Strong Correlation in Kohn-Sham Density Functional Theory

We use the exact strong-interaction limit of the Hohenberg-Kohn energy density functional to approximate the exchange-correlation energy of the restricted Kohn-Sham scheme. Our approximation corresponds to a highly nonlocal density functional whose functional derivative can be easily constructed, thus transforming exactly, in a physically transparent way, an important part of the electron-electron interaction into an effective local one-body potential. We test our approach on quasi-one-dimensional systems, showing that it captures essential features of strong correlation that restricted Kohn-Sham calculations using the currently available approximations cannot describe.

Figure 1

Self-consistent densities for $N=2$ electrons in Q1D [Hamiltonian of Eqs. (9, 10) with $b=0.1$ and $v_{\mathrm{ext}}(x)=\frac{1}{2}{\omega }^2{x}^2$], in units of the effective confinement length $L=2{\omega }^{-1/2}$ (here and in the following figures). The exact results are compared with KS LDA and KS SCE approximations. At large $L$ the KS LDA calculations do not converge, while KS SCE approaches the exact solution.

Figure 2

Self-consistent exchange-correlation potentials for the same system of Fig. 1. For clarity, the potentials for $L=29$ have been shifted by $-4$.

Figure 3

Self-consistent KS SCE densities for $N=4$ electrons in Q1D [Hamiltonian of Eqs. (9, 10) with $b=0.1$ and $v_{\mathrm{ext}}(x)=\frac{1}{2}{\omega }^2{x}^2$].

Figure 4

The KS eigenvalue obtained in the self-consistent KS LDA and KS SCE calculations for the same $N=2$ electron problem considered in Fig. 1, plotted against the negative of the exact ionization energy, $E_N-E_{N-1}$.