Reduced Bessel Functions as Atomic Orbitals: Some Mathematical Aspects and an LCAO-MO

Treatment of HeH++

E. 0. STEINBORN AND E. J. WENIGER Institut fur Chemie, Universitat Regensburg, 0-8400 Regensburg, West Germany

Abstract

The possibility of using reduced Bessel functions as basis functions is discussed, and with these functions an LCAO treatment of the heteronuclear system HeH++ is performed.

1. Introduction

In recent years, electronic structure calculations based on the LCAO-MO method have been performed mainly with the help of Gaussian-type orbitals (GTO’s), because then the molecular multicenter integrals, which occur, can be evaluated in a relatively easy way. However, it is generally accepted that the use of exponential-type orbitals (ETO’S) would be highly desirable because only relatively few ETO’S are required to yield a satisfactory approximation of the exact solution of the Schrodinger equation. Unfortunately, the problem of the economic evaluation of the molecular integrals over ETO’S has not yet been solved adequately.

Recently, we have been investigating a special class of ETO’S, the so-called “reduced Bessel functions” (RBF’S). These functions possess some remarkable mathematical properties, which make them suitable as basis functions. First, we discuss some of these aspects, relying on some mathematical and numerical results; second, the possible use of RBF’S as basis functions is discussed, and third, with their help, a simple LCAO calculation of the one-electron problem HeH++ is performed.

2. Some Mathematical Properties of Reduced Bessel Functions

We define the reduced Bessel function (RBF) k , of arbitrary order v by

k , (x ) = (2/7r)’/2X”KU(X) (1)

where K , is the modified Bessel function of the second kind. In the case of half-integral v = N - l/2, the RBF assumes the following form:

Shavitt [ 11 suggested the use of the RBF’S in quantum chemistry as the radial

International Journal of Quantum Chemistry: Quantum Chemistry Symposium 12, 103-108 (1978) 0 1978 by John Wiley & Sons, Inc. 0161-3642/78/0012-0103$01 .OO

104 STEINBORN AND WENIGER

part of a generalized type of orbital in atomic and molecular calculations, be- cause with the help of a Laplace transformation multicenter integrals over RBF’S can be reduced to integrals over GTO’s. However, this procedure requires mul- tidimensional numerical quadratures, which appear to be a serious obstacle for a general applicability. However, in recent work [2-51 it was shown that the RBF’S possess the following extraordinary intrinsic properties which strongly suggest their further investigation as basis functions in atomic and molecular calculations:

(1) Equation (2) can be inverted to yield a representation of Slater-type or- bitals (STO’S) as a linear combination of RBF’S [ 2 ] . Therefore, the molecular multicenter integrals over STO’S can be expressed by the corresponding integrals over RBF’S.

(2) The function k ~ - l p ( P l r - RI) possesses an addition theorem [3] which makes it possible to separate the variables r and R in spherical coordinates [ 2 ] . This addition theorem is much simpler than the addition theorem for a scalar STO (Pr’)N-le-Br’ with r’ = Ir - RI. The addition theorem of RBF’S offers the possibility of the analytical evaluation of multicenter integrals.

(3) The convolution products of two RBF’S can be expressed by surprisingly simple formulas [ 4 ] , e.g.,

Because the convolution product of two RBF’S can be represented in terms of, again, RBF’S, these relationships can be used to derive extremely compact for- mulas for various types of molecular integrals [ 5 ] .

( 4 ) Other kinds of orbitals and operators can be expanded in terms of RBF’S [ 4 ] . Therefore, molecular integrals over these functions and operators can be obtained analytically from integrals over RBF’S.

(5) The arguments given above still hold for nonscalar functions

( 4 ) B N - I / Z , L ( P r ) M = (Pr)LIZN-1/2(Pr)Y~(~2,)

for N = 0, 1,2, . . . and with arbitrary angular momentum quantum numbers L, M, with Y f being a surface spherical harmonic in Condon-Shortley phases

( 6 ) Because of point (l), any finite set of STO’S is connected by an invertible linear transformation with a finite set of B functions. Since STO’s form a com- plete basis set [6] in the space L2(R3) (i.e., Hilbert space of all square integrable functions defined on the three-dimensional Euclidean space R3), the system of B functions is also complete in this space. Therefore, it is possible to use B functions as a basis set in LCAO calculations from the very beginning without relying on STO’S.

Statement ( 6 ) implies that any scalar functionf(r) with finite norm [Jt dr r2 f2(r )]

141.

can be expanded in terms of RBF’S:

LCAO-MO TREATMENT OF HeH++ 105

It should be noted that Eq. (5) holds for all > 0 [6], i.e., the completeness of the system of RBF’S is not affected by the particular choice of the exponential parameter p.

(7) Another interesting feature of the RBF’S is the fact that the convolution- type integrals do not become much more complicated if higher orders or higher angular momentum quantum numbers are involved. However, these integrals become very simple [in some cases even trivial, e.g., Eq. (3)] if both functions in the integrand contain the same exponential parameter. This effect is even more pronounced in the case of the three-center nuclear attraction integrals 171, which can be evaluated analytically with the help of the addition theorem of RBF’S.

3. The Approximate Representation of Atomic Orbitals by RBF’S

In the LCAO-MO approach a molecular orbital *(r) is represented as a linear combination of one-center functions X A a ( r A ) ,

The summations run over the positions of the nuclei of the molecular system under consideration, denoted by A , and over all admissible sets of quantum numbers and parameters, denoted by a, that are required to characterize XAa(rA). Here, rA = r - RA. where RA denotes the position vector of the nucleus A.

If spherical coordinates are used, the one-center functions x A a (rA), that are called AO’S because of historical reasons, are represented in the following form:

XAa(fA) = 4 q a ( ~ A ) m T ( Q r A ) (7) Here, qa denotes the set of the remaining quantum numbers and parameters that are required to characterize 4 q a ( r ~ ) . The function &(rA) is now expressed in terms of the common basis functions used for molecular calculations. If a STO basis is used, one has

4 q a ( r A ) = rTfp-’e-EarA (8) whereas for a GTO basis, one has

$, , (rA) = rpe-Ear%

Another possibility would be the use of a RBF basis (9)

4q&A 1 = La- 1 / 2 ( E J A 1 (10)

Because the B functions defined by Eq. (4) form a complete set, any MO \k(r) can be approximated by these functions with arbitrary precision.

For practical reasons, we can use only a truncated basis. In order to determine *(r) as a linear combination of B functions, we have to optimize the linear and nonlinear variational parameters. The linear variational parameters of the trial

106 STEINBORN AND WENIGER

function can be determined with the help of standard techniques of linear al- gebra, whereas the optimization of the nonlinear variational parameters requires the application of time-consuming iterative procedures. Of course, in order to achieve good agreement with experimental results, it is desirable to optimize as many parameters as possible. As is well known, the nonlinear parameters especially are of great influence. However, we feel it is interesting to check how good results can be obtained by using only one nonlinear parameter and applying more linear parameters, i.e., more basis functions in the linear combination, instead. This problem is of particular interest if B functions are used because for these functions many of the matrix elements, i.e., the molecular integrals, become extremely simple especially for equal scaling parameters, such that even a very extended basis set is not prohibitive at all.

In a former investigation [S] we achieved promising results by using a RBF basis for the calculation of H2+. In order to test the aspects discussed above on a heteronuclear system, we did some test calculations on HeH++, because it is the simplest heteronuclear system, and the exact solution is available [9]. Of course, more complicated systems would require another choice of special basis functions from the B set; the use of basis functions with unequal scaling pa- rameters especially seems to be more advantageous. In this case, the matrix el- ements over B functions are still not much more complicated [5] than those for equal scaling parameters, but they require more computation, which must be done later.

4. An LCAO Calculation of HeH++

In this section we present the results of an LCAO calculation of the electronic energy of the lsa ground state of HeH++. The following class of trial functions, which are built up from scalar AO’S, are used:

N

i= 1 @N = C {AyLi-lp(@r) + B ~ L i - ~ p ( @ l r - RI)), N = 1 , . . . ,6 (11)

With this calculation we want to determine how many basis functions are re- quired to obtain a satisfactory accuracy of the electronic energy. We are thus able to gain some insight into the capability of the RBF’S to serve as a basis set in LCAO calculations.

Table I gives the electronic energies EN which were obtained by optimizing the linear parameters A?, B y and the nonlinear parameter p of @ N .

It can be seen from Table I, that only one function per center already yields more than 99% of the exact energy for all internuclear distances R. The opti- mized energies did not react sensitively on a variation of the common scaling parameter @. In fact, the energy values vary only very slightly over a rather broad range of p. It is interesting to ask for the intervals [Prnin, p,,,] within which at least a certain part, for instance 99.5%, of the exact energy is obtained. In Table I1 we list these parameter ranges for the trial functions @2, . . . ,@6 for the in- ternuclear distance R = 1 .O a.u. We chose R = 1 .O a.u. because for this distance we obtained the worst energy values. The best energy value E&30pt) = 3.01964

LCAO-MO TREATMENT OF HeH++ I07

TABLE I. Resultsa of the LCAO calculation of the ground state of HeH++.

R/ao -El -E2 3 -E4 5 -E6 -EBC -E -E

5.0 2.20000

4.5 2.22222

4.0 2.25001

3.5 2.28575

3.0 2.33354

2.5 2.40100

2.0 2.50421

1.5 2.61984

1.0 3.01295

0.75 3.28180

0.5 3.65280

0.25 4.13015

2.20000

2.22223

2.25002

2.28579

2.33369

2.401 5 0

2.50559

2.68398

3.0igia

3.2aaag

3.65134

4.13146

2.20000

2.22223

2.25003

2.28582

2.33372

2.40150

2.50578

2.68441

3.01 927

3.28899

3.65138

4.13147

2.20000

2.22224

2.25003

2.28583

2.33316

2.401 61

2.50589

2.68448

3.01931

3.28920

3.65816

4.13186

2.20000

2.22224

2.25004

2.28584

2.33378

2.40162

2.50593

2.68453

3.01 959

3.28938

3.65816

4.13187

2.20001

2.22224

2.25004

2.28585

2.33379

2.40163

2.50595

2 .ti8451

3.01 964

3.28940

3.65820

4.13193

2.20024

2.22259

2.25060

2.28679

2.33549

2.40489

2.5121 9

2.69546

3.03335

3.301 37

3.66554

4.13365

a The energies are in units of e*/uo. EBC refers to the Bates and Carson [9] energy calculation. Here the nuclear-nuclear repulsion energy is omitted.

TABLE 11. Energy dependencea on the scaling parameter 0 for R = 1.0 a.u.

T r i a l fct. Bmin Bmax

2.7 2.9

2.1 3.5

1.2 3.9

1.1 4 . 5

1.0 5.1

'2

'3

@ 4

'5

'6 a omin and omax are the values of the variational parameter P, for which at least 99.5% of the exact

energy is obtained.

a.u. is obtained for Popt = 3.8 a.u. At R = 1.0 a.u., E&) represents at least 99.5% of the exact energy for /3 between 1 .O and 5.1, respectively, as given in Table 11. It is interesting to note that E6(Popt) differs from all values E&) with 1 .o 5 P I 5.1 by only 0.05%. It can be seen from Table I1 that with an increasing number of functions the energy becomes less and less dependent on the value of the nonlinear variational parameter. This result confirms Eq. (5) numerically because for N - 00 the energy must become completely independent on p (provided > 0).

It is surprising that even a linear combination of RBF'S with equal exponential parameter as given by m. (1 1) applied to a heteronuclear system yields relatively

108 STEINBORN AND WENIGER

good energy values. With unequal scaling parameters and/or the inclusion of polarization functions a further improvement should be possible.

In recent years various other approximative treatments, using perturbation theory [ 10- 121, one-center expansions [ 13- 151, and floating one-center per- turbation theory [ 161, have been performed for the heteronuclear system HeH++. However, no LCAO-MO treatment is known to the authors.

5. Conclusion

The mathematical properties of the reduced Bessel functions make it possible to evaluate molecular integrals over exponential-type orbitals, especially STO’s, much more easily than so far expected. Because STO’S can be represented by linear combinations of RBF’S, it is of special interest whether RBF’S can profitably be used as AO’S in an LCAO-MO calculation. We have already shown that the ground state of H: can be approximated in a better way with RBF’S than with other common LCAO basis functions [8]. These conclusions are supported by the results of the LCAO calculation of HeH++ which we present in this paper. HeH++ is an asymmetric system which should be even more sensitive to the choice of the basis functions than the symmetric system H:.

Acknowledgments

The authors thank E. Filter for helpful discussions. One of them (E.O.S.) gratefully acknowledges support from the organizers of the 1978 Sanibel Symposium, where these results were presented.

Bibliography

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Revised May 8, 1978