Excitons and optical response in the excitonic insulator candidate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>TiSe</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math>

The origin of the charge-density-wave (CDW) phase in ${\mathrm{TiSe}}_{2}$ is a highly debated topic, with lattice and excitonic correlations proposed as the main driving mechanisms. One of the proposed scenarios is the excitonic insulator (EI) mechanism, where a soft electronic mode drives the phase transition. However, the existence of this purely electronic mode is controversial. Here, we perform fully ab initio calculations of the electron excitation spectra in ${\mathrm{TiSe}}_{2}$ with electron-hole excitonic effects included via Bethe-Salpeter equations. In the normal high-temperature phase, the excitation spectra are dominated by the exciton mode at 1.6 eV, while no well-defined soft electronic modes that could support the EI phase are present. In the CDW phase, the structural distortions induce a CDW band-gap opening between Ti-$d$ and Se-$p$ states, which supports the formation of the two low-energy excitonic modes in the optical spectrum at 0.4 eV and 80 meV. Close to the transition temperature ${T}_{\mathrm{CDW}}$, these two excitonic modes are softened and approach zero energy. These results suggest that the EI mechanism is not a main driving force in the formation of the CDW phase in ${\mathrm{TiSe}}_{2}$, but there is a region in the phase diagram near ${T}_{\mathrm{CDW}}$ where EI fluctuations could be relevant.