High-precision results are presented for calculations of the nonrelativistic energies, relativistic corrections, and quantum electrodynamic corrections for the $2{}^{2}S$, $2{}^{2}P$, and $3{}^{2}S$ states of Li and ${\mathrm{Be}}^{+}$, using nonrelativistic wave functions expressed in Hylleraas coordinates. Bethe logarithms are obtained for the states of ${\mathrm{Be}}^{+}$. Finite mass corrections are calculated with sufficient accuracy to extract the nuclear charge radius from measurements of the isotope shift for the $2{}^{2}S-2{}^{2}P$ and $2{}^{2}S-3{}^{2}S$ transitions. The calculated ionization potential for ${\mathrm{Be}}^{+}$ is $146882.923\pm 0.005{\mathrm{cm}}^{-1}$.

The critical nuclear charge $Z_c$ required to bind a nucleus plus two electrons in a heliumlike atom has recently been an area of active study, resulting in a disagreement with earlier calculations and with the value obtained from the radius of convergence $1/Z^{*}$ of a $1/Z$ expansion of the energy. In order to resolve the disagreement, have performed high-precision variational calculations in Hylleraas coordinates. With the double basis set method, we have been able to obtain good convergence for $Z$ very close to $Z_c$, which together with the Hellmann-Feynman theorem yields the value $Z_c=0.91102822407725573(4)$, corresponding to $1/Z_c=1.09766083373855980(5)$. This value is in agreement with the value obtained by Baker et al. [Phys. Rev. A 41, 1247 (1990)]. A significant feature of the results is that the outer electron remains localized near the nucleus, even at $Z=Z_c$, and the bound state evidently changes smoothly into a shape resonance for $Z<Z_c$. A qualitative polarization potential is proposed to account for the resonance, and the radial distribution function for the electron density is calculated.

In the ionization process, longitudinal momentum along the direction of propagation is transferred to the photoelectrons due to the action of the magnetic component of Lorentz force. In a recent experiment by Smeenk et al. [Phys. Rev. Lett. 106, 193002 (2011)], such a transfer is observed in the ionization of argon and neon atoms by circularly polarized light at 800 and 1400 nm in the intensity range of 1014\char21{}1015W/cm2. They accounted for the results by a purely classical model. We present a fully quantum-mechanical calculation of the transfer of longitudinal momentum to the photoelectrons. The results are in agreement with the observations of Smeenk et al. at high intensities, but clear evidence for additional Coulomb interactions emerges at low intensities.

The hyperfine structures of the $2{{}^{3}S}_1$ states of the ${}^6{\mathrm{Li}}^{+}$ and ${}^7{\mathrm{Li}}^{+}$ ions are investigated theoretically to extract the Zemach radii of the ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$ nuclei by comparing with precision measurements. The obtained Zemach radii are larger than the previous values of Puchalski and Pachucki [ Phys. Rev. Lett. 111, 243001 (2013)] and disagree with them by about 1.5 and 2.2 standard deviations for ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$, respectively. Furthermore, our Zemach radius of ${}^6\mathrm{Li}$ differs significantly from the nuclear physics value, derived from the nuclear charge and magnetic radii [ Phys. Rev. A 78, 012513 (2008)] by more than $6\sigma$, indicating an anomalous nuclear structure for ${}^6\mathrm{Li}$. The conclusion that the Zemach radius of ${}^7\mathrm{Li}$ is about 40% larger than that of ${}^6\mathrm{Li}$ is confirmed. The obtained Zemach radii are used to calculate the hyperfine splittings of the $2{{}^{3}P}_J$ states of ${{}^{6,7}\mathrm{Li}}^{+}$, where an order of magnitude improvement over the previous theory has been achieved for ${}^7{\mathrm{Li}}^{+}$.

We investigate the $2{}^3{S}_1–2{}^3{P}_J$ ($J=0$, 1, 2) transitions in ${}^6{\mathrm{Li}}^{+}$ using the optical Ramsey technique and achieve the most precise values of the hyperfine splittings of the $2{}^3{S}_1$ and $2{}^3{P}_J$ states, with smallest uncertainty of about 10 kHz. The present results reduce the uncertainties of previous experiments by a factor of 5 for the $2{}^3{S}_1$ state and a factor of 50 for the $2{}^3{P}_J$ states, and are in better agreement with theoretical values. Combining our measured hyperfine intervals of the $2{}^3{S}_1$ state with the latest quantum electrodynamic (QED) calculations, the improved Zemach radius of the ${}^6\mathrm{Li}$ nucleus is determined to be 2.44(2) fm, with the uncertainty entirely due to the uncalculated QED effects of order $m{\alpha }^7$. The result is in sharp disagreement with the value 3.71(16) fm determined from simple models of the nuclear charge and magnetization distribution. We call for a more definitive nuclear physics value of the ${}^6\mathrm{Li}$ Zemach radius.