  Superconducting transition temperatures ($T_c$'s) of MgB$_2$,
Mg$_{1-x}$Ca$_x$B$_2$, Mg$_{1-x}$Na$_x$B$_2$, and Mg$_{1-x}$Al$_x$B$_2$ are
studied within the McMillan approximation using electronic and structural
information obtained from density functional theory within the generalized
gradient approximation. The density of states and $T_c$ of MgB$_2$ are both
shown to be extremely sensitive to volume; in fact the density of states around
the Fermi level is found to rise with increasing volume because of a prominent
van Hove peak. Doping the Mg sublattice with small amounts of either Ca, which
substantially increases the unit cell volume, or Na, which removes an electron
from the unit cell while likewise increasing its volume, shifts the Fermi level
toward the peak and thus both types of doping are predicted to enhance $T_c$;
in Mg$_{1-x}$Al$_x$B$_2$, however, the combined effects of the additional
electron and decreasing average unit cell volume are shown to decrease $T_c$
with increasing Al concentration, consistent with recent experiments.
