Exact Quantization Rule to the Kratzer-Type Potentials: An Application to the Diatomic Molecules

For any arbitrary values of $n$ and $l$ quantum numbers, we present a simple exact analytical solution of the $D$-dimensional ($D\geq 2$) hyperradial Schr% \"{o}dinger equation with the Kratzer and the modified Kratzer potentials within the framework of the exact quantization rule (EQR) method. The exact energy levels $(E_{nl})$ of all the bound-states are easily calculated from this EQR method. The corresponding normalized hyperradial wave functions $% (\psi_{nl}(r))$ are also calculated. The exact energy eigenvalues for these Kratzer-type potentials are calculated numerically for the typical diatomic molecules $LiH,$ $CH,$ $HCl,$ $CO,$ $NO,$ $O_{2},$ $N_{2}$ and $I_{2}$ for various values of $n$ and $l$ quantum numbers. Numerical tests using the energy calculations for the interdimensional degeneracy ($D=2-4$) for $I_{2}, $ $LiH,$ $HCl,$ $O_{2},$ $NO$ and $CO$ are also given. Our results obtained by EQR are in exact agreement with those obtained by other methods.