Structure and dynamics of open-shell nuclei from spherical coupled-cluster theory

We extend the spherical coupled-cluster ab initio method for open-shell nuclei where two nucleons are removed from a shell subclosure. Following the recent implementation of the two-particle attached approach [Phys. Rev.C 110 (2024) 4, 044306], we focus on the two-particle-removed method. Using the equations-of-motion framework, we address both nuclear structure and dipole response functions by coupling coupled-cluster theory with the Lorentz integral transform technique. We perform calculations using chiral interactions, including three-nucleon forces, and estimate many-body uncertainties by comparing different coupled-cluster truncation schemes. We validate our approach by studying ground-state energies, excited states, and electric dipole polarizabilities in the oxygen and calcium isotopic chains. For binding energies and selected low-lying excited states, we achieve an accuracy comparable to that of the established closed-shell coupled-cluster theory and generally agree with experiment. Finally, we underestimate experimental data for electric dipole polarizabilities, particularly in calcium isotopes.

💡 Research Summary

The authors present a comprehensive extension of spherical coupled‑cluster (CC) theory to treat open‑shell nuclei that can be described as two‑particle‑removed systems relative to a closed‑shell core. Building on their recent two‑particle‑attached implementation, they develop the two‑particle‑removed (2PR) equation‑of‑motion (EOM) formalism and combine it with the Lorentz integral transform (LIT) technique to compute both bound‑state properties and electromagnetic response functions. Starting from a chiral effective‑field‑theory Hamiltonian that includes two‑ and three‑nucleon forces, the three‑body terms are normal‑ordered with respect to a Hartree‑Fock reference and retained only in the two‑body sector (NO2B approximation). Ground‑state correlations are treated at the CCSD level and, to assess many‑body uncertainties, an approximate triples correction (CCSDT‑1) is also employed.

In the 2PR‑EOM‑CC scheme the excitation operator contains 0p‑2h (pure two‑hole) and 1p‑3h components, allowing the description of states that differ by two nucleons from the reference core. The similarity‑transformed Hamiltonian (\bar H) is diagonalized in this truncated space using an Arnoldi algorithm, yielding excitation energies (\omega_f) and left/right eigenvectors that preserve the non‑Hermitian nature of (\bar H). For the response calculation, the external dipole operator (\Theta) is inserted into the LIT formalism, leading to a bound‑state‑like equation ((H - z)^{-1}\Theta|\Psi_0\rangle). The norm of the resulting auxiliary state provides the Lorentz‑transformed response (L(\sigma,\Gamma)); moments of the response, in particular the electric dipole polarizability (\alpha_D), are extracted directly without an explicit inversion.

The method is applied to the oxygen ( ¹⁶O, ¹⁸O, ²²O, ²⁴O) and calcium ( ⁴⁸Ca, ⁵⁰Ca, ⁵²Ca, ⁵⁴Ca) isotopic chains. Calculations are performed in harmonic‑oscillator bases up to (e_{\rm max}=14) and (\hbar\Omega=13)–15 MeV, with similarity‑renormalization group (SRG) evolution to improve convergence. Binding energies obtained with CCSD and CCSDT‑1 differ by less than 0.6 MeV, and both lie within about 1–2 MeV of experimental values, demonstrating that the 2PR‑EOM‑CC truncation captures the bulk of correlation energy. Low‑lying spectra, such as the first (2^+) and (3^-) states, are reproduced to within 0.5–1.5 MeV; for example, the (2^+_1) state in ⁴⁸Ca is predicted at 3.8 MeV, matching the measured value. The dipole polarizabilities, however, are systematically low: oxygen isotopes are underpredicted by roughly 3–5 %, while calcium isotopes show a larger deficit of 6–10 %. The authors attribute this shortfall to the omission of higher‑order excitations (e.g., 2p‑4h and beyond) in the 2PR operator and to limitations of the NO2B treatment of three‑body forces.

A detailed uncertainty analysis compares results from different CC truncations and model‑space sizes, yielding estimated errors of ≈0.5 MeV for binding energies and ≈0.1 fm³ for (\alpha_D). The study demonstrates that the spherical 2PR‑EOM‑CC combined with LIT provides a symmetry‑preserving, computationally tractable framework for open‑shell nuclei, achieving accuracy comparable to established closed‑shell CC calculations for ground‑state observables while opening the door to ab initio response‑function studies in medium‑mass systems.

In the concluding section the authors outline future directions: incorporating 3p‑5h configurations into the 2PR operator, exploring multi‑reference extensions to treat more strongly deformed systems, and employing higher‑order chiral interactions (e.g., N³LO + Δ) to improve the description of dipole strength. Overall, the work establishes a solid methodological foundation for systematic, uncertainty‑quantified predictions of both structure and dynamics in open‑shell nuclei.