A New Benchmark Set for Excitation Energy of Charge Transfer States: Systematic Investigation of Coupled-Cluster Type Methods
There are numerous publications on benchmarking quantum chemistry methods for excited states. These studies rarely include Charge Transfer (CT) states although many interesting phenomena in e.g. biochemistry and material physics involve transfer of electron between fragments of the system. Therefore, it is timely to test the accuracy of quantum chemical methods for CT states, as well. In this study we first suggest a set benchmark systems consisting of dimers having low-energy CT states. On this set, the excitation energy has been calculated with coupled cluster methods including triple excitations (CC3, CCSDT-3, CCSD(T)(a)* ), as well as with methods including full or approximate doubles (CCSD, STEOM-CCSD, CC2, ADC(2), EOM-CCSD( 2)). The results show that the popular CC2 and ADC(2) methods are much more inaccurate for CT states than for valence states. On the other hand, CCSD seems to have similar systematic overestimation of the excitation energies for both valence and CT states.Concerning triples methods, the new CCSD(T)(a)* method including non-iterative triple excitations preforms very well for all type of states, delivering essentially CCSDT quality results.
Introduction<!>Computational details 2.1 Excitation energy calculations<!>Characterization of CT states<!>Structures<!>New charge transfer benchmark set for two-component molecular complexes<!>Selection of the CT states<!>Mixing of different characters<!>Vertical excitation energies<!>Conclusions
<p>Charge transfer (CT) states are special types of electronically excited states, that play key role in processes related to molecular conductance and electron transfer properties. As these states tend to show up in larger molecules and complexes, approximate theoretical methods need to be invoked for their description. The popular time-dependent density functional theory (TDDFT) is, however, known to underestimate considerably the excitation energies of CT states, at least with standard functionals. 1 It was shown eg. in Refs. 1-4 that most functionals do not perform well for valence and CT type states at the same time, and only hybrid functionals which include a substantial amount of HF exchange (often close to 100 %) are capable of giving reasonable results for CT states. 4 Long range corrected and double hybrid models also cure the problem. 1,3,5,6 Therefore, there is a demand for wave function methods which are economic enough to treat large systems of chemical interest, or at least can provide reliable benchmark results to test and calibrate other methods.</p><p>Yet, the performance of quantum chemical methods has never been systematically examined for CT states. Typically, earlier studies included only one or two systems with only one state per system. [2][3][4] In these studies the methods tested included several DFT functionals, CIS and SAC-CI. The most systematic study has been published by Dutta et al. 7 which includes six states of two systems, and results for various Coupled-Cluster (CC) ansätze are presented. The importance of systematic study is emphasized by the fact that severe inconsistency of certain popular methods in the description of Rydberg-type states 8,9 poses a warning that not all techniques may be well suited for CT states, either.</p><p>In the last years, a large number of test calculations have been presented on excited state methods. Most of them used the benchmark set established by Thiel and co-workers 10 (often referred as the Mülheim set) and concentrated on vertical [8][9][10][11][12][13][14][15][16][17][18][19][20] or 0-0 excitation energies. [21][22][23][24] See a recent review for further details. 25 More recently, the scope of the benchmark studies has been extended to potential energy surfaces 26,27 as well.</p><p>Realising that the information on the performance of different methods on charge transfer states is rather sparse, 20,25 in this paper we aim at establishing a benchmark set from local and CT type states of two-component molecular complexes by providing high level (CCSDT and CCSDT-3) benchmark values. This set is then used to characterize the reliability of various methods of the Coupled-Cluster hierarchy for charge transfer excitation energies.</p><!><p>In this study, Coupled-Cluster type methods are used to calculate the excitation energies corresponding to CT states. In this respect, the CCSD level approximation termed either EOM-CCSD [28][29][30] (Equation of Motion Coupled-Cluster with Singles and Doubles) or CCSD-LR 31,32 (CCSD Linear Respnse) (no distinction between EOM and LR will be made here since these two give the same excitation energy, so for our purpose they are equivalent) represents a standard starting point which may be too expensive for larger applications, while not accurate enough for certain high accuracy demands. Therefore, on one hand, we include lower cost methods, such as the second-order approximations to CCSD, like the popular CC2 33 and ADC(2) 34,35 (2nd order Algebraic Diagrammatic Construction) methods, as well as EOM-CCSD(2) 36 (second order approximation to Equation-of-Motion Coupled Cluster Singles and Doubles, also known as EOM-MBPT(2) 37 ) and STEOM-CCSD (Similarity Transformed Equation-of-Motion Singles and Doubles). [38][39][40] The latter, though cheaper than CCSD, cannot be considered as an approximation thereof: 25 STEOM-CCSD aims at eliminating the doubles-singles (DS) block of the Hamiltonian matrix, thereby making possible to obtain excitation energies solely in the space of single excitations. This is done by a similarity transformation of the untruncated Hamiltonian matrix which changes also the SS block, and makes certain blocks (e.g. ST) smaller, in this respect including some contributions of connected triple excitations. For the present discussion an important formal property of STEOM-CCSD is that in the case of noninteracting fragments the CT excitation energy precisely equals the sum of the ionization energy and electron affinity of the source and destination fragments, respectively, calculated at the same level of theory. 40 On the other hand, we also go beyond the CCSD approximation by including the effect of connected triple excitations as the next step of hierarchical improvement. For some of the systems under inspection, calculations even at the CCSDT level 41 were possible. Several approximate triples methods (iterative and non-iterative) have been included in the study, e.g. the iterative CCSDT-3, 42 and the closely related CC3 43 variant, as well as the recent, non-iterative EOM-CCSD(T)(a)* method of Matthews and Stanton. 44 EOM-CCSD(T)(a)*, while economic since non-iterative, gives results very close to CCSDT, its error being similar to that of CCSDT-3. 8 As reference, we use CCSDT for the smaller systems, while when treating all complexes, CCSDT-3 results were used for this purpose. This reason for choice will be apparent from the discussions below, as CCSDT-3 is found to perform better than CC3 in the case of CT type states.</p><p>Several program systems have been used in this study. CC2, CCSD, CCSD, CCSDT, CCSDT-3, CC3 and CCSD(T)(a)* calculations have been performed by CFOUR; 45 for ADC(2) calculations TURBOMOLE 46 has been used with the resolution of identity (RI) approximation. Performing also the CC2 calculations this way, it was found that the RI approximation doesn't influence the excitation energies by more that 0.01 eV. STEOM-CCSD calculations were done by Orca. 47 The STEOM calculations require an active space both within the occupied and virtual spaces. The process of selecting this active space has already been made automatic using a process based on configuration interaction singles averaged densities. 48 In the present study, the default value (0.01) is used for both the occupied (OTHRESH) and virtual (VTHRESH) active space selection thresholds, in all but one cases. For the system containing a tetrafluor-ethylene and an ethyene molecule separated at 3.5 Å, both values are set to 0.001 to reach convergence with respect to the size of the active space.</p><p>In all excitation energy calculations the cc-pVDZ basis set of Dunning and co-workers 49 was used with the core electrons uncorrelated. Normally, the use of diffuse functions is also warranted while studying molecular complexes, as they might be important for the accurate description of intermolecular interaction energies. However, the presence of diffuse functions in the calculation will result in the appearance of Rydberg states in the same energy range of the spectrum as that of the valence states, making the proper analysis of the states rather difficult. Since we do not aim for the best absolute accuracy, rather just compare the results obtained with the different methods, the lack of diffuse function does not seem to be critical.</p><!><p>The identification and characterization of electronic states was performed using the analysis of the wave functions by inspecting the natural orbitals of the difference density of the ground and excited states, 50 as well as by examining the numerical descriptors defined by Plasser and co-workers. 51,52 The natural orbitals of the difference density can conveniently be used to graphically illustrate in the one-electron picture the dominant orbitals where the excited electron comes from ("from" orbital) and where it goes ("to" orbital). The corresponding "occupation numbers", which are negative for "from" orbitals and positive for the "to" orbitals, also inform about the weight of these orbitals in the wave function. In most cases there is only one orbital with a substantial occupation number for both types, presenting a nice simple description of the excitation corresponding to chemical intuition. However, when dealing with a large number of excited states, the inspection of these orbitals can be tedious, therefore in an automated procedure well-defined numerical descriptors are more useful.</p><p>The CT or local nature of excited states can be conveniently evaluated using the Ωdescriptors introduced by Plasser and co-workers, 51,52 based on the one-particle transition density. In this formalism, the CT character (ω CT ) is defined as the weight of configurations with charges separated on different fragments. An excitation is more local as ω CT is closer to 0 and more charge transfer with the character close to 1.</p><p>The average exciton position (ω P OS ) is given by the mean of ω P OSi and ω P OSf , which represent the average position of the initial and the final orbital, respectively. In a system consisting of just two fragments, ω P OS ranges from 1 to 2, with ω P OS ≈ 1 corresponding to a local excitation on the first fragment, ω P OS ≈ 2 to one on the second fragment, while ω P OS ≈ 1.5 values are typical for CT and ideal (one-to-one mixed) Frenkel-type excitations.</p><p>Similarly to ω P OS , ω P R is the arithmetic mean of ω P Ri and ω P Rf , which give the participation ratio of the fragments in the initial and final orbitals, respectively. Thus, ω P R ≈ 1 holds for both clear CT and local excitations, while in a two-component system, ω P R ≈ 2 is seen for ideal Frenkel-type excitations and charge resonance states. 51 This analysis tool depends on the definition of the fragments corresponding to the chromophores. In the present case, since the systems are molecular complexes, this definition is trivial.</p><p>The characters have been calculated with the TheoDORE program and libwfa library developed by Plasser et al. [53][54][55][56] at the CC2 and CCSD levels, utilizing the CFOUR 45 / libwfa interface developed by us. It has to be noted that, unlike the TheoDORE interface to the TURBOMOLE 46 program that approximates the one-particle transition density matrix with the single excitation part of the solution vector in the CC2 case, this analysis utilizes the entire one-particle transition density to obtain the descriptors.</p><!><p>With one exception, the structures used in the calculations have been obtained by a fulldimensional optimization for the ground states of the complexes, thereby all totally symmetric coordinates were allowed to change. These optimizations were performed at the CC2/cc-pVDZ level.</p><p>The pyrrole-pyrazine stacked structure was, for technical reasons, obtained as follows.</p><p>The structures of the monomers were taken from the Mülheim set, 10 and placed above each other such a way that their planes are parallel and their main axes perpendicular to their plain coincides. Moreover, the N-H bond of pyrrole was set to be perpendicular to the line defined by two nitrogen atoms of pyrazine. Then, only the distance of the two planes was optimized at the MP2/aug-cc-pVDZ level, resulting in a minimum-energy distance of 3.41 Å. The reason for this constrained optimization is that all attempts of fully optimizing the complex resulted in a structure where the two molecules end up in a single plane.</p><p>The structures are presented in the Supporting Information.</p><!><p>The new benchmark set we propose here consists of molecular complexes with low-lying charge transfer states. This time we do not consider well known examples of intramolecular CT states since in such cases the chromophores need to be well separated, resulting in quite large molecules which would prevent high level reference calculations.</p><p>To test quantum chemical methods on CT states, only a few systems had been suggested and used in the literature. The tetrafluoroethylene-ethylene (C 2 F 4 -C 2 H 4 ) and ammoniafluorine (NH 3 -F 2 ) complexes appear as test systems in several applications. 1,2,4,7 Using the former system, Dreuw and Head-Gordon 1,2 tested different TDDFT methods and compared the results to those obtained with the CIS method. Zhao and Truhlar 4 used the ammo-nia-fluorine system in their investigation of various TDDFT approaches. In these studies only one CT state per system was considered, thus only limited information on the performance of the methods could be obtained. Dutta et al. 7 also used these two systems to test STEOM-CCSD and its variants with respect to CCSDR(3) 57 reference values, but included no more than six states altogether. This set is clearly insufficient, in particular, since most of the states considered were higher lying ones involving non-valence orbitals. Recently, Mester and Kállay 58 tested approximate local CC approaches including also CT state of the tetrafluoroethylene-ethylene system.</p><p>When finding additional systems, we had two guiding principles: the molecules are required to have π bonds and lone pairs and the CT states should be energetically low lying. Starting with the traditional ammonia-fluorine complex, the ammonia and the fluorine molecules were replaced with other molecules showing lower ionization potential (IP) and electron affinity (EA) values, respectively. The resulting complexes include acetone-fluorine</p><p>. Finally, the pyrrolepyrazine pair, both in the stacked and the H-bonded forms, has also been considered to include ring systems. As we will discuss in more detail, for these systems 41 local and 14 CT states could be identified and included in the test set.</p><!><p>Table 1 shows the CCSD results on all states considered in this study. Beside the excitation energy and the assignment of the states, the table shows the different characters (ω CT , ω P R and ω P OS ), as well as the AEL (Approximate Excitation Level) values. 29 Examining the table, local and CT states can be classified. In most cases local states can easily be identified by ω P R being close to one and ω CT being close to zero, with ω P OS pointing to the appropriate fragment. On the other hand, about one third of the CT states are mixed with local states giving eventually CT characters as low as 0.6. As expected, for the clean CT states the ω P OS value is near 1.5, while ω P R is close to 1 and increases to 1.5 for mixed type ones. The only exception is the tetrafluoroethylene-ethylene complex, which will be discussed below.</p><p>The mixing of states of different character complicates the classification of the individual states to either local or CT. To have a clear definition, we decided that in the analysis below states showing ω CT > 0.5 (with ω CT evaluated at the CCSD level) are considered as charge transfer, while all others as local. As there is a certain degree of arbitrariness in this choice, further below we will also examine the accuracy of different methods as a function of ω CT .</p><p>An unfortunate consequence of the above definition is that none of the states of the tetrafluoroethylene-ethylene system are included in the CT set, although this complex is one of the most often used test example for CT type states. We investigate this system more closely in the next subsection, revealing that instead of the optimized complex, a structure represented by an interfragmental distance of 5 Å should be included in the statistical analysis. In this case, corresponding to the intuition, one finds among the low-lying states two local π − π * excitations on the fragments, and a clear CT state with ω CT = 0.99. This particular distance was used by Mester and Kállay 58 in their test calculation on approximate local CC approaches.</p><!><p>Local (valence) and CT excitations are very different in character. Due to the interaction of the fragments, local states might couple into Frenkel type excitations; the set studied here does not include typical Frenkel excitations, but these are out of the scope of the present study, anyway. It is the energy of the CT states rather than that of the local ones which seems to be much more dependent on the distance of the fragments. At infinite distance The ability of the different methods to describe this coupling will be studied in a subsequent publication. Here the focus is on the "vertical excitation energy", therefore we discuss only one aspect of this problem, namely that close to the "crossing" the assignment of the CT and local states is not trivial which hampers the definition of the two classes to be used in statistical analysis of this study.</p><p>When comparing the excitation energies obtained with different methods with a benchmarking goal, one has to select states from the different calculations to be compared. Around the crossing we have two states for each method; for the assignment there are two options:</p><p>either the states showing the same dominant character are compared, or one selects them by the order of their energies. The first option compares states of similar wave functions, while keeping the energy ordering will always result in smaller errors since one compares the energy of the lower state with the lower one, and vica versa. If the geometry of the calculation is on the same side of the "crossing" for both methods, the two options give the same results, otherwise one has to chose among them. Aiming to establish distinguished benchmark sets for CT states, in this study we compare the energy of those states which show the same dominant configurations, i.e., the first option above. The consequence of this choice is that in cases the geometry is on different side of the "crossing" for the two methods, we compare the energy of the lowest state obtained with one method to that of a higher-lying state calculated with the other. In such a case we will get larger errors than by an assignment according to the energy ordering and the procedure results in a pair of states with outlier errors in the positive and negative directions, respectively. By taking the second possibility for the assignment, this could be avoided, but we think that the possible wrong position of the "crossing" (i.e., the energy of the CT state decreasing too fast or too slow as the subsystems approach each other) is again an important indicator for the reliability of the methods which should not be hidden this way. In a subsequent paper we foresee to discuss this issue more closely by benchmarking different methods along the potential curve including the "crossing" region.</p><p>Since the Ω descriptors are available only for CCSD, CC2 and ADC(2), for other methods, the following procedure was used to determine the dominant character of the states: first, the ω CT characters have been calculated at the CCSD level to classify the states (see Table 1).</p><p>Then the dominant singles contributions of the eigenvectors obtained by the given method are compared to the CCSD one to get the assignment. In the discussion below, we will always mention if any discrepancy due to the assignment shows up and comment how the conclusions are affected. Situations of this kind were observed for two systems: ammoniapyrazine described by the CCSD(T)(a)*, CCSD and CCSD(2) methods, and the ammoniaoxygendifluoride complex at the CC3 and CC2 levels.</p><p>Another possibility to avoid the above problem is to perform the calculations at distances long enough so that such a strong coupling does not occur. However, with the exception of one case, we refrain from this solution, since we believe that from the point of view of practical applications the vicinity of the equilibrium structure of these complexes is more relevant than large intermolecular distances are.</p><p>It was the terafluoroethylene-ethylene complex in which case we, too, decided to include in the statistics a more distant pair of fragments instead of the equilibrium geometry. Here, the lowest-energy (valence) excitations of ethylene and tetrafluoroethylene are the respective π − π * transitions, in the latter case the involved orbitals also containing significant contribution on the fluorine atoms. Accordingly, the lowest CT state is expected to be the one dominated by an excitation from the π orbital of tetrafluoroethylene to the π * orbital of ethylene. As Table 2 shows, at an interfragment separation of 5 Å the corresponding three states can be identified by almost perfect ω characters: the two lowest B 1 excitations are the two local π − π * transitions, while the one at 10.87 eV (CCSD) is the CT excitation.</p><p>Figure 1 shows the "from" and "to" (difference density natural) orbitals of the corresponding three states of the complex at the equilibrium distance of about 3.5 Å. While the third state is essentially still a local π − π * excitation on tetrafluoroethylene with an excitation energy differing by just a few hundredths of an eV from the corresponding state at 5</p><p>Å at all levels of theory, the other two states are strong mixtures of local and CT character, sharing the "to" orbital as that of the π * orbital of ethylene. In Table 2 we observe ω CT characters of 0.24 and 0.39 for these two states, also reflecting this strong mixing, while the CT character of the third state is practically zero. Since the splitting of the energy of these two states is about 0.8 eV for all methods, we can conclude that there is no typical CT state at 3.5 Å and thus, none of the states of this complex at the equilibrium could be included in the test set. Since the terafluoroethylene-ethylene complex was used in earlier studies, we decided to use this complex with an intermolecular separation of 5 Å, where a typical CT state can be identified.</p><!><p>Vertical excitation energies of the complexes obtained with the various methods are given in table??? of the Supporting Information.</p><p>Although not the main objective of this paper, let us first shortly discuss local states.</p><p>In Table 3 statistics are presented for these local states using CCSDT-3 results as reference  -0.04 0.04 -0.13 0.12 CCSD(T)(a)* 0.01 0.05 -0.24 0.05 STEOM-CCSD -0.12 0.11 -0.38 0.10 Turning to the CT states, we have to realize that CCSDT results available for no more than seven states only, thus we have to select the best approximate triples method to use as reference for the entire set. The error of excitation energies calculated with the triples methods are presented in Table 4. It is the CCSDT-3 method which shows the smallest mean error as well as the smallest standard deviation. The error is typically no larger than +0.05 eV, oppositely signed error can only be observed for the ammonia-fluorine (-0.03 eV) and terafluoroethylene-ethylene (-0.01 eV) complexes. The errors of CCSD(T)(a)* are spread around 0, resulting in a small mean error. A large error of 0.24 eV can be observed in the case of the ammonia-pyrazine system which is due to the assignment uncertainty of close-lying states as discussed above. The large error essentially means that in this case the CCSD(T)(a)* results are on the other side of the crossing than the reference CCSDT is. If this state is not considered, CCSD(T)(a)* seems to give very similar statistical values as CCSDT-3, but even with this state included the method performs on average very well. Contrary to CCSDT-3, CC3 systematically underestimates the vertical excitation energy. The largest error is observed for ammonia-oxygendifluoride, but here, too, the strong coupling of a local and a CT state make the assignment ambiguous. If we disregard this state, the typical error of CC3 is around -0.10 eV. The final conclusion from this table is that CCSDT-3 seems to perform slightly better than other triples methods, so it will be used for reference in this study.</p><p>The accuracy of the methods for all CT states can be examined in Table 5 where the relative energies with respect to CCSDT-3 are shown. CCSD(T)(a)* gives results very close to CCSDT-3, the deviation is usually no more than a couple of hundredths of an eV.</p><p>The only exception is ammonia-pyrazine with an error of 0.20 eV, the cause of which has been discussed above. Inclusion or exclusion of this state does not influence the statistics significantly. In the case of CC3 we continue observing a regular underestimation, as it was the case for the smaller set with respect to CCSDT. The largest error is again for ammonia- oxygendifluoride which, as discussed above, is due to the assignment uncertainty, since CC3 gives different order of local and CT states not only with respect to CCSDT but also with respect to CCSDT-3. Irrespective of this, the average error in the case of CC3 is about -0.1 eV with a small SD.</p><p>Continuing with the doubles methods, CCSD is found to always overestimate the excitation energy of CT states, the mean error being less than 0.3 eV with an SD of 0.1 eV. This shows that CCSD is quite systematic, although the errors are somewhat larger than for valence states. 8 The CCSD(2) results are very close to CCSD with a slightly larger standard deviation. CC2, on the other hand, systematically underestimates, the error can be as large -0.7 eV (for acetone-fluorine, pyrazine-fluorine and ammonia-oxygendifluoride). For larger systems there is a tendency for somewhat smaller errors, but in most cases the inaccuracy is not less than 0.2 eV in absolute value. ADC(2) also underestimates the CT excitation energies systematically, even the individual excitation energies are almost indistinguishable from CC2, with only one exception (acetone-nitromethane), where the error is almost twice as large as that of CC2. To look for the cause of this unexpected discrepancy, we have observed that even for local states of acetone-nitromethane the difference between the CC2 and ADC(2) excitation energies are considerably larger (see Supporting Information). Analyzing the wave function it was found that the norm of the T 1 amplitudes in CC2 is about 0.15 in case while for the other systems investigated in this paper it is about 0.09 or smaller. The resonable explanation therefore is that the T 1 transformation included in CC2 but not in ADC(2) makes a difference here, contrary to the general situation. 9 STEOM-CCSD seems significantly more accurate than the other singles-doubles methods, in fact the statistical values resemble more those of the triples methods. This seems to be an indication that STEOM-CCSD indeed includes some important triples effects. 40 In Figure 2 the errors of the calculated excitation energies obtained with different methods are shown with respect to CCSDT-3, as a function of the excitation energy. The first observation is that for none of the methods there is an apparent correlation between the size of the error and the magnitude of the excitation energy, i.e. all methods perform independently of the excitation energy. Indeed, the correlation coefficient between the two quantities is in no case larger in absolute value than 0.3. negative errors observed for CCSD(2) above 7.5 eV perhaps seem to be exceptions.</p><p>The doubles methods give substantially larger error than the triples methods, it appears that triple excitations are needed to get the error down to 0.1 eV. CCSD, CCSD(2) and STEOM-CCSD are clearly more systematic than CC2 showing errors of the same sign in most cases. Note that systematic behavior is rather important to get the relative position of the bands in the spectrum correctly or to perform a reliable non-adiabatic dynamics.</p><p>An even more detailed picture can be obtained when considering the colors of Figure 2.</p><p>Here the states are assigned to three category: clean local states (ω CT < 0.1) marked with blue, clean CT states (ω CT ≥ 0.9) marked with yellow, while all other states are in the "mixed" category (0.1 ≤ ω CT < 0.9) marked with orange. This is a more detailed classification than the one we used for the statistics above where the local and CT states have been divided by the arbitrary value of ω CT = 0.5.</p><p>As mentioned before, CCSD seems to be quite systematic, but CT and mixed states clearly show a larger error than local ones. There is only one state with a substantial negative error, which is the highest considered state of the ammonia-pyrazine complex with ω CT = 0.27. Note that by 0.4 eV below there is another state of the same system (CT7) with a rather large positive error (ω CT = 0.63). These two states mix strongly and the discrepancy can again be explained by the assignment uncertainty caused by a close-lying crossing: the geometry used for the calculation is on the other side of the crossing with CCSD than with the reference CCSDT-3 method. The CCSD(2) figure is very similar to the CCSD one, but the systematically larger error is apparent. In the case of ammonia-pyrazine the pair of orange colored bars with opposite signed error shows up again, i.e. CCSD(2) reproduces even ). The color of the bar shows the character of the respective excited state: blue for local states (ω CT < 0.1), yellow for CT states (ω CT ≥ 0.9), and orange for states of mixed character (0.1 ≤ ω CT < 0.9).</p><p>the discrepancy observed for CCSD. The blue bars with negative sign have been mentioned already above and warn that CCSD(2) might perform worse for higher excitation energies.</p><p>Contrary to the systematic behavior of the errors of CCSD and in most part also CCSD(2), eye-catching is the discrepancy in case of the CC2 method: blue bars are small and positive, while yellow ones are negative and substantially larger. This shows again that CC2 systematically underestimates the excitation energy of the CT states. Since almost all orange bars are also on the negative side, one can conclude that even a partial CT contribution is enough to spoil the accuracy of CC2. A similar behavior has been observed for Rydberg states, 8 and might have the same cause: in both Rydberg and CT states longrange interaction of the active electrons needs to be described. It is apparent that CC2 is not balanced enough which is most probably due to the missing off-diagonal elements in the double-double block of the Jacobian. Note, however, that CC2, contrary to CCSD and CCSD(T)(a)*, gives the right order of the CT and local states for the ammonia-pyrazine, but, like CC3, gives the wrong order for ammonia-oxygendifluoride. The graph of ADC( 2)</p><p>is very similar to that of CC2, thus all said for CC2 also apply for ADC (2).</p><p>As of STEOM-CCSD, the yellow bars are very short on the respective panel of Figure 2 indicating that STEOM-CCSD is very accurate for CT states. On the other hand, the blue bars of the local states are more sizeable, in most cases showing underestimation. This is also true for the orange bars which are smaller than blue ones but larger than yellow ones.</p><p>This shows that the size of the error depends on the ratio of the local and CT characters in the given state. The largest negative orange bar belongs to one of two close-lying states of acetone-nitromethane with the usual uncertainty of its assignment. Thus, the cause for this outlier is again the relative position of the geometry used in the calculation with respect to the "crossing". (Note that contrary to the triples methods, CCSD and CCSD(2) also give the same order of the respective two states as STEOM-CCSD but the error of their energy is smaller therefore not resulting in outliers.) CCSD(T)(a)* performs well, its error is usually below 0.05 eV and scattered around 0.</p><p>Here, too, more blue is seen on the positive side and more yellow on the negative one. The pair of orange outliers belongs to the ammonia-pyrazine case discussed earlier: it seems that the non-iterative triples correction cannot fix the wrong order of the states seen at the CCSD level.</p><p>CC3 shows clearly larger differences with respect to CCSDT-3 than CCSD(T)(a)*. Yellow and orange bars are more negative than blue ones, therefore it seems that CC3 also shows the misbalance of CC2, although at a much smaller absolute value. Note that here CCSDT-3 is used as reference, but similar trend has been observed for cases where CCSDT results are also available (see above). The large orange bar at 7 eV together with the positive blue one at 6.8 eV corresponds to ammonia-oxygendifluoride, the pair of states for which CC3 (and CC2) gives the opposite energy ordering than all other methods (see the discussion above).</p><p>It has to be noted that no significant correlation between the error of excitation energies and the AEL value of the states could be observed for any of the methods. This finding is in agreement with the fact that all states included in this study are clean, singly excited states with AEL ≤ 1.14, thus the results are not biased by non-negligible double excitation characters.</p><!><p>The main goal of this study was to set up a benchmark set for charge transfer type excited states. Our proposal includes fourteen CT states of nine bi-molecular complexes and reference values obtained at the CCSDT-3 level with the cc-pVDZ basis. For the half of these states also CCSDT results are available, which show that CCSDT-3 is very accurate, the approximate treatment of connected triple excitations do not change the quality of the benchmark values.</p><p>This benchmark set with high level reference date has been used to test the accuracy of various CC-type methods. We emphasize here that the goal was not to find the most accurate technique relative to experiment, rather to test the intrinsic errors in the coupledcluster hierarchy.</p><p>Considering CCSD as the standard method of choice, we have studied approximate techniques which are more cost effective, as well as more advanced ones which include connected triple excitations. The results show that the second-order approximations to CCSD deteriorate the results on CT states: while CCSD(2) gives less systematic results with slightly larger mean error and standard deviation, CC2 and ADC(2) substantially underestimate the excitation energy of CT states. This is particularly problematic since these methods are quite accurate for local (valence) states, therefore the right order of the excited states of different character is not warranted. The failure of CC2 and ADC(2) for CT states shows some parallelity to the case of Rydberg states 8 and can likely be attributed to the inappropriate description of the electron moved far from its ground state position. STEOM-CCSD, which is, though cost effective, not an approximation to CCSD, shows some improvement and clearly gives the best statistics among singles-doubles methods for CT states. This behavior is attributed to the implicit inclusion of some triple excitation contributions in the STEOM-CCSD formalism. 40 Since the improvement for local (valence) states is somewhat smaller, here, too, some misbalance between different types of states can be observed. This method can be applied to all systems for which ground state CCSD(T) calculations are feasible.</p><p>In this paper we have chosen equilibrium structures for the calculations, i.e. the frag-</p>