  Despite the large number of theoretical III-V semiconductor studies reported
every year, our atomistic understanding is still limited. The limitations of
the theoretical approaches to yield accurate structural and electronic
properties on an equal footing, due to the unphysical self-interaction problem
that affects mainly the band gap and spin-orbit splitting (SOC) in
semiconductors and, in particular, III-V systems with similar magnitude of the
band gap and SOC. In this work, we will report a consistent study of the
structural and electronic properties of the III-V semiconductors employing the
screening hybrid-DFT framework, fitting the $\alpha$ parameters for 12
different III-V compounds, namely, AlN, AlP, AlAs, AlSb, GaN, GaP, GaAs, GaSb,
InN, InP, InAs, InSb, in order to minimize the deviation between the
theoretical and experimental values of the band gap and SOC. Structural
relaxation effects were also included. Except for AlP, whose $\alpha = 0.127$,
we obtained $\alpha$ values spreading from 0.209 to 0.343, deviating less than
0.1 from the universal value of 0.25. Our results for the lattice parameter and
elastic constants indicate that the fitting of the $\alpha$ does not affect
those structural parameters when compared with the HSE06 functional, where
$\alpha = 0.25$. Our analysis of the band structure based on the
$\textbf{k}{\cdot}\textbf{p}$ method shows that the effective masses are in
agreement with the experimental values, which can be attributed to the
simultaneous fitting of the band gap and SOC. Finally, we estimate the values
of $g$-factors, extracted directly from band structure, which are close to
experimental results indicating that the obtained band structure produced a
realistic set of $\textbf{k}{\cdot}\textbf{p}$ parameters.
