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Based on the second-order perturbation combining spin-orbit and nonadiabatic couplings, we derived an analytical formula for nonradiative decay rate between the triplet and singlet states by using the thermal vibration correlation function (TVCF) approach. Origin displacement, distortion, and Duschinsky rotation of the potential energy surfaces are taken into accounts within the multiple harmonic oscillator model. When coupled with first-principles calculation for the anthracene, the theoretical phosphorescence spectrum is in good agreement with the experiment. Furthermore, we found that the intersystem crossing from the first excited singlet state (S1) to the triplet states S1(Bu)→T2(Ag) is forbidden by direct spin-orbit coupling at the first-order perturbation but becomes allowed through combined spin-orbit and the nonadiabatic couplings at the second-order perturbation, and the rate is calculated to be 0.26 × 10(8) s(-1), in good agreement with the experiment. Such formalism is also applied to describe the phosphorescence quantum efficiency and the temperature dependent optical emission spectrum for fac-tris(2-phenylpyridine) iridium. We predict that the radiative decay rate is 6.36 × 10(5) s(-1), the nonradiative decay rate is 5.04 × 10(4) s(-1), and the phosphorescence quantum efficiency is found to be 92.7% from T1 to S0, which reproduce well the corresponding experimental measurements.
Peng et al. (Mon,) studied this question.