Benchmarking Ground-State Geometries and Vertical Excitation

Article pubs.acs.org/JPCA

Benchmarking Ground-State Geometries and Vertical Excitation Energies of a Selection of P‑Type Semiconducting Molecules with Different Polarity Charlotte Brückner† and Bernd Engels*,† †

Institut für Theoretische Chemie, Universität Würzburg, Emil-Fischer-Straße 42, 97074 Würzburg, Germany S Supporting Information *

ABSTRACT: A benchmark of TD-DFT, wave function-based and semiempiric methods was performed for the geometries and excitation energies of diverse molecular organic semiconductors with varying polarity. Geometries were benchmarked by means of RMSD (root-mean-square deviation) values and MAE (maximum absolute error) values of geometric parameters specific for the electronic structure of the respective molecule. MS-CASPT2 calculations were used to benchmark excitation energies with respect to a confidence interval around the values obtained with CASPT2. The effect of spin-component scaling (SCS) on several wave functionbased methods was thoroughly evaluated.

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INTRODUCTION The field of organic optoelectronics has evolved with the development of organic solar cells, organic field-effect transistors (OFETs),1,2 and particularly organic light-emitting diodes (OLEDs),3,4 the last of which have already come to widespread use.5 In comparison with organic polymers, optoelectronic devices relying on small organic molecules as the semiconducting layer possess more distinct and tunable structure−property relationships, because no distributions among different conjugation lengths and electronic properties are obtained as it is the case for polymers.6 However, the optoelectronic performances of these small semiconducting molecules are also determined by intermolecular interactions in the thin films or crystal aggregates.7 As these intermolecular interactions in thin films and crystals are largely dominated by molecular polarity, polarity has often been used as a selection criterion. Traditionally, apolar molecules like acenes and oligothiophenes are used in organic solar cells and organic field-effect transistors.8 Molecules with large local dipole moments but without an overall dipole moment like triarylamines, diketopyrrolopyrroles, and squaraines are also well established within the field of organic optoelectronics due to their high absorption coefficients in the visible region and their low ionization potentials. They can be considered as being of intermediate polarity.8 Moreover, recent investigations underlined the suitability of highly polar dyes like merocyanines for organic solar cells, a new area of application besides their original usage, the second-harmonic generation.9 Thereby, organic dyes used in optoelectronic devices span the whole polarity range from very apolar to highly polar. In the overwhelming majority of cases the optimization of the properties of organic semiconductors was performed © 2015 American Chemical Society

experimentally using trial-and-error approaches. Computations were only used to explain trends and support model considerations.10,11 Due to the vast amount of possible variations in the electronic structure and geometrical orientations as a function of the substitution pattern, reliable predictions are highly desirable. They are very difficult because the systems under considerations possess complicated electronic structures so that highly accurate methods are needed. Because the mutual orientation of the monomers strongly influences the aggregate properties, meaningful data are only achievable if the computed model systems are sufficiently large. Due to the necessary size the needed highly accurate methods are too expensive and only quite approximate methods can be employed. For the decision on which one among the approximate approaches is appropriate for which question, benchmark studies are needed. Extensive benchmarks of organic dyes have already been performed. We will only give a short overview because a complete overview is beyond the scope of this work. Vertical excitation energies of a series of large organic dyes (of low polarity) have been benchmarked by Grimme, focusing on the performance of double hybrid functionals or different versions of the CIS(D) methods, respectively.12,13 A TD-DFT study of similar molecules along with excited-state optimizations was thoroughly conducted by Mennucci et al.14 Excited-state gradients of a variety of molecules, among other small organic molecules, were also analyzed by Furche et al.15 Köhn et al. focused on triplet states of molecular organic semiconductors.16 Jacquemin et al. Received: October 21, 2015 Revised: December 1, 2015 Published: December 1, 2015 12876

DOI: 10.1021/acs.jpca.5b10315 J. Phys. Chem. A 2015, 119, 12876−12891

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The Journal of Physical Chemistry A

Figure 1. Selection of small organic molecules of p-type semiconduction used for the benchmark. Substituents colored in gray are cut off for benchmark CASPT2 calculations.

Merocyanines are the only polar organic p-type semiconductors of considerable importance. However, due to the limited number of model systems and their specific structural entities, performance differences of various methods do not necessarily rely on the differing polarity of the molecules, but rather on the structural elements typical for a group of semiconductors disposing a certain polarity. In view of the difficulty of obtaining accurate experimental data, we use the approach also employed by Thiel and coworkers28 and benchmark against methods being well-known for their accuracies. Concerning the geometries, we use the SCS-MP2/cc-pVTZ level of theory as a reference.29−32 Vertical excitation energies were compared with MS-CASPT2- and high-level ab initio methods such as SCS-CC2 and ADC(2).35−38 The effects of spin-component scaling on the transition energy were tested for several wave function based methods, among others for ADC(2). Although spin-component scaling is well established for CC2, its suitability has not been tested for ADC(2) to the best of our knowledge.

performed extensive TD-DFT benchmarks of excitation energies of organic dyes17 (see also TD-DFT review by Jacquemin et al.18). In cooperation with Truhlar et al. they also performed benchmarks with a special focus on pyrroles19 and cyanines.20 A multitude of further investigations focused on new methods rather than on their optoelectronic applications, among them, for example, GW methods,21 the Δ-SCF method,22,23 and the semiempiric OMx methods by Thiel et al.24 Moreover, many case studies exist, often in comparison with experimental data. Examples are studies about squaraines,25 pyrroles,26 and oligothiophenes.27 However, although benchmarks exist for organic molecules in general and in particular for special classes of organic semiconductors, to the best of our knowledge benchmarks about organic semiconductors with varying polarity are missing. This is the goal of the present work. To gain insight, we test the accuracies of various methods using the molecules depicted in Figure 1 as model systems. The test set includes the most important classes of organic molecular p-type semiconductors. For the apolar molecules, anthracene, a traditional acene, rubrene, a substituted acene, diindenoperylene (DIP), a perylene-based dye, and a dithiophene have been enclosed because comparable compounds are widely used as organic semiconductors.8 Triarylamines, squaraines, and diketopyrrolopyrroles are often employed in optoelectronic dyes due to their low ionization potentials and high tinctorial strengths.9 They possess large local dipole moments but no or only a very small overall dipole moment.6,8 Therefore, three differently substituted triarylamines, a squaraine, and a diketopyrrolopyrrole were included as model systems with intermediate polarity.

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COMPUTATIONAL DETAILS The methods benchmarked in this paper along with the employed basis sets are summarized in Table 1. In terms of wave function-based methods, besides MSCASPT2, CC2, ADC(2), CIS(D),39 and MP2 along with their spin-component scaled variants as well as HF are included.36 Spin-component scaling for MP2 was used according to Grimme.31 For all variants of spin-component and spinopposite scaling of CC2, CIS(D) and ADC(2), the parameters as implemented by Hättig et al. were used.36 The convergence 12877


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BLYP,40−42 the meta-GGA M06-L,44 and the stand-alone functional APFD45 were employed. As examples for the upper limit of Jacob’s ladder, the two different double hybrid functionals B2PYLPD354−56 and mPW2PLYPD3,57,58 were used. For all DFT functionals without any inherent dispersion correction, Grimme’s dispersion correction D3 was employed if available.43 For all DFT calculations, the ultrafine (99 590) grid was used. For the SCF cycles, the default tight convergence was applied. The default RMS criterion of 3 × 10−4 au was used during the geometry optimizations. To investigate the accuracy of semiempirical methods, the benchmark encloses the OMx methods with and without dispersion correction that were only applied to molecules containing no sulfur.60 Additionally, the qualities of AM1,63 the reparametrized form RM1,64 MNDO/H (particular parametrization for hydrogen bonds),62 and PM667 and PM7 (in part with dispersion or hydrogen bond correction) are tested.68 In the semiempirical calculations, the SCF convergence was set to 1 × 10−6 eV and the optimization threshold to 1 kcal/mol (default values). To compute excitation energies for semiempirical approaches, we performed CI calculations which include all triple excitations in a [16,16]-active space. For most approaches, we tested the qualities with respect to geometry and excitation energies. One exception is ZINDO. It was only used for excitation energies. All geometries were benchmarked against the SCS-MP2/ccpVTZ level of theory.29−32 Geometry optimizations started with the data of crystal structures, i.e., anthracene,79 diketopyrrolopyrrole, 80 dip, 81 dithiophene, 82 HB194, 9 HB238,6 MD353,9 rubrene,83 squaraine,84 TG008,85 triamine,86 “triamine-methoxy”,87 and “triamine-aldehyde”.88 For singlepoint calculations with SCS-MP2/cc-pVTZ, “relaxed” properties were calculated.89 In the calculations of the RMSD values, only the atoms within the rigid backbones of the semiconducting molecules are included. Flexible side chains were not taken into account. MAE (mean absolute error) values were calculated for the deviations of distinct geometric parameters such as angles or bond length alternations from the benchmark geometry. All benchmarked methods were used along with the cc-pVDZ basis sets.80 The vertical excitation energies were benchmarked against single-point MS-CASPT2 calculations at optimized SCS-CC2/ cc-pVDZ geometries. The necessity of diffuse functions was checked upon, but the use of augmented basis sets did not significantly influence the excitation energies, which is to be expected for valence excitations (Supporting Information).28 In these computations pyramidalized amino substituents were planarized to be able to exploit symmetry. For the CASPT2 calculations, in some cases smaller model systems (Figure 1) had to be used to exploit symmetry and to make the calculations feasible. The influence of the substituents was taken into account by a correction term, which is discussed below in detail. Within the computations, three states were included in the CI-matrix of the CASSCF wave function. The IPEA shift option (default value of 0.25) was applied.40 Multistate averaging was applied in the CASPT2 calculation for these states. In the case of intruder states, a level shift of 0.2 was applied.91 Apart from the anthracene and the dithiophene, all molecules are too large to include all π-electrons in the CASSCF computations. Therefore, the active space was stepwise increased starting from a [4,4]-space to a [14,14]-active space until the first three excited states were converged. For D2h-symmetric molecules,

Table 1. List of the Theoretical Methods Benchmarked and Basis Sets Used in the Present Paper (a) Theoretical Methods method

refs

Wave Function Based Methods multistate complete active space second-order perturbation theory (MS-CASPT2) second-order Møller−Plesset perturbation theory (MP2) spin-component scaled second-order Møller−Plesset perturbation theory (MP2) (SCS-MP2) second-order approximate coupled-cluster (CC2) spin-component scaled second-order second-order approximate coupled-cluster (SCS-CC2) algebraic diagrammatic construction through second order (ADC(2)) spin-component scaled algebraic diagrammatic construction through second order (SCS-ADC(2)) CI (configuration interaction) with a perturbation correction for connected double excitations (CIS(D)) spin-component scaled CI (configuration interaction) with a perturbation correction for connected double excitations (SCSCIS(D)) spin-opposite scaled CI (configuration interaction) with a perturbation correction for connected double excitations (SOSCIS(D)) Hartree−Fock (HF) Density Functional Based Approaches BLYP-D3 M06-L(-D3) APFD B3LYP-D3 SOGGA11X ωB97X-D CAM-B3LYP(-D3) (Coulomb-attenuated method of B3LYP) LC-wPBE (long-range corrected wPBE) long-range corrected functionals (LC-BLYP and LC-M06-L) B2PLYPD3 mPW2PLYPD3 Semiempiric Methods orthogonalization model 1 (OM1) orthogonalization model 2 (OM2) with dispersion correction (OM2-D3) modified neglect of diatomic overlap (MNDO/H) Austin model 1 (AM1) Recife model 1 (RM1) parameterization model 3 (PM3) parameterization model 6 PM6(-D3/DH2) parameterization model 7 PM7 Zerner’s intermediate neglect of differential overlap (ZINDO) (b) Employed Basis Sets method all methods SCS-MP2/SCSCC2/ωB97X-D

32−34 29, 30 31 35 36 37,38 36 39 31 30

40−43 44, 43 45 46, 43 47 48 43 50−52 53 54−56 57, 58 59, 60 61, 60 62 63 64 65, 66 67 68 69−77

basis sets

reference

cc-pVDZ (correlation consistent polarized valence double zeta) cc-pVTZ (correlation consistent polarized valence triple zeta)

78 78

of the SCF cycles and the density convergence were set to 1 × 10−7 to obtain well-converged orbitals. During the optimizations, an energy threshold of 1 × 10−6 and a gradient threshold of 1 × 10−3 were applied. To assess the reliability of DFT-based approaches, we tried to cover a broad and diverse selection of functionals. The benchmark comprises the long-range corrected functionals CAM-B3LYP49 and ωB97X-D48 and the hybrid functionals B3LYP46 together with the Minnesota functional SOGGA11X.47 As pure functionals, the LDA functional 12878


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Figure 2. RMSD values of the geometries in Å of organic semiconductors of different polarity calculated with the cc-pVDZ basis sets benchmarked against SCS-MP2/cc-pVTZ.

i.e., the dip and squaraine molecule, a [16,16]-space was employed. Only symmetric CASSCF-spaces were used. The necessity of asymmetric spaces for the inclusion of lone pairs was checked by employing dithiophene as an example. Even the charge-transfer excitations connected with in-plane π-orbitals, e.g, for aldehyde or nitrile groups, were converged for symmetric active spaces.

For the geometry optimizations and excitation energy calculations, the Gaussian09 (Revision D)92 (HF, DFT functionals), the Turbomole93 (all higher-level ab initio methods except for MS-CASPT2), the MNDO94 (all semiempirics but PM6 and PM7), and the Mopac95 (PM6,PM7) program packages were used, respectively. For the CASPT2 12879


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The Journal of Physical Chemistry A calculations, the MOLCAS 96−98 program package was employed.

that due to an error compensation, some density functionals yield more reliable geometries with smaller basis sets. Such a behavior has already been analyzed for B3LYP/6-31G(+) by Grimme.100 Please note that this error compensation is not valid for BLYP-D3, as can be seen from the positive deviations for BLYP-D3 in Figure 3. In this case, a larger basis set leads to an improved RMSD value. Moreover, the RMSD value for BLYP-D3 of 0.074 Å is almost equal to the error of 0.069 Å for HF (see below). In summary, our results confirm that if SCSMP2/cc-pVTZ is computationally too demanding for the system, DFT calculations yield more reliable geometries with smaller basis sets. Hence, they are recommended if larger basis sets are not affordable. Surprisingly, for the molecules of intermediate polarity, the RMSD values are generally higher. This may partly result due to the larger size of these molecules as well as their enhanced torsional flexibility. Indeed, the huge torsional flexibility of these molecules also led frequently to convergence problems. In general, for dyes of intermediate polarity, APFD, a modern functional including a dispersion correction, performs best (0.032 Å).45 This could be due to the fact that this functional is particularly optimized for weak interactions such as hydrogen bonds and interactions in noble-gas clusters. Such weak intermolecular interactions are especially important for the structures of the dyes of intermediate polarity as they are composed of several rather flexible parts. Apart from that, ωB97X-D yields also reliable geometries (0.042 Å). The good performance of ωB97X-D is again in line with the importance of intermolecular interactions for correct geometries of molecules of intermediate polarity. It has recently been shown by Grimme et al. that ωB97X-D is very suitable for the calculation of host−guest interactions.101 Moreover, for highly as well as intermediate polarities, the double hybrids yield very accurate results. The RMSD values of the apolar dyes are rather small, owing to the structural inflexibility of their rigid backbones and to the simplicity of their electronic structures. With respect to an average over the whole test set, ωB97X-D, SOGGA11X and M06-L show the smallest deviations. Please note that although the performance of ωB97X-D does not depend on the molecular polarity, both SOGGA11X and M06L are less reliable for molecules of intermediate polarity. The robust and good performance of ωB97X-D is in accordance with other findings in the literature for various systems with a complex electronic structures, for instance, metal complexes.102 Deviations for all semiempirical methods are considerably larger (∼0.15 Å) irrespective of the exact scheme, especially for dyes of high and intermediate polarity. For reliable predictions, not only the absolute deviations in geometrical parameters are important but also the predicted geometries have to reflect the correct electronic structures. This is best explained for the merocyanines. The electronic character of merocyanines is determined by the relative contributions of a neutral and a zwitterionic resonance structure. The ratio of both resonance structures is reflected in the bond length alternation (BLA) in the central carbon bridge connecting donating and accepting moieties, e.g., bonds 1−3 in the molecule HB238 depicted in Figure 4 (left-hand side). The BLA is defined in eq 1, and the involved bonds are depicted in Figure 4. Although merocyanines with purely neutral or zwitterionic characters have strongly alternating bonds (BLA > 0.5 or BLA < 0.5), a 50−50% composition of both resonance structures gives rise to equal bond lengths along the bridge

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RESULTS AND DISCUSSION Benchmark of Geometries and Ground-State Properties. Molecular p-type semiconductors play a crucial role in modern optoelectronic devices. A correct description of the involved intermolecular processes in these devices requires, on the one hand, large model systems but, on the other hand, also precise descriptions of molecular geometries. Hence, computationally methods with a favorable cost-accuracy ratio are needed. The performance of the different methods for geometries is first discussed by means of RMSD values of the geometries. The corresponding results for organic molecular semiconductors of high, intermediate and low polarity as well as the average over all compounds are given in Figure 2. From these data, it is evident that all methods perform rather well for apolar dyes, whereas significant deviations exist for molecules of high and intermediate polarity. This is especially true for the semiempirical methods (MNDO/H in particular) as well as HF and BLYP. Taking the SCS-MP2/cc-pVTZ-geometry as the benchmark geometry and employing the smaller cc-pVDZ basis sets, the geometry of polar dyes is best described by M06-L (RMSD = 0.024 Å) and ωB97X-D (0.028 Å). The M06-L meta-GGA Minnesota functional performs better without dispersion correction (RMSD (M06-L-D3) = 0.031 Å) as has already been stated by Truhlar et al.99 These deviations are about half of the basis set effects found for the SCS-MP2/cc-pVDZ approach (0.047 Å). CC2 and SCS-CC2 show comparable deviations (0.048−0.049 Å). Among the wave function-based methods, MP2 performs best as compared to the reference. However, this is due to an error compensation of the lacking spin-component scaling and the smaller basis sets. In comparison with these wave function-based methods, all density functionals including the double-hybrid approaches show smaller RMSD values, ranging from 0.024 to 0.039 Å. Exceptions are B3LYP-D3 (0.047 Å) and BLYP-D3 (0.074 Å). These smaller RMSD values can be explained by means of basis set effects. This is indicated by an analysis of the influence of the basis sets on the RMSD values of DFT functionals (Figure 3). As no polarity dependence of the basis set effects was observed, the values given in Figure 3 are averaged for all molecules. Although the RMSD value for SCS-MP2 decreases by about 0.05 Å upon increasing the basis sets to cc-pVTZ, the RMSD values obtained for the DFT functionals deteriorate by about 0.01 Å (Figure 3) as the basis set increases. This means

Figure 3. Decrease in the RMSD values if the basis sets are enlarged to cc-pVTZ (RMSD(cc-pVDZ) − RMSD(cc-pVTZ)). I.e., a negative sign indicates a deterioration of the RMSD values. The RMSD values are averaged over all molecules. 12880


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Figure 4. Bonds used for the calculation of the bond length alternation in each merocyanine. The MAE values were obtained by taking the absolute value of the difference between the BLA value of a specific geometry and the BLA value of the benchmark geometry.

Figure 5. MAE (mean average error) values for the computed BLAs of merocyanines (left-hand side) and torsional angles of organic dyes with intermediate polarity (right-hand side). For the definitions of the involved geometrical parameters see text.

effects between cc-pVTZ and cc-pVDZ. Upon decreasing the basis set size the BLA is underestimated but the deviation is nearly negligible. This is opposed to other findings with doublehybrid functionals in the literature.48 Although SCS-CC2 and SCS-MP2 gave nearly identical RMSD values for distances, they differ more strongly for the BLA. The deviation from the benchmark geometry doubles when going from SCS-CC2 to CC2 or from SCS-MP2 to MP2, showing the strong influence of the SCS correction on the computed BLA. This is in accordance with findings of Jacquemin for conjugated oligomers.103 Concerning the functionals, the CAM-B3LYPD3 prediction is extremely close to the benchmark value. Apparently, the long-range correction originally developed for excited states is important for predicting the correct amount of electron delocalization in the ground state. Also this is in agreement with the results of Jacquemin for conjugated oligomers.103 The trend that methods well-suited for the computation of excited states yield accurate BLA values can be expected because for merocyanines, the ground and the first excited state are closely related through the ratio of reference

(BLA = 0). Consequently, methods overestimating all bond lengths by an equal percentage would yield a large RMSD value but would still give a qualitatively correct characterization of the electronic characters of merocyanines. In contrast, methods that only differ in one bond length would possess smaller RMSD values but yet give a less accurate description of the electronic characters. Because the correct description of the electronic character is a prerequisite for reliable predictions of the properties of organic semiconductors, for merocyanines, for example, it is also necessary to check the mean absolute errors (MAE) of the various methods for the BLA value. We again use SCS-MP2/cc-pVTZ as the benchmark geometry because in a benchmark against CCSD(T) Jacquemin et al. showed that SCS-MP2 is reliable at predicting bond lengths.103,104 BLA =

0.5· (bond 3 + bond 1) − bond 2 ·100 bond 2

(1)

The computed MAEs (in %) are given in Figure 5 (left-hand side). The values given for SCS-MP2 reflect again only basis set 12881


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Table 2. Deviations of the Dipole Moments [D] of Merocyanines Calculated with Different Methods and the Geometries Optimized with the Respective Methods from the SCS-MP2/cc-pVTZ//SCS-MP2/cc-pVTZ valuea

a

merocyanine

ωB97X-D

CAM-B3LYP-D3

SOGGA11X

M06-L

B3LYP-D3

BLYP-D3

HF

PM7

OM2-D3

HB194 HB238 MD353 TG008

0.08 1.63 1.15 1.73

−0.12 1.50 0.95 1.58

0.47 2.36 1.59 2.38

0.91 2.82 1.51 2.84

0.34 2.38 1.25 2.42

0.49 2.43 1.06 2.46

−0.16 1.47 1.35 1.57

0.22 2.21 1.67 2.16

−2.37 1.04

Reference values from SCS-MP2/cc-pVTZ: HB194, 8.70 D; HB238, 13.47 D; MD353, 11.59 D; TG008, 13.39 D.

Table 3. Deviations of the Dipole Moments [D] of Merocyanines Calculated with SCS-MP2 at the Optimized Geometries of the Respective Methods from the SCS-MP2/cc-pVTZ//SCS-MP2/cc-pVTZ Value (See above) merocyanine

ωB97X-D

CAM-B3LYP-D3

SOGGA11X

M06-L

B3LYP-D3

BLYP-D3

HF

PM7

OM2-D3

HB194 HB238 MD353 TG008

−0.34 0.19 0.06 0.19

−0.42 0.24 0.02 0.24

−0.19 0.53 0.24 0.49

−0.10 1.17 0.51 1.13

−0.25 0.83 0.26 0.84

−0.24 1.13 0.40 1.17

−0.48 −0.44 −0.50 −0.44

−0.64 0.43 −0.62 0.08

−1.09 0.10

Figure 6. Geometric parameters used for the calculation of MAE values of the geometries of molecules of intermediate polarity (scaling factors different from 1: triarylamine, 10; squaraine, 100).

configurations.10,105 It is also valid for ωB97X-D, which also yields BLA values in good accordance with the benchmark geometry. Please note that opposed to the RMSD values, the BLA values calculated with M06-L(-D3) deviate more strongly, which means that absolute bond lengths values are correct, but their ratios do not fully correspond to those of the benchmark geometry. Furthermore, HF and BLYP-D3 show equal MAE values of about 1.5. However, although HF overestimates the bond length alternation, BLYP-D3 underestimates it severely, which is a well-known theoretical problem in the literature.106−108 This is in part due to the self-interaction error of DFT, which can be removed by adjusting the range separation.109 Similarly to the literature findings,48 the double hybrids perform well, especially mPW2PLYPD. Within the semiempirical methods the OMX approaches outperform all other approaches. This is in line with the findings that functionals that predict reliable excitation energies also compute better BLA values. All other semiempirical methods are considerably worse. Deviations for a given electronic property of a molecule also give insight how accurately a given method can describe the electronic structure. This is of special interest for the

merocyanines. Table 2 gives the deviations of the computed dipole moments of a given approach from the SCS-MP2/ccpVTZ value. The dipole moments are computed at the equilibrium geometries of the respective method. The values in Table 3 use the same geometries but employ the SCS-MP2/ccpVTZ method to compute the dipole moments. A comparison of both values allows us to differentiate between the influence of inaccuracies in the geometries on the calculated dipole moments and the influence of the electronic structure calculation. Obviously, all methods describe HB194 considerably better than all other merocyanines. CAM-B3LYP-D3 performs best with relative errors between 1% (HB194) and 12% (TG008) and an average error of 0.91 D. Unexpectedly, all functionals are worse than Hartree−Fock and PM7. The method OM2-D3 already fails to describe HB194. The trends found for the dipole moments only partly resemble the trends found for MAEs. As for the MAEs, CAM-B3LYP-D3 and ωB97X-D perform very well, whereas SOGGA11X and M06-L are worse. In contrast to the MAEs, Hartree−Fock performs considerably better for dipoles. This might be due to the exact inclusion of 12882


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CASPT2 and the SCS-CC2 calculations) are calculated with the cc-pVDZ basis sets using the SCS-CC2/cc-pVDZ geometry. Rydberg states are neglected because they are not important in organic semiconducting devices. As exemplified by Figure 7, the use of equal and rather small basis sets for the

exchange. The behavior of OM2-D3, which gave excellent MAEs, is also different. To differentiate between errors arising from the deviations in the geometries and from shortcomings arising from the electronic structure calculations, we repeated the calculation using SCS-MP2/cc-pVTZ single-point calculations, so that the dipole moments given in Table 3 only contain the error arising from the geometry. As expected, they correlate with the reliability of the methods in the prediction of the geometries. This shows that no error compensation takes place. Differences in the trends for RMSD, MAE values, and dipole moments suggest also an evaluation of the structures of molecules of intermediate polarity in terms of MAE values. In this compound class, dipole moments are less suited as test parameters because they are often zero due to symmetry. A problem arises because, due to the chemical diversity of these molecules, no single parameter qualitatively reflecting the electronic character exists. In four molecules, twisting motions can be taken as parameters because they are of major importance for the overall electron delocalization (see Figure 6): − In the diketopyrrolopyrrole the conjugation between the central pyrrole moiety and the donating side chains is determined by the two dihedral angles indicated in Figure 6. In the SCS-MP2/pVTZ-geometry, their average equals 9°. − In molecules possessing a three-coordinated nitrogen center the delocalization is strongly influenced by the twisting of the substituents with respect to the center. This is reflected by the dihedral angles (Figure 6). − In the aldehyde substituted triarylamines another bond length alternation can be defined. In the case of the planar squaraine, the ratios of the averaged C−C bond lengths in the central four-membered ring to the C−O bonds are indicative of the donating power of the substituents, of the electron delocalization in the ring, and especially of the zwitterionic character of the squaraine. Again, the deviations of the angles and ratios from the benchmark value are percentaged. Moreover, for visualization and averaging reasons, they are scaled by different factors (Figure 6) to obtain values of similar magnitudes. For a specific angle, the mean absolute errors (MAE) are defined as MAE =

Figure 7. Basis set effects on the excitation energies of anthracene calculated with SCS-CC2.

geometry and the transition energy calculation leads to an error compensation irrespective of the method (see also Supporting Information). Anthracene is chosen for the demonstration as it has two extremely close-lying bright excited states (La and Lb),110 making it a very difficult and highly sensitive test system. Basis set effects in other systems are usually smaller (Supporting Information). Spin-component scaling leads to a reordering of both states. Computing both the ground-state geometry and energies with the cc-pVDZ basis sets (cc-pVDZ//cc-pVDZ), SCS-CC2 predicts vertical energies of 3.81 eV (1B3u), 3.91 eV (1B2u), and 5.38 eV (1B1g). This deviates only 0.01, 0.07, and 0.03 eV, respectively, from the cc-pVTZ//cc-pVTZ-values; i.e., increasing the basis sets for geometry and vertical excitation energy calculations, the variation stays inside of the expected error bar of the method.13 If the basis sets are only increased for the calculations of the ground-state geometry, all states are shifted to higher energies by 0.08, 0.05, and 0.09 eV due to the more contracted ground-state geometry; i.e., the effect is larger than the overall basis set effect. If, however, only the vertical excitation energies are computed with a larger basis set (ccpVTZ//cc-pVDZ), all excited states are downshifted with respect to cc-pVDZ//cc-pVDZ by a similar amount. This downshift results because the electronic structures of the excited states are more complicated than the ground state so that the excited states profit more by larger basis sets than the ground state. This trend seems to be quite general because a similar proceeding with ωB97X-D yields similar results (Supporting Information). Deviations are comparable, but slightly smaller.

angle(benchmarked geometry) ·scaling factor angle(reference geometry)

The computed data are also given in Figure 5. For these geometric parameters, the double hybrid mPW2PLYPD performs best. Its MAE values are only half the MAE value of another double hybrid B2PLYPD3. SCSCC2, SCS-MP2, and MP2 yield all similar MAE values, whereas those of CC2 are larger. This is especially due to an overestimation of the twisting motion of the central moiety of the diketopyrrolopyrrole as predicted by CC2. As for the polar molecules, ωB97X-D is again the most accurate hybrid functional. The situation found for the semiempircal approaches resembles the situation found for the polar molecules. Investigations about the nonpolar compound class are not of interest because they possess quite simple electronic structures and vanishing dipole moments. Benchmark of Excitation Energies. For each molecule, the first three vertical excitation energies (as defined by the MS12883


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The Journal of Physical Chemistry A This is expected because the ωB97X-D/cc-pVDZ geometries deviate less from the SCS-MP2/cc-pVTZ values than their SCS-MP2/cc-pVDZ counterparts. Therefore, the cc-pVDZ basis sets are used for all excitation energy calculations. Basis set effects could be expected to be larger for molecules containing flexible moieties or disposing push−pull character. Furthermore, diffuse functions might be necessary to correctly describe excited states in these molecules. However, we found that the errors due to the limited basis set size are always rather small (for details see the Supporting Information). The basis set dependence of the CASPT2 calculations is also analyzed in the Supporting Information, which shows that the basis set errors are below the expected error of the method (±0.25 eV). Therefore, the smaller cc-pVDZ basis sets were used to reduce the computational effort. CASPT2 is ideally suited as a benchmark method for excitedstate calculations because it is well-known for its very accurate characterization of electronically excited states. However, various compounds in our set are too large for CASPT2 computations due to the size of their substituents. On the contrary, it can be expected that nonconjugated substituents (e.g., alkyl chains) will shift the vertical excitation energies quite uniformly. Indeed, test calculations showed that this shift is rather independent of the employed method. However, it depends on the character of the excitation. Hence, to estimate the influence of the substituents on the CASPT2 energies, we computed the influence of the substituents with various methods (SCS-CC2, CAM-B3LYP, ωB97X-D, LC-wPBE), averaged (although it is almost constant), and subtracted it from the excitation energies of the model systems to obtain approximated CASPT2 excitation energies for the whole molecules (Figure 1). This procedure was verified for the three lowest states of rubrene/tetracene test system. Rubrene should be a very sensitive test system because its phenyl rings are partly involved in the excitations. The results are given in Figure 8. The

compute the influence of the phenyl rings to 0.28, 0.23, 0.23, and 0.26 eV, respectively. This shows that all methods predict very similar shifts. Averaging over the predictions of the less sophisticated methods, we obtain a shift of 0.25 eV (blue part in Figure 8) for the 11Au state, which compares very well to the CASPT2 value of 0.3 eV. For the 11Bu and the 11Ag state, the differences in the excitation energies between both approaches are also only 0.05 eV. However, the influence of the substituents varies between 0.35 eV for 11Bu and 0.15 eV for 11Ag. The MAE values for the first three excitation energies for the different classes of molecules as well as for their average are shown in Figure 9. To take uncertainties resulting from CASPT2 itself and from the approximate inclusion of substituent effects into account, the MAE values are computed with respect to a ±0.2 eV uncertainty interval of the CASPT2 result. This interval is chosen to be slightly smaller than the error bar of CASPT2 along with substituent correction to differentiate more clearly between the performances of different methods. Comparison with the SCS-CC2 values served as an additional guideline for checking for converged CASPT2 spaces. The comparisons were performed to ensure that the active space covers all important excited states. Deviations from this interval were calculated as MAE (mean absolute averaged error) values for the respective methods. The benchmark against an uncertainty interval is based upon the fact that correct trends and a very good qualitative agreement are, from a user’s point of view, more important than numerically exact results. By choosing the uncertainty interval, methods with a constant but rather small offset would still obtain good results; i.e., the error of two methods would still be zero if their excitation energies deviate constantly by 0.4 eV from each other as long as they do not differ by more than 0.2 eV from the reference method. Figure 9 shows the differences between the various methods. Starting with the DFT-based approaches, SOGGA11X, CAMB3LYP, and ωB97X-D perform very well irrespective of the polarity of the molecules. The success of CAM-B3LYP and ωB97X-D is in accordance with previous findings of Jacquemin and Adamo et al., revealing that long-range corrected functionals constitute a further improvement compared to hybrid functionals.111 The finding that SOGGA11X is wellperforming for excitation energies although it contains no longrange correction is in accordance with benchmark data of Truhlar et al.47 However, this results from the fact that SOGGA11X contains 41% exact exchange, which is similar to the amount in CAM-B3LYP varying between 19% and 65%. The errors of these functionals amount to less than 0.05 eV. Using the SCS-CC2/cc-pVDZ geometry, the three functionals commonly underestimate vertical excitation energies slightly. Additional calculations using their own optimal geometries showed that the predicted excitation energies are about 0.1 eV higher in energy than the CASPT2 reference value, which is again irrespective of the polarity of the molecules. However, please note that the errors of 0.05 eV refer to an interval of ±0.2 eV. Therefore, they are very well in line with commonly cited TD-DFT errors of 0.15−0.30 eV calculated with respect to a certain CASPT2 value.47,111 In contrast to the polarity-independent performance of SOGGA11X, CAM-B3LYP, and ωB97X-D, the effect of the long-range correction in combination with M06-L or BLYP is polarity-dependent. Although it significantly improves the performance of these functionals for apolar molecules

Figure 8. Directly calculated CASPT2 excitation energies for tetracene and their comparison to indirectly calculated excitation energies via a CASPT2 calculation on rubrene and the addition of a substituent correction afterward.

[14,14]-MS-CASPT2 calculation for rubrene predicts vertical excitation energies of 2.7 eV (11Au), 3.3 eV (11Bu), and 3.8 eV (11Ag), respectively (red bars in Figure 8). For tetracene, the corresponding values are 3.0, 3.4, and 4.1 eV, respectively (green bars in Figure 8); i.e., the neglect of the phenyl rings increases the excitation energies by about 0.3 eV. For the 11Au state, SCS-CC2, CAM-B3LYP, ωB97X-D, and LC-wPBE 12884


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Figure 9. Averaged MAE values for the first three excitation energies of the different molecular classes as well as their average. The MAE values are taken with respect to an interval of ±0.2 eV of the CASPT2 result. For more information see text.

polarity, the long-range corrections yield an intermediate improvement (BLYP/LC-BLYP, 0.73 eV vs 0.24 eV; M06-L/ LC-M06-L, 0.50 eV vs 0.25 eV), thereby fitting well into the series of varying polarity. Moreover, ZINDO performs very well, especially for bright neutral excitations of polar dyes. The error amounts to only 0.08 eV for merocyanines. This is in good agreement with the fact that ZINDO computations of hyperpolarizabilities of cyanine dyes are in very good accordance with the corresponding experimental values.112,113 Averaged over all compounds, ZINDO has an error of 0.15 eV, which is equal to

(BLYP/LC-BLYP, 1.06 eV vs 0.1 eV; M06-L/LC-M06-L, 0.82 eV vs 0.12 eV), it does not lead to considerably better excitation energies for polar molecules (BLYP/LC-BLYP, 0.54 eV vs 0.33 eV; M06-L/LC-M06-L, 0.36 eV vs 0.35 eV). This is due to the fact that the long-range correction leads to an overestimation of the excitation energies, thus overcompensating the underestimation of the excitation energies by common DFT functionals for polar systems. Already Jacquemin and Truhlar et al. found that hybrid functionals are sufficient to describe valence excitations in cyanines, another very polar compound class.20 In the case of molecules of intermediate 12885


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they found that the OMx methods yield the correct order with errors of around 0.4 eV. Taking into account the interval of ±0.2 eV, this coincides with our results. Furthermore, AM1 and PM3 performed also reasonably well in their benchmark, although the errors differ significantly for different excitations. This corresponds to our findings.114 The overall agreement between the reference method CASPT2 and the wave function-based approaches CC2 and ADC(2) is excellent. The success depends slightly on the compound class, but the averaged MAE value is always below 0.1 eV with respect to the ±0.2 eV interval. For the polar compounds of our test set, CIS(D) is similarly accurate but deviates slightly stronger for nonpolar and intermediately polar compounds. The success of the spin-component scaling (SCS) corrections is well benchmarked for CIS(D) and CC2, but not for ADC(2). Figure 11 analyzes the influence of this correction

the error of LC-wPBE. ZINDO also predicts the right energy order of the states; i.e., the character of its three lowest states corresponds in most cases to those predicted by MS-CASPT2. In view of the necessary computational effort, this makes the method suitable for studying excitations in large molecules using geometries obtained with other methods. All other semiempirical methods (PM3, AM1, OMx) perform similarly for nonpolar molecules with a constant error of about 0.2−0.3 eV. For molecules with intermediate and high polarity, the differences increase. However, it should be pointed out that in spite of the slightly better numerical accuracy of the excitation energies obtained with PM3 and AM1 (average: 0.19 eV for both), the OMx methods (averaged error of 0.25−0.34 eV) deliver a considerably more reliable picture of a molecule’s excitation energies, especially for the higher states. Whereas for the OMx methods, the three lowest-lying excited states correspond mostly to the three lowest states predicted by CASPT2, this is not the case for PM3 and AM1. For both approaches, the severe underestimation of the energetic positions of the double excitations and a qualitatively wrong description of chargetransfer states lead to a wrong picture of the electronic excitations. Figure 10 shows the merocyanine HB194 and the

Figure 11. Differences of the error of the different methods with respect to the CASPT2 results are calculated, i.e., error(CC2) − error(SCS-CC2). In the case of positive results, spin-component scaling leads to an improvement of the calculated excitation energies.

in some more detail because for our test set, its success depends on the class of compound and the method under consideration. Positive signs indicate improvements, but negative signs reflect increasing differences from the MS-CASPT2 results. Please note that for Figure 11, we exceptionally compared with the respective MS-CASPT2 vertical excitation energies and not with respect to a ±0.2 eV uncertainty interval as used for the rest of this work. In this case, it was necessary to ignore the uncertainty interval because the differences between the methods would be invisible with an uncertainty interval. We only focus on the first excitation energies because the CASPT2space was expected to be most accurate for this excitation. Figure 11 indicates that spin-component scaling improves the performance of CC2 for polar and intermediate polar compounds whereas a slight deterioration is found for the class of nonpolar molecules. Despite this difference, the effect of the SCS correction on the excitation energy coincides for all classes: it always shifts the excitation energies by approximately 0.15 eV to higher values. The question if this improves or deteriorates the agreement with CASPT2 depends on the underlying method and the compound class. For highly and intermediately polar molecules, CC2 excitation energies are in general lower in energy than those obtained with MS-CASPT2. Hence, the increase in excitation energies induced by spincomponent scaling consequently improves the performance of CC2. The same holds true for the ADC(2) values. They are even lower than their CC2 counterparts and the SCS approximation gives slightly higher values in comparison to those from CC2. For our set of apolar molecules, the CC2 or

Figure 10. Correlation of the three lowest excitations (defined by the SCS-CC2 or MS-CASPT2 calculation) for two different semiempirical methods. Please note that the third excited state of HB194 corresponds to a charge-transfer excitation, and the second excited state of the triamine−aldehyde is a charge-transfer excitation (see Supporting Information also for oscillatory strengths).

aldehyde-substituted triarylamine as examples. The assignment of the states along with the molecular orbitals is given in the Supporting Information. From the correlation of the excited states, it is obvious that OM2 yields a correct energetic order but the excitation energies are systematically overestimated. This is not the case for AM1 (or PM3). The analysis reveals that AM1 and PM3 perform especially poorly for chargetransfer states, e.g., the third excitation energy for HB194 or the second excitation energy for triamine−aldehyde. Their excitation energies are considerably overestimated whereas the energies of doubly excited states are underestimated. However, within our test set the first excitation is generally also well described by AM1 or PM3. This is well in line with the findings of Thiel et al.114 Considering only valence excitations, 12886


The Journal of Physical Chemistry A ADC(2) vertical excitation energies still slightly underestimate the CASPT2 counterparts, but are already so close that the SCS correction leads to an overestimation of the excitation energies. This causes the negative deviations in Figure 11 for apolar dyes. However, it is important to note that the deviations never exceed 0.1 eV considered as the “chemical accuracy” for excited states.13 Only for the dithiophene is the CC2 value (4.01 eV) already higher than the CASPT2 prediction (4.0 eV). The SCS correction shifts the value to 4.06 eV. According to our findings, this is not the case for CIS(D). The perturbative inclusion of the doubles lowers the excitation energies significantly in comparison with CIS but the CIS(D) excitation energies are still higher than their MS-CASPT2 counterparts. Also in this case the application of the SCS correction leads to increasing excitation energies so that the overestimation of excitation energies is increased instead of decreased. With regard to the polarity dependence, spincomponent scaling is beneficial only for polar dyes for CIS(D). However, this might also be due to the fact that a CIS(D∞) calculation, i.e., a complete inclusion of the doubles, is computationally not feasible, but potentially necessary for converged values (see also ref 13). With respect to the influence of spin-component scaling on CC2 excitation energies, Hättig et al. found that SCS leads to an improvement of CC2 excitation energies using a benchmark set of small organic molecules.115 In contrast to this, Jacquemin et al. observed slightly larger errors of the excitation energies of medium- to large-sized molecules for both SCS-CC2 and SOSCC2.116 This suggests that, in line with our findings, the effects of spin-component scaling are rather system-dependent. The results of the effects of SCS on ADC(2) excitation energies complement the analysis of Dreuw and Wormit117 who could show that spin-opposite scaling of ADC(2) excitation energies also leads to an improvement. However, it might be important to mention that the SCS and SOS parameters used throughout our analysis were optimized by Hättig et al. for CC2.36 Therefore, they could be expected to be not perfectly suited for either ADC(2) or CIS(D). Other parametrizations optimized for ADC(2) or CIS(D) are available, for example, the SOS parameters for ADC(2) by Dreuw et al.118 Please note that both bright and dark excited states are among the first three excited states for the majority of the molecules (exceptions: dithiophene, “triamine”, “triaminemethoxy” with only bright states). The averaging over the first three excited states might conceal methods that are exclusively (in)accurate for a specific type of excitation. Therefore, a similar analysis was performed for the first bright and the first dark state (charge-transfer state or symmetryforbidden) for all molecules (Supporting Information). This analysis confirmed some important expectations. Generally, MAE values are larger for charge-transfer states as compared to neutral excitations. DFT functionals and also the semiempiric methods perform reasonably well for neutral excitations (and sometimes even better than range-separated hybrids), but they fail for charge-transfer excitations. However, on the one hand, overall performances do not differ between the averaging and the state-specific differentiation. On the other hand, we would like to define methods that perform equally well for both types of states as this is very important from a user’s point of view.

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CONCLUSION

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ASSOCIATED CONTENT

The present paper benchmarks the predictive power of a variety of wave function-based methods, DFT functionals, and semiempiric approaches for the characterization of p-type organic semiconductors commonly employed in optoelectronic devices. The test set contains very polar compounds as well as intermediately polar and nonpolar semiconductors. For this purpose, the study investigates their accuracies within the computations of geometries, electronic properties, and excitation energies. Although the geometrical data are tested with respect to the SCS-MP2/cc-pVTZ level of theory, for vertical excitation energies, MS-CASPT2/cc-pVDZ//SCSCC2/ccpVDZ data are used as a reference. In the latter calculations, the three lowest lying electronically excited states are included. The tested wave function approaches CC2/ccpVDZ and ADC(2)/cc-pVDZ perform very well for geometries but are worse than DFT/cc-pVDZ approaches. Hence, if the systems under consideration are too large for extended basis sets, DFT is the method of choice. Among the functionals, ωB97X-D, M06-L, and APFD perform best. SOGGA11X and CAM-B3LYP-D3 are only slightly worse. All these functionals performs very well irrespective of the polarity of the molecules. Hartree−Fock and BLYP-D3 fall below the quality of the other approaches. Semiempirical approaches are even worse, especially for molecules that are classified as intermediately polar or polar. With respect to the calculation of vertical excitation energies, SOGGA11X, CAM-B3LYP, and particularly ωB97X-D perform extremely well. For BLYP and M06-L, the success of the longrange correction depends on the character of the semiconductors. For valence excitations of semiconductors with intermediate and low polarity, the correction leads to considerably more precise values. For polar compounds, the improvement is smaller for BLYP and negligible for M06-L. As expected, the semiempiric method ZINDO yields reliable values for the excitation energies of all compounds (MAE = 0.15 eV), but especially for polar systems (MAE = 0.08 eV). All other semiempirical approaches are less successful but their averaged MAE values also stay below 0.3 eV. Nevertheless, the predictions of AM1 or PM3 often lead to a wrong energetic order of the electronic states. Especially the vertical excitation energies for charge-transfer states are overestimated whereas those of physically unimportant double excitations are underestimated. The wave function-based approaches CC2 and ADC(2) perform excellently as expected. For our test set, CIS(D) with and without SCS or SOS correction is less accurate than the best functionals. For the first bright vertical excitation, the effects of spin-component scaling were analyzed in more detail. For molecules with high or intermediate polarity, the SCS correction improves the accuracy of CC2 and ADC(2) by 0.1− 0.2 eV, but for nonpolar molecules, the predictions get slightly worse (∼0.1 eV). For CIS(D), only the vertical excitation energies for polar molecules improve. For these considerations, it has to be taken into account, however, that the variations are not larger than 0.1 eV.

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b10315. 12887


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Information on basis set effects on excitation energies, molecular orbitals of HB194, and triamine−aldehyde analysis of MAE values for specific excitation types; complete refs 92, 96, and 98 (PDF)

AUTHOR INFORMATION

Corresponding Author

*B. Engels. E-mail: [email protected]. Phone number: (+49) 931-31-85394. Fax number: (+49) 931-3185331. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the DFG in the framework of the FOR 1809 for financial support. C.B. thanks Christof Walter for proofreading this manuscript and fruitful discussions.

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