  The wurtzite phase group III-Nitrides (AlN, GaN, InN) have attracted great
interest due to their successful applications in the optoelectronics since the
90's. In this paper we perform a comprehensive study of AlN, GaN and InN
structural elastic and electronic properties using hybrid and conventional
Density Functional Theory, presenting a comparison of the features of the three
compounds. We perform a direct comparison of the features of their electronic
structures, including the inversion of the top valence band associated with a
negative crystal field splitting and its relation to the challenges of
acceptor-doping on AlN systems. With the determination of elastic constants and
the Young modulus we provide a simple model to connect a deformation energy
associated with the parameter $u$ and the effective crystal-field splitting,
showing a direct relation among internal strain and the crystal-field
splitting.
