Assessment of density-functional approximations: Long-range correlations and self-interaction effects

The complex nature of electron-electron correlations is made manifest in the very simple but nontrivial problem of two electrons confined within a sphere. The description of highly nonlocal correlation and self-interaction effects by widely used local and semilocal exchange-correlation energy density functionals is shown to be unsatisfactory in most cases. Even the best such functionals exhibit significant errors in the Kohn-Sham potentials and density profiles.

Figure 1

Percent errors $\Delta E=100\left(E^{\text{DFT}}-E^{\text{ex}}\right)∕E^{\text{ex}}$ for the exchange, correlation, and total energies as functions of the sphere radius $R$. All DFT results are obtained non-self-consistently over the exact density. Solid line, LDA; dashed line, GGA; dotted line, MGGA; dash-dotted line, $\text{EXX}+\mathrm{M}$.

Figure 2

Exchange and correlation potentials obtained from the exact density profile $\left(R=10\right)$. Thick solid line, exact results; thin solid line, LDA; dashed line, GGA; dotted line, MGGA; dash-dotted line, $\text{EXX}+\mathrm{M}$. Note that the shape of the LDA XC potential reproduces fairly well the exact one except in the region around the center of the sphere. This error is amplified by the $\text{EXX}+\mathrm{M}$ prescription, where the exchange part is exactly correct but the correlation potential is the same as the MGGA.

Figure 3

Kohn-Sham potential $v_{\mathrm{S}}\left(r\right)$ for the exact density profile $\left(R=10\right)$. Thick solid line, exact result; thin solid line, LDA; dashed line, GGA; dotted line, MGGA; dash-dotted line, $\text{EXX}+\mathrm{M}$. In order to allow an easier comparison, the potentials have been shifted in such a way that the corresponding first eigenvalue for each approximate potential is equal to zero. The behavior of the potentials near the sphere walls is less important due to the role played by the boundary condition $n\left(R\right)=0$.

Figure 4

Exact (solid line) and self-consistent GGA (dashes) and $\text{EXX}+\mathrm{M}$ (dash-dots) densities $n\left(r\right)$ and reduced radial densities $r^{2}n\left(r\right)$. All the quantities have been scaled with the radius of the confining sphere. The self-consistent LDA and MGGA densities are very similar to the GGA ones and have not been included in the figure. None of the approximate functionals is able to reproduce the correct behavior of the density in the center of the sphere, and their overall performance is very poor for $R⪢0$.