3.4: Rotation and Vibration of Diatomic Molecules

Last updated
Save as PDF

Page ID: 60516

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

For a diatomic species, the vibration-rotation \(\left(\dfrac{V}{R}\right)\) kinetic energy operator can be expressed as follows in terms of the bond length R and the angles \(\theta \text{ and } \phi\) that describe the orientation of the bond axis relative to a laboratory-fixed coordinate system:

\[ T_{V/R} = \dfrac{-\hbar^2}{2\mu}\left[ \dfrac{1}{R^2}\dfrac{\partial}{\partial R}\left( R^2\dfrac{\partial}{\partial R} \right) - \left( \dfrac{L}{R\hbar}\right)^2 \right], \nonumber \]

where the square of the rotational angular momentum of the diatomic species is

\[ L^2 = \hbar^2 \left[ \dfrac{1}{sin \theta} \dfrac{\partial}{\partial \theta} \left( sin \theta \dfrac{\partial}{\partial \theta} \right) + \dfrac{1}{sin^2 \theta} \dfrac{\partial^2}{\partial \phi^2} \right]. \nonumber \]

Because the potential \(E_j (R)\) depends on R but not on \(\theta \text{ or } \phi \text{ , the } \dfrac{V}{R} \text{ function } Xi^0_{j,m}\) can be written as a product of an angular part and an R-dependent part; moreover, because \(L^2\) contains the full angle-dependence of \(T_{V/R} , Xi^0_{j,n}\) can be written as

\[ \Xi^0_{j,n} = Y_{J,M}(\theta,\phi)F_{j,J,v}(R). \nonumber \]

The general subscript n, which had represented the state in the full set of 3M-3 R-space coordinates, is replaced by the three quantum numbers J,M, and v (i.e., once one focuses on the three specific coordinates \(R,\theta, \text{ and } \phi\), a total of three quantum numbers arise in place of the symbol n).

Substituting this product form for \(\Xi^0_{j,n}\) into the \(\dfrac{V}{R}\) equation gives:

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

as the equation for the vibrational (i.e., R-dependent) wavefunction within electronic state j and with the species rotating with \(J(J+1) \hbar^2\) as the square of the total angular momentum and a projection along the laboratory-fixed Z-axis of \(M\hbar.\) The fact that the \(F_{j,J,v}\) functions do not depend on the M quantum number derives from the fact that the \(T_{V/R}\) kinetic energy operator does not explicitly contain \(J_Z\); only \(J^2\) appears in \(T_{V/R}.\)

The solutions for which J=0 correspond to vibrational states in which the species has no rotational energy; they obey

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

The differential-operator parts of this equation can be simplified somewhat by substituting \(F= \dfrac{\chi}{R}\) and thus obtaining the following equation for the new function \(\chi:\)

\[ \dfrac{-\hbar^2}{2\mu} \dfrac{\partial}{\partial R} \dfrac{\partial}{\partial R} \chi_{j,0,v}(R) + E_j(R) \chi_{j,0,v}(R) = E^0_{j,0,v}\chi_{j,0,v}. \nonumber \]

Solutions for which \(J\neq 0\) require the vibrational wavefunction and energy to respond to the presence of the 'centrifugal potential' given by \(\frac{\hbar^2 J(J+1)}{2\mu R^2}\); these solutions obey the full coupled V/R equations given above.