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The hyperfine interaction is an important probe for understanding thestructure and symmetry of defects in a semiconductor. Density-functional theoryhas shown that it can provide useful first-principles predictions for both thehyperfine tensor and the hyperfine constants that arise from it. Recently therehas 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 importantapplications in quantum computing and quantum information science. In thispaper, we have calculated the hyperfine tensors for these X$V^-$ color centersusing the HSE06 screened Hartree-Fock hybrid exchange-correlation functionalwith the inclusion of core electron spin polarization. We have compared ourresults to calculations which only use the PBE exchange-correlation functionalwithout the inclusion of core electron spin polarization and we have found ourresults are in very good agreement with available experimental results.Finally, we have theoretically shown that these X$V^-$ color centers exhibit aJahn-Teller distortion which explains the observed anisotropic distribution ofthe hyperfine constants among the neighboring $^{13}$C nuclear spins.

Calculating the Hyperfine Tensors for Group-IV Impurity-Vacancy Centers in Diamond: A Hybrid Density-Functional Theory Approach