Hydrogenic Elliptic Orbitals, Coulomb Sturmian Sets, and Recoupling Coefficients Among Alternative Bases

摘要

The nonrelativistic Schrodinger equation for the Coulomb problem is separable in four different coordinate systems in configuration space: Alternative setse of orbitals for the hydrogen-like atoms correspond to each of them and permit to obtain Sturmian sets, useful as complete orthonomal expansion bases in atomic and molecular calculations. In this article the fundamental properties of the already known hydrogenic orbitals (the familier polar, the parabolic, and the rarely treated spheroidal sets) resumed; then, we discuss some properties of the spheroelliptic orbitals, which have been so far pratically ignored. We pay particular attention to the symmetries of the different orbital sets and to the relationships between them and order them in a complete scheme that exhibits passages from one to the other through explicity derived orthogonal transformations. Within this context we insert the study on the conservation of parity in the passage from the polar set to the parabolic one. We also show that - except for the spheroida set-all alternatives for hydrogenic wave functions induce irreducible respresentations of the point group D_2h, the "quaternation group." This has relevance for the discussion of connections between these sets and the corresponding ones in momentum sapce, presented in the following installment of this series.