Supersymmetry and the Hartmann potential of theoretical chemistry

Abstract

An exactly solvable ring-shaped potential in quantum chemistry given by $$V = \eta \sigma ^2 \varepsilon _{\text{o}} \left( {\frac{{2a_{\text{O}} }}{r} - \frac{{\eta a_{\text{O}}^2 }}{{r^2 {\text{sin}}^{\text{2}} \theta }}} \right)$$ was introduced by Hartmann in 1972 to describe ring-shaped molecules like benzene. In this article, the supersymmetric features of the Hartmann potential are discussed, We first review the results of a previous paper in which we rederived the eigenvalues and radial eigenfunctions of the Hartmann potential using a formulation of one-dimensional supersymmetric quantum mechanics (SUSYQM) on the half-line [0, ∞). A reformulation of SUSYQM in the full line (− ∞, ∞) is subsequently developed. It is found that the second formulation makes a connection between states having the same quantum number L but different values of ησ2 and quantum number N. This is in contrast to the first formulation, which relates states with identical values of the quantum number N and ησ2 but different values of the quantum number L.