Franck–Condon Factors for Diatomics: Insights and Analysis Using the Fourier Grid Hamiltonian Method

Abstract

Franck–Condon factors (FCFs) play a crucial role in determining the intensities of the vibrational bands in electronic transitions. In this article, a relatively simple method to calculate the FCFs is illustrated. An algorithm for the Fourier Grid Hamiltonian (FGH) method for computing the vibrational wave functions and the corresponding energy values for different electronic states of diatomics is outlined. For these computations, the Morse potential energy forms are used for constructing and diagonalizing the molecular Hamiltonians. Once the vibrational wave functions for the ground and the excited states are known, the vibrational overlap integrals are calculated by using the formula ∫ψ*v″ψv′ dτn, where dτn is the volume element for nuclear coordinate, v′ and v″ are the vibrational quantum numbers in the ground and exited electronic states, and ψv′ and ψv″ denote the vibrational wave functions. The effects of the changes in the location, height, and the nature of the excited state electronic energy curve (relative to the ground state) on the FCFs have been evaluated. The method is also illustrated for N2 and O2. This method can be effectively used for introducing the FCFs to the students of undergraduate molecular spectroscopy course and also for post-graduate students.