In a seminal paper in this journal, King has published a detailed discussion of atomic integrals containing explicit r_ij factors, entitled "Analysis of some integrals arising in the atomic four-electron problem." ¹ We have attempted to make computations based on this analysis, and have the following comments, which may be of use to others interested in the approach because it facilitates the use of King's important contribution.

The problem under consideration is the evaluation of integrals containing Slater-type s orbitals for each of four electrons and also arbitrary powers (–1) of each of the six interelectron distances r_ij. The integral is expressed in spherical polar coordinates of the four electrons by writing each factor r as an expansion² involving r_i, r_j, and P_l(cos _ij) (l = 0,,), where P_l is a Legendre polynomial and _ij is the angle between the vectors r_i and r_j. This process produces a sixfold infinite sum, with summation indices we will denote m,n,p,q,s,t, and with each term of the sum the product of an angular integral I and a radial integral I_R, i.e.,

The angular integral, which has the general form

is expressed by King, in his Eq. (36), in terms of a threefold summation (over indices additional to those shown here) of a product of four 3-j symbols. Since large ranges of the added indices (e.g., –30 to 30) will be needed to reach convergence in the expansions, these additional summations will consume significant computational resources. However, this difficulty can be avoided by recognizing that I is an angular-momentum coupling integral, and the additional threefold summation can be reduced to a single Wigner 6-j symbol.^3,4

The triple summation in King's Eq. (36) is only relevant for I when m + n + p, m + q + s, n + q + t, and p + s + t are all even, and in that case all the 3-j symbols in the equation are invariant under permutation of their columns. But a judicious column permutation enables the sum to be related to a standard contraction formula (cf. Ref. 4, p. 142), which is still valid even when m + n + p, etc., are not all even. The result is

leading to

Because m + n + p, n + q + t, etc., are even, the sign factor can be reduced to a less symmetric form such as (–1)^{n + s}, (–1)^{p + q}, or (–1)^{m + t}.

The formula given for I_R, King's Eq. (42), contains an explicitly written sum of 24 instances of an auxiliary function W₄, this function occurring each time with complicated, but different arguments. Unfortunately, the equation for I_R contains two errors (the seventh W₄ should have as its third argument K + ₄ + ₈ + ₁₁, and the 21st W₄ should have third argument I + ₂ + ₃ + ₆). We discovered the errors by identifying, and then programming, the logic that led to the equation. The 24 W₄ terms correspond to the 4! different size orderings of r₁, r₂, r₃, r₄, with each ordering identifiable by the permutation connecting it to a reference order such as r₁<r₂<r₃<r₄. If the variable names are reassigned to correspond to the reference order, then (in King's notation) the parameters a,b,c,d and the powers of the r_i (I,J,K,L) must be permuted similarly, while the parameters ₁–₁₂ must be transformed in a way corresponding to that of the index pairs with which they are associated, e.g., ₁ is associated with the pair (12), which, when 1–4 are permuted, transforms into some pair from the set {(12),(13),(14),(23),(24),(34)}. In place of the extreme detail, it would be useful to have a clear statement of the logic behind expressions such as King's Eq. (42), ideally in the context of a mathematical notation that captures the essence of the matter and is easily implemented.

To illustrate one way in which this might be done, let Î, â, , and stand for the respective ordered lists [I,J,K,L], [a,b,c,d], [₁,₃,₅,₇,₉,₁₁], and [₂,₄,₆,₈,₁₀,₁₂], let be a permutation of [1,2,3,4] and U() the matrix representation of the effect of on [(12),(13),(14),(23),(24),(34)]. For r₁<r₂<r₃<r₄ let

Then the sum of the 24 W₄ can be written

We acknowledge support from the Natural Sciences and Engineering Research Council of Canada and the U.S. National Science Foundation (Grant No. PHY-0303412).