Analytic Evaluation for Integrals of Product Gaussians with Different Moments of Distance Operators (Rc1-nRD1-m, RC1-nr12-mand r12-nr13-m with n, m=0,1,2), Useful in Coulomb Integrals for One, Two and Three-Electron Operators

In the title, where R stands for nucleus-electron and r for electron-electron distances in practice of computation chemistry or physics, 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, 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. The equations derived help to evaluate the important Coulomb integrals integral rho(r(1))R(C1)(-n)RD1(-m)dr(1), integral rho(r(1))rho(r(2))R(C1)(-n)r12(-m)dr(1)dr(2), integral rho(r(1))rho(r(2))rho(r(3))r12(-n)r13(-m)dr(1)dr(2)dr(3), where rho(r(i)), called one-electron density, is a linear combination of Gaussian functions of position vector variable ri, capable to describe the electron clouds in molecules, solids or any media/ensemble of materials.