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The NEO method was modified and extended to positron systems. NEO-MP2 energies and annihilation rates were calculated for PsH, and the effects of basis set size on correlation energies captured with the NEO-MP2 and NEO-FCI methods are compared and discussed. Equilibrium geometries and vibrational energy levels were computed for the LiX and e+LiX (X = H, F, Cl) systems at the MP2 and NEO-MP2 levels. It was found that anharmonicity plays a significant role, specifically in the differences between the vibrational energy levels of the LiX and e+LiX systems. The implications of these results with respect to VFR for these systems is discussed.