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Calculations of n→π* Transition Energies: Comparisons Between TDDFT, ADC, CC, CASPT2, and BSE/GW Descriptions Published as part of The Journal of Physical Chemistry virtual special issue “Manuel Yáñez and Otilia Mó Festschrift”. Cloé Azarias,† Chloé Habert,† Šimon Budzák,‡ Xavier Blase,¶,§ Ivan Duchemin,∥,§ and Denis Jacquemin*,†,⊥ J. Phys. Chem. A 2017.121:6122-6134. Downloaded from pubs.acs.org by EASTERN KENTUCKY UNIV on 08/13/18. For personal use only.

†

CEISAM, BP 92208, UMR CNRS 6230, Université de Nantes, 2, Rue de la Houssiniere, 44322 Nantes, Cedex 3, France Department of Chemistry, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, SK-97400 Banská Bystrica, Slovak Republic ¶ CNRS, Inst NEEL, F-38042 Grenoble, France § Univ. Grenoble Alpes, Inst NEEL, F-38042 Grenoble, France ∥ Univ. Grenobles Alpes, CEA, INAC-MEM, L_Sim, F-38000 Grenoble, France ⊥ Institut Universitaire de France, 1, rue Descartes, F-75231 Paris Cedex 05, France ‡

S Supporting Information *

ABSTRACT: Using a large panel of theoretical approaches, namely, CC2, CCSD, CCSDR(3), CC3, ADC(2), ADC(3), CASPT2, timedependent density functional theory (TD-DFT), and BSE/evGW, the two latter combined with different exchange-correlation functionals, we investigate the lowest singlet transition in 23 n→π* compounds based on the nitroso, thiocarbonyl, carbonyl, and diazo chromophores. First, for 16 small derivatives we compare the transition energies provided by the different wave function approaches to define theoretical best estimates. For this set, it surprisingly turned out that ADC(2) offers a better match with CC3 than ADC(3). Next, we use 10 functionals belonging to the “LYP” and “M06” families and compare the TD-DFT and the BSE/evGW descriptions. The BSE/evGW results are less sensitive than their TD-DFT counterparts to the selected functional, especially in the M06 series. Nevertheless, BSE/evGW delivers larger errors than TD-CAM-B3LYP, which provides extremely accurate results in the present case, especially when the Tamm−Dancoff approximation is applied. In addition, we show that, among the different starting points for BSE/evGW calculations, M06-2X eigenstates stand as the most appropriate. Finally, we confirm that the trends observed on the small compounds pertain in larger molecules.

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INTRODUCTION

theoretical chemists had the choice between two families of single-reference approaches to treat ES. On the one hand, one finds Time-Dependent Density Functional Theory (TDDFT),3 an extension of DFT to ES that enjoys several advantages, among which, computational efficiency thanks to both a formal 6(N 4) scaling with system size and the implementations of analytical first (gradient) and second (Hessian) derivatives.4,5 However, one of the most important limitations of TD-DFT is the dependency of the ES energies on the chosen exchange-correlation functional (XCF), a dependency that is generally much stronger than for GS properties, though it depends on the nature of the ES considered,6 at least when the popular adiabatic formulation of TD-DFT is used.7 In

Like their ground-state (GS) counterparts, the theoretical methods that can be applied to model electronically excitedstates (ES) can be roughly divided into two categories: singlereference and multireference approaches.1 If the latter methods can accurately account for static correlation effects, the former theories have the indisputable advantage of allowing more black-box applications. Indeed, in multireference schemes, such as in the Complete Active Space with Second-order Perturbation Theory (CASPT2),2 the selection of a chemically suitable active space and the follow-up of the calculations remain very consuming in terms of workforce. While singledeterminant theories are obviously unsuited for intrinsically multireference cases, for example, conical intersections, they are more straightforward and are still used in the vast majority of ES applications devoted to the modeling of absorption, fluorescence, and phosphorescence spectra. Up to now, © 2017 American Chemical Society

Received: May 30, 2017 Revised: July 23, 2017 Published: July 24, 2017 6122

DOI: 10.1021/acs.jpca.7b05222 J. Phys. Chem. A 2017, 121, 6122−6134

Article

The Journal of Physical Chemistry A practice, one could often find an XCF suited to treat a specific case,8 but TD-DFT tends to be less reliable when several ES of different natures should be described. For instance, it is not easy to simultaneously and accurately model local and chargetransfer (CT) ES in the same molecule,9−11 whereas the calculation of singlet−triplet gaps remains challenging.12,13 To circumvent the empirical choice of an XCF “ideal” for a given molecule or ES, several protocols have been developed, such as the optimal tuning of range-separated hybrids,14−16 but such optimization also requires a significant effort. On the other hand, one finds single-reference wave function-based approaches, such as the variants of the Coupled-Cluster (CC) and Algebraic Diagrammatic Construction (ADC) methods.17 Based on a Hartree−Fock determinant, these methods are, by construction, XCF-independent, and they also allow for a systematic checking of the quality of the results by increasing the expansion order, for example, by comparing the transition energies obtained using the CC2, 18,19 CCSD, 20−22 CCSDR(3),23 CC3,18,24 etc., or the ADC(2), ADC(3), etc.,17,25 series. This gives access to quality control, which is basically absent in TD-DFT, though some diagnostic tools have been developed.26−28 In addition, these wave function methods tend to be more consistent than TD-DFT when ES of different natures are considered and generally deliver quite accurate singlet−triplet gaps. However, these methods have several disadvantages, such as the lack of implementations of analytical second derivative and a rather large computational cost. Indeed, the two “cheapest” methods, that is, ADC(2) and CC2, present a formal 6(N 5) scaling with system size. Although the applications of Laplace transform or/and resolution of identity (RI) techniques allow ADC(2) and CC2 calculations on medium-sized systems (ca. 30−70 atoms), they become inapplicable for extended compounds for which TD-DFT calculations remain routine. Recently, a third alternative has been enjoying a rapidly increasing popularity in theoretical chemistry: the Bethe− Salpeter Equation (BSE/GW) method.29−31 Interestingly, in practical terms, BSE/GW can be seen as offering intermediate advantages (and disadvantages) compared to the above-cited single-reference methods. First, its formal cost is 6(N 4) as in TD-DFT, allowing applications on rather large systems,32 though the computational effort is increased compared to TDDFT.33 Second, when a partially self-consistent GW approach is used (the so-called BSE/evGW method), the dependency on the starting XCF is strongly (but not completely) reduced.34 Third, one can treat on an equal footing local and CT states, thanks to the use of a two-particle kernel,35,36 but the calculation of singlet−triplet gaps remains nevertheless quite problematic.37,38 In the list of drawbacks, one should also underline the lack of analytical derivatives for the BSE/GW theory,39 preventing geometry optimizations of the ES structures: only vertical properties are directly accessible to date with BSE/evGW. As stated above, the number of publications using the BSE/ GW approach to model molecular (i.e., finite) systems has been relatively reduced until 2010,40−47 but it has been exploding since then.32−38,48−83 In a recent contribution,83 we found that the accuracy provided by BSE/evGW was as high (actually even slightly superior for the tested set) as the one reached with the CASPT2 approach for (vertical) ES energies, whereas in a previous work devoted to 0−0 energies of large chromogens, we found that BSE/evGW, ADC(2), and CC2 similarly

reproduced experimental data with average errors of ca. 0.15 eV, that is, smaller than the one obtained with TD-DFT for the same data set.68 Despite these successes, the calculations of the n→π* of small molecules appeared rather difficult.34,83 First, in a work using BSE/evGW@PBE0, it was shown that the BSE transition energies of nitroso and thiocarbonyl dyes, two important n→π* chromogens, were significantly too small compared to the reference values.34 However, in a subsequent contribution comparing the BSE/evGW@M06 and BSE/ evGW@M06-2X descriptions,83 it was shown that the impact of the starting XCF on the TD-DFT and BSE/evGW results was comparable for the n→π* transitions of formaldehyde, acetaldehyde, and acetone and nitrogen-bearing cycles, contrasting with all other families of ES for which the XCF dependency was greatly reduced with BSE. For those n→π* ES, the BSE estimates were, however, quite accurate, that is, in good agreement with the available reference values. In light of these previous results, further investigations are obviously welcome, for example, to determine: (i) if these different outcomes are due to the consideration of different chromophores in the two studies; (ii) or, in contrast, related to the selection of different reference values (theory/experiment); (iii) if the impact of the XCF on the BSE/evGW results is larger for all n→π* transitions or only a subset of them; and (iv) what is the most adequate starting point for BSE/evGW calculations. We answer these questions in this contribution by first considering the set of 16 molecules displayed in Scheme 1: Scheme 1. Representation of the First Series of Compounds Investigated Herein

four nitroso derivatives, four thiocarbonyl compounds, four keto derivatives, and four diazo derivatives, so to encompass examples of all chemically relevant n→π* classes of dyes. Inspired by Thiel’s works,84−86 we perform high-level wave function calculations using ADC, CC, and CASPT2 formalisms for this set of compact compounds, in an effort to define reliable theoretical best estimates (TBE). Next, we perform TD-DFT and BSE/GW calculations starting with various XCF to quantify the impact on the starting point on the final results. We selected XCF of both the “LYP” and “M06” families. Indeed, at the TD-DFT level, previous investigations comparing the quality of the n→π* transitions obtained with several XCF87−89 concluded that the n→π* transitions are not strongly XCF-dependent with the former family87,88 but are strongly XCF-dependent with the latter.89 It is therefore interesting to determine if the use of BSE/GW also leads contrasted behaviors with these two series of XCF or not. In a 6123


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The Journal of Physical Chemistry A Table 1. Transition Energies Obtained for All Wavefunction Methods for the 12 Compounds of Scheme 1a I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI

CC2

CCSD

CCSDR(3)

CC3

ADC(2)

ADC(3)

CASPT2

literature

1.953 2.297 1.973 1.656 2.822 3.914 2.631 4.091 4.037 4.510 2.887 3.912 3.687 3.849 2.388 3.767

1.954 2.243 1.967 1.861 2.763 3.796 2.630 4.136 3.980 4.499 2.987 3.971 3.697 3.857 2.651 4.010

1.938 2.228 1.944 1.673 2.731 3.747 2.554 4.051 3.935 4.442 2.894 3.858 3.660 3.794 2.557 3.893

1.942 2.215 1.938 1.510 2.729 3.731 2.556 4.045 3.934 4.438 2.850 3.803 3.663 3.779 2.463 3.812

1.844 2.195 1.903 1.449 2.645 3.731 2.462 3.911 3.882 4.317 2.801 3.749 3.678 3.864 2.428 3.780

1.700 1.902 1.659 1.168 2.644 3.621 2.511 4.160 3.868 4.459 2.800 3.819 3.415 3.481 2.417 3.837

1.777 2.285 2.041 1.382 2.722 3.714 2.486 3.951 3.851 4.402 2.690 3.695 3.755 3.760 2.238 3.628

1.82b, 1.79c, 1.82d 2.10e, 2.21f, 2.08g 1.75h, 1.82I, 1.73j 1.87k 2.54l, 2.32m, 2.41n 3.52o 2.33p 4.28q 3.88r, 4.00s 4.38r, 4.43s 3.10t, 2.8u 3.83t, 3.63v, 3.74w 3.76x, 3.65y 3.89z, 3.46aa 2.46r, 2.25bb 3.85r, 3.30bb

a

All results were obtained with the aug-cc-pVTZ atomic basis set and are given in electronvolts. On the right-most column are reference values taken from the literature (see footnotes for details). Our TBE values are given in bold. See discussion in the text. bAdiabatic transition energy, at the MRAQCC/CAS(6−4)/cc-pVTZ level.113 cEstimated experimental 0−0 energy in vapor.114 dExperimental λmax in cyclohexane.115 eVertical MR-CI/631G(d)+spd//B3LYP/6-31G(d).116 fExperimental λmax in gas phase at 295 K.117 gExperimental λmax in CCl4.118 hAdiabatic transition energy, at the MR-AQCC/cc-pVTZ(-f) level.119 IExperimental λmax in vapor.120 jExperimental 0−0 energy in a jet-cooled experiment.121 kExperimental λmax in toluene.122 lResolved experimental λmax in hexane.123 mOrigin of the 410 band of the excitation spectrum in vapor.124 nPosition of the 210310410 vibronic band in gas.125 oExperimental λmax in gas phase.126 pExperimental 0−0 energy in vapor.127,128 qExperimental position of the shoulder in diethyl ether.129 rThiel TBE-2 value, that is, (frozen-core) CC3/aug-cc-pVTZ//MP2/6-31G(d) result.86 sExperimental value obtained from several experiments.130 tSAC−CI/[4s2p1d/2s] result.131 uExperimental electron-impact maximum in gas phase.132 vCASPT2/ANO-L resul.133 w Experimental λmax in 2-Me-THF.134 xCC3/aug-cc-pVDZ.135 yGas phase experiment, λmax.136 zCASPT2/6-31G(d) result.137 aaExperimental 0− 0 energy in gas phase.138 bbGas-phase experiment (VUV).139

final stage, we used a series of larger molecules to check if the trends noticed for the molecules of Scheme 1 pertain.

a.u. IPEA shift. Discussions about the impact of the IPEA shift on the computed transition energies can be found elsewhere,95 and it was observed that this IPEA value improves the match with reference data when large atomic basis sets are applied (see the Supporting Information for additional data for I). The CASPT2 calculations were performed with the aug-cc-pVTZ atomic basis set and no frozen orbitals to offer results directly comparable to their CC and ADC counterparts. To speed the CASPT2 calculations we applied Cholesky decomposition technique96 with the 1 × 10−4 decomposition threshold. Active spaces are described in the Supporting Information. We always applied state-averaged CASSCF, where both GS and ES had equal weights, followed by multistate variant of CASPT2.97 All TD-DFT/aug-cc-pVTZ calculations were performed with the Gaussian09 D01 program,90 using a 1 × 10−8 a.u. SCF convergence threshold and the large ultraf ine DFT integration grid. Tamm−Dancoff (TDA) calculations were also performed with the same program. BSE/evGW@DFT/aug-cc-pVTZ transition energies were determined with the Fiesta package98 applying the Coulomb-fitting resolution-of-identity (RI-V) technique using the corresponding auxiliary basis. We used here the so-called evGW scheme, that is, the occupied/virtual GW energy levels are self-consistently converged, whereas the input DFT eigenfunctions are frozen. We corrected at the GW level, all valence-occupied DFT levels and twice that amount of virtual levels (with maximal values of 20 and 40 for the valence and virtual, respectively), which is a choice guaranteeing convergence for low-lying n→π* ES that only involve a few orbitals around the gap. The BSE calculations were made with the “full” matrix; that is, we did not apply the TDA during the BSE step. Discussion regarding the impact of the application of the TDA on BSE transition energies for organic compounds can be found elsewhere.37 During the GW calculations, the

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COMPUTATIONAL DETAILS The GS structures of all compounds were first optimized at the PBE0/cc-pVTZ level using the highest possible point group symmetry, and it was checked, at this level of theory, that these structures led to no imaginary frequency. We next reoptimized all structures at the second-order Møller−Plesset level, that is, MP2/aug-cc-pVTZ. These calculations were performed with the Gaussian09 D01 program,90 tightening the energy and geometry convergence thresholds to 1 × 10−10 to 1 × 10−11 a.u and 1 × 10−5 a.u. (a.u. = atomic units) on the root-mean-square (rms) forces, respectively, and using the frozen-core option for the MP2 calculations. All calculations presented below were performed on these MP2/aug-cc-pVTZ geometries; the obtained Cartesian coordinates are given in the Supporting Information. The vertical CC2, CCSD, CCSDR(3), and CC3 calculations were performed with both the aug-cc-pVDZ and aug-cc-pVTZ atomic basis sets with the Dalton 2013 package91 using default convergence thresholds and correlating systematically all electrons, including the core ones. Note that we published before some CC values for a subset of the compounds treated here,34 but those data were determined using the less demanding frozen-core approach. The ADC(2) and ADC(3) transition energies were determined with the QChem 4.2 code,92 using the same two atomic basis sets and setting the self-consistent field (SCF) and integral convergence thresholds to 1 × 10−8 and 1 × 10−11 a.u, respectively. To be consistent with the CC calculations, the core electrons were also fully included in the ADC calculations. CASPT2 calculations were performed with the Molcas 8.0 software,93,94 and the active orbital energies were corrected by the usual 0.25 6124


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The Journal of Physical Chemistry A

To build a set of reference values, we first checked if the CASPT2 calculations revealed a dominant single-reference character. This was the case in general. A notable exception is IV in which the leading determinant presents a 0.74 weight only, so that the ADC and CC calculations are not built upon a reasonable reference. Unsurprisingly, for that compound, the changes between CCSD and CC3 values is sizable (>0.3 eV), and the difference between ADC(2) and ADC(3) transition energies is large as well. ADC(3) delivers a very small transition energy, but this model is known to be inadequate for molecules displaying a strong multireference character.25 Therefore, for IV, the CASPT2 value of 1.382 eV is probably the most accurate reference. The second borderline case is tetrazine, XV, for which the leading determinant in the CAS has a weight of 0.80. This molecule was treated before,25,84,86 and it is obviously uneasy to definitively decide if CASPT284 or CC386 or ADC(3)25 provides the most accurate reference. Given that ADC(2) and ADC(3) values are very consistent, intermediate between the CASPT2 and CC3 estimates as well as slightly above the experimental 0−0 value (as it should be), we selected the ADC(3) value as TBE for XV. In XVI, the situation is rather similar (CAS leading determinant weight of 0.82), with the ADC(3) and CC3 estimates being close from one another, and we selected the former as reference. The last case in which the CAS calculation indicates a possible multireference character is XI (0.80 weight), and one notices that ADC(3) is again intermediate between CC3 and CASPT2, with very similar ADC(2) and ADC(3) results, explaining our selection. For all other molecules, CASPT2 indicates a clear single-reference nature. For several compounds, one notices a well-behaved character, that is, the transition energies provided by all methods are consistent, and the CC results appear very stable when changing the expansion order. This is clearly the case for the three first thiocarbonyl derivatives, V−VII, and three carbonyl compounds, IX, X, and XII: the CCSDR(3)− CC3 discrepancies are generally trifling, and one additionally observes both excellent matches between CC3 and CASPT2 (V, VI, and X) and/or CC3 and ADC(3) (VII, X, and XII) accompanied by a small (ca. 0.1 eV) spread of the values determined with the most refined approaches. For these molecules, we therefore selected the CC3 estimates as TBE, though selecting the ADC(3) transition energies would have also been a rational choice, as advocated in ref 25. As can be seen from Table 1, for several compounds, it is however more difficult to find a consensus value among the tested approaches. The case of nitrosomethane (I), a tiny molecule with a wellseparated, chemically intuitive, low-lying ES, that was investigated at with several levels of theory,113,116,140,141 is rather puzzling. Indeed, our CASPT2 calculations reveal a clear monodeterminantal character for both the GS and the ES (weight of the leading term ≃0.9), which fits earlier investigations.113,140 This led to think that the CC expansion should be trustable, as one notices very stable CC estimates when changing the expansion order, that is, 1.94 ± 0.01 eV. However, both ADC(3) and CASPT2 deliver significantly lower estimates, a surprising outcome given that a consistent protocol was used. At such a stage, it is honestly hard to decide which method is the most suited here. We therefore analyzed the available experimental data to help decision-making. First, it is known that there is a relaxation (methyl rotation) in the ES,114,142 but it is related with a rather small barrier of ca. 0.05 eV, which is consistent with the fact that the measured λmax and 0−0 energies are similar: 1.82 and 1.79 eV, respectively.114,115

starting DFT eigenstates were generated with NWChem,99 applying the xf ine grid and 1 × 10−7/1 × 10−6 a.u. convergence thresholds for the total energies/densities. For both TD-DFT and BSE/evGW calculations, we performed calculations using the four XCF of the M06 series,100,101 that is, M06-L (0% of exact exchange), M06 (27% of exact exchange), M06-2X (54% of exact exchange), and M06-HF (100% of exact exchange), as well as a series of LYP XCF,102 namely, BLYP,103 OLYP,104 B3LYP,105 BH&HLYP,106 CAM-B3LYP,107 and LC-BLYP (ω = 0.33).108 The two former are GGA XCF that do not contain exact exchange, the two next are global hybrids containing, respectively, 20% and 50% of exact exchange, whereas the two latter are range-separated hybrids presenting 19% (65%) and 0% (100%) of short-range (long-range) exact exchange, respectively.

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RESULTS AND DISCUSSION

Wave Function Methods. The wave function results obtained are compared together and with available literature in Table 1. At this stage, our goal is to define reliable benchmark values to which TD-DFT and BSE/evGW results could be compared, that is, to obtain TBE. To this end, it is often unpractical to rely on “raw” experimental data. Indeed, we perform vertical calculations, whereas the most reliable experimental results are typically obtained for 0−0 energies. The differences between these two transition energies are strongly molecule-dependent but are typically in the 0.1−0.3 eV range for organic compounds.109−111 The computation of 0−0 energies requires the determination of the ES geometrical minimum and vibrations, which is possible, at least, with the “lower” levels of theory, for example, CC2 or ADC(2),112 but introduces a supplementary factor in the methodological analysis (the quality of the potential energy surfaces obtained with a given method), and we strive here to provide straightforward and consistent comparisons. In addition, BSE/evGW does not allow ES optimizations due to the absence of analytical gradient, which would introduce further uncertainties in our comparisons. As stated above, all results listed in Table 1 were obtained with the aug-cc-pVTZ atomic basis set. The interested reader will find in the Supporting Information a basis set study at both the CC and ADC levels for I, and it is clear that aug-cc-pVTZ is sufficient. For example, the CCSDR(3)/aug-cc-pVQZ and ADC(3)aug-cc-pVQZ results are within 0.003 eV of their tripleζ counterparts. We also provide the transition energies obtained with the more compact aug-cc-pVDZ in the Supporting Information, but they are generally significantly larger than the aug-cc-pVTZ ones; for example, an average shift of +0.051 eV is obtained with ADC(3), so that the use of a double-ζ atomic basis set would preclude the quality of further comparisons. In short, the selected basis set appears as adequate for our purposes. For some compounds, CCSDR(3) frozen-core results were previously published,34 and the values obtained here are typically slightly smaller, for example, by −0.005 eV for I and by −0.013 eV for V. Likewise, for IX, X, XV, and XVI (frozen-core) CC3/aug-cc-pVTZ and CASPT2/aug-cc-pVTZ values were reported previously on a slightly different geometry,85,86 and our values are, again, very close to these previous estimates. For example, we obtained a CC3 value of 3.93 eV for formaldehyde, only 0.05 eV above Thiel’s value (3.88 eV) determined on the MP2/6-31G(d) minimal structure. 6125


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Figure 1. Comparisons of CC (a) and ADC (b) transition energies with CC3 values (in eV) for the compounds in Table 1. The central diagonal line indicates a perfect match with CC3.

Table 2. Transition Energies Obtained at the TD-DFT and BSE Levels of Theory Using the M06 Series of Functionals for the Compounds Shown in Scheme 1a TD-DFT I II III IV V VI VII VIII IX X XI XII XIIII XIV XV XVI MSE MAE R2

BSE/evGW

M06-L

M06

M06-2X

M06-HF

M06-L

M06

M06-2X

M06-HF

TBE

2.116 2.324 2.033 1.510 2.986 4.004 2.762 4.112 4.256 4.676 2.546 3.550 3.694 3.619 2.116 3.502 0.037 0.200 0.944

1.597 1.874 1.526 1.474 2.694 3.650 2.515 3.848 3.822 4.355 2.528 3.561 3.282 3.271 2.080 3.444 −0.231 0.242 0.967

1.296 1.642 1.237 1.549 2.569 3.545 2.424 3.942 3.657 4.183 2.540 3.562 3.151 3.145 2.250 3.636 −0.305 0.326 0.939

0.861 0.349 0.884 1.350 2.167 3.092 2.061 3.779 3.046 3.433 2.065 3.101 2.431 2.664 2.157 3.456 −0.770 0.770 0.801

1.501 1.762 1.475 1.487 2.451 3.448 2.251 3.579 3.689 4.054 2.652 3.476 3.322 3.365 2.157 3.567 −0.311 0.324 0.977

1.531 1.843 1.525 1.453 2.483 3.467 2.289 3.650 3.579 3.905 2.689 3.542 3.339 3.383 2.209 3.593 −0.296 0.304 0.980

1.701 2.037 1.693 1.481 2.546 3.501 2.386 3.608 3.811 4.173 2.856 3.810 3.481 3.465 2.312 3.770 −0.161 0.181 0.979

2.424 2.806 2.145 1.697 2.839 3.860 3.208 4.949 4.927 5.435 3.331 4.654 4.625 4.335 2.604 4.386 0.564 0.564 0.954

1.942 2.215 1.938 1.382 2.729 3.731 2.556 4.045 3.934 4.438 2.800 3.803 3.663 3.779 2.417 3.837

a

All values are in electronvolts. At the bottom of the Table, a statistical analysis using the TBE as reference is provided: mean signed error (MSE), mean absolute error (MAE), and linear determination coefficient (R2).

The most refined method used to date, that is MR-AQCC,113 gives an adiabatic energy of 1.82 eV, in very good agreement with the measured 0−0 energy. We therefore determined the difference between vertical and 0−0 (including ZPVE correction) energies at the (EOM-)CCSD/def 2-TZVPP level, and it amounts to 0.17 eV (2.00−1.83 eV), the 0−0 energy being accurately given by CCSD. This is larger than the 0.27 eV difference between the adiabatic (1.49 eV, no ZPVE correction) and vertical (1.76 eV) energies obtained in a previous MSCASPT2(12e,9o) study.140 By adding the 0.17 eV CCSD correction to the experimental 0−0 energy, one reaches an estimate for the vertical energy of ca. 1.96 eV, very close to the CC3 value. Given this match, the stability of the CC values when increasing the expansion order, and the weight of the leading configuration according to CAS calculations, we therefore used the CC3 value as TBE. For III, one notices similar trends, that is, the MR-AQCC adiabatic estimate (1.75 eV)119 is very similar to the experimental 0−0 energy (1.73

eV).121 Applying (EOM-)CCSD/def 2-TZVPP, we obtained a vertical absorption energy (2.01 eV), 0.23 eV above the computed 0−0 value (1.78 eV), that is also in line with the experimental data. Therefore, the MR-AQCC vertical value should be ca. 1.98 eV, which is again close to our CC3 value, justifying that we used it as TBE. Given the chemical similarity with both I and III, we also considered the CC3 estimate as the best available to date for II, although in that case, the amount of previous theoretical and experimental evidences is too limited to definitively ascertain this choice. The only three remaining molecules, that is VIII, XIII, and XIV represent intermediate situations: one notices a significant though not dramatic spread of the theoretical estimates. Noticing that, for these compounds, the CC3 values are in between their ADC(2) and CASPT2 counterparts and that the differences between CC2 and CC3 results are significantly smaller than the gap separating the ADC(2) and ADC(3) results, we also selected the CC3 values as reference for these molecules. 6126


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Table 3. Transition Energies Obtained at the TD-DFT and BSE Levels of Theory Using the LYP Series of Functionals for the Compounds Shown in Scheme 1a TD-DFT I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI MSE MAE R2

BSE/evGW

BLYP

OLYP

B3LYP

BHH

CAM

LC

BLYP

OLYP

B3LYP

BHH

CAM

LC

TBE

1.825 2.045 1.731 1.258 2.637 3.560 2.428 3.644 3.791 4.203 2.195 3.168 3.443 3.321 1.890 3.510 −0.285 0.285 0.962

1.846 2.059 1.735 1.275 2.710 3.671 2.455 3.469 3.827 4.227 2.191 3.183 3.463 3.398 1.899 3.200 −0.288 0.288 0.941

1.803 2.044 1.732 1.493 2.711 3.654 2.549 3.890 3.891 4.373 2.522 3.542 3.479 3.385 2.250 3.563 −0.146 0.159 0.982

1.800 2.060 1.756 1.821 2.822 3.805 2.743 4.258 4.056 4.621 2.912 3.967 3.540 3.453 2.716 4.077 0.075 0.191 0.954

1.773 2.021 1.716 1.654 2.691 3.646 2.588 4.019 3.889 4.437 2.712 3.754 3.501 3.406 2.465 3.808 −0.071 0.115 0.977

1.729 1.971 1.664 1.639 2.596 3.546 2.511 3.941 3.756 4.349 2.658 3.697 3.482 3.370 2.404 3.743 −0.135 0.167 0.977

1.468 1.707 1.429 1.522 2.489 3.489 2.299 3.651 3.663 4.051 2.641 3.456 3.296 3.314 2.158 3.536 −0.315 0.333 0.970

1.462 1.696 1.419 1.494 2.433 3.434 2.228 3.512 3.632 3.998 2.631 3.440 3.292 3.316 2.155 3.554 −0.345 0.359 0.970

1.516 1.811 1.504 1.451 2.483 3.463 2.294 3.572 3.643 3.999 2.675 3.460 3.341 3.349 2.193 3.570 −0.305 0.314 0.979

1.669 2.041 1.677 1.444 2.500 3.464 2.342 3.502 3.706 3.952 2.794 3.650 3.478 3.483 2.320 3.723 −0.217 0.224 0.978

1.568 1.882 1.567 1.464 2.513 3.474 2.342 3.455 3.671 3.838 2.745 3.606 3.392 3.406 2.252 3.659 −0.273 0.284 0.969

1.576 1.862 1.563 1.495 2.567 3.516 2.425 3.827 3.778 4.290 2.800 3.756 3.420 3.432 2.263 3.723 −0.182 0.196 0.979

1.942 2.215 1.938 1.382 2.729 3.731 2.556 4.045 3.934 4.438 2.800 3.803 3.663 3.779 2.417 3.837

a

BHH, CAM, and LC stand for BH&HLYP, CAM-B3LYP, and LC-BLYP, respectively. All values are in electronvolts. At the bottom of the Table, a statistical analysis using the TBE as reference is provided: mean signed error (MSE), mean absolute error (MAE), and linear determination coefficient (R2).

Table 2. These results confirm the previously reported trend; for example, for II the transition energy goes from 2.32 eV (M06-L) to 0.35 eV (M06-HF). It is particularly striking that increasing the amount of exact exchange included in the XCF induces a decrease, rather than an increase, of the transition energy. This is an unusual behavior: in general the addition of exchange of HF form yields a more localized description and consequently a larger gap than in “pure” XCF.6 Additionally, the changes of the TD-DFT transition energies when varying the M06 XCF are large. Nevertheless, we note that the results obtained with the four M06 XCF become rather consistent with one another for the compounds presenting a more delocalized ES, for example, IV, VIII, XI, XII, XV, and XVI. As can be seen in Table 3, the TD-DFT results obtained with the LYP XCF behave more “classically”. Indeed, either the dependency on the XCF is quite small, e.g., for II, going from BLYP to LC-BLYP yields a variation smaller than 0.1 eV, or the transition energy increases with the amount of exact exchange included in the XCF, e.g., for XV, it goes from 1.89 eV (BLYP) to 2.72 eV (BH&HLYP). For the records, we note that using the Tamm−Dancoff approximation to TD-DFT increases the computed transition energies, as expected,144 but is not able to fully compensate the “odd” behavior of the M06 results (see Table S5: the TDA transition energy is 2.38 eV with M06-L and 1.45 eV with M06-2X for II). As can be deduced from Tables 2 and 3, BSE/evGW is able to restore a more standard behavior for the results, irrespective of the selection of M06 or LYP starting eigenstates. First, one notes that increasing the amount of exact exchange almost systematically increases the BSE transition energy, as one would expect. Second, at the exception of the M06-HF resultsM06HF is a rather “extreme” functional, relying on 100% of HF exchange, that would not be used normally in ES calculations of small chromophoresthe dependency on the XCF is much smaller with BSE/eVGW than with TD-DFT. For instance,

The data of Table 1 also offer a possibility to statistically compare the wave function results. While compared to Thiel’s set84,86 we considered a much smaller set of data, we underline that this set encompasses several low-lying excited states (0.980) and is particularly high within the CC methods. An outcome that was maybe less expected is that ADC(2) provides larger correlation with both CASPT2 (R2 = 0.994) and CC3 (R2 = 0.998) than with ADC(3) (R2 = 0.987). In Figure 1a we compare the CC values with one another. As can be seen, there are only two cases (XV and XVI) for which the CC2 estimate is smaller than the CC3 one, whereas CCSD systematically provides too large (compared to CC3) transition energies. The average absolute difference of CC2, CCSD, and CCSDR(3) with respect to CC3 are 0.075, 0.100, and 0.032 eV, and this result is in line with previous works.84,143 More impressive maybe is the fact that ADC(2) delivers transition energies closer from their CC3 counterparts (0.061 eV) than from ADC(3) (0.144 eV). Of course, this might be an outcome related to the selection of n→π* transitions. However, as can be seen in Figure 1b, it is clear that the magnitude of the deviation between ADC(3) and CC3 tends to increase when the transition energy gets smaller: the match between these two methods is generally nice above ca. 2.5 eV but tends to degrade for low-lying ES. Examining if this trends holds for π→π* is beyond our scope here, but it would be of interest. TD-DFT and BSE/evGW Methods. First, let us recall that previous TD-DFT works have found contrasted XCF dependency for n→π* transitions: strong in the M06 family89 but weak in the LYP family.87,88 The TD-DFT results obtained for the compounds of Scheme 1 with the four M06 XCF are listed in 6127


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Figure 2. MSE (top) and MAE (bottom) obtained with TD-DFT (left) and BSE/evGW (right) for the molecules of Scheme 1 using the TBE as references. All values are in electronvolts.

less accurate but provide MAE of ca. 0.2−0.3 eV, which remains a valuable performance by TD-DFT standards.6 In the M06 family, this TD-DFT accuracy can also be attained with both M06-L and M06, but the transition energies become significantly too small with M06-2X and M06-HF. Even more striking are the TDA results with CAM-B3LYP that are almost perfectly on the spot with MAE of 0.087 eV (see the Supporting Information). This is a logical outcome, as, on the one hand, TD-DFT energies are slightly too small (negative MSE) for this XCF, and, on the other hand, TDA increases the transition energies. With BSE/evGW, one obtains almost systematically too small transition energies compared to the reference values; that is, the MSE are negative with all XCF (but with M06-HF). At the BSE/evGW level, the MAE are rather independent of the starting XCF but tend to be larger than both their TD-DFT and TDA counterparts in most cases (see Figure 2). The most accurate BSE results are obtained with M06-2X that yields an MAE of 0.18 eV and an R2 of 0.98. This performance is inferior to the one of TD-CAM-B3LYP, which provides a smaller average deviation. Finally, the MAE values obtained at the BSE/evGW@M06-2X level for the nitroso, thiocarbonyl, carbonyl, and diazo chromophores are 0.191, 0.255, 0.113, and 0.167 eV, respectively; that is, BSE seems the most adequate for the carbonyl compounds. For comparison, the TD-CAM-B3LYP MAE values for those four families are 0.214, 0.043, 0.046, and 0.153 eV; i.e., the performance is

going from M06-L (0%) to M06-2X (54%) provokes an average absolute variation of the energy of 0.38 eV with TDDFT but 0.15 eV only with BSE/evGW considering the full list of compounds. On could think that this success is related to the specificities of the TD-M06 results for the present set, but this is incorrect. Indeed, going from BLYP (0%) to BH&HLYP (50%) gives an absolute variation of the transition energy equal to 0.36 eV with TD-DFT but 0.14 eV with BSE/evGW. In other words, the XCF dependency is indeed much reduced when using BSE, and the n→π* transitions are no exception to this general rule. This is a reassuring news: the fact that BSE/ evGW significantly washes out the XCF dependency seems true for many families of ES,34,83 including n→π* transitions. Let us now turn toward determining what is the most adequate approach to investigate the considered ES. Using the TBE determined above, we computed the mean signed error (MSE) and mean absolute error (MAE) for the 10 tested XCF with both TD-DFT and BSE/evGW approaches. These results are given at the bottom of Tables 2 and 3 and are represented in Figure 2. At the TD-DFT level, the MSE is generally small but XCF-dependent, whereas with BSE/evGW the MSE are in the −0.2 to −0.3 eV domain, M06-HF being an exception at both levels of theory. With TD-DFT, the most adequate XCF are B3LYP and CAM-B3LYP that delivers stunningly accurate results with an MAE of 0.16 and 0.11 eV, respectively, and R2 larger than 0.98. The other XCF of the LYP series are relatively 6128


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M06-L, BLYP, and OLYP, which led rather accurate TD-DFT estimates for the small compounds, produce larger deviations when the system size grows, which is an expected trend.6 In contrast, TD-CAM-B3LYP remains very accurate with an MAE of 0.131 eV and an R2 of 0.98. With BSE/evGW, the trends are consistent with the one noted above: (i) a more standard XCFdependency is again restored for the M06 series irrespective of the selected compound; (ii) the results are less dependent on the selected XCF, and the errors are therefore very significantly reduced with the pure XCF, compared to TD-DFT; (iii) the MSE are always negative (but with M06-HF); that is, there is a trend to underestimate the TBE; and (iv) the best starting point remains M06-2X, which delivers an MAE very similar to the one obtained with TD-CAM-B3LYP.

optimal for (thio)carbonyl derivatives but significantly worse for the two other families of dyes. Larger Compounds. To confirm that the trends obtained on the series of compact molecules pertain in larger derivatives, we used a second set of seven compounds represented in Scheme 2, which include well-known chromophores, for Scheme 2. Representation of the Second Series of Compounds Investigated Herein

■

SUMMARY

We have determined the lowest transition energies in a series of n→π* chromogens (nitroso, thiocarbonyl, carbonyl, and diazo chromophores) using several ab initio theories. First, using wave function approaches we have defined theoretical best estimates for 16 molecules. There was only one molecule, IV, for which a significant multireference character was detected by CASPT2 calculations. For several molecules, we found a good agreement between CASPT2, ADC(3), and CC3 transition energies, but this does not hold for nitroso derivatives, for which an analysis of the experimental and theoretical literature led us to conclude that CC3 transition energies are the most trustworthy. By comparing the wave function results, we noticed that ADC(2) provided more accurate estimates than ADC(3) with respective MAE of 0.061 and 0.144 eV as compared to the TBE. Indeed, for low-lying ES (ca. below 2.5 eV), the deviations between CC3 and ADC(3) appear quite significant. Turning to TD-DFT, it is important to consider separately the LYP and M06 results. For the former, TD-DFT yields (very) accurate results that are rather independent of the selected XCF; for example, TD-CAM-B3LYP (TD-B3LYP) provides a MAE as small as 0.12 eV (0.16 eV). For the latter, the results are more surprising; that is, in the most compact

example, diazobenzene (XX), 9,10-anthraquinone (XXIII), and coumarin (XXII). The TD-DFT and BSE/evGW results can be found in Tables 4 (M06 series) and 5 (LYP series). Of course, for derivatives of that size, it is not possible to perform CC3/aug-cc-pVTZ calculations, and compromises must be made in the calculation of the TBE, so that one should be careful with the statistical analysis given at the bottom of Tables 4 and 5. For the TD-DFT part, one notices that the odd behavior of the “M06” series described above, that is, the decrease of the transition energy with increasing amount of exact exchange, pertains for the molecules presenting an isolated chromophore but tends to disappear when the n→π* chromogen is included in a large π-conjugated system. Clearly, the pure XCF, that is,

Table 4. Transition Energies Obtained at the TD-DFT and BSE Levels of Theory Using the M06 Series of Functionals for the Compounds Shown in Scheme 2a TD-DFT XVII XVIII XIX XX XXI XXII XXIII MSE MAE R2

B1g Au B1g Au

BSE/evGW

M06-L

M06

M06-2X

M06-HF

M06-L

M06

M06-2X

M06-HF

TBE

2.351 3.846 3.429 2.504 2.258 2.417 4.034 2.662 2.941 −0.403 0.435 0.779

2.059 3.449 3.043 2.490 2.530 2.717 4.516 2.994 3.267 −0.334 0.334 0.985

1.910 3.369 2.946 2.502 2.697 2.905 4.814 3.146 3.399 −0.265 0.267 0.947

1.404 2.806 2.350 2.042 2.417 2.599 4.545 2.861 3.060 −0.665 0.665 0.876

1.881 3.521 3.146 2.465 2.620 2.758 4.476 2.944 3.165 −0.344 0.344 0.973

1.944 3.434 3.139 2.601 2.702 2.829 4.564 3.056 3.275 −0.281 0.281 0.976

2.000 3.533 3.271 2.865 2.946 3.082 4.856 3.352 3.577 −0.065 0.145 0.954

2.212 4.991 4.197 3.449 3.408 3.529 5.529 3.932 4.157 0.593 0.593 0.934

2.207b 3.859b 3.450c 2.902d 2.773e 2.897e 4.895f 3.444g 3.644g

a

See the caption of Table 2 for more details. bExtrapolated CC3 value from ref 34. cCC3/aug-cc-pVDZ value (3.493 eV) corrected for basis set effect from the difference between the CCSDR(3)/aug-cc-pVTZ (3.447 eV) and CCSDR(3)/aug-cc-pVDZ (3.490 eV) values (this work). dExtrapolated CCSDR(3) value from ref 34. eCC3/aug-cc-pVDZ values (2.771/2.884 eV) corrected for basis set effect from the difference between the CCSDR(3)/aug-cc-pVTZ (2.911/3.048 eV) and CCSDR(3)/aug-cc-pVDZ (2.909/3.035 eV) values (this work). fCCSDR(3)/aug-cc-pVDZ value (4.881 eV) corrected for basis set effect from the difference between the CCSD/aug-cc-pVTZ (5.063 eV) and CCSD/aug-cc-pVDZ (5.049 eV) values (this work). gCCSD/aug-cc-pVTZ value (this work). 6129


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Table 5. Transition Energies Obtained at the TD-DFT and BSE Levels of Theory Using the LYP Series of Functionals for the Compounds of Scheme 2a TD-DFT XVII XVIII XIX XX XXI XXII XXIII MSE MAE R2 a

B1g Au B1g Au

BSE/evGW

BLYP

OLYP

B3LYP

BHH

CAM

LC

BLYP

OLYP

B3LYP

BHH

CAM

LC

TBE

2.036 3.531 3.130 2.300 1.957 2.086 3.651 2.329 2.593 −0.718 0.718 0.753

2.066 3.606 3.205 2.279 1.941 2.087 3.682 2.315 2.587 −0.700 0.700 0.724

2.101 3.617 3.213 2.609 2.459 2.624 4.380 2.902 3.176 −0.332 0.332 0.975

2.200 3.729 3.326 2.921 3.082 3.261 5.172 3.538 3.768 0.103 0.161 0.949

2.093 3.643 3.242 2.767 2.840 3.035 4.877 3.280 3.525 −0.085 0.131 0.975

2.014 3.587 3.186 2.732 2.864 3.082 4.457 3.284 3.284 −0.176 0.237 0.948

1.899 3.468 3.096 2.458 2.582 2.720 4.473 2.923 3.156 −0.366 0.366 0.978

1.866 3.474 3.107 2.463 2.589 2.734 4.460 2.893 3.125 −0.373 0.373 0.973

1.921 3.499 3.135 2.573 2.658 2.791 4.473 2.999 3.222 −0.311 0.311 0.980

1.971 3.503 3.250 2.812 2.837 2.937 4.626 3.203 3.402 −0.170 0.193 0.970

1.977 3.468 3.175 2.724 2.785 2.916 4.648 3.159 3.379 −0.204 0.211 0.972

2.038 3.574 3.237 2.816 2.868 3.021 4.858 3.272 3.508 −0.098 0.146 0.969

2.207 3.859 3.450 2.902 2.773 2.897 4.895 3.444 3.644

See the footnotes of Tables 3 and 4 for more details.

■

ACKNOWLEDGMENTS C.A. and D.J. thank the ANR for support in the framework of the EMA grant. S.B. and D.J. are thankful to the Campus France and the Slovak Research and Development Agency for supporting their long-standing collaboration in the framework of Stefanik PHC program (BridgET project, Nos. 35646SE and SK-FR-2015-0003, respectively). S.B. is thankful to the Slovak Research and Development Agency Project No. APVV-150105. This research used resources of (i) the GENCI-CINES/ IDRIS; (ii) Centre de Calcul Intensif des Pays de Loire; (iii) a local Troy cluster, (iv) HPC resources from ArronaxPlus (Grant No. ANR-11-EQPX-0004 funded by the French National Agency for Research), and (v) High Performance Computing Center of the Matej Bel Univ. in Banska Bystrica (acquired in Project Nos. ITMS 26230120002 and 26210120002 of the Research and Development Operational Programme funded by the ERDF).

compounds the transition energies tend to decrease when increasing the amount of exact exchange, whereas in more delocalized compounds, a more standard behavior is obtained. As a consequence of the former compounds, the MSE significantly changes when going from TD-M06-L (0.037 eV) to M06-2X (−0.305 eV). Interestingly, BSE/evGW gives a limited XCF dependency with both series of XCF (the only exception is M06-HF), and the expected “logical” trend is recovered: the transition energies increase with the amount of exact exchange. This BSE/evGW success is however accompanied by a rather systematic underestimation of the reference transition energies, typically by −0.2 to −0.3 eV. The most accurate BSE/evGW results are reached starting with M06-2X eigenstates, a combination that provides a MAE of 0.18 eV and a R2 of 0.98, an error that is not strongly dependent on the considered subset of molecules. Indeed, the MAE obtained for the four families of compounds are in the 0.11−0.26 eV range. In the last stage of this work, we considered larger molecules, and most conclusions drawn for the set of compact dyes are found again, though the performances of pure functionals worsen significantly at the TD-DFT level.

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(1) Dreuw, A.; Head-Gordon, M. Single-Reference ab initio Methods for the Calculation of Excited States of Large Molecules. Chem. Rev. 2005, 105, 4009−4037. (2) 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. (3) Ullrich, C. Time-Dependent Density-Functional Theory: Concepts and Applications; Oxford Graduate Texts; Oxford University Press: New York, 2012. (4) van Caillie, C.; Amos, R. D. Geometric Derivatives of Excitation Energies Using SCF and DFT. Chem. Phys. Lett. 1999, 308, 249−255. (5) 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. (6) Laurent, A. D.; Jacquemin, D. TD-DFT Benchmarks: A Review. Int. J. Quantum Chem. 2013, 113, 2019−2039. (7) Ziegler, T.; Krykunov, M.; Seidu, I.; Park, Y. In Density-functional methods for excited states; Ferré, N., Filatov, M., Huix-Rotllant, M., Eds.; Topics in Current Chemistry; Springer Berlin Heidelberg, 2015; pp 1−35. (8) Laurent, A. D.; Adamo, C.; Jacquemin, D. Dye Chemistry with Time-Dependent Density Functional Theory. Phys. Chem. Chem. Phys. 2014, 16, 14334−14356. (9) Tozer, D. J. Relationship Between Long-Range Charge-Transfer Excitation Energy Error and Integer Discontinuity in Kohn-Sham Theory. J. Chem. Phys. 2003, 119, 12697−12699.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b05222. Active spaces used in the CASPT2 calculations. Geometries used in all calculations. Basis set effects. Pearson matrix for the wave function methods. TDA results. (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +33 (0) 2 51 12 55 64. ORCID

Ivan Duchemin: 0000-0003-4713-1174 Denis Jacquemin: 0000-0002-4217-0708 Notes

The authors declare no competing financial interest. 6130


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