  The electronic state around a single vacancy in silicon crystal is
investigated by using the Green's function approach. The triply degenerate
charge states are found to be widely extended and account for extremely large
elastic softening at low temperature as observed in recent ultrasonic
experiments. When we include the LS coupling $\lambda_{\rm Si}$ on each Si
atom, the 6-fold spin-orbital degeneracy for the $V^{+}$ state with the valence
+1 and spin 1/2 splits into $\Gamma_{7}$ doublet groundstates and $\Gamma_{8}$
quartet excited states with a reduced excited energy of $O(\lambda_{\rm
Si}/10)$. We also consider the effect of couplings between electrons and
Jahn-Teller phonons in the dangling bonds within the second order perturbation
and find that the groundstate becomes $\Gamma_{8}$ quartet which is responsible
for the magnetic-field suppression of the softening in B-doped silicon.
