Improved energy equations and thermal functions for diatomic molecules: a generalized fractional derivative approach

This work presents analytical expressions for ro-vibrational energy models of diatomic molecules by introducing fractional parameters to improve molecular interaction analysis. Thermodynamic models, including Helmholtz free energy, mean thermal energy, entropy, and isochoric heat capacity, are formulated for diatomic molecules such as CO (X ¹∑⁺), Cs₂ (3 ³∑_g⁺), K₂ (X ¹∑_g⁺), ⁷Li₂ (6 ¹Π_u), ⁷Li₂ (1 ³Δ_g), Na₂ (5 ¹Δ_g), Na₂ (C(2) ¹Π_u), and NaK (c ³∑⁺). The incorporation of fractional parameters improves predictive accuracy for vibrational energies, as shown by reductions in percentage average absolute deviations from 0.5511 to 0.2185% for CO. Findings indicate a linear decrease in Helmholtz free energy and an initial increase in heat capacity with rising temperature, providing valuable insights for characterizing materials and optimizing molecular processes in chemistry, material science, and chemical engineering. The results obtained show strong agreement with established theoretical predictions and experimental data, validating the robustness and applicability of the proposed models.

The energy equations are derived by solving the radial Schrödinger equation for a variant of the Tietz potential using the generalized fractional Nikiforov-Uvarov (GFNU) method in addition to a Pekeris-type approximation for the centrifugal term. The canonical partition function is derived using the modified Poisson series formula, which serves as a basis for calculating other thermodynamic functions. All computations are carried out using MATLAB programming software.