ABSTRACT:The nonrelativistic Schrö dinger equation for the Coulomb problem is separable in four different coordinate systems in configuration space: Alternative sets of orbitals for the hydrogen-like atoms correspond to each of them and permit to obtain Sturmian sets, useful as complete orthonormal expansion bases in atomic and molecular calculations. In this article the fundamental properties of the already known hydrogenic orbitals (the familiar polar, the parabolic, and the rarely treated spheroidal sets) are resumed; then, we discuss some properties of the spheroelliptic orbitals, which have been so far practically 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 explicitly 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 thatexcept for the spheroidal set-all alternatives for hydrogenic wave functions induce irreducible representations of the point group D 2h , the "quaternion group." This has relevance for the discussion of connections between these sets and the corresponding ones in momentum space, presented in the following installment of this series.
Momentum space hydrogenic orbitals can be regarded as orthonormal and complete Sturmian basis sets and explicitly given in terms of (hyper)-spherical harmonics on the 4-D hypersphere S 3 . Among the alternative coordinate systems that allow separation of variables, the usual ones involving parameterizations of the sphere S 3 by circular functions correspond to canonical subgroup reduction chains; we also investigate harmonic "elliptic" sets (as, e.g., obtained by parameterizations in terms of Jacobi elliptic functions). In this article we list the canonical hydrogenic Sturmian sets and the orthogonal transformations connecting them. The latter enjoy useful three-term recurrence relationships that allow their efficient calculations even for large strings. We also consider modifications needed when the conservation of the symmetry of Sturmians with respect to parity. Finally, we discuss some properties of elliptic hydrogenic Sturmians and their relations with canonical Sturmians. Because elliptic Sturmians cannot be expressed in closed form, it is important to find expansions in a suitable basis set and calculate the transformation coefficients. We derive three-term recursion relationships fulfilled by the coefficients of the transformation between elliptic Sturmians and canonical Sturmians. A concluding discussion on the connections between configuration space and momentum space hydrogenic Sturmians completes this article.
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