Self-consistency in frozen-density embedding theory based calculations.

The bi-functional for the non-electrostatic part of the exact embedding potential of frozen-density embedding theory (FDET) depends on whether the embedded part is described by means of a real interacting many-electron system or the reference system of non-interacting electrons (see [Wesolowski, Phys. Rev. A. 77, 11444 (2008)]). The difference δΔF(MD)[ρ(A)]/δρ(A)(r), where ΔF(MD)[ρ(A)] is the functional bound from below by the correlation functional E(c)[ρ(A)] and from above by zero. Taking into account ΔF(MD)[ρ(A)] in both the embedding potential and in energy is indispensable for assuring that all calculated quantities are self-consistent and that FDET leads to the exact energy and density in the limit of exact functionals. Since not much is known about good approximations for ΔF(MD)[ρ(A)], we examine numerically the adequacy of neglecting ΔF(MD)[ρ(A)] entirely. To this end, we analyze the significance of δΔF(MD)[ρ(A)]/δρ(A)(r) in the case where the magnitude of ΔF(MD)[ρ(A)] is the largest, i.e., for Hartree-Fock wavefunction. In hydrogen bonded model systems, neglecting δΔF(MD)[ρ(A)]/δρ(A)(r) in the embedding potential marginally affects the total energy (less than 5% change in the interaction energy) but results in qualitative changes in the calculated hydrogen-bonding induced shifts of the orbital energies. Based on this estimation, we conclude that neglecting δΔF(MD)[ρ(A)]/δρ(A)(r) may represent a good approximation for multi-reference variational methods using adequate choice for the active space. Doing the same for single-reference perturbative methods is not recommended. Not only it leads to violation of self-consistency but might result in large effect on orbital energies. It is shown also that the errors in total energy due to neglecting δΔF(MD)[ρ(A)]/δρ(A)(r) do not cancel but rather add up to the errors due to approximation for the bi-functional of the non-additive kinetic potential.

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