  Transition metal dichalcogenides (TMDs), whether in bulk or in monolayer
form, exhibit a rich variety of charge-density-wave (CDW) phases and stronger
periodic lattice distortions. While the actual role of nesting has been under
debate, it is well understood that the microscopic interaction responsible for
the CDWs is the electron-phonon coupling. The case of TiSe$_2$ is however
unique in this family in that the normal state above the critical temperature
$T_\mathrm{CDW}$ is characterized by a small quasiparticle bandgap as measured
by ARPES, so that no nesting-derived enhancement of the susceptibility is
present. It has therefore been argued that the mechanism responsible for this
CDW should be different and that this material realizes the excitonic insulator
phase proposed by Walter Kohn. On the other hand, it has also been suggested
that the whole phase diagram can be explained by a sufficiently strong
electron-phonon coupling. In this work, in order to estimate how close this
material is to the pure excitonic insulator instability, we quantify the
strength of electron-hole interactions by computing the exciton band structure
at the level of hybrid density functional theory, focusing on the monolayer. We
find that in a certain range of parameters the indirect gap at
$q_{\mathrm{CDW}}$ is significantly reduced by excitonic effects. We discuss
the consequences of those results regarding the debate on the physical
mechanism responsible for this CDW. Based on the dependence of the calculated
exciton binding energies as a function of the mixing parameter of hybrid DFT,
we conjecture that a necessary condition for a pure excitonic insulator is that
its noninteracting electronic structure is metallic.
