Molecules formed by hydrogen atoms as finite potential wells: numerical solution of the Schröndiger equation and proposal for didactic extension via mental models

Abstract

In this paper, we numerically solve the one-dimensional Schrödinger equation with a box-type potential (of finite depth) for systems of homonuclear and heteronuclear diatomic molecules. The analyzed molecules are composed of elements from the 1A group of the periodic table, aiming to estimate the depth of a finite potential box that best describes each of the molecules under study. We determine the values of the potential box depth and width (atomic diameter) that allow the model to reproduce the ground-state eigenenergies obtained using the ab-initio Hartree-Fock method with a def2-TVZP basis set. The numerical solutions found, which match the ab-initio results, correspond to potential depths V₀ ranging from –0.84 to –0.14 Ha and widths L (molecular diameter) ranging from 2.8 to 14.7 Bohr. In addition, we will aim to propose, as an extension, a possible didactic approach to this formulation, using Johnson-Laird’s theory of mental representation, by his mental models, as a theoretical framework, which will be justified by the phenomenological properties of the conceptual object.

Keywords:
Schrödinger equation; Hartree-Fock; Diatomic molecules; Box potential; Johnson-Laird’s mental models.