Title: Analytic evaluation of Coulomb integrals for one, two and three-electron distance operators, $R_{C1}^{-n}R_{D1}^{-m}$, $R_{C1}^{-n}r_{12}^{-m}$ and $r_{12}^{-n}r_{13}^{-m}$ with $n, m=0,1,2$,
Abstract: The state of the art for integral evaluation is that analytical solutions to
integrals are far more useful than numerical solutions. We evaluate certain
integrals analytically that are necessary in some approaches in quantum
chemistry. In the title, where R stands for nucleus-electron and r for
electron-electron distances, the $(n,m)=(0,0)$ case is trivial, the
$(n,m)=(1,0)$ and (0,1) cases are well known, fundamental milestone in
integration and widely used in computation chemistry, as well as based on
Laplace transformation with integrand exp(-$a^2t^2$). The rest of the cases are
new and need the other Laplace transformation with integrand exp(-$a^2t$) also,
as well as the necessity of a two dimensional version of Boys function comes up
in case. These analytic expressions (up to Gaussian function integrand) are
useful for manipulation with higher moments of inter-electronic distances, for
example in correlation calculations.