  In this work, we study the extended Falicov-Kimball model at half-filling
within the Hartree-Fock approach (HFA) (for various crystal lattices) and
compare the results obtained with the rigorous ones derived within the
dynamical mean field theory (DMFT). The model describes a system, where
electrons with spin-$\downarrow$ are itinerant (with hopping amplitude $t$),
whereas those with spin-$\uparrow$ are localized. The particles interact via
on-site $U$ and intersite $V$ density-density Coulomb interactions. We show
that the HFA description of the ground state properties of the model is
equivalent to the exact DMFT solution and provides a qualitatively correct
picture also for a range of small temperatures. It does capture the
discontinuous transition between ordered phases at $U=2V$ for small
temperatures as well as correct features of the continuous order-disorder
transition. However, the HFA predicts that the discontinuous boundary ends at
the isolated-critical point (of the liquid-gas type) and it does not merge with
the continuous boundary. This approach cannot also describe properly a change
of order of the continuous transition for large $V$ as well as various
metal-insulator transitions found within the DMFT.
