Th$^{4+}$ ion cluster structures and electric field gradients in $^{229}$Th:CaF$_2$

The low-energy nuclear isomeric transition of $^{229}$Th is a promising frequency reference for a solid-state nuclear optical clock. Wide-band-gap fluoride crystals such as CaF$_2$ offer transparent hosts for embedding Th nuclei at high density. Substitution of Th$^{4+}$ for Ca$^{2+}$ requires charge compensation, linking the local defect structure to nuclear quadrupole splitting. We use first-principles calculations to examine neutral Th dimers compensated by four interstitial F ions ($\mathrm{F}_{i}^{-}$). Screening different compensation geometries by their relative formation energies identifies a family of low-energy configurations with closely similar local structures. These configurations share large electric field gradients and small asymmetry parameters, consistent with the experimental high-EFG site 3. Their calculated quadrupole splittings are further compared with the measured C13 spectra. Together, the energetic and spectroscopic results support $\mathrm{F}_{i}^{-}$-compensated Th dimers as candidate microscopic structures for site 3 in Th:CaF$_2$.