Intensities of Vibration-Rotation Bands with Special Reference to Those of HCl

Abstract

Intensity formula for vibration bands. The intensities of vibration bands depend on certain matrix elements of the electric moment of the molecule. The electric moment of a diatomic molecule is a function of the nuclear separation and must be expanded in a power series about the equilibrium point. Matrix elements are calculated by perturbation methods for the fundamental and first two harmonic bands of the vibration spectrum, and it is found that, to a first approximation, for the ${\mathit{n}}^{\mathrm{th}}$ harmonic it is necessary to consider the ${(\mathit{n}+1)}^{\mathit{th}}$ power in the series expansion of the electric moment, and higher powers for better approximations. The formulas for the fundamental and first harmonic are given to a second approximation. The matrix elements are also calculated from wave functions due to Morse for a diatomic molecule, and it is shown that there is a negligible difference between the two methods of calculation for small quantum numbers. Formulas are also given for the ratio of the intensity of the first two harmonic bands to that of the fundamental.Application to HCl. The formulas derived in the first part are applied to the case of HCl which is the only molecule for which the intensities of the vibration bands have been measured with any precision. New data from wave-length measurements of Meyer and Levin are used and the value of the coefficient of the quadratic term in the power series expansion of the electric moment is found. It is found that two values of this coefficient would give the same intensities and no satisfactory way of resolving the ambiguity is available. Numerically it is found that if the electric moment $p={p}_{e}+p_{e}^{}{}_{}{}^{\ensuremath{'}}\ensuremath{\xi}+\frac{p_{e}^{}{}_{}{}^{\ensuremath{'}\ensuremath{'}}{\ensuremath{\xi}}^{2}}{2}$, then $p_{e}^{}{}_{}{}^{\ensuremath{'}\ensuremath{'}}=0.070\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18} or 4.56\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}$ e.s.u.