A method for calculating vibrational transition probabilities

Abstract

Relative vibrational transition probabilities for diatomic molecules can be calculated by methods utilizing band intensity observations or by methods not requiring such data. To date the latter class has been represented solely by the linear dipole moment function approximation. In this paper a second such method is developed. It makes use of the wave function expansion technique of Trischka and Salwen. Its critical assumption is the approximate equivalence of the Morse and simple harmonic potentials in the region about the potential minimum. This assumption is sufficient to remove all ambiguity in the relative signs of the matrix elements of the dipole moment. Further, it provides a relation through which the relative magnitudes of the matrix elements can be calculated. It is shown that this method is equivalent to using Morse eigenfunctions with a dipole moment function which departs from linearity in a manner that is qualitatively correct, at least in the region of the equilibrium configuration. The method is compared extensively with the linear approximation and with experiment. Insofar as the limited intensity data now available can show, it is generally superior to the linear approximation. The method can be of use in deriving Taylor coefficients for the expansion of the dipole moment function and in obtaining vibration-rotation interaction parameters. It also provides a new method for obtaining absolute values of transition probabilities. The experimental data required for this are values of the permanent dipole moment in two different vibrational levels. The procedure is illustrated by the calculation of the fundamental band strength of lithium hydride. The linear approximation and the present method are compared briefly with two methods which require intensity data. These are the quadratic and cubic approximations to the dipole moment function. It is suggested that the extent and the accuracy of the intensity data presently available is not sufficient for these methods to be of general use. Relative transition probabilities calculated by the present method are tabulated for all bands with v′ ≦ 9 for OH, HCl, and CO. The computational procedure used is outlined. It is not suitable for hand calculation but can be programmed for a high-speed computing machine with relative ease. On a large machine about one minute is required to compute the entire contents of each table.