorjai, Chem. Phys. Letters 2 (1968) 399. R. G. Parr and J. H. Weare, Prog. Theoret. Phys. (Kyoto) 36 (1966) 854. D. W. Widder, The Laplace Transform (Princeton University Press, 1946). M. E. Schwartz, Chem. Phys. Letters 1 (1967)269. C. A. Coulson, Valence. 2nd ed. (Oxford University Press, 1961) p. 60. G. Goertzal and N. Tralli, Some Mathematical Methods of Physics (McGraw-Hill, H. Basch and H. B. Gray, Theoret. Chim. Acta 4 (1966) ?67. E. Clementi. Chem. Rev. 68 (1968) 341.
1960) p. 98.
19
3. number
CHEMICAL
1
SLATER ANOTHER
PHYSICS
LETTERS
TRANSFORM
CLASS
OF
January
1969
FUNCTIONS.
APPROXIMATE
WAVE
D. D. SHILLADY
FUNCTLONS
*
Debartment of Chemistry, University of Virginia, Charlottesville, Virginia 22903, USA Received
16 December
1968
The construction form of a weighting
of a new class of approximate wave functions is described using the Laplace transfunction similar to a Slater Function. The proposed Slater Transform Functions (STF) are then tested variationally on the hydrogen atom ground state.
Recently in this journal [l] a discussion of a new type of approximate wave function pointed out that the two-parameter Hulthen functions used by Parr and coworkers [Z] can be considered as an infinite number of equally weighted screened 1s functions with orbital exponents ranging continuously in a variationally optimized interval
Somorjai suggested a more general form amenable to well known Laplace transform suggested the addition of an orbital exponent weighting function. Thus
F(r) = s,” e-xyftcy, where (Y and /3 are again variationally optimized constants. filnction could well be a sliding gaussian such as
operations 131; he
P, x) dx, The most conceptionally
desirable
envelope
f(a, P, x) = exp I-(x - P)2/4~2]f
(3)
but this leads, unfortunately, to an expression with a parameter-dependent might be avoided with some other envelope function, thus F(Y) = Lw eexy e -[!x -
,+/4a2J
&u
=
2a
e[&2 - P-1
limit of integration which
SW_e-x2ti_
ray-wwl
(4)
It is the purpose of this note to suggest another envelope function, so as to provide an infinite numType Orbitals (STO) with a parametric weighting of the orbital exponents_ It has Long been lrnown that a fewer number of ST0 functions are required as basis functions than for Gaussian Type Orbitals (GTO), but many calculations are more easily performed with a large number of GTO functions. It would obviously be desirable to use functions related to the central field eigenfunctions and stiIL have fewer functions required for an adequate basis. The GTO functions offer many computational advantages, but an undesirable fe8ture of GTO functions is their behavior near r = 0. As Schwartz [4] has shown very clearly, even a very good GTO function behaves poorly near the nucleus. Clearly a function should have a better representation of the cusp at Y = C! than do GTO functions. The Laplace transform functions suggested by Somorjai can represent the cusp, but we suggest here such a transform with a different weighting function, thus ber of Slater
s(n,a,x) * N. I. H. Predoctoral
Fellow
=zP
e-p%,
n 2.1.
(5)
1968/69. 17
Volume 3, number 1
CHEMICAL PHYSICS LETTERS
This weighting function is suggested by the Slater terest as follows:
Type
Orbitals
January 1969
themselves
and have properties
of in-
s(n, (Y, 0) = 0,
v 72,o!,
(64
s(72, a!, 4
V&a,,
(W
= 0,
Iim s(72,aopt, x) = 6(x- (Zme2/ti2)) , n-+-J
(64
where aopt is defined in eq. (14). Note that if iz is increased and./or (Y is decreased, the maximum of the envelope is shifted to larger values of X, which corresponds to more heavily weighting the large orbital exponent values in the radial function. We shall also show that 7t and ci --dlt: not independent of each other when the-minimization of energy is carried out. Thus we form the radial function.
R(n.
(Y,
7) =
irnem%'*s(n,
CY,
x) dr
=
Jrn e -(‘Y+ cY)Y %7Zdx
=
(7)
After normalization of the radial function we form the new Slater Transform cluding the angular momentum eigenfunctions,
Function (STF) by in-
As yet the multicenter integrals involving these functions have not been evaluated, but this note shows the results of this type of function when applied to the standard test case, namely the hydrogen atom, so that this work may be extended even as it extends that of Somorjai. Using the following Hamiltonian:
we will investigate
/+(I, 0, O,)z,
(Y)) by means of the variation
(+(1,0,0,?2,cY)I9&&(1,0,0,?2,cu))
= (4) Z>lo!
theorem.
[(2”-t()2(;;;)3)(pz)]
We find - (g)
(10)
Note that I @(l, 0. 0, ?Z,(Y)) = J-2 (212 1)(21Z-2j(212-3)(u(2”‘3) whence
-- 1 (r+(Y)
1 ?LGG9
(11)
(12) and (13) Setting
yields an optimum value for (Y, thus (14)
18
January 1969
CHEMICAL PHYSICS LETTERS
Volume 3, number 1
Table 1 Variationally determined energies of the hydrogen atom with STF orbitals Smin (a.u.)
n
2
-0.312 500
3
-0.437 500
6
-0.487500
10
-0.495833
20
-0.499013
100
-0.499962 _-
Using aopt in the energy expression Emin
we find
=(~I~I~)I,,,,=-[~)-6,(,3_,,
] (9).
(15)
Thus as n - 00, Emin --t -$ atomic units, We list some selected values of Emin for various values of n in table 1. We should remember that the best single gaussian function yields an energy of about -0.424 a-u. [5] whereas a single STF with ?Z= 3 yields an energy of -0.4375 a.u. and only one STF with 12= LOO yields an energy of -0.499 962 12 a.u. The main point c;f interest is the accuracy obtained by a single analytical expression which may be more easily treated in the multicenter case than are STff functions. This remains to be shown and work will continue here as time permits. However, the prize to be gained in improved behavior at the origin for these STF orbitals may be sufficient incentive to find the multicenter expressions. The value of IL could then be determined for arbitrary accuracy of each orbital and (Y evaluated from eq. (14). We should note that by using an infinite number of exponential functions, we have tried to find the ground state of the hydrogen atom which is actually a single exponential function. Hence we must use the limiting form of the envelope function which is a Dirac delta function [S]. However, in the case of Larger atoms with many electrons, Hartree-Fock functions can require two or more zeta values using exponential functions [7] or many more gaussian functions for proper representation of radial orbitam [a]. Using STF orbitals it may not be necessary to use as many basis functions, since for finite n and optimum CY values, the envelope function can select a continuous range of zeta values so as to require perhaps onLy one analytical expression per one-electron orbital. This is especially important for future work in accurate electronic calculations for large metal complexes, where one faces not only evaluation of multicenter integrals, but also a tremendous increase in the number of such multicenter integraIs for very large basis sets. Even so-called “minimum basis sets” can be in the hundreds quite easiIy for some orgar.ometallic compounds, and for this reason it is highly desirable to have efficient one-electron functions so as to avoid linear combinations as much as possible. It is hoped that the STF orbi+als can meet this need in the future.
Acknowledgement should be given to Dr. J. E, Bloor for allowing the aclthor some time for his project and to the author’s wife for typing the manuscript.
BEFEBENCES R. L. Somorjai, Chem. Phys. Letters 2 (1968) 399. R. G. Parr and J. H. Weare, Prog. Theoret. Phys. (Kyoto) 36 (1966) 854. D. W. Widder, The Laplace Transform (Princeton University Press, 1946). M. E. Schwartz, Chem. Phys. Letters 1 (1967)269. C. A. Coulson, Valence. 2nd ed. (Oxford University Press, 1961) p. 60. G. Goertzal and N. Tralli, Some Mathematical Methods of Physics (McGraw-Hill, H. Basch and H. B. Gray, Theoret. Chim. Acta 4 (1966) ?67. E. Clementi. Chem. Rev. 68 (1968) 341.
1960) p. 98.
19