3.4: Rotation and Vibration of Diatomic Molecules
- Page ID
- 60516
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.