A generating integral for the matrix elements of the Coulomb Green's function with the Coulomb wave functions

Abstract

We analytically evaluate the generating integral $K_{nl}(\beta ,{\beta }^{\prime })={\int }_0^{\infty }{\int }_0^{\infty }{e}^{-\beta r-{\beta }^{\prime }{r}^{\prime }}{G}_{nl}{r}^q{r}^{\prime q^{\prime }}drd{r}^{\prime }$ and integral moments $J_{nl}(\beta ,r)={\int }_0^{\infty }d{r}^{\prime }{G}_{nl}(r,r^{\prime })r^{\prime q}{e}^{-\beta r^{\prime }}$ for the reduced Coulomb Green's function $G_{nl}(r,r^{\prime })$ for all values of the parameters $q$ and $q^{\prime }$, when the integrals are convergent. These results can be used in second-order perturbation theory to analytically obtain the complete energy spectra and local physical characteristics such as electronic densities of multi-electron atoms or ions.

23 pages