  Accurate solution of the Schr\"odinger equation with Deng-Fan potential is
presented by means of Nikiforov-Uvarov method. A modified Pekeris-type
approximation is proposed for the centrifugal term, from a linear combination
of the $r \to 0$ and $r \to r_e$ limits. It can potentially offer a series of
approximations (depending on an adjustable parameter $\lambda$). The existing
approximations in the literature can then be recovered in certain special
cases. Its efficiency and feasibility is demonstrated by a critical comparison
of eigenvalues produced at various $\lambda$'s for four molecules, \emph{viz.},
H$_2$, LiH, HCl and CO. Analytical expressions are derived for energies,
eigenfunctions and the thermodynamic properties such as vibrational mean free
energy, vibrational free energy, vibrational entropy and vibrational specific
heat. The effect of quantum correction on partition function and thermodynamic
properties is discussed by including the correction up to 10th-order, for H$_2$
and LiH. The effect of $\lambda$ parameter on these properties is also studied.
