Incorporating Anharmonicity of Selected Modes into Fermi’s Golden Rule Rate Calculations for Large Molecular Systems

Properties of photofunctional materials are governed by nonadiabatic transitions, and theoretical methods to predict reaction rate constants for nonadiabatic transitions are desired. Fermi’s golden rule (FGR) is one of the most useful approaches for computing rate constants. In particular, FGR combined with the harmonic approximation to the potential energy surface (PES) is widely used, since it requires only geometry optimizations and Hessian calculations for the initial and final states. However, real PESs are inherently anharmonic, and such anharmonicity can affect the rate constant by altering vibrational energy levels and wave functions. In this study, we propose a method for computing rate constants that incorporates anharmonicity into the conventional harmonic-approximation framework. In this method, only a subset of important modes is treated anharmonically, whereas the remaining modes are treated harmonically. We applied the proposed method to two systems: the T1 → S0 nonradiative decay process of 4-pyrone molecule, and the S1 → T1 intersystem crossing process of 3-(9,9-dimethylacridin-10(9H)-yl)-9H-xanthen-9-one (ACRXTN) molecule. The calculation for 4-pyrone revealed that the rate constant decreases by 3 orders of magnitude compared with the harmonic rate when anharmonicity of all vibrational modes is taken into account, indicating the importance of considering the anharmonicity when calculating the rate constant. In addition, it was found that the fully anharmonic rate constant can be reproduced by considering the anharmonicity of only two vibrational modes. Further analysis based on the sum-over-states formalism revealed that this decrease in the rate constant originates from a reduction in the overlaps between the initial- and final-state vibrational eigenfunctions. It was also found that this decrease in the rate constant does not appear unless the anharmonicity of multiple vibrational modes is taken into account simultaneously, even though coupling between vibrational modes is not considered. The application to ACRXTN showed that the anharmonicity associated with dihedral torsional modes has little influence on the rate constant. Comparison of these results revealed that the anharmonicity of vibrational modes with high Huang–Rhys factors and high frequencies, or those whose harmonic frequencies are different between initial and final states, has an impact on the rate constant. The proposed method is expected to be useful for the theoretical design of new photofunctional materials.