Show that the fractional change in the equilibrium value of the internuclear distance of a diatomic molecule, as a result of rotation, is given by

\[
\frac{\Delta r_{0}}{r_{0}} \simeq\left(\frac{\hbar}{\mu r_{0}^{2} \omega}ight)^{2} J(J+1)=4\left(\frac{\Theta_{r}}{\Theta_{v}}ight)^{2} J(J+1)
\]

here, \(\omega\) is the angular frequency of the vibrational state in which the molecule happens to be. Estimate the numerical value of this fraction in a typical case.