Binding energy and geometry of e$^+$Na

We calculate the non-adiabatic binding energy and geometry of the weakly bound state of e$^+$Na. We use the Peach model potential to describe the e$^+$-Na$^+$ and e$^-$-Na$^+$ interactions and solve the effective three-body Schr\"odinger equation with the finite element method. Because the model potential gives rise to three spurious states (corresponding to 1s, 2s, and 2p), the true non-adiabatic ground state of e$^+$Na is embedded in a dense spectrum of spurious states. We developed a technique for extracting the correct ground state for e$^+$Na, even when the energy is nearly degenerate with a spurious level. This is the first calculation to include the quadrupole term in the polarization potential.