Stretched or noded orbital densities and self-interaction correction in density functional theory

Abstract

Semi-local approximations to the density functional for the exchangecorrelation energy of a many-electron system necessarily fail for lobed one-electron densities, including not only the familiar stretched densities but also the less familiar but closely-related noded ones. The Perdew-Zunger (PZ) self-interaction correction (SIC) to a semi-local approximation makes that approximation exact for all oneelectron ground-or excited-state densities and accurate for stretched bonds. When the minimization of the PZ total energy is made over real localized orbitals, the orbital densities can be noded, leading to energy errors in many-electron systems. Minimization over complex localized orbitals yields nodeless orbital densities, which reduce but typically do not eliminate the SIC errors of atomization energies. Other errors of PZ SIC remain, attributable to the loss of the exact constraints and appropriate norms that the semi-local approximations satisfy, and suggesting the need for a generalized SIC. These conclusions are supported by calculations for oneelectron densities, and for many-electron molecules. While PZ SIC raises and improves the energy barriers of standard generalized gradient approximations (GGA's) and meta-GGA's, it reduces and often worsens the atomization energies of molecules. Thus PZ SIC raises the energy more as the nodality of the valence localized orbitals increases from atoms to molecules to transition states. PZ SIC is applied here in particular to the SCAN meta-GGA, for which the correlation part is already self-interaction-free. That property makes SCAN a natural first candidate for a generalized SIC.

Self-interaction-free electric dipole polarizabilities for atoms and their ions using the Fermi-Löwdin self-interaction correction

“…For example, restricted to real orbitals, SIC improves the predictions of atomization energies for LSDA, but degrades them for the Perdew, Burke, and Ernzerhof [16] (PBE) generalized gradient approximation (GGA) and the strongly constrained and appropriately normed [17] (SCAN) meta-GGA functional. [18][19][20][21][22] (Using complex orbitals improves the performance of SIC-PBE over PBE, [23,24] but even with complex orbitals, SIC-SCAN fails to improve uncorrected SCAN for atomization energies. [22]) The GGA and SCAN functionals are constructed to fulfill more of the known constraints satisfied by the exact exchange-correlation functional than LSDA.…”

“…[18][19][20][21][22] (Using complex orbitals improves the performance of SIC-PBE over PBE, [23,24] but even with complex orbitals, SIC-SCAN fails to improve uncorrected SCAN for atomization energies. [22]) The GGA and SCAN functionals are constructed to fulfill more of the known constraints satisfied by the exact exchange-correlation functional than LSDA. Adding SIC to these functionals makes them one-electron self-interaction free, a property of the exact functional, but in the process may cause other constraints or norms to be violated.…”

“…This is reminiscent of the effect of SIC on atomization energies, where SIC improves LSDA, but worsens the already good performance of PBE and SCAN. A discussion of the effect of SIC on atomization energies in terms of noded or lobed orbitals can be found elsewhere [22]. IV.…”

Interpretation and Automatic Generation of Fermi‐Orbital Descriptors

“…Here, is the underlying exchange‐correlation functional evaluated for the one‐electron density and is the corresponding Coulomb term. Considering the one‐electron density limit, SIC calculations reproduce HF results, which are exact for this limit . For many‐electron systems, the application of the PZ scheme in any self‐consistent form offered significant improvements with respect to the underlying density functional, for example, for ionization potentials and (hyper)‐polarizabilities .…”

Self-interaction error overbinds water clusters but cancels in structural energy differences

“…More generally, for nonmetals (including water), any large error of SCAN is dominated by self-interaction. Improvements in the way the SIE is handled are under development (46)(47)(48), but here we will apply only a currently standard way.…”