Analytic evaluation for integrals of product Gaussians with different moments of 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), useful in Coulomb integrals for one, two and three-electron operators

摘要

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~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. The equations derived help to evaluate the important Coulomb integrals