On the Convergence of the Interpenetrating Bipolar Expansion for the Coulomb Potential

H.J. Silverstone,    Department of Chemistry, Johns Hopkins University, Baltimore, MD 21218, USA,    E-mail: hjsilverstone@jhu.edu

Abstract

In the interpenetrating region where , individual terms in the bipolar expansion for the Coulomb potential are shown to decay like (l1 + l2)-1, rendering convergence of the expansion at best conditional. Moreover, Laplace’s equation, which is satisfied term-by-term in the separated R > r1 + r2 region, is shown to be satisfied neither term-by-term nor pointwise.

Keywords

Convergence; Bipolar expansion; Coulomb operator; Coulomb potential; Multicenter integrals; Slater-type orbital; Exponential-type orbital

1 Introduction

The interacting-multipole formula for two distantly separated charge distributions (), as pictured of the left in Fig. 1.1,

(1)

(2)

(3)

is a direct consequence of the bipolar expansion for the Coulomb potential,

(4)

Convergence is “geometric” in powers of the two ratios and . Laplace’s equation is satisfied term-by-term.

Figure 1.1 Non-overlapping and overlapping charge distributions with sizes characterized by and .

When charge distributions overlap—that is, when the distances satisfy the triangle inequality , as pictured on the right in Fig. 1.1—the bipolar expansion has an additional summation index and a much more complicated dependence:

(5)

(i) In the non-overlapping case, ; a given can appear only in a finite number of terms. In the overlapping case, any pair of values that can “couple” to a given can occur; for each , the number of such pairs is infinite.

(ii) In the non-overlapping case, the radial factor is a constant times the single monomial . In the overlapping case, is a sum of terms with .

(iii) In the non-overlapping case, Laplace’s equation is satisfied term-by-term. In the overlapping case, Laplace’s equation is not satisfied term-by-term.

Compare the terms in the two cases:

(6)

(7)

(8)

(9)

In the overlapping case, the three radii appear democratically. Since all radii appear with similar roles in numerators and denominators and no single radius is dominant, how does the series converge? How is Laplace’s equation satisfied? These questions seem to have avoided investigation. We explore their answers through numerical calculations. Convergence turns out to be painfully slow and conditional. Laplace’s equation is not satisfied either term-by-term or pointwise.

2 Formulas for the bipolar expansion in the overlapping region

Formulas needed to numerically evaluate the and of Eq. (5) are given in this section.

2.1 Radial factors

Initially the existence of the overlapping region was missed, as in, e.g., Ref. 3. Buehler and Hirschfelder1,2 were the first to derive formulas for the overlapping region, integrating Laplace’s expansion termwise for specific cases. Sack6, exploiting recurrence formulas obtained via Laplace’s equation, showed that the general radial factor is an Appell function. Kay, Todd, and Silverstone extended a Fourier-transform formulation of Ruedenberg5 to obtain an explicit formula for that we find convenient to use here.4

(10)

(11)

Note that must be even. How each depends on , and is not obvious. For example, if is the smallest of , and , then the single contribution

(12)

in the first term in Eq. (11) increases without limit with increasing and . Yet if the series converges, the must tend to 0. Convergence evidently requires extensive cancellation.

2.2 Angular factor

The factor in Eq. (5) is the integral of a spherical-harmonic triple product; it evaluates to a product of two 3-j symbols,

(13)

(14)

3 Numerical explorations

We explore the convergence of Eq. (5) heuristically by using formulas (11) and (14) to evaluate terms in the series numerically.

3.1 Strategy

There are 9 variables: 6 angles; 3 radii. Some specific choice of values is necessary.

3.1.1 Angles

We assume that the numerical values of the angles are not important for controlling convergence, and for simplicity we take all angles to be 0, thereby linearizing the geometry and making just the reciprocal of .

(15)

Conveniently all spherical harmonics with vanish, eliminating summations over and in Eq. (5). The angular factor in Eq. (5) turns out to be relatively simple after using standard formulas for the 3-j symbols and that :

(16)

(17)

The ultimate linearized-geometry expansion (18) is much tighter than the general expansion:

(18)

where is given by Eq. (17) and is given by Eq. (11).

3.1.2 Order of summation in Eq. (18)

The summation indices in Eq. (18) run from 0 to . The optimal order of terms is not obvious. Two reasonable candidates are:

(i) replace each “” by a single , then increment —call this the “()-order”;

(ii) collect terms for which , then increment —call this the “()-order.”

(19)

(20)

We indicate below why the -order is advantageous.

3.1.3 Radii

Specification of values for , and R is more delicate. Only ratios are important, because the expansion is homogeneous of degree −1. It will be convenient to give at least one radius the value 1. From the viewpoint of convergence, it is not important which of three numerical values is assigned to which radius, because , and R enter the formulas almost indistinguishably. (If the expansion were for , they would enter indistinguishably.)

We pick four representative sets of radii. As shown in Figure 1.2, they are:

(i) All radii equal; this case is maximally radially independent:

(21)

(ii) Two radii equal, the third larger:

(22)

(iii) Two radii equal, the third smaller:

(23)

(iv) All radii different:

(24)

Figure 1.2 Four representative sets of radii with linear geometry.

Having specified angles and radii, we commence numerical experimentation.

3.2 Why the (l1 + l2 = lmax)-order is preferred over the (l1 ≤ lmax, l2 ≤ lmax)-order

We compare the two summation orders (19) and (20) in Figure 1.3 for the all-radii-equal case. For a fixed value of , there are fewer terms to calculate for the ()-order versus the ()-order, the oscillations are gentler, and the odd- terms have zero net contribution (when ). For these three reasons, we use only the ()-order for subsequent calculations.

Figure 1.3 Comparison of the convergence of the ()-order, Eq. (19) (gray diamonds), with the ()-order, Eq. (20) (black dots), when all the radii = 1. Both orderings converge toward 1. The ()-order has gentler oscillations and has zero contributions from odd .

3.3 Numerical evidence that the interpenetrating bipolar expansion converges pointwise

Since convergence will turn out to be conditional, we first verify numerically that the expansion does converge to the correct sum for each set of representative radii. The evidence is Figure 1.4.

Figure 1.4 Convergence of the bipolar expansion, organized by as in Eq. (20), for the four sets of representative radii. The pattern is most regular for all radii equal, least regular for all radii different.

The next question is then, how fast? A numerical answer can be inferred from Figure 1.5, in which the partial-sum increments (which were sequentially summed in Fig. 1.4), indexed by , are plotted, followed by log–log plots of the magnitudes of the nonzero increments. There are a variety of behaviors.

Figure 1.5 Partial sums and log–log plots of the magnitudes of the nonzero partial sums of the bipolar expansion terms with fixed for the four sets of representative radii. The pattern is most regular for all radii equal, least regular for all radii different. With equal radii, there are three “subseries” whose asymptotic slopes here are . The other three cases give some support for an approximate asymptotic behavior.

For the all-radii-equal case, and for , there are four asymptotic subsequences, one of which is identically zero, and three of which can be fit by straight lines of approximate slope . (The lines in the plot were determined by just the...