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A first-principles study of the electronic and superconducting properties ofthe Ni$_2$VAl Heusler compound is presented. The electron-phonon couplingconstant of $\lambda_{ep}$ = 0.68 is obtained, which leads to a superconductingtransition temperature of T$_c$ = $\sim$4 $K$ (assuming a Coulombpseudopotential $\mu^*$ = 0.13), which is a relatively high transitiontemperature for Ni based Heusler alloys. The electronic density of statesreveals a significant hybridization between Ni-$eg$ and V-$t_{2g}$ statesaround the Fermi level. The Fermi surface, consisting of two electron pocketsaround the X-points of the Brillouin zone, exhibits nesting and leads to a Kohnanomaly of the phonon dispersion relation for the transverse acoustic mode TA2along the (1,1,0) direction, which is furthermore found to soften withpressure. As a consequence, T$_c$ and $\lambda_{ep}$ vary non-monotonicallyunder pressure. The calculations are compared to similar calculations performedfor the Ni$_2$NbX (X = Al, Ga and Sn) Heusler alloys, which experimentally havebeen identified as superconductors. The experimental trend in T$_c$ is wellreproduced, and reasonable quantitative agreement is obtained. The calculatedT$_c$ of Ni$_2$VAl is larger than either calculated and observed T$_c$s of anyof the Nb compounds. The Fermi surfaces of Ni$_2$NbAl and Ni$_2$NbGa consist ofonly a single electron pocket around the X point, however under compressionsecond electron pocket similar to that of Ni2VAl emerges only in Ni2NbAl andthe Tc increases non monotonically in all the compounds. Fermi surface nestingand associated Kohn anomalies are a common feature of all four compounds,albeit weakest in Ni$_2$VAl.

Predicted superconductivity of Ni2VAl and pressure dependence of superconductivity in Ni2NbX (X = Al, Ga and Sn) and Ni2VAl