Isotope shifts and transition frequencies for the $S$ and $P$ states of lithium: Bethe logarithms and second-order relativistic recoil

Isotope shifts and total transition frequencies are calculated for the $2{}^{2}S-3{}^{2}S$ transition of the lithium isotopes ${}^6\mathrm{Li}, {}^7\mathrm{Li}, {}^8\mathrm{Li}, {}^9\mathrm{Li}$, and the halo nucleus ${}^{11}\mathrm{Li}$. The accuracy is improved for previously calculated relativistic and quantum electrodynamic corrections, and in particular a disagreement for the Bethe logarithm is resolved for the ground ${}^{2}S$ state. Our previous result is confirmed for the $2{}^{2}P$ state. We use the pseudostate expansion method to perform the sum over virtual intermediate states. Results for the second-order relativistic recoil term of order ${\alpha }^2{(\mu /M)}^2$ Ry are shown to make a significant contribution relative to the theoretical uncertainty, but because of accidental cancellations the final result for the isotope shift is nearly unchanged. However, the spin-orbit term makes an unexpectedly large contribution to the splitting isotope shift (SIS) for the $2{}^{2}P_{1/2}-2{}^{2}P_{3/2}$ fine structure, increasing the theoretical value for the ${}^6\mathrm{Li}-{}^7\mathrm{Li}$ isotopes to $0.55631\left(7\right)\pm 0.001$ MHz. A comparison is made with high-precision measurements and other calculations for the SIS and for the total $2{}^{2}S-3{}^{2}S$ transition frequency.