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

## TL;DR

This paper analytically evaluates integrals involving the Coulomb Green's function, enabling precise calculations of energy spectra and physical properties in multi-electron atomic systems.

## Contribution

It provides explicit formulas for generating integrals and moments of the Coulomb Green's function for all convergent parameter values, advancing analytical methods in quantum physics.

## Key findings

- Derived explicit expressions for integrals involving Coulomb Green's function.
- Applicable to second-order perturbation theory in atomic physics.
- Facilitates analytical computation of atomic energy levels and densities.

## Abstract

We analytically evaluate the generating integral $K_{nl}(\beta,\beta') = \int_{0}^{\infty}\int_{0}^{\infty} e^{-\beta r - \beta' r'}G_{nl} r^{q} r'^{q'} dr dr'$ and integral moments $J_{nl}(\beta, r) = \int_{0}^{\infty} dr' G_{nl}(r,r') r'^{q} e^{-\beta r'}$ for the reduced Coulomb Green's function $G_{nl}(r,r')$ 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.