We are going to evaluation the integral

Recalling the multi-pole expansion,

and the basis

Set an averaged basis

In the angular integrals, using Wigner 3-j symbol and the integral

we have

Thus,

The 3-j symbol restricted that

I guess it is the most simplified formula for the angular part.

The total integral is

The angular integral imposes condition for .

I am not sure this is a correct way to treat the problem.

First, the averaged basis is still an energy eigen state. It is not the eigen state for the angular part. So, this averaging could introduces an error and we should reminded that this is an approximation. But in the perturbation view point, this averaged basis is still valid.

Second thing is, the sum

is not symmetry for exchange of in general. For example,

This is a very uprising result that the mutual interaction dependent on the magnetic quantum number. Thus, in detail, we should use as a basis.

Third, the sum is depend on . The mutual interaction require us to sum all possible .

Fourth, the coupling between 1s2p triplet state, the total spin is , total L is , and the total angular momentum can be . In our treatment, we did not coupled the angular momentum in the calculation explicitly. In fact, in the integral of the spherical harmonic, the coordinate are integrated separately, and the coupling seem to be calculated implicitly. I am not sure how to couple two spherical harmonics with two coordinates.