Strictly Correlated Electrons density functional theory

Calculation of the co-motion functions and interaction energy of the SCE system

For a given density n ( r ) , the probability of finding one electron at a certain position r is the same as that of finding the i -th electron at f i [ n ] ( r ) , or, equivalently,

n ( r ) d r = n ( f i ( r ) ) d f i ( r ) .

The co-motion functions can be obtained from the integration of this equation. An analytical solution exists for 1D systems [2,3], but not for the general case.

The interaction energy of the SCE system for a given density n ( r ) can be exactly calculated in terms of the co-motion functions as [6]

V e e S C E [ n ] = ∫ d s n ( s ) N ∑ i = 1 N − 1 ∑ j = i + 1 N 1 | f i ( s ) − f j ( s ) | = 1 2 ∫ d s n ( s ) ∑ i = 2 N 1 | s − f i ( s ) | .

Notice that this is analogous to the Kohn-Sham approach, where the non-interacting kinetic energy is expressed in terms of the Kohn-Sham single-particle orbitals.

A very important property of the SCE system is the following one: since the position of one particle determines the position of the remaining ones, the total coulomb repulsion felt by a particle at a point r becomes a function of only r itself. This force can then be written as minus the gradient of some one-particle potential v S C E ( r ) [5,6]:

− ∇ v S C E ( r ) = F c o u l o m b ( r ) = ∑ i = 2 N r − f i [ n ] ( r ) | r − f i [ n ] ( r ) | 3 .

At the same time, it can be shown that the potential v S C E ( r ) satisfies the relation [4,6]

δ V e e S C E [ n ] δ n ( r ) = − v S C E ( r ) .

A promising route towards the application of the SCE approach to systems with general symmetry is the mass-transportation-theory reformulation of the approach [8]. This is based on the analogies between the SCE problem and the dual Kantorovich problem [8]. The SCE wave function is also very useful to set rigorous bounds for the constant appearing in the Lieb-Oxford inequality.

Combining the strictly-correlated-electron and the Kohn-Sham approaches

The one-body potential v S C E ( r ) can be used to approximate the Hartree-exchange-correlation (Hxc) potential of the Kohn-Sham DFT approach [4,5]. Indeed, one can see the analogy between the expression relating the functional derivative of V e e S C E [ n ] and v S C E ( r ) and the well-known one of Kohn-Sham DFT

δ E H x c [ n ] δ n ( r ) = v H x c ( r ) ,

which relates the Hartree-exchange-correlation (Hxc) functional and the corresponding potential.

The approximation (which becomes exact in the limit of infinitely strong interaction [5]) corresponds to writing the Hohenberg-Kohn functional as

F [ n ] = T s [ n ] + E H x c [ n ] ≈ T s [ n ] + V e e S C E [ n ] ,

where T s [ n ] is the non-interacting kinetic energy.

One has therefore v H x c ( r ) ≈ − v S C E ( r ) and this leads to the Kohn-Sham equations

( − ℏ 2 2 m ∇ 2 + v e x t ( r ) − v S C E ( r ) ) ϕ i ( r ) = ε i ϕ i ( r ) ,

which can be solved self-consistently.

Since the v S C E ( r ) potential is constructed from the exact properties of the SCE system [2-4], it is able to capture the effects of the strongly-correlated regime, as it has been recently shown in the first applications of this "KS-SCE DFT" approach to simple model systems [5-7]. In particular, the method has allowed to observe Wigner localization in strongly-correlated electronic systems without introducing any artificial symmetry breaking [5-7].