  The Bose-Einstein condensation of excitons continues to garner immense
attention as a prototypical example for observing emergent properties from
many-body quantum effects. In particular, Titanium Diselenide (TiSe$_2$) is a
promising candidate for realizing exciton condensation and was experimentally
observed only very recently. Surprisingly, the condensate was experimentally
characterized by a soft plasmon mode that only exists near the transition
temperature, $T_c$, of the charge density wave (CDW). Here, we characterize and
analyze the experimental spectra using linear-response time-dependent density
functional theory and find that the soft mode can be attributed to interband
electronic transitions. At the CDW state below $T_c$, the periodic lattice
distortions hamper the spontaneous formation of the exciton by introducing a
CDW gap. The band gap raises the soft mode and merges it into the regular
plasmon. Our surprising results contradict previous simplistic analytical
models commonly used in the scientific literature. In addition, we find that a
finite electronic temperature, $T_e$, introduces a dissipation channel and
prevents the condensation above $T_c$. The combined effect of the CDW and $T_e$
explains the fragile temperature-dependence of the exciton condensation. Taken
together, our work provides the first \textit{ab initio} atomic-level framework
for rationalizing recent experiments and further manipulating exciton
condensates in TiSe$_2$.
