Exchange-correlation energy in molecules: A variational quantum Monte Carlo study

We have used the combination of the coupling-constant integration procedure and the variational quantum Monte Carlo method to study the exchange-correlation (XC) interaction in small molecules: ${\mathrm{Si}}_2$, ${\mathrm{C}}_2{\mathrm{H}}_2$, ${\mathrm{C}}_2{\mathrm{H}}_4$, and ${\mathrm{C}}_2{\mathrm{H}}_6$. In this paper we report the calculated XC energy density, a central quantity in density functional theory, as deduced from the interaction between the electron and its XC hole integrated over the interaction strength. Comparing these “exact” XC energy densities with results using the local-density approximation (LDA), one can analyze the errors in this widely used approximation. Since the XC energy is an integrated quantity, error cancellation among the XC energy density in different regions is possible. Indeed we find a general error cancellation between the high-density and low-density regions. Moreover, the error distribution of the exchange contribution is out of phase with the error distribution of the correlation contribution. Similar to what is found for bulk silicon and an isolated silicon atom, the spatial variation of the errors of the LDA XC energy density in these molecules largely follows the sign and shape of the Laplacian of the electron density. Some noticeable deviations are found in ${\mathrm{Si}}_2$ in which the Laplacian peaks between the atoms, while the LDA error peaks in the regions “behind” atoms where a good portion of the charge density originates from an occupied $1{\sigma }_u$ antibonding orbital. Our results indicate that, although the functional form could be quite complex, an XC energy functional containing the Laplacian of the energy is a promising possibility for improving LDA.

Figure 1

Error of the binding energy for ${\mathrm{C}}_2{\mathrm{H}}_2$, ${\mathrm{C}}_2{\mathrm{H}}_4$, and ${\mathrm{C}}_2{\mathrm{H}}_6$ as compared with experimental values. Experimental data are taken from Ref. 20.

Figure 2

Various quantities (in atomic units) for the silicon pseudoatom as a function of radius, (a) the VMC total electron density; (b) difference of the XC energy density between LDA and VMC results, $e_{\mathit{xc}}^{\mathrm{LSDA}}\left(\mathbf{r}\right)-e_{\mathit{xc}}^{\mathrm{VMC}}\left(\mathbf{r}\right)$; and (c) the Laplacian of the electron density.

Figure 3

Various contour plots for ${\mathrm{Si}}_2$: (a) valence electron density; (b) Laplacian of the electron density; (c) XC energy density calculated with the VMC and coupling-constant integration methods; (d) error in the LSDA XC energy density, $e_{\mathit{xc}}^{\mathrm{LSDA}}\left(\mathbf{r}\right)-e_{\mathit{xc}}^{\mathrm{VMC}}\left(\mathbf{r}\right)$; (e) error in the LSDA exchange energy density, $e_x^{\mathrm{LSDA}}\left(\mathbf{r}\right)-e_x^{\mathrm{VMC}}\left(\mathbf{r}\right)$; and (f) error in the LSDA correlation energy density error, $e_c^{\mathrm{LSDA}}\left(\mathbf{r}\right)-e_c^{\mathrm{VMC}}\left(\mathbf{r}\right)$. All values are multiplied by 10 in (a) and (b) and by ${10}^4$ in (c)–(f). All quantities plotted are in atomic units.

Figure 4

Various contour plots for ${\mathrm{C}}_2{\mathrm{H}}_2$: (a) valence electron density; (b) Laplacian of the electron density; (c) XC energy density calculated with the VMC and coupling-constant integration methods; (d) error in the LDA XC energy density, $e_{\mathit{xc}}^{\mathrm{LDA}}\left(\mathbf{r}\right)-e_{\mathit{xc}}^{\mathrm{VMC}}\left(\mathbf{r}\right)$; (e) error in the LDA exchange energy density, $e_x^{\mathrm{LDA}}\left(\mathbf{r}\right)-e_x^{\mathrm{VMC}}\left(\mathbf{r}\right)$; and (f) error in the LDA correlation energy density error, $e_c^{\mathrm{LDA}}\left(\mathbf{r}\right)-e_c^{\mathrm{VMC}}\left(\mathbf{r}\right)$. All values are multiplied by 10 in (a) and (b) and by ${10}^4$ in (c)–(f). All quantities plotted are in atomic units.

Figure 5

Various contour plots for ${\mathrm{C}}_2{\mathrm{H}}_4$: (a) valence electron density; (b) Laplacian of the electron density; (c) XC energy density calculated with the VMC and coupling-constant integration methods; (d) error in the LDA XC energy density, $e_{\mathit{xc}}^{\mathrm{LDA}}\left(\mathbf{r}\right)-e_{\mathit{xc}}^{\mathrm{VMC}}\left(\mathbf{r}\right)$; (e) error in the LDA exchange energy density, $e_x^{\mathrm{LDA}}\left(\mathbf{r}\right)-e_x^{\mathrm{VMC}}\left(\mathbf{r}\right)$; and (f) error in the LDA correlation energy density error, $e_c^{\mathrm{LDA}}\left(\mathbf{r}\right)-e_c^{\mathrm{VMC}}\left(\mathbf{r}\right)$. All values are multiplied by 10 in (a) and (b) and by ${10}^4$ in (c)–(f). All quantities plotted are in atomic units.

Figure 6

Various contour plots for ${\mathrm{C}}_2{\mathrm{H}}_6$: (a) valence electron density; (b) Laplacian of the electron density; (c) XC energy density calculated with the VMC and coupling-constant integration methods; (d) error in the LDA XC energy density, $e_{\mathit{xc}}^{\mathrm{LDA}}\left(\mathbf{r}\right)-e_{\mathit{xc}}^{\mathrm{VMC}}\left(\mathbf{r}\right)$; (e) error in the LDA exchange energy density, $e_x^{\mathrm{LDA}}\left(\mathbf{r}\right)-e_x^{\mathrm{VMC}}\left(\mathbf{r}\right)$; and (f) error in the LDA correlation energy density error, $e_c^{\mathrm{LDA}}\left(\mathbf{r}\right)-e_c^{\mathrm{VMC}}\left(\mathbf{r}\right)$. All values are multiplied by 10 in (a) and (b) and by ${10}^4$ in (c)–(f). All quantities plotted are in atomic units.

Figure 7

Angle-averaged electron density of (a) ${\mathrm{Si}}_2$ and (b) ${\mathrm{C}}_2{\mathrm{H}}_2$ as a function of distance away from the maximum-density position.

Figure 8

Angle-averaged electron density of (a) ${\mathrm{Si}}_2$ and (b) ${\mathrm{C}}_2{\mathrm{H}}_2$ as a function of distance away from the atomic position.