What is the Key for Accurate Absorption and Emission Calculations

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What is the Key for Accurate Absorption and Emission Calculations ? Energy or Geometry ? Denis Jacquemin J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01224 • Publication Date (Web): 24 Jan 2018 Downloaded from http://pubs.acs.org on January 25, 2018

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Journal of Chemical Theory and Computation

What is the Key for Accurate Absorption and Emission Calculations ? Energy or Geometry ? Denis Jacquemin∗ Laboratoire CEISAM - UMR CNRS 6230, Université de Nantes, 2 Rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France E-mail: [email protected]

Abstract Using a hierarchy of wavefunction methods, namely ADC(2), CC2, CCSD, CCSDR(3) and CC3, we investigate the absorption and emission energies in a set of 24 organic compounds. For all molecules, reference values are determined at the CC3//CC3 or CCSDR(3)//CCSDR(3) levels and the energetic and geometric effects are decomposed considering all possible methodological combination between the five considered methods. For absorption, it is found that the errors are mainly energy-driven for ADC(2), CC2 and CCSDR(3), but not for CCSD. There is also an error compensation between the errors made on the geometries and transition energies for the two former approaches. For emission, the total errors are significantly larger than for absorption due to the significant increase of the structural component of the error. Therefore, the selection of a very refined method to compute the fluorescence energy will not systematically provide high accuracy if the excited-state geometry is not also optimized at a suitable level of theory. This is further demonstrated using results obtained from TD-DFT and hybrid TD-DFT/wavefunction protocols. We also found that, compared to full CC3, only CCSDR(3) is able to deliver errors below the 0.1 eV threshold, a

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statement holding for both absorption (mean absolute error: 0.033 eV) and emission (mean absolute error: 0.066 eV).

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1

Introduction

Over the years, many theoretical methods able to treat excited states (ES) have been developed and extended, and it clearly turned out that choosing an approach suited for describing ES is more difficult than for ground state (GS). Today, the two most popular ab initio approaches are probably Time-Dependent Density Functional Theory (TD-DFT) 1,2 and Complete Active Space Self-Consistent Field (CASSCF) 3,4 that allow to respectively account for dynamic and static electron correlation effects at reasonable computational costs. These methods also advantageously allow to determine ES structures as analytical gradients are accessible for both models in a wide variety of codes. 5–8 However, if the obtained results are not sufficiently accurate, the paths for improvement are either ill-defined and/or costly. For TD-DFT, one can certainly cite, amongst many possible strategies, the use of the socalled double-hybrids, 9,10 that provide more trustworthy results but come with significantly increased computational requirements compared to “conventional” TD-DFT relying on standard semi-local or hybrid functionals. For CASSCF, it is possible to include dynamical correlation effects using a second-order perturbative approach (CASPT2 method), 11,12 which greatly improves the quality of the computed transition energies, but becomes unpractically demanding for medium and large systems. To obtain more systematically improvable results, one can turn towards the Algebraic Diagrammatic Construction (ADC) 13 or Coupled-Cluster (CC) approaches. With these methods, one can increase the expansion order and follow the variations of the target property, e.g., one can compare the transition energies obtained with CC2, 14 CCSD, 15,16 CCSDR(3) 17 or (EOM-)CCSD(T), 18 CC3 14 or EOM-CCSDT-3, 19 etc. Nevertheless, this hierarchical strategy again comes with a steep increase of computational requirements, especially when contributions from triple excitations are included, as in the latter methods. To mitigate the cost of ES calculations, a popular approach is to select a less-demanding method to optimize both the GS and ES geometries, and next to determine the transition energies on these geometries using a more advanced theoretical model. There are countless 3



examples of this approach in the literature, and we only detail a short selection of these works here. In their famous papers defining a set of reliable vertical absorption energies, Thiel and coworkers performed CASPT2/aug-cc-pVTZ and CCn/aug-cc-pVTZ calculations but they systematically selected second-order Møller-Plesset (MP2)/6-31G(d) GS geometries. 20,21 In their assessment of single-reference methods for ES energies, Goerigk and Grimme first transformed the experimental 0-0 energies into vertical values using TD-PBE/TZVP structural and vibrational data, before computing transition energies with a large panel of methods, including the CC2/def2 -TZVPP approach. 22,23 For the calculation of the 0-0 energies of emissive fluoroborate dyes, one of us proposed to apply TD-DFT combined to the compact 6-31G(d) atomic basis set to determine the GS and ES structures as well as zero-point corrections, but to use much more refined methods to compute the adiabatic energies. 24,25 While such “composite” protocols have been successful in several cases, they nevertheless come with an often unchecked hypothesis: the geometries are viewed as less sensitive to the selected level of theory than the transition energies. For vertical absorption, this looks a priori a very reasonable assumption because cost-effective theoretical methods like DFT or MP2 generally provide GS structures that compare well with experiment. For emission (and consequently adiabatic properties), this hypothesis seems less factual: one does not have access to experimental ES structures in the vast majority of the cases; it is therefore hard to a priori understand why a large impact of the selected method on the computed ES energies would not reflect in similarly important effects for the ES geometries. In this framework, we underline that the suitability of “composite” protocols for 0-0 energies has been evaluated by several groups. 26–28 In Ref. 26, Winter et al. determined gas-phase 0-0 energies in 66 aromatic organic molecules, and found that the CC2 mean absolute deviation (MAD) with respect to experiment (0.07 eV) was only slightly degraded when neglecting corrections coming from single-point calculations with diffuse functions (0.08 eV) or when additionally using zero-point energies from TD-DFT (0.10 eV). In Ref. 27, it was showed that the MAD with respect to experiment obtained for 80 solvated dyes with TD-DFT (0.24 eV) could be

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significantly reduced by using ADC(2) (0.14 eV) or CC2 (0.13 eV) adiabatic energies determined on the TD-DFT structures. In Ref. 28, Oruganti, Fang and Durbeej investigated several mixed TD-DFT/CC2 approaches, and concluded that full CC2 0-0 energies, could be accurately and consistently estimated using TD-DFT structures and relaxation energies. Whilst all these results seem reassuring, we highlight that only TD-DFT and second-order approaches have been tested and that comparisons were made with experiments, so that it is uneasy to know if the deviations between calculated and measured 0-0 energies mainly originate in the inaccuracies of the transition energies or geometries. In addition, as comparisons are made directly with experiment, some error compensation mechanisms could be at play. Indeed, in a recent assessment of the quality of ADC(2), CC2 and CCSD for the calculation of ES geometries, we have showed that these three methods sometimes yield non-trifling errors compared to more advanced theories, such as CASPT2 or CC3. 29 For instance, the ES carbonyl bond lengths were found to be overestimated by 0.042 ˚ A and 0.029 ˚ A with ADC(2) and CC2, respectively, when compared to theoretical best estimates. 29 In the present contribution, we strive to resolve the article title’s question. To this end, we have selected a series of 24 molecules represented in Scheme 1 for which we determined the GS and ES structures as well as the transition energies with five single-reference wavefunction approaches of increasing complexity, namely ADC(2), CC2, CCSD, CCSDR(3) and CC3, considering all possible methodological combination (transition/geometry) for each compound. We additionally evaluate the performances of protocols relying on TD-DFT in the last part of this work. In this way, taking the “full” CC3 or CCSDR(3) results as references, one can rigorously quantify the errors originating from the energies and the structures for both absorption and emission. For Thiel’s set of compounds, CC3 transitions energies were shown by Watson and coworkers to be very close from EOM-CCSDT-3 values, 30 and by Kannar and Szalay to be also very similar to (EOM-)CCSDT results. 31,32 CC3 can therefore safely be used to obtain accurate reference transition energies.

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O H

H

H

Acetylene (1Au)

N

F

F

Carbonylfluoride (1A’’)

Cyanogen (1Σu-)

O

O

N N H

H

H Diacetylene (1Δu)

N

Cyanoacetylene (1Δ)

H H

H

N

Diazomethane (1A’’)

H

Cl

Formaldehyde (1A’’)

Formylchloride (1A)

H H

H

O

C H

F

H

Formylfluoride (1A)

N

Nitrosylcyanide

Nitrosomethane (1A’’)

O

Se

Se

Cl

Phosgene

(1A’’)

N H

N H

H

N N

F

F

S

S

N N

F

Cl

Tetrazine (1B3u)

Thiocarbonylchlorofluoride (1A)

S

S

S

H

Thioformaldehyde (1A2)

Cl

H

Thioformylchloride (1A)

H

Selenoformaldehyde (1A2)

Streptocyanine-1 (1B2)

H

H

Selenocarbonylfluoride (1A’’)

H H

N

Ketene (1A’’)

Cl (1A’’)

O

H 3C

H H Methylenecyclopropene (1B2)

O N

O

Cl

F

F

Thiocarbonylfluoride (1A’’) H

Cl

Thiophosgene (1A’’)

C S H Thioketene (1A’’)

Scheme 1: Representation of the 24 molecules investigated herein. The considered ES is indicated as well, the lowest ES of the indicated symmetry being systematically considered. For the blue and black compounds, reference values have been obtained at the CC3 and CCSDR(3) levels of theory, respectively. The blue molecules constitute the 12-compound set.

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2

Computational Details

The aug-cc-pVTZ atomic basis set was used throughout, for both energy calculations and geometry optimizations. Most of the structures used here have been extracted from Ref. 29. The interested reader will also find in that earlier work the results of the T1 diagnostic test, 33 and only one considered ES (diazomethane) present a T1 exceeding the 0.02 limit. In other words, but for that borderline case, the CC methods are adequate for the treated case. A few additional geometries had to be determined, and they have been obtained using Turbomole 34 for ADC(2) and CC2 methods, Gaussian 35 for CCSD, and Dalton 36 for CCSDR(3) and CC3. These geometry optimizations use exactly the same protocol as in Ref. 29, and we refer the reader to that contribution for further details. We underline that analytic gradients are not available for CCSDR(3) and CC3, and the structural minimizations relied on numerical gradients, which is obviously extremely computationally demanding at the CC3/aug-cc-pVTZ level. The ADC(2) transition energies were computed with the QChem package, 37 whereas all CC transition energies presented below were determined with the Dalton package 36 using default parameters but accounting for symmetry when possible. The DFT and TD-DFT calculations discussed in the last Section of this work have been performed with Gaussian 16, 35 using the PBE0 exchange-correlation functional. 38,39 During all calculations all electrons have been correlated, that is, the frozen-core approximation was not applied. In the following, we use the usual convention for composite calculations, i.e., “energy”//“geometry”. We recall that the GS methods corresponding to ADC(2) and CCSDR(3) are MP2 and CC(3), respectively, and that the latter is very close from the CCSD(T) “golden standard”.

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Results and Discussion

From the technical viewpoint, the main task needed for our purposes was to determine ES geometrical parameters at very high theoretical level, namely CCSDR(3)/aug-cc-pVTZ 7



and CC3/aug-cc-pVTZ. As stated above, we used to this end the data published in a recent database, 29 though, for three molecules, namely cyanoacetylene, methylenecyclopropene and thiocarbonylchlorofluoride, we have obtained improved geometries herein. There are described in the Supporting Information (SI). For the two first compounds, we have now obtained full CC3/aug-cc-pVTZ whereas only “basis-set extrapolated” values were previously available. The maximal discrepancy between the two sets of data is limited to 0.001 ˚ A for the N≡C bond of cyanoacetylene. For the later molecule, we have determined the CCSDR(3)/aug-cc-pVTZ GS and ES structural parameters, and we found that all bonds are slightly shorter than at the CCSDR(3)/def2 -TZVPP level (see Table S3 in the SI). In the following we start by discussing the wavefunction results obtained for both absorption (Section 3.1) and emission (Section 3.2) with ADC(2) and the four tested CC levels, before turning to TD-DFT approaches in Section 3.3.

3.1

Absorption

For absorption, all computed ADC and CC transition energies are listed in Table S4 to S26 in the SI. We do not discuss in the following the optimal geometrical parameters obtained with all methods, as these data are detailed in Ref. 29 and we focus on the vertical absorption energies. First, we found that the errors induced by the selection of a specific method for computing the transition energy are rather independent of the considered geometry. For instance, the CC2 transition energies of diacetylene are larger than their CC3 counterparts by +0.148, +0.150, +0.144, +0.147 and +0.147 eV when considering the MP2, CC2, CCSD, CC(3) and CC3 optimal GS structures, respectively (Table S8). Likewise, for thioketene, the CCSD absorption energies are larger than the CC3 values by +0.049 ± 0.001 eV for all five geometries (Table S24). The reverse also holds: the deviations related to the selection of a specific geometry are similar for all considered transition energy methods, e.g., in cabonylfluoride, the selection of a CC2 GS structure results in an absorption redshifted by -0.101, -0.092, -0.095, -0.094 and -0.094 eV compared to the CC3 structure, at the 8


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ADC(2), CC2, CCSD, CCSDR(3) and CC3 levels, respectively (Table S5). In other words, the methodological impacts obtained for absorption energies and geometries are nearly independent, and therefore nearly additive. Let us start by analyzing the results for the eleven molecules for which full CC3 estimates could be obtained (the compounds represented in blue in Scheme 1). The deviations with respect to CC3 are represented in Figure 1. In this Figure, the errors for transition energies and geometries have been computed considering the CC3 geometries and energies, respectively, whereas the total deviations are obtained by comparing ADC(2)//MP2, CC2//CC2, CCSD//CCSD and CCSDR(3)//CC(3) results to their CC3//C3 counterparts. As expected, the obtained methodological sensitivity is significantly molecule-dependent, e.g., no deviations larger than 0.1 eV can be found for selenoformaldehyde, whereas significantly bigger errors are observed for cyanogen. Globally, the molecules leading to the largest (and smallest) deviations are similar for the four tested levels of theory. Obviously, the CCSDR(3)//CC(3) approach provides very good approximations to the CC3//CC3 values, the maximal deviation being +0.068 eV for cyanogen (Table S7). It is also obvious from Figure 1 that the errors are mostly, but not exclusively, driven by the transition energies for this method, which is consistent with the fact that CC(3) provides GS geometries in very good agreement with their CC3 counterparts with average deviations on bond lengths as small as 0.001 ˚ A. 29 With CCSD//CCSD, all observed deviations are systematically positive and range from 0.079 eV (selenoformaldehyde) to 0.325 eV (methylenecyclopropene). Qualitatively, this means that CCSD provides a too localized description of the molecular properties, a conclusion already found previously by several groups for both transition energies 30,31,40 and geometries. 29 The electronic and geometric contributions to the total CCSD errors are also both positive and add up to lead to the total overestimation. Importantly and unexpectedly, the contribution of the geometry to the total deviation represents the largest share of the CCSD error in the half of the cases (6/12). With CC2//CC2, one also notices a systematic overestimation of the CC3 values in this 12-compound set, but with strongly different contributions from the

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Deviation (eV)

0.4

ADC(2)

0.3 0.2

Total Energy Geometry

0.1 0.0 -0.1 -0.2 -0.3

0.4

CC2

0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 0.4

CCSD

0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 0.4

CCSDR(3)

0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3

Acetylene Cyanoacetylene Diacetylene Formaldehyde MethyleneCP Thioform. Carbonylfluoride Cyanogen Diazomethane Ketene Selenoform. Thioketene

Figure 1: Errors obtained for the vertical absorption energies using the CC3 results as reference for the 12-compound set. The blue, red and green histograms for a given method (X) correspond to the total, energetic and geometric deviations, respectively, determined comparing the X//X, X//CC3 and CC3//X values to the CC3//CC3 data (see text). All 10 values are in eV. ACS Paragon Plus Environment

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transition energies (rather large positive errors in the 0.089–0.331 eV range) and geometries (either trifling deviations or large negative errors going up to -0.287 eV). Therefore, there exists, for several compounds, an error compensation between the two sources of deviations. This can be illustrated by diacetylene for which the CC3//CC3 absorption energy (5.609 eV) is very accurately reproduced by CC2//CC2 (5.629 eV), but not by CC3//CC2 (5.479 eV) nor CC2//CC3 (5.757 eV), see Table S8. Eventually with ADC(2), the error pattern becomes more erratic with positive and negative deviations found for all contributions depending on the considered molecule. It is nevertheless clear from Figure 1 that the origin of the errors is mostly to be found in the transition energy method, i.e., MP2 geometries are quite accurate. Finally, one also notices that the ADC(2) errors are of the same order of magnitude as their CC2 counterparts. For the data of Figure 1, we have tried to relate the errors made, with a given method, for energies and geometries, but obtained very weak linear correlation coefficient (R): -0.25, -0.56, 0.02 and 0.49 for ADC(2)//MP2, CC2//CC2, CCSD//CCSD and CCSDR(3)/CC(3), respectively. This indicates that the errors made for the transition energy are not a relevant guideline to estimate the errors due to the geometry. We further determined the Pearson correlation matrix between the errors made by all methods. The results are displayed in Tables S52–S54 in the SI. For the errors related to the level selected for transition energies, one finds quite large R between the different methods with values in the 0.70–0.91 range, 41 e.g., a transition hard to describe with CC2 is also challenging with CCSDR(3). For the errors due to geometries, the obtained |R| are also larger than 0.7 but in one case. However, negative correlations are found between, one one side, ADC(2) and CC2 and, on the other side, CCSD and CCSDR(3), which is consistent with Figure 1. As a result of error compensation, all R become small for the total errors (< 0.4), but for ADC(2) versus CC2 (0.85) and CCSD versus CCSDR(3) (0.86). These pairs of methods are therefore significantly correlated, but a small deviation obtained with CC2//CC2 is not indicative of a small CCSD//CCSD error. The average errors are listed in Table 1. In this Table, we report both the statistical data

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obtained for the 12-compound set for which CC3//CC3 values are used as reference, and the average deviations calculated for the full 24-compound set for which CCSDR(3)//CC(3) values serve as benchmarks. We note that the latter method provides very accurate results on the small set and that the largest set of compounds is, of course, more representative. The deviations obtained with all methods are rather small, the total MAD obtained with the two cheapest methods, namely ADC(2)//MP2 and CC2//CC2, are 0.125 (0.090) eV and 0.127 (0.090) eV, respectively, for the 12-compound (24-compound) set. The fact that ADC(2) delivers about the same accuracy as CC2 (for a smaller computational cost) is consistent with previous benchmarks performed by Dreuw’s group. 42 In addition, the magnitudes of these deviations are in line of previous works evaluating the performances of these approaches with respect to reference experimental values for low-lying ES. 23,26,27 These errors are also smaller than what can be expected from simpler methods, like TD-DFT that generally yields deviations of ca. 0.20–0.30 eV for valence ES. 2 Interestingly, one notices that CCSD provides less accurate results than CC2, whereas CCSDR(3) values are very close from their CC3 counterparts, that is the non-iterative inclusion of triple excitations corrects most of the CCSD error. These two conclusions are in line with previous findings for Thiel’s set of compounds. 30,31,40,43 For both ADC(2) and CC2, the MAD related to the transition energy is ca. 2–3 times larger than the geometric MAD, whereas they have alike amplitudes for CCSD. Overall, the data listed in Table 1 hint that performing CCSDR(3) absorption calculations on MP2 geometries might be a valuable compromise between accuracy and computational cost. Indeed, this CCSDR(3)//MP2 composite model yields a MSD of 0.004 eV and a MAD of 0.044 eV compared to CC3//CC3 (12-compound set). In other words, for absorption, on can indeed put the largest share of the computational effort in the calculation of energies, which is consistent with expectations (see Introduction).

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Table 1: Top: statistical analysis for the absorption energies of Figure 1: mean signed and absolute deviations (MSD and MAD, in eV) considering the CC3//CC3 results as reference. Bottom: statistical analysis of the absorption energies carried our for all compounds of Scheme 1 considering CCSDR(3)//CC(3) values as reference. 12-compound set – CC3 reference MSD MAD Method Total Energy Geometry Total Energy Geometry ADC(2)–MP2 0.065 0.082 -0.018 0.125 0.137 0.044 CC2 0.117 0.191 -0.078 0.117 0.191 0.080 0.166 0.088 0.079 0.166 0.088 0.079 CCSD 0.022 0.010 0.033 0.024 0.010 CCSDR(3)–CC(3) 0.032 24-compound set – CCSDR(3) reference MSD MAD Method Total Energy Geometry Total Energy Geometry ADC(2)–MP2 -0.018 -0.008 -0.011 0.090 0.098 0.036 0.068 0.126 -0.060 0.090 0.140 0.062 CC2 CCSD 0.108 0.055 0.052 0.108 0.055 0.054

3.2

Emission

The computed ADC and CC emission energies determined for all compounds are listed in Tables S28 to S51 in the SI. As a first remark, we underline that several of the reported compounds are not fluorescent experimentally because a non-radiative pathway is more favored, and that the reported transition energies simply correspond to difference between the energies of the considered excited state and the ground state at the optimal geometry of the former. We have checked if the near-independent character of the geometric and energetic methodological effects, noticed for absorption, holds. This is indeed often the case. For instance, in ketene the CCSD fluorescence energy exceeds its CC3 counterpart by +0.135, +0.138, +0.131, +0.136 and +0.137 eV for the ADC(2), CC2, CCSD, CCSDR(3) and CC3 optimal ES structures, respectively (Table S37). Likewise, for the same compound, using the CCSD instead of the CC3 optimal geometries induce blueshifts of +0.189, +0.188, +0.185, +0.186 and +0.191 eV when computing the energies at the same five levels of theory. There are nevertheless slightly more important deviations for some molecules especially with ADC(2). Notably, for formaldehyde the discrepancy between the ADC(2) and CC3 transition energies attains -0.230 eV on the ADC(2) geometry but only -0.134 eV on the CC3 structure (Table 13



S34, see also below). As such exceptions remain rare in the treated set of compounds we nevertheless used the same metric as for absorption here, that is we computed the errors for transition energies and geometries considering the CC3 geometries and energies, respectively. As emission has been much less investigated than absorption, we will first discuss three representative examples, in an effort to correlate the geometric parameters to the computed fluorescence energies. First, for selenoformaldehyde (Table S43) that does not undergo puckering in the ES, the only key geometrical parameter is the C=Se distance, that attains 1.843 ˚ A with CC3. CC2 and CCSDR(3) provide very accurate C=Se bond lengths (1.843 ˚ A and 1.838 ˚ A), whereas ADC(2) overestimates it (1.863 ˚ A) and CCSD delivers the opposite error (1.813 ˚ A). 29 The errors obtained when using these ADC(2), CC2, CCSD and CCSDR(3) ES structures to compute the CC3 fluorescence energies are -0.049, -0.003, +0.073 and +0.012 eV, respectively. There is therefore an obvious correlation between both the sign and magnitude of the errors made for the C=Se distance and emission energy. In cyanogen (Table S31), the selected ES can be fully defined with the C-C and C≡N bond lengths that respectively attain 1.237 ˚ A and 1.299 ˚ A at the CC3/aug-cc-pVTZ level, corresponding to a bond length alternation (BLA) of 0.062 ˚ A. 29 The CCSDR(3) distances are very similar (1.232 ˚ A and 1.302 ˚ A, BLA of 0.070 ˚ A) but using the CCSDR(3) geometry still induces a significant +0.072 eV upshift of the emission energy. With the three remaining methods, the BLA is either significantly too small (0.037 ˚ A with ADC(2) and 0.020 ˚ A with CC2) or too large (0.079 ˚ A with CCSD). This logically translates into too small transition energies for the two former approaches with errors of -0.217 and -0.368 eV deviations for CC3//ADC(2) and CC3//CC2, respectively, but a too large value with the CC3//CCSD method (+0.199 eV). For cyanogen, one can therefore qualitatively relate the computed ES BLA to the errors on the fluorescence energy: a too small BLA, indicating a too conjugated electronic cloud, yields an underestimated emission energy. Of course, such relationship can be found only in the “simplest” systems. Indeed, for formaldehyde (Table S34), ADC(2) yields a significantly longer C=O bond (1.380 ˚ A) and a smaller puckering angle (η=18.9o ) than CC3 (1.326 ˚ A and

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38.2o , respectively), 29 and these large deviations translate into a rather modest error for the CC3//ADC(2) emission energy, that is only -0.056 eV redshifted compared to the full CC3 result. The CC3//CC2 value is even closer to the spot (error of -0.028 eV) although the C=O bond length and dihedral angle are still significantly overshot and underestimated, respectively, with value of 1.353 ˚ A and 29.5o for the CC2 ES structure. In contrast, CCSD that provides almost the same puckering angle as CC2 (30.9o ) but a too short bond length (1.300 ˚ A) yields much exaggerated emission energy (CC3//CCSD: +0.255 eV). For comparison, for absorption the CC3//CC2 and CC3//CCSD errors are -0.050 and +0.050 eV, respectively. In other words, it is more difficult to directly correlate the magnitude of the errors made on geometrical parameters to discrepancies on the emission energies for formaldehyde than for the two other systems. The errors obtained with respect to CC3 for emission energies are displayed in Figure 2 for the same 12 compounds as in Figure 1, allowing direct comparisons. As previously, CCSDR(3) provides accurate results, though there is a clear trend to overestimate the CC3 values. Indeed, the only two negative CCSDR(3) errors are the energetic contributions for thioformaldehyde and selenoformaldehyde that are both as small as -0.002 eV. Compared to absorption, the deviations are clearly larger, e.g., the CCSDR(3)//CCSDR(3) error is +0.149 eV for the emission of cyanogen, more than twice the deviation noticed for absorption (+0.068 eV). With CCSD, one also finds that the deviations are often larger than for absorption, particularly strong overestimations being observed for carbonylfluoride and methylenecyclopropene. Nevertheless, the errors remain systematically positive, confirming the tendency of CCSD to yield a too localized picture. Importantly, the CCSD inaccuracies can be mainly be ascribed to the geometrical component that dominates in the majority of cases (9/12). In this 12-compound set, the CC2//CC2 fluorescence energies are always too large, and the energetic contribution, that is always positive as well, is always the largest component of the total error. As for absorption, one finds significant error compensations at the CC2 level for cyanoacetylene, cyanogen and diacetylene. The total deviations computed

15



with ADC(2) are positive in six-out-of-eleven cases and errors of any sign can be found for all components. We determined the Pearson correlation matrices for the errors on emission energy obtained at different levels of theory considering the 12-compound set (Tables S55–S57 in the SI). The only significant correlations (R > 0.6) that could be found were between ADC(2) and CC2 that present similar error patterns for the two contributions to the error (> 0.8) and a significant relationship for the total error (R = 0.73) as well. In other words, a large relative error obtained with ADC(2) for a fluorescence energy would also yield a large relative error with CC2. For the other methods, there is no clear correlation, so that climbing the CC ladder does not necessarily translate in alike improvements of the emission energies for all compounds. We have also investigated the correlation between the total errors made for absorption and emission. For ADC(2), CC2, CCSD and CCSDR(3) we obtained positive R of 0.77, 0.73, 0.39 and 0.79, indicating that a large deviation for absorption hints at a large error for the emission as well, but at the CCSD level. This CCSD result is clearly surprising, and to check that it was not originating from an artifact, we have computed the correlation between the CCSD absorption and emission deviations with respect to CCSDR(3) for the 24-compound set and obtained a very poor R as well (0.17). The statistical analysis obtained for the emission energies on both the 12-compound and 24-compound sets are given in Table 2. For the former set, one notices that the average error obtained with CCSDR(3) is twice the one determined for absorption (0.066 eV versus 0.032 eV), and this comes mainly from the the tripling of the structural error compared to absorption, that now has a magnitude close to the one of the energy error. With CCSD, one obtains large positive errors for all parameters and we note that the fact that they are slightly smaller for the 24-compound group is probably due to the use of a CCSDR(3) reference that provides slightly too large transition energies. For both sets, the CCSD deviation is dominated by the geometry, which delivers approximatively two third of the total error. For ADC(2) and CC2, the errors remain mostly driven by the transition energies, but the

16


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Deviation (eV)

0.5

ADC(2)

0.4

Total Energy Geometry

0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4

0.5

CC2

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4

0.5

CCSD

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4

0.5

CCSDR(3)

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3

-0.4Acetylene Cyanoacetylene Diacetylene Formaldehyde MethyleneCP Carbonylfluoride Cyanogen

Diazomethane

Ketene

Thioform. Selenoform. Thioketene

Figure 2: Errors obtained for the vertical emission energies using the CC3 results as reference for the 12-compound set. See caption of Figure 1 for details. Note the different Y scales compared to the absorption case. 17



contributions of the geometry to the total error has increased significantly compared to absorption. Indeed, for the small (large) set of compounds the geometry corresponds 35% (40%) of the energy error with ADC(2) for absorption but 71% (53 %) for the emission. It is also valuable to note that the mean signed deviations are rather small for ADC(2), especially for the small compound set: on average ADC(2) is close to the spot but with non-trifling dispersion. Eventually, paralleling the results obtained in the previous Section, we have investigated the errors obtained with the CCSDR(3)//ADC(2) protocol. For the 12-compound set, this approach delivers a MSD of 0.019 eV and a MAD of 0.115 eV, both significantly above the values obtained for absorption. Table 2: Top: statistical analysis for the emission energies of Figure 2: mean signed and absolute deviations (MSD and MAD, in eV) considering the CC3//CC3 results as reference. Bottom: statistical analysis of the emission energies carried our for all compounds of Scheme 1 considering CCSDR(3)//CCSDR(3) values as reference. 12-compound set – CC3 reference MSD MAD Method Total Energy Geometry Total Energy Geometry ADC(2) -0.014 0.025 -0.020 0.165 0.149 0.117 0.148 0.208 -0.063 0.148 0.208 0.101 CC2 CCSD 0.270 0.104 0.169 0.270 0.104 0.169 0.038 0.029 0.066 0.039 0.029 CCSDR(3) 0.066 24-compound set – CCSDR(3) reference MSD MAD Method Total Energy Geometry Total Energy Geometry ADC(2) -0.177 -0.104 -0.031 0.239 0.163 0.127 CC2 0.023 0.105 -0.077 0.135 0.123 0.121 0.224 0.073 0.153 0.224 0.078 0.155 CCSD

3.3

TD-DFT and hybrid TD-DFT/Wavefunction protocols

As stated in the Introduction, the most popular approach for ES calculation remains TDDFT. It is therefore interesting to determine if the TD-DFT errors are mostly driven by energy of by the geometry for both absorption and emission. Choosing PBE0 as functional, we have therefore determined the transition energies, as well as the GS and ES geometries with DFT and TD-DFT for all considered compounds. 44 Similarly to what has been done 18


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above, various combinations have been performed with CC3 for the 12-compound set and with CCSDR(3) for the 24-compound set. Complete results are given in the SI (Tables S58–S81). It is not our goal here to discuss the merits of this specific functional as many TD-DFT benchmarks are already available in the literature, 45 but rather to compare the absorption and emission discrepancies for energies and geometries. The statistical results are listed in Table 3. As can be seen from the MAD, for absorption, the total error is mainly driven by the energy. Indeed, for both sets, the geometry error is ca. 24% of its energy counterpart. When turning to emission, one notices that the MAD related to energy slightly increases by ca. 20% (from 0.256 eV to 0.302 eV for the 12-compound set, and from 0.194 to 0.236 eV for the 24-compound set), whereas the geometry errors become much larger, by doubling (from 0.064 eV to 0.134 eV) or tripling (from 0.046 eV to 0.145 eV) for the small and large set, respectively. For emission, the structural component of the error cannot be neglected anymore, consistently with the results obtained when considering CC methods only. These data clearly hint that performing a CC calculation on top of a (TD-)DFT geometry is a reasonable approach for absorption, but a less pertinent choice for emission. Indeed, the MAD obtained from CC2//TD-DFT calculations, a popular protocol nowadays (see Introduction) attains 0.180 eV for absorption but 0.268 eV (+49%) for fluorescence for the 24-compound set, using the CCSDR(3)//CCSDR(3) values as reference. Table 3: Statistical analysis obtained with (TD-)PBE0, using either the CC3 (top) or CCSDR(3) values as reference. All values are given in eV. See caption of Table 1 for more details. 12-compound set – CC3 reference MSD MAD Method Total Energy Geometry Total Energy Geometry Absorption -0.166 -0.230 0.064 0.223 0.256 0.064 Emission -0.108 -0.237 0.134 0.348 0.302 0.134 24-compound set – CCSDR(3) reference MSD MAD Method Total Energy Geometry Total Energy Geometry Absorption -0.098 -0.139 0.040 0.181 0.194 0.046 44 Emission 0.007 -0.113 0.131 0.345 0.236 0.145

19



4

Conclusions

In an effort to disentangle the contributions of the energy and the geometry to the total errors obtained for absorption and fluorescence energies, we have computed the transition energies energies for a set of 24 compounds selecting the aug-cc-pVTZ atomic basis set and considering a group of five increasingly accurate methods going from the relatively simple ADC(2) approach to the reference CC3 method. For this set of organic derivatives, we have systematically obtained CCSDR(3)//CCSDR(3) reference values and, for a subset of 12 derivatives, CC3//CC3 results as well. Using these data as references, we have determined the transition energies by degrading step-by-step either the method used to compute the transition energy, the geometry or both. For both absorption and emission, we found that the energetic and geometric contributions to the total error are nearly additive for a given compound, which facilitated the analysis of the errors. Some clear methodological trends emerged. First, CCSD tends to provide a too localized description of the considered compounds, which consequently yields to positive deviations for both components of the total error: the transition energies are always overestimated. Second, CCSDR(3) allows to correct most of the CCSD error and to bring the absorption and emission energies much closer to the CC3 benchmark values, though a slight overestimation pertains in the vasty majority of cases. These two conclusions are consistent with the results obtained previously for Thiel’s set of derivatives for absorption. 30,31,40,43 Third, CC2 gave positive deviations compared to CC3 for the transition energies considered herein, whereas positive and negative geometric errors could be found. For several compounds, there is therefore a clear compensation of energetic and geometric errors at CC2 level, so that a good match between, e.g., a CC2//CC2 emission energy and a measured fluorescence spectrum is probably not a definitive proof that this approach is well suited. With ADC(2), one can find positive and negative deviations for both terms (energy/geometry), and for both absorption and emission. Nevertheless, on average the deviations obtained with ADC(2) are of the same order of magnitude as their CC2 counterparts, which fits previous findings. 42 20


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For absorption, we noticed that the geometric contribution to the total error is quite small with respect to its energetic counterpart for the ADC(2)//MP2, CC2//CC2 and CCSDR(3)//CC(3) methods, but is of the same order of magnitude for CCSD//CCSD. The errors obtained at ADC(2) and CC2 levels are significantly correlated for absorption, and this holds for CCSD and CCSDR(3) deviations. Using an approach like CCSDR(3)//MP2 in which a more refined theoretical level is used for computing transition energies than for determining GS geometrical parameters delivers very satisfying result with a MAD of 0.044 eV compared to CC3//CC3. However, the situation is significantly different for emission, because it is challenging to obtain accurate excited-state geometrical parameters with a “cheap” method. Indeed, for fluorescence, the typical deviations due to the method selected for computing the transition energies remain of the same order of magnitude as for absorption, but the geometric contribution to the total error becomes significantly bigger, irrespective of the selected level of theory. As a consequence, the CCSDR(3)//ADC(2) approach gives a MAD of 0.115 eV for fluorescence, 2.6 times larger than for absorption. For emission, the correlation between the errors determined at different levels of theory are also small, at the notable exception of the ADC(2)–CC2 pair, so that crystalballing the accuracy of CCSDR(3) from the ADC(2) results is probably not possible. In contrast, we found that there are reasonable correlations between the errors made on absorption and emission for a given method, but for CCSD. These conclusions were additionally corroborated by TD-DFT and CC2//TD-DFT calculations: it is more difficult to obtain accurate emission energies than absorption values, because the geometric error becomes significantly larger for the latter. In short, it appears clearly from this study that while the method used to compute the transition energy is the key to accurate absorption calculations, this does not necessarily holds for emission, for which the quality of the selected level for determining ES geometry also plays a key role in the final accuracy of the results.

21



Supporting Information Available Geometrical parameters for cyanoacetylene, methylenecyclopropene, and thiocarbonylchlorofluoride. Full list of transition energies for both absorption and emission. Pearson correlation matrices.

Acknowledgement The author is indebted to Prof. C. Adamo (ENSCP, Paris) for enlightening discussions. This research used resources of i) the GENCI-CINES/IDRIS; 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).

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What is the Key for Accurate Absorption and Emission Calculations

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