Analytic and Algebraic Evaluation of Franck−Condon Overlap Integrals

Abstract

An analytic and algebraic evaluation of Franck−Condon overlap integrals for harmonic oscillators displaced by an amount Δ and of different frequencies (ω, ω‘) is presented. The results are extended to Morse oscillators to first order in effective anharmonicities.

The vibronically-resolved emission spectrum of disulfur monoxide (S2O): An algebraic calculation and quantitative interpretation of Franck–Condon transition intensities

“…͑12͒ entail the eigenstates of one-dimensional Morse ͑or Pöschl-Teller͒ potentials and can be evaluated by a variety of methods. Rather than resorting to a purely numerical procedure, the ensuing FranckCondon calculations make use of an analytical formula 48,68 derived through perturbative correction of the analogous overlap expression for two harmonic oscillators characterized by different frequencies, (Ј), and concavities, ␣ (␣Ј), as well as relative spatial displacement, ⌬:…”

“…͑12͔͒ reveal the changes in molecular structure incurred through electronic excitation. 48,68 As demonstrated by the results compiled in Table IV, the precise relationship between ͑local͒ anharmonic oscillators comprising the ground and excited states can be ascertained from the analysis of FranckCondon intensities, thereby enabling the extraction of equilibrium geometries and related structural constants.…”

Application of the Nikiforov-Uvarov Method in Quantum Mechanics

“…However, the power series method has more details to reach the solution. The algebraic methods based on Lie algebra (Adams, 1994;Iachello & Levine, 1995;Iachello & Oss, 1996;Iachello & Ibrahim, 1998) are another tool to solve the SE in the framework of quantum mechanics. To constitute a suitable Lie algebra, the quantum system we are trying to find an exact solution has to be displayed a dynamical symmetry.…”

A quantitative study of non-Condon effects in the S2O C̃→X̃ emission spectrum

“…͑1͔͒ already has been reported. 12 The present incorporation of non-Condon effects ͓Eq. ͑2͔͒ builds upon an explicit analytical formula for the quantity ͗v͉M ͉vЈ͘ as derived for two harmonic oscillators characterized by different frequencies, (Ј), and concavities, ␣(␣Ј), as well as relative spatial displacement, ⌬:…”

“…When ␤ϭ0, this expression reduces to that employed for Franck-Condon analyses. 12 In the non-Condon case (␤ 0), Eq. ͑3͒ can be used to investigate anharmonic oscillators ͑e.g., Morse͒ by exploiting the same perturbative corrections introduced in our previous work, 4,5 ␣ϭ␣ 0 ͑ 1Ϫv ͒, ␣Јϭ␣ 0 Ј͑1ϪЈvЈ͒, first-order corrections to the width ͑i.e., concavity͒ and coordinate origin, respectively, for the harmonic oscillators embodied in Eq.…”