Extensive TD-DFT Benchmark Singlet-Excited States of Organic Molecules

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2420 J. Chem. Theory Comput. 2009, 5, 2420–2435 Extensive TD-DFT Benchmark: Singlet-Excited States of Organic Molecules Denis Jacquemin,*,† Valerie Wathelet,† Eric A. Perpete,† and Carlo Adamo*,‡ ´ ` Groupe de Chimie-Physique Theorique et Structurale, Facultes UniVersitaires ´ ´ Notre-Dame de la Paix, rue de Bruxelles, 61, B-5000 Namur, Belgium, and Ecole Nationale Superieure de Chimie de Paris, Laboratoire Electrochimie et Chimie ´ Analytique, UMR CNRS-ENSCP no. 7575, 11, rue Pierre et Marie
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  Extensive TD-DFT Benchmark: Singlet-Excited States ofOrganic Molecules Denis Jacquemin,* ,† Vale´rie Wathelet, † Eric A. Perpe`te, † and Carlo Adamo* ,‡ Groupe de Chimie-Physique The´orique et Structurale, Faculte´s Uni V ersitaires Notre-Dame de la Paix, rue de Bruxelles, 61, B-5000 Namur, Belgium, and Ecole Nationale Supe´rieure de Chimie de Paris, Laboratoire Electrochimie et Chimie Analytique, UMR CNRS-ENSCP no. 7575, 11, rue Pierre et Marie Curie,F-75321 Paris Cedex 05, France Received June 10, 2009 Abstract: Extensive Time-Dependent Density Functional Theory (TD-DFT) calculations havebeen carried out in order to obtain a statistically meaningful analysis of the merits of a largenumber of functionals. To reach this goal, a very extended set of molecules ( ∼ 500 compounds, > 700 excited states) covering a broad range of (bio)organic molecules and dyes have beeninvestigated. Likewise, 29 functionals including LDA, GGA, meta  -GGA, global hybrids, and long-range-corrected hybrids have been considered. Comparisons with both theoretical referencesand experimental measurements have been carried out. On average, the functionals providingthe best match with reference data are, one the one hand, global hybrids containing between22% and 25% of exact exchange (X3LYP, B98, PBE0, and mPW1PW91) and, on the otherhand, a long-range-corrected hybrid with a less-rapidly increasing HF ratio, namely LC- ω PBE(20).Pure functionals tend to be less consistent, whereas functionals incorporating a larger fractionof exact exchange tend to underestimate significantly the transition energies. For most treatedcases, the M05 and CAM-B3LYP schemes deliver fairly small deviations but do not outperformstandard hybrids such as X3LYP or PBE0, at least within the vertical approximation. With theoptimal functionals, one obtains mean absolute deviations smaller than 0.25 eV, though theerrors significantly depend on the subset of molecules or states considered. As an illustration,PBE0 and LC- ω PBE(20) provide a mean absolute error of only 0.14 eV for the 228 states relatedto neutral organic dyes but are completely off target for cyanine-like derivatives. On the basisof comparisons with theoretical estimates, it also turned out that CC2 and TD-DFT errors are ofthe same order of magnitude, once the above-mentioned hybrids are selected. 1. Introduction Developing methodological approaches able to accuratelydeliver the transition energies corresponding to electronicallyexcited-states remains a major challenge for theoreticalchemists. Historically, the first computational schemesdeveloped relied on semiempirical theories. 1 The mostsuccessful model, namely ZINDO, 2 was purposed-designedto allow quick estimates of the main features of UV/visiblespectra and remains popular today. However, the quantitativeaspect of the obtained results (absorption wavelengths andtransition probabilities) was found to be highly system-dependent, a problematic feature. 3 - 5 More recently, calcula-tions carried out for organic dyes have indicated that PM5 6 could be a promising approach, 7 but such a claim remainsto be tested on a broader set of transitions and molecules.At the other extreme of the theoretical palette, one findshighly correlated ab initio approaches such as SAC-CI,EOM-CC, MR-CI, or CAS-PT2 that allow very accurate * Corresponding author e-mail: denis.jacquemin@fundp.ac.be(D.J.); carlo-adamo@enscp.fr (C.A.). † FUNDP, Namur. ‡ ENSCP, Paris.  J. Chem. Theory Comput. 2009, 5, 2420–2435 2420 10.1021/ct900298e CCC: $40.75  2009 American Chemical Society Published on Web 08/11/2009  estimates but are limited to rather small systems due to theirextreme computational cost. Two of the most extensiveinvestigations performed with such high-accuracy approaches(CC3 and CCSDR) have been published very recently byThiel and its co-workers. 8,9 Although these contributionscertainly represent a very large computational effort, theyhave been “limited” to molecules of about 15 atoms(naphthalene was the largest compound treated with CC3)and relied on a diffuse-free polarized triple- ζ basis set. Whileit is true that CAS-PT2 and CC2, the two “lighter” wavefunction approaches, could possibly be applied to moleculescontaining about 40 atoms, 10 - 18 the current implementationsof these ab intio theories often do not permit a systematicinclusion of medium effects. This is a problematic drawback,as it is well-known that excited-state properties tend to bemore solvent-sensitive than their ground-state counterparts. 19 Clearly, the difficulties to apply the highly correlatedapproaches to a broad set of molecules in a real-lifeenvironment have yet not been completely solved. In termsof computational cost, one finds an intermediate betweensemiempirical theories and wave function approaches, namelytime-dependent density functional theory (TD-DFT). 20 - 23 TD-DFT is the most widely applied ab initio tool formodeling the electronic spectra of organic and inorganicmolecules 24,25 and can be extended to incorporate environ-mental effects either through a modeling of the bulk environment 19,26 - 29 or through a variety of QM/MMapproaches. 30 - 35 Despite its successes and versatility, TD-DFT is limited and suffers an important drawback: the qualityof the obtained results is profoundly functional-dependent.Indeed, the appropriate selection of the exchange-correlationform is often crucial to grasp chemically sound conclusions.For most excited-states, hybrid functionals that incorporatea fraction of exact exchange (EE) tend to provide moreaccurate estimates than pure functionals. Anyway, transitionwavelengths to excited-states presenting a doubly excitedcharacter or a significant charge-transfer nature are tradition-ally poorly estimated, as are the electronic spectrum of molecules having a strong multideterminantal nature. For surethese deficiencies are related to the approximate nature of today’s implementation, as illustrated by the recently devel-oped long-range-corrected hybrids (LCH), 36 - 44 that appearto correctly appraise the charge-transfer properties. Contraryto the global hybrids (GH), LCH presents an EE percentagedepending on the interelectronic distance, allowing a physi-cally correct asymptotic behavior when the two electronsare far apart.It is quite astonishing that only a limited number of contributions collated the pros and cons of functionals inthe TD-DFT framework, for a significant set of molecules.In Table 1, we summarize the selected methodologies andtraining sets for twelve investigations tackling this question.One can certainly find many other TD-DFT studies usingGH or LCH but often specific to a specific class of molecules. 5,45 - 53 As can be seen, not only the training setbut also the details of the methodologies selected forbenchmarking (including the size of the basis set and thepossible modeling of solvent effects) differ significantly fromone work to the other. We believe it is especially strikingthat most studies include only a very small number of functionals (typically three) and that only four works usedmore than 100 excitations to obtain statistically meaningfulconclusions. Considering the different training sets andprocedures, it is to be expected that the conclusions of theseinvestigations are not perfectly uniform. While the obtainedmean absolute deviation (MAE) for the “best” functional istypically close to 0.25 eV, the actual findings are in factpartly antagonistic, making it difficult to appreciate the“general” functional performance in the TD-DFT framework:1. Tozer and co-workers concluded that CAM-B3LYP 40 leads to much smaller deviations than B3LYP 54 for a varietyof transitions of medium-size chromogens. 55 The averageB3LYP error being completely unacceptable ( > 1.0 eV) forboth Rydberg and charge-transfer states. 55,56 For valencetransitions, all tested functionals (PBE, B3LYP, and CAM-B3LYP) provided similar MAE (0.27, 0.26, and 0.27 eV,respectively). 55 2. Rohrdanz and Herbert found that an accurate descriptionof both the ground-state and excited-state properties of largemolecules was uneasy with common LCH functionals 57 andsubsequently design a LCH functional working for bothground- and excited-states. 58 This new LCH functionalprovided a MAE of about 0.3 eV. 58 3. For the λ max related to π  f π  f transitions in 100 organicdyes, we found, within the vertical approximation, that PBE0outperforms LCH and provides a MAE close to 0.15 eV, 59 the errors being of the same order of magnitude for n f π  f transitions. 60,61 For the same set of  π  f π  f transitions,CAM-B3LYP provided significantly larger deviations (0.26eV). 59 4. Thiel’s group used BP86, B3LYP, and BHHLYP andthey obtained MAE of 0.52 eV, 0.27, and 0.50 eV,respectively, for more than 100 transitions in small mol-ecules, 62 using their own “best theoretical estimates” 8 asreference values.5. Dierksen and Grimme concluded from an extensivevibronic investigation of (mainly) hydrocarbons that theoptimal global hybrid should contain between 30% and 40%of EE. 63 Comparing their vertical (0 - 0) TD-DFT data totheir solvent-corrected experimental references, we calculatedMAE of 0.43 (0.57) eV, 0.21 (0.34) eV, and 0.31 (0.18) eVfor BP86, B3LYP, and BHHLYP, respectively.6. Very recently, 64 Goerigk et al. used the CAS-PT2 resultof ref 8 to benchmark double-hybrid functionals 65 and founda MAE of 0.22 eV for B2PLYP and B2GPPLYP, signifi-cantly smaller than with B3LYP (0.30 eV) and confirmedthis finding on a set of five large chromophores.Consequently, given an arbitrary molecule, it remainsdifficult to know without testing what is (are) “reasonably”the most adequate functional(s) to evaluate the electronicspectra. Should one choose a GH or a LCH? Would the errorbe much larger with a GGA than with a GH? What is the“expected” accuracy with today’s computational procedure?Are ab intio functionals outperforming (or not) parametrizedfunctionals? Should the chosen functional vary for moleculesof different size? Of course, all these questions have beentackled in part in the above-mentioned works, but with nogeneric answer embraced by a large community. Here, we Extensive TD-DFT Benchmark  J. Chem. Theory Comput., Vol. 5, No. 9, 2009 2421  have performed benchmarks that are more complete than anypreviously published data, both from the point of view of the number of molecules considered and of the set of pureand hybrid functionals incorporated. 2. Methodology 2.1. Strategy. As can be seen in Table 1, two philosophiescan be used to benchmark TD-DFT functionals: versusexperiment (VE) or versus theory (VT). Both approacheshave advantages and disadvantages. Trying to closely matchexperiment (VE) is generally desired in most practicalapplications and allows to include in the training set a widerange of molecules and compounds. On the other hand, onewould normally need to compute the full vibronic spectra(and not “simply” vertical transitions) and to perfectly modelthe experimental setup (pressure, temperature, full environ-mental effects, ...), both tasks being impossible for a largeset of solvated molecules. Additionally, it is not alwaysstraightforward to pinpoint the theoretical transition actuallycorresponding to the experimental measures, especially forhighly excited states. Comparisons with accurate wavefunction estimates (VT) allows straightforward and physicallymeaningful comparisons (same conditions, same transitions)but is obviously limited by the availability of theoretical data,i.e. only small molecules can be included. In many cases,CC2 results have been used as reference values for mediumsize molecules, a strategy that we think unsatisfying. Indeed,we computed a MAE of 0.27 eV (0.30 eV) between the CC2/ TZVP and the CAS-PT2/TZVP (“best estimates”) values forthe 103 singlet-excited excited-states of ref 8. 66 Even forlow-lying excited-states, CC2 is often off the theoretical limitby 0.1 eV, 8 a value equal to one-half or one-third of thetypical TD-DFT error.In the following, we will use both philosophies so to beas general as possible. In what concerns the versus theoryscheme, we have selected Thiel’s set (VT set in thefollowing) and mimic exactly the computational procedure(basis set and geometry). For the VE set, we have used acomputational strategy that is at the limit of today’s pos-sibilities for such a set of molecules, trying to circumventthe possible limitations of our computational procedure. Forthe sake of consistency, we have chosen to use a uniformmethodology (basis set, solvent effects, ...) for all VEmolecules. 2.2. General Computational Procedure. All calculationshave been performed with the Gaussian suite of programs,using both the commercial and development versions 67,68 with a tight self-consistent field convergence threshold (10 - 8 to 10 - 10 au). For the VE set, we have followed a well-established three-step approach: 25 i) the ground-state geom-etry of each compound has been optimized until the residualmean force is smaller than 1.0 × 10 - 5 au (so-called tight  threshold in Gaussian); ii) the vibrational spectrum isanalytically determined to confirm that the structure is a trueminimum; and iii) the vertical transition energies to thevalence excited states are computed with TD-DFT. For theVT set, the geometries have been taken from ref 8 and stepiii) directly performed.As the majority of experimental data are obtained incondensed phase, we have included bulk solvent effects inour VE model (all VT calculations are in gas-phase). Thiswas performed at each stage, including geometry optimiza-tions and Hessian calculations, using the well-known Po-larizable Continuum Model (PCM), 19 that is able to obtaina valid approximation of solvent effects as long as no specificinteractions link the solute and the solvent molecules.Typically solvent - solute hydrogen bonds tend to influencemore significantly the n f π  f transitions than their π  f π  f counterparts, and we have tried to select aprotic solvent forthe former, at least when different experimental values areavailable. The list of solvent selected is given in theSupporting Information. The default PCM Gaussian param-eters have generally been used, though for a few calculationsif was necessary to change the atomic raddi (UAKS insteadof UA0) or to switch off the presence of smoothing sphere(NoAddSph) to converge the force minimizations. For therecords, note that some default PCM parameters might differbetween the two versions of the program used. All TD-DFTcalculations have been performed within the nonequilibriumapproximation, valid for absorption spectra. 19 2.3. Functionals and Basis Sets. As we want to assessthe pros and cons of a series of DFT approaches, a veryextended set of functionals has been used. Apart from theTime-Dependent Hartree - Fock approach (TD-HF, refereedto as HF in the following), the selected functionals can beclassified in five major categories: LDA, GGA, meta -GGA,GH, and LCH. In the first category, that is expected to bethe less efficient we have selected only one functional,SVWN5. 69,70 We have chosen four GGAs, namely BP86, 71,72 BLYP, 71,73 OLYP 73,74 and PBE, 75 whereas we have pickedup three popular meta -GGA: VSXC, 76 τ  -HCTH 77 andTPSS. 78 Twelve global hybrids have been used: TPSSh(10%), 79 O3LYP (11.61%), 80 τ  -HTCHh (15%), 77 B3LYP(20%), 54,81 X3LYP (21%), 82 B98 (21.98%), 83 mPW1PW91(25%), 84 PBE0 (25%), 85,86 M05 (28%), 87 BMK (42%), 88 BHHLYP (50%), 89 and M05 - 2X (56%). 90 The LCH con-stitute the last category and use a growing fraction of EEwhen the interelectronic distance increases. This is formallyperformed by partitioning the two-electron operator as 36,40,91 The first term of the rhs of this equation describes the so-called short-range effect and is modeled through DFTexchange, whereas the second term corresponds to the long-range contribution calculated with the HF exchange formula.In eq 1 ω is the range separation parameter, while R and R + β define the EE percentage at r  12 ) 0 and r  12 ) ∞ ,respectively. The LC model uses R ) 0.00, β ) 1.00, and ω ) 0.33 au - 1 in eq 1 37,38 and has been applied to bothGGA and meta -GGA to give LC-BLYP, LC-OLYP, LC-PBE, LC- τ  -HCTH, and LC-TPSS. The approach designedby Vydrov and Scuseria, 42,43 namely LC- ω PBE, with ω ) 0.40 au - 1 and R ) 0, β ) 1, has been used as well. Notethat in LC- ω PBE, the short-range exchange functional canbe rigorously derived 41,92 by integration of the model1 r  12 ) 1 - [ R + β erf( ω r  12 )] r  12 +R + β erf( ω r  12 ) r  12 (1) Extensive TD-DFT Benchmark  J. Chem. Theory Comput., Vol. 5, No. 9, 2009 2423
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