3.5: Separation of Vibration and Rotation

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It is common, in developing the working equations of diatomic-molecule rotational/vibrational spectroscopy, to treat the coupling between the two degrees of freedom using perturbation theory as developed later in this chapter. In particular, one can expand the centrifugal coupling \(\hbar^2 \frac{J(J+1)}{2\mu R^2}\) around the equilibrium geometry \(R_e\) (which depends, of course, on \(J\)):

\[ \hbar^2 \dfrac{J(J+1)}{2\mu R^2} = \hbar^2 \dfrac{J(J+1)}{2\mu [R_e^2 (1+\Delta R)^2]} \nonumber \]

\[ \hbar^2 \dfrac{J(J+1)}{2\mu R_e^2}[1 - 2\Delta R + ...], \nonumber \]

and treat the terms containing powers of the bond length displacement \(\Delta R^k\) as perturbations. The zeroth-order equations read:

\[ \dfrac{-\hbar^2}{2\mu}\left[ \dfrac{1}{R^2}\dfrac{\partial}{\partial R}\left( R^2\dfrac{\partial}{\partial R} \right) \right] F_{j,J,v}^0 (R) + E_j(R) F^0_{j,J,v}(R) + \hbar^2 \dfrac{J(J+1)}{2\mu R_e^2}F^0_{j,J,v} F^0_{j,J,v}, \nonumber \]

and have solutions whose energies separate

\[ E^0_{j,J,v} = \hbar^2\dfrac{J(J+1)}{2\mu R^2_e} + E_{j,v} \nonumber \]

and whose wavefunctions are independent of \(J\) (because the coupling is not R-dependent in zeroth order)

\[ F^0_{j,J,v}(R) = F_{j,v}(R). \nonumber \]

Perturbation theory is then used to express the corrections to these zeroth order solutions as indicated in Appendix D.