Franck-Condon factors and ladder operators

Within the framework of operator algebra methods, perhaps the simplest approach to treating the FC integral is to consider the ladder operators associated with each of the wells, i.e., the ground (G) and excited (E) states whose minima are shifted with respect to each other by a distance 

Although there exist several relations involving these operators, they are the only ones that lead to recurrence relations which are useful in spectroscopy calculations. The derivation of such relations is much simpler than by using the traditional method, i.e., through recurrence relations between Hermite polynomials, since we need to use only the well-known properties of ladder operators  

These are the familiar recurrence relations originally derived by Wagner [11] and Ansbacher [12], Closed formula can also be obtained by means of these techniques, for which we need to introduce the well-known Cauchy relation in the complex variable theory 

The ladder operators uq and can be written using equation (3) and two complex variables and, after substitution in equation (4) and some algebra. 

Although this formula could hardly be of any practical use in spectroscopy, its importance lies in the fact that the FC overlap has an aniytic closed formnla for the harmonic oscillator potential. It was derived for the first time by Ansbacher with a minor mistake, which has inspired some authors [13,14] to derive analogous formulae for other potentials. The above equation is beautiful and elegant because it represents an exact and closed expression for the FC overlap, with the restriction of being valid only for the special case of the harmonic oscillator. The application of this method to matrix elements of monomial, exponential and Gaussian operators is straightforward and has been published elsewhere [15,16]. 

---