  A new state of matter, an excitonic insulator (EI) state, was predicted to
emerge from Bose-Einstein condensation of electron-hole pairs. Some candidate
materials were suggested but it has been elusive to confirm its existence.
Recent works gave renewed support for the EI picture of the charge density wave
(CDW) state below the critical temperature $T_c \approx 200 $ K of
$1T$-TiSe$_{2}$. Yet, an important link to its establishment is to show that a
majority fraction of the measured $T_c$ indeed follows from the Coulomb
interaction alone, while a quantitative match of the $T_c$ may require
assistance from the electron-lattice coupling. This will establish that the CDW
is formed predominantly by the Coulomb interaction and help confirm the EI view
for TiSe$_2$. Here, we provide such calculations by solving the exciton gap
equation with material specific electronic structures. We obtain, with no
fitting parameters, $T_c \approx 135 \pm 27$ K for the normal state gap of $E_g
\approx 74 \pm 15$ meV. It seems that the calculated $T_c$ from Coulomb
interaction gives a majority fraction of experimental $T_c$ for recently
determined values of $E_g$. The measured doping dependence of $T_c$ was
satisfactorily reproduced as well. Also in agreement with experiments are the
same set of calculations of the photoemission spectroscopy and density of
states. The semiconducting state above and EI below $T_c$ together should give
a coherent picture of $1T$-TiSe$_2$.
