Prediction of nuclear clock transition frequency difference between 229Th3+ and 229Th4+ via ab initio self-consistent field theory

The ²²⁹Th isotope is a promising candidate for nuclear clocks. However, the clock transition varies due to the electron-induced nuclear frequency shift. To achieve the accuracy required for developing a nuclear clock, this effect must be precisely determined. In this work, we employed a non-perturbative multi-configuration Dirac-Hartree-Fock (MCDHF) method, in contrast to the perturbation theory used previously, to resolve the electron-induced field shift effect. As a more internally consistent ab initio method, this calculation accounts for subtle differences in the nuclear potential while considering the ²²⁹Th isotope in both its ground and isomeric states. Consequently, the nuclear clock transition frequency difference between ²²⁹Th³⁺ and ²²⁹Th⁴⁺ was determined to be −639 MHz with computational convergency down to 1 MHz. Given recently measured transition frequency of ²²⁹Th⁴⁺ in ²²⁹Th-doped CaF₂ [Nature 633, 63 (2024)], here the transition frequency of isolated ²²⁹Th³⁺ is predicted to be \(2020407009(1)_{\text{comp.}}(77)_{\delta\langle{r}^{2}\rangle}(70)_{\text{ext.}}\) MHz, with brackets indicating uncertainties stemming from our atomic structure computations, the nuclear charge radius difference from the literature, and the influence of the crystal environment as reported in the literature. This provides valuable guidance for direct laser excitation of isolated ²²⁹Th³⁺ based on ion trap experiments.