  Defects in silicon carbide are of intense and increasing interest for
quantum-based applications due to this material's properties and technological
maturity. We calculate the multi-particle symmetry adapted wave functions of
the negatively charged silicon vacancy defect in hexagonal silicon carbide via
use of group theory and density functional theory and find the effects of
spin-orbit and spin-spin interactions on these states. Although we focused on
$\textrm{V}_{\textrm{Si}}^-$ in 4H-SiC, because of its unique fine structure
due to odd number of active electrons, our methods can be easily applied to
other defect centers of different polytpes, especially to the 6H-SiC. Based on
these results we identify the mechanism that polarizes the spin under optical
drive, obtain the ordering of its dark doublet states, point out a path for
electric field or strain sensing, and find the theoretical value of its
ground-state zero field splitting to be 68 MHz, in good agreement with
experiment. Moreover, we present two distinct protocols of a spin-photon
interface based on this defect. Our results pave the way toward novel quantum
information and quantum metrology applications with silicon carbide.
