The nuclear electric quadrupole moment of $^{87}$Sr from highly accurate molecular relativistic calculations

The nuclear electric quadrupole moment (NQM) of $^{87}$Sr has recently been revisited using high-precision relativistic atomic calculations [B. Lu et al., Phys. Rev. A 100, 012504 (2019)], indicating that the currently accepted value should be revised and that their result may serve as a new reference. In the present work, we determine the NQM of $^{87}$Sr from the molecular method, by combining the experimentally measured nuclear quadrupole coupling constants (NQCCs) of SrO and SrS with highly accurate relativistic calculations of the electric field gradient (EFG) at the Sr nucleus. Electronic correlation is treated at the CCSD(T), CCSD-T and CCSD$\tilde{\text{T}}$ levels. The iterative T contribution of the latter, composite scheme was obtained using a newly implemented parallel scheme where the distributed memory tensor library Cyclops Tensor Framework (CTF) was made available to the DIRAC code for relativistic molecular calculations through TAPP, the new community standard for tensor operations. All correlated calculations are performed using the exact two-component molecular mean-field Hamiltonian (X2C$\mathrm{mmf}$). The Gaunt two-electron interaction is incorporated, an even-tempered optimized quadruple-$ζ$ quality basis set is employed, and vibrational corrections are accounted for. Our best result is $Q($$^{87}$Sr$) = 0.33666 \pm 0.00258$ b, which is about 10% larger than currently accepted standard value, while it is in excellent agreement with recent determinations [Y.-B. Tang, arXiv:2512.07603 [physics.atom-ph] (2025)].

💡 Research Summary

The paper presents a rigorous determination of the nuclear electric quadrupole moment (NQM) of the isotope ^87Sr by exploiting a molecular approach that combines experimentally measured nuclear quadrupole coupling constants (NQCCs) for SrO and SrS with state‑of‑the‑art relativistic quantum‑chemical calculations of the electric field gradient (EFG) at the strontium nucleus. The authors begin by emphasizing that the NQM, denoted Q, couples to the EFG generated by surrounding electrons, and that the product Q·q determines the observable NQCC via the relation NQCC = 234.9647 × Q × q (with q expressed in atomic units). While atomic methods have traditionally been used to extract Q, the molecular method offers the advantage of using multiple NQCCs, thereby reducing systematic errors associated with a single‑molecule calculation.

All electronic‑structure calculations were performed with the DIRAC program suite, employing the exact two‑component molecular mean‑field (X2C‑mmf) Hamiltonian. The four‑component Hartree–Fock reference includes scalar relativistic effects, spin‑orbit coupling, the Gaunt two‑electron term, and the SS integrals. A Gaussian nuclear charge distribution model was adopted for all atoms. The basis set strategy started from the dyall.ae4z set for all elements, and the Sr basis was augmented with two very tight d‑functions (exponents 1.45205990 × 10⁴ and 6.52893664 × 10³) in an even‑tempered fashion, yielding an effective quadruple‑ζ quality.

Electron correlation was treated at three increasingly sophisticated levels: (i) CCSD(T), the conventional perturbative triples correction; (ii) CCSD‑T, which adds a broader class of disconnected triples; and (iii) a composite CCSD˜T scheme. In the latter, a fully iterative CCSDT calculation is performed with a smaller basis (dyall.v3z) and a reduced active space; the resulting “T‑correction” is added to a CCSD calculation performed with a larger, near‑complete basis and an active space sufficient for property convergence. The CCSDT calculations were enabled by the ExaCorr module, which uses the tenpi code generator to produce unrestricted CC equations. Correlation was limited to the four highest‑lying occupied Kramers pairs and 97 virtual Kramers pairs, corresponding to an energetic cutoff of roughly 3.8–4.3 Eh for SrO and SrS, respectively.

A major technical achievement of the work is the implementation of massively parallel tensor contractions via the Tensor Algebra Processing Primitives (TAPP) interface and the Cyclops Tensor Framework (CTF). By extending TAPP to handle distributed‑memory tensors (through UUID‑based metadata) and reconciling DIRAC’s master‑worker MPI model with CTF’s symmetric parallelism, the authors achieved scalable CCSDT and CCSDTQ calculations across hundreds of nodes on the French Adastra supercomputer. This infrastructure allowed the otherwise prohibitive iterative triples calculations to be performed efficiently.

EFG values were obtained using a finite‑field approach with an external field strength ε = ±10⁻⁷ a.u., ensuring linear response while avoiding numerical noise. The correlation contribution to the EFG was extracted as q_corr ≈