Koopmans’ condition in self-interaction-corrected density-functional theory

We investigate from a practitioner's point of view the computation of the ionization potential (IP) within density-functional theory (DFT). DFT with (semi)local energy-density functionals is plagued by a self-interaction error which hampers the computation of the IP from the single-particle energy of the highest occupied molecular orbital (HOMO). The problem may be cured by a self-interaction correction (SIC) for which there exist various approximate treatments. We compare the performance of the SIC proposed by Perdew and Zunger with the very simple average-density SIC (ADSIC) for a large variety of atoms and molecules up to larger systems such as carbon rings and chains. Both approaches to the SIC provide a large improvement to the quality of the IP if calculated from the HOMO level. The surprising result is that the simple ADSIC performs even better than the original Perdew-Zunger SIC in the majority of the studied cases.

Figure 1

Illustration of the ground-state energy $E$ as a function of fractional occupation number $\nu$. The IP from the HOMO level, $I_{\varepsilon }$, corresponds to the left-handed derivative (slope $=$ thin black line) of the energy at $\nu =N$. The IP from energy differences, $I_{\Delta }$, is associated with the exactly linear behavior (red, heavy solid line). The LDA result provides a smooth curve (green, dotted line). The Hartree-Fock result (blue, dashed line) also has a discontinuous derivative at $\nu =N$ as the exact trend, but tends to overestimate the kink.

Figure 2

Ionization potentials $I_{\Delta }$ from Eq. (1) and $I_{\varepsilon }$ from Eq. (3) for neutral atoms from hydrogen to argon and different approaches to the self-interaction correction: Average density SIC (squares), Perdew-Zunger SIC (diamonds), and the uncorrected local-density approximation (open circles). Experimental data are displayed as closed circles [50].

Figure 3

Errors in calculated IPs compared with experimental values for the series of atoms depicted in Fig. 2. Top and middle panels: errors from $I_{\varepsilon }$ and $I_{\Delta }$, respectively, according to Eq. (5). Bottom panel: non-Koopman's energy defined in Eq. (4).

Figure 4

As in Fig. 2, but for a set of simple molecular systems.

Figure 5

As in Fig. 3, but for a set of simple molecular systems.

Figure 6

As in Fig. 3, but for families of molecules with systematically varied properties.

Figure 7

Non-Koopmans' energies (bottom) and ionization potentials $I_{\Delta }$ (from energy differences, middle) and $I_{\varepsilon }$ (from the HOMO, top), computed in various schemes for carbon rings of various size ($5\le N_{\mathrm{atoms}}\le 32$). In the top panel, $I_{\Delta }$ from the LDA (see middle panel) is superimposed on the $I_{\varepsilon }$ calculated in the LDA, ADSIC, and PZSIC.

Figure 8

As in Fig. 7 but for linear carbon chains ($3\le N_{\mathrm{atoms}}\le 11$). The data are complemented by results obtained from bare exchange-only Hartree-Fock calculations (triangles).