  The hyperfine interaction is an important probe for understanding the
structure and symmetry of defects in a semiconductor. Density-functional theory
has shown that it can provide useful first-principles predictions for both the
hyperfine tensor and the hyperfine constants that arise from it. Recently there
has been great interest in using group-IV impurity-vacancy color centers X$V^-$
(where X = Si, Ge, Sn, or Pb and $V$ is a carbon vacancy) for important
applications in quantum computing and quantum information science. In this
paper, we have calculated the hyperfine tensors for these X$V^-$ color centers
using the HSE06 screened Hartree-Fock hybrid exchange-correlation functional
with the inclusion of core electron spin polarization. We have compared our
results to calculations which only use the PBE exchange-correlation functional
without the inclusion of core electron spin polarization and we have found our
results are in very good agreement with available experimental results.
Finally, we have theoretically shown that these X$V^-$ color centers exhibit a
Jahn-Teller distortion which explains the observed anisotropic distribution of
the hyperfine constants among the neighboring $^{13}$C nuclear spins.
