PHYSICAL REVIEW B 86, 035130 (2012)

Semilocal dynamical correlation with increased localization

Lucian A. Constantin,1 Eduardo Fabiano,2 and Fabio Della Sala1,2

1Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia (IIT), Via Barsanti, I-73010 Arnesano (LE), Italy2National Nanotechnology Laboratory (NNL), Istituto Nanoscienze-CNR, Via per Arnesano 16, I-73100 Lecce, Italy

(Received 18 May 2012; published 20 July 2012)

Using a localization technique for the correlation energy density and a linear response constraint, we constructa localized semilocal model for dynamical correlation. The model is incorporated into a meta-generalizedgradient approximation functional which is very accurate for jellium surfaces, Hooke’s atom at all regimes,and works well both in conjunction with a semilocal exchange and together with local and nonlocal exactexchange.

DOI: 10.1103/PhysRevB.86.035130 PACS number(s): 71.10.Ca, 71.15.Mb, 71.45.Gm

Kohn-Sham (KS) density functional theory (DFT),1 themost popular method for electronic calculations in quantumchemistry and condensed-matter physics, provides in principlean exact description of many-electron systems. However, itsaccuracy depends on the approximation of the exchange-correlation (XC) energy, which describes the quantum many-body effects beyond the Hartree approximation. Thus, alarge number of XC functionals of increasing complexityhave been developed over the years,2 starting from thesimple local density approximation (LDA),1 the generalizedgradient approximations (GGAs),2,3 to more recent meta-GGAfunctionals4,5 using the noninteracting kinetic energy densityas an ingredient.

In most cases the remarkable accuracy of XC functionalsdepends on a large error cancellation between the exchange andcorrelation parts.6 More recent hyper-GGA functionals,7–11

based on the exact exchange (EXX), aimed to reduce thiserror cancellation and proved the possibility of constructingaccurate “pure” density functional correlation approxima-tions. Nevertheless, all these functionals contain an importantnumber of empirical parameters, denoting the difficulty ofsuch a task. Indeed, the goal of constructing pure correlationfunctionals is attractive in two ways. First, because a clearseparation of exchange and different correlation effects wouldgreatly aid the comprehension and functional development.Secondly, because such a correlation functional would be(more) compatible with EXX. The latter feature would allowone to solve the problems related with the Coulomb self-interaction error12 and to improve the description of numerousproperties, such as charge-transfer excitation energies.11,13

In this work we show that an important step in this directioncan be done by considering a semilocal correlation modelwith an increased localization in the energy density. Thismodel is employed to build an accurate meta-GGA dynamicalcorrelation functional which is tested both with exact andmeta-GGA exchange. Note that in the former case, becausewe do not consider here nondynamical and/or long-rangecorrelation, only simple small systems near equilibrium willbe considered.

I. THEORY

To start our analysis, we consider the widely usedPerdew-Burke-Ernzerhof (PBE)3 correlation energy per parti-

cle εPBEc = εLDA

c + H (rs,φ,t), with

H (rs,φ,t) = γφ3 ln

(1 + β

γ

t2 + At4

1 + At2 + A2t4

), (1)

where A = β/γ (exp{−εLDAc /(γφ3)} − 1)−1, εLDA

c is the cor-relation energy per particle of the uniform electron gas,14

φ = [(1 + ζ )2/3 + (1 − ζ )2/3]/2 is a spin scaling factor withζ = (n↑ − n↓)/n being the relative spin polarization,3 t =|∇n|/[φ4(3/π )1/6n7/6], γ is a constant,3 and β is fixed bythe slowly varying density behavior. In fact for t → 0 we have(by construction) εPBE

c → εLDAc + βφ3t2. In the original PBE,

β = βPBE = 0.066 725 as derived in the high-density limit,15

whereas in the revTPSS correlation functional5 the density-dependent β = βrevTPSS(rs), based on the true second-ordergradient expansion (GE2) of the correlation energy of Huand Langreth,16 was employed. Several different variants ofthe PBE correlation also exist.17–22 (Unless otherwise stated,atomic units are used throughout, i.e., e2 = h = me = 1.)

However, little attention has been devoted in literature to therapidly varying density limit (t → ∞) of the PBE correlationfunctional, beside the fact that it should go to zero.3 By Taylorexpansion in the large t limit, we find that

εPBEc → Q/β2(1/t)4 + O(1/t6), (2)

with

Q = γ 3φ3(eεLDA

c /(γφ3) − 1)3

e−2εLDAc /(γφ3) < 0 . (3)

Thus, εPBEc displays a slow decaying behavior23 depending

on β: This seems an artifact of the PBE correlation functionalform, as there is no physical reason for this β dependence att → ∞. Instead, setting

β(rs,t) = β0 + at2f (rs), (4)

with β0 the constant determining the slowly varying densitybehavior, we obtain

εPBElocc → Q/[af (rs)]

2(1/t)8 + O(1/t10), (5)

i.e., the correlation energy density decays much faster (i.e.,with a doubled power of 1/t) and the leading 1/t8 term isnot related to β0. In Eq. (4), a is a constant (to be found) andf (rs) is an appropriate function to modulate our correctionin the high- and low-density limits (see below). The PBEcorrelation functional with β given by Eq. (4) is named PBEloc(for localized correlation).

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CONSTANTIN, FABIANO, AND DELLA SALA PHYSICAL REVIEW B 86, 035130 (2012)

In considering the ansatz of Eq. (4) we require the followingconstraints:

(i) Under the uniform scaling [n(r) → γ 3n(γ r), rs ∼ γ −1,and t ∼ γ 1/2] to the high- (γ → ∞) and low- (γ → 0) densitylimits, β should recover β0, as in the PBE construction.Thus, we must have f (rs → 0) ∼ rk

s with k > 1 and f (rs →∞) must not diverge. These conditions imply a strongerlocalization of the correlation since, for a Z-electron sphericalatom, β is large in the tail of the density, where both rs and t

are large, while it recovers β0 near the nucleus where rs → 0and t ∼ Z.

(ii) Because εPBElocc is more localized than εPBE

c , it isreasonable to suppose that it must be larger (more negative)than εPBE

c in some regions, so that the two energy densitiesintegrate to roughly the same result. Thus, the two functionsmust cross for some value of t . Indeed, it is simple to seethat the cross occurs when t =

√(βPBE − β0)/af (rs) and

therefore the following condition must be satisfied: βPBE > β0

(to be used later).To satisfy all these requirements we propose the simple

ansatz f (rs) = 1 − e−r2s , and we select

β0 = βGE2−LR = 3μGE2x

π2= βrevT PSS(∞) = 0.0375, (6)

which comes from the accurate LDA linear response ofthe GE2 of exchange energy,3 and was already widelyused.3,18,21,24 Note that in this way the PBEloc functionalwill not have the exact second-order correlation coefficient.However, previous results19,21 indicate that this is not animportant constraint.

Finally, we fix the parameter a = 0.08 by fitting thePBEloc correlation functional to the jellium surface correlationenergies (see Fig. 1). Reference values are revTPSS results,5

which are close to the exact values.25,26 The final functionalperforms remarkably well for this problem, outperformingother GGAs, with a mean absolute relative error (MARE)of only 0.3%. The data in Fig. 1 also show that a PBE-likecorrelation functional with a constant β too large (βPBE) ortoo small (PBEloc at a = 0) yields quite bad jellium surfacecorrelation energies. Better results are obtained for mediumvalues of β [PBEsol (Ref. 19) and PBEint (Ref. 20) withβ = 0.046 and β = 0.052, respectively]. The ansatz in Eq. (4)

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16a

0.00

0.05

0.10

MA

RE

PBE

PBEsol

PBEint

FIG. 1. (Color online) Mean absolute relative errors of thejellium surface correlation energies (with respect to revTPPS ones),considering the bulk parameters rs = 2, 3, 4, and 6, for PBEloc asa function of the parameter a, as well as reference values for PBE,PBEsol, and PBEint.

0 20 40 60 80 100r

0

0

0.1

0.2

0.3

0.4

|(Ec-E

cappr

ox)E

c|

PBElocPBETPSSlocJSTPSSrevTPSS

FIG. 2. (Color online) Absolute relative errors of several correla-tion functionals, for the Hooke’s atom at different classical electrondistances r0 = (ω2/2)−1/3, with ω the frequency of the isotropicharmonic potential.

can achieve even better results than PBEint, modulating thevalues of β in different regions.

Having at hand the PBEloc correlation functional we canproceed further and build a corresponding meta-GGA func-tional. To do so we consider the TPPS functional form4 andsimply replace PBE with PBEloc in it: The resulting functionalis named TPSSloc. The spin-independent parameters d andC(0,0) are fixed, as was done in the case of the JS (JelliumSurface) meta-GGA functional,26 by fitting to jellium surfacesand the Hooke’s atom,27 finding d = 4.5 and C(0,0) = 0.35.This procedure guarantees that the functional is very accuratefor jellium surfaces (MARE of only 2%) as well as for theHooke’s atom at any frequency ω (see Fig. 2). We note that inthe Hooke’s atom, the electrons are strongly correlated at largevalues of r0 and tightly bound at small values of r0. Moreover,the density is more diffuse than in the He atom28 providing amodel for chemical bonding (in the tightly bounded regime)or for low-density strongly correlated systems (at high valuesof r0). The simultaneous description of all these regimes posesa hard challenge for density functionals. At GGA level it is notpossible to have accurately both jellium surfaces and Hooke’satoms. This can be instead obtained at the meta-GGA level.26

Another important feature of both PBEloc and TPSSloc isthat, unlike the other functionals, their performance is betterfor small r0, i.e., for high confinement, which is consistentwith the increased localization.

The localization effect in the PBEloc and TPSSloc corre-lation functionals is visualized in Fig. 3, where we report aradial plot of εc for a Ne atom. Both PBEloc and TPSSlocare similar to PBE and TPSS for r < 1, then show a strongerlocalization at larger distances. The correlation energies areinstead all similar with respect to the reference one (−0.3905Ha): −0.3544 (TPSS), −0.3539 (TPPSloc), −0.3513 (PBE),−0.3652 (revTPSS), and −0.3629 (PBEloc).

II. RESULTS

We verified the performance of the PBEloc and TPSSlocfunctionals together with Hartree-Fock (HF) and the localizedHartree-Fock (LHF)29 exchange, for atomization energies[AE6 test (Ref. 30], barrier heights [BH6 test (Ref. 30],kinetics [K9 test (Refs. 30 and 31)], and hydrogen bonds

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SEMILOCAL DYNAMICAL CORRELATION WITH . . . PHYSICAL REVIEW B 86, 035130 (2012)

0 0.3 0.6 0.9 1.2 1.5 1.8Radial distance (bohr)

-0.08

-0.06

-0.04

-0.02

0

ε c (H

a) PBElocPBETPSSlocTPSS

FIG. 3. (Color online) Radial plot of εc for the Ne atom fordifferent functionals. The revTPSS curve (not reported) is almostsuperimposable with the TPSS one.

[HB6 (Ref. 32)]. We recall that the LHF method employs analmost exact EXX local KS potential, while HF uses a nonlocalexchange operator. LHF and HF yield very close exchange-only total energies (difference of about 5–10 kcal/mol forthe systems considered in this work). All calculations wereperformed with the TURBOMOLE program33 using the def2-TZVPP basis set34 and full-self-consistent densities. Theresults for the mean absolute error (MAE) are reported inTable I, together with other reference GGAs and meta-GGAs.For the AE6 test, comparing (L)HF + PBE and (L)HF +PBEloc, we see that the localization reduces the MAE by25% (for both HF and LHF). The improvement with respect toLee-Yang-Parr (LYP) GGA (Ref. 36) (explicitly constructedfor post-HF correlation) is even more significant (≈35%for both HF and LHF). Note that the MAE obtained byHF + PBEloc for the AE6 test is almost the absolute minimumfor this functional. In fact, a reoptimization of a to minimizethe MAE of AE6 yields a = 0.1 and MAE = 23.5 kcal/mol,validating the use of the jellium surface reference model forthe parametrization.

For barrier heights and kinetics all functionals are extremelyaccurate (as they are based on the HF density8,37) outper-forming meta-GGA XC approaches and competing with con-

TABLE I. Mean absolute errors in kcal/mol for atomization en-ergy (AE6), barrier heights (BH6), kinetics (K9), and hydrogen bond(HB6) tests, as computed with Hartree-Fock exchange complementedwith different GGA correlation functionals. The best value, for eachline, is shown in boldface. In parentheses the results of differentsemilocal correlation functionals plus the localized Hartree-Fockexchange are reported. Full results are reported in Ref. 35.

Test set (L)HF + LYP (L)HF + PBE (L)HF + PBEloc

AE6 38.2 (41.3) 31.9 (35.4) 24.0 (26.7)BH6 5.3 (7.2) 5.6 (7.8) 4.4 (7.4)K9 6.0 (7.0) 5.7 (7.4) 4.7 (6.9)HB6 2.3 (1.7) 1.5 (1.0) 1.7 (1.2)

Test set (L)HF + TPSS (L)HF + revTPSS (L)HF + TPSSlocAE6 29.1 (31.4) 30.0 (31.0) 25.5 (27.4)BH6 4.7 (6.7) 5.1 (6.9) 3.9 (6.6)K9 5.6 (6.7) 6.1 (6.8) 4.3 (6.4)HB6 1.4 (0.9) 1.5 (1.0) 1.6 (1.1)

ventional hybrids.30,38 Nevertheless, significantly improvedresults are obtained for the localized PBEloc functional whencomparing with the PBE one. In a similar way, HF + TPSSlocimproves over HF + TPSS and HF + revTPSS (see secondhalf of Table I) for AE6, BH6, and K9. Note that for AE6,BH6, and K9 the MAE for LHF is only a few kcal/mol largerthan HF; the full EXX results can be expected to be in betweenLHF and HF.29

For hydrogen bonds, where longer bond distances areinvolved, the localization might be expected to be less accurate.Instead, the PBEloc and TPSSloc functionals perform verysimilarly to the other ones (but LYP, which is the worst),with differences of only 0.1–0.2 kcal/mol, i.e., below theexpected accuracy of the test. This shows the robustness ofthe localization model.

The results in Table I indicate that PBEloc and TPSSlochave a better compatibility than other (meta)GGAs withexact exchange. However, errors in atomization energies arestill quite high. In addition, bond lengths or systems withtransition metals are poorly described (see also Ref. 39).These facts originate from the neglect of most nondynamicalcorrelation, which is instead (approximatively) included inGGA/meta-GGA functionals through the exchange part.6

Even if the exchange part of an XC functional is defined40

by Ex = limγ→∞ γ −1Exc[nγ ], where nγ (r) = γ 3n(γ r) is theuniformly scaled density, it may contain the nondynamicalcorrelation that uniformly scales as exchange.

Concerning PBEloc we found, as expected, that it is nomore compatible with PBE exchange: the MAE of AE6 in-creases to 23.0 kcal/mol (PBEx + PBEc gives 14.5 kcal/mol).However, using the revPBE exchange,41 which reproducesbetter exchange-only calculations, we found 7.1 kcal/mol(with PBEloc) with respect to 9.0 kcal/mol (with PBEc). Thisresult confirms that PBEloc works better in combination with(more) accurate exchange functionals.

Then we tested the TPSSloc correlation functional inconjunction with the revTPSS exchange,5 which is one of thebest nonempirical semilocal models for exact exchange, andis also accurate for jellium exchange surface energies, thusthe XC TPSSloc functional is accurate for jellium surfaces.In this case we considered, in addition to the AE6, BH6,and HB6 tests, also more difficult tests, such as transitionmetal atomization energies [TM10 (Ref. 38)], bond lengths[MGBL19 (Ref. 42)], kinetics [K9 (Refs. 30 and 31)],ionization potential [IP13 (Ref. 39)], proton affinities [PA12(Ref. 43)], and reaction energies [OMRE (Ref. 38)]. TPSSlocsystematically improves over the revTPSS or TPSS for systemswhere the exact XC hole is localized (atomization energies,kinetics, and bond lengths of small molecules), being slightlyworse than revTPSS only for barrier heights (but still as goodas TPSS). These results indicate that the TPSSloc meta-GGAcorrelation functional is a useful tool, and can be further usedin functional developments.

III. DISCUSSION AND CONCLUSIONS

The results of Table I indicate how the localization proce-dure improves the compatibility of the correlation functionalwith the exact exchange, significantly reducing the errorsfor most tests. This can be traced back to reduced error

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CONSTANTIN, FABIANO, AND DELLA SALA PHYSICAL REVIEW B 86, 035130 (2012)

TABLE II. MAE in kcal/mol (mA for MGBL19) for several testsets, as computed at the meta-GGA level. The best values are shownin boldface. Full results are reported in Ref. 35.

Test set TPSS revTPSS TPSSloc

AE6 5.4 6.6 3.9BH6 8.3 7.4 8.6HB6 0.6 0.6 0.6K9 7.0 7.2 6.5TM10 10.7 11.1 10.5MGBL19 6.9 7.4 6.8IP13 3.1 2.9 3.0PA12 4.7 4.8 3.8OMRE 8.0 10.2 7.9

cancellation between the HF energy density, which is longrange, and the increased localization of the correlation. Infact, the very similar performance of PBEloc and TPSSloc,when used with HF exchange, indicates that in this sensethe localization construction is even more important than theone-electron self-correlation error.

Moreover, the results in Table II indicate that, whencombined with a meta-GGA exchange, our TPSSloc is avery promising correlation functional, as it improves overrevTPSS for many properties. Thus, the localized model can

be used to construct a meta-GGA correlation functional, withminimal empiricism, that is significantly more compatiblewith Hartree-Fock exchange than other popular approxi-mations and very accurate for many properties includingjellium surfaces and the Hooke’s atom at all confinementregimes. Further improvement of atomization energies canbe obtained considering a spin-dependent gradient correction(as in Ref. 22).

Before concluding we must stress that the present workis concerned only with short-range dynamical correlation andcannot be expected to account for long-range (tail) correlationeffects.12 In fact, the semilocal XC hole model44 is localized,and thus for finite systems where the XC hole becomesdelocalized (e.g., stretched molecules far from the equilibriumpoint), semilocal functionals fail.12 Nevertheless, the presentapplication to small organic systems indicates that the local-ization construction may open a path towards a nonempirical(or less empirical) semilocal correlation functional for theexact exchange, as well to further the improvement of hybridfunctionals.

ACKNOWLEDGMENTS

This work was partially funded by the ERC Starting GrantFP7 Project DEDOM (No. 207441). We thank TURBOMOLEGmbH for the TURBOMOLE program package.

1W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).2G. E. Scuseria and V. Staroverov, in Theory and Applicationsof Computational Chemistry:The First Forty Years, edited byC. Dykstra et al. (Elsevier, New York, 2005).

3J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865(1996).

4J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys.Rev. Lett. 91, 146401 (2003).

5J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin, andJ. Sun, Phys. Rev. Lett. 103, 026403 (2009).

6N. C. Handy and A. J. Cohen, Mol. Phys. 99, 403 (2001); E. J.Baerends and O. V. Gritsenko, J. Phys. Chem. A 101, 5383 (1997).

7J. Tao, V. N. Staroverov, G. E. Scuseria, and J. P. Perdew, Phys.Rev. A 77, 012509 (2008); J. P. Perdew, V. N. Staroverov, J. Tao,and G. E. Scuseria, ibid. 78, 052513 (2008).

8R. Haunschild, M. M. Odashima, G. E. Scuseria, J. P. Perdew, andK. Capelle, J. Chem. Phys. 136, 184102 (2012).

9A. D. Becke and E. R. Johnson, J. Chem. Phys. 127, 124108 (2007).10A. J. Cohen, P. Mori-Sanchez, and W. Yang, J. Chem. Phys. 126,

191109 (2007).11Y. Zhao and D. G. Truhlar, J. Phys. Chem. A 110, 13126 (2006).12A. J. Cohen, P. Mori-Sanchez, and W. Yang, Chem. Rev. 112, 289

(2012).13A. Dreuw, J. L. Weisman, and M. Head-Gordon, J. Chem. Phys.

119, 2943 (2003).14J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).15S.-K. Ma and K. A. Brueckner, Phys. Rev. 165, 18 (1968).16C. D. Hu and D. Langreth, Phys. Scr. 32, 391 (1985).17X. Xu and W. A. Goddard III, J. Chem. Phys. 121, 4068 (2004);

M. Swart, M. Sola, and F. M. Bickelhaupt, ibid. 131, 094103(2009); A. J. Thakkar and S. P. McCarthy, ibid. 131, 134109 (2009);L. Chiodo, L. A. Constantin, E. Fabiano, and F. Della Sala, Phys.Rev. Lett. 108, 126402 (2012).

18L. S. Pedroza, A. J. R. da Silva, and K. Capelle, Phys. Rev. B 79,201106 (2009); P. Haas, F. Tran, P. Blaha, L. S. Pedroza, A. J. R. daSilva, M. M. Odashima, and K. Capelle, ibid. 81, 125136 (2010).

19J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E.Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett.100, 136406 (2008).

20E. Fabiano, L. A. Constantin, and F. Della Sala, Phys. Rev. B 82,113104 (2010).

21L. A. Constantin, E. Fabiano, S. Laricchia, and F. Della Sala, Phys.Rev. Lett. 106, 186406 (2011).

22L. A. Constantin, E. Fabiano, and F. Della Sala, Phys. Rev. B 84,233103 (2011).

23J. P. Perdew, J. Tao, V. N. Staroverov, and G. E. Scuseria, J. Chem.Phys. 120, 6898 (2004).

24E. Fabiano, L. A. Constantin, and F. Della Sala, J. Chem. TheoryComput. 7, 3548 (2011).

25B. Wood, N. D. M. Hine, W. M. C. Foulkes, and P. Garcia-Gonzalez,Phys. Rev. B 76, 035403 (2007).

26L. A. Constantin, L. Chiodo, E. Fabiano, I. Bodrenko, and F. DellaSala, Phys. Rev. B 84, 045126 (2011).

27M. Taut, Phys. Rev. A 48, 3561 (1993).28J. Katriel, S. Roy, and M. Springborg, J. Chem. Phys. 124, 234111

(2006).29F. Della Sala and A. Gorling, J. Chem. Phys. 115, 5718

(2001).

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SEMILOCAL DYNAMICAL CORRELATION WITH . . . PHYSICAL REVIEW B 86, 035130 (2012)

30B. J. Lynch and D. G. Truhlar, J. Phys. Chem. A 107, 8996 (2003).31B. J. Lynch and D. G. Truhlar, J. Phys. Chem. A 107, 3898 (2003).32Y. Zhao and D. G. Truhlar, J. Chem. Theory Comput. 1, 415

(2005).33TURBOMOLE V6.3, 2011, a development of University of Karlsruhe

an Forschungszentrum Karlsruhe GmbH, 1989–2007, TURBOMOLE

GmbH since 2007 [http://www.turbomole.com].34F. Weigend, F. Furche, and R. Ahlrichs, J. Chem. Phys. 119, 12753

(2003); F. Weigend and R. Ahlrichs, Phys. Chem. Chem. Phys. 7,3297 (2005).

35See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.86.035130 for the results of all tests.

36C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988).37B. G. Janesko and G. E. Scuseria, J. Chem. Phys. 128, 244112

(2008).38E. Fabiano, L. A. Constantin, and F. Della Sala, Int. J. Quantum

Chem. (2012), doi:10.1002/qua.24042.39Y. Zhao and D. G. Truhlar, Theor. Chem. Acc. 120, 215 (2008).40M. Levy, Phys. Rev. A 43, 4637 (1991).41Y. Zhang and W. Yang, Phys. Rev. Lett. 80, 890 (1998).42Y. Zhao and D. G. Truhlar, J. Chem. Phys. 125, 194101 (2006).43L. Goerigk and S. Grimme, J. Chem. Theory Comput. 6, 107 (2010).44L. A. Constantin, J. P. Perdew, and J. M. Pitarke, Phys. Rev. B 79,

075126 (2009).

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