Polarizabilities of two-electron positive ions and Rydberg levels of lithium

We have carried out precision calculations of various second-order sums for the two-electron positive ions ${\mathrm{Li}}^{+}$, ${\mathrm{Be}}^{2+}$, and ${\mathrm{B}}^{3+}$. These have the form ${\mathit{tsum}}_{\mathit{N}}$〈0‖V‖N〉〈N‖V‖0〉/(${\mathit{E}}_{\mathit{N}}$-${\mathit{E}}_0$ ${)}^{\mathit{k}}$, where k=1,2,3. The perturbation V is in each case a multipole-potential term, and for k=1 the sum corresponds to the usual definition of multipole polarizability. The initial state 0 and the intermediate states N are constructed from two-particle Hylleraas basis sets of the appropriate symmetry and angular momentum. Although the set N does not contain any true continuum functions, the pseudostates represent the continuum very well, and excellent convergence is obtained. These two-body quantities are then used to evaluate the energy levels of three-electron systems in high Rydberg states by the asymptotic-optical-potential method previously used for two-electron systems, retaining all terms up to order ${\mathit{x}}^{\mathrm{-}6}$. The results are good, but they can be still further improved by including higher-order terms, which will require multiple summation over intermediate states.