Self-interaction (SI) error, which results when exchange-correlation contributions to the total energy are approximated, limits the reliability of many density functional approximations. The Perdew-Zunger SI correction (PZSIC), when applied in conjunction with the local spin density approximation (LSDA), improves the description of many properties, but overall, this improvement is limited. Here, we propose a modification to PZSIC that uses an iso-orbital indicator to identify regions where local SICs should be applied. Using this local-scaling SIC (LSIC) approach with LSDA, we analyze predictions for a wide range of properties including, for atoms, total energies, ionization potentials, and electron affinities and, for molecules, atomization energies, dissociation energy curves, reaction energies, and reaction barrier heights. LSIC preserves the results of PZSIC-LSDA for properties where it is successful and provides dramatic improvements for many of the other properties studied. Atomization energies calculated using LSIC are better than those of the Perdew, Burke, and Ernzerhof generalized gradient approximation (GGA) and close to those obtained with the strongly constrained and appropriately normed meta-GGA. LSIC also restores the uniform gas limit for the exchange energy that is lost in PZSIC-LSDA. Further performance improvements may be obtained by an appropriate combination or modification of the local scaling factor and the particular density functional approximation.

The Kohn-Sham (KS) formulation of density functional theory (DFT) has become the most popular approach for studying the electronic, structural, and other properties of molecular and condensed systems.1 KS-DFT is a formally exact theory1,2 to obtain the ground-state energy and electron density, but its practical realization requires an approximation to the exchange-correlation density functional. The enormous popularity of DFT is due to the combined appeal of sufficiently accurate density functional approximations (DFAs), favorable scaling with respect to the number of atoms, and numerically accurate and efficient implementations that have resulted in numerous easy-to-use codes. The local spin density approximation (LSDA),1,3,4 based on the uniform electron gas model, was an early and simple DFA. The success of LSDA in describing the electronic properties of solids made DFT popular in the physics community. Careful analysis attributed this success to the spherical exchange-hole of LSDA being a good approximation to the spherical average of the exact exchange-hole and to the satisfaction of the exchange correlation hole sum rule.3,5,6 Subsequent improvements beyond the LSDA were obtained7–16 by including information about the local electron density gradient in generalized gradient approximations (GGAs), and also the Laplacian and kinetic energy density in meta-GGAs. The nonempirical functionals among these are designed to satisfy various constraints and norms of the exact functional, including the uniform electron gas limit.17 

Extensive work has shown that local and semilocal DFAs work well when the exact exchange-correlation hole density is localized around the electron, as is usually the case near equilibrium configurations in molecules and solids. But these functionals can fail dramatically in stretched-bond situations such as in the transition states of chemical reactions and molecular dissociation,17 causing the underestimation of barrier heights in chemical reactions and the incorrect dissociation of radical and heteroatomic molecules. This failure can be traced to electron self-interaction errors (SIEs) caused by the incomplete cancellation of the self-Coulomb energy with the approximate self-exchange-correlation energy for one electron densities. This was recognized long ago, and attempts to remove the SIE were pursued.18–22 One widely used approach to mitigating the effect of the SIE, introduced by Becke, is combining Hartree-Fock exchange with semilocal functionals.23 As the Hartree-Fock approximation is self-interaction free and introduces errors that are often of opposite sign to those of semilocal functionals,24 this approach can overcome a number of deficiencies of semilocal DFAs. The formal justification for such mixing can be obtained by an adiabatic connection5,6,25 between the real interacting system and the noninteracting Kohn-Sham system. Global hybrids,23 local hybrids,26 and range-separated hybrids27 are all approximations that add Hartree-Fock exchange using various criteria. A majority of these functionals, however, still suffer from nonzero SIEs.

A systematic procedure for eliminating one-electron self-interaction error was given by Perdew and Zunger (PZ) in 1981.28 In the Perdew-Zunger self-interaction correction (PZSIC) approach, the SIE of a DFA is removed from the total energy in an orbital-by-orbital fashion by redefining the energy as

EPZSICDFA=EDFA[ρ,ρ]i,σoccU[ρiσ]+EXCDFA[ρiσ,0],
(1)

where i is the orbital index, σ is the spin index, U[ρ] is the exact self-Coulomb energy, and EXCDFA[ρiσ,0] is the approximate self-exchange and correlation energy. PZSIC-DFA is exact for any one-electron density and gives no correction to the exact functional.

One of the features of PZSIC is that EPZSICDFA is not invariant to the choice of orbitals used to represent the total electron density. Different orbitals that give the same total density yield different total energies so that finding the minimum energy formally requires searching over all sets of orbitals that span the correct density. It can be shown that the variational minimum energy corresponds to ρ = |ϕ|2 for orbitals ϕ that satisfy the set of conditions known as the Pederson or localization equations,29,30

ϕiσ|ViσSICVjσSIC|ϕjσ=0.
(2)

In traditional PZSIC, a unitary transformation of the KS orbitals is performed to construct the local orbitals. Optimizing the local orbitals to satisfy Eq. (2) requires tuning the O(N2) elements of the transformation matrix, which is computationally expensive.

PZSIC provided a way to go beyond the LSDA, but the computational difficulties mentioned above deterred practitioners from following this path31 and only a relatively few implementations of PZSIC have been reported.32–70 A review31 by Pederson and Perdew nicely summarizes this and related work. A handful of studies involved PZSIC combined with semilocal approximations.43–45,60,71,72 These found that while PZSIC improves properties like the dissociation pathway of heteroatomic molecules, it worsens the good description of semilocal functionals for near-equilibrium properties such as atomization energies due to overcorrection.44,73 This has come to be known as the paradox of SIC.74 A few approaches have been proposed to rectify this behavior based on scaling down the SIC contribution to the energy [second terms in the right hand side of Eq. (1)]. Reference 11 proposed to use the ratio between the von Weizsäcker and the total kinetic energy densities to identify one- and two-electron regions for meta-GGAs, and Tsuneda et al.48 first proposed to use this ratio to identify one-electron regions where SIC is expected to be important. Reference 48 replaced the DFA energy density in these regions with an expression based on the exchange energy of hydrogenic orbitals. Later, Vydrov et al.45 used a selective orbital-by-orbital scaling down of the SIC contribution to the energy, and more recently, Jónsson et al.75 proposed to globally reduce the SIC energy by 50%. The Jónsson group also pioneered76 the use of complex orbitals in PZSIC, which work well with PZSIC-Perdew, Burke, and Ernzerhof (PBE). The scaling approaches, which are discussed in more detail below, achieve success for selected properties, but, in general, they destroy the desirable −1/r asymptotic form of the potential seen by an electron in a localized system such as a neutral atom in a PZSIC calculation.45 This unphysical behavior has important consequences for properties like charge transfer.

Considerable effort has been spent trying to understand the origin of the PZSIC paradox. A recent study found that PZSIC raises the total energy as the nodality of the valence local orbitals increases from atoms to molecules to transition states.77 More recently, it was shown that, unlike the nonempirical semilocal functionals, PZSIC violates the uniform electron gas norm for the exchange and correlation energies.78 The implication of this is that adding PZSIC breaks the correct behavior of these functionals for slowly varying densities.

In this work, we propose an approach that adjusts the PZSIC locally, that is, at each point in space, by adjusting the magnitude of the correction using an iso-orbital indicator. We call this approach local-scaling SIC (LSIC). It is implemented in the FLOSIC code79,80 and applied perturbatively to self-consistent PZSIC solutions obtained using the Fermi-Löwdin Orbital SIC (FLO-SIC) method.81,82 As discussed further below, the method applies SIC at full strength for a density with a single-orbital character and turns it off for a uniform density. We assessed the predictions of this approach for a number of properties including, for atoms: total energies, ionization potentials, and electron affinities and, for molecules: atomization energies, reaction energies, dissociation energy curves, and reaction barrier heights. We find significant improvement for properties that PZSIC typically worsens while retaining the successful predictions of PZSIC in situations where removing the SIE is critical. The proposed LSIC method, unlike semilocal functionals and most earlier PZSIC implementations, provides a good description of both near-equilibrium properties and properties associated with stretched-bond situations. LSIC thus appears to resolve the paradox of PZSIC and opens the door to designing universally accurate DFAs.

The application of PZSIC worsens the quality of equilibrium properties when used with semilocal functionals.43–45,72,76,83,84 Attempts have been made to restore the accuracy of semilocal functionals used in combination with PZSIC by reducing the size of the corrections. For example, Jónsson and co-workers used a scaled-down version of PZSIC in which the SIC is reduced by 50%.75 Such a diminished correction, when applied with the PBE functional, resulted in overall improvement of atomization energies, but significant absolute errors still remained. Instead of using a fixed constant scaling factor, Vydrov and co-workers45 had earlier proposed setting a scaling factor for each local orbital i in the following way:

Xiσk=τσWτσkρiσ(r)dr,
(3)

where τσW(r) is the von Weizsäcker kinetic energy density and τσ(r) is the Kohn-Sham kinetic energy density. This scaling factor is subsequently used to attenuate the Coulomb and exchange-correlation parts of the SIC as follows:

EscaledSIC=iσoccXiσk(U[ρiσ]+EXCDFA[ρiσ,0]).
(4)

We shall refer to this approach as exterior orbital scaling. Like PZSIC, and unlike the 50% global scaling, this approach is exact for all one-electron densities and, with integer k ≥ 1, for all uniform densities.

Vydrov et al. noted that while increasing k above zero satisfies some additional exact constraints, the correct −1/r asymptotic behavior of the one electron potential is lost if k > 0. Vydrov and Scuseria also proposed a simpler method47 of moderating SIC with a scaling factor given as

Wiσm=ρiσρσmρiσdr=ρiσm+1ρσmdr.
(5)

This factor depends on the ratio of overlaps of orbital density ρ and the total density ρσ for a given spin σ. The authors noted that the SIC-PBE with m = 1 performs consistently well for the benchmark tests, but a larger value of m, such as m = 4, is needed for SIC-LSDA.

PZSIC improves results where semilocal functionals fail drastically on account of SIE,45,77,85 but it overcorrects and worsens the description of near-equilibrium properties such as molecular atomization energies. Based on this observation, we propose a modification of PZSIC in such a way that the self-interaction correction is enforced only where it is necessary. This can be done locally, or pointwise in space, that is, it can be applied only in the regions where self-interaction is expected to be strong. Tsuneda and co-workers48 defined these to be regions where the density has one-electron character, and they used the ratio zσ(r)=τσW(r)/τσ(r) to identify these regions. Here, the noninteracting (Kohn-Sham) kinetic energy density τσ for a spin σ is given as

τσ(r)=12i|ψiσ(r)|2,
(6)

and τσW is given as

τσW(r)=|ρσ(r)|28ρσ(r).
(7)

Since τσW is the single orbital limit of τσ and vanishes for a uniform density, zσ(r) varies between zero and one, with zero corresponding to uniform densities and one to one-electron densities. In their regional SIC scheme, Tsuneda and co-workers48 used this ratio to replace the conventional DFT expression for the exchange and correlation potential in regions where z is close to one by a simple model expression intended to mimic the exchange potential of a single hydrogenic orbital. They used this scheme to study reaction barriers, where they found significant improvement over conventional DFT calculations. Following Tsuneda et al., we propose the following modification to the PZSIC energy expression:

EXCLSICDFA=EXCDFA[ρ,ρ]i,σoccULSIC[ρiσ]+EXCLSIC[ρiσ,0],
(8)

where

ULSIC[ρiσ]=12dr{zσ(r)}kρiσ(r)drρiσ(r)|rr|
(9)

and

EXCLSIC[ρiσ,0]=dr{zσ(r)}kρiσ(r)ϵXCDFA([ρiσ,0],r).
(10)

The LSIC-DFA of Eq. (8) recovers the PZSIC [Eq. (1)] for k = 0. The k limit, on the contrary, zeroes out the SIC and reduces thereby to a standard DFA, except in fully one-electron regions. In the present work, we use k = 1. This is the simplest choice of scaling factor based on zσ. It smoothly interpolates between uniform density regions and one-electron regions. In the rest of this section, we provide the computational details.

We implemented the LSIC approach in the FLOSIC code which is based on the UTEP-NRLMOL79,80 software package. UTEP-NRLMOL is a modern version of the Gaussian-orbital-based NRLMOL code.86–88 We use the NRLMOL default basis sets89 that are of approximately quadruple zeta quality.90 For a better description of atomic anions, we added long range s, p, and d single Gaussian orbitals to the default NRLMOL basis set. The exponents for the additional functions were obtained from the relation β(N+1)=β(N)β(N1)β(N), where β(N) is the exponent of the Nth Gaussian in the basis for a given atom. NRLMOL’s variational integration mesh87 adapts to the range of basis functions so that integrals are computed to a specified accuracy.

We use the FLO-SIC approach proposed by Pederson et al.81,82 to implement the PZSIC and LSIC methods. In FLO-SIC, the local orbitals used to evaluate the PZSIC total energy are based on Fermi orbitals constructed from the density with parameters known as Fermi orbital descriptors (FODs), M positions in 3-dimensional space for M occupied orbitals. Using these FODs, one can write the Fermi orbitals as

Fiσ(r)=jMψjσ*(aiσ)ψjσ(r)ρσ(aiσ),
(11)

where aiσ is the FOD position for the orbital i of spin σ, ρσ is the electron spin density, and ψ is one of the M occupied orbitals. Since Fermi orbitals are generally not orthogonal, Löwdin orthogonalization is performed to transform the F into the orthonormal local orbital ϕ. The FLO-SIC approach is unitarily invariant because any set of orbitals spanning the occupied orbital space can be used in Eq. (11) to generate the Fermi orbitals. To minimize the PZSIC energy, the 3M FOD positions must be optimized. This is done using gradients of the energy with respect to the FOD positions91 in a manner analogous to a molecular geometry optimization. This is a computationally simpler process than determining the O(M2) parameters required to define the local orbital transformation in traditional PZSIC. We follow the self-consistency procedure of Ref. 92. Iteration averaging was performed for the DFA potentials, using either Broyden mixing or a simple mixing scheme to accelerate convergence. A self-consistency convergence tolerance of 10−6 Ha in the total energy was used for all calculations. For PZSIC calculations using FLO-SIC, an FOD force tolerance of 10−3 Ha/bohr was used to ensure optimized FOD positions. LSIC-LSDA total energies [Eq. (8)] were computed using the corresponding self-consistent PZSIC-LSDA density and optimized local orbitals. Both LSIC and FLO-SIC calculations have similar computational costs. The only additional quantity needed for LSIC is the scaling factor, which requires the evaluation of the kinetic energy densities whose computational cost is negligible. The FLO-SIC methodology has been recently employed to study atomic energies,93 atomic polarizabilities,83 and magnetic exchange couplings.94 

1. Total energy

We compared the total energy E of atoms for atomic numbers Z = 1–18 computed using different methods against the accurate nonrelativistic energies (Eaccu) reported by Chakravorty et al.95 The total energy differences EEaccu are shown in Fig. 1 for LSDA, PZSIC-LSDA, and LSIC-LSDA. In general, LSDA overestimates the total energies and applying SIC shifts the energies in the proper direction. But the corrections are too large and PZSIC-LSDA predicts atomic energies that are too low. The LSIC-LSDA total energies, by contrast, are very close to the reference energies. LSIC reduces the mean absolute errors (MAEs) in the energies by an order of magnitude compared to PZSIC-LSDA. The MAEs are 0.726, 0.381, and 0.041 Ha for LSDA, PZSIC-LSDA, and LSIC-LSDA, respectively. Results for atomic total energies for a variety of methods are summarized in Table I. The LSIC-LSDA results are better than PBE results, but slightly worse than strongly constrained and appropriately normed (SCAN) results.

FIG. 1.

LSDA, PZSIC-LSDA, and LSIC-LSDA total energies of atoms Z = 1–18, relative to reference energies (Eaccu) from Ref. 95.

FIG. 1.

LSDA, PZSIC-LSDA, and LSIC-LSDA total energies of atoms Z = 1–18, relative to reference energies (Eaccu) from Ref. 95.

Close modal
TABLE I.

MAEs of atomic total energies Z = 1–18 (Ha).

MethodMAE
LSDAa 0.726 
PBEa 0.083 
SCANa 0.019 
PZSIC-LSDAa 0.381 
PZSIC-PBEa 0.159 
PZSIC-SCANa 0.147 
LSIC-LSDA 0.041 
MethodMAE
LSDAa 0.726 
PBEa 0.083 
SCANa 0.019 
PZSIC-LSDAa 0.381 
PZSIC-PBEa 0.159 
PZSIC-SCANa 0.147 
LSIC-LSDA 0.041 
a

Reference 72.

2. Ionization potential

Since the ionization potential is the energy required to remove an electron from the outermost orbital, this quantity is sensitive to the asymptotic behavior of the potential and can be expected to be affected by SIC, especially for large systems. We calculated the ionization potential (IP) for the atoms He–Kr using the ΔSCF approach as follows:

EIP=EcationEneutral.
(12)

Figure 2 shows a comparison of calculated IPs against experimental values96 for LSDA, PZSIC-LSDA, and LSIC-LSDA. The MAEs are presented in Table II, along with results for other methods. From LSDA to PZSIC, the IPs improve noticeably, with the MAE dropping from 0.458 to 0.364 eV. LSIC further improves the IPs, reducing the MAE down to 0.170 eV. Because the LSIC-LSDA total energies for the neutral atoms are very close to the reference energies, the accurate IP values imply that the LSIC-LSDA cation energies are also quite accurate. In Table II, we show the results for the atoms from He to Ar and for all atoms in separate columns, to distinguish the performance for light vs heavy atoms. PBE and SCAN perform well for the lighter atoms, but less so for the heavier ones. LSIC-LSDA, on the contrary, performs equally well for all atoms. LSIC-LSDA performs better than both PBE and SCAN (MAEs 0.253 and 0.273 eV, respectively) for the 35 IPs.

FIG. 2.

Ionization potential of atoms Z = 2–36. LSDA, PZSIC-LSDA, and LSIC-LSDA are shown.

FIG. 2.

Ionization potential of atoms Z = 2–36. LSDA, PZSIC-LSDA, and LSIC-LSDA are shown.

Close modal
TABLE II.

MAEs of ΔSCF ionization potentials (eV).

MethodZ = 2–18 (17 IPs) MAEZ = 2–36 (35 IPs) MAE
LSDA 0.275 0.458 
PBE 0.159 0.253 
SCAN 0.175 0.273 
PZSIC-LSDA 0.248 0.364 
PZSIC-PBE 0.405 0.464 
PZSIC-SCAN 0.274 0.259 
LSIC-LSDA 0.206 0.170 
MethodZ = 2–18 (17 IPs) MAEZ = 2–36 (35 IPs) MAE
LSDA 0.275 0.458 
PBE 0.159 0.253 
SCAN 0.175 0.273 
PZSIC-LSDA 0.248 0.364 
PZSIC-PBE 0.405 0.464 
PZSIC-SCAN 0.274 0.259 
LSIC-LSDA 0.206 0.170 

3. Electron affinity

The electron affinities (EAs) of atoms from H to Br were also investigated. As in the case of the IPs, the EAs were calculated using the ΔSCF method, taking the total energy differences of the neutral atoms and their anions. We considered the 20 atoms in the first three rows (H, Li, B, C, O, F, Na, Al, Si, P, S, Cl, K, Ti, Cu, Ga, Ge, As, Se, and Br) with stable anions, for which experimental EAs are available.96Figure 3 shows the results for each atom for LSDA, PZSIC-LSDA, and LSIC-LSDA. In Table III, we again analyze the performance of various methods, dividing the results into two groups, the first containing the 12 EAs corresponding to first and second row atoms and the second containing all 20 EAs. The MAEs relative to the experiment are shown in the table. We note that although the ΔSCF approach yields positive EAs for the DFAs, the eigenvalue corresponding to the added electron becomes positive in all DFA anion calculations, indicating that the extra electron is not actually bound in the complete basis set limit. This problem is due to the incorrect asymptotic form of the potential in the DFA calculations. SIC rectifies this,28 leading to bound states for the HOMO in the anions. Nevertheless, we include the EAs of DFA calculations based on the ΔSCF approach in Table III for comparison purposes. We note that these results compare well with PZSIC results of Vydrov and Scuseria.45 

FIG. 3.

Electron affinity of atoms Z = 1–35. LSDA, PZSIC-LSDA, and LSIC-LSDA are shown where experimental values are reported.

FIG. 3.

Electron affinity of atoms Z = 1–35. LSDA, PZSIC-LSDA, and LSIC-LSDA are shown where experimental values are reported.

Close modal
TABLE III.

MAEs of ΔSCF electron affinities (eV).

Method12 EA MAE20 EA MAE
LSDAa,b 0.349 0.362 
PBEa,b 0.167 0.172 
SCANa,b 0.115 0.148 
PZSIC-LSDAa 0.151 0.189 
PZSIC-PBEa 0.534 0.531 
PZSIC-SCANa 0.364 0.341 
LSIC-LSDA 0.097 0.102 
Method12 EA MAE20 EA MAE
LSDAa,b 0.349 0.362 
PBEa,b 0.167 0.172 
SCANa,b 0.115 0.148 
PZSIC-LSDAa 0.151 0.189 
PZSIC-PBEa 0.534 0.531 
PZSIC-SCANa 0.364 0.341 
LSIC-LSDA 0.097 0.102 
a

Reference 72.

b

DFA results are based on total energies. The eigenvalue of the extra electron becomes positive.

Overall, LSDA overestimates the values of the EAs, whereas PZSIC-LSDA tends to underestimate them, particularly for O, F, and Ti. The LSIC-LSDA method improves the EA values so that they consistently fall within ±0.2 eV of the experimental values. The MAE of 20 EAs is reduced from 0.362 eV for LSDA, to 0.189 eV for PZSIC-LSDA, to 0.102 eV for LSIC-LSDA.

The atomization energy (AE) of a molecule is defined as

AE=iNatomEiEmol>0,
(13)

where Ei is the energy of an individual atom and Emol is the energy of the molecule. We computed AEs for a diverse set consisting of 37 molecules. The majority of the molecules were taken from the G2/97 test set.97 We also included the six systems from the AE6 test set.98 All molecular geometries were taken from Ref. 96 [B3LYP and the 6-31G(2df,p) basis] except O2, CO, CO2, C2H2, Li2, CH4, NH3, and H2O, which were obtained using PBE and the default NRLMOL basis set.

The percentage errors in calculated AEs relative to the experiment are shown in Fig. 4 for LSDA, PZSIC-LSDA, and LSIC-LSDA. LSDA significantly overestimates the AEs. PZSIC-LSDA tends to improve them, but in most cases, it still overestimates their values. LSIC-LSDA reduces the AEs further, bringing them into better agreement with experimental values. Mean absolute percentage errors (MAPEs) for a variety of methods are compared in Table IV. The MAPE for the full set of molecules is 24.21% for LSDA, 13.42% for PZSIC-LSDA, and 6.94% for LSIC-LSDA. The performance of the LSIC-LSDA falls between that of PBE (8.64%) and SCAN (5.22%). We also list results for the AE6 test set in Table IV, showing both the MAE and MAPE. The AE6 molecules are intended to give a good representative of atomization energy performance. Here, also, it can be seen that LSIC-LSDA has a performance that is better than that of PBE, though not as good as that of SCAN.

FIG. 4.

Percentage errors for calculated atomization energies relative to experimental values. Results for the LSDA, PZSIC-LSDA, and LSIC-LSDA methods are shown.

FIG. 4.

Percentage errors for calculated atomization energies relative to experimental values. Results for the LSDA, PZSIC-LSDA, and LSIC-LSDA methods are shown.

Close modal
TABLE IV.

Atomization energies: AE6 errors (MAE and MAPE) and errors for the full set (MAPE) are shown.

AE6 MAEAE637 molecules’
Method(kcal/mol)MAPE (%)MAPE (%)
LSDA 74.26 15.93 24.21 
PBE 13.43 3.31 8.64 
SCAN 2.85 1.15 5.22 
PZSIC-LSDA 57.97 9.37 13.42 
PZSIC-PBE 18.83 6.82 9.67 
PZSIC-SCAN 16.31 5.64 10.24 
LSIC-LSDA 9.95 3.20 6.94 
AE6 MAEAE637 molecules’
Method(kcal/mol)MAPE (%)MAPE (%)
LSDA 74.26 15.93 24.21 
PBE 13.43 3.31 8.64 
SCAN 2.85 1.15 5.22 
PZSIC-LSDA 57.97 9.37 13.42 
PZSIC-PBE 18.83 6.82 9.67 
PZSIC-SCAN 16.31 5.64 10.24 
LSIC-LSDA 9.95 3.20 6.94 

Recently, Sharkas et al.85 used the FLO-SIC methodology to study the performance of the PZSIC for a test set consisting of dissociation energies99 (SIE4×4) and reaction energies100 (SIE11) that are expected to be strongly affected by self-interaction errors in DFAs. They found that PZSIC significantly decreases the errors of LSDA and PBE relative to reference calculations using the coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)] method. We studied the same test sets using LSIC-LSDA. The SIE4×4 set consists of four symmetric dimer cations at four different dimer separations R relative to the respective equilibrium separations Re: R/Re = 1, 1.25, 1.5, and 1.75. The SIE11 set consists of six cationic reactions (of which four are the dimer cations from the SIE4×4 data set at their equilibrium geometries) and five neutral reactions. We use the atomic geometries and FOD positions found in the supplementary information of Ref. 85 as starting points for our FLO-SIC-LSDA calculations. We reoptimized the FOD positions to ensure the FOD forces were below the 10−3 Ha/bohr threshold.

Results for the individual cases in the test sets are shown in Table V for LSDA, PZSIC-LSDA, and LSIC-LSDA. Results are given as signed errors (in kcal/mol) relative to accurate reference values (also shown). For the SIE4×4 case, PZSIC-LSDA and LSIC-LSDA clearly improve on the results of LSDA for all separations. The PZSIC results are generally better than LSIC results for R/Re > 1, though the differences are small compared to the scale of the self-interaction corrections. Conversely, the LSIC results are consistently better near Re. The mean average error (MAE) is slightly smaller in LSIC than in PZSIC, 2.6 vs 3.0 kcal/mol, respectively. For the SIE11 test set, the signed errors are typically somewhat smaller in LSIC than in PZSIC. In the case of the dissociation of (CH3)2CO+, there is a dramatic reduction in the signed error. This drops the MAE for the SIE11 cationic reactions from 14.83 for PZSIC to 2.31 kcal/mol for LSIC. The MAE for the SIE11 neutral reactions also shows an improvement from 9.01 to 6.31 kcal/mol.

TABLE V.

SIE4×4 binding energy curves and SIE11 reaction energies (kcal/mol). Reference energies and signed errors are shown.

ReactionR/ReReferenceaPZSIC-LSDAbLSIC-LSDA
H2+ H + H+ 1.0 64.4 0.0 0.0 
 1.25 58.9 0.0 0.0 
 1.5 48.7 0.0 0.0 
 1.75 38.3 0.1 −0.1 
He2+ He + He+ 1.0 56.9 5.8 −1.7 
 1.25 46.9 1.9 −2.7 
 1.5 31.3 −0.4 −3.0 
 1.75 19.1 −1.6 −3.0 
(NH3)2+NH3+NH3+ 1.0 35.9 11.7 1.6 
 1.25 25.9 7.3 6.7 
 1.5 13.4 4.0 8.1 
 1.75 4.9 3.4 6.7 
(H2O)2+H2O + H2O+ 1.0 39.7 5.8 0.4 
 1.25 29.1 −1.4 4.2 
 1.5 16.9 −2.7 1.5 
 1.75 9.3 −1.5 1.8 
C4H10+C2H5+C2H5+ 35.28 11.44 −6.00 
(CH3)2CO+CH3+CH3CO+ 22.57 39.39 1.86 
ClFCl → ClClF −1.01 4.63 4.37 
C2H4…F2 → C2H4 + F2 1.08 −0.23 −2.82 
C6H6…Li → Li + C6H6 9.50 10.19 −13.50 
NH3…ClF → NH3 + ClF 10.50 5.60 −4.56 
NaOMg → MgO + Na 69.56 28.56 11.45 
FLiF → Li + F2 94.36 −4.82 −1.18 
MAE SIE4×4 3.0 2.6 
MAE SIE11 cationic 14.83 2.31 
MAE SIE11 neutral 9.01 6.31 
ReactionR/ReReferenceaPZSIC-LSDAbLSIC-LSDA
H2+ H + H+ 1.0 64.4 0.0 0.0 
 1.25 58.9 0.0 0.0 
 1.5 48.7 0.0 0.0 
 1.75 38.3 0.1 −0.1 
He2+ He + He+ 1.0 56.9 5.8 −1.7 
 1.25 46.9 1.9 −2.7 
 1.5 31.3 −0.4 −3.0 
 1.75 19.1 −1.6 −3.0 
(NH3)2+NH3+NH3+ 1.0 35.9 11.7 1.6 
 1.25 25.9 7.3 6.7 
 1.5 13.4 4.0 8.1 
 1.75 4.9 3.4 6.7 
(H2O)2+H2O + H2O+ 1.0 39.7 5.8 0.4 
 1.25 29.1 −1.4 4.2 
 1.5 16.9 −2.7 1.5 
 1.75 9.3 −1.5 1.8 
C4H10+C2H5+C2H5+ 35.28 11.44 −6.00 
(CH3)2CO+CH3+CH3CO+ 22.57 39.39 1.86 
ClFCl → ClClF −1.01 4.63 4.37 
C2H4…F2 → C2H4 + F2 1.08 −0.23 −2.82 
C6H6…Li → Li + C6H6 9.50 10.19 −13.50 
NH3…ClF → NH3 + ClF 10.50 5.60 −4.56 
NaOMg → MgO + Na 69.56 28.56 11.45 
FLiF → Li + F2 94.36 −4.82 −1.18 
MAE SIE4×4 3.0 2.6 
MAE SIE11 cationic 14.83 2.31 
MAE SIE11 neutral 9.01 6.31 
a

Reference 100.

b

Reference 72.

The LSIC-LSDA results are also as good as or better than PZSIC-PBE results,85 which have MAE of 2.3, 8.7, and 7.9 kcal/mol for SIE4×4, SIE11 cationic, and SIE11 neutral, respectively.

Figure 5 shows ground-state dissociation curves for H2+ and He2+. These give useful comparisons of the overall behavior of PZSIC-LSDA and LSIC-LSDA with that of LSDA and PBE both near and far from the equilibrium separations. In both cases, the DFA calculations produce qualitatively incorrect energy curves as the interatomic separation increases, featuring a slight energy barrier at large separations on the way to a too-low energy for the dissociation products, two H+0.5 or He+0.5 fragments. Both PZSIC and LSIC calculations restore the correct qualitative shape to the dissociation curves. In the case of H2+, PZSIC and LSIC give identical and exact results because the iso-orbital indicator zσ is exactly one everywhere for this one-electron case. For He2+, LSIC and PZSIC give the same results in the dissociation limit of He and He+, since zσ is again one everywhere in this limit (He has one-electron of each spin). Near the equilibrium separation, however, LSIC reduces the size of the self-interaction correction, resulting in a binding energy that is close to that of PBE.

FIG. 5.

Ground-state dissociation curves for (a) H2+ and (b) He2+. The CCSD(T)/cc-pVQZ results from Ref. 101 and local hybrid results from Ref. 102 are also shown for comparison.

FIG. 5.

Ground-state dissociation curves for (a) H2+ and (b) He2+. The CCSD(T)/cc-pVQZ results from Ref. 101 and local hybrid results from Ref. 102 are also shown for comparison.

Close modal

To investigate the performance of LSIC for barrier heights in chemical reactions, we used the BH6 test set. This is a representative subset of the BH24 set.103 The reactions included in BH6 are OH + CH4 → CH3 + H2O, H + OH → H2 + O, and H + H2S → H2 + HS. Total energies at the left hand side, the right hand side, and the saddle point of these chemical reactions were evaluated, and the barrier heights of the forward (f) and reverse (r) reactions were obtained by taking the relevant energy differences. We used the geometries provided in Ref. 103 and reference values from Ref. 98. The results for various methods are summarized in Table VI.

TABLE VI.

BH6 forward (f) and reverse (r) barrier heights (kcal/mol). Signed errors are shown.

DFAPZSICLSIC
ReactionBarrierReferenceaLSDAPBESCANLSDAPBESCANLSDA
OH + CH4 → CH3 + H26.7 −23.6 −12.2 −8.3 −2.2 5.7 4.6 2.6 
 19.6 −17.4 −10.7 −7.8 −12.5 −10.3 −7.1 −0.2 
H + OH → H2 + O 10.7 −11.8 −2.2 −7.5 −1.1 2.3 0.0 −0.6 
 13.1 −25.3 −9.9 −11.0 −4.8 2.9 1.8 1.2 
H + H2S → H2 + HS 3.6 −10.3 −4.8 −6.3 −1.7 1.7 −1.9 −1.3 
 17.3 −17.2 −8.1 −6.2 −7.0 −2.1 −2.2 2.2 
ME −17.6 −8.0 −7.9 −4.9 0.0 −0.8 0.7 
MAE 17.6 8.0 7.9 4.9 4.2 3.0 1.3 
DFAPZSICLSIC
ReactionBarrierReferenceaLSDAPBESCANLSDAPBESCANLSDA
OH + CH4 → CH3 + H26.7 −23.6 −12.2 −8.3 −2.2 5.7 4.6 2.6 
 19.6 −17.4 −10.7 −7.8 −12.5 −10.3 −7.1 −0.2 
H + OH → H2 + O 10.7 −11.8 −2.2 −7.5 −1.1 2.3 0.0 −0.6 
 13.1 −25.3 −9.9 −11.0 −4.8 2.9 1.8 1.2 
H + H2S → H2 + HS 3.6 −10.3 −4.8 −6.3 −1.7 1.7 −1.9 −1.3 
 17.3 −17.2 −8.1 −6.2 −7.0 −2.1 −2.2 2.2 
ME −17.6 −8.0 −7.9 −4.9 0.0 −0.8 0.7 
MAE 17.6 8.0 7.9 4.9 4.2 3.0 1.3 
a

Reference 98.

DFAs such as LSDA, PBE, and SCAN underestimate barrier heights45 by giving transition state energies that are too low compared to the reactant and product energies. An accurate description of the stretched bonds in the transition states requires full nonlocality in the exchange-correlation potential that the semilocal functionals cannot provide. Use of PZSIC reduces the overall errors, but in PZSIC-LSDA, the barriers are still too small compared to reference values. This can be seen in the negative signed errors of all six barrier heights in Table VI. Applying LSIC-LSDA improves the barrier heights in almost every case. The MAE of the barrier heights improves from 17.6 kcal/mol for LSDA, to 4.9 in PZSIC-LSDA, to only 1.3 kcal/mol in LSIC-LSDA. Remarkably, as seen in the table, LSIC-LSDA has a smaller MAE than any of the methods listed, including PZSIC-PBE and PZSIC-SCAN.

The LSIC method uses a pointwise scaling of SIC terms [Eqs. (8)–(10)] to reduce the effect of self-interaction in many-electron regions while applying SIC at full strength in one-electron regions. We can think of this as interior orbital scaling, in comparison with the exterior orbital scaling of Eq. (4). We showed in Sec. III that using LSIC-LSDA results in significant performance gains for all the common electronic properties tested, as compared to both LSDA and PZSIC-LSDA. LSIC-LSDA improves on PZSIC for barrier heights and the SIE test sets where SIC is critical for getting physically reasonable results. For near-equilibrium properties where PZSIC degrades the performance of semilocal DFAs, LSIC-LSDA gives results that are better than PBE results and nearly as good as SCAN results. This is remarkable, given the relative simplicity of LSDA compared to the semilocal functionals. It is worth comparing and contrasting LSIC-LSDA results with results using an exterior orbital scaling method, such as that presented in Ref. 45. These authors suggest k = 2 as the best overall choice for use in Eqs. (3) and (4). With this choice, the exterior orbital scaling method used with LSDA gives an MAE of 8.6 kcal/mol for the AE6 atomization energies. This is slightly better than the LSIC-LSDA result of 9.95 kcal/mol shown in Table IV. For the BH6 barrier heights, the exterior scaling method gives an MAE of 4.7 kcal/mol, compared to 1.3 kcal/mol for LSIC-LSDA (Table VI). While these results are similar, one should note that the exterior scaling approach causes the asymptotic form of the one-electron potential to differ from the −1/r form expected for the exact functional and maintained by PZSIC. This has an impact on properties that are sensitive to the nature of the potential in this region. For the HOMO eigenvalues of the atoms H–Kr, for example, our investigation shows that the MAE for PZSIC-LSDA is 0.672 eV and that for the orbitalwise scaling approach of Eqs. (3) and (4) is 1.034 eV (k = 1) when compared to the experimental IPs. Equality of the HOMO eigenvalue and the IP is a consequence of the linear variation of the total energy between adjacent integer numbers of electrons. This many-electron self-interaction freedom101 is exact for the exact functional and approximately true for PZSIC. It has been argued elsewhere101 that this property requires a full Hartree self-interaction correction term and thus should not be true for exterior orbital scaling corrections (or even for LSIC). A similar problem involves dissociating heteroatomic molecules to the correct neutral atom limits.101 LSDA and the exterior orbital scaling method fail to do this in many cases, while PZSIC-LSDA succeeds. It is not yet clear how pointwise local scaling will affect such properties in general, since examining this requires fully self-consistent LSIC calculations. Preliminary quasi-self-consistent LSIC calculations on the atomic systems using the weighted average of SIC potentials show that the self-consistency in fact slightly improves the LSIC results. The MAE in the HOMO eigenvalues of quasi-self-consistent LSIC results is 0.363 eV, compared to 0.672 eV for perturbative LSIC and 1.034 for exterior orbital scaling [cf. Eq. (4)]. A fully self-consistent implementation of LSIC-LSDA has been formulated and is implemented into the FLOSIC code.

Recently, Santra and Perdew78 showed that uniform electron gas norms satisfied by semilocal functionals are violated by the corresponding PZSIC-DFAs. To show how functionals behave in this limit, they fitted the calculated results for the exchange energy for neutral noble gas atoms using an exact large-Z expansion of EX as a fitting function. We used the same approach to test LSIC. We computed the LSIC-LSDA exchange energy of Ne, Ar, Kr, and Xe and then fitted these energies using the function

EXapproxEXexactEXexact×100%=a+bx2+cx3,
(14)

where x = Z−1/3 and a, b, and c are fit parameters. The result is shown in Fig. 6. The value of a corresponds to the limit where Z−1/3 → 0 which corresponds to the uniform density limit. a should vanish for the nonempirical LSDA, PBE, and SCAN functionals that are exact in this limit. The reported values of a are −0.18, −0.06, and −0.28 for LSDA, PBE, and SCAN and 5.79, −3.30, and −3.63 for PZSIC-LSDA, PZSIC-PBE, and PZSIC-SCAN.78 The small residual values of a for LSDA, PBE, and SCAN are due to errors in the extrapolations. For LSIC-LSDA, we obtain a = −0.62. We note that the scaling factor zσ(r)=τσW(r)/τσ(r) we have chosen vanishes for a uniform density and that LSIC would therefore give no correction to LSDA in this limit. This may not be the case for a different choice of scaling factor. A constraint of the exact functional that is lost in PZSIC is thus recovered by LSIC (as by the exterior orbital scaling approach of Ref. 45).

FIG. 6.

Percentage errors of the approximated exchange energies using the exact exchange energies as a reference. In LSIC-LSDA, the Z−1/3 → 0 limit is dramatically improved over PZSIC-LSDA.

FIG. 6.

Percentage errors of the approximated exchange energies using the exact exchange energies as a reference. In LSIC-LSDA, the Z−1/3 → 0 limit is dramatically improved over PZSIC-LSDA.

Close modal

We introduced the LSIC-LSDA method that incorporates a pointwise scaling of self-interaction corrections based on a simple iso-orbital descriptor zσ(r)=τσW(r)/τσ(r). The essential idea is to retain the benefit of PZSIC in the regions where the self-interaction is expected to be strong while reducing its effect in other regions. We showed the results of LSIC-LSDA for a number of properties, including atomic total energies, IPs, and EAs for the atoms up to Kr and atomization energies for a subset of G2 molecules and AE6 molecules, dissociation and reaction energies for the SIE4×4 and SIE11 test sets, and chemical reaction barriers for the BH6 data set. In nearly all cases, the performance of LSIC is dramatically improved compared to that of both pure LSDA and PZSIC-LSDA. LSIC-LSDA even performs better than PBE for atomization energies and is competitive with SCAN in many cases while keeping the benefits of PZSIC for properties like barrier heights, where the semilocal functions do poorly. We also showed that LSIC-LSDA restores the correct uniform density limit of the exchange energy that is lost in PZSIC. In all, LSIC-LSDA brings the full nonlocality of the PZSIC method to bear for cases like stretched bonds where SIE effects are dominant while maintaining the already good description of near equilibrium properties provided by semilocal functionals.

It is interesting to compare LSIC-LSDA with advancements made on the well-trodden path of creating and correcting more sophisticated semilocal functionals.31 A major development along the latter was the introduction of a fraction of Hartree-Fock exchange which resulted in mitigating many deficiencies of the pure density functional approaches. As mentioned earlier, the formal justification for such mixing was the adiabatic connection between the real interacting system and the noninteracting KS system. Because the exact exchange-correlation energy is an integral over the coupling constant from 0 to 1, it could include some fraction of exact exchange, which is the correct integrand at the limit of the zero coupling constant. It is interesting to see the parallels between this path and SIC. Because typical real systems are part-way between slowly varying density and one-electron density limits, the exact exchange-correlation energy could include some fraction of PZSIC, which is exact for any one-electron density. The 50% scaling approach used by Jónsson and co-workers75 can be considered as a global (orbital-independent) hybrid of DFA and PZSIC-DFA, in analogy to the traditional global hybrids. On the other hand, the present LSIC approach is analogous to local hybrids. Understanding obtained in the development of hybrid functionals may be beneficial in the further development of the LSIC method.

The authors dedicate this paper to Dr. Mark Pederson on the occasion of his 60th birthday. R.R.Z. and Y.Y. acknowledge Dr. Luis Basurto and Dr. Jorge Vargas for discussions and technical assistance. This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, as part of the Computational Chemical Sciences Program under Award No. DE-SC0018331. The work of R.R.Z. was supported in part by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0006818. Support for computational time at the Texas Advanced Computing Center through NSF Grant No. TG-DMR090071 and at NERSC is gratefully acknowledged. R.R.Z. and Y.Y. conceived the LSIC concept and prepared the first draft of the manuscript. The other authors contributed to calculations, figures, tables, references, discussions, and revision of the manuscript.

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