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 delta Delta F-MD[rho(A)]/delta rho(A)((r) over right arrow), where Delta F-MD[rho(A)] is the functional bound from below by the correlation functional E-c[rho(A)] and from above by zero. Taking into account Delta F-MD[rho(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 Delta F-MD[rho(A)], we examine numerically the adequacy of neglecting Delta F-MD[rho(A)] entirely. To this end, we analyze the significance of delta Delta F-MD[rho(A)]/delta rho(A)((r) over right arrow) in the case where the magnitude of Delta F-MD[rho(A)] is the largest, i.e., for Hartree-Fock wavefunction. In hydrogen bonded model systems, neglecting delta Delta F-MD[rho(A)]/delta rho(A)((r) over right arrow) 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 delta Delta F-MD[rho(A)]/delta rho(A)((r) over right arrow) 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 delta Delta F-MD[rho(A)]/delta rho(A)((r) over right arrow) do not cancel but rather add up to the errors due to approximation for the bi-functional of the non-additive kinetic potential. (C) 2011 American Institute of Physics. [doi:10.1063/1.3624888]