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

We analytically evaluate the generating integral Knl(β,β′)=∫0∞∫0∞e−βr−β′r′Gnl(r,r′)rqr′q′drdr′ and integral moments Jnl(β,r)=∫0∞dr′Gnl(r,r′)r′qe−βr′ for the reduced Coulomb Green’s function Gnl(r, r′) for all values of the parameters q and q′, 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.We analytically evaluate the generating integral Knl(β,β′)=∫0∞∫0∞e−βr−β′r′Gnl(r,r′)rqr′q′drdr′ and integral moments Jnl(β,r)=∫0∞dr′Gnl(r,r′)r′qe−βr′ for the reduced Coulomb Green’s function Gnl(r, r′) for all values of the parameters q and q′, 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.