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Frozen Density Embedding (FDE) represents a versatile embedding scheme todescribe the environmental effect on the electron dynamics in molecularsystems. The extension of the general theory of FDE to the real-timetime-dependent Kohn-Sham method has previously been presented and implementedin plane-waves and periodic boundary conditions (Pavanello et al. J. Chem.Phys. 142, 154116, 2015). In the current paper, we extend our recentformulation of real-time time-dependent Kohn-Sham method based on localizedbasis set functions and developed within the Psi4NumPy framework (De Santis etal. J. Chem. Theory Comput. 2020, 16, 2410) to the FDE scheme. The latter hasbeen implemented in its "uncoupled" flavor (in which the time evolution is onlycarried out for the active subsystem, while the environment subsystems remainat their ground state), using and adapting the FDE implementation alreadyavailable in the PyEmbed module of the scripting framework PyADF. Theimplementation was facilitated by the fact that both Psi4NumPy and PyADF, beingnative Python API, provided an ideal framework of development using the Pythonadvantages in terms of code readability and reusability. We demonstrate thatthe inclusion of the FDE potential does not introduce any numerical instabilityin time propagation of the density matrix of the active subsystem and in thelimit of weak external field, the numerical results for low-lying transitionenergies are consistent with those obtained using the reference FDEcalculations based on the linear response TDDFT. The method is found to givestable numerical results also in the presence of strong external field inducingnon-linear effects.

Environmental effects with Frozen Density Embedding in Real-Time Time-Dependent Density Functional Theory using localized basis functions