Position-dependent mass Schrödinger equation for exponential-type potentials

In quantum chemical calculations, there are two facts of particular relevance: the position-dependent mass Schrödinger equation (PDMSE) and the exponential-type potentials used in the theoretical study of vibrational properties for diatomic molecules. Accordingly, in this work, the treatment of exactly solvable PDMSE for exponential-type potentials is presented. The proposal is based on the exactly solvable constant mass Schrödinger equation (CMSE) for a class of multiparameter exponential-type potentials, adapted to the position-dependent-mass (PDM) kinetic energy operator in the O von Roos formulation. As a useful application, we consider a PDM distribution of the form m(x)=c21±be−λx−r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m(x)=c^{2}\left (1\pm \text {be}^{-\lambda x}\right )^{-r}$\end{document}, where the different parameters can be adjusted depending on the potential under study. The principal advantage of the method is that solution of different specific PDM exponential potential models are obtained as particular cases from the proposal by means of a simple choice of the involved exponential parameters. This means that is not necessary resort to specialized methods for solving second-order differential equations as usually done for each specific potential. Also, the usefulness of our results is shown with the calculation of s-waves scattering cross-section for the Hulthén potential although this kind of study can be extended to other specific potential models such as PDM deformed potentials.