PyGraSO: Analytical Nuclear Derivatives of Spin-Orbit Coupling for Intersystem Crossing Dynamics Simulations

We present PyGraSO, an open-source program for the efficient and accurate calculation of spin-orbit coupling matrix elements (SOCMEs) and their analytical nuclear derivatives within the framework of time-dependent density functional theory. The SOCME derivatives are crucial physical quantities for capturing spin-vibronic coupling (SVC) effects associated with the induced SOC values via molecular vibrations. These SVC effects are often crucial for describing spin-forbidden or spin-flip processes, such as intersystem crossing (ISC), reverse ISC (RISC), and phosphorescence. Previously, these derivatives could only be obtained through numerical differentiation. With our new implementation, they can now be computed based on analytical formulas, underpinning a significant advance toward more accurate and efficient dynamics simulations of spin-forbidden processes where SVC effects are vital. Although developing the analytical nuclear derivative method is challenging, our implementation achieves substantial efficiency gains over numerical approaches, yielding up to a 36-fold speedup in benchmark tests. To showcase its applicability, we performed excited-state dynamics simulations based on thermal vibration correlation function (TVCF) formalism, incorporating SVC effects via SOCME derivatives, to predict RISC rate constants ($k_\mathrm{RISC}$) for a large set of 126 thermally activated delayed fluorescence (TADF) molecules. The results demonstrated that the inclusion of SVC effects improves agreement with experimental data, particularly for molecules with small reorganization energies, where the other methods that do not incorporate SOCME derivatives tend to fail.