  We investigate the electronic states around a single vacancy in silicon
crystal by using the Green's function approach. The triply degenerate vacancy
states within the band gap are found to be extended over a large distance
$\sim20 {\rm \AA}$ from the vacancy site and contribute to the reciprocal
temperature dependence of the quadrupole susceptibility resulting in the
elastic softening at low temperture. The Curie constant of the quadrupole
susceptibility for the trigonal mode ($O_{yz},O_{zx},O_{xy}$) is largely
enhanced as compared to that for the tetragonal mode ($O_{2}^{0},O_{2}^{2}$).
The obtained results are consistent with the recent ultrasonic experiments in
silicon crystal down to 20 mK. We also calculate the dipole and octupole
susceptibilities and find that the octupole susceptibilities are extremely
enhannced for a specific mode.
