Angular reduction of the Coulomb interaction

The first step in reducing the MBPT expressions into a form suitable for numerical evaluation is a decomposition of the Coulomb integrals Uy w into sums of products of angular momentum coupling coefficients and radial integrals. To accomplish this decomposition, we first expand the kernel of the Coulomb integrals l/ri2 as 

With the aid of this expansion, one finds Vabed = 2 L abcd) XL abcd), 

The factor J abcd) in Eq. (94) contains the entire dependence on magnetic quantum numbers. With the aid of the identity 

Substituting (94) and (100) into Eq. (90) and carrying out the sums over magnetic quantum numbers, one can express the second-order energy in terms of radial integrals as 

To evaluate the sums over virtual states in Elqs. (102) and (103), we make use of the B-spline basis functions described later in this section. 

---