Matrix Elements of Coulomb Interaction (gfun)

Matrix Elements of Coulomb Interaction

In this part we look in detail on the simplest possible case, when the basis states are limited to the target only. Also, the basis set is independent of time. We illustrate how complicated the problem is even in this simplest case. On the other hand, we also illustrate it's conceptual simplicity.
We calculate matrix element of Coulomb interaction

between the basis states, i.e.

using the multipole expansion of the Coulomb potential

the well known notation

This separates the radial and angular variables. The separation leads to a simple notation

where the G-functions are

and the integration over the angular parts, which is the integral over three spherical harmonics, gives

The symbols

are Wigner 3j-symbols, related to Clebsch-Gordan Coefficients.