Host-Dependent Frequency Offsets in ²²⁹Th Nuclear Clockwork.

Recent advances in laser excitation of the low-energy nuclear isomer transition in ²²⁹Th have opened avenues for developing nuclear clocks, a novel quantum technology with exceptional performance and sensitivity to exotic physics. Here, we explore the host dependence of the nuclear clock frequency, focusing on the isomer shift induced by the difference in the nuclear charge distribution between the ground and excited nuclear states. We combine relativistic many-body methods of atomic structure with periodic density functional theory to evaluate the isomer shifts in solid-state hosts. We elucidate the critical importance of the "relaxation" effect in evaluating the isomer shifts. Our analysis predicts nuclear clock frequencies for various solid-state and trapped ion platforms: ωclk(solid state)=2 020 407 384(40) MHz, ωclk(²²⁹Th⁴⁺)=2 020 407 648(70) MHz, and ωclk(²²⁹Th³⁺)=2 020 407 114(70) MHz. We also determine the nuclear transition energy for the bare ²²⁹Th nucleus to be ωnuc=8.272(22) eV. Our calculated valence band isomer shifts for different host materials constrain the nuclear transition frequencies to an 80-MHz-wide frequency window, aiding experimental searches for the ²²⁹Th nuclear transition in novel materials.