Valence excitation energies of alkenes, carbonyl compounds, and azabenzenes by time-dependent density functional theory: Linear response of the ground state compared to collinear and noncollinear spin-flip TDDFT with the Tamm-Dancoff approximation Miho Isegawa and Donald G. Truhlar Citation: J. Chem. Phys. 138, 134111 (2013); doi: 10.1063/1.4798402 View online: http://dx.doi.org/10.1063/1.4798402 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i13 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors THEJOURNALOFCHEMICALPHYSICS138,134111(2013) Valence excitation energies of alkenes, carbonyl compounds, and azabenzenes by time-dependent density functional theory: Linear response of the ground state compared to collinear and noncollinear spin-flip TDDFT with the Tamm-Dancoff approximation MihoIsegawaandDonaldG.Truhlar DepartmentofChemistry,ChemicalTheoryCenter,andSupercomputingInstitute,UniversityofMinnesota, Minneapolis,Minnesota55455-0431,USA (Received3February2013;accepted13March2013;publishedonline3April2013) Time-dependent density functional theory (TDDFT) holds great promise for studying photochem- istry because of its affordable cost for large systems and for repeated calculations as required for direct dynamics. The chief obstacle is uncertain accuracy. There have been many validation stud- ies, but there are also many formulations, and there have been few studies where several formu- lations were applied systematically to the same problems. Another issue, when TDDFT is applied with only a single exchange-correlation functional, is that errors in the functional may mask suc- cesses or failures of the formulation. Here, to try to sort out some of the issues, we apply eight formulations of adiabatic TDDFT to the first valence excitations of ten molecules with 18 density functionals of diverse types. The formulations examined are linear response from the ground state (LR-TDDFT), linear response from the ground state with the Tamm-Dancoff approximation (TDDFT-TDA), the original collinear spin-flip approximation with the Tamm-Dancoff (TD) ap- proximation (SF1-TDDFT-TDA), the original noncollinear spin-flip approximation with the TDA approximation(SF1-NC-TDDFT-TDA),combinedself-consistent-field(SCF)andcollinearspin-flip calculations in the original spin-projected form (SF2-TDDFT-TDA) or non-spin-projected (NSF2- TDDFT-TDA), and combined SCF and noncollinear spin-flip calculations (SF2-NC-TDDFT-TDA andNSF2-NC-TDDFT-TDA).ComparingLR-TDDFTtoTDDFT-TDA,weobservedthattheexci- tationenergyisraisedbytheTDA;thisbringstheexcitationenergiesunderestimatedbyfulllinear responseclosertoexperiment,butsometimesitmakestheresultsworse.Forethyleneandbutadiene, theexcitationenergiesareunderestimatedbyLR-TDDFT,andtheerrorbecomessmallermakingthe TDA.NeitherSF1-TDDFT-TDAnorSF2-TDDFT-TDAprovidesalowermeanunsignederrorthan LR-TDDFT or TDDFT-TDA. The comparison between collinear and noncollinear kernels shows that the noncollinear kernel drastically reduces the spin contamination in the systems considered here, and it makes the results more accurate than collinear spin-flip TDDFT for functionals with a low percentage of Hartree-Fock exchange and sometimes for functionals with a higher percent- ageofHartree-Fockexchange,butityieldslessaccurateresultsthanground-stateTDDFT.©2013 AmericanInstituteofPhysics.[http://dx.doi.org/10.1063/1.4798402] occupiedmolecularorbital(HOMO)inthegroundstatewhen I. INTRODUCTION localfunctionalsareemployed.6 Time-dependent density functional theory1 (TDDFT) is Another problematic issue for LR-TDDFT and the a potentially transformative methodology for applications to TDDFT-TDA is the description of doubly excited states. electronic spectroscopy and photochemical dynamics due to The correct description of doubly excited states would thelowcomputationalcostofdensityfunctionalcalculations. require a frequency-dependent xc functional,7,8 but most ConventionalTDDFTisbasedonthelinearresponse(LR)of attempts to make TDDFT practical so far have used the ground state to a time-dependent perturbation and may frequency-independentxcfunctionals,whichiscalledtheadi- be called LR-TDDFT. The Tamm-Dancoff approximation2 abatic approximation, and which allows one to use xc func- (TDA) to TDDFT eliminates the coupling of the occupied– tionalsthathavealreadybeenheavilyvettedforgroundstates. vacanttothevacant–occupiedblocksinCasida’sequations,1,3 Socalleddoublyexcitedstatesareknowntoplayaprominent whichreducestheinstabilitynearaconicalintersection.With roleinpolyenes,9–12whichareveryimportantinbiochemistry localexchange-correlation(xc)functionals,suchasthelocal andtechnology,butthenatureofthesestatesisnowknownto spin density approximation or the generalized gradient ap- bemorecomplicatedthanisimpliedbydescribingthemasre- proximation(GGA),bothLR-TDDFTandTDDFT-TDAsuf- sultingfromdoubleexcitations.Moleculesforwhichdoubly ferfrominaccuratetreatmentofRydbergexcitedstatescom- excitedstatesarealow-energyexcitationalsohavesignificant pared with valence excited states;4,5 this has been explained contributions from doubly excited configurations in the asarisingfromsystematicerrorsinthepositionofthehighest groundstate,sothatoneshouldconsidersuchmoleculestobe 0021-9606/2013/138(13)/134111/13/$30.00 138,134111-1 ©2013AmericanInstituteofPhysics 134111-2 M.IsegawaandD.G.Truhlar J.Chem.Phys.138,134111(2013) examplesofmultireferencestates,whereasingleSlaterdeter- gestedbyLiuetal.,15–17 althoughextracomputationalcostis minant is not a good description even in the ground state. It required. has been suggested that a better description is that in excit- Our recent study of the performance of LR-TDDFT for ingfromthegroundstatetoasocalleddoublyexcitedstate, various types of density functionals18 showed that for most oneswitchesfromawavefunctiondominatedbytheHartree– functionals the performance for valence states is superior to Fock determinant and one or more double excitations to an- that for Rydberg states; including the results by Caricato other(orthogonal)linearcombinationwithalargerweighton etal.,19theaverageabsoluteerroris0.38eVforvalencestates thedoubleexcitations.4 and0.98eVforRydbergstatesfor56densityfunctionalsWe InLR-TDDFT,thereferencestateisthegroundstate,ob- also found that the best performing xc functionals are M06- tainedbyarestrictedKohn-Sham(RKS)calculation.Oneat- 2X and ωB97X-D (M06-2X is also seen to perform well on tempt to make TDDFT more accurate, especially for states anotherdatabase20).Wefoundthatthemostsuccessfulfunc- with significant double excitation character, is called spin- tionalsoverallhavearelativelyhighpercentage(about40%) flip TDDFT (SF-TDDFT). In this method, an unrestricted ofHartree-Fockexchange.Theinclusionofahealthyamount Kohn–Sham (UKS) spin state with higher multiplicity than ofHartree-Fockexchangeoftenremovesthespuriousstates. the ground-state singlet is taken as a reference; in the cases Thepurposeofthepresentstudyistoexploresevenad- ofinterestherethisisasinglyexcitedtripletstatewithM = ditional formulations of the TDDFT method on some of the S +1, where M is the spin projection quantum number. One samemoleculesforwhichwepreviouslystudiedLR-TDDFT S then calculates the “excitation” to both lower- and higher- to see if they offer any systematic improvement in accuracy. energy states byspinflips,both withand without orbitalex- In particular, we compare results by LR-TDDFT to those citations;oneobtainsthesingletgroundstateandbothsingly by TDDFT-TDA, collinear spin-flip TDDFT-TDA, and non- and doubly excited singlet and triplet states (only the M = collinear spin-flip TDDFT-TDA for the first valence excita- S 0componentsofthelatter)asalinearresponseofthetriplet tionenergiesofalkenes,carbonyls,andazabenzenes.Forthe reference state. One advantage of spin-flip TDDFT is that it lattertwomethodswetesttheoriginalspin-flipmethod,here should be better for treating a system in which the HOMO called SF1, and wealsotestascheme21 forcombining spin- and the lowest unoccupied molecular orbital are nearly projected spin-flip TDDFT for states not well described by degenerate. a single Slater determinant with self-consistent-field (SCF) A significant disadvantage of spin-flip TDDFT is that, calculations for states that are well described; this is called just like ground-state TDDFT, it tends to generate a number spin-flip method 2 (SF2). To gain insight we also test SF2 of spurious low-energy states as well as the real low-energy withoutthespinprojection,whichiscallednonprojectedSF2 states (this would not occur if one used the unknown, exact, orNSF2.TheSF1andNSF2methodsdonotcorrectforspin nonadiabatic xc functional), and these spurious states make contamination,butSF2does. the state assignments difficult. Actually, it is not clear if one WitheightformulationsofTDDFTunderconsideration, shouldcallthesespuriousstatesorspuriouslyloweredstates; thenamesandeventheabbreviationsrequiredtodistinguish wewillusetheformerlanguageforsimplicity.Anotherprob- the formulations are rather long. Therefore, we introduce lem,sharedbyallunrestrictedKohn–Shammethods,13isthat short names for the formulations in Table I, and we will use thecalculatedexcitedstatesarenoteigenstatesofthesquare theseintherestofthisarticle. Sˆ2 of the electron spin operator; rather they correspond to a Here, we only test frequency-independent functionals. mixture of the singlets and higher spin states. The restricted Sincetheunknown,exact,frequency-dependentxcfunctional open-shellKohn-Sham(ROKS)methodissometimesusedto foreachcase(singletandtriplet)wouldyieldthecorrectex- avoid this problem,14 but the ROKS equation is ad hoc and citation energies, this is, in principle, a test of the xc func- is often difficult to converge. An apparently better solution tionals rather than of the formulations of TDDFT. However, forthisproblemisthespin-adaptedSF-TDDFTmethodsug- someways ofusingTDDFTarelesssensitivethanothersto TABLEI. AbbreviationsforformulationsofTDDFTstudiedinthisarticle. Method Longabbreviation Shortabbreviationa Linearresponsefromthegroundstate LR-TDDFT LR LinearresponsefromthegroundstatewithTamm-Dancoffapproximation TDDFT-TDA TDA Collinearspin-flipmethodwithTamm-Dancoffapproximationb SF1-TDDFT-TDA SF1 Noncollinearspin-flipmethodwithTamm-Dancoffapproximationb SF1-NC-TDDFT-TDA NC-SF1 CombinedSCFandcollinearspin-flipcalculationsc SF2-C-TDDFT-TDA SF2 NonprojectedcombinedSCFandcollinearspin-flipcalculationsd NSF2-C-TDDFT-TDA NSF2 CombinedSCFandnoncollinearspin-flipcalculationsc SF2-NC-TDDFT-TDA NC-SF2 NonprojectedcombinedSCFandnoncollinearspin-flipcalculationsd NSF2-NC-TDDFT-TDA NC-NSF2 aWhenweusetheabbreviationSF,itrefersgenericallytoSF1,SF2,andNSF2ortoelementsthatarethesameinallthreemethods.Similarly,NC-SFrefersgenericallytoNC-SF1, NC-SF2,andNC-NSF2ortoelementsthatarethesameinallthreemethods. bInSF1methods,theenergyofS0isbasedontheMS=0stategeneratedbyspin-flipexcitationfromT1. cInSF2methods,theenergyofS0isbasedontheMS=0RKSSCFgroundstate. dInnonprojectedSF2methods;noprojectionprocedureisemployedinSF2orNC-SF2. 134111-3 M.IsegawaandD.G.Truhlar J.Chem.Phys.138,134111(2013) the inexactness of the xc functionals, and some approximate clude,inprincipal,theclosed-shellgroundstateS ,thelowest 0 functionalsaremoresuitablethanothersforusewithagiven excitedsingletstateS ,manyothersingletstates,andM =0 1 S formulation; and these are additional issues about which the components of triplet states. Due to the approximate nature presenttestsrevealinformation. of the available density functionals and to the adiabatic ap- proximation, these states might be mixed; that is, the calcu- lated final states might be linear combinations of more than II. THEORETICALBACKGROUND onerealstate.Ofspecialconcernisthatthecalculatedstates Throughoutthearticle,weusetheadiabaticapproxima- mightcorrespondtolinearcombinationsofsingletandtriplet tion,whichmeansthattheexchange-correlationpotentialad- states. justsinstantaneouslytoanychangeinthedensity.1 Aconse- Thematrixelements ofAaregiven intermsofthespin quenceofthisapproximationisthatthedensityfunctionalis orbitalsofthereferencefunctionby4,23–25 (cid:4) independentoffrequency,andwefollowtheusualprocedure of using the approximate density functionals developed for Aaσiσ(cid:3),bτjτ(cid:3) =δστδσ(cid:3)τ(cid:3) δijδab(εaσ −εiσ(cid:3)) ground-state calculations. To make the methods we employ (cid:5) clear,wefirstbrieflyreviewthekeyformulas. X First,weconsidercollinearunrestrictedTDDFT.Leti,j +δσσ(cid:3)δττ(cid:3)Kai,bj − 100(ij|ab) , (3) denoteoccupiedorbitalsanda,bdenoteunoccupiedorbitals, (cid:2) (cid:3) andletσ andτ representspincomponents(αorβ).IntheLR X approximation, the excitation energies ω are determined by Kai,bj =(ai|jb)+ 1− (ai|w|jb), (4) 100 first carrying out a SCF calculation on the ground state and thensolvingtheLRequationsderivedbyCasida:3 where εaσ is the ground-state molecular orbital energy of (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) orbital a with spin σ, (...|...) is a two-electron integral in A B X =ω 1 0 X , (1) theMullikennotation,and(...|w|...)isacontributionfrom B∗ A∗ Y 0 −1 Y the exchange-correlation functional; note that the kernel where A is a matrix with elements Aaσiσ(cid:3),bτjτ(cid:3) coupling a oftheintegrals(...|w|...)containstheHessianwithrespect one-electron excitation aσ ← iσ(cid:3) to one-electron excitation to the spin densities of the exchange–correlation energy. We bτ ← jτ(cid:3), B is a matrix with elements coupling the excita- seethattheσ (cid:5)=σ(cid:3) andτ (cid:5)=τ(cid:3) blocksofAarenonzeroonly tion aσ ← iσ(cid:3) to the de-excitation jτ(cid:3) ← bτ, and X and Y if the percentage X of Hartree–Fock exchange in the density arevectorsofexcitationandde-excitationamplitudes,respec- functional employed is nonzero. Also note that the term in- tively,inparticular,Xaσiσ(cid:3)andYbτjτ(cid:3).ThematrixelementsofB volvingtheHessianoftheexchange-correlationenergydoes arethesameasthosebetweenthegroundanddoublyexcited notcontributeintheSFtheory. states,22 so the effect of nonzero B is sometimes considered We will let E denote an energy calculated by a toaccountforcorrelationinthegroundstate. Kohn–Sham SCF calculation; ω(V|U) will denote an energy In the Tamm-Dancoff approximation to LR,2 all of the differenceofstateVfromareferenceUasobtaineddirectly elements of matrix B are set to zero, and the non-Hermitian fromEqs.(1)or(2),and(cid:9)Ewilldenoteanapproximationto eigenvalue equation (1) is reduced to the Hermitian eigen- theexcitationenergyfromS0toS1. valueequation: In the present SF calculations, the reference state is al- ways taken as the lowest triplet state (αα,M = 1). We esti- AX=ωX. (2) S matetheexcitationenergyinthreeways.Thefirstway,called The TDA is simpler (although computationally about the SF1, is the conventional method. In this method the energy same cost as LR), yields a straightforward approximation to differencebetweenthesingletgroundstateS0 andthesinglet theexcited-statedensitymatrix,andismorestableforthecal- excitedstateS1 isobtainedasalinearresponseofthetriplet culationofpotentialenergysurfaces.1 state: The matrix elements of A and B are evaluated using (cid:9)ESF1 =ω(S |T )−ω(S |T ). (5) 1 1 0 1 the orbitals of a reference SCF state calculated by the UKS method,or—asaspecialcase—theRKSmethod. The second way of using SF-TDDFT, called NSF2 (nonpro- In the usual ground-state LR and TDA approximations, jected SF2), uses the difference in SCF energies to get the the orbitals come from a ground-state RKS self-consistent excitationenergyofthetriplet: fieldcalculation,andonlythespin-conservingblocks(ααand (cid:9)ENSF2 =ω(S |T )−[EUKS(S )−ERKS(T )]. (6) ββ)ofXandYareallowedtobenonzero.Therefore,thespin 1 1 0 1 componentisconservedintheorbitalexcited,andonedeals In the third method, called SF2 and introduced previously,21 onlywithM =0states. one uses Yamaguchi’s formula26–30 to correct for spin con- S Inspin-flipTDDFT-TDA,onestarts(fortheapplications taminationoftheexcitedsingletstate: (cid:4) (cid:5) considered in the present article) with the MS = +1 compo- 2(ω(S |T )) nentofthelowest-energytripletstate,andonlytheαβandβα (cid:9)ESF2 =(cid:9)ENSF2− ω(S1|T1)− (cid:6)S2(cid:7)T1 −1(cid:6)S12(cid:7)S1 , (7) blocksofXareconsidered,sotheexcitationalwaysinvolves achangeinthespincomponent.Thecalculatedexcitationen- where (cid:6)S2(cid:7) denotes the expectation value of the spin opera- ergies correspond to transitions from a single M = 1 triplet torSˆ2 intheUKSreferenceT stateorthespinflipS state. S 1 1 state(T )toasetofstateswithM =0;thesefinalstatesin- (We know that (cid:6)S2(cid:7) calculated using the Slater determinants 1 S 134111-4 M.IsegawaandD.G.Truhlar J.Chem.Phys.138,134111(2013) is not the same as its value for the real wave function,13 but TABLEII. Densityfunctionalsstudiedinthisarticle. weuseitasanalgorithmictooltocorrectforspincontamina- Functional Type Xa Reference(s) tioninthereferencestate;looselyspeaking,thiscorrectionis calledspinprojection.)Thiscorrectionforspincontamination BLYP GGA 0 44and45 islesscomplicatedandlesswelljustifiedthanthesymmetry- PW91 GGA 0 42 adaptedapproachofLiuetal.15–17butitissimpler,andithas OLYP GGA 0 43and44 frequentlybeenappliedtotransitionmetalsystems.31–36Note revTPSS Meta-GGA 0 46 thatif(cid:6)S2(cid:7)T1 is2and(cid:6)S2(cid:7)S1 is0,thenEq.(7)reducestoEq. B3LYP HybridGGA 20 47and48 (6).Veryoftenthough,(cid:6)S2(cid:7)S1 iscloserto1,whichmeansthat PBE0 HybridGGA 25 49 thefinalstateoftheTDDFTcalculationisamixtureofsinglet X3LYP HybridGGA 21.8 50 M06 Hybridmeta-GGA 27 59 andtriplet,andEq.(7)attemptstocorrectforthat. M05 Hybridmeta-GGA 28 57 In noncollinear SF-TDDFT,37 one still employs a BHHLYP HybridGGA 50 44and45 collinearSlaterdeterminantforthereferencestate,thatis,ev- M08-HX Hybridmeta-GGA 52.23 61 eryspinfunctioninthereferencestateiseitherpureαorpure M06-2X Hybridmeta-GGA 54 59 β, but one employs density functionals whose (...|w|...) M05-2X Hybridmeta-GGA 56 58 matrix elements (the noncollinear kernels) couple pure spin M08-SO Hybridmeta-GGA 56.79 61 functionstononcollinearones,thatis,spinfunctionsthatare CAM-B3LYP Range-separatedhybridGGAb 19–65 52 linearcombinationsofαandβ.37,38 ωB97X Range-separatedhybridGGAb 15.77–100 53 M11 Range-separatedhybridGGAb 42.8–100 54 M06-HF Hybridmeta-GGA 100 60 III. COMPUTATIONALDETAILS aXisthepercentageofHartree-Fockexchangeinthedensityfunctional. The lowest singlet valence excited states are calculated bArange-separatedhybridGGAmightalsobecalledahybridNGA(hybridnonsepara- blegradientapproximation). for ten molecules, in particular, two alkenes: ethylene (D ) 2h andtrans-1,3-butadiene(C ),threecarbonyls:formaldehyde 2h (C ), acetaldehyde (C ), and acetone (C ), and five az- 2v s 2v tional.InaGGA,thedensityfunctionaldependsonelectron abenzenes: pyridine (C ), pyrazine (D ), pyrimidine (C ), 2v 2h 2v density and electron density gradient. In the meta-GGAs in pyrazine (C ), and symmetric tetrazine (D ), where the 2v 2h Table II, the density functional depends on electron density, pointgroupofthenuclearcoordinatesisgiveninparentheses. electron density gradient, and kinetic energy density. In the AllgeometriesareoptimizedbyMP2/6-311+G**.Allexci- range-separated density functionals, the interelectronic sep- tationenergies arecalculated usingthe6-311(2+,2+)G(d,p) aration is divided smoothly into two ranges. In the spectro- basis set,39 in which the angular momentum quantum num- scopic context, the motivation for range separation is to use bersandexponentialparametersoftheadditionaldiffusesub- a larger percentage of Hartree-Fock exchange at long range shells are s (H), 0.00108434; sp (C), 0.0131928; sp (O), inordertopartiallyremedytheself-interactionerrorofalo- 0.025451869;sp(N),0.0192470.Thisisawell-balancedba- cal density functional at large interelectronic separation and sis set for the description of either valence excitations or therebyreduceoreliminatethefailuresoflocaldensityfunc- Rydbergexcitations.40 tionalsforlong-rangechargetransferexcitations.4,56,64 The reference values of the excitation energy are exper- For noncollinear SF-TDDFT-TDA, we examined a imental values taken from the database of Caricato et al.19 smaller number of density functionals, in particular, BLYP, We note a recent study41 that suggests that the correct value B3LYP,PBE0,ωB97X,M06,M06-2X,andM06-HF. for the excitation energy of butadiene is 6.3 eV, not 5.9 eV, whichistheexperimentalvalueofthepeakintheabsorption spectrum,butnotnecessarilythebestestimateofthevertical excitation energy. Previously, the literature has used 5.9 eV IV. RESULTSANDDISCUSSION asthestandard,andweretainthatvaluehereforconsistency IV.A. Assignments withRefs.18and19. LR, TDA, and collinear spin-flip calculations are per- Theexcitationswestudyinthisarticlearethelowestva- formed with the GAMESS42 software package and non- lence excitation of each of the ten organic molecules men- collinear spin-flip calculations are performed with the Q- tioned above. The assignments and state symmetries are as CHEM43 software package. In these packages Casida’s follows:B1u (π →π∗)forethylene,Bu (π →π∗)fortrans- equations are solved by the Davidson algorithm.22 The grid 1,3-butadiene,A (n→π∗)forformaldehyde,A(cid:3)(cid:3) (n→π∗) 2 usedforthenumericalintegrationofthexcfunctionalhas99 for acetaldehyde, 1A (n → π∗) for acetone, B (n → π∗) 2 1 radial shells and 590 angular points per shell. The number for pyridine, 1B (n → π∗) for pyrazine, B (n → π∗) for 3u 1 of grid points is enough to avoid the numerical instability in pyrimidine, B (n → π∗) for pyridazine, and B (n → π∗) 1 3u noncollinearSF-TDDFTnotedinpreviouswork.38 fors-tetrazine. We test 18 density functionals. These are listed and ex- In the previous assessment18 of density functionals for plained in Table II.44–63 A key quantity in Table II is X, the LR calculations, we observed spurious low-energy states (or percentage of Hartree-Fock exchange. Local density func- spuriouslyloweredrealchargetransferstates),especiallyfor tionals have no Hartree–Fock exchange, and hybrid density the local density functionals, and following Caricato et al.’s functionals have finiteX. Table II alsoliststhe type of func- study,19 we checked the orbital shape for the valence states. 134111-5 M.IsegawaandD.G.Truhlar J.Chem.Phys.138,134111(2013) We found that making assignments based on orbital shape For spin-flip calculations on all azabenzenes, the lowest reduced the deviation from experiment, and we therefore calculatedsingletstateisassignedasthefirstsingletexcited chose to use this procedure for LR and TDA calculations in state;ineverycasethereisatripletbelowthis. the present study as well. For the LR and TDA calculations, wedonotfindanyspuriousstatesforthefirstvalenceexcita- tion for any of the molecules and density functionals we ex- IV.B. Criteriaforsuccess amined in this study; therefore, the lowest excitation energy In our previous article, we classified density function- corresponding to the irreducible representation of interest is als with mean unsigned errors (MUEs) of 0.30 eV or less selectedforallexcitedstates. as successful and those with MUEs in the range 0.31–0.36 In SF-TDDFT (for the case that the reference state is a eVasmoderatelysuccessful.Tomakeiteasiertospottrends triplet,|αα(cid:7)),thestatesproducedintheresponseincludeboth in tables, in the present article we place density functionals singlets, √1 (|αβ(cid:7)−|βα(cid:7)),andtriplets, √1 (|αβ(cid:7)+|βα(cid:7)),as 2 2 andtheirMUEsinboldfontiftheycorrespondtoaMUEof wellasmixedsinglet-triplets,|αβ(cid:7)and|βα(cid:7),whereM =0 S 0.36eVorsmallerforthattable. forallstates.Wewillsimplyconsideranystatewith(cid:6)S2(cid:7)<1 tobeanapproximationtoasingletandanystatewith(cid:6)S2(cid:7)>1 to be an approximation to a triplet. Only the states with IV.C. LRandTDA (cid:6)S2(cid:7)<1shouldbeconsideredinmakingassignmentsforthe TableIIIshowsthefirstvalenceexcitationenergycalcu- lowest excited singlet state. We next summarize the proce- lated by LR, and Table IV shows the results for TDA. For dures used to assign the final states for spin-flip calculations the ethylene molecule, all of the density functionals give an byacollinearkernel. excitation energy lower than the experimental one, 7.65 eV, The lowest B state of ethylene has high spin contami- nation;(cid:6)S2(cid:7)is∼2.10u;therefore,weregardthisasatriplet,and and TDA gives a larger value than LR by ∼0.90 eV except forM05-2X.Thistrendisobservedforalldensityfunctionals thelowestsingletstateistakenforallofthedensityfunction- andmoleculesexceptfors-tetrazine.Asaconsequence,TDA als except for M11 and M06-HF. For these two functionals, (cid:6)S2(cid:7)ofthelowestsingletstate((cid:6)S2(cid:7)<1)ismorethantwice makessomeenergeticimprovementformoleculesforwhich the excited state energy is underestimated LR; however, it ashighasthatofthenextlowestsingletstate,andthesecond makes the error larger in other cases. The MUEs, therefore, lowestsingletstateischosen. For most of the density functionals, (cid:6)S2(cid:7) is ∼2.0 for the depend on the composition of the test set. Nevertheless, the testsetusedhereisrepresentativeofmanyapplicationsinor- lowestspin-flipB stateofbutadiene;andthesecondaswell u ganicandbiologicalchemistry,andwebelievethattheMUE asthethirdlowestspin-flipB states,althoughbothsinglets, u isusefulfororganizingthediscussion.TablesIIIandIVshow are both characterized mainly as electron transitions from thatmostofthedensityfunctionalsstudiedherearesuccess- the second highest occupied α-orbital to the lowest unoc- fulwithLRandTDA. cupied β-orbital and from highest occupied α-orbital to the thirdlowestunoccupiedβ-orbitalortoanenergeticallyhigher β-orbital. Except for M08-HX, M08-SO, M11, and M06- IV.D. CollinearSF-TDA:ComparisonsofSF1,SF2, HF, the lowest singlet B state of butadiene is chosen. For u andNSF2 M08-HX, M08-SO, M11, and M06-HF, the second lowest singlet state is chosen because it has smaller spin contami- Table V shows the excitation energy evaluated by nationthanthefirstone;(cid:6)S2(cid:7)is0.49,0.54,0.36,and0.50for collinearSF-TDDFT,wheretheexcitationenergiesarecalcu- the second lowest state against 0.70, 0.64, 0.94, 0.91 for the latedusingSF1andSF2asshowninEqs.(5)and(7).Fairly loweststate.Asaconsequence,theerroroftheS -S excita- highspincontaminations((cid:6)S2(cid:7)>0.4)areobservedforevery 0 1 tionenergyestimatedbytheconventionalway(SF1)islower density functional at least for one molecule except for M05- forM08-HXandM11,butishigherforM08-SOandM06-HF 2XandM06-2X. comparedwiththechoiceofthelowestsingletstate. We first compare the results given by Eqs. (5) and (7). For spin-flip calculations on all carbonyl compounds The MUE tends to be large when the spin-contamination is (formaldehyde, acetaldehyde, and acetone), the lowest sin- large, and we found that SF2 showed a larger MUE for all glet state shows the minimum (cid:6)S2(cid:7) value for the irreducible densityfunctionalsexceptM05-2X.Forseveraldensityfunc- representation of interest (A for formaldehyde, A(cid:3)(cid:3) for ac- tionals,theresultswithSF2areterrible.Figures1(a)–1(c)plot 2 etaldehyde, and A for acetone) for all of the density func- theabsoluteerrorofexcitationenergybySF1,SF2,andNSF2 2 tionals. The lowest state of these irreducible representations vs.(cid:6)S2(cid:7)value.TheerrorofexcitationenergybySF1(Fig.1) istripletforallcases.Theelectronictransitionfromtherefer- seems to have no relationship with the expectation value of encetripletstatetothelowestsingletstateofeachirreducible (cid:6)S2(cid:7);ontheotherhand,theerrorbySF2isclearlylargerfor representationisstronglycharacterizedbythespin-flipelec- large(cid:6)S2(cid:7).WeconcludefromthetablethatSF2shouldnotbe trontransitionwithinthelowestsinglyoccupiedmolecularor- appliedwhenthereisalargedegreeofspin-contamination. bitalfordensityfunctionalswith50%orlowerHartree-Fock One can also see in the tables that, except for M06-2X, exchange(B3LYP,X3LYP,PBE0,M06,M05,andBHHLYP); SF1islessaccuratethanLR. on the other hand, the contributions of the other orbitals in- OneoftheunexpectedresultsisthefairlylargeMUEof creaseforthedensityfunctionalswithhigher percentages of theωB97Xfunctional(0.60eVand1.51eVforSF1andSF2, Hartree-Fockexchange. against 0.27 eV for TDA), because this functional was one 134111-6 M.IsegawaandD.G.Truhlar J.Chem.Phys.138,134111(2013) TABLEIII. Thelowestexcitationenergies(ineV)ofvalencestatescalculatedbyLR. Ethylene Butadiene HCHO CH3CHO Acetone Pyridine Pyrazine Pyrimidine Pyridazine s-Tetrazine 1B1u 1Bu 1A2 A(cid:3)(cid:3) 1A2 B1 B3u B1 B1 B3u Functional Xa π→π∗ π→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ MUE BLYP 0 7.03 5.01 3.83 4.16 4.22 4.40 3.61 3.83 3.17 1.90 0.32 OLYP 0 6.96 5.25 3.86 4.20 4.24 4.44 3.63 3.85 3.22 1.91 0.28 PW91 0 7.20 5.36 3.81 4.14 4.22 4.37 3.57 3.80 3.14 1.85 0.29 revTPSS 0 7.34 5.52 4.06 4.31 4.39 4.58 3.75 4.00 3.35 2.02 0.15 B3LYP 20 7.31 4.94 3.92 4.26 4.38 4.81 3.97 4.29 3.59 2.26 0.23 X3LYP 21.8 7.31 5.54 3.92 4.26 4.39 4.84 4.00 4.32 3.62 2.29 0.18 PBE0 25 7.47 5.65 3.92 4.27 4.41 4.88 4.01 4.35 3.66 2.29 0.16 M06 27 6.94 5.37 3.87 4.23 4.35 4.75 3.88 4.23 3.48 2.08 0.24 M05 28 7.16 5.44 3.89 4.26 4.36 4.73 3.80 4.16 3.41 1.90 0.22 BHHLYP 50 7.44 5.77 4.07 4.43 4.61 5.32 4.41 4.89 4.10 2.73 0.41 M08-HX 52.23 7.32 5.64 3.61 4.01 4.21 4.94 4.08 4.50 3.79 2.47 0.31 M06-2X 54 7.51 5.79 3.61 3.99 4.13 4.92 4.03 4.46 3.71 2.33 0.26 M05-2X 56 7.59 5.83 3.62 4.01 4.15 4.99 4.07 3.31 3.74 2.33 0.25 M08-SO 56.79 7.23 5.61 3.55 3.95 4.08 4.87 3.99 4.45 3.64 2.28 0.30 CAM-B3LYP 19–65 7.46 5.74 3.91 4.27 4.43 5.08 4.20 4.57 3.84 2.48 0.25 ωB97X 15.77–100 7.60 5.90 3.95 4.29 4.47 5.20 4.28 4.68 3.95 2.55 0.27 M11 42.8–100 7.33 5.71 3.54 3.93 4.07 4.96 4.07 4.51 3.76 2.40 0.33 M06-HF 100 7.42 5.86 2.95 3.34 3.40 4.81 3.95 4.53 3.64 2.38 0.45 7.65 5.91 4.00 4.28 4.43 4.59 3.83 3.85 3.60 2.25 aXisthepercentageofHartree-Fockexchangeinthedensityfunctional. of the best functionals in our previous LR study for the ex- formance, comparable to that of M08-SO, with the LR and citationenergyforadatabasethatincludesbothvalenceand SF1methods. Rydberg states. This unexpectedly poor behavior of ωB97X In Table S1 in supplementary material,65 we show the with SF1 was also observed in the study by Bernard et al.14 singlet (S )-singlet (S ) excitation energies calculated based 0 1 AnotherunexpectedresultisseenfortheM11functional;the on NSF2. The NSF2 method usually has lower MUEs than SF2 calculation with this functional shows a fairly large de- SF2; however, the error is larger than for SF1. The latter viationfromexperiment,eventhoughM11showedgoodper- is an unexpected result, because the error of the excitation TABLEIV. Thelowestexcitationenergies(ineV)ofvalencestatescalculatedbyTDA. Ethylene Butadiene HCHO CH3CHO Acetone Pyridine Pyrazine Pyrimidine Pyridazine s-Tetrazine 1B1u 1Bu 1A2 A(cid:3)(cid:3) 1A2 B1 B3u B1 B1 B3u Functional Xa π→π∗ π→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ MUE BLYP 0 7.15 5.14 3.84 4.17 4.23 4.44 3.66 3.86 3.23 1.97 0.27 OLYP 0 7.08 5.43 3.87 4.21 4.26 4.48 3.67 3.88 3.28 1.98 0.23 PW91 0 7.41 5.67 3.83 4.16 4.24 4.42 3.62 3.82 3.20 1.92 0.21 revTPSS 0 7.78 6.08 4.13 4.49 4.66 5.40 4.50 4.95 4.21 2.83 0.46 B3LYP 20 7.56 5.16 3.94 4.28 4.40 4.86 4.02 4.32 3.66 2.34 0.20 X3LYP 21.8 7.78 6.00 3.95 4.29 4.43 4.94 4.07 4.39 3.74 2.38 0.17 PBE0 25 7.59 5.84 3.94 4.28 4.41 4.89 4.05 4.35 3.69 2.37 0.14 M06 27 7.84 6.07 3.93 4.29 4.46 5.14 4.27 4.62 3.93 2.57 0.29 M05 28 7.60 5.90 3.95 4.29 4.47 5.20 4.28 4.68 3.95 2.55 0.27 BHHLYP 50 7.50 5.85 4.07 4.32 4.40 4.61 3.79 4.02 3.40 2.07 0.10 M08-HX 52.23 7.53 5.78 3.94 4.31 4.41 4.81 3.88 4.21 3.52 2.04 0.13 M06-2X 54 8.06 6.24 3.73 4.10 4.25 5.11 4.21 3.31 3.90 2.52 0.34 M05-2X 56 7.15 5.58 3.90 4.26 4.38 4.82 3.95 4.27 3.57 2.19 0.19 M08-SO 56.79 7.89 6.12 3.69 4.07 4.21 5.02 4.15 4.54 3.85 2.50 0.31 CAM-B3LYP 19–65 7.85 6.17 3.27 3.64 3.70 5.10 4.26 4.79 4.00 2.79 0.54 ωB97X 15.77–100 7.64 5.87 3.69 4.08 4.29 5.04 4.19 4.57 3.92 2.61 0.29 M11 42.8–100 7.56 5.87 3.65 4.04 4.17 4.99 4.12 4.54 3.79 2.46 0.28 M06-HF 100 7.61 5.89 3.63 4.01 4.14 5.07 4.19 4.59 3.91 2.58 0.32 7.65 5.91 4.00 4.28 4.43 4.59 3.83 3.85 3.60 2.25 aXispercentageofHartree-Fockexchangeinthedensityfunctional. 134111-7 M.IsegawaandD.G.Truhlar J.Chem.Phys.138,134111(2013) TABLEV. Thelowestexcitationenergies(ineV)ofvalencestatescalculatedbycollinearSF1andSF2. Ethylene Butadiene HCHO CH3CHO Acetone Pyridine Pyrazine Pyrimidine Pyridazine s-Tetrazine 1B1u 1Bu 1A2 A(cid:3)(cid:3) 1A2 B1 B3u B1 B1 B3u Functional Xa π→π∗ π→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ MUE B3LYP 20 SF1 6.41 4.72 4.83 5.13 5.25 5.73 4.14 4.84 4.18 2.47 0.82 SF2 7.42 5.32 6.31 6.39 6.42 7.14 4.48 5.40 5.07 2.92 1.41 (cid:6)S2(cid:7) 0.15 0.08 0.79 0.74 0.73 0.86 0.16 0.57 0.66 0.28 X3LYP 21.8 SF1 6.49 4.78 4.80 5.10 5.22 5.71 4.17 4.84 4.17 2.49 0.80 SF2 7.41 5.36 6.21 6.32 6.35 7.05 4.49 5.37 5.00 2.93 1.37 (cid:6)S2(cid:7) 0.14 0.09 0.78 0.73 0.72 0.85 0.14 0.55 0.62 0.27 PBE0 25 SF1 6.68 4.97 4.72 5.06 5.20 5.70 4.19 4.82 4.16 2.52 0.75 SF2 7.88 5.65 6.08 6.26 6.37 7.34 4.48 5.25 4.96 2.98 1.34 (cid:6)S2(cid:7) 0.15 0.10 0.70 0.67 0.68 0.85 0.07 0.37 0.57 0.28 M06 27 SF1 6.45 4.85 4.65 4.95 5.05 5.32 4.03 4.56 3.84 2.37 0.62 SF2 7.94 5.43 5.45 5.58 5.63 6.57 4.10 4.64 3.98 2.57 0.85 (cid:6)S2(cid:7) 0.39 0.21 0.75 0.71 0.66 0.81 0.02 0.05 0.22 0.40 M05 28 SF1 6.80 4.95 4.66 4.98 5.05 5.26 4.01 4.66 3.83 2.27 0.57 SF2 7.02 5.41 5.43 5.53 5.56 5.89 4.00 5.01 3.87 2.34 0.79 (cid:6)S2(cid:7) 0.07 0.10 0.69 0.61 0.56 0.62 0.03 0.60 0.11 0.33 BHHLYP 50 SF1 7.56 5.79 4.12 4.44 4.58 5.20 4.54 4.81 4.07 2.85 0.40 SF2 8.25 6.52 5.02 5.28 5.40 5.79 4.86 5.27 4.51 3.37 0.99 (cid:6)S2(cid:7) 0.26 0.33 0.57 0.54 0.52 0.35 0.06 0.15 0.16 0.26 M08-HX 52.23 SF1 8.09 6.25 3.91 4.30 4.40 4.97 4.36 4.71 3.96 2.73 0.35 SF2 9.24 7.07 3.54 3.92 4.10 4.86 4.15 4.42 3.62 2.63 0.55 (cid:6)S2(cid:7) 0.52 0.49 0.02 0.06 0.06 0.15 0.06 0.06 0.07 0.10 M06-2X 54 SF1 7.63 5.80 3.87 4.25 4.34 4.84 4.34 4.64 3.93 2.64 0.26 SF2 7.54 6.00 3.38 3.71 3.88 4.84 4.05 4.30 3.52 2.40 0.31 (cid:6)S2(cid:7) 0.11 0.28 0.06 0.02 0.02 0.20 0.04 0.06 0.11 0.07 M05-2X 56 SF1 8.03 6.08 3.90 4.29 4.39 5.09 4.45 4.78 4.02 2.70 0.36 SF2 7.27 5.92 3.46 3.82 3.99 4.74 4.12 4.42 3.57 2.45 0.31 (cid:6)S2(cid:7) 0.04 0.13 0.04 0.02 0.02 0.13 0.05 0.12 0.10 0.11 M08-SO 56.79 SF1 7.19 6.41 3.87 4.26 4.34 4.95 4.29 4.54 3.84 2.58 0.33 SF2 7.79 7.27 3.04 3.34 3.49 4.45 3.94 4.17 3.31 2.28 0.52 (cid:6)S2(cid:7) 0.52 0.54 0.22 0.30 0.33 0.40 0.04 0.12 0.18 0.07 CAM-B3LYP 19–65 SF1 7.39 5.66 4.15 4.53 4.71 5.10 4.30 4.69 3.87 2.58 0.36 SF2 8.60 6.97 4.84 5.33 5.58 5.95 5.13 5.85 4.65 3.50 1.20 (cid:6)S2(cid:7) 0.18 0.23 0.30 0.41 0.48 0.19 0.09 0.42 0.07 0.18 ωB97X 15.77–100 SF1 9.35 7.23 4.29 4.56 4.71 5.50 4.16 4.36 3.93 2.34 0.60 SF2 7.42 6.51 0.41 0.74 0.95 3.79 3.35 3.37 2.54 1.45 1.51 (cid:6)S2(cid:7) 0.04 0.19 0.82 0.84 0.86 0.91 0.14 0.83 0.49 0.12 M11 42.8–100 SF1 8.13 6.31 3.85 4.26 4.39 4.89 4.37 4.69 3.81 2.64 0.34 SF2 10.55 8.96 5.69 6.27 6.64 6.83 6.03 6.88 5.51 4.37 2.33 (cid:6)S2(cid:7) 0.32 0.36 0.37 0.48 0.57 0.24 0.06 0.43 0.12 0.07 M06-HF 100 SF1 9.64 7.90 3.17 3.40 3.20 4.78 4.62 4.46 4.08 3.11 0.98 SF2 8.06 7.08 −0.30 −0.12 −0.18 2.62 3.50 3.10 2.32 2.14 1.93 (cid:6)S2(cid:7) 0.38 0.50 0.74 0.72 0.70 0.74 0.10 0.28 0.35 0.35 7.65 5.91 4.00 4.28 4.43 4.59 3.83 3.85 3.60 2.25 aXispercentageofHartree-Fockexchangeinthedensityfunctional. energy is expected to be reduced by the more accurate de- IV.E. CollinearSFvs.noncollinearSF scriptions of ground states obtained by RKS than those by SF-TDDFT. Table VI gives the excitation energies by NC-SF1 for Table V does not show results for BLYP, OLYP, and several density functionals. The noncollinear excitation en- PW91becausethelocalfunctionalsgivesuchhighspincon- ergiesobtainedwithPBE0,B3LYP,M06,M06-2X,ωB97X, tamination with collinear spin-flip TDDFT that state assign- andM06-HFcanbecomparedwiththecollinearonesforthe ments of singlets are ambiguous. Such a failure of local samedensityfunctionalsinTableV. density functionals is disappointing, because local density The most remarkable difference between the collinear functionals are much less expensive than nonlocal ones for and noncollinear treatments is that the degree of spin- largesystems,andthismotivatestheexaminationofthenon- contamination is drastically reduced by applying the non- collinearmethods. collinear kernel; the spin expectation value of squared spin 134111-8 M.IsegawaandD.G.Truhlar J.Chem.Phys.138,134111(2013) nel,ontheotherhandtheerrorbecomeslargeforthedensity functionals with high percentage of Hartree-Fock exchange; 0.26, 0.60, and 0.98 eV for M06-2X, ωB97X, and M06-HF withcollinearkernelagainst0.88,0.66,and1.61eVwithnon- collinearkernel. Overall, the calculated error by NC-SF2 is comparable tothatobtainedwithNC-SF1formostcases,andtheenergy differencebetweenthepredictionsofNC-SF1andNC-SF2is rathersmallcomparedwiththecaseofcollinearxckernelfor the systems with low spin contamination. The SF1 and SF2 noncollinear excitation energies are very different for M06- HFforalkenes,whichhavelarge(cid:6)S2(cid:7). Table S2 in supplementary material65 shows results for NC-NSF2.ComparisonofTableVIandTableS2showsthat theexcitationenergiescalculatedNC-SF2andNC-NSF2are verysimilar. Tables V and VI show that when (cid:6)S2(cid:7) is small, as for M05-2X and M06-2X, SF1 and SF2 show similar errors; on the other hand, when (cid:6)S2(cid:7) is larger than 0.4, SF2 has a sig- nificantly larger error. This trend is especially strong in the noncollinear case. This tendency suggests that, without spe- cialjustification,oneshouldavoidapplyingSF2forsystems with(cid:6)S2(cid:7)>0.4. Another notable conclusion from Table VI is that there islessdifferenceamongthedifferentkindsoffunctionals,as longas(cid:6)S2(cid:7)issmall.However,noneofthefunctionalstested give an error below 0.43 eV with NC-SF1 or below 0.41 eV withNC-SF2. IV.F. Energydifferencesofthesingletsfromthe triplets TheSF1formulahastwocomponents:theenergydiffer- ences of S –T and T –S as shown in Eq. (5). To ascertain 0 1 1 1 whichoftheenergycomponentsprovidethelargercontribu- tionstotheerrorsinSF1,weshowtheseenergycomponents inTablesVIIandVIII. TableVIIliststheS –T energydifference.TheSF2rows 0 1 givetheenergydifferencesbetweenS andT whichareob- 0 1 tained by solving the RKS and UKS equations, respectively, andtheSF1rowsgivetheenergydifferencebetweenthesin- glet spin state obtained by a SF-TDDFT calculation and the tripletspinstateobtainedbysolvingUKS;ω(S |T ).Theref- 0 1 erence used to compute the MUEs is the energy difference between ground singlet and triplet state as calculated in the present work by CCSD(T) with the 6-311(2+,2+)G** basis set. OnecanseeinTableVIIthattheSF-TDDFTexcitation energies showlargerMUEs(oftenmorethantwiceaslarge) FIG.1. Absoluteerrorsofelectronicexcitationenergy(ineV)calculatedby thanareobtainedbyRKSandUKS.Theenergygapbetween (a)SF1,(b)SF2,and(c)NSF2plottedvs.(cid:6)S2(cid:7). T andS stateisbestestimatedbySF2calculationsofden- 1 0 sity functionals with a high percentage of Hartree-Fock ex- operator of NC-SF2 is under 0.4 except for M06-HF which change and one of the range-separated density functionals: includes 100% Hartree-Fock exchange. The comparison of M08-HX, M06-2X, M05-2X, M08-SO, ωB97X, and M06- MUEsbetweenSFandNC-SFshowsthattheMUEisreduced HF. On the other hand, the energy gap is underestimated for byapplyingthenoncollinearkernelforthedensityfunctionals density functionals with a low percentage of Hartree-Fock withalowpercentageofHartree-Fockexchange;0.55,0.68, exchange. and0.43eVforB3LYP,PBE0,andM06withthenoncollinear AlthoughTableVIIshowsthatthegroundstateobtained kernelagainst0.82,0.75,and0.62eVwiththecollinearker- bySF-TDDFTdoesnothaveasignificantspincontamination 134111-9 M.IsegawaandD.G.Truhlar J.Chem.Phys.138,134111(2013) TABLEVI. Thelowestexcitationenergies(ineV)ofvalencestatescalculatedbyNC-SF1andNC-SF2. Ethylene Butadiene HCHO CH3CHO Acetone Pyridine Pyrazine Pyrimidine Pyridazine s-Tetrazine 1B1u 1Bu 1A2 A(cid:3)(cid:3) 1A2 B1 B3u B1 B1 B3u Functional Xa π→π∗ π→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ n→π∗ MUE BLYP 0 NC-SF1 6.27 4.35 4.26 4.60 4.70 4.97 3.67 4.26 3.72 1.96 0.52 NC-SF2 6.73 4.58 4.29 4.59 4.67 4.93 3.73 4.18 3.61 2.04 0.41 (cid:6)S2(cid:7)b 0.08 0.03 0.03 0.01 0.01 0.09 0.01 0.02 0.05 0.03 B3LYP 20 NC-SF1 6.89 5.03 4.09 4.43 4.56 6.51 4.01 4.91 3.83 2.31 0.55 NC-SF2 7.29 5.24 4.20 4.52 4.63 6.73 4.10 4.91 3.85 2.44 0.56 (cid:6)S2(cid:7) 0.12 0.08 0.04 0.03 0.03 0.06 0.02 0.04 0.05 0.07 PBE0 25 NC-SF1 7.28 5.37 3.84 4.20 4.36 6.82 4.01 6.68 3.79 2.36 0.68 NC-SF2 7.73 5.55 3.95 4.31 4.46 7.03 4.09 6.84 3.81 2.46 0.67 (cid:6)S2(cid:7) 0.20 0.11 0.02 0.02 0.02 0.09 0.05 0.07 0.05 0.07 M06 27 NC-SF1 7.43 5.05 4.04 4.51 4.57 4.76 5.13 4.85 3.94 2.26 0.43 NC-SF2 8.25 5.57 4.00 4.44 4.50 5.13 5.14 4.86 3.91 2.27 0.44 (cid:6)S2(cid:7) 0.33 0.37 0.06 0.03 0.03 0.28 0.07 0.19 0.18 0.11 M06-2X 54 NC-SF1 7.09 5.41 4.60 4.89 5.01 7.00 4.43 5.63 4.22 2.74 0.88 NC-SF2 8.28 6.23 4.46 4.55 4.72 7.53 4.32 5.50 4.01 2.74 0.80 (cid:6)S2(cid:7) 0.40 0.33 0.48 0.35 0.11 0.17 0.05 0.37 0.09 0.20 ωB97X 15.77–100 NC-SF1 7.54 7.83 4.06 4.37 4.50 5.01 4.36 6.68 3.77 2.60 0.66 NC-SF2 7.79 7.93 4.15 4.44 4.57 5.38 4.50 6.69 3.98 2.78 0.78 (cid:6)S2(cid:7) 0.09 0.26 0.02 0.02 0.04 0.18 0.04 0.05 0.06 0.09 M06-HF 100 NC-SF1 6.57 5.16 5.76 5.73 5.73 5.58 6.06 8.07 4.74 3.42 1.61 NC-SF2 8.37 7.02 4.85 4.74 4.87 5.60 6.41 7.51 4.49 3.24 1.27 (cid:6)S2(cid:7) 0.51 0.72 0.27 0.05 0.49 0.59 0.17 0.18 0.70 0.12 7.65 5.91 4.00 4.28 4.43 4.59 3.83 3.85 3.60 2.25 aXisthepercentageofHartree-Fockexchangeinthedensityfunctional. b(cid:6)S2(cid:7)isgivenfortheS0stateobtainedbyspin-flipTDDFT. problem, thedeviation oftheexcitation energy fromtheref- In previous work on triplet states, it was already dis- erence is fairly large. This indicates the large error observed cussed that hybrid functionals with similar values of X may in S -S energy is not only due to the spin-mixing problem give quite different errors,67 and this was later discussed in 0 1 butitalsoseemstoreflectinaccuracyinthexcfunctional.A termsoftripletinstabilityinthegroundstate.68,69 relatedfindingwasreportedinthepreviousstudy,38 inwhich the authors compared the performance of SF1-TDDFT with SF-EOM-CCSD66 method and found that the larger error of SF1-TDDFTisduetothexcfunctionalratherthanthespin- IV.G. EffectofpercentageofHartree-Fockexchange flipapproachperse. infunctional Table VIII gives the S –T energy difference; ω(S |T ), 1 1 1 1 The role of Hartree-Fock exchange is especially impor- where the reference value used to compute MUEs is deter- tant for improving the description of two electrons that are minedbyusingtheT –S energygapandtheS -S excitation 1 0 0 1 widely separated (e.g., in delocalized interactions, Rydberg energyfromexperiment: states,long-rangechargetransfer,anddiradicalsobtainedby partialbonddissociation);however,inclusionofHartree-Fock (ES1 −ET1)ref. =((cid:9)ES1−S0)exp−(ET1 −ES0)CCSD(T). exchange introduces static correlation error. It seems to be a (8) consequence of compromising these two demands, one for high X and one for low X, that the density functionals with (We used the same basis set for the coupled cluster calcula- best performance have about 40% Hartree–Fock exchange tions.)AnegativevalueinTableVIIIindicatesthatthetriplet whenoneconsidersbothvalencestatesandRydbergstates.18 isincorrectlypredictedtolieabovethesinglet.Thisincorrect However,whenoneconsidersonlyvalencestatesinLRcalcu- predictionoccursforfourmoleculeswithM06-2X,fivewith lations,thebestfunctionalsarethosewith20%–30%Hartree- M08-SO,andeightwithωB97XandM06-HF. Fock exchange,5 and even the local density functionals can OnecanseethatmostoftheerrorsinTableVIIIarelarger competereasonablywellwiththehybridones18,70 (theerrors than those that we saw for the T –S energy difference in ofthelocaldensityfunctionalsareunacceptablefortheRyd- 1 0 TableVII;inparticular,nineofthe14densityfunctionalsgive bergstatesbutnotnecessarilyforthevalencestates).Onecan agreatermeanunsignederrorforS –T thanforT –S .Thus, seefromTableIII,whichcontainsonlyvalencestates,thatno 1 1 1 0 theerroroftheS –S excitationenergyispredominantlydue remarkable difference is observed between the type of den- 0 1 to the error of the T –S excitation energy for most density sityfunctionalandthepercentage ofHartree-Fock exchange 1 1 functionals. forLRandTDAcalculations.