The strictly-correlated-electrons density functional theory (SCE DFT) approach, originally proposed by Michael Seidl in 1999, is a formulation of density functional theory (DFT), alternative to the DFT approach based on Kohn–Sham equations, especially aimed at the study of strongly correlated materials. The essential difference between the two approaches is the choice of the auxiliary system (having the same density n ( r ) {\displaystyle n(\mathbf {r} )} as the real, physical one). In Kohn–Sham DFT this system is composed by non-interacting electrons, for which the kinetic energy can be calculated exactly and the interaction term has to be approximated. In SCE DFT, instead, the starting point is totally the opposite one: the auxiliary system has infinite electronic correlation and zero kinetic energy.
Strictly-correlated-electrons reference system To understand how the systems of strictly-correlated electrons (SCE) is constructed, it is useful to first think in terms of a simple example. Consider a collection of N {\displaystyle N} identical classical charges (with repulsive Coulomb interaction) confined in some container with a given shape. If let alone, the charges will distribute themselves within the container until they reach the spatial configuration that minimizes their interaction energy (in equilibrium, their kinetic energy is zero). Of course, the equilibrium position of the charges will depend on the shape of the container. Suppose now that in this classical system one of the N {\displaystyle N} charges, which we can label as number "1", is pinned at some arbitrary position r 1 = r {\displaystyle \mathbf {r} _{1}=\mathbf {r} } inside the container. Clearly, the equilibrium position of the other N − 1 {\displaystyle N-1} charges will now not only depend on the shape of the container, but also on the position
r {\displaystyle \mathbf {r} } of the pinned charge. Thus, for a given confining geometry, one can write the position of the i {\displaystyle i} -th particle ( i = 2 , . . , N ) {\displaystyle (i=2,..,N)} , r i {\displaystyle \mathbf {r} _{i}} , as a function of r {\displaystyle \mathbf {r} } : r i = f i ( r ) {\displaystyle \mathbf {r} _{i}=\mathbf {f} _{i}(\mathbf {r} )} . In the SCE system, as in the classical example described above, the position r 1 = r {\displaystyle \mathbf {r} _{1}=\mathbf {r} } of a reference electron determines the position of the remaining ones. The analogue role of the confining container is now played by the condition that the density at each point must be the same as that of the real system, n ( r ) {\displaystyle n(\mathbf {r} )} : the electrons will always try to be as far apart from each other as possible, in order to minimize their repulsion, but always restricted by this condition. The positions r i = f i ( r 1 ) {\displaystyle \mathbf {r} _{i}=\mathbf {f} _{i}(\mathbf {r} _{1})} are called co-motion functions and play a fundamental role in the SCE formalism, analogue to the Kohn–Sham single-particle orbitals in Kohn–Sham DFT.
Calculation of the co-motion functions and interaction energy For a given density n ( r ) {\displaystyle n(\mathbf {r} )} , the probability of finding one electron at a certain position r {\displaystyle \mathbf {r} } is the same as that of finding the i {\displaystyle i} -th electron at f i [ n ] ( r ) {\displaystyle \mathbf {f} _{i}[n](\mathbf {r} )} , or, equivalently, n ( r ) d r = n ( f i ( r ) ) d f i ( r ) {\displaystyle n(\mathbf {r} )d\mathbf {r} =n(\mathbf {f} _{i}(\mathbf {r} ))d\mathbf {f} _{i}(\mathbf {r} )} . The co-motion functions can be obtained from the integration of this equation. An analytical solution exists for 1D systems, but not for the general case. The interaction energy of the SCE system for a given density n ( r ) {\displaystyle n(\mathbf {r} )} can be exactly calculated in terms of the co-motion functions as
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