Highly charged ions (HCIs) are attractive candidates for next-generation optical clocks because their compact electronic orbitals strongly suppress environmental perturbations while retaining high sensitivity to possible variations of fundamental constants. Ni^12+ has a 3s^2 3p^4 ground configuration and supports the ultranarrow ^3P_2-^3P_0 clock transition. The electric quadrupole moment of its ^3P_2 ground state directly determines the coupling to stray electric-field gradients and is therefore an essential input for evaluating the quadrupole shift. Its accurate determination is nevertheless demanding because the property depends on the combined effects of relativistic interactions, multielectron correlation, basis-set completeness, and quantum electrodynamic (QED) corrections.
We calculate the electric quadrupole moment of the Ni^12+ ground state (^3P_2, |M_J|=2) by combining a four-component Dirac-Coulomb-Gaunt Hamiltonian, the relativistic Fock-space coupled cluster (FSCC) method, and a finite-field treatment. Ni^10+ is used as the closed-shell Fermi vacuum, and the target Ni^12+ states are described in the two-hole sector. The quadrupole moment is obtained from the first-order response of the correlated energy to an external electric-field gradient. Calculations with the dyall.aaeXz (X=2,3,4) and aug-cc-pV5Z basis sets are used to establish basis-set convergence. Single and double excitations are included in the large orbital space, whereas explicit triple excitations are evaluated with dyall.aae2z and dyall.aae3z basis sets and a virtual-orbital cutoff of 100 a.u. The QED contribution is estimated using effective potentials that account for Uehling vacuum polarization and model self-energy effects.
For the quadrupole moment, the aae4z and aug-cc-pV5Z results differ by only 5.4\times10^-6\,ea_0^2, demonstrating that the one-electron basis is well converged. The effective QED correction is -5.96\times10^-5\,ea_0^2. In comparison, the triple-excitation correction reaches -2.09\times10^-4\,ea_0^2 at the aae3z level and is therefore substantially larger than both the residual basis-set and QED contributions. By combining the converged FSCCSD value with these corrections, we recommend
\theta(3s^2 3p^4\,^3P_2, |M_J|=2)=0.07290(5)\,ea_0^2.
The quoted uncertainty is dominated by the basis-set dependence of the triple-excitation contribution.
These results show that explicit higher-order electron correlation is the leading refinement required for a reliable quadrupole moment of Ni^12+. The present finite-field FSCC strategy provides a systematically improvable route to electric multipole moments of open-shell HCIs and supplies a high-accuracy theoretical input for evaluating quadrupole shifts in the Ni^12+ optical clock.