In this work, the improved q-deformed Scarf II oscillator (IQSO) is formulated through the Varshni conditions for diatomic molecule potential. Equation for bound state energy eigenvalues of the IQSO is obtained by the SUSYQM method, a Pekeris-like approximation scheme is used to model the centrifugal term of the Schrödinger equation. Expressions for molar entropy and Gibbs free energy are deduced for the IQSO from equations of energy eigenvalues and partition function within the context of Poisson summation formula. The IQSO is used to model experimental Rydberg-Klein-Rees (RKR) data of six diatomic molecules including: KRb (B ¹ Π), CO ⁺ (X ² Σ ⁺ ), Cl ₂ (X ¹ Σ _g ⁺ ), ICl (X ¹ Σ _g ⁺ ), NbO (X ¹ Σ ⁺ ), and K ₂ (X ¹ Σ _g ⁺ ). The average absolute deviation and mean absolute percentage deviation are used as accuracy indicators. Computed data revealed that the IQSO is an excellent model in fitting the RKR data of the diatomic molecules examined in this work. Numerical values of molar entropy and Gibbs free energy calculated for ICl (X ¹ Σ _g ⁺ ) molecule are almost indistinguishable from existing experimental data for gaseous ICl (X ¹ Σ _g ⁺ ) molecules. This suggests that the IQSO is a near perfect model for representing the internal vibrations of the ICl (X ¹ Σ _g ⁺ ) molecule.

In this paper, Scarf II potential is used to construct the improved Scarf II potential energy function (ISPEF) for applications to diatomic molecules. Conditions to be satisfied by a diatomic molecule potential are applied in the derivation of potential parameters of the ISPEF. Equation of bound state ro-vibrational energies is obtained for the ISPEF via a Pekeris-type approximation to the centrifugal term of the Schrödinger equation. The ISPEF and expression for eigen energies are used to model spectroscopic data of five diatomic molecules: ${\text{N}}_{\text{2}}^{+}({\text{X}}^{\text{1}}\sum _{\text{g}}^{+})$, Na₂ (c¹ Π_u), ScI (B¹ Π), NaK (c³ Σ⁺) and CO (X¹ Σ⁺). Average absolute deviation from dissociation energy (AAD) and mean absolute percentage deviation from Rydberg-Klein-Rees data (MAPD) are used to analyse data. On the basis of AAD as goodness-of-fit index, the ISPEF represents a near-perfect model for the diatomic molecules analysed and also a better model than the existing literature data of improved Tietz potential for the Na₂ (c¹ Π_u) molecule. On the other hand, with the MAPD as goodness-of-fit index, the available literature results show that eigen energies of ISPEF are equivalent to ro-vibrational energies of Morse and improved generalised Pöschl-Teller potentials in fitting spectroscopic data of CO (X¹ Σ⁺) molecule. GRAPHICAL ABSTRACT

In this paper, the formalism of improved quantization rule is used to obtain analytical expression for bound state ro-vibrational energies of the improved Pöschl–Teller oscillator (IPTO). In modeling the centrifugal term of the effective potential, a Pekeris-like approximation scheme is used. Expression for analytical normalized eigenfunctions is obtained by ansatz solution technique. The equation obtained for ro-vibrational energy is then used to derive relations for some important thermodynamic functions for the IPTO; these include ro-vibrational: partition function , Helmholtz free energy , mean energy , entropy and specific heat capacity . Using the spectroscopic data of RbH (X ¹ Σ ⁺ ) diatomic molecule, pure vibrational state energies for the IPTO are calculated. The results obtained are in good agreement with experimental Rydberg–Klein–Rees data and are better than energies of the RbH (X ¹ Σ ⁺ ) diatomic molecule obtained via improved generalized Pöschl–Teller potential. Studies have also shown that the ro-vibrational partition function increases monotonically with increase in either temperature ( T ) or upper bound vibrational quantum number ( λ ). On the other hand, increase in any of T or λ results in monotonic decrease in ro-vibrational Helmholtz free energy of the system.