Locally Scaled Self-Interaction Corrected Energy Functionals with Complex Optimal Orbitals

TL;DR

The paper tackles the self-interaction error (SIE) inherent in KS-DFT by introducing a fully variational locally scaled self-interaction correction (LSSIC) based on complex optimal orbitals and a local iso-orbital indicator $z_\sigma(\mathbf{r})$ with $0 \le z_\sigma(\mathbf{r}) \le 1$, implemented in GPAW via UP-AW; the SIC energy is formulated as $E^{SIC} = E^{KS}[n] - \sum_{k\u03c3} ( E_C[n_{k\u03c3}, z_\u03c3[n_{k\u03c3}]] + E_{xc}[n_{k\u03c3},0, z_\u03c3[n_{k\u03c3}] ] )$, and the kinetic-energy based expressions are extended to complex orbitals with current-density terms. The approach yields an exact dissociation limit for $H^+_2$ and improves electron affinities for C, N, and O, while producing reasonable bond energies and equilibrium geometries for C$_2$, N$_2$, and O$_2$, with one-shot and gradient-phase variants providing different levels of accuracy. Overall, LSSIC offers a general, tunable framework that preserves SIC benefits in localized regions while mitigating overcorrection in delocalized regions, with broad applicability to atoms, molecules, and solids in both ground and excited states.

Abstract

We present a fully variational locally scaled self-interaction corrected (SIC) energy functional using complex optimal orbitals. This represents an important milestone for fully variational SIC energy functionals, which have been shown to improve the prediction of the properties of atomic, molecular and solid state systems in general, in both ground and excited states. However, it depends on the system and property of the system whether it is beneficial to scale the SIC correction by a factor of one-half, which makes the application of SIC inconsistent. In the limit of a single electron the SIC exactly cancels the self interaction error, but overcorrects the error in regions of high density where there is large overlap between occupied orbitals. The newly implemented local scaling function, $z(\mathbf{r})$, which is based on an iso-orbital indicator derived from considering the kinetic energy density in the iso-electron and many electron case, and takes into account that the orbitals are complex, naturally scales the SIC correction from $0\leq z(\mathbf{r}) \leq 1$ in regions of high and low (isolated orbital) electron density. The locally scaled and fully variational SIC framework is general and applicable to atomic, molecular and solid-state systems.

Figures (5)

• Figure 1: Binding energy of H$^+_2$ calculated with $\frac{1}{2}\text{SIC}$ and SIC (yellow and green) and $\text{LSSIC}$ (orange). The base functional is PBE (blue). The zero of energy corresponds to the dissociation limit H+H$^+$. Note that the SIC (green) and $\text{LSSIC}$ (orange) curves perfectly overlap, since they are equivalent in the one-electron limit. The experimental estimate is marked on the plot (red star).
• Figure 2: Local scaling function evaluated with complex-valued pseudo-wavefunctions and pseudo-electron density of the H$^+_2$ molecule. The local scaling function is shown for different distances between the hydrogen and the hydrogen cation (indicated by the filled circles). In all cases the local scaling is approximately 1.0 in all space where the electron density is above a set threshold of $n_\sigma(\mathbf{r}) > 1e-12$ [au].
• Figure 3: Left: bar plot showing the deviation of the calculated electron affinity from the experimental result. Right: bar plot showing the deviation of the calculated ionization energy from the experimental result. In both plots, the numerical values in electron-volts are given on the left y-axis, and the relative deviation is given on the right y-axis.
• Figure 4: Localscaling function evaluated for the carbon dimer at two distances - 1.24 Å, which is close to the experimental bond length, and 3.24 Å. Both spin channels are presented and are denoted alpha (blue) and beta (orange). It is clear that the spin densities are symmetric in all space when the distance is close to the experimental bond length, whereas the spin densities are spatially different as we approach the dissociation limit, with one spin being preferentially conentrated on one carbon vs the other. The green dots denote the position of the carbon nuclei along the axis which is parallel to the bond.
• Figure 5: Potential energy of the nitrogen dimer as a function of distance. The experimental estimate is marked on the plot (red star).