Ab initio calculations of $^{229}$Th band-to-band internal conversion rate in $^{229}$ThO$_2$

We present an ab initio calculation of the band-to-band internal-conversion rate of the $\hbar\omega_{\rm nuc} \approx 8.35$ eV isomeric transition in $^{229}$ThO$_2$. Because the nuclear transition energy exceeds the electronic band gap of ThO$_2$, the isomer can decay nonradiatively by resonantly promoting a valence electron into the conduction band. We formulate this process as a Brillouin-zone sum over vertical interband transitions weighted by local Th-centered hyperfine matrix elements, which are evaluated directly from all-electron full-potential linearized augmented-plane-wave Bloch spinors. A finite nuclear magnetization model is included to regularize the short-range hyperfine interaction and to account for the Bohr-Weisskopf effect. After applying scissor shifts to span the experimentally reported ThO$_2$ band gaps, we find calculated internal-conversion lifetimes in the range of $1-16~\mu{\rm s}$. The lifetime increases strongly as the band gap approaches $\omega_{\rm nuc}$ because the resonant interband phase space at the nuclear transition energy is reduced. For the larger reported ThO$_2$ gaps, the calculated lifetime is comparable to the measured conversion-electron M\"ossbauer lifetime [Nature 648, 300 (2025)]. Our analysis implies that choosing solid-state hosts with band-gap values slightly lower than $\omega_{\rm nuc}$ can optimize solid-state nuclear clock performance with internal-conversion electron readout.