Direct All-Electron Relativistic Framework for Nuclear-Clock Field Shifts: 229Th in CaF2

Recent progress toward nuclear clocks based on 229Th-doped crystals has made it increasingly important to understand how the local crystal environment affects the nuclear transition frequency. Based on the quasi-four-component (Q4C) relativistic density-functional method we have developed, we establish a direct all-electron framework for calculating nuclear-clock field shifts in periodic solids: by combining a finite-nucleus model with a Hellmann-Feynman formulation, the method directly evaluates the electronic response to a change in nuclear size. Q4C can describe both the large and small spinor components near a heavy nucleus while reducing the final eigenvalue problem to two-component dimensions, making fully relativistic calculations feasible for large defect supercells. We systematically validate the method through benchmarks on one-electron atoms, many-electron atoms, and molecules; we then apply it to 229Th-doped CaF2, focusing on the widely discussed but still controversial charge-compensation configurations, i.e., F-90 and F-180. The calculated field-shift difference is δν(180°) - δν(90°) = -16.8 × (1.00 ± 0.27) MHz. The predicted frequency separation provides direct theoretical evidence for distinguishing these two local structures and places an important constraint on the assignment of the observed 229Th:CaF2 spectra. More generally, the present work provides a practical first-principles relativistic tool for solid-state nuclear-clock studies.