Anharmonic and damped correction to displaced Franck–Condon factors for simulating molecular absorption and fluorescence spectra

The anharmonic correction to displaced Franck–Condon (FC) factors introduces a first-order perturbative correction to the Huang–Rhys factors, enabling accurate simulation of vibronic spectra in the gas phase. In contrast, the damped correction to displaced FC factors provides a non-perturbative modification to the Huang–Rhys factors for simulating vibronic spectra in solution, where the damped local-mode motions of the solute molecule effectively represent strong solute–solvent interactions. Practically, the damped FC method is implemented by transforming the mass-weighted unperturbed Hessian matrix (gas phase) into a perturbed Hessian matrix (solution phase) through effective scaling of atomic masses. The anharmonic FC method is applied to simulate the absorption and fluorescence spectra of the pyridine molecule. The results reveal an enhancement in absorption intensity and a corresponding weakening of fluorescence in the first excited state, which are in good agreement with experimental observations. The damped FC method is further employed to reproduce solvent-enhanced absorption and fluorescence spectra for perylene and carbazole molecules in solution. For perylene in benzene, a single-group mass-scaling parameter successfully reproduces the observed spectral enhancement. For carbazole in n-hexane, however, multiple-group mass-scaling parameters are required to capture the extremely enhanced spectral features. Overall, the present anharmonic and damped corrections to displaced Franck–Condon factors provide an analytically tractable and efficient framework for simulating vibronic spectra of large and complex molecular systems.