Electron avoidance: A nonlocal radius for strong correlation

We present here a model of the exchange-correlation hole for strongly correlated systems using a simple nonlocal generalization of the Wigner-Seitz radius. The model behaves similarly to the strictly correlated electron approach, which gives the infinitely correlated limit of density functional theory. Unlike the strictly correlated method, however, the energies and potentials of this model can be presently calculated for arbitrary geometries in three dimensions. We discuss how to evaluate the energies and potentials of the nonlocal model, and provide results for many systems where it is also possible to compare to the strictly correlated electron treatment.

Figure 1

Nonlocal radius $R\left(r\right)$ (32) for the exact densities of the spherically symmetric atoms helium, beryllium, and neon; exact data from Refs. [82, 83, 84]. To see the asymptotic behavior, the $y=x$ line is also plotted.

Figure 2

Nonlocal radius $R\left(\mathbf{r}\right)$ (32) for the exact density of the ${\mathrm{H}}_2$ molecule with bond length $R=6$, in cylindrical coordinates $(z,\rho )$. Exact density is the Full-CI result from GAMESS-US [85] within the aug-cc-pV6Z [86] basis set. The bond axis is along the $z$ coordinate, and the nuclei are at $z=\pm 3$.

Figure 3

The interaction XC energy per particle (41) as a function of the radius $r$ for the exact 3D helium atom density [49]. Inset: $r{w}_{\mathrm{XC}}\left(\mathbf{r}\right)$ to see asymptotic behavior. The exact $w_{\mathrm{XC}}\left(\mathbf{r}\right)\to -1/\left(2r\right)$ as $r\to \infty$.

Figure 4

The interaction XC energy per particle (41) as a function of the radius $r$ for the exact 3D ${\mathrm{H}}^{-}$ atom density [49]. Inset: $r{w}_{\mathrm{XC}}\left(\mathbf{r}\right)$.

Figure 5

The interaction XC energy per particle (41) as a function of the radius $r$ for the exact 3D beryllium atom density [49]. Inset: $r{w}_{\mathrm{XC}}\left(\mathbf{r}\right)$. Notice that the SCE energy density physically has a kink near $r=1$, which has to do with classical electrons disappearing to infinity in the SCE method [49].

Figure 6

3D ${\mathrm{H}}_2$ comparing exact energies from Ref. [93] with non-self-consistent KS-NLR energies on exact densities; densities from GAMESS-US [85] Full-CI calculations within the aug-cc-pV6Z [86] basis set, with fits thanks to S. Vuckovic. Lines are cubic spline interpolations through data (at markers).

Figure 7

Self-consistent results for the KS-SCE and KS-NLR (52) energy functionals of the 3D helium atom: plotted are the densities and HXC potentials, with comparisons to $1/r$ $\left(2/r\right)$, the asymptotic behavior of the exact (LDA) HXC potential. Exact results thanks to C. Umrigar [82].

Figure 8

Self-consistent energies of the KS-NLR, KS-SCE, and KS-LDA methods for the hydrogen atom with a variable number of electrons $N$, performed within a spin-restricted framework. KS-SCE and KS-LDA data from Ref. [43].

Figure 9

Self-consistent energies and HOMO eigenvalues of the KS-NLR and KS-SCE functionals for the 3D helium isoelectronic series $(N=2,Z$ variable), compared with the exact behavior (cubic spline interpolation of $Z=1$, 2, 4, 10 values from Ref. [82]), KS-SCE data from Ref. [43].

Figure 10

Densities of the $2k_F\to 4k_F$ transition for $N=4$ soft-Coulomb interacting electrons in a parabolic external potential $v\left(x\right)=k{x}^2/2$. Exact density from DMRG [109].

Figure 11

KS potentials for the $k={10}^{-5}$ system of Fig. 10.

Figure 12

The total molecular energy of 1D ${\mathrm{H}}_2$ with soft-Coulomb interactions between all particles. Exact and KS-LDA data from Ref. [51]. The horizontal dotted line is twice the energy of a KS-LSDA hydrogen atom.

Figure 13

(Lack of) binding for the 1D soft-Coulomb helium dimer. KS-SCE predicts a weakly bound state. Horizontal lines indicate twice the energy of a single isolated helium atom within each method.

Figure 14

The total molecular energy of 1D LiH with soft-Coulomb interactions between all particles. Horizontal lines indicate the sum of energies of an isolated Li atom and an isolated H atom within each method.