  To clarify the origin of a charge-density wave (CDW) phase in 1$T$-TiSe$_2$,
we study the ground state property of a half-filled two-band Hubbard model in a
triangular lattice including electron-phonon interaction. By using the
variational Monte Carlo method, the electronic and lattice degrees of freedom
are both treated quantum mechanically on an equal footing beyond the mean-field
approximation. We find that the cooperation between Coulomb interaction and
electron-phonon interaction is essential to induce the CDW phase. We show that
the "pure" exciton condensation without lattice distortion is difficult to
realize under the poor nesting condition of the underlying Fermi surface.
Furthermore, by systematically calculating the momentum resolved hybridization
between the two bands, we examine the character of electron-hole pairing from
the viewpoint of BCS-BEC crossover within the CDW phase and find that the
strong-coupling BEC-like pairing dominates. We therefore propose that the CDW
phase observed in 1$T$-TiSe$_2$ originates from a BEC-like electron-hole
pairing.
