Entanglement and density-functional theory: Testing approximations on Hooke’s atom

We present two methods of calculating the spatial entanglement of an interacting electron system within the framework of density-functional theory. These methods are tested on the model system of Hooke’s atom for which the spatial entanglement can be exactly calculated. We analyze how the strength of the confining potential affects the spatial entanglement and how accurately the methods that we introduced reproduce the exact trends. We also compare the results to the outcomes of standard first-order perturbation methods. The accuracies of energies and densities when using these methods are also considered.

Figure 1

Linear entropy of the reduced density matrix versus $\omega$.

Figure 2

Ratio of the expectation of the Coulomb interaction to the expectation of the external potential versus $\omega$.

Figure 3

Von Neumann entropy of the reduced density matrix versus $\omega$.

Figure 4

Comparison of $S$ and $L$ scaled by 3.75 versus $\omega$.

Figure 5

Position-space information entropy $S_n$ versus $\omega$.

Figure 6

Exact $E_{\text{xc}}$ versus $\omega$.

Figure 7

$\left|E_{\text{xc}}/E\right|$ versus $\omega$ as a possible entanglement indicator.

Figure 8

Comparison of the LDA density (dashed lines) and exact density (solid lines) plotted against $r$ for three different values of $\omega$.

Figure 9

Relative error in the LDA density compared to the exact density versus $\omega$.

Figure 10

Comparison of the LDA density (dashed line) to the density arising from ${\psi }_{\text{LDA},\text{EA}1}$ (solid line) plotted against $r$ for three different values of $\omega$.

Figure 11

(Color online) Comparison of $L$ for the exact wave function and the approximate interacting wave functions (${\psi }_{\text{LDA},\text{EA}1}$ and ${\psi }_{\text{LDA},\text{EA}2}$) against $\omega$. Inset: relative error in the density match $f$ versus $\omega$.

Figure 12

Position-space information entropy $S_n$ for the LDA density (crosses) and the exact density (solid line) plotted against $\omega$.

Figure 13

(Color online) Relative error of the approximate energy from LDA, first-order perturbation of the LDA KS equations, first-order perturbation of the exact KS equations, and standard first-order perturbation, compared to the exact energy plotted against $\omega$.

Figure 14

(Color online) Comparison of the relative errors of first- and second-order perturbations of the LDA KS equations together with the relative error in the LDA energy plotted against $\omega$. Inset: comparison of the relative errors of the zeroth- (dashed line) and first-order (solid line) perturbations of the LDA KS equations compared to the exact energy plotted against $\omega$.

Figure 15

(Color online) Entanglement quantified with $L$ of the exact solution, first-order perturbation of the KS equations, first-order perturbation of the LDA KS equations, and standard first-order perturbation, plotted against $\omega$.

Figure 16

(Color online) Comparison of the relative error in the reproduction of the density by approximate methods versus $\omega$. Inset: comparison of the relative error in the reproduction of the density between perturbation of the exact KS equations and perturbation of the LDA KS equations versus $\omega$.

Figure 17

(Color online) Overview of $L$ for the approximate methods and exact solution plotted against $\omega$.

Figure 18

(Color online) Overview of $S$ for the approximate methods and exact solution plotted against $\omega$.