Full-scale analytical perturbation calculation of He-isoelectronic series with Hydrogenic bound states via Green's function expansion (monopole) of Coulomb interactions

Abstract

Increasing number of carrier electrons and nuclear charge in Hydrogenic systems associated with bound and ionised (scattered) states invites long-standing divergence issue of Schrödinger's equations due to Coulomb interactions. In this regard, He-isoelectronic series represents the benchmark for atoms and molecules within non-relativistic quantum limit. Schrödinger equations for such complex systems being transformed into either Whittaker-M or Associated Laguerre Polynomial, furnish multipole operators of Green's function expansion of Coulomb interaction a robust framework of lower and upper incomplete Gamma functions. The resulting monopole integral gives terminating, finitely summed, simple and analytical forms of Lauricella functions. This novel form succinctly remedies the paradox of Energy Contribution or Correction occurring due to divergent length scales among Hydrogenic bound states. Both singly and doubly excited spherically symmetric states of two-electron systems ( ) are examined to register full-scale third-order perturbation for monopole factor within 0.2–8.2% deviation of ground-state energies from experimental values. The deviation urges for further analytical treatment to dipole factor. Moreover, exchange of electronic coordinates by virtue of exchanging quantum numbers exhibits invariant integrals of multipoles which confirms reciprocity condition of Green's function. This methodology has achieved a new paradigm of employing generic H-like orbitals for electronic structure calculations.