Parametrization of Linear Vibronic Coupling Models for Degenerate Electronic States

Linear vibronic coupling (LVC) models have proven to be effective in describing coupled excited-state potential energy surfaces of rigid molecules. However, obtaining the LVC parameters in molecules with many degrees of freedom and a large number of, possibly (near-)degenerate, electronic states can be challenging. In this paper, we discuss how the linear intra- and interstate couplings can be computed correctly using a numerical differentiation scheme, requiring a phase correction and sufficient numerical precision in the involved electronic structure calculations. The numerical scheme is applied to three test systems with symmetry-induced state degeneracies: SO₃, [PtBr₆]^2-, and [Ru(bpy)₃]²⁺. The first two systems are employed to validate the performance of the parametrization scheme. LVC potentials for SO₃ are shown to reproduce the trigonal symmetry of the potential energy surfaces. The integration of the LVC potentials for [PtBr₆]^2- with the surface-hopping trajectory method illustrates how spurious parameters lead to erroneous trajectory behavior. In the transition metal complex [Ru(bpy)₃]²⁺, extensive nonadiabatic simulations using LVC potentials are compared to those conducted with direct on-the-fly potentials. The simulations with LVC potentials demonstrate excellent agreement with the on-the-fly results while incurring costs that are 5 orders of magnitude lower. Further, the simulations evidence that intersystem crossing in [Ru(bpy)₃]²⁺ occurs at a slightly slower rate than luminescence decay, underscoring the importance of simulating the actual experimental observable when comparing computed time constants with experimental time constants. Lastly, the initial nuclear response to excitation involves a rapid, short-lived, and small elongation of the Ru-N bonds, with no charge localization occurring on a sub-ps time scale.