Energy Distribution Model Using Modified Hylleraas Potential (MHsP) Based Non-Relativistic Schrödinger Equation

Abstract

Solving the Schrödinger equation for systems with multiple particles is particularly challenging due to the intricacies of inter-particle interactions. In this study, we utilize the Modified Hylleraas Potential (MHsP), which provides a more precise depiction of these interactions. Incorporating this refined potential into the non-relativistic Schrödinger equation enables a deeper insight into atomic and molecular structures, as well as their dynamic behaviour. The study further explores how changes in the dimensionless potential parameters  influence the predicted energy spectrum. It was found that for  values between 0.01 and 0.03, the molecular interactions intensify, leading to a higher energy spectrum. However, this effect diminishes at  when evaluated against ​. Similarly, a significant energy spectrum is noted for λ values ranging from 1 to 3, but this decreases at λ = 4 when analysed in conjunction with ​ under varying ​. Results derived using Maxwell-Boltzmann, Fermi-Dirac, and Boltzmann statistical models indicate that at lower values of the temperature parameter (β), intermolecular interactions are stronger. Conversely, higher β values correspond to weaker interactions and a lower energy spectrum. Moreover, plotting the energy spectrum against β under different dimensionless parameters shows that energy increases as these parameters rise. The behaviour of the partition function mirrors these trends—initially increasing at low β values and decreasing at higher temperatures, thereby also impacting molecular interactions.