Design of kinetic functionals for many-body electron systems: Combining analytical theory with Monte Carlo sampling of electronic configurations

In a previous work [L. Delle Site, J. Phys. A: Math. Theor. 40, 2787 (2007)], the derivation of an analytic expression for the kinetic functional of a many-body electron system has been proposed. Though analytical, the formula is still nonlocal (multidimensional) and thus not ideal for numerical applications. In this work, by treating the test case of a uniform gas of interacting spinless electrons, we propose a computational protocol which combines the previous analytic results with the Monte Carlo sampling of electronic configurations in space. This, we show, leads to an internally consistent scheme to design well founded local kinetic functionals.

Figure 1

The three panels show the values of $\gamma$ that minimize the functional $I\left[\rho \left({\mathbf{r}}_1\right)\right]+C\left[\rho \left({\mathbf{r}}_1\right)\right]$ at three representative densities (for $N=100$ electrons). Energies and densities are expressed in a.u.

Figure 2

Calculated values of $I\left[\rho \left({\mathbf{r}}_1\right)\right]$ for $N=50$, 100, and 250 electrons. The continuous line is the fit we propose (see text).