Vibronic intensity distributions in the electronic spectra of polyatomic molecules

Intensity distributions in progressions in non-totally symmetric vibrations are discussed. It is shown that these distributions can be expressed in concise form as the terms in certain simple series of the form (1 - x)^-n/2, where n is a positive integer, x = {(ν" - ν')/(ν" + ν')}² and ν', ν" are the vibration frequencies in upper and lower states respectively. For the progressions v^l <- l^l v = l, l+2, l+4,... in electronically allowed systems, the intensity of the band v^l <- l^l relative to that of the band l^l <- l^l is given by the {½(v - l + 2)}th term in the series (1 - x)^{-(2[vertical line]l[vertical line]+d)/2}, where d is the degeneracy of the vibration, v is the vibrational quantum number and [vertical line]l[vertical line] = 0 (d = 1) or v, v - 2,..., 0 or 1 (d = 2, 3). It is also shown that (intensity of band l^l <- l^l)/(Σ_ν intensity of band v^l <- l^l) = q^{2[vertical line]l[vertical line]+d} where q = (ν'ν")^½/½(ν' + ν").

Systems which are electronically forbidden are discussed in terms of the Herzberg-Teller series expansion of the transition moment, and it is shown that for the forbidden progression v¹ <- 0⁰ v = 1, 3, 5... the intensity of the band v¹ <- 0⁰ relative to that of the band 1¹ <- 0⁰ is given by the {½(v + 1)}th term in the series (1 - x)^-(d+2)/2. Also, the ratio (intensity of band 1¹ <- 0⁰)/(Σ_ν intensity of band ν¹ <- 0⁰) = q^d+2. The results are applied to the 2600 Å system of benzene; the agreement between experiment and theory is poor. The application of the results to the infra-red spectrum is indicated, and finally the effects of mechanical and electrical anharmonicity are briefly discussed.