Analytic Evaluation of some 2-, 3- and 4- Electron Atomic Integrals Containing Exponentially Correlated Functions of $r_{ij}$

A simple method is outlined for analytic evaluation of the basic 2-electron atomic integral with integrand containing products of atomic s-type Slater orbitals and exponentially correlated function of the form $r_{ij} exp(-\lambda_{ij}r_{ij})$, by employing the Fourier representation of $exp(-\lambda_{ij}r_{ij})/r_{ij}$ without the use of either the spherical harmonic addition theorem or the Feynman technique. This method is applied to obtain closed-form expressions, in a simple manner, for certain other 2-,3- and 4-electron atomic integrals with integrands which are products of exponentially correlated functions and atomic s-type Slater orbitals.