  In order to characterize in detail the charge density wave (CDW) transition
of 1$T$-VSe$_2$, its electronic structure and lattice dynamics are
comprehensively studied by means of x-ray diffraction, angle resolved
photoemission (ARPES), diffuse and inelastic x-ray scattering (IXS), and
state-of-the-art first principles density functional theory calculations.
Resonant elastic x-ray scattering (REXS) does not show any resonant enhancement
at either V or Se K-edges, indicating that the CDW peak describes a purely
structural modulation of the electronic ordering. ARPES identifies (i) a
pseudogap at T$>$T$_{CDW}$, which leads to a depletion of the density of states
in the $ML-M'L'$ plane at T$<$T$_{CDW}$, and (ii) anomalies in the electronic
dispersion reflecting a sizable impact of phonons on it. A diffuse scattering
precursor, characteristic of soft phonons, is observed at room temperature (RT)
and leads to the full collapse of the low-energy phonon ($\omega_1$) with
propagation vector (0.25 0 -0.3) r.l.u. We show that the frequency and
linewidth of this mode are anisotropic in momentum space, reflecting the
momentum dependence of the electron-phonon interaction (EPI), hence
demonstrating that the origin of the CDW is, to a much larger extent, due to
the momentum dependence EPI with a small contribution from nesting. The
pressure dependence of the $\omega_1$ soft mode remains nearly constant up to
13 GPa at RT, with only a modest softening before the transition to the
high-pressure monoclinic $C2/m$ phase. The wide set of experimental data are
well captured by our state-of-the art first-principles anharmonic calculations
with the inclusion of van der Waals (vdW) corrections in the
exchange-correlation functional. The description of the electronics and
dynamics of VSe$_2$ reported here adds important pieces of information to the
understanding of the electronic modulations of TMDs.
