Performance of DFT: The good, the bad, and the ugly

Performance of DFT: The good, the bad, and the ugly

Structures and frequencies, electronic densities;Thermochemical properties: reaction energies and barriers;Potential energy curves (for bond breaking);Charged systems (H2+ example);Weak interactions and dispersion;Excited states (will discuss later).

Thursday, November 20, 2014

TABLE 3: Summary of the Benchmarking Studies on Density Functionals Published during the Last 4 Years, with Major Emphasis Being Given to Performance of B3LYP and ItsRanking among the Density Functionals Tested, and to the Best Functional for Each Property in Each Study

the studyB3LYP

MUE rankingbest results

functional (MUE) dataset size dataset characteristicsav. value inthe dataseta basis set ref

StructureA. bond lengths (Å) 0.008 3/6 BPW91 (0.007) 26 bond lengths 17 first-row closed shell molecules 1.493 Aug-cc-PV5Z 105

0.06 12/12 MPW1K (0.01) 10 bond lengths 5 saddle-point geometries of H-transfer reactions 1.17 MG3S 700.008 1/5 B3LYP (0.008) 44 bond lengths 32 first-row closed shell molecules 1.186 6-311+G** 1060.014 5/42 MPW3LYP (0.013) 13 bond lengths 13 metal compounds 1.689 DZQ/TZQ 670.16 27/42 SVWN3 (0.05) 8 bond lengths 8 metal compounds 2.11 DZQ/TZQ 1070.007 2/37 VSXC (0.006) 71 bond lengths 44 small organic molecules n/a Aug-cc-pVQZ 108

B. angles (deg) 0.75 1/6 B3LYP (0.75) 10 bond angles 10 first-row closed shell molecules 108.99 Aug-cc-pV5Z 1051.94 5/6 c-SVWNV (1.82) 16 bond angles 12 first-row closed shell molecules 110.67 6-311+G** 1061.20 11/37 PBE1PBE (1.11) 34 bond angles 27 small organic molecules n/a Aug-cc-pVQZ 108

C. H bonds (Å) 0.02 7/15 MPW3LYP (0.01) 4 distances 4 H-bonding dimers 2.81 MG3S 70D. weak interactions (Å) 1.02 15/15 B97-1 (0.08) 4 distances 4 rare-gas dimers 3.27 MG3S 70

Kinetics (kcal/mol)A. barrier heights 4.7 11/22 MPW1K (1.4) 6 barrier heights representative dataset of 6 H-transfer reactions 11.85 MG3S 109

4.31 14/15 BB1K (1.16) 42 barrier heights mostly open-shell H-transfer reactions 13.96 MG3S 703.04 15/25 BB1K (1.50) 76 barrier heights 38 H-transfer reactions + 38 non-H-transfer reactions 18.12 MG3S 804.73 17/42 BB1K (1.14) 6 barrier heights A representative dataset of 6 H-transfer reactions 11.85 MG3S 1074.50 17/29 BB1K (1.37) 76 barrier heights 38 H-transfer reactions + 38 non-H-transfer reactions 18.12 MG3S 684.30 7/37 BB1K (1.05) 23 barrier heights 23 small radical transition-state reactions 12.29 Aug-cc-pVTZ 1083.10 2/37 B1LYP (2.58) 6 barrier heights 6 large singlet transition-state reactions 27.55 Aug-cc-pVTZ 108

Thermochemistry (kcal/mol)A. atomization energies 2.19 1/6 B3LYP (2.19) 17 compounds 17 first-row closed shell molecules 270.15 Aug-cc-pV5Z 105

0.7 5/22 VSXC (0.5) 6 compounds 6 organic molecules 517.22 MG3S 1090.90 9/15 X1B95 (0.52) 109 compounds 109 organic and inorganic molecules 497.65 MG3S 700.91 12/25 PW6B95 (0.40) 109 compounds 109 organic and inorganic molecules 497.65 MG3S 800.61 4/42 MPW3LYP (0.43) 6 compounds 6 organic molecules 517.22 MG3S 10716.7 29/42 BLYP (5.3) 9 compounds 9 metal dimers 56.8 DZQ/TZQ 10726.3 27/42 BLYP (5.8) 9 compounds 9 metal dimers 56.8 DZQ/TZQ 67

B. binding energies 6.5 10/42 TPSS1KCIS (5.4) 21 compounds 21 transition metals 82.7 DZQ/TZQ 6712.0 7/7 M05 (7.8) 18 compounds 18 transition metals 65.5 QZVP 110

C. ionization potentials 4.72 14/15 MPWB1K (2.05) 13 compounds 6 atoms and 7 molecules 253.84 MG3S 704.72 18/25 MPWB1K (2.05) 13 compounds 6 atoms and 7 molecules 253.84 MG3S 803.8 1/6 B3LYP (3.8) 88 compounds 88 atoms and molecules from the G3 test set 253.13 6-311+G** 1067.2 27/42 OLYP (3.1) 7 compounds 7 atoms (including 5 metals) 202.6 DZQ/TZQ 675.1 13/37 B1B95 (4.25) 37 compounds derived from the Gaussian G2/97 test set n/a Aug-cc-PVTZ 108

D. electron affinities 2.29 4/15 B98 (1.84) 13 compounds 6 atoms and 7 molecules 38.16 MG3S 702.29 7/25 PW6B95 (1.78) 13 compounds 6 atoms and 7 molecules 38.16 MG3S 803.4 3/5 c-SVWNV (2.6) 58 compounds 58 atoms and molecules from the G3 test set 32.55 6-311+G** 1063.5 3/37 B98 (3.2) 25 compounds derived from the Gaussian G2/97 test set n/a Aug-cc-PVTZ 108

E. heats of formation 27.1 30/42 mPWPW91 (9.5) 372 compounds 372 containing 5 or fewer heavy atoms 45.9 MIDI! 11117.8 2/6 BLYP (15.0) 223 compounds 223 molecules from the G3 test set 50.9 6-311+G** 1067.0 8/37 B3PW91 (3.95) 156 compounds derived from the Gaussian G2/97 test set n/a Aug-cc-PVTZ 1083.31 3/23 B98 (2.90) 148 compounds 148 molecules from the G2/97 test set 49.0 6-311+G(3d2f,2p) 112

E. isomerizations 6.24 17/25 M05 (1.84) 3 pairs of compounds 3 pairs of cumulenes and poly-ynes Isomers 8.17 6-311+G(2df,2p) 113

ReviewArticle

J.Phys.Chem

.A,Vol.111,No.42,2007

10445

Dow

nloa

ded

by U

NIV

JOSE

PH F

OU

RIER

GRE

NO

BLE

on S

epte

mbe

r 25,

200

9 | h

ttp://

pubs

.acs

.org

P

ublic

atio

n D

ate

(Web

): A

ugus

t 25,

200

7 | d

oi: 1

0.10

21/jp

0734

474


mean

unsignederror(M

UE)w

asobtainedwithBPW

91(0.007

Å,for

theaug-cc-pV

5Zbasis

set),closelyfollow

edbyBP86

andB3LY

P(MUE)0.008

Å).

Zhaoand

Truhlar 70have

evaluatedthe

performance

of12

densityfunctionals

inthe

determination

ofsaddlepointgeom

-etries,considering

adatasetoffive

hydrogen-transferreactions(10

bondlengthsin

total),forwhich

very-highlevelcalculations

ofsaddle

pointgeom

etrieswere

available.Inthis

study,thebestresults

inthe

determination

ofbondlengths

(MG3Sbasis

set)were

obtainedwith

MPW

1K(MUE

)0.01

Å).X

B1K,

BB1K,MPW

B1Kalso

renderedvery

goodresults(M

UE)0.02

Å),w

hereasB3LYP(MUE)0.06

Å)gave

theworstM

UEof

thesetof12

densityfunctionals

tested.Riley

etal. 106havecom

paredsixdensity

functionals(Slater,SVWNV,BLY

P,B3LYP,and

c-SVWNV)in

thedeterm

inationofbond

lengths,considering

adataset

of32

small

neutralmolecules(44

experimentally

determined

bondlengths).In

thisstudy

(basisset6-311+G**),B3LY

Pgave

thebestresults,w

ithaMUEof0.008

Å,and

wasfollow

edbyc-SV

WNVand

BLYP

(MUEof0.013

and0.015

Å).

Theperform

anceof42functionals

inthe

calculationof13

metal-

ligandbond

lengthswasrecently

analyzedbySchultz

etal. 67The

molecules

usedfor

bondlength

comparison

were

AgH,BeO

,CoH,CoO

+,FeO,FeS,LiCl,LiO

,MgO,RhC,V

O,

andVS.In

thedetermination

ofmetal-

ligandbond

lengthswith

bothdouble-!

quality(DZQ)and

atriple-!

quality(TZQ

)basissets,the

bestresultswere

obtainedwithMPW

3LYP,X

3LYP,

TPSSh,andTPSS1K

CIS(MUE)0.013

Å),closely

followed

byB3LY

P(MUE)0.014

Å),w

hichwasranked

fifthinthe

test.Schultzetal. 107

havetested

theability

of42

densityfunctionalsin

geometry

determination

for8metaldim

ers(Ag2 ,

Cr2 ,Cu2 ,CuA

g,Mo2 ,N

i2 ,V2 ,and

Zr2 )withDZQ

andaTZQ

basissets.

Thelow

estaverage

MUEinbond

lengthswas

obtainedwith

SPWLand

SVWN3(MUE

)0.05

Å).In

theGGAmethods,BP86,G

96LYP,m

PWLYP,and

X3LY

Pwere

themostsuccessful(M

UE)0.07

Å),w

hereasBB95cam

efirst

among

theM-GGAs(MUE)0.07

Å),and

B97-2among

theH-GGAs(MUE)0.15

Å).B1B95,TPSSh,and

TPSS1KCIS

gavethe

bestresultsamong

theHM-GGAs(MUE)0.14

Å).

B3LYPresulted

inanMUEof0.16

Å(27th

positionoutof42

densityfunctionalsin

thetest).These

resultshighlightamarked

differencebetw

eenthe

MUEstypically

encounteredforsm

allorganic

molecules

(asdescribed

inWang

andWilson

105),andthe

onescalculatedform

etalcompoundsSchultz

etal. 107Eventhough

thesetwosetsofresultsare

notdirectlycom

parable,asthe

basissets

employed

largelydiffer

(aug-cc-pV5Z

inWang

andWilson,and

DZQ

andTZQ

inSchultz

etal.),the

errorinvolved

ingeom

etrydeterm

inationofm

etalsystemsisnorm

allyhigherthan

theoneassociatedwithcalculationsofsm

allorganicmolecules.Arecentstudy

byRiley

etal. 108has

compared

theperfor-

mance

of37density

functionalsinthe

determination

of71bond

lengthsof44smallorganic

moleculesw

hosestructuresare

well

characterizedexperim

entally.Inthisstudy,thebestperform

ancewiththeaug-cc-pV

QZwasobtained

withVSXC(MUE)0.006

Å),with

B3LYPclosely

following

(MUE

)0.007

Å)and

yieldingthe

secondbestresult.

Angles.Wang

andWilson

105haveevaluated

theperform

anceofsix

well-established

densityfunctionals

(B3LYP,B3PW

91,B3P86,

BLYP,BPW

91,and

BP86)inthe

determination

ofmoleculargeom

etries.Inparticular,a

totalof10experim

entallyderived

bondangles

offirst-row

closed-shellmolecules

ofrelevance

inatm

osphericchem

istrywere

considered.Thebest

TABLE 3 (Continued)the study

B3LYPMUE ranking

best resultsfunctional (MUE) dataset size dataset characteristics

av. value inthe dataseta basis set ref

Nonbonded Interactions Ib (kcal/mol)A. H bonding 0.45 4/15 B97-1 (0.26) 4 binding energies 4 H-bonding dimers 9.38 MG3S 70

1.94 6/6 PWB6K (0.46) 5 binding energies 5 π H-bonding systems 4.49 MG3S 1140.77 21/44 B3P86 (0.46) 6 binding energies 6 H-bonding dimers 8.37 DIDZ/aug-cc-pVTZ/MG3S 710.76 18/25 PBE1PBE (0.34) 6 binding energies 6 H-bonding dimers 8.37 MG3S 800.65 9/37 MPWLYP (0.31) 10 binding n energies 10 H-bonding systems 7.15 Aug-cc-pVTZ 108

B. charge transfer 0.80 9/44 MPWB1K (0.50) 7 binding energies 7 charge-transfer complexes 4.63 DIDZ/aug-cc-pVTZ/MG3S 710.63 9/25 PWB6K (0.21) 7 binding energies 7 charge-transfer complexes 4.63 MG3S 80

C. dipole interactions 0.78 29/44 PBE1KCIS (0.36) 6 binding energies 6 dipole-interaction complexes 3.05 DIDZ/aug-cc-pVTZ/MG3S 710.86 20/25 PWB6K (0.28) 6 binding energies 6 dipole-interaction complexes 3.05 MG3S 80

Nonbonded Interactions IIc (kcal/mol)A. weak interactions 0.23 15/15 B97-1 (0.02) 4 binding energies 4 rare-gas dimers 0.08 MG3S 70

0.60 32/44 B97-1 (0.19) 9 binding energies 9 weak-interaction complexes 0.47 DIDZ/aug-cc-pVTZ/MG3S 710.35 21/25 B97-1 (0.10) 7 binding energies 7 weak-interaction complexes 0.22 MG3S 80

M05-2X (0.12) 10 binding energies 10 rare-gas dimers 0.16 Aug-cc-pVTZ 115B. π-π interactions 8.52 6/6 PWB6K (1.86) 11 binding energies 6 nucleic acid bases complexes + 5 amino acid pairs 7.53 DIDZ 72

3.06 24/25 PWB6K (0.90) 5 binding energies 5 π-π stacking complexes 2.02 MG3S 80a Average value of the property in the dataset used for evaluation. b Nonbonded interactions in which electrostatic or orbital-orbital interactions are dominant and the dispersion contribution is small.

c Nonbonded interactions in which the dispersion contribution is dominant.

10446J.Phys.C

hem.A,Vol.111,N

o.42,2007Sousa

etal.

Downloaded by UNIV JOSEPH FOURIER GRENOBLE on September 25, 2009 | http://pubs.acs.org Publication Date (Web): August 25, 2007 | doi: 10.1021/jp0734474


their respective minima and drawn to large interatomic distances,as it is done in the two panels in Figure 1. These show the(essentially correct) curves obtained by UHF (H2•+) or CCSD-(T) calculations (He2•+) at the top and those obtained by theB(LYP) functional at the bottom. The two curves in the middlecorrespond to admixture of increasing amounts of HF exchangedensity (B3LYP and BeckeH&H, respectively). After tracingthe bottom of the potential energy wells rather accurately, theDFT “dissociation” curves go through fictitious transition states(cf. Tables 1 and 2) before converging very slowly to acompletely unrealistic fragment energy, which may even liebelow the binding energy of the original cation.Inspection of the spin and charge distribution from the DFT

calculations reveals that it remains completely uniform fromthe equilibrium distance to infinity, thus confirming our recentexperience with the acetylene dimer cation where we had notedthe same odd behavior.2 In contrast, localization of spin andcharge in one of the atoms of He2•+ occurs between theinflection point of the potential energy curve and the dissociationlimit.10 Interestingly, when comparing the charge and spindistribution obtained by UHF and by a high-level correlatedmethod (QCISD), one finds, as expected, that correlation delaysthe onset of charge and spin localization, as found previouslyfor acetylene dimer cation.2 It therefore appears that DFTmethods continue to correlate the motions of the electrons wherethis should cease to be of energetic advantage.However, if a DFT calculation is started from a UHF density

matrix for a localized wave function, it converges also to thatlocalized solution for large interatomic distances. Series of suchcalculations result in curves that give a qualitatively correctpicture; i.e., they converge to the dissociation limit for eachmethod, but these curves lie invariably at higher energy thanthose depicted in the two plots. Hence, for large interatomicdistances, the (converged!) localized Kohn-Sham wave func-tions are unstable with regard to equalization of spin and charge

and thus DFT methods appear to be prone to what might becalled “inverse symmetry breaking”.But even if we disregard the strange dissociation behavior

depicted in Figure 1, the De values obtained by taking the energydifference between the fragment atoms or ions and the molecularradical cations are in much poorer agreement with experimentthan HF (H2•+) or CCSD(T) (He2•+). In Tables 1 and 2 we listre, ωe, and D0 values obtained by the different methods as wellas rT and ωT, the distances and imaginary frequencies at thefictitious transition states obtained by DFT. The very poorprediction of D0 obtained at the BLYP level is improvedsomewhat by admixture of increasing amounts of HF densitybut is still about 0.5 eV from the experimental value when thedensity is 50% HF.We also performed some exploratory calculations to delineate

the scope of the problem we encountered with the DFT methods.It occurs likewise in homonuclear radical anions (F2-), but notin unsymmetric radical ions (LiH•+, CH3OH•+ dissociating intoOH• + CH3+) or closed-shell cations (HeH+) or unchargedradicals (BeH•). It seems that a strong inherent dissymmetryputs a strongly delocalized form at a disadvantage even in DFTmethods. However, our experience with larger organic radicalcations shows that if the dissymmetry is not pronounced thetendency of DFT for delocalization of spin and charge prevails.11

TABLE 1: Equilibrium Bond Lengths (re), HarmonicFrequencies (ωe), and Bond Dissociation Energies (D0) asWell as Bond Lengths (rT) and Harmonic Frequencies (ωT)of the “Fictitious Transition States” of H2•+ with thecc-pVQZ Basis Setmethod re/Å ω�/cm-1 D0/kcal‚mol-1 rT/Å ωT/cm-1

exptla 1.052 2321.7 61.13HF 1.0571 2323.8 61.04bBLYP 1.1357 1876.7 66.42 2.9586 -459.7B3LYP 1.1142 1997.4 65.02 3.2861 -387.9BH&H 1.0890 2135.1 63.38 3.8554 -272.4

a Huber, K. P.; Herzberg, G. Constants of Diatomic Molecules; VanNostrand Reinhold: New York, 1979. bMost of the difference betweenthe experimental and calculated D0 is due to the neglect of anharmo-nicity.

TABLE 2: Equilibrium Bond Lengths (re), HarmonicFrequencies (ωe), and Bond Dissociation Energies (D0) asWell as Bond Lengths (rT) and Harmonic Frequencies (ωT)of the “Fictitious Transition States” of He2•+ with thecc-pVQZ Basis Set

method re/Å ω�/cm-1D0/

kcal‚mol-1 rT/Å ωT/cm-1

exptla 1.0806 1698.5 54.62bUHF 1.0757 1737.7 43.03UCCSD(T) 1.0806 1701.2 54.33BLYP 1.1830 1192.8 81.58 2.1586 -372.3B3LYP 1.1454 1359.9 75.46 2.3677 -320.4BH&H 1.1062 1555.9 67.00 2.7601 -227.7

a Huber, K. P.; Herzberg, G. Constants of Diatomic Molecules; VanNostrand Reinhold: New York, 1979. b Combination of the bestcalculated value for De5 with the experimental zero-point energy.

Figure 1. Dissociation curves for H2•+ (upper panel) and He2•+ (lowerpanel) as obtained at the levels indicated under the curves. Allcalculations were done with Dunning’s cc-pVQZ basis set. In the lowerpanel, the solid and dashed line mark the percentage of spin and chargeon one of the two He atoms as a function of the interatomic distance(50% indicates full delocalization, 0% full localization) as obtained atthe UHF (dashed) and QCISD levels (solid).

7924 J. Phys. Chem. A, Vol. 101, No. 43, 1997 Letters

LETTERS

Incorrect Dissociation Behavior of Radical Ions in Density Functional Calculations

Thomas Bally* and G. Narahari SastryInstitut de Chimie Physique de l’UniVersite, Perolles, CH-1700 Fribourg, Switzerland

ReceiVed: July 23, 1997; In Final Form: September 17, 1997X

The current lineup of popular density functional theories, in particular those based on Becke’s exchangefunctionals, fail to predict a correct dissociation behavior in radical ions where charge and spin must beseparated (model: H2•+) or where both must be localized on one fragment (model: He2•+). The repercussionsof this on the location of certain transition states on radical ion potential energy surfaces are pointed out.

1. IntroductionIn the course of our ongoing work on rearrangement of

organic radical cations,1 we encountered unusual difficulties inlocating transition states for certain simple reactions when usingdensity functional theory (DFT) based methods that otherwiseproved to be very successful in these types of applications. Asthe same transition states were often relatively easy to find byHartree-Fock based methods, we began to ask ourselves aboutthe origin of these difficulties. A closer examination revealeda common feature of such elusive structures, i.e., that theyrequired either a separation of spin and charge in different (say,allylic) moieties or a localization of spin and charge in one part(say, a double bond) of a radical ion. Apparently, the DFTmethods available to us “refused” to carry out such separationsand localizations in cases where these were clearly required.Independently, we found that a similar feature of DFT

methods comes to bear when radical ions are dissociated intofragments, either by separating spin and charge or by localizingthem on one of the fragments. Thus, we were unable to modelthe loosely bound region of the acetylene dimer cation becauseeven at very large distances, spin and charge remainedcompletely delocalized over both acetylene moieties.2 This ledus to investigate the nature of this problem on the example ofthe most simple model cases, i.e., H2•+ and He2•+, the formerof which has the additional advantage that electron correlationeffects are entirely absent. This Letter reports the results ofthis model study, which should serve to attract the attention ofthe authors of exchange functionals to this particular problemin the hope of remedying it in future versions.

2. MethodsReference calculations for H2•+ were performed by the

unrestricted Hartree-Fock (UHF) method using Dunning’squadruple-� basis set,3 which yields De in excellent agreementwith experiment (cf. Table 1). For He2•+, dynamic electroncorrelation was accounted for by the coupled cluster methodincluding all single and double excitations augmented by anoniterative estimate of the contribution of connected tripleexcitations (CCSD(T)).4 This reproduces re and ωe withinexperimental error and gives a value of De only 0.3 kcal/molabove a recent large MRCI calculation.5 DFT calculations werecarried out using Becke’s gradient-corrected exchange func-tional6 as well as the hybrid three-parameter (B3) and the so-called “half-and-half” functionals (BH&H), which incorporate20% or 50% Hartree-Fock exchange density, respectively.7 ForHe2•+, these were used in conjunction with the Lee, Yang, andParr correlation functional.8 Calculations were carried out withthe standard 6-31G* and with Dunning’s cc-pVnZ basis sets,3but for the sake of consistency we report only the resultsobtained with the cc-pVQZ basis set (qualitatively similar resultswere obtained with all basis sets). All calculations were donewith the Gaussian 94 set of programs.9 The effect of spincontamination (which is absent in H2•+ as there is only oneelectron) is practically negligible for He2•+ as indicated by the!S2" values, which are less than 0.767 at any given internucleardistance.

3. Results and DiscussionThe problem becomes most clearly evident when the dis-

sociation curves for the two model cations are normalized toX Abstract published in AdVance ACS Abstracts, October 15, 1997.

© Copyright 1997 by the American Chemical Society VOLUME 101, NUMBER 43, OCTOBER 23, 1997

S1089-5639(97)02378-5 CCC: $14.00 © 1997 American Chemical Society

Self-interaction error: H2+ example


LRC func)onals

5

-‐ DFT has problems with separa)ng the charge (ar)ficial charge delocaliza)on) due to SIE (Examples: Diss-‐n of charged dimers, failure of Koopmans picture, too low TS).

-‐ DFT has issues when dealing with charge transfer excited states because it requires full descrip)on of non-‐local Hartree-‐Fock exchange.

-‐ Hybrid func)onals like B3LYP and PBE0 employ 20% and 25% long-‐range Hartree-‐Fock, respec)vely and when these are used within the TD-‐DFT formalism for excited states, they fail to capture the correct distance dependence of charge-‐transfer excited states.

-‐ For large molecules, this can result in a near-‐con)nuum of spurious, low-‐lying charge transfer states.

The idea of LRC-‐DFT is to maintain the form of par6cular func6onal at short range, while including Hartree-‐Fock exchange at long range.


LRC: Cont-‐d

6

-‐ One way to accomplish this goal is to have “Coulomb-‐aYenuated” func)onals that makes use of the erf func)on by spli[ng up the Coulomb poten)al as follows:

-‐ The first term is short range because it decays to zero quickly (roughly as 1/ω).

-‐ Consider the following generic exchange-‐correla)on func)onal (not LRC):

where CHF represents the frac)on of Hartree-‐Fock exchange.

For GGA, CHF = 0; CHF = 0.20 for B3LYP, and CHF = 0.25 for PBE0.

! !

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! >2&!<0?1*!*&?-!01!12+?*!?(%3&!8&,(71&!0*!:&,()1!*+!@&?+!A70,=/)!B?+732/)!(1!"C!D

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LRC: Cont-‐d

7

-‐ The LRC version takes on the following form

where SR = short range and LR = long range;-‐ As ω → 0 one obtains the original func)onal; as ω → ∞ one obtains new func)onal that includes 100% Hartree-‐Fock exchange;-‐ This is inappropriate for most generalized gradient approxima)on (GGA) func)onals, so one has two op)ons for choosing ω.

1. Use a blanket omega value that comes from the literature. For example, the LRC-‐ωPBEH has sta)s)cally-‐op)mal parameters of CHF = 0.20 and ω = 0.20 bohr-‐12. One can “tune” the parameter for a specific property such as ver)cal detachment energies or ioniza)on poten)als (see Baer, ARPC 61, 85 2010).

Some func)onals do not need the ω value tuned such as the wB97X and M11 families. BNL is designed to have omega tuned to match Koopmans IP condi)on.

! !

! "#$!%&'!($)*+,-!./0$*!,-!.#$!1,22,3+-4!1,)5

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! M*$!/!<2/-0$.!,5$4/!(/2=$!.#/.!>,5$*!1),5!.#$!2+.$)/.=)$F!!D,)!$E/5G2$L!.#$!%&'C!NOPB!#/*!*./.+*.+>/22QC,G.+5/2!G/)/5$.$)*!,1!'

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Systematic optimization of long-range corrected hybrid density functionalsJeng-Da Chaia! and Martin Head-Gordonb!

Department of Chemistry, University of California and Chemical Sciences Division, Lawrence BerkeleyNational Laboratory, Berkeley, California 94720, USA

!Received 26 November 2007; accepted 3 January 2008; published online 27 February 2008"

A general scheme for systematically modeling long-range corrected !LC" hybrid density functionalsis proposed. Our resulting two LC hybrid functionals are shown to be accurate in thermochemistry,kinetics, and noncovalent interactions, when compared with common hybrid density functionals.The qualitative failures of the commonly used hybrid density functionals in some “difficultproblems,” such as dissociation of symmetric radical cations and long-range charge-transferexcitations, are significantly reduced by the present LC hybrid density functionals. © 2008American Institute of Physics. #DOI: 10.1063/1.2834918$

I. INTRODUCTION

In the last two decades, density functional theory1 !DFT"based on the Kohn–Sham !KS" approach2,3 has been attract-ing considerable attention.4,5 Due to its favorable scalingwith system size and reasonable accuracy in many applica-tions, KS-DFT has been regarded as one of the most power-ful theoretical tools for studying both electronic and dynamicproperties of medium to large ground-state systems. Re-cently, the development of time-dependent density functionaltheory !TDDFT" for treating excited-state systems has alsobeen making considerable progress.6,7

In KS-DFT, the exact exchange-correlation energy func-tional Exc#!$, however, remains unknown, and needs to beapproximated. Functionals based on the local spin densityapproximation !LSDA" have been successful for nearly-free-electron systems.4,5 However, for molecular systems, whereelectron densities are highly nonuniform, the severeoverbinding tendency of LSDA means it is not sufficientlyaccurate for most quantum chemical applications.

Functionals based on the semilocal generalized gradientapproximations !GGAs" have considerably reduced the er-rors associated with the LSDA and have shown reasonableaccuracy for atomization energies of many strongly boundsystems.8 For some weakly bound systems, such ashydrogen-bonded systems, GGAs are still reasonable for theenergetics and geometries. However, GGAs can completelyfail for van der Waals systems. For such systems, GGAs giveinsufficient binding or even unbound results. Moreover,GGAs tend to give predicted barrier heights of chemical re-actions that are usually seriously underestimated.

Both of the LSDA and GGAs !commonly denoted asDFAs for density functional approximations" are based onthe localized model exchange-correlation holes. The exactexchange-correlation hole is, however, fully nonlocal. There-fore, the success of DFAs is commonly believed to be due toa cancelation of errors between the DFAs for exchange and

correlation.4,5 In situations where the cancelation of errors isnot complete, the DFAs can produce erroneous results. No-ticeably, some of these situations occur in the asymptoticregions of molecular systems, where the electron densitiesdecay exponentially. In such regions, due to the severe self-interaction error !SIEs" of DFAs, the DFA exchange-correlation potential exhibits an exponential decay, instead ofthe correct −1 /r decay. This leads to many qualitative fail-ures for problems such as dissociation of cations with oddnumber of electrons or even the alkali halides.9,10 In time-dependent DFT, SIE causes dramatic failures for long-rangecharge-transfer excitations of two well-separatedmolecules.11–13 The spatially localized nature of DFAs alsoleads to the absence of London forces, which are a long-range correlation effect. Therefore, to circumvent the abovedifficulties, it seems necessary to incorporate part or all ofthe nonlocality of the exchange-correlation hole into theDFAs.

Hybrid DFT methods, which combine KS-DFT withwave function theory !WFT", are promising as a cost-effective way to incorporate nonlocality of the exchange-correlation hole into the DFAs. They can provide reasonableaccuracy for treating large-scale systems. In fact, the mostwidely used density functionals in quantum chemistry are allhybrid functionals! This happy marriage of KS-DFT andWFT was first proposed by Becke,14 who argued that mixinga small fraction of the exact Hartree–Fock !HF" exchange!associated with the Kohn–Sham reference wave function"with DFAs will provide the desired nonlocality and therebygenerally improve the DFA results. The general form of ahybrid density functional can be written as

Exc = cxExHF + Exc

DFA, !1"

where cx is a small fractional number, typically ranging from0.2 to 0.25 for thermochemistry,14 and from 0.4 to 0.6 forkinetics.15

Indeed, a remarkable accuracy has been achieved by hy-brid density functionals. For example, one of the mostwidely used hybrid density functionals, B3LYP,14,16 a com-bination of the B88 exchange functional17 and the LYP cor-relation functional,18 has achieved a better accuracy for

a"Electronic mail: [email protected]"Author to whom correspondence should be addressed. Electronic mail:

[email protected].

THE JOURNAL OF CHEMICAL PHYSICS 128, 084106 !2008"

0021-9606/2008/128"8!/084106/15/$23.00 © 2008 American Institute of Physics128, 084106-1

Downloaded 05 Sep 2012 to 128.125.134.34. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

We perform unrestricted calculations with the aug-cc-pVQZ basis set and a high-quality EML!250,590" grid. TheDFT results are compared with results from HF theory, andthe very accurate CCSD!T" theory !coupled-cluster theorywith iterative singles and doubles and perturbative treatmentof triple substitutions".76,77

As shown in Table XI, the predicted bond length andbinding energy of !B97X and !B97 are better than that ofB97-1, B3LYP, and BLYP. Furthermore, the !B97X and!B97 functionals predict no spurious barriers on the disso-ciation curves and provide much improved results for H2

+ andAr2

+ !see Fig. 1" relative to the common hybrid functionals.However, their results for He2

+ !see Fig. 2" and Ne2+ are less

satisfactory, as shown in Table XII.To circumvent this, one needs to correct the remaining

SIE by either using a larger ! !to approach full HF exchangefaster" and/or by using a larger fraction of the Ex

SR-HF !toreduce the remaining SR SIE". For both cases, a correlationfunctional more nonlocal than the Ec

B97 correlation is needed.Work along these lines is in progress.

F. Long-range charge-transfer excitations

It has been shown by Dreuw et al. that pure DFA func-tionals qualitatively fail to describe long-range charge-transfer !CT" excitations between a donor and anacceptor.11–13 Following Dreuw et al., we perform TDDFTcalculations for the lowest CT excitations between ethyleneand tetrafluoroethylene, when separated by a distance R.

During this process, an electron transfers from the highestoccupied molecular orbital of tetrafluoroethylene to the low-est unoccupied molecular orbital of ethylene. The experi-mental geometry for each monomer C2H4 and C2F4 with theintermolecular distance R=8 Å was taken from Zhao andTruhlar.78 Similarly, the geometries at different values of Rare obtained by varying the distance between their centers ofmass without further reoptimzations. Calculations were per-formed using the 6-31G* basis set with the extrafine grid,EML!99,590". For comparison, high-level SAC-CI resultsare taken from Tawada et al.38

The correct CT excitation energy !CT!R" has the follow-ing asymptote:11

!CT!R → "" # −1R

+ IPC2F4− EAC2H4

, !33"

where IPC2F4and EAC2H4

are the ionization potential of tet-rafluoroethylene and the electron affinity of ethylene, respec-tively.

TABLE X. Statistical errors !in Å" of EXTS !Ref. 70".

Error !B97X !B97 B97-1 B3LYP BLYP

MSE −0.003 −0.002 0.004 0.003 0.018MAE 0.009 0.010 0.008 0.008 0.019rms 0.014 0.015 0.013 0.013 0.024Max!#" −0.084 −0.085 −0.071 −0.078 −0.064Max!$" 0.055 0.059 0.065 0.065 0.103

TABLE XI. Dissociation energies of symmetric radical cations, De=E!X"+E!X+"−E!X2+ ,Re" !in kcal/mol",

where Re !in Å" is the equilibrium bond length. The reference values are taken from Ref. 71 for H2+, from Ref.

72 for He2+, from Ref. 73 for Ne2

+, and from Ref. 74 for Ar2+.

Molecule Ref. !B97X !B97 B97-1 B3LYP BLYP

Re

H2+ 1.057 1.102 1.100 1.107 1.114 1.136

He2+ 1.081 1.137 1.141 1.146 1.146 1.184

Ne2+ 1.765 1.784 1.780 1.831 1.827 1.924

Ar2+ 2.423 2.457 2.445 2.519 2.536 2.621

MSE 0.039 0.035 0.069 0.074 0.134MAE 0.039 0.035 0.069 0.074 0.134

De

H2+ 64.4 68.4 68.5 68.9 67.9 69.1

He2+ 57.0 71.5 71.9 71.5 77.4 83.2

Ne2+ 32.2 54.0 56.0 58.5 59.4 73.1

Ar2+ 30.8 38.6 36.9 44.1 43.3 49.0

MSE 12.0 12.2 14.7 15.9 22.5MAE 12.0 12.2 14.7 15.9 22.5

FIG. 1. Dissociation curve of Ar2+ curve. Zero level is set to E!Ar"

+E!Ar+" for each method.

084106-12 J.-D. Chai and M. Head-Gordon J. Chem. Phys. 128, 084106 !2008"


We perform unrestricted calculations with the aug-cc-pVQZ basis set and a high-quality EML!250,590" grid. TheDFT results are compared with results from HF theory, andthe very accurate CCSD!T" theory !coupled-cluster theorywith iterative singles and doubles and perturbative treatmentof triple substitutions".76,77

As shown in Table XI, the predicted bond length andbinding energy of !B97X and !B97 are better than that ofB97-1, B3LYP, and BLYP. Furthermore, the !B97X and!B97 functionals predict no spurious barriers on the disso-ciation curves and provide much improved results for H2

+ andAr2

+ !see Fig. 1" relative to the common hybrid functionals.However, their results for He2

+ !see Fig. 2" and Ne2+ are less

satisfactory, as shown in Table XII.To circumvent this, one needs to correct the remaining

SIE by either using a larger ! !to approach full HF exchangefaster" and/or by using a larger fraction of the Ex

SR-HF !toreduce the remaining SR SIE". For both cases, a correlationfunctional more nonlocal than the Ec

B97 correlation is needed.Work along these lines is in progress.

F. Long-range charge-transfer excitations

It has been shown by Dreuw et al. that pure DFA func-tionals qualitatively fail to describe long-range charge-transfer !CT" excitations between a donor and anacceptor.11–13 Following Dreuw et al., we perform TDDFTcalculations for the lowest CT excitations between ethyleneand tetrafluoroethylene, when separated by a distance R.

During this process, an electron transfers from the highestoccupied molecular orbital of tetrafluoroethylene to the low-est unoccupied molecular orbital of ethylene. The experi-mental geometry for each monomer C2H4 and C2F4 with theintermolecular distance R=8 Å was taken from Zhao andTruhlar.78 Similarly, the geometries at different values of Rare obtained by varying the distance between their centers ofmass without further reoptimzations. Calculations were per-formed using the 6-31G* basis set with the extrafine grid,EML!99,590". For comparison, high-level SAC-CI resultsare taken from Tawada et al.38

The correct CT excitation energy !CT!R" has the follow-ing asymptote:11

!CT!R → "" # −1R

+ IPC2F4− EAC2H4

, !33"

where IPC2F4and EAC2H4

are the ionization potential of tet-rafluoroethylene and the electron affinity of ethylene, respec-tively.

TABLE X. Statistical errors !in Å" of EXTS !Ref. 70".

Error !B97X !B97 B97-1 B3LYP BLYP

MSE −0.003 −0.002 0.004 0.003 0.018MAE 0.009 0.010 0.008 0.008 0.019rms 0.014 0.015 0.013 0.013 0.024Max!#" −0.084 −0.085 −0.071 −0.078 −0.064Max!$" 0.055 0.059 0.065 0.065 0.103

TABLE XI. Dissociation energies of symmetric radical cations, De=E!X"+E!X+"−E!X2+ ,Re" !in kcal/mol",

where Re !in Å" is the equilibrium bond length. The reference values are taken from Ref. 71 for H2+, from Ref.

72 for He2+, from Ref. 73 for Ne2

+, and from Ref. 74 for Ar2+.

Molecule Ref. !B97X !B97 B97-1 B3LYP BLYP

Re

H2+ 1.057 1.102 1.100 1.107 1.114 1.136

He2+ 1.081 1.137 1.141 1.146 1.146 1.184

Ne2+ 1.765 1.784 1.780 1.831 1.827 1.924

Ar2+ 2.423 2.457 2.445 2.519 2.536 2.621

MSE 0.039 0.035 0.069 0.074 0.134MAE 0.039 0.035 0.069 0.074 0.134

De

H2+ 64.4 68.4 68.5 68.9 67.9 69.1

He2+ 57.0 71.5 71.9 71.5 77.4 83.2

Ne2+ 32.2 54.0 56.0 58.5 59.4 73.1

Ar2+ 30.8 38.6 36.9 44.1 43.3 49.0

MSE 12.0 12.2 14.7 15.9 22.5MAE 12.0 12.2 14.7 15.9 22.5

FIG. 1. Dissociation curve of Ar2+ curve. Zero level is set to E!Ar"

+E!Ar+" for each method.

084106-12 J.-D. Chai and M. Head-Gordon J. Chem. Phys. 128, 084106 !2008"


Modern long-range corrected functionals


Figure 3: Independent comparison of an established GGA (BLYP) against an established hybrid(B3LYP), a recent range-separated hybrid (ωB97X), and a range-separated hybrid that includes anempirical long-range dispersion correction (ωB97X-D). MAE (mean average errors) are computedfor the atomization energies (48 reactions comprising the G3/05 test set) and weak interactions (25intermolecular complex binding energies).23,24

Table 2: Timings for CCSD energy, properties, and gradient calculations for a nucleobase dimer(C9H10N8O1, C1 symmetry) with the 6-31+G(d) basis set (362 basis functions, frozen core, no RI).2×6-core Intel Xeon node (3.06GHz, 128 Gb RAM).

Energya Propertiesb Gradientc

1 core 101.5 hrs4 cores 26.8 hrs (3.8x) 31.0 hrs 6.4 hrs8 cores 16.8 hrs (6.0x) 22.0 hrs 4.0 hrs

a CCSD equations.b Λ-equations and non-relaxed density matrices.c calculation of orbital response and relaxed density matrices.

Gradients and properties calculations are implemented for a number of methods including RI-MP2, SOS-MP2, CCSD and OO-CCD.

3.3 Open-shell and electronically excited species

One of the distinguishing features of Q-Chem is a broad set of methods for electronically excited andopen-shell species, in particular, approaches that comply with the Pople definition of a theoreticalmodel chemistry.

As illustrated in Fig. 4, equation-of-motion CC theory,40–43 EOM-CC, allows one to treat avariety of multi-configurational wave functions within a strictly single-reference formalism. In EOM,target states Ψex are described as excitations from a reference state Ψ0:

Ψex = RΨ0 = ReTΦ0, (1)

where R is a general excitation operator, T is a coupled-cluster operator for the reference state, andΦ0 is the reference Slater determinant. Different EOM models are defined by choosing the referenceand the form of R (Fig. 4). Q-Chem features the following EOM models:40 EOM-EE (excitationenergies), EOM-IP (ionization potentials), EOM-EA (electron affinities), EOM-SF (spin-flip, fortriplet and quartet references44), EOM-2SF (double SF, for quintet references45), and EOM-DIP

5

wB97X and wB97X-D


LRC: Cont-‐d

10

This is an example of tuning the LRC-‐μBOP func)onal func)onal for an anionic water hexamer cluster Using the “tuned” value for μ, one can then use the func)onal for TD-‐DFT calcula)ons for excited states

In this example

! !

! "#$%!$%!&'!()&*+,(!-.!/0'$'1!/#(!2345!678!.0'9/$-'&,!.0'9/$-'&,!.-:!&'!&'$-'$9!;&/(:!#()&*(:!9,0%/(:!

! <%$'1!/#(!=/0'(>?!@&,0(!.-:!AB!-'(!9&'!/#('!0%(!/#(!.0'9/$-'&,!.-:!"C5CD"!9&,90,&/$-'%!.-:!()9$/(>!%/&/(%

! E'!/#$%!()&*+,(

! !"#"$%&'()*+!"# #,*%-.%$%$*%-.%

!"# #,*%-.%$

! 5"#$%$

&$%!/#(!('(:1F!-.!/#(!2<G7!.:-*!&!9&,90,&/$-'!-.!/#(!&'$-'$9!9,0%/(:!

*0,/$+,$(>!HF!'(1&/$@(!-'(!IJ--+*&'K%!"#(-:(*L

! "#(!:(>!,$'(!$%!&!44MCI"L!H('9#*&:N!.-:!5"#$%$

&%-!-'(!9&'!/0'(!!!/-!H(!/#(!

@&,0(!;#(:(!/#(!CD"!5"#$%$

&$%!(O0&,!/-!/#(!44MCI"L!:(%0,/!IA!P!QRSS!H-#:5TL

! E.!%09#!>&/&!&:(!'-/!&@&$,&H,(B!-'(!9&'!/#('!/0'(!$/!/-!%09#!/#&/!

'M4D!U!5"#$%$

&

!"#"$%&'(&')$*+,$-"."$!*/0(10+2!"#!$%&'#!(%)*#!+$!!"2$34456734489$:;633<$! !

! "#$%!$%!&'!()&*+,(!-.!/0'$'1!/#(!2345!678!.0'9/$-'&,!.0'9/$-'&,!.-:!&'!&'$-'$9!;&/(:!#()&*(:!9,0%/(:!

! <%$'1!/#(!=/0'(>?!@&,0(!.-:!AB!-'(!9&'!/#('!0%(!/#(!.0'9/$-'&,!.-:!"C5CD"!9&,90,&/$-'%!.-:!()9$/(>!%/&/(%

! E'!/#$%!()&*+,(

! !"#"$%&'()*+!"# #,*%-.%$%$*%-.%

!"# #,*%-.%$

! 5"#$%$

&$%!/#(!('(:1F!-.!/#(!2<G7!.:-*!&!9&,90,&/$-'!-.!/#(!&'$-'$9!9,0%/(:!

*0,/$+,$(>!HF!'(1&/$@(!-'(!IJ--+*&'K%!"#(-:(*L

! "#(!:(>!,$'(!$%!&!44MCI"L!H('9#*&:N!.-:!5"#$%$

&%-!-'(!9&'!/0'(!!!/-!H(!/#(!

@&,0(!;#(:(!/#(!CD"!5"#$%$

&$%!(O0&,!/-!/#(!44MCI"L!:(%0,/!IA!P!QRSS!H-#:5TL

! E.!%09#!>&/&!&:(!'-/!&@&$,&H,(B!-'(!9&'!/#('!/0'(!$/!/-!%09#!/#&/!

'M4D!U!5"#$%$

&

!"#"$%&'(&')$*+,$-"."$!*/0(10+2!"#!$%&'#!(%)*#!+$!!"2$34456734489$:;633<$

εSOMO is the energy of the LUMO from a calcula)on of the anionic cluster mul)plied by nega)ve one (Koopmans’ theorem)

The red line is a CCSD(T) benchmark for -‐εSOMO so one can tune μ to be the value where the DFT -‐εSOMO is equal to the CCSD(T) result (μ ~ 0.33 bohr-‐1)If such data are not available, one can then tune it to such that ΔSCF = -‐εSOMO

Q-‐Chem has a script for op6mizing ω either using IP (or IP-‐EA) Koopmans condi6on


LRC: Exercise

11

1. Consider two (H2O)6-‐ isomers: h2o_6_it.xyz and h2o_6_i2.xyz. 2. Using these geometries, calculate the first excited state with TD-‐DFT and these LRC-‐func)onals: LRC-‐μBOP, LRC-‐ωPBEh, and ωB97X at the 6-‐31(+,+)G* level.For LRC-‐μBOP use μ = 0.330 bohr-‐1, for LRC-‐ωPBEh use ω = 0.200 bohr-‐1.

Tips: -‐ These will be anionic calcula)ons, set change the “0 1” under $molecule to “-‐1 2” which means charge of -‐1 and spin mul)plicity of 2-‐ Set CIS_N_ROOTS = 1-‐ One also needs to set LRC_DFT = TRUE and set OMEGA = XXX, depending on the func)onal (use OMEGA even for LRC-‐μBOP)-‐ In the Q-‐Chem Setup menu, look for “user-‐defined” in the Exchange drop down menu which creates the $xc_func)onal sec)on-‐ For LRC-‐μBOP add “c (B88)OP 1.0” and “x muB88 1.0” in $xc_func)onal-‐ For LRC-‐wPBEh add “c PBE 1.0”, “x wPBE 0.80”, and “x HF 0.20”

Each “x blah blah” or “c blah blah” is a new line in $xc_func)onal


LRC: Analysis

! !

!"#$%&'()')*$+,'#-.*,#/'+,0,#'1#23

4+(5#$'6758#$

9:;<)7".,*("0=> ?

!"#$%&'() !"#!$# !"%&&'

!"#$*)'+, !"%$'$ %"$(%!

-%./0'123 !"!&)$ %"$$$(

EOM-‐EE-‐CCSD/6-‐31(+,+)G*: 1.1477 eV and 0.9751 eV


Documents

Performance of DFT: The good, the bad, and the ugly