  A first-principles study of the electronic and superconducting properties of
the Ni$_2$VAl Heusler compound is presented. The electron-phonon coupling
constant of $\lambda_{ep}$ = 0.68 is obtained, which leads to a superconducting
transition temperature of T$_c$ = $\sim$4 $K$ (assuming a Coulomb
pseudopotential $\mu^*$ = 0.13), which is a relatively high transition
temperature for Ni based Heusler alloys. The electronic density of states
reveals a significant hybridization between Ni-$eg$ and V-$t_{2g}$ states
around the Fermi level. The Fermi surface, consisting of two electron pockets
around the X-points of the Brillouin zone, exhibits nesting and leads to a Kohn
anomaly of the phonon dispersion relation for the transverse acoustic mode TA2
along the (1,1,0) direction, which is furthermore found to soften with
pressure. As a consequence, T$_c$ and $\lambda_{ep}$ vary non-monotonically
under pressure. The calculations are compared to similar calculations performed
for the Ni$_2$NbX (X = Al, Ga and Sn) Heusler alloys, which experimentally have
been identified as superconductors. The experimental trend in T$_c$ is well
reproduced, and reasonable quantitative agreement is obtained. The calculated
T$_c$ of Ni$_2$VAl is larger than either calculated and observed T$_c$s of any
of the Nb compounds. The Fermi surfaces of Ni$_2$NbAl and Ni$_2$NbGa consist of
only a single electron pocket around the X point, however under compression
second electron pocket similar to that of Ni2VAl emerges only in Ni2NbAl and
the Tc increases non monotonically in all the compounds. Fermi surface nesting
and associated Kohn anomalies are a common feature of all four compounds,
albeit weakest in Ni$_2$VAl.
