Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities

We reformulate the strong-interaction limit of electronic density functional theory in terms of a classical problem with a degenerate minimum. This allows us to clarify many aspects of this limit, and to write a general solution, which is explicitly calculated for spherical densities. We then compare our results with previous approximate solutions and discuss the implications for density-functional theory.

Figure 1

The radial co-motion functions $f_2\left(r\right)$ and $f_3\left(r\right)$ for the Li atom density. Hartree atomic units are used.

Figure 2

The positions of the three electrons in the SCE state for the Li atom density. The two big circles are the projections of the spheres containing, on average in the quantum-mechanical problem, one electron (radius $a_1$) and two electrons (radius $a_2$). The position $r$ of the first electron is varied along the vertical $z$ axis (a) from 0 to $a_1$, (b) from $a_1$ to $a_2$, and (c) from $a_2$ to $\infty$. The other two electrons have distances from the center given, respectively, by $f_2\left(r\right)$ and $f_3\left(r\right)$ of Eqs. (37, 38), and angular positions given by the minimizing angles ${\theta }_2[r,f_2\left(r\right),f_3\left(r\right)]$ and ${\theta }_3[r,f_2\left(r\right),f_3\left(r\right)]$. In each panel, the starting position of the three electrons is represented by a full circle (●) and their final position by an empty circle (엯). The dashed curves represent the trajectories of the three electrons, and the arrows their direction. The three panels are actually equivalent (see text). Distances are in atomic units.

Figure 3

The external potential $v\left(r\right)$ of Eq. (9) for the SCE state in the case of the Li atom. The asymptotic expansion for large $r$, $v(r\to \infty )=-\frac{N-1}{r}$, is also reported. All quantities are in hartree atomic units.

Figure 4

The radial co-motion functions $f_2\left(r\right)$, $f_3\left(r\right)$, and $f_4\left(r\right)$ for the Be atom density. Hartree atomic units are used.

Figure 5

The external potential $v\left(r\right)$ of Eq. (9) for the SCE state in the case of the Be atom. The asymptotic expansion for large $r$, $v(r\to \infty )=-\frac{N-1}{r}$, is also reported. All quantities are in hartree atomic units.

Figure 6

The radial co-motion functions for the sphericalized B atom density. Hartree atomic units are used.

Figure 7

The radial co-motion functions for the sphericalized C atom density. Hartree atomic units are used.

Figure 8

The radial co-motion functions for the Ne atom density. Hartree atomic units are used.

Figure 9

The external potential $v\left(r\right)$ of Eq. (9) for the SCE state in the case of the Ne atom and of the sphericalized B and C atoms. All quantities are in hartree atomic units.