Intersystem Crossing as Vibronically-Induced Phonon Emission and Absorption Processes: A Unified View of Nonradiative Transitions in a Molecule

An analytical expression for the nonradiative rate constant is derived based on Fermi's golden rule within the mixed-spin crude adiabatic (CA) approximation. The mixed-spin CA basis is defined by a set of eigenstates for the electronic Hamiltonian that comprises the nonrelativistic electronic Hamiltonian and spin-orbit coupling clumped at the reference nuclear configuration. The mixed-spin basis differs from the pure-spin basis defined by a set of eigenstates for the nonrelativistic electronic Hamiltonian. The mixed-spin CA representation provides a unified view of the nonradiative transition; both internal conversion and intersystem crossing (ISC) are regarded as vibronically-induced phonon emission and absorption processes. The analytical expression enables us to determine important vibrational modes responsible for phonon emission/absorption (promoting modes) and accepting excitation energy (accepting modes) according to the selection rule of vibronic coupling. An advantage of the CA representation is that the spatial distribution of vibronic coupling is elucidated based on its density form, i.e., vibronic coupling density, which can be applied to theoretical molecular design with controlled nonradiative processes. The calculated ISC rate constant of tetracene reproduces the experimental result well.