Comparison of vertical and adiabatic harmonic approaches for the calculation of the vibrational structure of electronic spectra

The calculation of the vibrational structure associated to electronic spectra in large molecules requires a Taylor expansion of the initial and final state potential energy surface (PES) around some reference nuclear structure. Vertical (V) and adiabatic (A) approaches expand the final state PES around the initial-state (V) or final-state (A) equilibrium structure. Simplest models only take into account displacements of initial- and final-state minima, intermediate ones also allow for difference in frequencies and more accurate models introduce the Dushinsky effect through the computation of the Hessians of both the initial and final state. In this contribution we summarize and compare the mathematical expressions of the complete hierarchy of V and A harmonic models and we implement them in a numerical code, presenting a detailed comparison of their performance on a number of prototypical systems. We also address non-Condon effects through linear expansions of the transition dipole as a function of nuclear coordinates (Herzberg–Teller effect) and compare the results of expansions around initial and final state equilibrium geometries. By a throughout analysis of our results we highlight a number of general trends in the relative performance of the models that can provide hints for their proper choice. Moreover we show that A and V models including final state PES Hessian outperform the simpler ones and that discrepancies in their predictions are diagnostic for failure of harmonic approximation and/or of Born–Oppenheimer approximation (existence of remarkable geometry-dependent mixing of electronic states).