Host-Dependent Frequency Offsets in ^{229}Th Nuclear Clockwork

Recent advances in laser excitation of the low-energy nuclear isomer transition in ^{229}Th 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: ω_{clk}(solid state)=2 020 407 384(40)  MHz, ω_{clk}(^{229}Th^{4+})=2 020 407 648(70)  MHz, and ω_{clk}(^{229}Th^{3+})=2 020 407 114(70)  MHz. We also determine the nuclear transition energy for the bare ^{229}Th nucleus to be ω_{nuc}=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 ^{229}Th nuclear transition in novel materials.