  Methods based on propagation of the one-body reduced density-matrix hold much
promise for the simulation of correlated many-electron dynamics far from
equilibrium, but difficulties with finding good approximations for the
interaction term in its equation of motion have so far impeded their
application. These difficulties include the violation of fundamental physical
principles such as energy conservation, positivity conditions on the density,
or unchanging natural orbital occupation numbers. We review some of the recent
efforts to confront these problems, and explore a semiclassical approximation
for electron correlation coupled to time-dependent Hartree-Fock propagation. We
find that this approach captures changing occupation numbers, and excitations
to doubly-excited states, improving over TDHF and adiabatic approximations in
density-matrix propagation. However, it does not guarantee $N$-representability
of the density-matrix, consequently resulting sometimes in violation of
positivity conditions, even though a purely semiclassical treatment preserves
these conditions.
