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 (kRISC) 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.