By David K. Zhang

Consider a system of particles moving in -dimensional space.

- is an matrix with nonnegative integer entries.
- is an matrix with nonnegative integer entries.
- is a real symmetric matrix.
- is a real symmetric matrix.
- is an -dimensional vector whose components are -dimensional real vectors.
- is an -dimensional vector whose components are -dimensional real vectors.
- is an -dimensional vector whose components are -dimensional real vectors.

We define the following functions:

(1)

(2)

We make the following definitions:

(3)

The overlap integral of the generating functions is then given by

(4)

We now define a real symmetric matrix and a -dimensional vector whose entries are -dimensional vectors.

(5)

We now rewrite the overlap integral of the generating functions, as follows:

(6)

Here, we have defined a new matrix , which is an upper-triangular matrix whose diagonal is one-half times the diagonal of , and whose strictly upper-triangular part is identical to that of .

(7)

Here we have defined as the set of all -tuples of symmmetric matrices whose entries are nonnegative integers. Thus if , then denotes the -entry of the symmetric matrix .

(Note that the preceding sum depends only on values of for which ; thus, only the upper-triangular part of each matrix is accessed, and we might as well define the lower-triangular part as the transpose of the upper-triangular part.)

Suppose now that we are given an array and an integer . We wish to examine the term

(8)

For each integer , we ask the following question: how many times does appear in the preceding expression?

- It appears in each term with and , where its exponent is .
- It appears in the term, with exponent .
- It appears in each term with and , with exponent .

Thus, the exponent of in the overall expression is

(9)

which, by using the symmetry of , can be rewritten as

(10)

This shows that

(11)

and by generalizing to all values of , we see that

(12)

Now, we need to take derivatives with respect to the parameters .

(13)

Finally, we need to set . Observe that the preceding expression is nonzero only if , that is, if

(14)

for all and . In this case, each factor in the product becomes . Assuming that this condition is satisfied, the expression (13) simply reduces to

(15)

This leads us naturally to consider the following combinatorial problem:

**Definition:** Let be a matrix with nonnegative integer entries. We denote by the set of all -tuples of symmetric , again having nonnegative integer entries, satisfying the condition

(16)

for all and .

We will return to solving this problem later. For now, let us assume that we know how to compute the set . The overlap matrix element of two ECG basis functions is then given by

(17)

(18)

(19)