Quantum critical benchmark for electronic structure theory

Two electrons at the threshold of ionization represent a severe test case for electronic structure theory. Variational methods can yield highly accurate energies, but may be less accurate for other operators of interest. A pseudospectral method yields very accurate expectations values for the two-electron ion with nuclear charge at and close to the critical value which yields zero ionization energy. As an illustration, the ground-state density is calculated and an extremely accurate parametrization is given. Other components in Kohn-Sham density functional theory are also calculated, and the efficacy of approximations is discussed.

Figure 1

Exact (dots) and asymptotic ($A_1=5.528\times {10}^{-3},A_2=1.517$, and $B=0.1375$) densities (solid line) for $Z=1,2$, and $Z_c$.

Figure 2

Exact and approximate correlation energies, evaluated with the exact densities.

Figure 3

Error in approximations evaluated with the exact densities. For PBE at $Z=Z_c$, the error is 0.6 millihartree.

Figure 4

Exact density-weighted holes for $Z=Z_c$ and 1. The magnitude of the exchange hole decreases as $Z\to Z_c$, allowing for the error cancellation.

Figure 5

The correlation potential times $r^4$ calculated with our numerical density (exact, black line), Eq. (8) (fit, dashed red line), and using the asymptotic form (7) for the local noninteracting kinetic energy and the fit for the exchange energy (dotted blue line). Only by combining the asymptote with our exact data can we extract the behavior for all $r$.

Figure 6

Relative error in the asymptotic densities [Eqs. (4) and (7)] compared to the exact densities (points) from zeroth to fourth order in $1/r$. The leading-order correction to these formulas is calculated with higher-order terms and plotted as dark solid lines, except for the fourth-order error which is made with a fit. These lines indicate whether or not the errors have reached the asymptotic regime. Note that for $Z=1$, the zeroth, first, and second order are the same. There does not appear to be a general pattern. Deviations from limits at very large $r$ are due to the limits in machine-precision arithmetic in calculating $\Psi$ in the tail.