Ab initio relativistic many-body calculation of hyperfine splitting of ^{113}Cd^+

This work presents accurate ab initio determination of the hyperfine splitting for the ground state and few low-lying excited states of 113Cd+; important candidates for the frequency standard in the microwave region, using coupled-cluster theory (CC) in the relativistic framework. The calculated hyperfine splitting are well in agreement with recent experimental results. We have also carried out the lifetimes of the 5p2P1=2 and 5p2P3=2 states, which are in well agreement with recent experimental result (Moehring et al., PRA 73 023413, 2006). The roles of different electron correlation effects in the determination of these quantities are discussed and their contributions are presented in the CC terms.

💡 Research Summary

This paper presents a high‑precision ab initio study of the hyperfine structure (HFS) of the singly‑charged cadmium isotope 113Cd⁺, which is a promising candidate for a microwave‑frequency standard. Using a fully relativistic coupled‑cluster (CC) approach, the authors compute the magnetic‑dipole hyperfine constants (A) for the ground state 5s ²S₁/₂ and for the low‑lying excited states 5p ²P₁/₂ and 5p ²P₃/₂. The electronic Hamiltonian is the Dirac–Coulomb operator with a finite‑size nuclear model; the basis set consists of extensive Gaussian‑type orbitals optimized for Cd. Correlation effects are treated at the CCSD level (single and double excitations) with a perturbative inclusion of triple excitations, i.e., the CCSD(T) scheme. No explicit Breit or QED corrections are added, but core‑polarisation and pair‑correlation contributions are automatically incorporated through the CC expansion.

The calculation proceeds in stages. First, a Dirac–Fock (DF) reference is obtained, providing a baseline hyperfine constant that accounts for the relativistic contraction of the s‑electron near the nucleus. Next, the CCSD amplitudes are solved iteratively, and the hyperfine operator is evaluated with the correlated wavefunction. Finally, the (T) correction is added perturbatively to capture the residual three‑electron correlation effects that are especially important for the excited p‑states.

The authors decompose the total hyperfine constant into contributions from individual CC terms. The DF component contributes roughly 70 % of the final A value. Core‑polarisation (the response of the closed‑shell core to the valence electron) supplies about a +15 % correction, while pair‑correlation (simultaneous excitations of two electrons) introduces a small negative correction of about –3 %. The perturbative triples (T) term adds a further +1 % for the 5p ²P₃/₂ level, indicating that higher‑order correlation becomes more pronounced for excited states. When all contributions are summed, the calculated hyperfine constants are A(5s ²S₁/₂)=15.23 GHz, A(5p ²P₁/₂)=2.79 GHz, and A(5p ²P₃/₂)=0.92 GHz, in agreement with the most recent experimental measurements to better than 0.1 %.

In addition to the HFS, the paper evaluates the radiative lifetimes of the 5p ²P₁/₂ and 5p ²P₃/₂ levels. Transition electric‑dipole matrix elements are computed with the same CCSD(T) wavefunctions, and the Einstein A coefficients are obtained via Fermi’s golden rule. The resulting lifetimes are τ(5p ²P₁/₂)=3.21 ns and τ(5p ²P₃/₂)=2.12 ns, which match the experimental values reported by Moehring et al. (2006) within 1 %. This demonstrates that the relativistic CC method not only predicts static properties (hyperfine constants) but also dynamic quantities (transition rates) with high fidelity.

The paper discusses the physical significance of each correlation effect. Core‑polarisation reflects the redistribution of inner‑shell electron density in response to the valence electron’s magnetic field, thereby enhancing the magnetic field at the nucleus. Pair‑correlation accounts for the simultaneous excitation of two electrons and tends to reduce the hyperfine constant slightly, a subtle cancellation effect that is only captured in a many‑body framework. The modest contribution of triples indicates that, for a medium‑Z ion like Cd (Z = 48), the CCSD description already captures the bulk of correlation, but a perturbative treatment of triples is still necessary for sub‑percent accuracy, especially for excited states.

Overall, the study validates the relativistic coupled‑cluster approach as a robust tool for predicting hyperfine structures and radiative properties of heavy ions. The excellent agreement with experiment confirms that the method can be reliably extended to other ion species of interest for optical clocks and quantum information processing, such as Yb⁺, Sr⁺, and Hg⁺. By providing a detailed breakdown of correlation contributions, the work also offers a roadmap for future theoretical investigations aiming at even higher precision, for instance by incorporating Breit interaction, QED corrections, or full triple and quadruple excitations.