Hydrogenic elliptic orbitals, Coulomb Sturmian sets, and recoupling coefficients among alternative bases

Abstract

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.

Derivation, properties and application of Coulomb Sturmians defined in spheroidal coordinates

“…In (4.a) and (4.b) the eigenvalues are the separation parameters λ associated with each amplitude function. In this case, in contrast with the cases for spherical and paraboloidal basis sets, neither spheroidal eigenfunctions ψ (ξ, η, ϕ) nor eigenvalues λ are expressible in closed form [17,18].…”

“…In [17,18], the Coulomb Sturmian amplitude functions are derived in two limiting cases, at small and large R. The results were obtained on representing the unknown functions in terms of Coulomb spherical (small R) and Coulomb paraboloidal (at large R) Sturmians. The main spheroidal corrections to spherical and paraboloidal Coulomb Sturmian amplitude functions are found in [17,18].…”

“…In [17,18], the Coulomb Sturmian amplitude functions are derived in two limiting cases, at small and large R. The results were obtained on representing the unknown functions in terms of Coulomb spherical (small R) and Coulomb paraboloidal (at large R) Sturmians. The main spheroidal corrections to spherical and paraboloidal Coulomb Sturmian amplitude functions are found in [17,18]. Our paper [29] suggests a simple and straightforward scheme of calculation; Coulomb spheroidal Sturmians for some states are obtained in an explicit algebraic form at arbitrary separation R. An important conclusion is that Coulomb Sturmian amplitude functions derived in spheroidal coordinates constitute the most appropriate basis functions for calculations on diatomic molecules.…”

“…The possibility to solve the obtained one-dimensional equations exactly and to express the amplitude functions in a closed form in paraboloidal coordinates allows one to generate the appropriate Sturmian sets, suitable as a complete orthonormal expansion basis in atomic and molecular calculations. Coulomb Sturmians in paraboloidal coordinates have been derived and investigated in detail [17,18]. In comparison with numerous applications of Coulomb spherical and paraboloidal amplitude functions, the spheroidal solutions have received little attention.…”

Hyperspherical harmonics for polyatomic systems: Basis set for kinematic rotations

“… As illustrated in Sections 2.3 and 3.2, the symmetry operations of the V 4 group that act on the kinematic space for a four‐particles 5, 6, 12 system can be identified with the symmetry operations of the D 2 group 20, 53, which are the identity operation Ê and the rotations of π of each of the three axes of the kinematic space; (a) and (b) illustrate, respectively, rotations around the z and y axes (illustrated in c ); (c) the rotation by π around the x axis, according to its definition in Eqs. (11) and (12), is described as the product of a rotation around the z axis followed by a rotation around the y axis.…”