The Interaction Between Vibration and Rotation for Symmetrical Molecules

The explanation first proposed by Teller of the anomalous fine structure of the infrared bands of symmetrical molecules is discussed. The fine structure is closely related to an internal angular momentum of magnitude $\frac{\ensuremath{\zeta}h}{2\ensuremath{\pi}}$ which is due to the vibration of the molecule and arises from the degenerate character of the motion. A simple derivation is given for the spacing constant $\frac{[\frac{(1\ensuremath{-}\ensuremath{\zeta})}{C}\ensuremath{-}\frac{1}{A}]h}{4{\ensuremath{\pi}}^{4}}$ of axially symmetric molecules and for the spacing constant $\frac{(1\ensuremath{-}\ensuremath{\zeta})h}{4{\ensuremath{\pi}}^{2}A}$ of tetrahedral molecules. A detailed calculation is made of the internal angular momenta to be associated with the $\ensuremath{\perp}$ frequencies ${\ensuremath{\nu}}_{2}$ and ${\ensuremath{\nu}}_{4}$ of the axial molecule Y${\mathrm{X}}_{3}$. The resulting $\ensuremath{\zeta}$'s are found as functions of the moments of inertia and of the potential constants. The sum ${\ensuremath{\zeta}}_{2}+{\ensuremath{\zeta}}_{4}$ is shown to have the value $\frac{C}{2A}\ensuremath{-}1$. These results are applied to the molecules N${\mathrm{H}}_{3}$ and N${\mathrm{D}}_{3}$ and the line spacings of ${\ensuremath{\nu}}_{2}$ and ${\ensuremath{\nu}}_{4}$ are computed. The axial molecule ZY${\mathrm{X}}_{3}$ is treated and it is proved that the sum of the $\ensuremath{\zeta}$'s characterizing the three $\ensuremath{\perp}$ bands ${\ensuremath{\nu}}_{2}$, ${\ensuremath{\nu}}_{4}$, and ${\ensuremath{\nu}}_{6}$ is equal to $\frac{C}{2A}$. The moment of inertia $C$ of the methyl halides is computed and found to be, 5.61, 5.35, 5.44, and 5.44\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}40}$ for methyl fluoride to methyl iodide, respectively. The error is estimated to be around 5 percent. A calculation is made of the $\ensuremath{\zeta}$'s to be associated with the overtones of axial molecules possessing threefold symmetry. It is found that the $\ensuremath{\zeta}$ appropriate for the overtone $2{\ensuremath{\nu}}_{i}$ of a $\ensuremath{\perp}$ frequency is $\ensuremath{-}2{\ensuremath{\zeta}}_{i}$ while the $\ensuremath{\zeta}$ for $3{\ensuremath{\nu}}_{i}$ is ${\ensuremath{\zeta}}_{i}$ itself. The combination of two $\ensuremath{\perp}$ frequencies ${\ensuremath{\nu}}_{2}+{\ensuremath{\nu}}_{4}$ is next treated and the $\ensuremath{\zeta}$ is proved to be $\ensuremath{-}({\ensuremath{\zeta}}_{2}+{\ensuremath{\zeta}}_{4})=1\ensuremath{-}\frac{C}{2A}$. An application of these formulae to the observed spacings of the overtone bands of N${\mathrm{H}}_{3}$ and C${\mathrm{H}}_{3}$Cl results in a very satisfactory agreement. Expressions are obtained for the two $\ensuremath{\zeta}$'s which determine the line spacings of the active fundamentals ${\ensuremath{\nu}}_{3}$ and ${\ensuremath{\nu}}_{4}$ of the tetrahedral molecule Y${\mathrm{X}}_{4}$. It is shown that ${\ensuremath{\zeta}}_{3}+{\ensuremath{\zeta}}_{4}=\frac{1}{2}$. The positions of the fundamental bands of methane together with the line spacings of ${\ensuremath{\nu}}_{3}$ and ${\ensuremath{\nu}}_{4}$ yield the moment of inertia $A=5.47\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}40}$ together with the five potential constants describing the molecule. A good agreement is found between these constants and those obtained by Ginsburg and Barker from the spectrum of methyl deuteride. The $\ensuremath{\zeta}$'s suitable for the overtones of Y${\mathrm{X}}_{4}$ are next treated. It is found that the $\ensuremath{\zeta}$ to be used with $2{\ensuremath{\nu}}_{i}$, $i=3,4$ is $\ensuremath{-}{\ensuremath{\zeta}}_{i}$ while the $\ensuremath{\zeta}$ belonging with the combination band ${\ensuremath{\nu}}_{3}+{\ensuremath{\nu}}_{4}$ is $\ensuremath{-}\frac{1}{2}({\ensuremath{\zeta}}_{3}+{\ensuremath{\zeta}}_{4})=\ensuremath{-}\frac{1}{4}$. The observations on the overtones of methane are in very good accord with these expressions. Less satisfactory agreement is obtained in the case of the silane spectrum where it appears that the higher order perturbation terms play a larger role.