A function of the type f(R)Y^M_L(Θ,Φ) around a specific center B is expanded about another center A as f(R)Y^M_L(Θ,Φ) =∑^∞_{#= ‖ M ‖} (1/r)α_#(fLM ‖ a,r) x Y^M_#(ϑ,φ), where Y^M_L(Θ,Φ) and Y^M_#(ϑ,φ) are the complex spherical harmonics around the respective centers, which have been assumed to be quantized in the direction of A to B. The distances from the respective centers are R and r, and the separation between the two centers is a. The expansion coefficient α_#( fLM ‖ a,r), which is usually called Löwdin's α-function, can be expressed in a remarkably simple form by using Silverstone and Moats' formula. Owing to the simplicity, the properties of the coefficients b_{K k}(LM ‖ #) appearing in the α-function are manifested, and the recurrence formulas for evaluating b_{K k}(LM ‖ #) are derived. Furthermore, when r≊0, the asymptotic expression for the α-function is given in a compact form.

This paper subsequent to the one [J. Math. Phys. 25, 1133 (1984)] (referred to as Part I) presents the following new results: It is found out that for M=L and L-1 the coefficients b_{K k}(LM ‖ #) in Löwdin's α-function have properties other than manifested in Part I. The expression for b_{K k}(LM ‖ #) is shown to be equivalent to the one into which Sharma's expression, obtained in a different manner from that in Part I, is simplified by Rashid. The use of Rashid's expression leads to the recurrence formula for b_{K k}(LM ‖ #) with respect to M only. This formula and the expression for the b_{K k}(LM ‖ #) with M=L provide an easier procedure for successively evaluating b_{K k}(LM ‖ #) than in Part I. Furthermore, it is proved that the coefficients h_n,2n-i(LM ‖ #)> in the asymptotic form of the α-function vanish for i<#+M and for n<#.

When r/a≊0, Löwdin's α function (1/r)α_#(fLM ‖ a,r) is expressed as (1/r)α_# ( fLM ‖ a,r) =2γ(LM ‖ #)a^L-1∑^∞_n=# [∑ⁱ_maxi=#+M[h_n,2n-i (LM ‖ #)/(2n-i)!]a^2n-ig^(2n-i) (a)](r/a)^2n-#, where a is a separation between a new center placed on the origin and the old center located at a given point on the z axis that is the origin of the coordinate system defining a function f(R)Y^M_L(Θ,Φ) to be expanded around the new center [f(R) is an arbitrary radial function, and Y^M_L(Θ,Φ), a complex spherical harmonics], and r is a distance from the new center. Here, i_max=min{2n,2(L+#)}, and the function g^(j)(a) is expressed as g^(j)(a)=[(d/dR)^(j) (f(iR)/R^L-1)]_R=a.

The closed form of the coefficients h_n,2n-i(LM ‖ #) is given by Eq. (27) in J. Math. Phys. 26, 3193 (1985) (referred to as Part II). Since they are expressed in terms of the different coefficients b_{K k}(LM ‖ #) independent of the form of f(R), h_n,2n-i(LM ‖ #) constitute a set of universal constants as do b_{K k}(LM ‖ #). The explicit expression for the factor γ(LM ‖ #) in front of the summation symbol is given by Eq. (2.9) in J. Math. Phys. 25, 1133 (1984). Hereafter the coefficients expressed by h_n,2n-i(LM ‖ #)/(2n-i)! are symbolized by #_n,2n-i(LM ‖ #). Although they are expressed in a complex form with a triple sum, that expression can be reduced to two considerably simple forms by changing the summation indices to others by using the addition theorem for binomial coefficients and the condition for the sum in this theorem to vanish. Moreover, in the two special cases that i=#+M and i=2(L+#) or 2(L+#)-1, those two expressions are reduced to two single-term forms, respectively.

Introducing into the expression for #_n,2n-i(LM ‖ #) in terms of b_{K k}(LM ‖ #) the recursion formula for b_{K k}(LM ‖ #) in only M given by Eq. (24) in Part II, leads to the recursion formula for #_n,2n-i(LM ‖ #) with respect to M. On the other side, the recursion formulas for #_n,2n-i(LM ‖ #) with M=L and # as to i are obtained through a skillful manipulation. By connecting those recursion formulas for #_n,2n-i(LM ‖ #) to each other, we can obtain a procedure for calculating #_n,2n-i(LM ‖ #) successively. As the result of actual performance of the procedure, all the formulas expressing #_n,2n-i(LM ‖ #) with restriction 0≤M≤min{L,#}≤max{L,#}≤2 are presented as functions of the parameter n in a table.

By using the explicit expression for the coefficients b_{K k}(LM ‖ #)[Eq. (16) in J. Math. Phys. 26, 3193 (1985), which is referred to hereafter as Part II], a closed expression for the coefficients c_ij(NLM ‖ #) appearing in Löwdin's alpha function (1/r)α_#(NLM ‖ a,r) for a Slater-type atomic orbital is obtained. By manipulating its expression mathematically, it was manifested that c_ij(NLM ‖ #) vanishes if i≥N+L-M+#. For the specific cases that each of i and j takes 0 or 1, c_ij(NLM ‖ #) take the same value. From the symmetry relation on b_{K k}(LM ‖ #) [Eq. (6) in Part II], it was proved that c_ij(NLM ‖ #) = c_ji(NLM ‖ #), provided M=L. Through some manipulation, the recursion formula for c_ij(NLM ‖ #) in N was derived, while that about M was obtained by introducing the recursion formula for b_{K k}(LM ‖ #) in M [Eq. (24) in Part II] into the expression for c_ij(NLM ‖ #). For the special cases where N and M both are equal to L, the expression for c_ij(NLM ‖ #) is given in a compact and symmetric form. The use of this expression leads to a recursion formula for c_ij(LLL ‖ #) in L. Especially, c_ij(LLL ‖ #) for L=0 is expressed by a single term, and a recursion formula for c_ij(000 ‖ #) in # was obtained in a very simple form. The systematic use of these recursion formulas leads to a procedure for evaluating c_ij(NLM ‖ #) successively. Finally, c_ij(NLM ‖ #) evaluated by this procedure, with the restriction that 2≥N≥L≥0 and 2≥min{L,#}≥M≥0, are given in a matrix form.

Abstract

I have used MATHEMATICA to solve several problems that relate to the symbolic calculation of the 'molecular integrals' that are a mainstay of computational chemistry.

This work has provided many new results of chemical and mathematical interest, and it has led to a powerful programming methodology that I call MATHSCAPE that uses a novel open ended set of macros. Some further work on molecular integrals is presented here, largely as an introduction to MATHSCAPE. I discuss (1) the immediate mathematical problem of the 'J' integrals, (2) the key features of MATHSCAPE, (3) a novel reduction of the J integrals, (4) the chemical context of this work, and (5) the computer science context.