  Recently a novel approach to find approximate exchange-correlation
functionals in density-functional theory (DFT) was presented (U. Mordovina et.
al., JCTC 15, 5209 (2019)), which relies on approximations to the interacting
wave function using density-matrix embedding theory (DMET). This approximate
interacting wave function is constructed by using a projection determined by an
iterative procedure that makes parts of the reduced density matrix of an
auxiliary system the same as the approximate interacting density matrix. If
only the diagonal of both systems are connected this leads to an approximation
of the interacting-to-non-interacting mapping of the Kohn-Sham approach to DFT.
Yet other choices are possible and allow to connect DMET with other DFTs such
as kinetic-energy DFT or reduced density-matrix functional theory. In this work
we give a detailed review of the basics of the DMET procedure from a DFT
perspective and show how both approaches can be used to supplement each other.
We do so explicitly for the case of a one-dimensional lattice system, as this
is the simplest setting where we can apply DMET and the one that was originally
presented. Among others we highlight how the mappings of DFTs can be used to
identify uniquely defined auxiliary systems and auxiliary projections in DMET
and how to construct approximations for different DFTs using DMET inspired
projections. Such alternative approximation strategies become especially
important for DFTs that are based on non-linearly coupled observables such as
kinetic-energy DFT, where the Kohn-Sham fields are no longer simply obtainable
by functional differentiation of an energy expression, or for reduced
density-matrix functional theories, where a straightforward Kohn-Sham
construction is not feasible.
