Kohn-Sham calculations with the exact functional

As a proof of principle, self-consistent Kohn-Sham calculations are performed with the exact exchange-correlation functional. Finding the exact functional for even one trial density requires solving the interacting Schrödinger equation many times. The density matrix renormalization group method makes this possible for one-dimensional, real-space systems of more than two interacting electrons. We illustrate and explore the convergence properties of the exact KS scheme for both weakly and strongly correlated systems. We also explore the spin-dependent generalization and densities for which the functional is ill defined.

Figure 1

The KS scheme.

Figure 2

Density inversion of arbitrary four-electron density for noninteracting and interacting potentials. Solving either the interacting Schrödinger equation in the potential $v\left[n\right]\left(x\right)$ or solving the noninteracting Schrödinger equation in the potential $v_{\mathrm{S}}\left[n\right]\left(x\right)$ yields the density in the top panel.

Figure 3

To determine the $E_{\mathrm{HXC}}\left[n\right]$ and $v_{\mathrm{HXC}}\left[n\right]\left(\mathbf{r}\right)$, our exact calculation requires a computationally demanding inversion algorithm to find the one-body potential $v\left[n\right]\left(\mathbf{r}\right)$ of the interacting system whose density is $n\left(\mathbf{r}\right)$, with KS orbitals $\varphi \left(\mathbf{r}\right)$, in addition to a noninteracting inversion to find $v_{\mathrm{S}}\left[n\right]\left(\mathbf{r}\right)$. In case of degeneracy, mixed states should be used instead of pure-state wave functions in both noninteracting and interacting inversions [7, 54]. The right-hand side differs from the left in that it describes an interacting inversion.

Figure 4

KS procedure for a moderately correlated four-electron system (four hydrogen atoms separated by an interatomic spacing of $R=3$), using a fixed $\lambda =0.3$ and showing the first few iterations of (a) differences in the trial output densities from the ground-state density (shown in Fig. 5) and (b) trial KS potentials. Data are taken from Ref. [6].

Figure 5

External, KS, Hartree, and XC potentials, as well as the ground-state density, for a moderately correlated four-electron system (four hydrogen atoms separated by an interatomic spacing of $R=3$).

Figure 6

A single step in the KS scheme for a weakly correlated system (H${}_4$ with $R=2$) away from two different initial densities: noninteracting electrons in the external potential (NI) and a pseudouniform electron density (PU). These initial densities are the dashed curves in (a) and (b), and the solid curves are the output densities for each KS step; for comparison, the dotted curve is the exact density. Panel (c) plots Eq. (22), the energy of the system as it interpolates from the input to the output density.

Figure 7

A single step in the KS scheme for a strongly correlated system (H${}_4$ with $R=4$) away from two different initial densities: noninteracting electrons in the external potential (NI) and a pseudouniform electron density (PU). These initial densities are the dashed curves in (a) and (b), and the solid curves are the output densities for each KS step; for comparison, the dotted curve is the exact density. Panel (c) plots Eq. (22), the energy of the system as it interpolates from the input to the output density.

Figure 8

Taking a second step in the KS scheme for a strongly correlated system (H${}_4$ with $R=4$). Panel (a) shows the input density which is near to the exact density (the $\lambda =42\%$ density of the NI input density of Fig. 7) and the resulting output density, which is far from the ground state. Panel (b) plots Eq. (22), and the inset (c) magnifies the small $\lambda$ region.

Figure 9

The most important density response functions $M_{\beta }\left(x\right)$ from Eq. (29) and their response amplitudes $a_{\beta }$ for the weakly correlated system ($R=2$) in the top panel and the strongly correlated system ($R=4$) in the bottom panel. The locations of the atoms are shown in solid gray circles.

Figure 10

The number of iterations required to converge an LDA calculation to $\eta <{10}^{-8}$ (12), as a function of $\lambda$, for various bond lengths $R$ of the H${}_2$ molecule, starting with an initial density of H${}^{-}$ on the left atom. The asymptotic form for small $\lambda$ can be well approximated by $7/\lambda$ for the data shown.

Figure 11

Plotting ${\lambda }_c$ for an LDA calculation as a function of the bond length $R$ of a stretched hydrogen molecule, starting with the exact H${}^{-}$ density on one atom, as well as KS gaps for both the LDA and exact systems.

Figure 12

The first few steps (numbered) in the KS scheme from some arbitrary starting density for H${}_4$ with $R=4$ in the LDA approximation. Numerical precision makes the energy data noisy. Metric distances are compared with the LDA ground state. Panel (a) plots the energy and panel (b) the metric as a function of the accumulated $\lambda$ step. The density $n_{\lambda }\left(x\right)$ with the lowest metric distance is not the energetic minimum, but they are fairly close.

Figure 13

Starting an exact KS calculation of stretched H${}_2$ with a spin-polarized density still converges to the correct spin-singlet density. Through the iterations $i$, we plot (a) the polarization density $n_{\uparrow }\left(x\right)-n_{\downarrow }\left(x\right)$ and (b) the up KS potentials $v_{\mathrm{S},\uparrow }\left(x\right)$; the down potentials are the mirror images.

Figure 14

Inverting $n_{\gamma }\left(r\right)$ of Eq. (49) for the KS potential. As $\gamma$ becomes smaller and smaller, the changes in potential near $r=1$ become larger and larger.

Figure 15

Kinetic energy convergence of $n_{\mathrm{P}}\left(r\right)$ from Eq. (49) by smoothing out the kink as in Eq. (36). While the von Weizsäcker kinetic energy (vW) may be evaluated and integrated using only one derivative of the density, higher-order derivatives of the density develop nonintegrable features.