Ionic Dipole and Quadrupole Matrix Elements from Non Adiabatic Core Polarization

The radial matrix elements connecting the ionic Ba$^+$ state to low lying excited $6p$ and $5d$ states can be extracted from the K splittings of the bound $6sn\ell$ states. We develop an expression for the K splitting by a pair of expansions. Comparison of the contributions from different ionic states confirms that all but the lowest two may be safely ignored. Finally we extract the radial Ba$^+$ matrix elements $\left<6s|r|6p\right>$ and $\left<6s|r^2|5d\right>$.