Perdew-Zunger Self-interaction Correction Research Articles

Ensemble Generalization of the Perdew-Zunger Self-Interaction Correction: A Way Out of Multiple Minima and Symmetry Breaking.

The Perdew-Zunger (PZ) self-interaction correction (SIC) is an established tool to correct unphysical behavior in density functional approximations. Yet, the PZ-SIC is well-known to sometimes break molecular symmetries. An example of this is the benzene molecule, for which the PZ-SIC predicts a symmetry-broken electron density and molecular geometry, since the method does not describe the two possible Kekulé structures on an even footing, leading to local minima [Lehtola et al. J. Chem. Theory Comput. 2016, 12, 3195]. The PZ-SIC is often implemented with Fermi-Löwdin orbitals (FLOs), yielding the FLO-SIC method, which likewise has issues with symmetry breaking and local minima [Trepte et al. J. Chem. Phys. 2021, 155, 224109]. In this work, we propose a generalization of the PZ-SIC─the ensemble PZ-SIC (E-PZ-SIC) method─which shares the asymptotic computational scaling of the PZ-SIC (albeit with an additional prefactor). The E-PZ-SIC is straightforwardly applicable to various molecules, merely requiring one to average the self-interaction correction over all possible Kekulé structures, in line with chemical intuition. We showcase the implementation of the E-PZ-SIC with FLOs, as the resulting E-FLO-SIC method is easy to realize on top of an existing implementation of the FLO-SIC. We show that the E-FLO-SIC indeed eliminates symmetry breaking, reproducing a symmetric electron density and molecular geometry for benzene. The ensemble approach suggested herein could also be employed within approximate or locally scaled variants of the PZ-SIC and its FLO-SIC versions.

A numerical Poisson solver with improved radial solutions for a self-consistent locally scaled self-interaction correction method

The universal applicability of density functional approximations is limited by self-interaction error made by these functionals. Recently, a novel one-electron self-interaction-correction (SIC) method that uses an iso-orbital indicator to apply the SIC at each point in space by scaling the exchange-correlation and Coulomb energy densities was proposed. The locally scaled SIC (LSIC) method is exact for the one-electron densities, and unlike the well-known Perdew–Zunger SIC (PZSIC) method recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to PZSIC method as a special case when isoorbital indicator is set to the unity. Here, we present a numerical scheme that we have adopted to evaluate the Coulomb potential of the electron density scaled by the iso-orbital indicator required for the self-consistent LSIC calculations. After analyzing the behavior of the finite difference method (FDM) and the green function solution to the radial part of the Poisson equation, we adopt a hybrid approach that uses the FDM for the Coulomb potential due to the monopole and the GF for all higher-order terms. The performance of the resultant hybrid method is assessed using a variety of systems. The results show improved accuracy than earlier numerical schemes. We also find that, even with a generic set of radial grid parameters, accurate energy differences can be obtained using a numerical Coulomb solver in standard density functional studies.

How well do one-electron self-interaction-correction methods perform for systems with fractional electrons?

Recently developed locally scaled self-interaction correction (LSIC) is a one-electron SIC method that, when used with a ratio of kinetic energy densities (zσ) as iso-orbital indicator, performs remarkably well for both thermochemical properties as well as for barrier heights overcoming the paradoxical behavior of the well-known Perdew-Zunger self-interaction correction (PZSIC) method. In this work, we examine how well the LSIC method performs for the delocalization error. Our results show that both LSIC and PZSIC methods correctly describe the dissociation of H2+ and He2+ but LSIC is overall more accurate than the PZSIC method. Likewise, in the case of the vertical ionization energy of an ensemble of isolated He atoms, the LSIC and PZSIC methods do not exhibit delocalization errors. For the fractional charges, both LSIC and PZSIC significantly reduce the deviation from linearity in the energy vs number of electrons curve, with PZSIC performing superior for C, Ne, and Ar atoms while for Kr they perform similarly. The LSIC performs well at the endpoints (integer occupations) while substantially reducing the deviation. The dissociation of LiF shows both LSIC and PZSIC dissociate into neutral Li and F but only LSIC exhibits charge transfer from Li+ to F- at the expected distance from the experimental data and accurate abinitio data. Overall, both the PZSIC and LSIC methods reduce the delocalization errors substantially.

Self-interaction correction schemes for non-collinear spin-density-functional theory

We extend some of the well-established self-interaction correction (SIC) schemes of density-functional theory—the Perdew–Zunger SIC and the average-density SIC—to the case of systems with noncollinear magnetism. Our proposed SIC schemes are tested on a set of molecules and metallic clusters in combination with the widely used local spin-density approximation. As expected from the collinear SIC, we show that the averaged-density SIC works well for improving ionization energies but fails to improve more subtle quantities like the dipole moments of polar molecules. We investigate the exchange-correlation magnetic field produced by our extension of the Perdew–Zunger SIC, showing that it is not aligned with the local total magnetization, thus producing an exchange-correlation torque.

Orbital-optimized density functional calculations of molecular Rydberg excited states with real space grid representation and self-interaction correction.

Density functional calculations of Rydberg excited states up to high energy are carried out for several molecules using an approach where the orbitals are variationally optimized by converging on saddle points on the electronic energy surface within a real space grid representation. Remarkably good agreement with experimental estimates of the excitation energy is obtained using the generalized gradient approximation (GGA) functional of Perdew, Burke, and Ernzerhof (PBE) when Perdew-Zunger self-interaction correction is applied in combination with complex-valued orbitals. Even without the correction, the PBE functional gives quite good results despite the fact that corresponding Rydberg virtual orbitals have positive energy in the ground state calculation. Results obtained using the Tao, Perdew, Staroverov, and Scuseria (TPSS) and r2SCAN meta-GGA functionals are also presented, but they do not provide a systematic improvement over the results from the uncorrected PBE functional. The grid representation combined with the projector augmented-wave approach gives a simpler and better representation of diffuse Rydberg orbitals than a linear combination of atomic orbitals with commonly used basis sets, the latter leading to an overestimation of the excitation energy due to confinement of the excited states.

Downward quantum learning from element 118: Automated generation of Fermi-Löwdin orbitals for all atoms.

A new algorithm based on a rigorous theorem and quantum data computationally mined from element 118 guarantees automated construction of initial Fermi-Löwdin-Orbital (FLO) starting points for all elements in the Periodic Table. It defines a means for constructing a small library of scalable FLOs for universal use in molecular and solid-state calculations. The method can be systematically improved for greater efficiency and for applications to excited states such as x-ray excitations and optically silent excitations. FLOs were introduced to recast the Perdew-Zunger self-interaction correction (PZSIC) into an explicit unitarily invariant form. The FLOs are generated from a set of N quasi-classical electron positions, referred to as Fermi-Orbital descriptors (FODs), and a set of N-orthonormal single-electron orbitals. FOD positions, when optimized, minimize the PZSIC total energy. However, creating sets of starting FODs that lead to a positive definite Fermi orbital overlap matrix has proven to be challenging for systems composed of open-shell atoms and ions. The proof herein guarantees the existence of a FLOSIC solution and further guarantees that if a solution for N electrons is found, it can be used to generate a minimum of N - 1 and a maximum of 2N - 2 initial starting points for systems composed of a smaller number of electrons. Applications to heavy and super-heavy atoms are presented. All starting solutions reported here were obtained from a solution for element 118, Oganesson.

How Do Self-Interaction Errors Associated with Stretched Bonds Affect Barrier Height Predictions?

Density functional theory (DFT) suffers from self-interaction errors (SIEs) that generally result in the underestimation of chemical reaction barrier heights. This is commonly attributed to the tendency of density functional approximations to overstabilize delocalized densities that typically occur in the stretched bonds of transition state structures. The Perdew-Zunger self-interaction correction (PZSIC) and locally scaled self-interaction correction (LSIC) improve the prediction of barrier heights of chemical reactions, with LSIC giving better accuracy than PZSIC on average. These methods employ an orbital-by-orbital correction scheme to remove the one-electron SIE. In the context of barrier heights, this allows an analysis of how the self-interaction correction (SIC) for each orbital contributes to the calculated barriers using Fermi-Löwdin orbitals (FLOs). We hypothesize that the SIC contribution to the reaction barrier comes mainly from a limited number of orbitals that are directly involved in bond-breaking and bond-making in the reaction transition state. We call these participant orbitals (POs), in contrast to spectator orbitals (SOs) which are not directly involved in changes to the bonding. Our hypothesis is that ΔETotalSIC ≈ ΔEPOSIC, where ΔETotalSIC is the difference in the SIC corrections for the reactants or products and the transition state. We test this hypothesis for the reaction barriers of the BH76 benchmark set of reactions. We find that the stretched-bond orbitals indeed make the largest individual SIC contributions to the barriers. These contributions increase the barrier heights relative to LSDA, which underpredicts the barrier. However, the full stretched-bond hypothesis does not hold in all cases for either PZSIC or LSIC. There are many cases where the total SIC contribution from the SOs is significant and cannot be ignored. The size of the SIC contribution to the barrier height is a key indicator. A large SIC correction is correlated to a large LSDA error in the barrier, showing that PZSIC properly gives larger corrections when corrections are needed most. A comparison of the performance of PZSIC and LSIC shows that the two methods have similar accuracy for reactions with large LSDA errors, but LSIC is clearly better for reactions with small errors. We trace this to an improved description of reaction energies in LSIC.

Self-consistent implementation of locally scaled self-interaction-correction method.

Recently proposed local self-interaction correction (LSIC) method [Zope et al., J. Chem. Phys. 151, 214108 (2019)] is a one-electron self-interaction-correction (SIC) method that uses an iso-orbital indicator to apply the SIC at each point in space by scaling the exchange-correlation and Coulomb energy densities. The LSIC method is exact for the one-electron densities, also recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to the well-known Perdew-Zunger SIC (PZSIC) method as a special case. This article presents the self-consistent implementation of the LSIC method using the ratio of Weizsäcker and Kohn-Sham kinetic energy densities as an iso-orbital indicator. The atomic forces as well as the forces on the Fermi-Löwdin orbitals are also implemented for the LSIC energy functional. Results show that LSIC with the simplest local spin density functional predicts atomization energies of the AE6 dataset better than some of the most widely used generalized-gradient-approximation (GGA) functional [e.g., Perdew-Burke-Ernzerhof (PBE)] and barrier heights of the BH6 database better than some of the most widely used hybrid functionals (e.g., PBE0 and B3LYP). The LSIC method [a mean absolute error (MAE) of 0.008 Å] predicts bond lengths of a small set of molecules better than the PZSIC-LSDA (MAE 0.042 Å) and LSDA (0.011 Å). This work shows that accurate results can be obtained from the simplest density functional by removing the self-interaction-errors using an appropriately designed SIC method.

Spin-state gaps and self-interaction-corrected density functional approximations: Octahedral Fe(II) complexes as case study.

Accurate prediction of a spin-state energy difference is crucial for understanding the spin crossover phenomena and is very challenging for density functional approximations, especially for local and semi-local approximations due to delocalization errors. Here, we investigate the effect of the self-interaction error removal from the local spin density approximation (LSDA) and Perdew-Burke-Ernzerhof generalized gradient approximation on the spin-state gaps of Fe(II) complexes with various ligands using recently developed locally scaled self-interaction correction (LSIC) by Zope et al. [J. Chem. Phys. 151, 214108 (2019)]. The LSIC method is exact for one-electron density, recovers the uniform electron gas limit of the underlying functional, and approaches the well-known Perdew-Zunger self-interaction correction (PZSIC) as a particular case when the scaling factor is set to unity. Our results, when compared with reference diffusion Monte Carlo results, show that the PZSIC method significantly overestimates spin-state gaps favoring low spin states for all ligands and does not improve upon density functional approximations. The perturbative LSIC-LSDA using PZSIC densities significantly improves the gaps with a mean absolute error of 0.51eV but slightly overcorrects for the stronger CO ligands. The quasi-self-consistent LSIC-LSDA, such as coupled-cluster single double and perturbative triple [CCSD(T)], gives a correct signof spin-state gaps for all ligands with a mean absolute error of 0.56eV, comparable to that of CCSD(T) (0.49eV).

Unification of Perdew-Zunger self-interaction correction, DFT+U, and Rung 3.5 density functionals.

This Communication presents a unified derivation of three different approximations used in density functional theory (DFT): the Perdew-Zunger self-interaction correction (PZSIC), the Hubbard correction DFT+U, and the Rung 3.5 density functionals. All three approximations can be derived by introducing electron self-interaction into the Kohn-Sham (KS) reference system of noninteracting electrons. The derivation uses the Adiabatic Projection formalism: one projects the electron-electron interaction operator onto certain states, introduces the projected operator into the reference system, and defines a density functional for the remainder. Projecting onto individual localized KS orbitals recovers our previous derivation of the PZSIC [B. G. Janesko, J. Phys. Chem. Lett. 13, 5698-5702 (2022)]. Projecting onto localized atom-centered orbitals recovers a variant of DFT+U. Projecting onto localized states at each point in space recovers Rung 3.5 approaches. New results include an "atomic state PZSIC" that does not require localizing the KS orbitals, a demonstration that typical Hubbard U parameters reproduce a scaled-down PZSIC, and a Rung 3.5 variant of DFT+U that does not require choosing atom-dependent states.

Systematically Improvable Generalization of Self-Interaction Corrected Density Functional Theory.

Perdew-Zunger self-interaction correction (PZSIC) reintroduces an exact constraint to approximate density functional theory (DFT). However, PZSIC can paradoxically degrade performance, and standard DFT approximations (with or without PZSIC) are not systematically improvable. We use the adiabatic projection formalism (Janesko, B. G. J. Chem. Phys. 2022, 156, 014111, https://doi.org/10.1063/5.0076144) to derive PZSIC in terms of a reference system experiencing only electron self-interaction. Generalization to a "self-and-some-others" interaction introduces correlation into the reference system, systematically bridging from PZSIC to exact wave function theory without the double counting of correlation. Minimal active spaces accurately treat nearly one electron, near-equilibrium, and strongly correlated model systems at modest computational expense.

Fermi-Löwdin orbital self-interaction correction of adsorption energies on transition metal ions.

Density functional theory (DFT)-based descriptions of the adsorption of small molecules on transition metal ions are prone to self-interaction errors. Here, we show that such errors lead to a large over-estimation of adsorption energies of small molecules on Cu+, Zn+, Zn2+, and Mn+ in local spin density approximation (LSDA) and Perdew, Burke, Ernzerhof (PBE) generalized gradient approximation calculations compared to reference values computed using the coupled-cluster with single, doubles, and perturbative triple excitations method. These errors are significantly reduced by removing self-interaction using the Perdew-Zunger self-interaction correction (PZ-SIC) in the Fermi-Löwdin Orbital (FLO) SIC framework. In the case of FLO-PBE, typical errors are reduced to less than 0.1eV. Analysis of the results using DFT energies evaluated on self-interaction-corrected densities [DFT(@FLO)] indicates that the density-driven contributions to the FLO-DFT adsorption energy corrections are roughly the same size in DFT = LSDA and PBE, but the total corrections due to removing self-interaction are larger in LSDA.

Study of Self-Interaction Errors in Density Functional Calculations of Magnetic Exchange Coupling Constants Using Three Self-Interaction Correction Methods.

We examine the role of self-interaction error (SIE) removal on the evaluation of magnetic exchange coupling constants. In particular, we analyze the effect of scaling down the self-interaction correction (SIC) for three nonempirical density functional approximations (DFAs) namely, the local spin density approximation, the Perdew-Burke-Ernzerhof generalized gradient approximation, and the recent SCAN family of meta-GGA functionals. To this end, we employ three one-electron SIC methods: Perdew-Zunger SIC [Perdew, J. P.; Zunger, A. Phys. Rev. B, 1981, 23, 5048.], the orbitalwise scaled SIC method [Vydrov, O. A. et al. J. Chem. Phys. 2006, 124, 094108.], and the recent local scaling method [Zope, R. R. et al. J. Chem. Phys. 2019, 151, 214108.]. We compute the magnetic exchange coupling constants using the spin projection and nonprojection approaches for sets of molecules composed of dinuclear and polynuclear H···He models, organic radical molecules, and chlorocuprate and compare these results against accurate theories and experiment. Our results show that for the systems that mainly consist of single-electron regions, PZSIC performs well, but for more complex organic systems and the chlorocuprates, an overcorrecting tendency of PZSIC combined with the DFAs utilized in this work is more pronounced, and in such cases, LSIC with kinetic energy density ratio performs better than PZSIC. Analysis of the results in terms of SIC corrections to the density and to the total energy shows that both density and energy correction are required to obtain an improved prediction of magnetic exchange couplings.

Study of self-interaction-errors in barrier heights using locally scaled and Perdew-Zunger self-interaction methods.

We study the effect of self-interaction errors on the barrier heights of chemical reactions. For this purpose, we use the well-known Perdew-Zunger self-interaction-correction (PZSIC) [J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981)] as well as two variations of the recently developed, locally scaled self-interaction correction (LSIC) [Zope et al., J. Chem. Phys. 151, 214108 (2019)] to study the barrier heights of the BH76 benchmark dataset. Our results show that both PZSIC and especially the LSIC methods improve the barrier heights relative to the local density approximation (LDA). The version of LSIC that uses the iso-orbital indicator z as a scaling factor gives a more consistent improvement than an alternative version that uses an orbital-dependent factor w based on the ratio of orbital densities to the total electron density. We show that LDA energies evaluated using the self-consistent and self-interaction-free PZSIC densities can be used to assess density-driven errors. The LDA reaction barrier errors for the BH76 set are found to contain significant density-driven errors for all types of reactions contained in the set, but the corrections due to adding SIC to the functional are much larger than those stemming from the density for the hydrogen transfer reactions and of roughly equal size for the non-hydrogen transfer reactions.

Chemical bonding theories as guides for self-interaction corrected solutions: Multiple local minima and symmetry breaking.

Fermi-Löwdin orbitals (FLOs) are a special set of localized orbitals, which have become commonly used in combination with the Perdew-Zunger self-interaction correction (SIC) in the FLO-SIC method. The FLOs are obtained for a set of occupied orbitals by specifying a classical position for each electron. These positions are known as Fermi-orbital descriptors (FODs), and they have a clear relation to chemical bonding. In this study, we show how FLOs and FODs can be used to initialize, interpret, and justify SIC solutions in a common chemical picture, both within FLO-SIC and in traditional variational SIC, and to locate distinct local minima in either of these approaches. We demonstrate that FLOs based on Lewis theory lead to symmetry breaking for benzene-the electron density is found to break symmetry already at the symmetric molecular structure-while ones from Linnett's double-quartet theory reproduce symmetric electron densities and molecular geometries. Introducing a benchmark set of 16 planar cyclic molecules, we show that using Lewis theory as the starting pointcan lead to artifactual dipole moments of up to 1D, while Linnett SIC dipole moments are in better agreement with experimental values. We suggest using the dipole moment as a diagnostic of symmetry breaking in SIC and monitoring it in all SIC calculations. We show that Linnett structures can often be seen as superpositions of Lewis structures and propose Linnett structures as a simple way to describe aromatic systems in SIC with reduced symmetry breaking. The role of hovering FODs is also briefly discussed.

Method for Calculating Excited Electronic States Using Density Functionals and Direct Orbital Optimization with Real Space Grid or Plane-Wave Basis Set.

A direct orbital optimization method is presented for density functional calculations of excited electronic states using either a real space grid or a plane-wave basis set. The method is variational, provides atomic forces in the excited states, and can be applied to Kohn-Sham (KS) functionals as well as orbital-density-dependent (ODD) functionals including explicit self-interaction correction. The implementation for KS functionals involves two nested loops: (1) An inner loop for finding a stationary point in a subspace spanned by the occupied and a few virtual orbitals corresponding to the excited state; (2) an outer loop for minimizing the energy in a tangential direction in the space of the orbitals. For ODD functionals, a third loop is used to find the unitary transformation that minimizes the energy functional among occupied orbitals only. Combined with the maximum overlap method, the algorithm converges in challenging cases where conventional self-consistent field algorithms tend to fail. The benchmark tests presented include two charge-transfer excitations in nitrobenzene and an excitation of CO to degenerate π* orbitals where the importance of complex orbitals is illustrated. The application of this method to several metal-to-ligand charge-transfer and metal-centered excited states of an FeII photosensitizer complex is described, and the results are compared to reported experimental estimates. This method is also used to study the effect of the Perdew-Zunger self-interaction correction on valence and Rydberg excited states of several molecules, both singlet and triplet states, and the performance compared to semilocal and hybrid functionals.

Fermi-Löwdin-orbital self-interaction correction using the optimized-effective-potential method within the Krieger-Li-Iafrate approximation

Perdew-Zunger self-interaction correction (PZ-SIC) offers a route to remove self-interaction errors on an orbital-by-orbital basis. A recent formulation of PZ-SIC by Pederson, Ruzsinszky and Perdew proposes restricting the unitary transformation to localized orbitals called Fermi-L\"owdin orbitals. This formulation, called the FLOSIC method, simplifies PZ-SIC calculations and was implemented self-consistently using a Jacobi-like (FLOSIC-Jacobi) iteration scheme. In this work we implement the FLOSIC approach using the Krieger-Li-Iafrate (KLI) approximation to the optimized effective potential (OEP). We compare the results of present FLOSIC-KLI approach with FLOSIC-Jacobi scheme for atomic energies, atomization energies, ionization energies, barrier heights, polarizability of chains of hydrogen molecules etc. to validate the FLOSIC-KLI approach. The FLOSIC-KLI approach, which is within the realm of Kohn-Sham theory, predicts smaller energy gaps between frontier orbitals due to the lowering of eigenvalues of the lowest unoccupied orbitals. Results show that atomic energies, atomization energies, ionization energy as an absolute of highest occupied orbital eigenvalue, and polarizability of chains of hydrogen molecules between the two methods agree within 2%. Finally the FLOSIC-KLI approach is used to determine the vertical ionization energies of water clusters.

Exploring and enhancing the accuracy of interior-scaled Perdew-Zunger self-interaction correction.

The Perdew-Zunger self-interaction correction (PZ-SIC) improves the performance of density functional approximations for the properties that involve significant self-interaction error (SIE), as in stretched bond situations, but overcorrects for equilibrium properties where SIE is insignificant. This overcorrection is often reduced by local scaling self-interaction correction (LSIC) of the PZ-SIC to the local spin density approximation (LSDA). Here, we propose a new scaling factor to use in an LSIC-like approach that satisfies an additional important constraint: the correct coefficient of the atomic number Z in the asymptotic expansion of the exchange-correlation (xc) energy for atoms. LSIC and LSIC+ are scaled by functions of the iso-orbital indicator zσ, which distinguishes one-electron regions from many-electron regions. LSIC+ applied to the LSDA works better for many equilibrium properties than LSDA-LSIC and the Perdew, Burke, and Ernzerhof generalized gradient approximation (GGA), and almost close to the strongly constrained and appropriately normed (SCAN) meta-GGA. LSDA-LSIC and LSDA-LSIC+, however, fail to predict interaction energies involving weaker bonds, in sharp contrast to their earlier successes. It is found that more than one set of localized SIC orbitals can yield a nearly degenerate energetic description of the same multiple covalent bond, suggesting that a consistent chemical interpretation of the localized orbitals requires a new way to choose their Fermi orbital descriptors. To make a locally scaled down SIC to functionals beyond the LSDA requires a gauge transformation of the functional's energy density. The resulting SCAN-sdSIC, evaluated on SCAN-SIC total and localized orbital densities, leads to an acceptable description of many equilibrium properties including the dissociation energies of weak bonds.

Implementation of Perdew-Zunger self-interaction correction in real space using Fermi-Löwdin orbitals.

Most widely used density functional approximations suffer from self-interaction error, which can be corrected using the Perdew-Zunger (PZ) self-interaction correction (SIC). We implement the recently proposed size-extensive formulation of PZ-SIC using Fermi-Löwdin Orbitals (FLOs) in real space, which is amenable to systematic convergence and large-scale parallelization. We verify the new formulation within the generalized Slater scheme by computing atomization energies and ionization potentials of selected molecules and comparing to those obtained by existing FLOSIC implementations in Gaussian based codes. The results show good agreement between the two formulations, with new real-space results somewhat closer to experiment on average for the systems considered. We also obtain the ionization potentials and atomization energies by scaling down the Slater statistical average of SIC potentials. The results show that scaling down the average SIC potential improves both atomization energies and ionization potentials, bringing them closer to experiment. Finally, we verify the present formulation by calculating the barrier heights of chemical reactions in the BH6 dataset, where significant improvements are obtained relative to Gaussian based FLOSIC results.

Density-related properties from self-interaction corrected density functional theory calculations.

The Perdew-Zunger self-interaction correction (PZ-SIC) removes unphysical electron self-interaction from calculations employing standard density functional approximations. Doing so improves many computed properties, bringing them into better agreement with experimental observations or with results from high-level quantum chemistry calculations. However, while PZ-SIC generally corrects in the right direction relative to the corresponding reference values, in many cases, it over-corrects. For this reason, scaled-down versions of PZ-SIC have been proposed and investigated. These approaches have mostly employed exterior scaling in which SIC correction terms are scaled in the same way at every point in space. Recently, a new local, or interior, scaling SIC method was proposed on non-empirical grounds to restore a property of the exact, but unknown, density functional that is broken in PZ-SIC. In this approach, the scaling at each point depends on the character of the charge density at that point. However, the local scaling can be done in various ways while still restoring the behavior of the exact functional. In this work, we compare and contrast the performance of various interior scaling approaches for addressing over-corrections of calculated molecular dipole moments and atomic polarizabilities and properties that reflect the nature of the electronic charge density.