In this post, we explain how to calculate the Nilsson orbital using perturbation method by compute the matrix elements using spherical-spin function. In that post, I said I will give the calculation for the perturbation element. Here we go for the diagonal elements

The diagonal matrix element,

where the spherical-spin wave-function is

First, the radial part is

The radial integral is easy, we can use the integration formula. The radial integration is,

When using the integration formula, one has to be careful when changing variable . Since , we have to pull out to properly do the change of variable.

Set , , and

Thus,

The angular-spin part is

This contains spatial and spin part.

The dot-product of the spin part restricted the

And since and , therefore ,

And the integration of spherical harmonic gives,

Sum up everything,

Note: I haven’t numerically check the formula. ( may be later )

For the off-diagonal element. The angular-spin part should be similar. The difficulty is the radial part, we have to evaluate the most general orthogonal relation of the Laguerre polynomial with weighting .

We only knew the Laguerre polynomil is orthogonal with respect to , i.e.

But not this.

or this

But we are quite sure the last one, with $\alpha \neq \beta$ will not give zero, otherwise, The Nilsson orbital will be very simple and boring.