Pauli potential from Heilmann-Lieb electron density obtained by summing hydrogenic closed-shell densities over the entire bound-state spectrum

Independently, in the mid-1980s, several groups proposed to bosonize the density-functional theory (DFT) for fermions by writing a Schrödinger equation for the density amplitude $\rho (\mathbf{r}){}^{1/2}$, with $\rho (\mathbf{r})$ as the ground-state electron density, the central tool of DFT. The resulting differential equation has the DFT one-body potential $V(\mathbf{r})$ modified by an additive term $V_P(\mathbf{r})$ where $P$ denotes Pauli. To gain insight into the form of the Pauli potential $V_P(\mathbf{r})$, here, we invoke the known Coulombic density, ${\rho }_{\infty }(r)$ say, calculated analytically by Heilmann and Lieb (HL), by summation over the entire hydrogenic bound-state spectrum. We show that $V_{P\infty }(r)$ has simple limits for both $r$ tends to infinity and $r$ approaching zero. In particular, at large $r$, $V_{P\infty }(r)$ precisely cancels the attractive Coulomb potential $-{\mathit{Ze}}^2/r$, leaving $V(r)+V_{P\infty }(r)$ of $O(r^{-2})$ as $r$ tends to infinity. The HL density ${\rho }_{\infty }(r)$ is finally used numerically to display $V_{P\infty }(r)$ for all $r$ values.

Figure 1

Form of the Pauli potential $V_{P\infty }(r)$ as calculated from Eq. (5). Input for Eq. (5) is the HL density ${\rho }_{\infty }(r)$, ${\rho }_{s\infty }(r)$, thus, obtained from Eq. (3), with $\epsilon$ finally set equal to zero.

Figure 2

Form of ${\mathit{rV}}_{\mathit{PN}}(r)$ for some closed-shell cases derived for ${\rho }_N$ of Eq. (6). The curves are shown for $N=2,3,4,5,6,7,10,\infty$ from bottom to top, respectively.