A general analytical expression for the two-dimensional Franck–Condon integral and simulation of the photoelectron spectra of nitrogen dioxide

Generalized Analytic Approach for Determination of Multidimensional Franck-Condon Factors: Simulated Photoelectron Spectra of Polynuclear Aromatic Hydrocarbons

An enhanced generalized analytic approach for determination of multidimensional Franck-Condon Factors (FCFs) enables efficient computational prediction of photoelectron spectra for large-dimensional systems. Incorporation of the automated assignment of Cartesian coordinate handedness and coordinate superposition between the ground and excited electronic states satisfies the Eckart conditions and allows evaluation of the Duschinsky effect. The model shows excellent agreement with experiments for the determination of FCFs and photoelectron spectra of a series of increasing dimensions polynuclear hydrocarbons (PAHs), including naphthalene, anthracene, phenanthrene, and pyrene. In addition, a high-resolution prediction of the PES for the 84-dimensional PAH corannulene provides motivation for an additional experimental study. For FCFs, coordinate transformation between the initial and final states rather than the dimension of the systems more greatly influences the complexity of the spectral band shapes.

Analytical Evaluation for Calculation of Two-Center Franck–Condon Factor and Matrix Elements

The Franck–Condon (FC) factor is defined as squares of the Franck–Condon (FC) overlap integral and represents one of the principle fundamental factors of molecular physics. The FC factor is used to determine the transition probabilities in different vibrational levels of the two electronic states and the spectral line intensities of diatomic and polyatomic molecules. In this study, new analytical formulas were derived to calculate Franck–Condon integral (FCI) of harmonic oscillators and matrix elements (xη, e−2cx, and e−cx2) including simple finite summations of binomial coefficients. These formulas are valid for arbitrary values. The results of formulas are in agreement with the results in the literature.