Ricardo Méndez-Fragoso

Rotations of Asymmetric Molecules and the Hydrogen Atom in Free and Confined Configurations

Abstract The contribution to the 50th Sanibel Symposium, “Lame Spheroconal Harmonics in atoms and molecules,” is the connecting reference among previous investigations on rotations of free asymmetric molecules and the Hydrogen atom, and recent results on rotations of the same systems in situations of confinement by elliptical cones. This chapter for Advances in Quantum Chemistry is based on both sets of works, presenting them as a review of the exact formulations and evaluations for the rotations of the free systems in Section 2, and as a guide for the results of the confined situations because of the new boundary conditions including elliptical cones in Section 3. Additionally, we report some tools and advances in developing the theory of angular momentum in bases of spheroconal harmonics in Section 4. The list of contents indicates some of the details in the respective sections, of which brief descriptions of the successive references are given next, commenting about their sequence and relationships with emphasis on the open problems and advances in each stage.

Energy-level structure of the hydrogen atom confined by a penetrable cylindrical cavity

The bound-state energy spectrum and its evolution for a hydrogen atom located along the axis of a standard cylindrical confining cavity with either impenetrable or penetrable confining boundaries are obtained by solving the stationary Schrodinger equation using a finite differences approach. New results are first presented for a nuclear-centered position for the penetrable case as the barrier height and cavity size change. Special attention is then given to the energy-level dependence on the nuclear position along the cylinder axis, where proper identification for the evolution of states from the nuclear-centered position (centered states) up to the cylinder cap (cap states) is proposed, while the corresponding state evolution for intermediate nuclear positions (intermediate states) remains consistent with node conservation and symmetry. It is found that in general the energy levels evolve with an increasing value as the nuclear position is shifted from the central position up to a cylinder cap. As the barrier height (and cavity size) are reduced, a progressive extinction of bound states appears in the order cap states, intermediate states and centered states. Finally, a predominance of cavity-size over barrier-height effects on the energy level shift is found.

Shift operators and recurrence relations for individual Lamé polynomials

Our contribution to Gr31 “Review of the development and application of spheroconal theory of angular momentum” included a mention of our current work on individual Lame polynomials. This written version reports original results on shift operators and recurrence relations connecting a Lame polynomial with angular momentum l, species [A] and excitation n, with neighbouring polynomials with

Properties of the spectra of asymmetric molecules: matrix evaluation in bases of spherical harmonics and common generating function

EQUATION

The hydrogen atom in a semi-infinite space with an elliptical cone boundary

l’ = l±1, species [A’] where the prime indicates a derivative, and excitations n’ in their different possibilities, for the derivative operator. Other operators involve multiplication by another Lame polynomial, starting with the singularity removing factors Ai = 1, sn, cn, dn, sncn, sndn, cndn, sncndn as monomials. The successive and complementary use of these operators for Lame polynomials in their two respective degrees of freedom connects with the ladder and shift operator actions of cartesian components of the angular momentum and linear momentum on the product of those polynomials as rotational eigenstates. The identification of the operators for individual Lame polynomials fills a gap in the study of their properties and connections.