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