Local-spin-density functional for multideterminant density functional theory

Based on exact limits and quantum Monte Carlo simulations, we obtain, at any density and spin polarization, an accurate estimate for the energy of a modified homogeneous electron gas where electrons repel each other only with a long-range Coulombic tail. This allows us to construct an analytic local-spin-density exchange-correlation functional appropriate to new, multideterminantal versions of the density functional theory, where quantum chemistry and approximate exchange-correlation functionals are combined to optimally describe both long- and short-range electron correlations.

Figure 1

The size dependence of the VMC energy for the Coulomb potential at $r_s=1$. Empty symbols: VMC energy $E_N$; filled symbols: $E_N+T_{\infty }-T_N$. The curves show the best-fit $1∕N$ dependence of the latter, according to Eq. (25): circles and solid line refer to $\zeta =0$ (left scale), triangles and dashed line to $\zeta =1$ (right scale). The cross is the $\zeta =0$ result obtained in the thermodynamic limit by Ref. 43, which appears fully consistent with the present calculation. The statistical errors on the data points are much smaller than the symbol sizes. The ${\chi }^2$ is 11.8 for $\zeta =0$, and 0.4 for $\zeta =1$ (much poorer values would be obtained by just fitting $E_{\mathrm{VMC}}$, i.e., without including the kinetic-energy size correction).

Figure 2

Our DMC data for the correlation energy $(•)$ of the electron gas with long-range interaction $\mathrm{erf}\left(\mu r\right)∕r$ are compared with the fitting function (lines) of Eq. (26) for the unpolarized case (upper panel) and the fully polarized case (lower panel). The statistical errors on the DMC data are comparable with the symbol size.

Figure 3

A sample of our DMC pair-distribution functions. Upper panel: for $r_s=2$ and $\zeta =0$, $g^{\mathrm{LR}}$ is shown for $\mu =0.25,0.5,1,2$, and for the Coulomb gas $(\mu =\infty )$. Lower panel: for $r_s=10$ and $\zeta =1$, $g^{\mathrm{LR}}$ is shown for $\mu =0.05,0.1,0.2,0.4$, and for the Coulomb gas $(\mu =\infty )$.

Figure 4

The numerical evaluation of Eq. (A1), as a function of $x=\mu \sqrt{r_s}∕{\varphi }_2\left(\zeta \right)$, and multiplied by $\left[{\varphi }_2\right(\zeta ){]}^{-3}$. If the scaling of Eq. (19) were exact, all the values corresponding to different $\zeta$ would lie on the solid curve. The value $\zeta =0.86$ shown in the figure corresponds to the maximum deviation from the scaling of Eq. (A1).