Klein-Gordon equation and machine learning-enhanced functional analysis: Insights into diatomic molecular systems via analytical and predictive modeling approaches

This study investigates the behavior of spinless particles under the influence of scalar and vector potentials by analytically solving the Klein-Gordon equation using the Nikiforov-Uvarov functional analysis method, coupled with the Hellmann and modified Kratzer potentials, employing the Greene-Aldrich approximation for the centrifugal term. The analytical energy eigenvalues and eigenfunctions were utilised to examine the energy spectra of specific diatomic molecules (CO, NO, N₂, and CH), demonstrating the correlation of these properties with potential parameters and quantum numbers. In addition to the analytical results, machine learning techniques like Random Forest and Neural Network regressors were used to model and predict the energy spectra based on the calculated data. This made it possible to swiftly explore energy landscapes. The ML models showed great agreement with the analytical results and were better at extrapolating to new quantum numbers and molecular types. This hybrid analytical-ML approach is a strong way to speed up the study of diatomic molecular systems. It combines the rigour of quantum mechanics with data-driven predictions and makes it possible to efficiently screen molecular energy spectra in theoretical and computational chemistry.