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