King and second-King plots with optimized sensitivity for lithium ions

King plots constructed from combinations of mass-weighted atomic isotope shifts provide a sensitive technique to search for electron-neutron interactions beyond the standard model mediated by a light boson. Using high-precision variational wave functions in Hylleraas coordinates, we present a comprehensive survey of all possible King plots arising from states of ${\mathrm{Li}}^{+}$ up to principal quantum number $n=10$ and angular momentum $L=7$ in order to identify the ones most sensitive to new physics. A major limitation in previous work due to second-order mass polarization is eliminated by the introduction of a second-King plot defined in terms of second differences. The residual theoretical uncertainty is then of the order of ${\ensuremath{\alpha}}^{2}{(\ensuremath{\mu}/M)}^{3}\ensuremath{\sim}0.4$ Hz, where $\ensuremath{\alpha}$ is the fine-structure constant and $\ensuremath{\mu}$ is the reduced electron mass for a nucleus of mass $M$. Test results are presented for the ${}^{A}{\mathrm{Li}}^{+}$ isotope sequence with $A=6,7,8,9,11$ and are compared with other methods, including the ${\mathrm{Yb}}^{+}$ case recently studied both experimentally and theoretically. It is shown that the second-King plots for ${\mathrm{Li}}^{+}$ have about the same sensitivity to new physics as the ${\mathrm{Yb}}^{+}$ case for boson masses up to about 10 keV, and nuclear size uncertainties (including nuclear polarization) are suppressed. This greatly extends the sensitivity to new physics for light two-electron systems.