  We address the question of the origin of the recently discovered chiral
property of the charge-density-wave phase in 1$T$-TiSe$_2$ which so far lacks a
microscopic understanding. We argue that the lattice degrees of freedom seems
to be crucial for this novel phenomenon. We motivate a theoretical model that
takes into account one valence and three conduction bands, a strongly screened
Coulomb interaction between the electrons, as well as the coupling of the
electrons to a transverse optical phonon mode. The Falicov-Kimball model
extended in this way possesses a charge-density-wave state at low temperatures,
which is accompanied by a periodic lattice distortion. The charge ordering is
driven by a lattice deformation and electron-hole pairing (excitonic)
instability in combination. We show that both electron-phonon interaction and
phonon-phonon interaction must be taken into account at least up to quartic
order in the lattice displacement to achieve a stable chiral charge order. The
chiral property is exhibited in the ionic displacements. Furthermore, we
provide the ground-state phase diagram of the model and give an estimate of the
electron-electron and electron-phonon interaction constants for 1$T$-TiSe$_2$.
