# 3.5: Separation of Vibration and Rotation

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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.