Non-Local Translationally Invariant {\it Ab Initio} One-Body Density Matrices*

A derivation of microscopic effective interactions (optical potentials) between nucleons and a nucleus requires in first order aside from the NN interaction a non-local {\it ab initio} one-body density matrix for the target nucleus which is translationally invariant. We obtained non-local one-body density matrices from no-core shell-model (NCSM) calculations based on the JISP16 NN interaction for $^4$He, $^6$He, and $^6$Li as well as from a symmetry-adapted NCSM for $^{12}$C using the same interaction. We present how the center-of-mass inherent in the NCSM calculation is removed exactly for non-local one-body density matrices. We further explore the nature of the non-locality and compare with commonly used approximations for this non-locality.