  We study the in-plane and out-of-plane density ordering instabilities of
quasi-two-dimensional fermionic polar molecules in single-layer and multi-layer
configurations. We locate the soft modes by evaluating linear response
functions within the conserving time-dependent Hartree-Fock (TDHF). The
short-range exchange effects are taken into account by solving the
Bethe-Salpeter integral equation numerically. An instability phase diagram is
calculated for both single-layer and multi-layer systems and the unstable
wave-vector is indicated. In all cases, the in-plane density wave instability
is found to precede the out-of-plane instability. The unstable wave-vector is
found to be approximately twice the Fermi wave-vector of one of the subbands at
a time and can change discontinuously as a function of density and dipolar
interaction strength. In multi-layer configurations, we find a large
enhancement of density wave instability driven by dilute quasiparticles in the
first excited subband. Finally, we provide a simple qualitative description of
the phase diagrams using a RPA-like approach. Compared to previous works done
within the RPA approximation, we find that inclusion of exchange interactions
stabilize the normal liquid phase further and increase the critical dipolar
interaction strength corresponding to the onset of density-wave instability by
over a factor of two.
