Introduction

Franck-Condon factors are the squares of the overlap integrals of vibrational wave functions. Given a transition between two electronic, spin or charge states, the Franck-Condon factors represent the probabilities for accompanying vibrational transitions. As such, they can be used to predict the relative intensities of absorption or emission lines in spectroscopy or excitation lines in transport measurements.

When a molecule makes a transition to another state, the equilibrium position of the nuclei changes, and this will give rise to vibrations. To determine which vibrational modes will be active, we first have to express the displacement of the nuclei in the normal modes:

k=L'^Tm^1/2(B₀x₀-x'₀)

Here, k is the displacement vector, L is the normal mode matrix, m is a matrix with the mass of the nuclei on the diagonal, B is the zero-order axis-switching matrix and x₀ is the equilibrium position of the nuclei. For free molecules, depending on symmetry constraints, the geometries of both states may have been translated and/or rotated with respect to each other. To remove the six translational and rotational degrees of freedom, we can center the equilibrium positions around the center of mass and rotate one of the states to provice maximum overlap. The latter is included with the zero-order axis-switching matrix B, implemented according to [19].

When we have obtained the displament vector, it is trivial to calculate the dimensionless electron-phonon couplings. They are given by:

λ=(Γ/2)^1/2k

Here, Γ=2πω/h is a vector containing the reduced frequencies. [20]

When the displacement vector k, the reduced frequencies Γ and Γ', and the Duschinsky rotation matrix J=L'^TB₀L have been obtained, the Franck-Condon factors can be calculated using the two-dimensional array method of Ruhoff and Ratner.[20]

There is one Franck-Condon factor for every permutation of the vibrational quanta over both states. Since they represent transition probabilities, all Franck-Condon factors of one state which respect to one vibrational state of the other state must sum to one. Since the total number of possible vibrational quanta, and hence the total number of permutations, is infinite, in practice we will calculate the Franck-Condon factors until those sums are sufficiently close to one. Since the number of permutations rapidly increases with increasing number of vibrational quanta, it is generally possible to already stop after the sum is greater than about two thirds. The remaining one third will be distributed over so many Franck-Condon factors that their individual contributions are negligible.

In the limiting case of one vibrational mode, with the same frequency in both states, the expression for the Franck-Condon factors of transitions from the ground vibrational state to an excited vibrational state are given by the familiar expression:

|I_0,n|²=e^-λ2λ²ⁿ/n!