Accuracy of electron densities obtained via Koopmans-compliant hybrid functionals

We evaluate the accuracy of electron densities and quasiparticle energy gaps given by hybrid functionals by directly comparing these to the exact quantities obtained from solving the many-electron Schrödinger equation. We determine the admixture of Hartree-Fock exchange to approximate exchange-correlation in our hybrid functional via one of several physically justified constraints, including the generalized Koopmans' theorem. We find that hybrid functionals yield strikingly accurate electron densities and gaps in both exchange-dominated and correlated systems. We also discuss the role of the screened Fock operator in the success of hybrid functionals.

Figure 1

Upper: The variation in hybrid ionization energy $I\left(3\right)[=A(2\left)\right]$, exact $I\left(3\right)\mathrm{[}=A\left(2\right)]$, ${\varepsilon }_3\left(3\right)$, and ${\varepsilon }_3\left(2\right)$ with $\alpha$ are illustrated for three electrons in an harmonic oscillator with $\omega =0.25$, an exchange-dominated system. Energies are in Hartree atomic units. There are three “crossing points”: (A) $A$-LUMO, (B) HOMO-LUMO, and (C) $I$-HOMO. Center: The integrated absolute error in the density $\mathrm{\Delta }\rho$ is shown for each value of $\alpha$. This is defined as $\int \left|{\rho }^{\text{EXT}}\left(x\right)-{\rho }^{\text{HYB}}\left(x\right)\right|dx$, where the ${\rho }^{\text{EXT}}$ and ${\rho }^{\text{HYB}}$ correspond to the exact and hybrid densities. Lower: The densities for crossings (A) and (C) are benchmarked against the exact, LDA, and HF cases; the hybrid, HF, and exact curves lie close together.

Figure 2

As Fig. 1, for three electrons in an atomlike external potential $\mathrm{[}{V}_{\mathrm{ext}}\left(x\right)=-1/\left(0.05\left|x\right|+1\right)]$. The system is correlated as HF fails to predict the exact density and energy.

Figure 3

The derivative of energy with respect to total number of electrons $N,\partial E/\partial N$, for a number of approximations. The external potential and $\alpha$ values chosen are the same as that of Fig. 2. We verified that the $\partial E/\partial N$ curve lies exactly on that of the HOMO eigenvalue within each approach. Each node at integer numbers of electrons corresponds to the HOMO and LUMO, with the lower-energy value being the HOMO.

Figure 4

Upper: Densities for various approximations are shown for an exchange-dominated asymmetric double-well potential. The dashed line, illustrating the potential (scaled by 0.15), shows that the two wells are asymmetric. The HF case follows the exact one, placing one electron in each well of a strongly localized system. The LDA predicts that an additional 0.1 electrons are present in the deeper well. The GKT yields $\alpha \approx 1$, effectively HF. We show the density for $\alpha =0.2$, which places the correct charge in each well, but has an incorrect density shape. Lower: The integrated charge of the left (shallower) well is shown for a range of $\alpha$ values.