  Frozen Density Embedding (FDE) represents a versatile embedding scheme to
describe the environmental effect on the electron dynamics in molecular
systems. The extension of the general theory of FDE to the real-time
time-dependent Kohn-Sham method has previously been presented and implemented
in plane-waves and periodic boundary conditions (Pavanello et al. J. Chem.
Phys. 142, 154116, 2015). In the current paper, we extend our recent
formulation of real-time time-dependent Kohn-Sham method based on localized
basis set functions and developed within the Psi4NumPy framework (De Santis et
al. J. Chem. Theory Comput. 2020, 16, 2410) to the FDE scheme. The latter has
been implemented in its "uncoupled" flavor (in which the time evolution is only
carried out for the active subsystem, while the environment subsystems remain
at their ground state), using and adapting the FDE implementation already
available in the PyEmbed module of the scripting framework PyADF. The
implementation was facilitated by the fact that both Psi4NumPy and PyADF, being
native Python API, provided an ideal framework of development using the Python
advantages in terms of code readability and reusability. We demonstrate that
the inclusion of the FDE potential does not introduce any numerical instability
in time propagation of the density matrix of the active subsystem and in the
limit of weak external field, the numerical results for low-lying transition
energies are consistent with those obtained using the reference FDE
calculations based on the linear response TDDFT. The method is found to give
stable numerical results also in the presence of strong external field inducing
non-linear effects.
