  At low temperature TiSe2 undergoes a charge density wave instability.
Superconductivity is stabilized either by pressure or by Cu intercalation. We
show that the pressure phase diagram of TiSe2 is well described by
first-principles calculations. At pressures smaller than 4 GPa charge density
wave ordering occurs, in agreement with experiments. At larger pressures the
disappearing of the charge density wave is due to a stiffening of the
short-range force-constants and not to the variation of nesting with pressure.
Finally we show that the behavior of Tc as a function of pressure is entirely
determined by the electron-phonon interaction without need of invoking
excitonic mechanisms. Our work demonstrates that phase-diagrams with competing
orders and a superconducting dome are also obtained in the framework of the
electron-phonon interaction.
