The statistical ensemble of q-deformed hyperbolic modified P\"oschl-Teller potential for certain diatomic molecules through Euler-Maclaurin  approach

The statistical properties are essentially needed to understand the macroscopic behaviours of atomic molecules, which is a crucial aspect of physics and chemistry research. In this work, q-deformed hyperbolic modified P"oschl-teller was used to obtain the statistical properties of $\mathrm{H_{2}}$, $\mathrm{HCl}$, $\mathrm{LiH}$, and $\mathrm{CO}$ using the energy eigenvalues that were gotten from the potential. The Nikiforov-Uvarov approach was used to elucidate the Schr"odinger equation using the potential, while the partition functions were obtained through the Euler-MacLaurin technique. The result shows that both the partition functions and each of the statistical properties increase with an increase in the Boltzmann factor. The analytical results of vibrational internal energy, vibrational entropy, vibrational free energy, and vibrational specific heat capacity were obtained for the limit $0<q\leq 1$. The study comes up with a new way to predict how potential deformation will affect the thermodynamic properties of diatomic molecules and the existence critical temperature which explain the phase transition of the diatomic particles. It also gives us useful information about the statistical properties of diatomic molecules, which helps us understand how they behave at the macro level. This is a big step forward for molecular physics.