A new functional form for multi-body expansions of potential energy surfaces and basis functions for correlated adiabatic and fully non-adiabatic variational energy calculations is presented. N-body explicitly correlated Gaussians with pre-multiplying factors consisting of products of powers of internal distance coordinates are utilized in a dual role to analytically represent isotropic potentials and energy eigen-functions in the same internal coordinate system. Practical aspects of this new methodology are presented. The ideas and methods are prototyped and illustrated with two simple diatomic examples; the Morse potential and an accurate H-2 potential for which essentially exact results are obtained for vibrational energy levels.