  We introduce a new method for deriving the time-dependent Hartree or
Hartree-Fock equations as an effective mean-field dynamics from the microscopic
Schroedinger equation for fermionic many-particle systems in quantum mechanics.
The method is an adaption of the method used in [Pickl, Lett. Math. Phys.,
97(2):151-164, 2011] for bosonic systems to fermionic systems. It is based on a
Gronwall type estimate for a suitable measure of distance between the
microscopic solution and an antisymmetrized product state. We use this method
to treat a new mean-field limit for fermions with long-range interactions in a
large volume. Some of our results hold for singular attractive or repulsive
interactions. We can also treat Coulomb interaction assuming either a mild
singularity cutoff or certain regularity conditions on the solutions to the
Hartree(-Fock) equations. In the considered limit, the kinetic and interaction
energy are of the same order, while the average force is subleading. For some
interactions, we prove that the Hartree(-Fock) dynamics is a more accurate
approximation than a simpler dynamics that one would expect from the subleading
force. With our method we also treat the mean-field limit coupled to a
semiclassical limit, which was discussed in the literature before, and we
recover some of the previous results. All results hold for initial data close
(but not necessarily equal) to antisymmetrized product states and we always
provide explicit rates of convergence.
