Accuracy of TD-DFT Geometries: A Fresh Look - ACS Publications

Article pubs.acs.org/JCTC

Cite This: J. Chem. Theory Comput. 2018, 14, 3715−3727

Accuracy of TD-DFT Geometries: A Fresh Look Eric Brémond,† Marika Savarese,‡ Carlo Adamo,*,§ and Denis Jacquemin*,∥ †

Sorbonne Paris Cité, ITODYS, UMR CNRS 7086, Université Paris Diderot, 15 rue Jean-Antoine de Baı̈f, F-75013 Paris, France D3 Compunet, Istituto Italiano di Tecnologia via Morego, 30 16163 Genova, Italy § Chimie ParisTech, CNRS, Institut de Recherche de Chimie Paris, PSL Research University, 11 rue Pierre et Marie Curie, F-75005 Paris, France ∥ Laboratoire CEISAM, UMR CNRS 6230, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Cedex 3 Nantes, France ‡

J. Chem. Theory Comput. 2018.14:3715-3727. Downloaded from pubs.acs.org by UNIV OF SUSSEX on 08/15/18. For personal use only.

S Supporting Information *

ABSTRACT: We benchmark a panel of 48 DFT exchange− correlation functionals in the framework of TD-DFT optimizations of the geometry of valence singlet excited states. To this end, we use a set of 41 small- and medium-sized organic molecules for which reference geometries were obtained at high level of theory, typically, CC3 or CCSDR(3), with the aug-cc-pVTZ atomic basis set. For the ground-state parameters, the tested functionals provide average deviations that are small (0.010 Å and 0.5° for bond lengths and valence angles) and not very sensitive to the selected (hybrid) functional, but the errors are larger for the most polarized bonds (CO, CN, and so on). Nevertheless, DFT has a tendency to provide too compact distances, a trend slightly enhanced for functionals including a large amount of exact exchange. The average errors largely increase when going to the excited-state for most bond types, that is, TD-DFT delivers less accurate excited-state distances than DFT for ground state. In particular TD-DFT combined with hybrid functionals provides significantly too short CO and CS/CSe bonds with respective average errors in the −0.026/−0.052 Å and −0.015/−0.082 Å ranges, depending on the selected hybrid functional. For the carbonyl bonds, the sizes of the TD-DFT deviations obtained when selecting standard hybrid functionals are of the same order of magnitude as the EOM-CCSD ones.

1. INTRODUCTION For the computations of the properties of electronically excited states (ES), time-dependent density functional theory (TDDFT)1−3 stands as the de facto single-reference quantum chemical standard. Indeed, despite the constant improvements in both efficient wave function-based approaches [e.g., the second-order algebraic diagrammatic construction, ADC(2),4 and the second-order coupled-cluster, CC2, methods5] and low-scaling “beyond-DFT” schemes [e.g., the Bethe-Salpeter equation, BSE/GW6], TD-DFT remains extremely popular owing to several favorable characteristics. Indeed, beyond its computational efficiency allowing one to treat extended molecular systems, TD-DFT enjoys the availability of both first7−10 and second11−13 analytical geometrical derivatives needed for fast ES geometry optimization and vibrational frequency calculations, which, in turn, give access to several important spectroscopic quantities such as 0−0 energies and band shapes. Nevertheless, in practice, performing a TD-DFT calculation requires choosing an exchange−correlation functional (XCF), a task that is far from trivial as this choice tends to significantly impact the accuracy. To help in the selection of a given XCF, many TD-DFT benchmarks have appeared during the past 15 years, but the majority of them are focused on vertical and adiabatic transition energies; see a review on the topic.14 This is because a large amount of highly accurate theoretical and experimental data are available for transition © 2018 American Chemical Society

energies, allowing for extensive assessments of the pros and cons of a given XCF. For instance, Thiel and his co-workers defined a popular data set,15−17 which encompasses numerous transition energies determined at both CASPT2 (complete active space with second-order perturbation theory)18,19 and CC3 (third-order coupled cluster)20,21 levels of theory for 28 small representative molecules. This data set was used in subsequent benchmarks focused on TD-DFT22−25 or other low-cost methods.26−29 As recently pointed out,30 much less investigation has been devoted to the benchmarking of ES properties, e.g., dipole moments and geometries, to cite two important ES properties, respectively related to solvatochromism and emission wavelengths. This scarcity can probably be attributed to the lack of very accurate reference ES data, whereas, in contrast, Barone and co-workers have showed how accurate (GS) geometries can be obtained by combining experimental and theoretical data.31−33 On the one hand, experimental ES geometries were obtained for a few (very) small molecules only, and these structures are deduced through extensive post-treatments of the measured vibrational patterns of the emission bands, implying complex fitting procedures and, often, frozen parameters.34 On the other hand, theoretical Received: March 29, 2018 Published: June 8, 2018 3715

DOI: 10.1021/acs.jctc.8b00311 J. Chem. Theory Comput. 2018, 14, 3715−3727

Article

Journal of Chemical Theory and Computation

300 valence angles for 60 singlet and 40 triplet ES of 32 medium-sized molecules.46 They determined the ES structures with the TZVP atomic basis set with both TD-B3LYP and CC2 as well as several semiempirical approaches. For the bond distances, the mean absolute deviation (MAD) obtained when comparing the two ab initio approaches are 0.011, 0.003, 0.038, 0.020, and 0.006 Å for CC, CH, CO, CN, and NH bonds, respectively. Other works also compared TD-DFT structures to the geometries obtained from computationally efficient wave function methods, e.g., CC2, but as can be deduced from several works,40,46,47 this method can hardly be viewed as error-free for ES geometries. As can be seen from this literature survey, despite notable efforts by several groups, the quality of previous TD-DFT structural benchmarks is generally limited by the accuracy and/or size of the reference data set. Indeed, several relied on wave function results obtained with rather compact basis set, which introduces a bias, as the basis set dependencies are known to significantly differ at TD-DFT and post-HF levels, e.g., CASPT2 and CC2 levels. To date, the most consistent investigation probably remains ref 37 in which VMC structures are used as benchmarks, but this work treats five compounds only. This is why we present herein a new evaluation of the accuracy of TD-DFT geometries. We use as reference the geometries determined in ref 40 with large atomic basis sets (def2-TZVPP and aug-cc-pVTZ) and highly accurate wave function approaches, as well as additional structures determined herein. Indeed, in this recent work, one of us obtained GS and ES structures for 35 small molecules (39 ES) using CCSDR(3),48 CC3,20,21 and CASPT2.18,19 Interestingly, when selecting active spaces encompassing all valence electrons for the CASPT2 calculations, CC3 and CASPT2 bond distances and valence angles are very close to one another, with typical deviations of ca. 0.003 Å for the former parameter, CCSDR(3) values being also very similar, giving confidence in the accuracy of the reference values.40 In Tables S22 and S23 we compare the ground and excited state geometries of acetylene and formaldehyde obtained with various theoretical models. For acetylene, previous high-level CC calculations including quadruples have been performed,49 and it turns out that all CCSDR(3), CC3, and CASPT2 bond distances are within 0.005 Å of those reference values. For the ground state of formaldehyde, one finds a similarly good agreement between the geometries obtained with the methods used herein and a recently published CCSDTQ structure, with a maximal deviation of 0.003 Å.50 We have optimized the excited-state geometry of formaldehyde at the CCSDT/aug-ccpVTZ level and also obtained bond distances not differing by more than 0.002 Å compared to the CASPT2 and CC3 estimates. As expected from our previous work, the CCSDR(3) CO bond length is slightly too compact in that case,40 though the deviation attains −0.005 Å only as compared to CCSDT. The accuracy of CCSDR(3) and CC3 ES geometries was additionally demonstrated by Hättig, who reported MAE smaller than 0.005 Å for both methods in his comparison of experimental and wave function bond lengths of diatomics.47 In contrast, ADC(2), CC2, and EOM-CCSD geometrical parameters are significantly less accurate,47 especially for polarized bonds.40 Importantly, the compounds treated in ref 40 also encompass several heteroatoms (sulfur, selenium, fluorine, chlorine, and so on) allowing one to obtain conclusions beyond the usually considered CNOH molecules.

ES geometries determined using highly reliable wave function schemes remain rare as well.35−40 Let us review previous TD-DFT investigations aiming at pinpointing the most adequate XCF for ES geometry optimizations of minima, though at least one benchmark exists for conical intersections as well.41 To our knowledge, the first TD-DFT benchmark of molecular ES geometries is due to Furche and Ahlrichs,9 who compared the performances of six XCF (LDA, BLYP, BP86, PBE, B3LYP, and PBE0) for 40 bond lengths, 15 valence angles, and 3 dihedral angles of (mostly) tiny molecules, using experimental data as reference. For this set, B3LYP leads the smallest mean absolute errors (MAE), 0.013 Å and 2.1°, for bond lengths and valence angles, respectively, but the other XCF provide very similar results. Sets of experimental structures, closely related to the one of Furche and Ahlrichs, were used by Liu and co-workers42 and by Hellweg et al.43 to respectively assess the merits of the ωB97 XCF and CC2 schemes, including the SOS-CC2 approach. It is noteworthy that these authors reported MAE of 0.029 and 0.026 Å for the CC2 and SOS-CC2 bond lengths, respectively, that is, average deviations twice larger than their B3LYP counterpart. Of course, theoretical benchmarks of TDDFT geometries were also reported, but they were generally limited by the quality of the reference values. In 2010, Guido and co-workers used the CASPT2/6-31G(d) Page and Olivucci’s set35 to assess the merits of six hybrid XCF (B3LYP, B3P86, PBE0, BMK, BH&HLYP, and CAMB3LYP).44 They found TD-DFT errors rather insensitive to the selected XCF, but significantly dependent on the bond nature with errors of 0.010−0.021 Å for CC bond lengths, but as high as 0.063−0.082 Å and 0.031−0.038 Å for CO and CN distances, respectively. In ref 37, the authors used VMC (variational Monte Carlo) geometries determined for five small molecules (acetone, trans- and cis-acrolein, methylenecyclopropene, and the propanoic acid anion) to assess the merits of several other theories, i.e., CASPT2, NEVPT2 (N-electron valence state perturbation theory), CC2, SACCI (symmetry adapted cluster configuration interaction), and TD-DFT (B3LYP, PBE0, M06-2X, CAM-B3LYP, and LCBLYP). They found that CASPT2 and NEVPT2 provide bond lengths close to the VMC reference, whereas both CC2 and SACCI deliver larger deviations, especially for the CO and CC bonds, respectively. Similarly to the conclusions of the work of Guido and co-workers, Guareschi and Filippi found that TDDFT is rather poor for the CO bond length and that no definitive conclusions regarding the optimal choice of a XCF could be reached, but for the fact that LC-BLYP provides unsatisfying results. Using their SACCI/D95(d,p) data as reference, Bousquet et al. found that TD-PBE0 provides reasonably accurate ES structures for the nine heteroaromatic cycles they studied, though the MAE was significantly compound and ES dependent.38,39 There are also works comparing TD-DFT and CC2 geometries. For instance, in 2013, Guido and co-workers determined the ES structures with CC2/aug-cc-pVTZ and TD-DFT (B3LYP, PBE0, and CAMB3LYP) for a set of 15 compounds, including both the molecules from Page and Olivucci’s set and larger push−pull dyes.45 They found that no obvious pattern relates the CC2 ES structures to their CASPT2 counterparts for the former group, whereas, for the latter group, CAM-B3LYP clearly outperforms B3LYP and PBE0 for ES geometries, though the reverse is found for the ground-state geometries. Recently, Tuna et al. defined a very extended set of more than 500 bond lengths and 3716


Article


2. COMPUTATIONAL DETAILS All our DFT and TD-DFT computations are performed with the Gaussian16 code, 13 using default thresholds and algorithms, except for those detailed below. For all molecules, we optimize both the GS and ES, at DFT and TD-DFT levels, respectively, starting with the structures obtained at the reference level of theory, e.g., CC3. These optimizations are performed until the residual mean square force becomes smaller than 1 × 10−5 au (tight threshold in Gaussian). We subsequently compute analytically the vibrational frequencies to ascertain the nature of the optimized structure for eight representative XCF (see section 3.2). All these calculations rely on an improved DFT integration grid (at least the 99,590 pruned ultraf ine grid), use a stricter-than-default energy convergence threshold (ca. 10−9 au), and use the full-response TD-DFT approach. Discussions of the impact of the TammDancoff approximation (TDA) on ES structures and vibrations can be found elsewhere.12,51,52 To be consistent with the data of ref 40, we apply the aug-cc-pVTZ atomic basis set, except when noted. We evaluate here the performances of 48 XCF including 13 semilocal, 24 global-hybrid, and 11 rangeseparated hybrid XCF. The complete list of XCF considered is given in Table 1. Note that LC-BLYP and LC-PBE use the latter parametrization,53 that is, rely on an attenuation parameter of 0.47 bohr−1. The reported values are numerically accurate up to 1 × 10−3 Å for bond lengths, 0.1° for valence angles and 0.2° for dihedral angles. We also underline that the XCF including empirical dispersion corrections possess dispersion parameters optimized for GS properties only and that dispersion effects significantly differ in the ES.54 The interested reader can find a more complete discussion regarding excited state dispersion effects in ref 55 and references therein. Most of our reference data are taken from refs 40 and 110. Nevertheless, we extend this previous data set below by (i) providing for molecules treated in these earlier works’ improved estimates; and (ii) adding several additional molecules to the data set. These new geometries are briefly described below, and details can be found in the first part of the Supporting Information (SI). To perform these calculations, we follow the same protocol as in ref 40; i.e., we first optimize the structures and compute the vibrational spectra at the (EOM-)CCSD/def2-TZVPP level with Gaussian16,13 before reoptimizing the obtained structures at CCSDR(3) and, when technically feasible, CC3 levels using the Dalton code.111 For the first part, the energy and geometry convergence thresholds are systematically tightened, with requested convergence of 10−10−10−11 au for the SCF energy and a residual mean smaller than 10−5 au for the forces. During the (EOM-)CCSD calculations, all electrons including the core ones are correlated (so-called f ull option) and the CCSD energy convergence threshold is set to 10−8−10−9 au, whereas the EOM-CCSD energy convergence is tightened to 10−7− 10−8 au. Analytical gradients are available in Gaussian16 for EOM-CCSD, and therefore these gradients are differentiated numerically to obtain the vibrational frequencies. The CCSDR(3) and CC3 optimizations use default convergence thresholds of Dalton and all electrons are correlated. We recall that analytical gradients are not available for these two levels of theory in Dalton, so that the optimizations are performed purely numerically.

Table 1. List of the Exchange−Correlation Density Functionals Considered in This Worka name

year

typeb

% EXX

refs

SVWN BLYP PBE B97-D B97-D3 SOGGA11 N12 HCTH407 TPSS M06-L M11-L MN12-L MN15-L TPSSh O3LYP B3PW91 B3LYP mPW3PBE X3LYP B97-1 B97-2 B98 APF APFD PBE0 mPW1PW91 mPW1LYP mPW1PBE M05 M06 SOGGA11-X BMK MN15 BH&HLYP M08-HX M06-2X M05-2X HISSb LC-BLYP LC-PBE ωB97 CAM-B3LYP ωB97X ωB97X-D HSE06 N12-SX MN12-SX M11

1980 1988 1996 2006 2011 2011 2012 1997 2002 2006 2011 2012 2016 2002 2001 1992 1993 1998 2004 1998 2001 1998 2012 2012 1999 1997 1998 1998 2005 2008 2011 2004 2016 1993 2008 2008 2005 2008 2001 2001 2008 2004 2008 2008 2006 2012 2012 2011

LDA GGA GGA GGA+D GGA+D GGA NGA GGA mGGA mGGA mGGA mNGA mNGA GH-mGGA GH-GGA GH-GGA GH-GGA GH-GGA GH-GGA GH-GGA GH-GGA GH-GGA GH-GGA GH-GGA+D GH-GGA GH-GGA GH-GGA GH-GGA GH-mGGA GH-mGGA GH-GGA GH-mGGA GH-mGGA GH-GGA GH-mGGA GH-mGGA GH-mGGA RSH-GGA RSH-GGA RSH-GGA RSH-GGA RSH-GGA RSH-GGA RSH-GGA+D RSH-GGA RSH-NGA RSH-mNGA RSH-mGGA

0 0 0 0 0 0 0 0 0 0 0 0 0 10 12 20 20 20 21 21 21 22 23 23 25 25 25 25 27 27 40 42 44 50 52 54 56 0−60−0 0−100 0−100 0−100 19−65 16−100 22−100 25−0 25−0 25−0 43−100

56−58 61−63 65 67 68 70 72 74−76 78 80 82 84 85 87 89 91 91−94 66 97 74 99 106 108 108 59, 60 64 66 66 69 71 73 77 79 81 83 71 86 88 53, 61−63, 90 53, 65, 90 95 96 95 98 100−105 107 107 109

a

EXX denotes the percentage of exact-like exchange energy (rounded). bList of acronyms: LDA, local density approximation; GGA, generalized gradient approximation; mGGA, meta-GGA; NGA, nonseparable gradient approximation; mNGA, meta-NGA; +D, addition of empirical dispersion corrections; GH, global-hybrid; and RSH, range-separated hybrid.

At this stage, it is important to underline that only valence singlet ES of closed-shell molecules are considered here. This choice is justified as those states are the most generally considered in ES optimizations of “real-life” systems. In addition, to our knowledge, no highly accurate reference 3717


Article


Difluorodiazirine. The ES structure of this compound, a precursor of the CF2 carbene, was previously investigated both theoretically116,117 and experimentally,118−120 and we present here the first CC3 results. All levels of theory predict an increase of the C−N distances as well as a contraction of the C−F bonds when going from the GS to the ES, which is consistent with the precursor nature of this compound. For the elongation of the NN bond after absorption, our CC3 estimate of +0.038 Å fits well the most recent measurement (+0.036 Å),120 as well as a previous GVVPT2 value (+0.031 Å),116 but is surprisingly much larger than a QCISD estimate (+0.002 Å).117 Isocyanogen. This elusive compound was synthesized in 1988 only,121 and its GS and ES structures were previously investigated at the EOM-CCSD level by the Sherill’s and Crawford’s groups.122 We provide CC3 geometries for this molecule in Table S17 of the SI. The lowest excited state breaks the cylinder symmetry and presents a terminal CN bond very strongly elongated compared to the ground state, our best estimate being +0.217 Å for this lengthening. For that molecule, the CC model used has a significant influence on the computed ES parameters, consistently with the results of the T1 diagnostic that hints at a non-negligible multireference character (see also below). Pyrrole. As for cyclopentadiene, the lowest ES of pyrrole is out-of-plane (A″ in the Cs point group). The computed CCSDR(3) geometrical parameters are rather close from previous CASPT2,123 Mk-MRCCSD,36 and SACCI38 estimates (Table S18), with an elongation of the CC bonds (+0.072 Å) and a shortening of the C−C bond (−0.053 Å) upon electronic transition. Streptocyanine-C3. For this short compound, the C2v symmetry needs to be enforced in the ES to avoid the twisting leading to a conical intersection.124 One can notice from Table S19 that both EOM-CCSD and CCSDR(3) foresee similar elongations of the CN and CC bonds upon absorption. This cyanine derivative was added in the set, because it is known that TD-DFT often delivers rather poor transition energies for these dyes, irrespective of the applied XCF.124−128 Thiocarbonyl Bromochloride. This compound is a member of the formaldehyde family,129 which undergoes a significant elongation of the CS bond (+0.126 Å) and a puckering (33.9°) in the ES according to CCSDR(3). 3.2. Selected Examples. The reader can find in parts III and IV of the SI DFT and TD-DFT parameters computed for all molecules modeled with a panel of eight popular hybrid functionals, namely, B3LYP, PBE0, M06, BMK, M06-2X, CAM-B3LYP, ωB97X, and ωB97X-D (Tables S24−S68). This set was selected because the majority of TD-DFT optimizations of molecules are performed with these XCF and because this set is representative: it contains global hybrids having small and large ratios of exact-like exchange (EXX), as well as three range-separated hybrids with different attenuation parameters. Before performing a statistical analysis, let us first discuss a few representative molecules as described by these XCF, using as reference the most accurate geometries available. For all investigated compounds, the DFT optimizations of the GS geometries are, as expected, unproblematic. In contrast, a few TD-DFT ES optimizations fail and this outcome is dependent on both the XCF and the compound. These failures appear because (i) a conical intersection-like point is reached, that is, the transition energy becomes vanishingly small as the minimization of the ES energy proceeds; (ii) no chemically

CC3 geometries are available yet for Rydberg states or openshell molecules. As it should be recalled that the behavior of XCF differs for valence and Rydberg states,112 it is clear that our conclusions will be limited to low-lying singlet valence ES only. In the following, we select as reference the most accurate geometries available; the interested reader can find them at the top of the tables listing the TD-DFT parameters in the SI.

3. RESULTS AND DISCUSSION 3.1. Revised and Extended Reference Database. Although the reference database of refs 40 and 110 is rather large, we choose to extend the database in three directions: (i) adding CC3 geometries for some compounds; (ii) obtaining structures with a basis set containing diffuse orbitals, namely, aug-cc-pVTZ, when possible; and (iii) adding extra molecules not considered in the original set. Complete data are given in the SI. For some molecules already treated in our previous work,40 CC3 geometries are now available in lieu of CCSDR(3) structures. This includes the following: (i) cyanoacetylene for which CC3 estimates are obtained for the lowest A″ ES; (ii) cyanoformaldehyde in its lowest ES; (iii) the lowest Au of diacetylene; (iv) the n → π* ES of glyoxal; (v) the ES of phosgene, for which extrapolated CC3 values are now given; (vi) the ES geometry of selenocarbonyl difluoride; (vii) tetrazine for which we provide the full CC3/aug-cc-pVTZ values rather than the basis set extrapolated ones for both states; (viii) the ES of thioformyl chloride; and (ix) thiophosgene. The differences between the CCSDR(3) and the CC3 geometrical parameters are generally small, as they are in the (1−4) × 10−3 Å range for the bond lengths. There are however exceptions: the central C−C distance in glyoxal, the CO distance in phosgene, and the CS bond lengths in thioformyl chloride and thiophosgene, that respectively attain 1.476, 1.322, 1.728, and 1.740 Å with CC3, and are therefore significantly modified compared to the corresponding CCSDR(3) values of 1.485, 1.314, 1.721, and 1.732 Å. In addition, for acetaldehyde, acrolein and pyrazine CCSDR(3)/ aug-cc-pVTZ results given in the SI replace the previous CCSDR(3)/def2-TZVPP values. All bonds are slightly more contracted with the diffuse-containing basis set, which fits earlier findings.40 The data corresponding to these 13 molecules can be found in Tables S1−S13 of the SI. CC geometrical parameters for seven additional molecules are given in the SI, and we briefly discuss the results below. Benzoquinone. We optimized the lowest state of Au symmetry, enforcing the D2h point group, and the results are displayed in Table S14 in the SI. This symmetry yields a stable minimum at the CIS level.38 The CCSDR(3)/def2-TZVPP geometrical parameters are in good agreement with previous SACCI/D95(d,p) estimates,38 but differ significantly from CASSCF results.113 As expected from previous benchmarks,37,40,45−47,114 CC2 gives a significantly too long CO bond in the excited state compared to CCSDR(3) for benzoquinone (1.311 Å versus 1.273 Å). Cyclopentadiene. This cyclic derivative presents a puckered π → π* excited state in which the CH2 group is significantly out of the π plane. As can be seen in Table S15, our CCSDR(3) parameters are very close to those obtained previously by Jagau and Gauss with the Mk-MRCCSD approach,36 whereas the experimental information is very limited.115 3718


Article


M06 predict that the lowest ES belongs to the C2h (Au) rather than the D2h (B3u) point group obtained with the other six XCF or with EOM-CCSD.40 We therefore performed a CCSDR(3)/def2-TZVPP ES minimization starting from the B3LYP minium and it led back to the high-symmetry structure. For pyrazine B3LYP, PBE0, and M06 D2h ES structures present an imaginary frequency, absent with the other XCF. Likewise, for the Au ES of benzoquinone, B3LYP does not yield a true minimum in the D2h point group, in contrast to the other XCF. In these two latter cases, it was unfortunately impossible to run CCSDR(3) geometry optimizations in a lower point group to obtain a more definitive conclusion. Let us now turn toward quantitative comparisons for a few representative derivatives. For formaldehyde (Table S36), the DFT GS CO bond length tends to be too small by ca. −0.010 Å with all XCF. In the ES, the error presents the same sign but is quantitatively much larger, as the TD-DFT CO bond lengths are underestimated by values ranging from −0.034 Å (B3LYP) to −0.046 Å (CAM-B3LYP, M06-2X, and ωB97X), with a clear trend of increasing discrepancies when XCF with large EXX are selected. As a consequence, the GS to ES CO elongation, which attains +0.118 Å with CC3,40 and +0.116 Å34 or +0.117 Å135 experimentally, is always too small with TD-DFT, the best estimate being given by B3LYP (+0.092 Å). We recall that, for this parameter, CC2 is quite inaccurate as well, with an overestimated CO elongation of +0.136 Å.40 For the puckering angle in the ES, the TD-DFT estimates are in the 30.9° and 36.5° range with no clear pattern in terms of EXX. For comparisons, the CC3 value is 36.8°, whereas ADC(2), CC2, and EOM-CCSD, respectively, provide estimates of 18.9°, 29.5° and 30.9°;40 that is, TDDFT with typical hybrid XCF is at least as accurate as these three popular wave function approaches for this dihedral angle. In thioformaldehyde and selenoformaldehyde, two molecules that remain planar in their lowest n → π* singlet ES,136,137 the TD-DFT ES CS and CSe bond lengths are always shorter than their CC3 counterparts, and the TD-DFT estimate of the elongation of these bonds upon photon absorption is too small as well (Tables S47 and S43, respectively). In both molecules the most accurate ES bond distances are given by B3LYP: 1.681 and 1.825 Å for H2CS and H2CSe, respectively. These values can be compared to their CC3 counterparts of 1.709 and 1.843 Å. In contrast, ωB97X delivers much too contracted ES bonds with an absolute error exceeding 0.050 Å (1.653 and 1.789 Å). A similar tendency is obtained in most of the treated halogenated (thio or seleno)formaldehyde derivatives: the TD-DFT ES CX bond distances are always smaller than the reference values, the only exception being F2CSe for which the B3LYP value (1.913 Å) slightly exceeds the CC3 reference (1.908 Å). We also underline that, in these halogenated compounds, there is generally no clear evolution of the CX bond distances with increasing EXX, as illustrated by the data obtained for phosgene, for which the most accurate XCF is M06-2X (1.280 Å, see Table S44). According to CC3, the bond length alternation (BLA, the difference between single and multiple bond lengths) of diaceytlene goes from 0.158 to 0.022 Å upon transition from the GS to the lowest Δu ES (Table S33). For the GS, the DFT BLA are accurate (in the 0.160−0.182 Å range) and they are larger for the XCF presenting the largest share of EXX, an expected trend.138,139 The same holds in the ES, with TD-DFT BLA going from 0.021 Å (M06) to 0.033 Å (M06-2X), the residual discrepancies with respect to CC3 being mainly

sound minimum is found on the ES potential energy surface; or (iii) an ES obviously different from the one delivered by CC3, CCSDR(3), or CASPT2 is reached with TD-DFT, irrespective of the starting point. The first problem is encountered for diazomethane, nitrosomethane and trifluoronitrosomethane with both BMK and M06-2X, and for cyclopentadiene and ketene with M06-2X, whereas the other XCF are able to locate a reasonable ES minimum for these derivatives. The second issue is noticed for formyl chloride: the TD-B3LYP optimization leads to the breaking of the C−Cl bond with both the cc-pVTZ and aug-cc-pVTZ atomic basis sets. The third difficulty is found for isocyanogen for which three XCF (B3LYP, PBE0, and ωB97X-D) yield a bent minimal structure with a strong elongation of the CN bond, as in CC3, whereas the five other XCF incorrectly foresee that the lowest ES conserves the GS cylinder symmetry with significant variations of the three bond distances (Table S40). We underline however that according to the CCSD T1 diagnostic,130 the ES of isocyanogen could present a significant multiconfigurational character and, as no CASPT2, MRCI, or alike estimates are available for that molecule, it is removed from the statistical analysis in the next section. Therefore, among the eight above-listed XCF, only two (PBE0 and ωB97X-D) are able to provide qualitatively sound ES structures for the full set of compounds considered herein. Nevertheless, we note that, for pyrrole, using diffuse orbitals during the TD-DFT computations yields strong state-mixing preventing ES geometry optimizations and that cc-pVTZ has to be applied instead of aug-cc-pVTZ for the TD-DFT minimizations of that compound. In addition, even when the TD-DFT optimizations lead to a meaningful minimum, geometries significantly differing from the reference can be obtained. A first interesting and challenging case is thioformyl chloride (ClHCS, Table S51). For this formaldehyde homologue, both CCSDR(3)40 and CC3 foresee a puckered ES, which is consistent with experiment.131 Our CC3 value for the out-of-plane dihedral angle of the hydrogen is 23.2°, a good match with MR-AQCC (25°)132 and CASPT2 (25.8°)40 estimates. With TD-DFT, a planar ES structure is reached with four XCF (M06-2X, CAMB3LYP, ωB97X, and ωB97X-D), whereas the four others correctly foresee a puckered ES, though with a too small dihedral angle (from 8.8° to 16.9° depending on the XCF). Therefore, the XCF including the largest share of EXX tend to produce a too flat ES structure for this compound. Interestingly, EOM-CCSD, a method that provides, like HF, a too localized description of the ES15,133,134 suffers from the same problem for thioformyl chloride.40 For difluorodiazirine (Table S35), both TD-BMK and TD-M06-2X yield an imaginary frequency in the C2v point group. The optimization in the Cs point group gives a true minimum for these two XCF, but the final structure is strongly asymmetric, the ES C−N bond lengths being 1.355 and 1.656 Å with BMK and 1.389 and 1.521 Å with M06-2X. This outcome is contradictory with both our EOM-CCSD result and previous ab initio estimates116,117 that deliver a stable C2v ES structure. However, to ascertain that this result is not an artifact of EOM-CCSD as in thioformyl chloride, a CCSDR(3)/def2-TZVPP optimization was performed starting from the TD-M06-2X C s minimum, and it led back to the C2v ES described in Table S17, hence confirming that BMK and M06-2X provide a qualitatively incorrect result. A few other disagreements on the point group of the ES are obtained. For tetrazine, B3LYP and 3719


Article

Journal of Chemical Theory and Computation ascribable to the triple bond length. For comparisons, (EOM)CCSD gives a slightly too large BLA in both the GS (0.172 Å) and the ES (0.034 Å);40 that is, the (EOM-)CCSD results are further away from the CC3 references than TD-PBE0’s values. Qualitatively similar conclusions are reached for the Σ−u ES of cyanogen: the DFT and TD-DFT BLA are rather close to their CC3 counterparts, though they tend to be slightly overestimated. For the shortest streptocyanine (Table S47), the DFT GS C−N bonds are slightly too contracted and the same trend is found in the ES but with larger quantitative deviations. At the exception of B3LYP, the obtained bond lengths for that compound are almost unaffected by the functional selected within our eight-XCF set. These trends of a too localized TDDFT description and a limited XCF sensitivity are consistent with literature reports for the transition energies of cyanine derivatives, 124−128 and are also found in the longer streptocyanine structure containing three carbon atoms capped by two amino groups (Table S63). For difluorodiazirine (Table S35), beyond the above-mentioned symmetry difference with BMK and M06-2X, we found that the CC3 geometries are generally well-reproduced by the other XCF, ωB97X-D being especially accurate for the ES C−N bond length. Nevertheless, in CF2N2 all XCF giving the correct point group symmetry produce slightly too contracted NN bonds in both the GS and the ES. For the ES of acrolein (Table S56), the ES CO bond length is always too short with TD-DFT, whereas the C−C and CC bond distances are more accurately described, especially with PBE0. 3.3. Statistical Analysis. Data obtained with all the 48 XCF listed in Table 1 are reported in part V of the SI (Tables S69−S113). To perform our statistical analyses, we select CC3 or CCSDR(3) values as references, choosing the former when possible. However, for the Bu state of acetylene and the lowest ES of diazomethane, CASPT2 benchmarks are selected since significant multireference characters were previously noticed for these two states.40 This is also why the lowest ES of isocyanogen is excluded from the statistics: a non-negligible multireference character is foreseen and the coupled-cluster values might be less accurate. Although our focus in this contribution is not the ground electronic state, let us first briefly discuss the statistical results obtained for the GS parameters, as they can be next compared to the error patterns obtained for the ES values. The results are reported in Tables S114 and S116 in the SI for the molecules for which the CC3 and CCSDR(3) values are used as references, respectively. For all XCF, the statistical trends are highly similar, and we therefore only discuss the results relying on the highest level of theory as benchmark, i.e., CC3 references. As reported in a preliminary investigation dealing with GS structure evaluation,140 the vast majority of the considered XCF accurately describe the bond distances with MAE typically ranging from 0.005 to 0.010 Å (Table S115). Out of 48 XCF considered, only seven deliver MAE larger than 0.010 Å: SVWN, BLYP, M11-L, BH&HLYP, HISSb, LCBLYP, and LC-PBE, but these XCF are quite rarely used for geometry optimizations of molecules. The clear tendency of both global and range-separated hybrids is nevertheless to deliver slightly too short bonds, but for the CH bonds, and only the XCF with rather low EXX show negative mean signed error (MSE; see Table S114). Graphical representations of the mean absolute and signed errors for each bond type are given in Figures 1 and 2, respectively. These plots show cumulative errors; that is, the MAE obtained for the different bond types

Figure 1. Cumulative mean absolute errors for (left) ground and (right) excited states made for six types of bond lengths considering the CC3 or CASPT2 reference values. All values are in angstroms. See Figures S1 and S2 in the SI for comparisons with the other sets of compounds.

are added in Figure 1, and the same holds in Figure 2 for each error signs. Plots for other sets, as well as representation of the maximal errors and histograms, showing the statistical dispersion obtained with all XCF can be found in the SI. For a given hybrid XCF, the largest errors are typically obtained for the CO, CN, and CS/CSe bonds. Indeed, the more the bond is polarized, the more difficult it is for DFT to describe it. This trend is not found with semilocal XCF. The results obtained with hybrids are similar to the ones obtained with CC2, though the sign of the error is reversed with DFT.40 This means that benchmarking DFT using CC2 values would artificially yield larger errors for DFT, and this finding underlines the need of robust references.33 Globally, going from semilocal to hybrid XCF improves the quality of the estimates. O3LYP delivers the most trustworthy results with no MAE larger than 0.006 Å for all types of bonds and a limited statistical dispersion of the errors (see Figure S9), but 22 XCF give MAE not larger than the 0.010 Å threshold for all bond types. Like in one of our previous works,140 M06-L, PBE0, and 3720


Article


mitigated at the CCSDR(3)/def2-TZVPP level, due to an error compensation mechanism, this hints that it is probably safer to select, as for the GS, the compounds for which CC3 or CASPT2 reference values have been obtained. A second question is the statistical impact of the few molecules for which some XCF are unable to deliver a meaningful ES minimum. In Tables S126−S131 in the SI, we report the MSE and MAE determined with all XCF, removing the compounds for which at least one XCF fails. The trends are globally similar to those computed without excluding any compound. In Figures 1 and 2, the cumulative errors obtained for the ES structures are shown for the cases with the most accurate benchmarks (representation for other sets are available in the SI). At the exception of the CH bond lengths, that are almost unaffected by the change of electronic state, the errors are larger for the ES parameters than for the corresponding GS distances. Overall, this effect is rather small for the CC and CX (X = F, Cl, or Br) bonds and significant for the CN distances and becomes large for both CO and CS/CSe bond lengths, which are crucial in many chromophores. Indeed, while the MAE of M06-2X attain 0.010 and 0.012 Å for GS CO and CS/CSe bonds, the corresponding ES errors are much larger: 0.042 and 0.043 Å, respectively. For most XCF, one indeed notices 3−4 times larger deviations in the ES than in the GS for these polarized bonds, which tend to be too short with TD-DFT, as shown by the MSE of Figure 2. The tendency of TD-DFT to provide too compact ES bonds for the present set is also obvious from the histograms displayed in Figures S12−S15 in the SI. As can be seen in Figure 1, there is a general correlation between the accuracies obtained for the GS and ES parameters: XCF that yield inaccurate ground-state geometries, e.g., M11-L and LC-PBE, are also providing the least accurate excited-state parameters. For the considered molecular set, the five most accurate XCF for ES geometries (smallest overall MAE) are B97-D, B97-D3, TPSSh, O3LYP, and B3LYP. These functionals also deliver rather small maximal deviations (see Figure S5 in the SI) and show quite limited statistical dispersion, especially B97-D3 (Figure S12 in the SI). As two of these five XCF include dispersion corrections, we evaluated, for the latter functional, the impact of the addition of extra dispersion corrections using three empirical models (B3LYPD, B3LYP-D3, and B3LYP-D3BJ).68,141 However, irrespective of the selected dispersion approach, no significant variations of the obtained MSE and MAE compared to the standard B3LYP XCF were found for both GS and ES parameters (see Tables S132 and S133 in the SI). As found for transition energies,14,142 the global hybrids encompassing up to 30% of EXX produce similar results, demonstrating the strong importance of the single hybridization scheme with respect to the semilocal approximations for the evaluation of excitedstate structures. For the valence angles, the errors given by TDDFT are also larger than in the GS, with MAE going from 1.0° (TPSSh and B3LYP) to 4.2° (HISSb), most XCF providing average deviations of ca. 1.1°−2.5°, i.e., also 3−4 times larger than in the GS. The errors are quite large for the ES dihedral angles (here puckering angles of formaldehyde homologues), for which the vast majority of XCF provide too small values, with MAE ranging from 1.5° (M05) to 10.8° (LC-BLYP), the XCF including the largest EXX ratio delivering the least accurate results. In Figures S16−S21, we provide histograms showing the errors of the 48 XCF for each molecule of the CC3 set. It is obvious that the statistical dispersions are larger for ES than for

Figure 2. Cumulative mean signed errors for (left) ground and (right) excited states. See caption of Figure 1 for more details.

SOGGA11-X are confirmed as excellent candidates for GS geometry optimizations. Several other XCF are also very satisfying if CS/CSe and CX bonds are not considered. Indeed, as depicted by Figure 1, the former seem problematic for XCFs having a large faction of EXX, whereas the latter are inaccurately described by most semilocal XCF. For the valence angles, all approaches are very accurate with MAE not exceeding 0.5°. Lets us now turn toward ES parameters. As for the GS, we determined the statistical errors for the full set of compounds (Tables S124 and S125) and for a subset for which the most reliable benchmark geometries have been obtained (Tables S120 and S121). The overall trends for both sets are very similar. Only the CO bond length statistics slightly differ from one set to the other; e.g., M06 gives a MAE of 0.035 and 0.040 Å, for the first and second sets, respectively. This outcome is partially related to the limits of the CCSDR(3) method that is very accurate but nevertheless yields its largest error for CO bonds.40 Indeed, the CCSDR(3)/aug-cc-pVTZ CO distances are shorter than the reference data by −0.006 Å on average for a very similar set of compounds.40 Though this error is 3721


Article


Figure 3. Correlation plot for all tested XCF between the MAE for the excited-state structures (Å) and adiabatic energies (eV). Structural and energy deviations are computed with the all bond criteria and adiabatic excitations, respectively, using the CC3 values as references. The right panel corresponds to the gray zone of the left panel, zoomed.

the selected reference value. We also considered several compounds with sulfur atoms and various halogen centers, so to extend the conclusions beyond the usual “CNOH” systems. To our view, these choices naturally imply a restriction on the size of the considered compounds that can be considered, which is the main limitation of the present contribution. While, for the GS structures, DFT systematically allowed one to attain a meaningful minimum, qualitative TD-DFT problems, strongly dependent on the chosen XCF, were detected in several cases, typically because the minimization yields to a conical-intersection-like point or because the final geometry does not belong to the point group obtained with the reference method. Among a set of eight popular hybrid functionals (B3LYP, PBE0, M06, BMK, M06-2X, CAM-B3LYP, ωB97X, and ωB97X-D), only two, PBE0 and ωB97X-D, systematically provided physically sound structures for all tested compounds. Overall, the vast majority of XCF deliver accurate ground-state parameters with typical deviations smaller than 0.010 Å and 0.5° for bond lengths and valence angles (compared to CC3). It was nevertheless concluded that DFT, relying on global and range-separated hybrids, tends to produce slightly too compact bonds (but for CH), and that the deviations are often larger for the most polarized bonds. The average deviations largely increase in the ES, especially for the polar bonds that tend to be significantly too short with TD-DFT on a nearly systematic basis. Indeed, the CO and CS/CSe distances are respectively underestimated by −0.026/−-0.052 Å and −0.015/−0.082 Å on average, with hybrid functionals, the XCF with low EXX ratio tending to deliver the smallest deviations for both types of bonds. Interestingly, for the CO bond, the previously reported mean signed errors are +0.042, +0.029, and −0.025 Å with ADC(2), CC2, and EOM-CCSD, respectively, for a very similar set of molecules.40 This indicates, on the one hand, that the TD-DFT absolute discrepancies for this key parameter are of the same order of magnitude as the one obtained with these three methods, and, on the other hand, that selecting CC2

GS, i.e., the ES are more sensitive on average to the selected XCF than the GS. Interestingly, one notes a quite tight histogram for the shortest streptocyanine; i.e., the geometries of this challenging compound are relatively insensitive to the selected functional. In contrast the ES geometry of several formaldehyde analogues (e.g., formyl chloride, phosgene, and thiophosgene) are particularly XCF dependent. Since it is common practice to evaluate excited-state structures and transition energies at the same level of theory, we have reported in Figure 3 the errors for these two properties. Several assessments of the accuracies of TD-DFT and second-order wave function approaches for adiabatic and 0−0 energies are available in the literature for organic molecules,143−149 so that they are not discussed in detail herein. We nevertheless underline that CC3 was again reported to be an accurate reference for adiabatic energies.47 In Figure 3, it is reassuring that a significant number of tested functionals are in the left down corner of the graphic, demarked by deviations in the 0.25−0.40 eV range for energies and (0.5−2) × 10−2 Å range for structures. There XCF have a “coherent” behavior; that is, they show a good degree of accuracy on both ES energies and geometries. Functionals outside this zone are more specialized and should be used for one of these two properties. This is the case, for instance, of B97-2 which provides accurate structures but relatively large errors on excitation energies, or LC-PBE which shows the opposite behavior.

4. CONCLUSIONS We assessed the performance of TD-DFT for reproducing the geometries of low-lying singlet valence excited-states of closedshell molecules. To this end, we selected, as benchmark references, accurate structural parameters obtained at the CCSDR(3), CC3, or CASPT2 level with an extended atomic basis set (aug-cc-pVTZ), therefore allowing comparisons in which the main error originates from TD-DFT and not from 3722


Article


New York, 2012; DOI: 10.1093/acprof:oso/ 9780199563029.001.0001. (2) Casida, M. E.; Huix-Rotllant, M. Progress in Time-Dependent Density-Functional Theory. Annu. Rev. Phys. Chem. 2012, 63, 287− 323. (3) Jacquemin, D.; Adamo, C. In Density-Functional Methods for Excited States; Ferré, N., Filatov, M., Huix-Rotllant, M., Eds.; Springer: Cham, Switzerland, 2016; Vol. 368, pp 347−375, DOI: 10.1007/ 128_2015_638. (4) Dreuw, A.; Wormit, M. The Algebraic Diagrammatic Construction Scheme for the Polarization Propagator for the Calculation of Excited States. WIREs Comput. Mol. Sci. 2015, 5, 82−95. (5) Hättig, C.; Weigend, F. CC2 Excitation Energy Calculations on Large Molecules Using the Resolution of the Identity Approximation. J. Chem. Phys. 2000, 113, 5154−5161. (6) Blase, X.; Duchemin, I.; Jacquemin, D. The Bethe-Salpeter Equation in Chemistry: Relations with TD-DFT, Applications and Challenges. Chem. Soc. Rev. 2018, 47, 1022−1043. (7) van Caillie, C.; Amos, R. D. Geometric Derivatives of Excitation Energies Using SCF and DFT. Chem. Phys. Lett. 1999, 308, 249−255. (8) van Caillie, C.; Amos, R. D. Geometric Derivatives of Density Functional Theory Excitation Energies Using Gradient-Corrected Functionals. Chem. Phys. Lett. 2000, 317, 159−164. (9) Furche, F.; Ahlrichs, R. Adiabatic Time-Dependent Density Functional Methods for Excited States Properties. J. Chem. Phys. 2002, 117, 7433−7447. (10) Scalmani, G.; Frisch, M. J.; Mennucci, B.; Tomasi, J.; Cammi, R.; Barone, V. Geometries and Properties of Excited States in the Gas Phase and in Solution: Theory and Application of a Time-Dependent Density Functional Theory Polarizable Continuum Model. J. Chem. Phys. 2006, 124, 094107. (11) Liu, J.; Liang, W. Z. Analytical Hessian of Electronic Excited States in Time-Dependent Density Functional Theory With TammDancoff Approximation. J. Chem. Phys. 2011, 135, 014113. (12) Liu, J.; Liang, W. Z. Analytical Approach for the Excited-State Hessian in Time-Dependent Density Functional Theory: Formalism, Implementation, and Performance. J. Chem. Phys. 2011, 135, 184111. (13) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; Li, X.; Caricato, M.; Marenich, A. V.; Bloino, J.; Janesko, B. G.; Gomperts, R.; Mennucci, B.; Hratchian, H. P.; Ortiz, J. V.; Izmaylov, A. F.; Sonnenberg, J. L.; Williams-Young, D.; Ding, F.; Lipparini, F.; Egidi, F.; Goings, J.; Peng, B.; Petrone, A.; Henderson, T.; Ranasinghe, D.; Zakrzewski, V. G.; Gao, J.; Rega, N.; Zheng, G.; Liang, W.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Throssell, K.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J. J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Keith, T. A.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Millam, J. M.; Klene, M.; Adamo, C.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.; Foresman, J. B.; Fox, D. J. Gaussian16, Revision A.03; Gaussian: Wallingford, CT, USA, 2016. (14) Laurent, A. D.; Jacquemin, D. TD-DFT Benchmarks: A Review. Int. J. Quantum Chem. 2013, 113, 2019−2039. (15) Schreiber, M.; Silva-Junior, M. R.; Sauer, S. P. A.; Thiel, W. Benchmarks for Electronically Excited States: CASPT2, CC2, CCSD and CC3. J. Chem. Phys. 2008, 128, 134110. (16) Silva-Junior, M. R.; Sauer, S. P. A.; Schreiber, M.; Thiel, W. Basis Set Effects on Coupled Cluster Benchmarks of Electronically Excited States: CC3, CCSDR(3) and CC2. Mol. Phys. 2010, 108, 453−465. (17) Silva-Junior, M. R.; Schreiber, M.; Sauer, S. P. A.; Thiel, W. Benchmarks of Electronically Excited States: Basis Set Effecs Benchmarks of Electronically Excited States: Basis Set Effects on CASPT2 Results. J. Chem. Phys. 2010, 133, 174318.

(EOM-CCSD) to benchmark TD-DFT carbonyl bond lengths would yield to incorrectly overestimated (underestimated) TD-DFT errors. Overall, the MAE obtained with TD-DFT for the ES bond lengths are in the 0.009−0.028 Å range depending on the XCF, with 26-out-of-48 XCF providing MAE of 0.015 Å or smaller, the five most accurate functionals being B97-D, B97-D3, TPSSh, O3LYP, and B3LYP for the set of small compounds considered herein. This 0.015 Å threshold can again be compared to ADC(2), CC2, and EOM-CCSD with respective MAEs of 0.016, 0.010, and 0.013 Å, obtained for a similar set of compounds: if one avoids the most problematic XCF, the TD-DFT accuracy on structural parameters is not far from the one of the second-order wave function approaches. Clearly, in a single-reference context, only wave function approaches incorporating corrections for the triples can be regarded as sufficiently accurate to reliably assess TD-DFT’s performances. Finally, given that many XCF provide reasonably accurate GS and ES geometries with DFT and TD-DFT, respectively, one can probably select a specific XCF on the basis of the quality of the obtained transition energies rather than structures, if the least accurate XCF for structural parameters are discarded, which can be straightforwardly done with the help of Figures 1 and 3. This in turn, indicates that the “hybrid” protocols used to determined 0−0 energies, in which the structures and vibrations are obtained with TD-DFT but the transition energies with a more accurate model143,146,148−152 are indeed valuable compromises.

■

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.8b00311. Extended set of reference CC values, list of GS and ES parameters obtained at DFT and TD-DFT levels, additional statistical analyses, and impact of the addition of statistical dispersion corrections on B3LYP statistics (PDF)

■

AUTHOR INFORMATION

Corresponding Authors

*(C.A.) E-mail: [email protected]. *(D.J.) E-mail: [email protected]. ORCID

Eric Brémond: 0000-0002-8646-9365 Marika Savarese: 0000-0002-4609-9603 Carlo Adamo: 0000-0002-2638-2735 Denis Jacquemin: 0000-0002-4217-0708 Funding

D.J. acknowledges the Région des Pays de la Loire for financial support. This research used resources of (i) the GENCICINES/IDRIS (Projects AP010810360 and A0040810359); (ii) CCIPL (Centre de Calcul Intensif des Pays de Loire); (iii) a local Troy cluster; and (iv) HPC resources from ArronaxPlus (Grant ANR-11-EQPX-0004 funded by the French National Agency for Research). Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Ullrich, C. Time-Dependent Density-Functional Theory: Concepts and Applications; Oxford Graduate Texts; Oxford University Press: 3723


Article

Journal of Chemical Theory and Computation (18) Andersson, K.; Malmqvist, P. A.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. Second-Order Perturbation Theory With a CASSCF Reference Function. J. Phys. Chem. 1990, 94, 5483−5488. (19) Andersson, K.; Malmqvist, P.-A.; Roos, B. O. Second-Order Perturbation Theory With a Complete Active Space Self-Consistent Field Reference Function. J. Chem. Phys. 1992, 96, 1218−1226. (20) Christiansen, O.; Koch, H.; Jørgensen, P. The Second-Order Approximate Coupled Cluster Singles and Doubles Model CC2. Chem. Phys. Lett. 1995, 243, 409−418. (21) Christiansen, O.; Koch, H.; Jørgensen, P. Response Functions in the CC3 Iterative Triple Excitation Model. J. Chem. Phys. 1995, 103, 7429−7441. (22) Silva-Junior, M. R.; Schreiber, M.; Sauer, S. P. A.; Thiel, W. Benchmarks for Electronically Excited States: Time-Dependent Density Functional Theory and Density Functional Theory Based Multireference Configuration Interaction. J. Chem. Phys. 2008, 129, 104103. (23) Jacquemin, D.; Wathelet, V.; Perpète, E. A.; Adamo, C. Extensive TD-DFT Benchmark: Singlet-Excited States of Organic Molecules. J. Chem. Theory Comput. 2009, 5, 2420−2435. (24) Goerigk, L.; Moellmann, J.; Grimme, S. Computation of Accurate Excitation Energies for Large Organic Molecules with Double-Hybrid Density Functionals. Phys. Chem. Chem. Phys. 2009, 11, 4611−4620. (25) Jacquemin, D.; Perpète, E. A.; Ciofini, I.; Adamo, C.; Valero, R.; Zhao, Y.; Truhlar, D. G. On the Performances of the M06 Family of Density Functionals for Electronic Excitation Energies. J. Chem. Theory Comput. 2010, 6, 2071−2085. (26) Silva-Junior, M. R.; Thiel, W. Benchmark of Electronically Excited States for Semiempirical Methods: MNDO, AM1, PM3, OM1, OM2, OM3, INDO/S, and INDO/S2. J. Chem. Theory Comput. 2010, 6, 1546−1564. (27) Harbach, P. H. P.; Wormit, M.; Dreuw, A. The Third-Order Algebraic Diagrammatic Construction Method (ADC(3)) for the Polarization Propagator for Closed-Shell Molecules: Efficient Implementation and Benchmarking. J. Chem. Phys. 2014, 141, 064113. (28) Bruneval, F.; Hamed, S. M.; Neaton, J. B. A Systematic Benchmark of the Ab Initio Bethe-Salpeter Equation Approach for Low-Lying Optical Excitations of Small Organic Molecules. J. Chem. Phys. 2015, 142, 244101. (29) Jacquemin, D.; Duchemin, I.; Blase, X. Benchmarking the Bethe-Salpeter Formalism on a Standard Organic Molecular Set. J. Chem. Theory Comput. 2015, 11, 3290−3304. (30) Mewes, S. A.; Plasser, F.; Krylov, A.; Dreuw, A. Benchmarking Excited-state Calculations Using Exciton Properties. J. Chem. Theory Comput. 2018, 14, 710−725. (31) Penocchio, E.; Piccardo, M.; Barone, V. Correction to Semiexperimental Equilibrium Structures for Building Blocks of Organic and Biological Molecules: The B2PLYP Route. J. Chem. Theory Comput. 2016, 12, 3001−3001. (32) Mendolicchio, M.; Penocchio, E.; Licari, D.; Tasinato, N.; Barone, V. Development and Implementation of Advanced Fitting Methods for the Calculation of Accurate Molecular Structures. J. Chem. Theory Comput. 2017, 13, 3060−3075. (33) Puzzarini, C.; Barone, V. Diving for Accurate Structures in the Ocean of Molecular Systems with the Help of Spectroscopy and Quantum Chemistry. Acc. Chem. Res. 2018, 51, 548−556. (34) Clouthier, D. J.; Ramsay, D. A. The Spectroscopy of Formaldehyde and Thioformaldehyde. Annu. Rev. Phys. Chem. 1983, 34, 31−58. (35) Page, C. S.; Olivucci, M. Ground and Excited State CASPT2 Geometry Optimizations of Small Organic Molecules. J. Comput. Chem. 2003, 24, 298−309. (36) Jagau, T.-C.; Gauss, J. Ground and Excited State Geometries via Mukherjee’s Multireference Coupled-Cluster Method. Chem. Phys. 2012, 401, 73−87.

(37) Guareschi, R.; Filippi, C. Ground- and Excited-State Geometry Optimization of Small Organic Molecules with Quantum Monte Carlo. J. Chem. Theory Comput. 2013, 9, 5513−5525. (38) Bousquet, D.; Fukuda, R.; Maitarad, P.; Jacquemin, D.; Ciofini, I.; Adamo, C.; Ehara, M. Excited-State Geometries of Heteroaromatic Compounds: A Comparative TD-DFT and SAC-CI Study. J. Chem. Theory Comput. 2013, 9, 2368−2379. (39) Bousquet, D.; Fukuda, R.; Jacquemin, D.; Ciofini, I.; Adamo, C.; Ehara, M. Benchmark Study on the Triplet Excited-State Geometries and Phosphorescence Energies of Heterocyclic Compounds: Comparison Between TD-PBE0 and SAC-CI. J. Chem. Theory Comput. 2014, 10, 3969−3979. (40) Budzák, Š .; Scalmani, G.; Jacquemin, D. Accurate Excited-State Geometries: a CASPT2 and Coupled-Cluster Reference Database for Small Molecules. J. Chem. Theory Comput. 2017, 13, 6237−6252. (41) Huix-Rotllant, M.; Filatov, M.; Gozem, S.; Schapiro, I.; Olivucci, M.; Ferré, N. Assessment of Density Functional Theory for Describing the Correlation Effects on the Ground and Excited State Potential Energy Surfaces of a Retinal Chromophore Model. J. Chem. Theory Comput. 2013, 9, 3917−3932. (42) Liu, F.; Gan, Z.; Shao, Y.; Hsu, C. P.; Dreuw, A.; Head-Gordon, M.; Miller, B. T.; Brooks, B. R.; Yu, J. G.; Furlani, T. R.; Kong, J. A Parallel Implementation of the Analytic Nuclear Gradient for TimeDependent Density Functional Theory Within the Tamm−Dancoff Approximation. Mol. Phys. 2010, 108, 2791−2800. (43) Hellweg, A.; Grün, S. A.; Hättig, C. Benchmarking the Performance of Spin-Component Scaled CC2 in Ground and Electronically Excited States. Phys. Chem. Chem. Phys. 2008, 10, 4119−4127. (44) Guido, C. A.; Jacquemin, D.; Adamo, C.; Mennucci, B. On the TD-DFT Accuracy in Determining Single and Double Bonds in Excited-State Structures of Organic Molecules. J. Phys. Chem. A 2010, 114, 13402−13410. (45) Guido, C. A.; Knecht, S.; Kongsted, J.; Mennucci, B. Benchmarking Time-Dependent Density Functional Theory for Excited State Geometries of Organic Molecules in Gas-Phase and in Solution. J. Chem. Theory Comput. 2013, 9, 2209−2220. (46) Tuna, D.; Lu, Y.; Koslowski, A.; Thiel, W. Semiempirical Quantum-Chemical Orthogonalization-Corrected Methods: Benchmarks of Electronically Excited States. J. Chem. Theory Comput. 2016, 12, 4400−4422. (47) Hättig, C. In Response Theory and Molecular Properties (A Tribute to Jan Linderberg and Poul Jørgensen); Jensen, H. J. Å., Ed.; Advances in Quantum Chemistry; Academic Press, 2005; Vol. 50; pp 37−60, DOI: 10.1016/S0065-3276(05)50003-0. (48) Christiansen, O.; Koch, H.; Jørgensen, P. Perturbative Triple Excitation Corrections to Coupled Cluster Singles and Doubles Excitation Energies. J. Chem. Phys. 1996, 105, 1451−1459. (49) Kállay, M.; Gauss, J. Calculation of Excited-State Properties Using General Coupled-Cluster and Configuration-Interaction Models. J. Chem. Phys. 2004, 121, 9257−9269. (50) Morgan, W. J.; Matthews, D. A.; Ringholm, M.; Agarwal, J.; Gong, J. Z.; Ruud, K.; Allen, W. D.; Stanton, J. F.; Schaefer, H. F. Geometric Energy Derivatives at the Complete Basis Set Limit: Application to the Equilibrium Structure and Molecular Force Field of Formaldehyde. J. Chem. Theory Comput. 2018, 14, 1333−1350. (51) Chantzis, A.; Laurent, A. D.; Adamo, C.; Jacquemin, D. Is the Tamm-Dancoff Approximation Reliable for the Calculation of Absorption and Fluorescence Band Shapes? J. Chem. Theory Comput. 2013, 9, 4517−4525. (52) Robinson, D. Accurate Excited State Geometries within Reduced Subspace TDDFT/TDA. J. Chem. Theory Comput. 2014, 10, 5346−5352. (53) Song, J. W.; Hirosawa, T.; Tsuneda, T.; Hirao, K. Long-Range Corrected Density Functional Calculations of Chemical Reactions: Redetermination of Parameter. J. Chem. Phys. 2007, 126, 154105. (54) Li, J.; Cramer, C. J.; Truhlar, D. G. Two-Response-Time Model Based on CM2/INDO/S2 Electrostatic Potentials for the Dielectric 3724


Article


(75) Boese, A. D.; Doltsinis, N. L.; Handy, N. C.; Sprik, M. New Generalized Gradient Approximation Functionals. J. Chem. Phys. 2000, 112, 1670−1678. (76) Boese, A. D.; Handy, N. C. A New Parametrization of Exchange-Correlation Generalized Gradient Approximation Functionals. J. Chem. Phys. 2001, 114, 5497−5503. (77) Boese, A. D.; Martin, J. M. L. Development of Density Functionals for Thermochemical Kinetics. J. Chem. Phys. 2004, 121, 3405−3416. (78) Tao, J. M.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Climbing the Density Functional Ladder: Nonempirical MetaGeneralized Gradient Approximation Designed for Molecules and Solids. Phys. Rev. Lett. 2003, 91, 146401. (79) Yu, H. S.; He, X.; Li, S. L.; Truhlar, D. G. MN15: A KohnSham Global-Hybrid Exchange-Correlation Density Functional with Broad Accuracy for Multi-Reference and Single-Reference Systems and Noncovalent Interactions. Chem. Sci. 2016, 7, 5032−5051. (80) Zhao, Y.; Truhlar, D. G. A New Local Density Functional for Main-Group Thermochemistry, Transition Metal Bonding, Thermochemical Kinetics, and Noncovalent Interactions. J. Chem. Phys. 2006, 125, 194101. (81) Becke, A. D. A New Mixing of Hartree-Fock and Local DensityFunctional Theories. J. Chem. Phys. 1993, 98, 1372−1377. (82) Peverati, R.; Truhlar, D. G. M11-L: A Local Density Functional That Provides Improved Accuracy for Electronic Structure Calculations in Chemistry and Physics. J. Phys. Chem. Lett. 2012, 3, 117− 124. (83) Zhao, Y.; Truhlar, D. G. Exploring the Limit of Accuracy of the Global Hybrid Meta Density Functional for Main-Group Thermochemistry, Kinetics, and Noncovalent Interactions. J. Chem. Theory Comput. 2008, 4, 1849−1868. (84) Peverati, R.; Truhlar, D. G. An Improved and Broadly Accurate Local Approximation to the Exchange−Correlation Density Functional: The MN12-L Functional for Electronic Structure Calculations in Chemistry and Physics. Phys. Chem. Chem. Phys. 2012, 14, 13171− 13174. (85) Yu, H. S.; He, X.; Truhlar, D. G. MN15-L: A New Local Exchange-Correlation Functional for Kohn−Sham Density Functional Theory with Broad Accuracy for Atoms, Molecules, and Solids. J. Chem. Theory Comput. 2016, 12, 1280−1293. (86) Zhao, Y.; Schultz, N. E.; Truhlar, D. G. Design of Density Functionals by Combining the Method of Constraint Satisfaction with Parametrization for Thermochemistry, Thermochemical Kinetics, and Noncovalent Interactions. J. Chem. Theory Comput. 2006, 2, 364−382. (87) Staroverov, V. N.; Scuseria, G. E.; Tao, J.; Perdew, J. P. Comparative Assessment of a New Nonempirical Density Functional: Molecules and Hydrogen-Bonded Complexes. J. Chem. Phys. 2003, 119, 12129−12137. (88) Henderson, T. M.; Izmaylov, A. F.; Scuseria, G. E.; Savin, A. Assessment of a Middle-Range Hybrid Functional. J. Chem. Theory Comput. 2008, 4, 1254−1262. (89) Cohen, A. J.; Handy, N. C. Dynamic Correlation. Mol. Phys. 2001, 99, 607−615. (90) Iikura, H.; Tsuneda, T.; Yanai, T.; Hirao, K. A Long-Range Correction Scheme for Generalized-Gradient-Approximation Exchange Functionals. J. Chem. Phys. 2001, 115, 3540−3544. (91) Becke, A. D. Density-Functional Thermochemistry. 3. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652. (92) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields. J. Phys. Chem. 1994, 98, 11623−11627. (93) Barone, V.; Orlandini, L.; Adamo, C. Proton Transfer in Model Hydrogen-Bonded Systems by a Density Functional Approach. Chem. Phys. Lett. 1994, 231, 295−300. (94) Stephens, P. J.; Devlin, F. J.; Frisch, M. J.; Chabalowski, C. F. Ab initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields. J. Phys. Chem. 1994, 98, 11623−11627.

Polarization Component of Solvatochromic Shifts on Vertical Excitation Energies. Int. J. Quantum Chem. 2000, 77, 264−280. (55) Fabrizio, A.; Corminboeuf, C. How do London Dispersion Interactions Impact the Photochemical Processes of Molecular Switches? J. Phys. Chem. Lett. 2018, 9, 464−470. (56) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864−B871. (57) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133− A1138. (58) Vosko, S. H.; Wilk, L.; Nusair, M. Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: A Critical Analysis. Can. J. Phys. 1980, 58, 1200−1211. (59) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods Without Adjustable Parameters: the PBE0 Model. J. Chem. Phys. 1999, 110, 6158−6170. (60) Ernzerhof, M.; Scuseria, G. E. Assessment of the Perdew− Burke−Ernzerhof Exchange-Correlation Functional. J. Chem. Phys. 1999, 110, 5029−5036. (61) Becke, A. D. Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 3098−3100. (62) Lee, C.; Yang, W.; Parr, R. G. Development of the ColleSalvetti Correlation-Energy Formula Into a Functional of the Electron-Density. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 785−789. (63) Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H. Results Obtained with the Correlation-Energy Density Functionals of Becke and Lee, Yang and Parr. Chem. Phys. Lett. 1989, 157, 200−206. (64) Adamo, C.; Barone, V. Toward Reliable Adiabatic Connection Models Free from Adjustable Parameters. Chem. Phys. Lett. 1997, 274, 242−250. (65) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (66) Adamo, C.; Barone, V. Exchange Functionals with Improved Long-Range Behavior and Adiabatic Connection Methods Without Adjustable Parameters: The mPW and mPW1PW Models. J. Chem. Phys. 1998, 108, 664−675. (67) Grimme, S. Semiempirical GGA-Type Density Functional Constructed with a Long-Range Dispersion Correction. J. Comput. Chem. 2006, 27, 1787−1799. (68) Grimme, S.; Ehrlich, S.; Goerigk, L. Effect of the Damping Function in Dispersion Corrected Density Functional Theory. J. Comput. Chem. 2011, 32, 1456−1465. (69) Zhao, Y.; Schultz, N. E.; Truhlar, D. G. Exchange-Correlation Functionals with Broad Accuracy for Metallic and Nonmetallic Compounds, Kinetics, and Noncovalent Interactions. J. Chem. Phys. 2005, 123, 161103. (70) Peverati, R.; Zhao, Y.; Truhlar, D. Generalized Gradient Approximation That Recovers the Second-Order Density-Gradient Expansion with Optimized Across-the-Board Performance. J. Phys. Chem. Lett. 2011, 2, 1991−1997. (71) Zhao, Y.; Truhlar, D. G. The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Functionals. Theor. Chem. Acc. 2008, 120, 215−241. (72) Peverati, R.; Truhlar, D. G. Exchange-Correlation Functional with Good Accuracy for Both Structural and Energetic Properties while Depending Only on the Density and Its Gradient. J. Chem. Theory Comput. 2012, 8, 2310−2319. (73) Peverati, R.; Truhlar, D. A Global Hybrid Generalized Gradient Approximation to the Exchange-Correlation Functional That Satisfies the Second-Order Density-Gradient Constraint and Has Broad Applicability in Chemistry. J. Chem. Phys. 2011, 135, 191102. (74) Hamprecht, F. A.; Cohen, A. J.; Tozer, D. J.; Handy, N. C. Development and Assessment of New Exchange-Correlation Functionals. J. Chem. Phys. 1998, 109, 6264−6271. 3725


Article

Journal of Chemical Theory and Computation (95) Chai, J. D.; Head-Gordon, M. Systematic Optimization of Long-Range Corrected Hybrid Density Functionals. J. Chem. Phys. 2008, 128, 084106. (96) Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid ExchangeCorrelation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51−56. (97) Xu, X.; Goddard, W. A., III The X3LYP Extended Density Functional for Accurate Descriptions of Nonbond Interactions, Spin States, and Thermochemical Properties. Proc. Natl. Acad. Sci. U. S. A. 2004, 101, 2673−2677. (98) Chai, J. D.; Head-Gordon, M. Long-range Corrected Hybrid Density Functionals with Damped Atom−Atom Dispersion Corrections. Phys. Chem. Chem. Phys. 2008, 10, 6615−6620. (99) Wilson, P. J.; Bradley, T. J.; Tozer, D. J. Hybrid ExchangeCorrelation Functional Determined from Thermochemical Data and Ab Initio Potentials. J. Chem. Phys. 2001, 115, 9233−9242. (100) Heyd, J.; Scuseria, G. Efficient Hybrid Density Functional Calculations in Solids: The HS-Ernzerhof Screened Coulomb Hybrid Functional. J. Chem. Phys. 2004, 121, 1187−1192. (101) Heyd, J.; Scuseria, G. E. Assessment and Validation of a Screened Coulomb Hybrid Density Functional. J. Chem. Phys. 2004, 120, 7274−7280. (102) Heyd, J.; Peralta, J. E.; Scuseria, G. E.; Martin, R. L. Energy Band Gaps and Lattice Parameters Evaluated with the Heyd-ScuseriaErnzerhof Screened Hybrid Functional. J. Chem. Phys. 2005, 123, 174101. (103) Izmaylov, A. F.; Scuseria, G. E.; Frisch, M. J. Efficient Evaluation of Short-Range Hartree-Fock Exchange in Large Molecules and Periodic Systems. J. Chem. Phys. 2006, 125, 104103. (104) Krukau, A. V.; Vydrov, O. A.; Izmaylov, A. F.; Scuseria, G. E. Influence of the Exchange Screening Parameter on the Performance of Screened Hybrid Functionals. J. Chem. Phys. 2006, 125, 224106. (105) Henderson, T. M.; Izmaylov, A. F.; Scalmani, G.; Scuseria, G. E. Can Short-Range Hybrids Describe Long-Range-Dependent Properties? J. Chem. Phys. 2009, 131, 044108. (106) Schmider, H. L.; Becke, A. D. Optimized Density Functionals from the Extended G2 Test Set. J. Chem. Phys. 1998, 108, 9624− 9631. (107) Peverati, R.; Truhlar, D. G. Screened-Exchange Density Functionals with Broad Accuracy for Chemistry and Solidstate Physics. Phys. Chem. Chem. Phys. 2012, 14, 16187−16191. (108) Austin, A.; Petersson, G. A.; Frisch, M. J.; Dobek, F. J.; Scalmani, G.; Throssell, K. A Density Functional with Spherical Atom Dispersion Terms. J. Chem. Theory Comput. 2012, 8, 4989−5007. (109) Peverati, R.; Truhlar, D. Improving the Accuracy of Hybrid Meta-GGA Density Functionals by Range Separation. J. Phys. Chem. Lett. 2011, 2, 2810−2817. (110) Jacquemin, D. What is the Key for Accurate Absorption and Emission Calculations ? Energy or Geometry? J. Chem. Theory Comput. 2018, 14, 1534−1543. (111) Aidas, K.; Angeli, C.; Bak, K. L.; Bakken, V.; Bast, R.; Boman, L.; Christiansen, O.; Cimiraglia, R.; Coriani, S.; Dahle, P.; Dalskov, E. K.; Ekström, U.; Enevoldsen, T.; Eriksen, J. J.; Ettenhuber, P.; Fernández, B.; Ferrighi, L.; Fliegl, H.; Frediani, L.; Hald, K.; Halkier, A.; Hättig, C.; Heiberg, H.; Helgaker, T.; Hennum, A. C.; Hettema, H.; Hjertenæs, E.; Høst, S.; Høyvik, I.-M.; Iozzi, M. F.; Jansík, B.; Jensen, H. J. A.; Jonsson, D.; Jørgensen, P.; Kauczor, J.; Kirpekar, S.; Kjærgaard, T.; Klopper, W.; Knecht, S.; Kobayashi, R.; Koch, H.; Kongsted, J.; Krapp, A.; Kristensen, K.; Ligabue, A.; Lutnæs, O. B.; Melo, J. I.; Mikkelsen, K. V.; Myhre, R. H.; Neiss, C.; Nielsen, C. B.; Norman, P.; Olsen, J.; Olsen, J. M. H.; Osted, A.; Packer, M. J.; Pawlowski, F.; Pedersen, T. B.; Provasi, P. F.; Reine, S.; Rinkevicius, Z.; Ruden, T. A.; Ruud, K.; Rybkin, V. V.; Sałek, P.; Samson, C. C. M.; de Merás, A. S.; Saue, T.; Sauer, S. P. A.; Schimmelpfennig, B.; Sneskov, K.; Steindal, A. H.; Sylvester-Hvid, K. O.; Taylor, P. R.; Teale, A. M.; Tellgren, E. I.; Tew, D. P.; Thorvaldsen, A. J.; Thøgersen, L.; Vahtras, O.; Watson, M. A.; Wilson, D. J. D.; Ziolkowski, M.; Ågren, H. The Dalton Quantum Chemistry Program System. WIREs Comput. Mol. Sci. 2014, 4, 269−284.

(112) Isegawa, M.; Peverati, R.; Truhlar, D. G. Performance of Recent and High-Performance Approximate Density Functionals for Time-Dependent Density Functional Theory Calculations of Valence and Rydberg Electronic Transition Energies. J. Chem. Phys. 2012, 137, 244104. (113) Weber, J.; Malsch, K.; Hohlneicher, G. Excited Electronic States of p-Benzoquinone. Chem. Phys. 2001, 264, 275−318. (114) Köhn, A.; Hättig, C. Analytic Gradients for Excited States in the Coupled-Cluster Model CC2 Employing the Resolution-Of-TheIdentity Approximation. J. Chem. Phys. 2003, 119, 5021−5036. (115) Sabljić, A.; McDiarmid, R. Analysis of the Absorption Spectrum of the NV1 Transition of Cyclopentadiene. J. Chem. Phys. 1990, 93, 3850−3855. (116) Pandey, R. R.; Khait, Y. G.; Hoffmann, M. R. Ground and Low-Lying Excited Electronic States of Difluorodiazirine. J. Phys. Chem. A 2004, 108, 3119−3124. (117) Terrabuio, L. A.; Haiduke, R. L. A.; Matta, C. F. Difluorodiazirine (CF2N2): A Comparative Quantum Mechanical Study of the First Triplet and First Singlet Excited States. Chem. Phys. Lett. 2016, 655−656, 96−102. (118) Lombardi, J. R.; Klemperer, W.; Robin, M. B.; Basch, H.; Kuebler, N. A. Optical Spectra of Small Rings. I. The n → π Transition of Difluorodiazirine. J. Chem. Phys. 1969, 51, 33−44. (119) Hepburn, P.; Hollas, J. The 352 nm Absorption Spectrum of Difluorodiazirine. J. Mol. Spectrosc. 1974, 50, 126−141. (120) Sieber, H.; Riedle, E.; Neusser, H. Doppler-Free Two-Photon Spectrum of the 000 Band of the Ã 1B1 ← X̃ 1 A1 Transition in Difluorodiazirine, F2CN2. Chem. Phys. Lett. 1990, 169, 191−197. (121) van der Does, T.; Bickelhaupt, F. Diisocyanogen. Angew. Chem., Int. Ed. Engl. 1988, 27, 936−938. (122) Ringer, A. L.; Sherrill, C. D.; King, R. A.; Crawford, T. D. Low-Lying Singlet Excited States of Isocyanogen. Int. J. Quantum Chem. 2008, 108, 1137−1140. (123) Celani, P.; Werner, H.-J. Analytical Energy Gradients for Internally Contracted Second-Order Multireference Perturbation Theory. J. Chem. Phys. 2003, 119, 5044−5057. (124) Send, R.; Valsson, O.; Filippi, C. Electronic Excitations of Simple Cyanine Dyes: Reconciling Density Functional and Wave Function Methods. J. Chem. Theory Comput. 2011, 7, 444−455. (125) Grimme, S.; Neese, F. Double-Hybrid Density Functional Theory for Excited Electronic States of Molecules. J. Chem. Phys. 2007, 127, 154116. (126) Moore, B., II; Autschbach, J. Longest-Wavelength Electronic Excitations of Linear Cyanines: The Role of Electron Delocalization and of Approximations in Time-Dependent Density Functional Theory. J. Chem. Theory Comput. 2013, 9, 4991−5003. (127) Zhekova, H.; Krykunov, M.; Autschbach, J.; Ziegler, T. Applications of Time Dependent and Time Independent Density Functional Theory to the First π to π* Transition in Cyanine Dyes. J. Chem. Theory Comput. 2014, 10, 3299−3307. (128) Le Guennic, B.; Jacquemin, D. Taking Up the Cyanine Challenge with Quantum Tools. Acc. Chem. Res. 2015, 48, 530−537. (129) Simard, B.; Steer, R. P.; Judge, R. H.; Moule, D. C. Vibrational Analysis of the Low Resolution ã ← X̃ Absorption Spectra of BrClCS and Br2CS. Can. J. Chem. 1988, 66, 359−366. (130) Lee, T. J.; Taylor, P. R. A Diagnostic for Determining the Quality of Single-Reference Electron Correlation Methods. Int. J. Quantum Chem. 1989, 36, 199−207. (131) Judge, R.; Moule, D. Thiocarbonyl Spectroscopy: The Ã 1A″ ← X̃ 1A′ and ã3A″ ← X̃ 1A′ Electronic Transitions in Thioformyl Chloride, CHCIS. J. Mol. Spectrosc. 1985, 113, 77−84. (132) Bokarev, S. I.; Dolgov, E. K.; Bataev, V. A.; Godunov, I. A. Molecular Parameters of Tetraatomic Carbonyls X2CO and XYCO (X, Y = H, F, Cl) in the Ground and Lowest Excited Electronic States, Part 1: A Test of Ab Initio Methods. Int. J. Quantum Chem. 2009, 109, 569−585. (133) Watson, T. J.; Lotrich, V. F.; Szalay, P. G.; Perera, A.; Bartlett, R. J. Benchmarking for Perturbative Triple-Excitations in EE-EOMCC Methods. J. Phys. Chem. A 2013, 117, 2569−2579. 3726


Article


(152) Santoro, F.; Jacquemin, D. Going Beyond the Vertical Approximation with Time-Dependent Density Functional Theory. WIREs Comput. Mol. Sci. 2016, 6, 460−486.

(134) Kánnár, D.; Szalay, P. G. Benchmarking Coupled Cluster Methods on Valence Singlet Excited States. J. Chem. Theory Comput. 2014, 10, 3757−3765. (135) Job, V.; Sethuraman, V.; Innes, K. The 3500 Å 1A2 - X̃ 1A1 Transition of Formaldehyde-h2, d2, and hd: Vibrational and Rotational Analyses. J. Mol. Spectrosc. 1969, 30, 365−426. (136) Clouthier, D. J.; Judge, R.; Moule, D. Selenoformaldehyde: Rotational Analysis of the Ã 1A2-X̃ 1A1 735 nm Band System of H2C78Se, H2C80Se, and D2C78Se from High-Resolution Laser Fluorescence Excitation Spectra. J. Mol. Spectrosc. 1990, 141, 175− 203. (137) Dunlop, J. R.; Karolczak, J.; Clouthier, D. J.; Ross, S. C. Pyrolysis Jet Spectroscopy: The S1 − S0 Band System of Thioformaldehyde and the Excited-State Bending Potential. J. Phys. Chem. 1991, 95, 3045−3062. (138) Peach, M. J. G.; Tellgren, E.; Salek, P.; Helgaker, T.; Tozer, D. J. Structural and Electronic Properties of Polyacetylene and Polyyne from Hybrid and Coulomb-Attenuated Density Functionals. J. Phys. Chem. A 2007, 111, 11930−11935. (139) Jacquemin, D.; Adamo, C. Bond Length Alternation of Conjugated Oligomers: Wave Function and DFT Benchmarks. J. Chem. Theory Comput. 2011, 7, 369−376. (140) Brémond, É .; Savarese, M.; Su, N. Q.; Pérez-Jiménez, Á . J.; Xu, X.; Sancho-García, J. C.; Adamo, C. Benchmarking Density Functionals on Structural Parameters of Small-/Medium-Sized Organic Molecules. J. Chem. Theory Comput. 2016, 12, 459−465. (141) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate ab initio Parametrization Of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. J. Chem. Phys. 2010, 132, 154104. (142) Brémond, E.; Ciofini, I.; Sancho-García, J. C.; Adamo, C. Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists. Acc. Chem. Res. 2016, 49, 1503−1513. (143) Goerigk, L.; Grimme, S. Assessment of TD-DFT Methods and of Various Spin Scaled CISnD and CC2 Versions for the Treatment of Low-Lying Valence Excitations of Large Organic Dyes. J. Chem. Phys. 2010, 132, 184103. (144) Send, R.; Kühn, M.; Furche, F. Assessing Excited State Methods by Adiabatic Excitation Energies. J. Chem. Theory Comput. 2011, 7, 2376−2386. (145) Jacquemin, D.; Planchat, A.; Adamo, C.; Mennucci, B. A TDDFT Assessment of Functionals for Optical 0−0 Transitions in Solvated Dyes. J. Chem. Theory Comput. 2012, 8, 2359−2372. (146) Winter, N. O. C.; Graf, N. K.; Leutwyler, S.; Hattig, C. Benchmarks for 0−0 Transitions of Aromatic Organic Molecules: DFT/B3LYP, ADC(2), CC2, SOS-CC2 and SCS-CC2 Compared to High-Resolution Gas-Phase Data. Phys. Chem. Chem. Phys. 2013, 15, 6623−6630. (147) Jacquemin, D.; Moore, B.; Planchat, A.; Adamo, C.; Autschbach, J. Performance of an Optimally Tuned Range-Separated Hybrid Functional for 0−0 Electronic Excitation Energies. J. Chem. Theory Comput. 2014, 10, 1677−1685. (148) Jacquemin, D.; Duchemin, I.; Blase, X. 0−0 Energies Using Hybrid Schemes: Benchmarks of TD-DFT, CIS(D), ADC(2), CC2 and BSE/GW formalisms for 80 Real-Life Compounds. J. Chem. Theory Comput. 2015, 11, 5340−5359. (149) Oruganti, B.; Fang, C.; Durbeej, B. Assessment of a Composite CC2/DFT Procedure for Calculating 0−0 Excitation Energies of Organic Molecules. Mol. Phys. 2016, 114, 3448−3463. (150) Boulanger, P.; Chibani, S.; Le Guennic, B.; Duchemin, I.; Blase, X.; Jacquemin, D. Combining the Bethe−Salpeter Formalism with Time-Dependent DFT Excited-State Forces to Describe Optical Signatures: NBO Fluoroborates as Working Examples. J. Chem. Theory Comput. 2014, 10, 4548−4556. (151) Chibani, S.; Laurent, A. D.; Le Guennic, B.; Jacquemin, D. Improving the Accuracy of Excited State Simulations of BODIPY and aza-BODIPY Dyes with a Joint SOS-CIS(D) and TD-DFT Approach. J. Chem. Theory Comput. 2014, 10, 4574−4582. 3727


Accuracy of TD-DFT Geometries: A Fresh Look - ACS Publications

Recommend Documents