Analytical two-center one-electron overlap and exchange integrals for \(^{1}\Sigma\) states: Lah number guided Coulomb Green function of H-like s-orbitals

Theoretical studies of two-center one-electron (2c-1e) small microcluster are associated with hurdles in Schr\(\ddot{o}\)dinger equation (SE) born out of divergence of Coulomb interactions and nuclear separation (R). The SE deals with morphologically altered H-like AOs, Slater type orbitals (STO), Gaussian type orbitals (GTO), B-spline, Sturmian function and etc in both VBT and MOT calculations. Few elegant computational and analytical methods are available for STO, GTO and other square integrable trial wavefunction under Born-Oppenheimer approximation. Even so, analytical treatment for H-like AOs has become very necessary. Utilizing Sheffer identity in associated Laguerre polynomial/Whittaker-M H-like AOs and adopting elliptic coordinates provide exact, analytical and simple 2c-1e Coulomb exchange interactions (Ks) and overlap integrals as functions of R with different scaling factors associated with electrons. The energetics of diatomic molecule is evident to be the function of R with extrema as Lah number moderated \(L_n^{-1}\) for nuclear coordinates.