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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(-at). The rest of the cases are new and need the other Laplace transformation with integrand exp(-at) 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 (r1)RC1 -nRD1 -mdr1, (r1)(r2)RC1 -nr12 -mdr1dr2, (r1)(r2)(r3)r12 -nr13 -mdr1dr2dr3, where (ri), 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. The full version is published in American Institute of Physics, Conference Proceedings, 2018 May, # 032 in ICNAAM-2017.

aip conference proceedings

Analytic evaluation for integrals of product Gaussians with different moments of distance operators (RC1−nRD1−m, RC1−nr12−m and r12−n r13−m with n, m=0,1,2), useful in Coulomb integrals for one, two and three-electron operators