  Time-dependent Hartree-Fock (TDHF) is one of the fundamental
post-Hartree-Fock (HF) methods to describe excited states. In its Tamm-Dancoff
form, equivalent to Configuration Interaction Singles, it is still widely used
and particularly applicable to big molecules where more accurate methods may be
unfeasibly expensive. However, it is rarely implemented in real space, mostly
because of the expensive nature of the exact-exchange potential in real space.
Compared to widely used Gaussian-type orbitals (GTO) basis sets, real space
often offers easier implementation of equations and more systematic convergence
of Rydberg states, as well as favorable scaling, effective domain
parallelization, flexible boundary conditions, and ability to treat model
systems. We implemented TDHF in the Octopus real-space code as a step toward
linear-response hybrid time-dependent density-functional theory (TDDFT), other
post-HF methods, and ensemble density-functional theory methods involving exact
exchange. Calculation of HF's non-local exact exchange is very expensive in
real space. We overcome this limitation with Octopus' implementation of
Adaptively Compressed Exchange (ACE), and find the appropriate mixing and
starting point to complete the ground-state calculation in a practical amount
of time, to enable TDHF. We compared our results to those from GTOs on a set of
small molecules and confirmed close agreement of results, though with larger
deviations than in the case of semi-local TDDFT. We find that convergence of
TDHF demands a finer real-space grid than semi-local TDDFT. We also present the
subtleties in benchmarking a real-space calculation against GTOs, relating to
Rydberg and vacuum states.
