  Results of FLMTO-GGA (full-potential linear muffin-tin orbital -- generalized
gradient approximation) calculations of the band structure and boron electric
field gradients (EFG) for the new medium-T$_c$ superconductor (MTSC), MgB$_2$,
and related diborides MB$_2$, M=Be, Al, Sc, Ti, V, Cr, Mo and Ta are reported.
The boron EFG variations are found to be related to specific features of their
band structure and particularly to the M-B hybridization. The strong charge
anisotropy at the B site in MgB$_2$ is completely defined by the valence
electrons - a property which sets MgB$_2$ apart from other diborides. The boron
EFG in MgB$_2$ is weakly dependent of applied pressure: the B p electron
anisotropy increases with pressure, but it is partly compensated by the
increase of core charge assymetry. The concentration of holes in bonding
$\sigma$ bands is found to decrease slightly from 0.067 to 0.062 holes/B under
a pressure of 10 GPa. Despite a small decrease of N(E$_F$), the Hopfield
parameter increases with pressure and we believe that the main reason for the
reduction under pressure of the superconducting transition temperature, T$_c$,
is the strong pressure dependence of phonon frequencies, which is sufficient to
compensate the electronic effects.
