We
present an efficient method for propagating the time-dependent
Kohn–Sham equations in free space, based on the recently introduced
Fourier contour deformation (FCD) approach. For potentials which are
constant outside a bounded domain, FCD yields a high-order accurate
numerical solution of the time-dependent Schrödinger equation
directly in free space, without the need for artificial boundary conditions.
Of the many existing artificial boundary condition schemes, FCD is
most similar to an exact nonlocal transparent boundary condition,
but it works directly on Cartesian grids in any dimension, and runs
on top of the fast Fourier transform rather than fast algorithms for
the application of nonlocal history integral operators. We adapt FCD
to time-dependent density functional theory (TDDFT), and describe
a simple algorithm to smoothly and automatically truncate long-range
Coulomb-like potentials to a time-dependent constant outside of a
bounded domain of interest, so that FCD can be used. This approach
eliminates errors originating from the use of artificial boundary
conditions, leaving only the error of the potential truncation, which
is controlled and can be systematically reduced. The method enables
accurate simulations of ultrastrong nonlinear electronic processes
in molecular complexes in which the interference between bound and
continuum states is of paramount importance. We demonstrate results
for many-electron TDDFT calculations of absorption and strong field
photoelectron spectra for one and two-dimensional models, and observe
a significant reduction in the size of the computational domain required
to achieve high quality results, as compared with the popular method
of complex absorbing potentials.