  In the present work, we introduce a Self-Consistent Density-Functional
Embedding technique, which leaves the realm of standard energy-functional
approaches in Density Functional Theory and targets directly the
density-to-potential mapping that lies at its heart. Inspired by the Density
Matrix Embedding Theory, we project the full system onto a set of small
interacting fragments that can be solved accurately. Based on the rigorous
relation of density and potential in Density Functional Theory, we then invert
the fragment densities to local potentials. Combining these results in a
continuous manner provides an update for the Kohn-Sham potential of the full
system, which is then used to update the projection. The scheme proposed here
converges to an accurate approximation for the density and the Kohn-Sham
potential of the full system. Convergence to exact results can be achieved by
increasing the fragment size. We find, however, that already for small embedded
fragments accurate results are obtained. We benchmark our approach for
molecular bond stretching in one and two dimensions and demonstrate that it
reproduces the known steps and peaks that are present in the exact
exchange-correlation potential with remarkable accuracy.
