We present three formulas for calculating intersystem crossing rates in the Condon approximation to the golden rule by means of a time-dependent approach: an expression using the full time correlation function which is exact for harmonic oscillators, a second-order cumulant expansion, and a short-time approximation of this expression. While the exact expression and the cumulant expansion require numerical integration of the time correlation function, the integration of the short-time expansion can be performed analytically. To ensure convergence in the presence of large oscillations of the correlation function, we use a Gaussian damping function. The strengths and weaknesses of these approaches as well as the dependence of the results on the choice of the technical parameters of the time integration are assessed on four test examples, i.e., the nonradiative S₁ $⇝$ T₁ transitions in thymine, phenalenone, flavone, and porphyrin. The obtained rate constants are compared with previous results of a time-independent approach. Very good agreement between the literature values and the integrals over the full time correlation functions are observed. Furthermore, the comparison suggests that the cumulant expansion approximates the exact expression very well while allowing the interval of the time integration to be significantly shorter. In cases with sufficiently high vibrational density of states also the short-time approximation yields rates in good agreement with the results of the exact formula. A great advantage of the time-dependent approach over the time-independent approach is its excellent computational efficiency making it the method of choice in cases of large energy gaps, large numbers of normal modes, and high densities of final vibrational states.

In this work, we present the generalization of a time-dependent method for the calculation of intersystem crossing (ISC) rates in the Condon approximation. When ISC takes place between electronic states with the same orbital type, i.e., when the transition is forbidden according to the El-Sayed rules, it is necessary to go beyond the Condon approximation. Similar to the Herzberg-Teller expansion of the vibronic interaction, the electronic spin–orbit matrix elements are assumed to depend linearly on the nuclear coordinates. The ISC rate is then a sum of three contributions: a direct, mixed direct-vibronic, and vibronic term. The method, presented in this work, is based on the generating function formalism and the multi-mode harmonic oscillator approximation. In addition to the zero-temperature case, we implemented formulae for finite-temperature conditions assuming a Boltzmann population of vibrational levels in the initial state. Tests have been carried out for a variety of molecules for which literature data were available. We computed vibronic one-photon spectra of free-base porphyrin and free-base chlorin and calculated ISC rates for xanthone, thioxanthone, thionine, as well as free-base porphyrin and found excellent agreement with previous results. Quantitative rates for triplet formation in rhodamine A have been determined theoretically for the first time. We find the S1↝ T2 channel to be the major source of triplet rhodamine formation in the gas phase.

Cinnoline (1,2-diazanaphthalene) is of particular interest among the diazanaphthalenes. Its triplet quantum yield upon photoexcitation depends strongly on the temperature and the solvent environment. At the beginning of this study, the properties of the lowest triplet electronic state were not understood either. To elucidate the photophysics of cinnoline, we implemented algorithms based on the time-dependent approach for calculating intersystem crossing rates and one-photon spectra of thermally equilibrated vibronic levels. Our quantum chemical investigations reveal that the triplet formation in hydrocarbon solutions at low temperatures is an El-Sayed forbidden process. At higher temperatures and in hydroxylic solutions an additional El-Sayed allowed channel opens up, increasing the intersystem crossing rate substantially. Furthermore, we have solved the old puzzle concerning the character of the lowest triplet state of cinnoline. In the gas phase the electronic structure has mainly nπ* character with additional contributions from ππ* configurations since the nuclear arrangement in the pyridazine ring is not planar. In hydroxylic solvents, the electronic structure of the T₁ state is altered. The simulation of the triplet emission shows that the experimentally observed phosphorescence of cinnoline in ethanol most certainly stems from the ³(ππ*) emission.