Associated Legendre Polynomials - Gaunt's Formula

Gaunt's Formula

The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials. For instance, this turns out to be necessary when doing atomic calculations of the Hartree–Fock variety where matrix elements of the Coulomb operator are needed. For this we have Gaunt's formula

This formula is to be used under the following assumptions:

1. the degrees are non-negative integers
2. all three orders are non-negative integers
3. is the largest of the three orders
4. the orders sum up
5. the degrees obey

Other quantities appearing in the formula are defined as

The integral is zero unless

1. the sum of degrees is even so that is an integer
2. the triangular condition is satisfied