  Recent experiments suggest that excitonic degrees of freedom play an
important role in precipitating the charge density wave (CDW) transition in
$1T$-$\mathrm{TiSe}_2$. Through systematic calculations of the electronic and
phonon spectrum based on density functional perturbation theory, we show that
the predicted critical doping of the CDW phase overshoots the experimental
value by 1 order of magnitude. In contrast, an independent self-consistent
many-body calculation of the excitonic order parameter and renormalized band
structure is able to capture the experimental phase diagram in extremely good
qualitative and quantitative agreement. This demonstrates that
electron-electron interactions and the excitonic instability arising from
direct electron-hole coupling are pivotal to accurately describe the nature of
the CDW in this system. This has important implications to understand the
emergence of superconductivity within the CDW phase of this and related
systems.
