## Analytic evaluation of Coulomb integrals for one, two and three-electron distance operators

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^{2}t^{2}). The
rest of the cases are new and need the other Laplace transformation with
integrand exp(-a^{2}t) 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.