Band Spectra Intensities for Symmetrical Diatomic Molecules

Abstract

The intensity of a spectral line may be calculated in the new quantum mechanics by evaluating the integral of the product of the electric moment and the wave functions of the initial and of the final states. The complete wave function may be written approximately as a product of a nuclear wave function and an electronic wave function. Furthermore, the electric moment can be approximately written as the sum of a function of the nuclear coordinates only and a function of the electronic coordinates only. For symmetrical diatomic molecules the term in the electric moment which is a function of the nuclear coordinates only is zero so that the intensity integral reduces to a double integral which may be written as a product of an electronic integral and a nuclear integral. The electronic integral is constant for all the lines corresponding to the same electronic transition and the integrand of the nuclear integral consists only of the product of the radial wave functions for the initial and the final states. The integration of the nuclear integral is carried out using harmonic oscillator wave functions whose origins are shifted due to the change of the nuclear separation during the electronic transition.A comparison is made between the intensities calculated in the above manner and those obtained by experiment. In the case of ${\mathrm{Na}}_{2}$ good quantitiative agreement is obtained between the experimental and the calculated results. Accurate values of the moments of inertia in the initial and final states are not known for ${\mathrm{K}}_{2}$ but it is possible to show that if a value of the change in the nuclear separations during the transition is assumed, agreement is possible for all of the lines corresponding to low quantum numbers. Good qualitative agreement is obtained for ${\mathrm{I}}_{2}$. Calculations are made for the absorption spectra and the ${3}^{\ensuremath{'}}B\ensuremath{-}{2}^{\ensuremath{'}}S$ emission band system of ${\mathrm{H}}_{2}$ but the agreement in these cases is not as good as for ${\mathrm{Na}}_{2}$ and ${\mathrm{K}}_{2}$.