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Self-Interaction Error in Density Functional Theory: An Appraisal Junwei Lucas Bao, Laura Gagliardi, and Donald G. Truhlar J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b00242 • Publication Date (Web): 06 Apr 2018 Downloaded from http://pubs.acs.org on April 6, 2018

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The Journal of Physical Chemistry Letters 1

prepared for JPC Lett., 4/2/2018

Self-Interaction Error in Density Functional Theory: An Appraisal Junwei Lucas Bao, Laura Gagliardi,* and Donald G. Truhlar* Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455-0431, United States ABSTRACT: Self-interaction error (SIE) is considered to be one of the major sources of error in most approximate exchange-correlation functionals for Kohn-Sham density-functional theory (KS-DFT), and it is large with all local exchange-correlation functionals and with some hybrid functionals. In this work, we consider systems conventionally considered to be dominated by SIE. for these systems, we demonstrate that by using multiconfiguration pair-density functional theory (MC-PDFT), the error of a translated local density-functional approximation is significantly reduced (by a factor of 3) when using an MCSCF density and on-top density, as compared to using KS-DFT with the parent functional; the error in MC-PDFT with local on-top functionals is even lower than in some popular KS-DFT hybrid functionals. Density-functional theory, either in MC-PDFT form with local on-top functionals or in KS-DFT form with some functionals having 50% or more nonlocal exchange has smaller errors for SIE-prone systems than does CASSCF, which has no SIE.

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It is widely accepted that one of the key failings of Kohn-Sham density functional theory1,2 (KS-DFT) with approximate density functionals in common use is self-interaction error. In wave function theory, one includes Coulomb interactions only for two-particle interactions; an electron does not interact with itself. In KS-DFT, the energy is a functional of the single-particle density (or the single-particle spin densities) so there is no way to precisely distinguish two-body Coulomb interactions from self-interaction. One therefore includes the interaction of each electron with the entire electron density (including its own density) as a Coulomb energy, and one attempts to remove self-interaction from the Coulomb energy by an approximate density functional, called the exchange-correlation functional. For a one-electron system, the self-interaction error (SIE) is equal to the entire Coulomb energy, and for a many-electron system whose density is described by a single Slater determinant (as in KS-DFT), one can define the self-interaction energy as the sum of the energies of each orbital interacting with itself.3 However, that definition is not compelling from all points of view, and it has been shown that “For systems with more than one electron, the SIE definition is not all that obvious.”4 Regardless of the difficulty of defining it exactly or precisely separating it from other sources of error, it is recognized that approximate density functionals in common use do not totally remove the self-interaction error, and there is general agreement as to the kind of systems where SIE dominates. The classic example is a collection of weakly interacting identical or nearly identical fragments from which one removes a single electron; the energy required in an accurate quantum mechanical description is expected to be approximately equal to the energy of removing it from the most easily ionized fragment, but KS-DFT may delocalize the hole over many fragments and yield a very poor approximation to the correct energy. Yang and coworkers5 have proposed that this delocalization is an even more fundamental issue than self-interaction, and, in light of the difficulty of precisely defining SIE, they have proposed centering attention on delocalization error. They defined a test for delocalization error in which one removes an electron from a cluster of weakly interacting He atoms. Self-interaction-error is clearest for one-electron systems but is clearly present and important in many-electron systems as well.6 Many workers have considered that delocalization error is a consequence of self-interaction error, but Yang and coworkers7,8 have pointed out that some exchange-correlation functionals that are considered to be SIE-free still have delocalization error. Although Hartree-Fock exchange might be considered to correct one-electron SIE, mixing some



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amount of Hartree-Fock exchange into the exchange-correlation functional (as is done in hybrid functionals) can involve incomplete cancellation of local-exchange delocalization error and Hartree-Fock-exchange localization error. In a previous paper, we addressed the issue of delocalization error independent of self-interaction considerations, but in the present paper we address the related issue of SIE by considering some cases that have traditionally been held up as the kind of system most likely to suffer from SIE. Pederson and coworkers9 recently proposed an extension of the Perdew-Zunger self-interaction correction (SIC) correction10 to KS-DFT in which a size-extensive construction of SIC orbitals makes SIC computationally efficient. Yang and coworkers recently proposed a localized orbital scaling correction for systematic elimination of delocalization error that uses additional ingredients beyond those in mainstream KS-DFT functionals to eliminate or minimize the SIEs of mainstream KS-DFT functionals.5 Hartree-Fock nonlocal exchange eliminates SIE in Hartree-Fock theory, and we will discuss below the extent to which it reduces errors in systems traditionally considered to be dominated by SIE in Kohn-Sham theory. Multi-configuration pair-density functional theory11,12 (MC-PDFT) is an alternative to KS-DFT in which the electronic subsystem is described not just by the one-particle density ρ(rj=r), where rj is the coordinate of electron j, but also by a local functional of the diagonal part (r1=r, r2=r) of the two-particle density ρ(r1, r2). In a previous study we showed that,13 as judged by the helium cluster ionization test, MC-PDFT has no delocalization error, whereas KS-DFT has a very large one with most functionals. Here we return to the older concept of SIE, and we test MC-PDFT for three classes of problem where the errors of mainstream KS-DFT functionals have traditionally been considered14,15,16,17,18,19,20,21 to be dominated by SIE: (i) the potential energy curves for dissociation of rare gas cation dimers; (ii) the dissociation energies of dimeric radical cations; (iii) neutral reactions that are considered20 “extremely prone to the SIE”. For class (i), we consider the potential curves for He2+ → He + He+ and ArKr+ → Ar + Kr+. For class (ii) we consider the SIE4×4 test set of Goerigk et al.,21 which consists of the dissociation energies of four positively charged dimers (H2+, He2+, (NH3)2+, and (H2O)2+), each calculated at 1.0, 1.25, 1.5, and 1.75 times their equilibrium inter-monomeric distance. (Note that, as detailed in Supporting Information, the H2+ calculations are performed in the presence of a spectator He atom.) For class (iii) we consider the six neutral reactions in the SIE11 test set of an older paper


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by Goerigk et al.20 The MC-PDFT calculations presented in this paper are based on complete active space self-consistent-field22 (CASSCF) reference wave functions (For the cases of H2+ and He2+ dissociation curve computed with full-valence active space, our CASSCF calculations reduce to restricted open-shell Hartree-Fock23 (ROHF), and we will also present results for ROHF.) We allow spatial symmetry breaking,24,25 but all reference wave functions are eigenfunctions of S2, where S is total electron spin. Details of the choice of the active spaces (by the full-valence and the correlated-participating-orbitals26,27,28 schemes) and basis sets employed in this work for MC-PDFT calculations are given in Supporting Information (SI). Our results are compared to benchmark-quality (coupled-cluster calculations) results obtained by wave function theory to test the accuracy of the MC-PDFT calculations and to CASPT2 calculations and KS-DFT calculations with mainstream functionals to judge the relative accuracy of various methods. Details of these comparison calculations are also presented in SI. In MC-PDFT, we use on-top density functionals, which at the present stage of development of the theory are obtained by translation11 of parent exchange-correlation functionals developed for Kohn-Sham theory, and in the present article the parent functional is the PBE one, in which the ingredients are the total electron density, the spin magnetization density, and their gradients. In the original translation, called tPBE, where “t” denotes “translated,” the ingredients of the on-top functional are total density, its gradient, and the on-top pair density. In the second form of translation, denoted ftPBE, where “ft” denotes “fully-translated,29 we add the gradient of the on-top pair density as another ingredient. Furthermore, the tPBE functional has a discontinuous derivative, and this is smoothed in the ftPBE functional. Dissociation curves of He2+ and ArKr+. Generating qualitatively correct dissociation curves for He2+ is a well-known challenging task for local exchange-correlation functionals. KS-DFT yields reasonable results when the internuclear distance R is at its equilibrium value (Re), but approximate exchange-correlation functionals can greatly underestimate the energy when R is large, due to spurious delocalization – potentially caused by self-interaction. Figure 1(a) shows that PBE calculations (i.e., unrestricted KS-DFT calculations with the PBE30 exchange-correlation density functional) predict a potential curve that is physical for R near Re but rapidly becomes too low in energy as R increases. We also see that tPBE calculations (by which we mean MC-PDFT calculations with the translated PBE on-top density functional11) yield a



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monotonically increasing potential curve in agreement with the shape of the benchmark calculations by CCSD(T). Figure 1 also shows results for dissociation of the asymmetric system ArKr+. Because the first ionization potential of Ar (15.8 eV) is only slightly higher than that of Kr (14.0 eV), KS-DFT with the PBE functional delocalizes the positive hole and again predicts an unphysical potential curve, whereas MC-PDFT with the tPBE functional produces a qualitatively correct dissociation curve. For comparison, in the SI we present additional calculations comparing the He2+ dissociation curves in Fig. 1 to CASSCF and other methods. The quantitative accuracy of MC-PDFT is much better than the corresponding functional in UKS-DFT (by a factor of 3 when averaged over all the data in Table 1) and than the corresponding hybrid functional in UKS-DFT (by a factor of 1.8 in Table 1). However, although tPBE has a much smaller error than PBE, the error is still not negligible. We assume that the error can be reduced further by optimizing the functional. Tests against the SIE4×4 cationic dataset. To more quantitatively assess the ability of MC-PDFT for reducing the SIE, we next compute the reaction energies for four molecular cation dissociation processes, each of which includes dissociation starting from four different internuclear distances R. In both our KS-DFT and our MC-PDFT calculations, the dissociation limit is a supermolecule of the two fragments that are separated by a large distance; for H2+ we also add a spectator He atom at 6 Å from one H. See the SI for details. Table S2 in the SI shows that for all of the species considered in this test, the weight (square of the configuration interaction coefficient) of the dominant configuration is 0.95 or greater, and the M diagnostic26 is 0.05 or less, indicating that the multireference character of these systems is small. The errors are therefore not dominated by static correlation errors, confirming that they are good tests for isolating SIE, as has been assumed in the literature. Table 1 shows that for KS-DFT, the KS-DFT calculations with the PBE30 functional – which is the parent functional of tPBE11 and ftPBE29 – has a mean unsigned error of 45 kcal/mol for the computed dissociation energies of molecular cation dissociations, while the errors of MC-PDFT with the tPBE and ftPBE functionals are only about one third as large (about 14 kcal/mol). Hybrid functionals (both global hybrid functionals and range-separated hybrid functionals) reduce SIE by including some nonlocal Hartree-Fock (HF) exchange. However, the PBE0 functional,31 which replaces 25% of PBE’s local exchange with nonlocal HF exchange, only reduces the MUE to 26 kcal/mol. Tables S3 and S4 of the SI show that MC-PDFT even shows better performance than


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the highly successful M06-2X32 functional, which has 54% HF exchange and gives an MUE of 16 kcal/mol. It also shows that the M06-HF33 and HFLYP34,35,36 functionals, which have 100% HF exchange, bring the error in KS-DFT down to the 5–6 kcal/mol range, but 100% HF exchange can lead to large errors for systems with high static correlation37,38 so it is not used in the most popular functionals. Tests against the SIE11 neutral reactions. Five of the neutral molecule reactions involve dissociation, and the two products are again computed as a supermolecule in which the product molecules are separated by a large distance. For this test set, MC-PDFT with the tPBE or the ftPBE functional has an MUE of only 3 kcal/mol, which is much better than KS-DFT with the parent PBE functional (MUE = 10 kcal/mol). For the neutral molecule reactions, MC-PDFT even shows similar performance as the M06-HF functional with 100% HF exchange (MUE = 3 kcal/mol) and much better performance than HFLYP with 100% HF exchange (MUE = 10 kcal/mol). Overall appraisal. In addition to the functionals we have discussed so far, we made tests against the cationic dissociations of test set SIE4×4 and the neutral dissociations of SIE11 for other density functionals and for unrestricted35 and restricted open-shell Hartree-Fock,23 and unrestricted MP2,39 and we computed the average MUE for each method, where the average MUE is defined as the average of the cationic MUE and the neutral MUE, i.e., [MUE(cationic)+MUE(neutral)]/2. The results are shown in Table 3, which provides an overall assessment (individual errors and the two individual MUEs are in the SI). The tests in Table 3 include the two hybrid functionals (BHLYP40 and M08-HX41) that were found to do best for SIE4×4 in ref. 21 and the local functional (MN12-L42) that did best of all local functionals in the tests of that reference; plus, they include unrestricted MP2, five range-separate hybrid functionals, and other functionals we added to aid in the appraisal. Note that UHF, UMP2, and the KS-DFT calculations minimize delocalization in the cation tests by sacrificing spin symmetry to put the extra majority-spin electron on a single monomer. Tables 1–3 taken together show that MC-PDFT with a purely local functional and reference wave functions with the correct spin symmetry reduce the self-interaction error that causes large errors in KS-DFT when it is used with exchange-correlation functionals having less than 50% nonlocal HF exchange, and sometimes it even does better than KS-DFT with functionals that have 100% nonlocal HF exchange. We see that the average MUE for MC-PDFT is in the 8–9 kcal/mol range, depending on functional,



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whereas KS-DFT with the parent functionals have average MUEs in the range 27–28 kcal/mol. Notice that the popular B3LYP hybrid functional, with 20% nonlocal exchange, has an average MUE of 18 kcal/mol, only slightly better than that of the best local functional, 20 kcal/mol. Some KS-DFT functionals with 25–54% HF exchange bring the average error down to the 9-16 kcal/mol range, and some KS-DFT functionals with 100% HF exchange (either globally as in M06-HF and HFLYP, or at large interelectronic separation as in the five range-separated hybrid functions) bring the average MUE down to 5–9 kcal/mol. Because the on-top energy of MC-PDFT is added to the classical Coulomb energy, which manifestly includes self-interaction, the local on-top functionals we have used must remove self-interaction energy just as exchange-correlation functionals must do so in KS-DFT, and the tests presented here show how well this is done without using nonlocal functionals. We must, however, view this success in light of the diffculty5 – already remarked – of precisely separating SIE from other sources of error. We observe that UHF, ROHF, and CASSCF, none of which has any self-interaction because only inter-particle Coulomb interactions are used, have average MUEs of 12–15 kcal/mol, and all of the MC-PDFT functionals and some of the KS-DFT functionals have average MUEs smaller than this, so errors in dynamic electron correlation cannot be discounted, and SIE is not necessarily the dominant error in these methods, although for the systems studied here it has been considered the dominant error in mainstream KS-DFT calculations when they do not include nonlocal exchange, and the larger errors we see for such calculations (up to 28 kcal/mol) are not inconsistent with this interpretation. The delocalization error in KS functionals is a different and more fundamental issue than the self-interaction error.5 The delocalization error can be isolated by using the widely-separated He cluster tests proposed by Yang and coworkers5 that we used in our previous work to demonstrate that MC-PDFT does not suffer from delocalization error. However, unlike delocalization error, the many-electron self-interaction error is not well defined and cannot be unambiguously separated from the other sources of errors in a density-functional calculation (including both KS-DFT and MC-PDFT). By showing that MC-PDFT with local functionals can have a smaller error than some methods in which self-interaction error is completely removed by using 100% Hartree-Fock exchange for all interelectronic separations (as in HFLYP), the present work shows that the SIE is not the dominant error in some systems in which the error has been traditionally attributed to mainly to SIE. We also show that the errors for these systems can be reduced by using


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Kohn-Sham theory with other exchange-correlation functionals having 100% nonlocal exchange at all or some interelectronic separations; however, introducing nonlocal exchange in Kohn-Sham density functional theory also brings in static correlation error,37 which is avoided by using local functionals in MC-PDFT. The functionals currently used in MC-PDFT are obtained by translation of functionals developed for other uses (with no re-parametrization for MC-PDFT); their current good performance (as compared to KS-DFT with functionals originally defined for KS-DFT and refined since the 1980s) is especially encouraging in showing that the formalism itself, rather than functionals optimized for this purpose, reduces the self-interaction errors below the level of the error in CASSCF. Kohn-Sham theory with 50% or more nonlocal exchange can also reduce the errors for SIE-prone test sets to below the error in CASSCF. We conclude that density functional theory, either in MC-PDFT form with local functionals or in KS-DFT form with 50% or more nonlocal exchange, does not suffer from the large SIEs that appear in tests of KS-DFT with local functionals and even with popular hybrid functionals like B3LYP.

Figure 1. Dissociation curves of (a) He2+ and (b) ArKr+ computed by tPBE (MC-PDFT), PBE (KS-DFT), and CCSD(T) (wave function theory) with the aug-cc-pVTZ basis. Each curve is separately referenced to have its zero of energy at its minimum energy point.



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Table 1. Benchmark reaction energies (in kcal/mol), signed errors for the computed reaction energies (in kcal/mol), and mean unsigned errors (MUEs) for ionic dissociation energies in SIE4×4. Reaction +

H2 → H + H

+

He2+ → He + He+

(NH3)2+→ NH3 + NH3+

(H2O)2+ → H2O + H2O+

Error

Bench-

R/Re

mark

1.0 1.25 1.5 1.75 1.0 1.25 1.5 1.75 1.0 1.25 1.5 1.75 1.0 1.25 1.5 1.75

CASSCF

CASPT2

tPBE

ftPBE

PBE

PBE0

64.4 58.9 48.7 38.3 56.9 46.9 31.3 19.1 35.9 25.9 13.4 4.9 39.7 29.1 16.9 9.3

-0.1 0.0 0.0 -0.1 -3.4 -1.9 -1.7 -2.1 -16.8 -14.0 -13.4 -13.9 -13.0 -11.3 -7.7 -3.8

-0.1 0.0 0.0 -0.1 -0.3 0.0 0.1 0.1 1.2 1.1 1.1 1.2 0.8 0.9 -5.1 -3.1

3.7 7.2 10.9 14.6 14.7 22.9 30.8 38.4 7.1 11.5 16.3 21.2 12.3 19.3 -1.9 -2.6

3.7 7.2 10.9 14.6 15.7 23.4 30.7 37.9 6.4 10.7 15.2 19.7 11.2 18.0 -2.0 -2.5

-54.8 -50.9 -46.7 -42.3 -68.3 -58.9 -49.5 -41.2 34.0 40.7 47.1 52.5 22.0 30.7 38.1 43.7

-40.2 -37.4 -34.5 -31.4 -49.1 -43.1 -36.7 -30.8 -14.6 -11.0 -7.0 -3.3 -25.8 -20.9 -16.1 -12.3

0

6.4

1.0

14.7

14.4

45.1

25.9

MUE(cationic)

Table 2. Benchmark reaction energies (in kcal/mol), signed errors of the computed reaction energies, and mean unsigned errors (MUEs) for neutral reaction energies in SIE11 Reaction

Bench-

Error

mark

CASSCF

CASPT2

tPBE

ftPBE

PBE

PBE0

ClFCl → ClClF C2H4...F2 → C2H4 + F2 C6H6Li → C6H6 + Li NH3...ClF → NH3 + ClF NaOMg → Na + MgO FLiF → Li + F2

1.0 1.1 5.7 10.5 69.6 94.4

22.4 -1.3 -38.8 -8.5 17.8 -12.2

5.0 1.9 -1.5 3.1 -2.4 1.0

-0.3 -0.8 -2.1 1.1 -4.3 -10.3

0.5 -0.7 -2.0 1.6 -2.9 -8.6

-23.0 1.5 0.2 6.1 -5.3 22.6

-10.8 0.0 0.0 2.9 -0.6 19.9

MUE (neutral)

0

16.8

2.5

3.2

2.7

9.8

5.7


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Table 3. Average mean unsigned errors (in kcal/mol) on the two SIE test sets for various methods and density functionals ref.a 35 39 23 22 43 44

Xb 100 100 100 100 100 100

Average MUEc 13.8 6.3 14.6 11.6 1.3 1.7

34,45 30 46 47 42 34,48,49

0 0 0 0 0 20

28.1 27.4 27.5 25.7 20.0 18.2

PBE0 BHLYPe ,f M08-HXf M06-2X CAM-B3LYP g LC-𝜔PBE g LC-BLYP g 𝜔B97X g M11 g HFLYP M06-HF

50 40 41 32 51 52,53 54 55 56 34,35 33

25 50 52.23 54 19–65 0–100 0–100 15.8–100 42.8–100 100 100

15.8 9.1 10.7 10.3 9.9 4.8 5.0 7.2 7.4 7.6 4.6

tPBE tBLYP ftPBE ftBLYP trevPBE ftrevPBE

11 11 29 29 57 57

0 0 0 0 0 0

8.9 9.5 8.5 9.6 9.4 8.9

Category

Method/Functional

wave-function theory

UHF UMP2 ROHF CASSCF CASPT2-0 CASPT2

KS-DFT

BLYP PBE revPBE M06-L MN12-Ld B3LYP

MC-PDFT

a reference for method or functional; b percentage of nonlocal exchange for wave function methods;

percentage of nonlocal exchange in the density functional for density functional methods; c average

of the MUE for ionic reactions of Table 1 and the MUE for neutral reactions of Table 2; d best local functional for SIE4×4 out of 27 local functionals tested in Ref. 21; e also known as BHandH; f one of the two best hybrid functionals for SIE4×4 out of 44 hybrid functionals tested in Ref. 21.

grange-separated function (for such functionals the X column shows the range, with the first number

being percentage of nonlocal exchange at small interelectronic separation, and the second number being that for large interelectronic separation.



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ASSOCIATED CONTENT

Supporting Information. Computational details, Cartesian coordinates of the molecules, and complete results including other density functionals. The Supporting Information is available free of charge on the ACS Publications website at DOI/ AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] *E-mail: [email protected] ACKNOWLEDGMENT

The authors are grateful to Aron Cohen, Paula Mori-Sanchez, and Weitao Yang for helpful discussions. This work was supported in part by the Air Force Office of Scientific Research by grant no. FA9550-16-1-0134. J. L. Bao acknowledges a Doctoral Dissertation Fellowship (DDF) provided by the University of Minnesota. References 1

Kohn, W.; Sham, L. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133−1138. 2 Kohn, W.; Becke, A. D.; Parr, R. G. Density Functional Theory of Electronic Structure. J. Phys. Chem. 1996, 100, 12974−12980. 3 Engel, E.; Dreizler, R. M. Density Functional Theory, Theoretical and Mathematical Physics; Springer-Verlag: Berlin, 2011; pp. 112–113. 4 Zhang, Y.; Yang, W. A Challenge for Density Functionals: Self-Interaction Error Increases for Systems with a Noninteger Number of Electrons. J. Chem. Phys. 1998, 109, 2604−2608. 5 Li, C.; Zheng, X.; Su, N. Q.; Yang, W. Localized Orbital Scaling Correction for Systematic Elimination of Delocalization Error in Density Functional Approximations. National Science Rev. 2017, DOI: 10.1093/nsr/nwx111. 6 Tsuneda, T.; Kamiya, M.; Hirao, K. Regional Self-Interaction Correction of Density Functional Theory. J. Comput. Chem. 2003, 24, 1592-1598. 7 Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Many-electron self-interaction error in approximate density functionals. J. Chem. Phys., 2006, 125, 201102. 8 Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Localization and Delocalization Errors in Density Functional Theory and Implications for Band-Gap Prediction. Phys. Rev. Lett., 2008, 100, 146401. 9 Pederson, M. R.; Ruszinsky, A.; Perdew, J. P. Communication: Self-Interaction Correction with Unitary Invariance in Density Functional Theory. J. Chem. Phys. 2014, 140, 121103.


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