Non-Additive Kinetic Energy and Potential in Analytically Solvable Systems and Their Approximated Counterparts

The one-electron equation for orbitals embedded in frozen electron density (Eqs. 20-21 in [Wesolowski and Warshel, J. Phys. Chem, 97 (1993) 8050]) in its exact and approximated version is solved for an analytically solvable model system. The system is used to discuss the role of the embedding potential in preventing the collapse of a variationally obtained electron density onto the nucleus in the case when the frozen density is chosen to be that of the innermost shell. The approximated potential obtained from the second-order gradient expansion for the kinetic energy prevents such a collapse almost perfectly but this results from partial compensation of flaws of its components. It is also shown that that the quality of a semi-local approximation to the kinetic-energy functional, a quantity needed in orbital-free methods, is not related to the quality of the non-additive kinetic energy potential - a key component of the effective embedding potential in one-electron equations for embedded orbitals