Intensity Distribution in a Band System of Symmetrical Triatomic Molecules

Abstract

The Franck-Condon theory has been extended to a study of band intensities of a triatomic molecule of the general type $X{Y}_2$. The most probable transitions are found to be ${v_1}^{\prime }=\{A_1\pm B_1{v_1}^{\frac{1}{2}}\pm C_1{v_2}^{\frac{1}{2}}+D_1{v}_1+E_1{v}_2-F_1{v}_1-G_1{v}_2+H_1{\left(v_1{v}_2\right)}^{\frac{1}{2}}\}\{A_1\pm B_1{v_1}^{\frac{1}{2}}\mp C_1{v_2}^{\frac{1}{2}}+D_1{v}_1+E_1{v}_2-F_1{v}_1-G_1{v}_2-H_1{\left(v_1{v}_2\right)}^{\frac{1}{2}}\}$ ${v_2}^{\prime }=\{A_2\pm B_2{v_1}^{\frac{1}{2}}\pm C_2{v_2}^{\frac{1}{2}}+D_2{v}_1+E_2{v}_2-F_2{v}_1-G_2{v}_2+H_2{\left(v_1{v}_2\right)}^{\frac{1}{2}}\}\{A_2\pm B_2{v_1}^{\frac{1}{2}}\mp C_2{v_2}^{\frac{1}{2}}+D_2{v}_1+E_2{v}_2-F_2{v}_1-G_2{v}_2-H_2{\left(v_1{v}_2\right)}^{\frac{1}{2}}\}$ ${v_3}^{\prime }=\left\{\right(\frac{{\nu }_3}{{{\nu }_3}^{\prime }}\left)v_3\right\}\left\{\right(\frac{{{\nu }_3}^{\prime }}{{\nu }_3}\left)v_3\right\}$ where the $v\text{'}\mathrm{s}$ with and without a prime refer to the vibrational quantum numbers of the upper and lower states, respectively, and the values of the coefficients depend on the atomic masses, normal frequencies, force constants, and molecular dimensions. Two special cases which occur when the three atoms become equal and when the three atoms lie along a straight line are also considered. A wave mechanical treatment is outlined. It is found that for a triangular model the integrals, which measure the transition probabilities, corresponding to the (${v_3}^{\prime }$, 0) transitions when ${v_3}^{\prime }$ is an odd integer all vanish, and that for a linear model, in addition to these, all integrals pertaining to the (${v_2}^{\prime }$, 0) transitions when ${v_2}^{\prime }$ is an odd integer become zero. These results are, however, not to be interpreted as selection rules; they are simply consequences of the assumption of particular force fields. The results have been discussed in connection with the band intensities of Cl${\mathrm{O}}_2$. It is shown that a knowledge of the structure of the excited molecule is essential to test quantitatively the results of the present work.