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 \(J\)):

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

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

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

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

and have solutions whose energies separate

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

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

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

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

Contributors and Attributions

Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry and Jeff A. Nichols (Oak Ridge National Laboratory)