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Analytical treatment to Helium isoelectronic ions via Green function expansion of Coulomb interactions: perturbation calculations of ground-state energies with Hydrogenic orbitals

Authors : Shivalika Sharma, Priyanka Aggarwal, Harsimran Kaur and Ram Kuntal Hazra*



Inseparability of Coulomb interactions into generic coordinates calls hues of dichotomy in solving Schrödinger equation of multi-electron quantum systems. Especially, Hydrogenic systems have both bound and scattered (ionized) states to achieve completeness relation of respective Hilbert space. Consequently, variational and perturbation calculations experience nontrivial Coulomb (exchange) integrals. Albeit, Born-Oppenheimer approximation simplifies the hamiltonian for electrons of atoms under frozen nucleii, the electrostatic Green function expansion of interactions becomes inevitable prospect. Each pole of Green function expansion is composed of operators of two electrons as a composite of source and test charges. Non-relativistic quantum equations of the source and the test electrons are solved in Whittaker-function and Associated Laguerre polynomial forms respectively. It furnishes all operators of the above mentioned expansions a template of lower and upper incomplete Gamma functions with integer arguments for interior integrals and finally terminable, exact, finitely summed and easily calculable Lauricella functions in exterior integrals in analytical treatment of Coulomb correlations. However, exchange of coordinates among test and source electrons leading to identical integrals ensures symmetric nature of Green function expansion. As a benchmark, we have calculated ground state energies of He and isoelectronic ions through perturbation calculations of first, second and higher orders with bound states only. It clearly shows that percentage contribution upto second order perturbation calculation of bound excited states increases with increasing nuclear charge (atomic number, Z) of isoelectronic series. The theoretical development also ensures unquestionable future for analytical treatment of size-extended quantum systems.



Helium isoelectronic ions, Coulomb interactions, Green's function expansion, Hydrogenic bound states, first and second order perturbation calculations.