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Relativistic and QED corrections to one-bond indirect nulcear spin-spin couplings in X$_2^{2+}$ and X$_3^{2+}$ ions (X = Zn, Cd, Hg)

The indirect nuclear spin-spin coupling tensor, $\mathbf J$, between mercury nuclei in Hg-containing systems can be of the order of few kHz and one of the largest measured. We conduct an analysis of the physics behind the electronic mechanisms that contribute to the one- and two-bond couplings $^n {\mathbf J}_{\mathrm{Hg}-\mathrm{Hg}}$ ($n=1, 2$). We performed calculations for $J$-couplings in X$_2^{2+}$ and $X_3^{2+}$ ions ($X$ = Zn, Cd, Hg), within polarization propagator theory, using the random phase approximation (RPA) and the pure zeroth order approximation (PZOA), with Dirac-Hartree-Fock (DHF) and Dirac-Kohn-Sham (DKS) orbitals, both at four-component and ZORA levels. We show that the "paramagnetic-like" mechanism contribute with more than 99.98\% to the total isotropic component of the coupling tensor. By means of an analysis of the molecular and atomic orbitals involved in the total value of the response function, we find that the $s$-type valence atomic orbitals have a predominant role in the description of the coupling. This fact allows us to develop an effective model from which quantum electrodynamics (QED) effects on $J$-coupling in the aforementioned ions can be estimated. The estimated QED corrections were found in the interval $(0.7; ~ 1.7)$\% of the total relativistic effect on isotropic one-bond $^1 {\mathbf J}$ coupling and from the interval $(-0.2; ~ -0.4)$\%, in Zn-containing ions, to $(-0.8; ~ -1.2)$\%, in Hg-containing ions, of the total isotropic coupling constant in the studied systems. We also show that estimated QED corrections cast a visible dependence on the nuclear charge $Z$ of each atom $X$ in the form of a power-law $\propto Z^5$.