Semi-analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian expansion of Distance Operators W= R$_{C1}^{-n}$R$_{D1}^{-m}$, R$_{C1}^{-n}$r$_{12}^{-m}$, r$_{12}^{-n}$r$_{13}^{-m}$

Abstract: The equations derived help to evaluate semi-analytically (mostly for k=1,2 or 3) the important Coulomb integrals Int rho(r1)...rho(rk) W(r1,...,rk) dr1...drk, where the one-electron density, rho(r1), is a linear combination (LC) of Gaussian functions of position vector variable r1. It is capable to describe the electron clouds in molecules, solids or any media/ensemble of materials, weight W is the distance operator indicated in the title. R stands for nucleus-electron and r for electron-electron distances. The n=m=0 case is trivial, the (n,m)=(1,0) and (0,1) cases, for which analytical expressions are well known, are widely used in the practice of computation chemistry (CC) or physics, and analytical expressions are also known for the cases n,m=0,1,2. The rest of the cases - mainly with any real (integer, non-integer, positive or negative) n and m - needs evaluation. We base this on the Gaussian expansion of |r|^-u, of which only the u=1 is the physical Coulomb potential, but the u.ne.1 cases are useful for (certain series based) correction for (the different) approximate solutions of Schrodinger equation, for example, in its wave-function corrections or correlation calculations. Solving the related linear equation system (LES), the expansion |r|^-u about equal to SUM(k=0toL)SUM(i=1toM) Cik r^2k exp(-Aik r^2) is analyzed for |r| = r12 or RC1 with least square fit (LSF) and modified Taylor expansion. These evaluated analytic expressions for Coulomb integrals (up to Gaussian function integrand and the Gaussian expansion of |r|^-u) are useful for the manipulation with higher moments of inter-electronic distances via W, even for approximating Hamiltonian.