The critical role of negative-energy states in the Landé $g$-factor of lithium-like ions

We report relativistic many-body calculations of the interelectronic-interaction correction to the Landé $g$-factor of the $2s_{1\mathrm{/}2}$, $2p_{1\mathrm{/}2}$, $2p_{3\mathrm{/}2}$, and $3s_{1\mathrm{/}2}$ states in lithium-like ions with nuclear charge $Z=4-20$. Starting from the Dirac-Coulomb-Breit Hamiltonian, we treat positive-energy contributions using the coupled-cluster method with single and double excitations and include negative-energy contributions through third-order perturbation theory. We observe that negative-energy states give a state-dependent correction whose magnitude and sign vary with both Z and the state; for $2p_{1\mathrm{/}2}$, the correction from the negative-energy states reaches 30% of the total interelectronic-interaction contribution at $Z=20$. Agreement with previous high-precision calculations is better than $0.1\mathrm{\%}$, confirming the reliability of the present approach. This work may serve as a valuable reference for future precise calculations of $g$-factors for many-electron atomic systems.