Relativistic coupled-cluster-theory analysis of unusually large correlation effects in the determination of $g_j$ factors in ${\mathrm{Ca}}^{+}$

We investigate roles of electron correlation effects in the determination of the $g_j$ factors of the $4s{}^2{S}_{1/2}$, $4p{}^2{P}_{1/2}$, $4p{}^2{P}_{3/2}$, $3d{}^2{D}_{3/2}$, and $3d{}^2{D}_{5/2}$ states, representing different parities and angular momenta, of the ${\mathrm{Ca}}^{+}$ ion. Correlation contributions are highlighted with respect to the mean-field values evaluated using the Dirac-Hartree-Fock method, relativistic second-order many-body theory, and relativistic coupled-cluster (RCC) theory with the single- and double-excitation approximation considering only the linear terms and also accounting for all the nonlinear terms. This shows that it is difficult to achieve results below ${10}^{-5}$ precision employing an approximate perturbative approach. We also find that contributions through the nonlinear terms and higher-level excitations such as triple excitations, estimated perturbatively in the RCC method, are found to be crucial to attain precise values of the $g_j$ factors in the considered states of the ${\mathrm{Ca}}^{+}$ ion.

Figure 1

Trends of electron correlation effects among the $4s{}^2{S}_{1/2}$, $3d{}^2{D}_{3/2}$, $3d{}^2{D}_{5/2}$, $4p{}^2{P}_{1/2}$, and $4p{}^2{P}_{3/2}$ states for the evaluation of $g_j^D$ values in ${\mathrm{Ca}}^{+}$. We plot the relative $(g_j^D-\mathrm{DHF})/(g_j-\mathrm{DHF})$ values to highlight the roles of correlation effects through different many-body methods. We consider values from the DHF, MBPT(2), MBPT(3), LCCSD, and CCSD methods in a sequence on the $X$ axis in an arbitrary unit distance.

Figure 2

Some of the important contributing Goldstone diagrams appearing through the $T_2^{†}O{S}_{3v}^{\mathrm{pert}}$ RCC term. Lines going up and down represent the virtual and occupied orbitals of ${\mathrm{Ca}}^{+}$. Lines with double arrows correspond to the valence orbital, dashed horizontal lines denote Coulomb interaction, and solid horizontal lines correspond to the all-order Coulomb interactions appearing through the $T_2$ and $S_{2v}$ operators.