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Accuracy of Coupled Cluster excitation energies in diffuse basis sets Dániel Kánnár, Attila Tajti, and Peter G. Szalay J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00875 • Publication Date (Web): 13 Dec 2016 Downloaded from http://pubs.acs.org on December 18, 2016

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Journal of Chemical Theory and Computation

Accuracy of Coupled Cluster excitation energies in diffuse basis sets Dániel Kánnár, Attila Tajti, and Péter G. Szalay∗ Laboratory of Theoretical Chemistry, Institute of Chemistry, Eötvös University, P. O. Box 32, H-1518, Budapest 112, Hungary E-mail: [email protected]

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Abstract In this paper, we present a comprehensive statistical analysis on the accuracy of various excited state Coupled Cluster methods, accentuating the the effect of diffuse basis sets on vertical excitation energies of valence and Rydberg type states. Many popular approximate doubles and triples methods are benchmarked with basis sets up to aug-cc-pVTZ, with high level EOM-CCSDT results used as reference. The results reveal a serious deficiency of CC2 Linear Response and CIS(D) techniques in the description of Rydberg states, a feature not shown by the EOM-CCSD(2) and EOMCCSD variants. The CC3 theory proves to be an accurate choice among the iterative approximate triples methods, while the novel perturbation-based CCSD(T)(a)* variant turns out to be the best way to include the effect of triple excitations in a non-iterative way.

Keywords excited states, Coupled Cluster, basis set, CC2, CCSD(2), CCSD, CCSD(T), CCSDT, Rydberg states, valence states, Equation of Motion, diffuse functions

Introduction Over the past decades, quantum chemical methods have successfully been extended to describe excited electronic states of molecules. Nowadays, spectroscopic and dynamic investigation of excited states can also be performed using quantum chemical calculations. 1 Due to the detailed and accurate description they provide at the molecular level, theoretical methods could successfully augment the experimental results and provide a deeper insight into important processes. 2–6 However, the treatment of electronically excited states by quantum chemical methods is much more challenging than that of the ground state, and still far from being suited for black-box applications. To achieve this, the reliability of different methods 2


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need to be investigated from many points of view. A well known benchmark series for excited states has been put forward by Thiel and co-workers, 7–12 which includes over 100 excited states of 28 molecules, often referred as the M¨ ulheim benchmark set. They compared results for the CC2-CCSD-CCSDR(3) series of Coupled Cluster (CC) methods to CC3 and CASPT2 reference data 7,8 using the TZVP 13 basis set. Later, the investigation was extended using the aug-cc-pVTZ basis to estimate the basis set effect for the CC2, CCSDR(3) and CC3 methods. 9 However, in this study, CC3 was only used for a limited number of cases, and the respective data for the CCSD method was omitted completely. Basis set effects have also been benchmarked on CASPT2 results. 10 Furthermore, the performance of B3LYP, BP86 and BHLYP TD-DFT functionals, as well as that of the DFT/MRCI method 11 and semi-empirical methods 12 have been investigated. A common point in most of these studies is that the high level CC3/aug-cc-pVTZ and CASPT2/aug-cc-pVTZ results are presented as the ”theoretical best estimate”, and the other methods are benchmarked against these values. The results showed that the CC2 method performs better than CCSD for vertical excitation energies, and the non-iterative CCSDR(3) method is a cost-effective approximation of CC3 with small mean error in both basis sets. In the augmented basis, the inclusion of diffuse functions was found to lower the excitation energy by about 0.2 eV, but without affecting much the overall performance of Coupled Cluster methods. The methods containing triple excitations have also been benchmarked by Watson and co-workers, 14 using the TZVP triple-ζ basis set for the molecules forming the M¨ ulheim set. This study was recently extended in our laboratory to be more systematic. As using CASPT2 results as reference can be a controversial choice, the CC3 reference data has been completed, including also the larger molecules. To also check the reliability of CC3, CCSDT calculations were performed for some of the smaller members of the test set, and the CC3 oscillator strengths have been calculated for the whole set. 15 There, we concluded that the CC2 method gives results closer to the high level CC3 than CCSD, which overestimates the

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excitation energies by about 0.2 eV. At the same time, however, CCSD was found to be more systematic throughout the set of the treated excited states. In this paper, we investigate these methods from a different perspective and in more detail. The basis set dependence of CC methods will be benchmarked through the vertical excitation energies calculated by the doubles (CIS(D), CC2, CCSD(2), CCSD) and the approximate triples methods (CCSD(T), CCSD(T)(a)*, CCSDT-3, CC3), using the high level CCSDT values as reference. With the inclusion of diffuse functions in the basis set, we extend the set of studied excited states to the often disregarded Rydberg states, and characterize the performance of CC methods for these states, as well. The importance of such an investigation is emphasized by many recent studies on small to medium sized organic biomolecules 4,6 that reportedly possess many low-lying Rydberg states as well as mixed ones with a significant Rydberg component. 16

Methods One of the main advantages of Coupled Cluster (CC) theory is its hierarchical nature providing approximations to the Full-CI limit: by truncating the expansion at different excitation levels (singles, doubles, triples), the accuracy can be increased systematically, obviously for an increasing computational cost. To extend Coupled Cluster theory towards describing excited states of molecules, two separate formalisms were established: the Equation of Motion (EOM) 17–19 and Linear Response Theory (LRT) 20 variants. Both approaches form a straightforward and consistent extension of the corresponding ground state treatment to that of the excited state and - at the same level of truncation - both provide the same excitation energies, but slightly different transition properties. For example, EOM-CCSD 21,22 provides the description of the excited state at the CCSD level, while EOM-CCSDT 23–25 is the excited state counterpart of the higher level CCSDT method. (Throughout the rest of this paper, whenever referring

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to excitation energies, we simply use the name of the corresponding ground state method, omitting the denotation for Equation of Motion or Linear Response Theory.) Although the EOM or LRT extensions of CC methods retain the computational scaling of their ground state counterparts, it has to be kept in mind that the computational cost grows linearly with the number of desired excited states. For this reason, it is inevitable to check the performance of the cost-effective approximate methods which allow the treatment of many excited states of large systems. One way to reduce computational cost is the use of perturbation theory. One may either formulate higher order perturbational energy correction non-iteratively to a lower order excitation energy expression, or, alternatively, truncate the equations at lower order terms and obtain the result in an iterative way. The CIS(D) method 26 gives a second-order a posteriori energy correction to the CIS excitation energy, which can be seen as a non-iterative approximation to the CCSD excitation energy. The iterative CC2, 27 EOM-CCSD(2) 28,29 and Partitioned EOM-MBPT(2) 29 (P-EOM-MBPT(2)) methods are designed as approximations to the CCSD method, based on different arguments. While CC2 is defined as an iterative ground state method and describes excited states in the linear response framework, 27 EOMCCSD(2) and P-EOM-MBPT(2) use an MBPT(2) ground state and formulate the excited states by second order perturbational approximation with (P-EOM-MBPT(2)) or without (EOM-CCSD(2)) matrix partitioning of the CCSD hamiltonian. The non-iterative EOM˜ 31 CCSDR(3) 32,33 and the recently derived CCSD(T)(a)* 34 CCSD(T), 30 EOM-CCSD(T), triples methods correct the CCSD excitation energy to approximate the CCSDT 24 result. The latter method is special in the sense that it applies a perturbative triples correction to both the ground state and the excited state wave functions. The approximation of CCSDT is also possible in the iterative way: the CC3, 35 CCSDT-1 30 and CCSDT-3 31 methods were worked out for this purpose. In this work, beside the TZVP basis, we have also used the cc-pVDZ and cc-pVTZ 36,37 valence basis sets for the calculation of excitation energies. In contrast to our previous pa-

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per, 15 we aim to study not only singlet valence but also Rydberg states, thereby investigating the reliability of the methods from a new perspective. To describe at least the low lying Rydberg states properly, we applied augmented basis sets (aug-cc-pVDZ, aug-cc-pVTZ) 36,37 throughout the calculations. To be able to calculate excitation energies even at the CCSDT level in the aug-cc-pVTZ basis set, a smaller set of test molecules had to be chosen. Ethylene, acetylene, formaldehyde, formaldimine and formamide were selected for the investigation. Although these molecules belong to the smaller ones in the M¨ ulheim set and the resulting population of states will also be limited in size, the ability to compare against high-accuracy CCSDT results should provide us more reliable and representative statistics than a comparison to experimental results that are often biased by assignation issues and the unclear definition of experimental vertical excitation energy. To keep the calculations including connected triple excitations computationally tractable, the core electrons were frozen in all calculations presented here. The geometries were identical to those in the works of Thiel and co-workers i.e. optimized at the MP2/6-31G* level. The optimized structures can be found in the Supporting Information. For the representation of the character of excited states, the natural orbitals of the difference density of the ground and excited states 38 were used. These natural orbitals, along with the corresponding occupation numbers, allow an unbiased characterization of σ − π ∗ , π − π ∗ , n − π ∗ and also Rydberg states. All calculations have been performed using the CFOUR 39 program package.

Results and discussion The results of all calculations can be found in the Supporting Information. Excitation energies are listed, which were obtained at the CIS(D), CC2, CCSD(2), CCSD, CCSD(T), CCSD(T)(a)*, CCSDT-3, CC3 and CCSDT levels of theory, using the TZVP as well as the

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correlation consistent double- and triple-ζ basis sets with and without diffuse functions. The deviation of the calculated excitation energies to CCSDT results has been analyzed statistically and the results are summarized in Table 1: mean errors, standard deviation (SD), as well as largest positive (+) and negative (-) deviations are listed. Valence and Rydberg states are considered separately since it turned out that certain methods do not describe these states with the same accuracy. 40 Altogether, a total of 46 excited states were treated, from which 16 are transitions to valence states (π − π ∗ , n − π ∗ , σ − π ∗ ), and 30 are transitions to Rydberg states (π − R, n − R). A graphical representation of this analysis is presented in Figure 1. The top panel of Figure 1 shows the mean errors in case of the valence states with the respective standard deviations represented as error bars. Let us consider first the CCSD method as a “standard” choice. As already noted in several studies, 40,41 CCSD overestimates the excitation energies, for the given set this overestimation is about 0.1 eV, with a standard deviation of about the half of that. This conclusion depends only slightly on the basis set, showing that CCSD is a very balanced approximation. Compared to CCSD, all approximate triples methods reduce the mean error considerably. The CCSD(T) method seems to show the largest deviation, but even this method cuts the error approximately to half of that of CCSD. The mean errors for the other methods are hardly larger than 0.02 eV. Very promising is the new non-iterative method CCSD(T)(a)*, 34 showing practically the same accuracy as the iterative techniques. As of the approximate doubles methods, we observe larger mean errors with substantially larger error bars and non-systematic behavior with different basis sets. The mean error for CIS(D) is about 0.2 eV, with a standard deviation almost this large. Upon including diffuse basis functions the error is cut down to below 0.1 eV, but this is most probably due to error cancellation which is also suggested by the growing standard deviation: for several states the excitation energy is already underestimated when using augmented basis sets. The error of CC2 is a bit larger than that of CCSD; this is in contrast with the conclusion obtained

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for the complete set of all 28 molecules. 15 The basis set dependence of CC2 errors is quite similar to that of CIS(D): the mean error reduces when including diffuse basis functions, but at the same time the standard deviation grows. The CCSD(2) approximation performs rather strangely for valence states: the results depend very much on the basis set. The mean error is almost zero with the TZVP basis, but with large (0.12 eV) standard deviation showing sign of an error cancellation. Indeed, with the double-ζ basis set the excitation energies are underestimated by about 0.06 eV, while they become overestimated by about the same amount in the triple-ζ quality basis sets. The standard deviation is, on the other hand, similar to that of CCSD with values of 0.08-0.11 eV. Some of the unsystematic behavior of the results can be understood by studying also the Rydberg states (see the bottom panel of Figure 1). First of all, we need to make clear that, without diffuse functions in the basis set, the Rydberg states can not be described properly, one mostly gets Rydberg-like artifact states. In what follows, for the fair comparison we focus on the results calculated in the augmented basis sets. As there is usually a certain Rydberg component in the valence states as well, the underestimation of the excitation energies of Rydberg states can also lead to a basis set dependence of valence excitation energies. Therefore, for the accurate description of the valence states, it seems desired to describe also the Rydberg states in a balanced way. This requirement does not seem to be satisfied with the CIS(D) and CC2 methods: when inspecting the bottom panel of Figure 1, the very first observation is the huge underestimation of the Rydberg excitation energies by CIS(D) and CC2. CCSD(2) seems to describe Rydberg states in a balanced way though, providing similar mean errors for both types of states. This error, however, changes considerably with the basis set size. In case of CCSD, there is no substantial difference between the error associated with valence and Rydberg states, either, although in the latter case the mean error decreases somewhat when diffuse functions are included in the basis. In Figure 2 the errors of the excitation energies are plotted against the CCSDT values for

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the doubles Coupled Cluster methods, while Figure 3 shows the same representation for the performance of the triples CC techniques. The results on these figures are calculated with the cc-pVTZ and aug-cc-pVTZ basis sets and are presented separetely for the valence and Rydberg states. In this way, we are able to compare the effect of the diffuse functions in the basis sets and the performance of Coupled Cluster methods in the description of different transition types. In both basis sets, the same trends can be observed for all doubles CC methods by comparing the results of the valence states (left panels of Figure 2). The CCSD method systematically overestimates the CCSDT results with the maximum error of 0.28 eV in the cc-pVTZ and 0.19 eV in the aug-cc-pVTZ basis sets, but the results typically posses an error less than 0.1 eV. The CCSD(2) results scatter around the CCSDT values in both basis sets in the error range from -0.16 to 0.30 eV, and tend to follow the CCSD values from below. The CIS(D) and CC2 valence excitation energies are usually over the CCSD values, but this trend can change above the 7.5 eV where the Rydberg states are starting to appear in the augmented basis set (see e.g. the bottom right panel of Figure 2) and the Rydberg components in certain valence states can become significant. The mixing of transitions with different character for the states obtained with these methods could be confirmed by inspecting the EOM-vectors of the calculated excited states. In those valence states where the valence character is still dominating but Rydberg-type components are also significant, the opposite sign and magnitude of the error can even result in a smaller average deviation with respect to the CCSDT value. Inspecting the results for Rydberg states (right panels of Figure 2) we can see that the CCSD(2) and CCSD methods perform similarly for these states as for the valence ones. The maximum error for the CCSD method is 0.35 eV in the cc-pVTZ and 0.39 eV in the aug-ccpVTZ basis. These errors belong to 1 A′ states of formamide, where strong valence-Rydberg mixing is taking place. Most errors are, however, still under 0.2 eV in the cc-pVTZ and 0.1 eV in the aug-cc-pVTZ basis. Similar accuracy can be observed for the CCSD(2) method:

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the maximum error - for the same states as above - is found to be 0.40 eV and 0.38 eV in the cc-pVTZ and aug-cc-pVTZ basis sets, respectively. However, more points are found to lie outside the ±0.1 eV error range than for CCSD. Regarding the reliability of the CIS(D) and CC2 methods, it can be pointed out that the inclusion of diffuse functions largely influences the performance of these techniques: in the augmented basis, the excitation energies are mostly underestimated. The maximum deviation in the aug-cc-pVTZ basis set is -0.77 eV and -0.81 eV for CIS(D) and CC2, respectively. Nevertheless, the mean error in this basis set is -0.23 eV for the CIS(D) and -0.29 eV for the CC2 method, and also it can be seen from the figure that there are points in both the small and in the large error domains. It has to be noted that we also observed some correlation between transition type and the accuracy of CC2, as the underestimation for π − R type transitions remains in the range from 0.03 to 0.20 eV, while for the n − R type transitions it was found to be around 0.40.8 eV. A similar behavior of the CC2 method was observed for valence excited states of nucleobases. 41 Our conclusion was that the CC2 method has a tendency to underestimate the excitation energies of n − π ∗ transitions and overestimates the π − π ∗ ones with respect to the CC3 results. In the view of the present results, this behavior can be attributed to the valence-Rydberg mixing in those states, and the failure of CC2 in describing n − R type transitions. The same conclusion can be made for the CIS(D) method, which follows the trend of CC2. All triples Coupled Cluster methods, as it is clearly shown in Table 1, perform very well for Rydberg states in the augmented basis sets, their mean error being under 0.05 eV with a small deviation of up to 0.04 eV. Similar trend can be observed for Rydberg states as for valence ones. With the mean deviation of 0.01 eV, the CCSD(T)(a)* 34 is the better noniterative method, even performing better than the iterative CCSDT-3: for this latter method the mean error is as large as 0.04 eV in the aug-cc-pVDZ basis. Among the triples methods, the accuracy of the iterative CC3 method is the best with the mean error under 0.005 eV in

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case of the aug-cc-pVTZ basis set. Figure 3 shows the performance of the triples methods with the error of the excitation energies plotted against the CCSDT values. As expected, the inclusion of triples terms significantly increases the accuracy of the applied methods with respect to the doubles CC variants. It can be seen that the non-iterative triples method CCSD(T)(a)* outperforms its CCSD(T) counterpart in all cases and the accuracy of this former method is in line with the iterative ones. CCSD(T)(a)* gives even closer results to the CCSDT ones than the CCSDT-3 method.

These results seem to show CCSD(T)(a)*

being more accurate than expected in the original publication (Ref. 34), thereby making it a very attractive choice among the non-iterative variants. The maximum deviations are -0.17 eV, 0.10 eV, -0.16 eV and -0.05 eV for the CCSD(T), CCSD(T)(a)*, CCSDT-3 and CC3 methods, respectively. Surprising behavior of CC3 is that it often provides excitation energies below the CCSDT values, while the CCSDT-3 results just approach them. These tendencies are shown even more articulately with the Rydberg states, resembling the case of the CC2 method. The observed underestimation of the CCSDT energies may thus be a common property of the simplifications employed in both methods to reduce computational scaling by approximating the effect of highest order excitations.

Conclusions In this paper, we have investigated the effect of diffuse basis sets on the performance of the doubles and triples Coupled Cluster methods by calculating the excitation energies for singlet valence and Rydberg states. For the inspection of the reliability, the high level CCSDT results with basis sets up to aug-cc-pVTZ have been used as reference. For this benchmark to be possible, a smaller set of test molecules had to be used with the total number of 46 excited states. One has to keep in mind that the treated systems belong to the smaller side of the M¨ ulheim set which, due to the clear character of excited states, might cause a better absolute error for all methods than what one might experience in real life applications

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on medium sized systems. Nevertheless, this sample is expected to be representative for characterizing the relative performance of the investigated theories. The CCSD(2) and CCSD methods prove to be more systematic than the other doubles methods, providing the excitation energy for the valence and Rydberg states, as well as for the various transition types with a similar accuracy. However, in case of the CCSD(2) method, some error cancellation is taking place which shows a dependence on the cardinal number of the basis set. As of the triples methods, the CCSD(T)(a)* is clearly the better non-iterative method, performing as good as the iterative CCSDT-3 variant. The accuracy of CC3 - both its mean error and standard deviation - is found to be even better than that of CCSDT-3, making it the best of the investigated approximate triples techniques. The CIS(D) and CC2 methods perform similarly in all investigated basis sets: these approaches are sensitive to the inclusion of diffuse functions, since both methods fail to describe the Rydberg states properly. They tend to overestimate the excitation energy of valence states, and underestimate that of the Rydberg ones. Furthermore, transition type dependence can be observed in the case of these methods as the errors for n − R transitions are significantly larger than for the π − R ones. Although CIS(D) and CC2 seem to be attractive approaches to treat valence excitations, there is a rising error bar for these types of states upon inclusion of diffuse functions. This raises concerns that even minor Rydberg components in valence excited states may compromise these techniques’ reliability through the bad description of these contributions.

Acknowledgement This work has been supported by the Postdoctoral Research Program of the Hungarian Academy of Sciences (MTA Posztdoktori Program) and the Országos Tudományos Kutatási Alap (OTKA; Grant No. 104672). The authors thank Prof. John Stanton for useful discus-

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sions and access to the novel CCSD(T)(a)* method.

Supporting Information Available Optimized structures of the investigated molecules, as well as the singlet valence and Rydberg excitation energies that form the benchmark set of this study.

This material is available

free of charge via the Internet at http://pubs.acs.org/.

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(16) Szalay, P. G.; Watson, T.; Perera, A.; Lotrich, V.; Bartlett, R. J. Benchmark Studies on the Building Blocks of DNA. 3. Watson-Crick and stacked base pairs. Journal Of Physical Chemistry A 2013, 117, 3149–3157. (17) Emrich, K. An extension of the coupled cluster formalism to excited states I. Nucl. Phys. A 1981, 351, 379–396. (18) Monkhorst, H. J. Calculation of Properties with Coupled-Cluster Method. Int. J. Quantum Chem. 1977, 421–432. (19) Sekino, H.; Bartlett, R. J. A Linear Response, Coupled-Cluster Theory for ExcitationEnergy. Int. J. Quantum Chem. 1984, 255–265. (20) Koch, H.; Jørgensen, P. Coupled cluster response functions. J. Chem. Phys. 1990, 93, 3333. (21) Stanton, J. F.; Bartlett, R. J. The Equation of Motion Coupled-Cluster Method A Systematic Biorthogonal Approach to Molecular-Excitation Energies, TransitionProbabilities, and Excited-State Properties. J. Chem. Phys. 1993, 98, 7029–7039. (22) Comeau, D. C.; Bartlett, R. J. The Equation-of-Motion Coupled-Cluster Method Applications to Open-Shell and Closed-Shell Reference States. Chem. Phys. Lett. 1993, 207, 414–423. (23) Kowalski, K.; Piecuch, P. The active-space equation-of-motion coupled-cluster methods for excited electronic states: Full EOMCCSDt. J. Chem. Phys. 2001, 115, 643–651. (24) Kucharski, S. A.; Wloch, M.; Musial, M.; Bartlett, R. J. Coupled-cluster theory for excited electronic states: The full equation-of-motion coupled-cluster single, double, and triple excitation method. J. Chem. Phys. 2001, 115, 8263–8266. (25) Hirata, S.; Nooijen, M.; Bartlett, R. J. High-order determinantal equation-of-motion

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(34) Matthews, D. A.; Stanton, J. F. A New Approach to Approximate Equation-of-Motion Coupled Cluster with Triple Excitations. J. Chem. Phys. 2016, 145, 124102. (35) Christiansen, O.; Koch, H.; Jørgensen, P. Response Functions in the CC3 Iterative Triple Excitation Model. J. Chem. Phys. 1995, 103, 7429–7441. (36) Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007. (37) Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron-Affinities of the 1st-Row Atoms Revisited - Systematic Basis-Sets and Wave-Functions. J. Chem. Phys. 1992, 96, 6796–6806. (38) Szalay, P. G.; Watson, T. J., Jr.; Perera, A.; Lotrich, V. F.; Fogarasi, G.; Bartlett, R. J. Benchmark Studies on the Building Blocks of DNA: 2. Effect of Biological Environment on the Electronic Excitation Spectrum of Nucleobases. J. Phys. Chem. A 2012, 116, 8851–8860. (39) CFOUR, Coupled-Cluster techniques for Computational Chemistry, a quantumchemical program package by J.F. Stanton, J. Gauss, M.E. Harding, P.G. Szalay with contributions from A.A. Auer, R.J. Bartlett, U. Benedikt, C. Berger, D.E. Bernholdt, Y.J. Bomble, L. Cheng, O. Christiansen, M. Heckert, O. Heun, C. Huber, T.-C. Jagau, D. Jonsson, J. Jusélius, K. Klein, W.J. Lauderdale, F. Lipparini, D.A. Matthews, T. Metzroth, L.A. M¨ uck, D.P. O’Neill, D.R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiffmann, W. Schwalbach, C. Simmons, S. Stopkowicz, A. Tajti, J. Vázquez, F. Wang, J.D. Watts and the integral packages MOLECULE (J. Almlöf and P.R. Taylor), PROPS (P.R. Taylor), ABACUS (T. Helgaker, H.J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van W¨ ullen. For the current version, see http://www.cfour.de (accessed july 2016). (40) Goings, J. J.; Caricato, M.; Frisch, M. J.; Li, X. Assessment of low-scaling approxima17



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tions to the equation of motion coupled-cluster singles and doubles equations. J. Chem. Phys. 2014, 141, 164116. (41) Kánnár, D.; Szalay, P. G. Benchmarking coupled cluster methods on singlet excited states of nucleobases. J Mol Model 2014, 20, 2503.

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Figure 2: The error of excitation energies (in electron volts) with respect to the CCSDT results plotted against the CCSDT excitation energies (in eV) for the doubles Coupled Cluster methods CIS(D), CC2, CCSD(2) and CCSD. Left panels represent the results for valence states, while the right panels represent the Rydberg ones. Top panels stand for the cc-pVTZ results and the bottom panels for the aug-cc-pVTZ results. (Note here, that Rydberg states are only described properly in the aug-cc-pVTZ basis set.) Valence states, cc-pVTZ

Rydberg states, cc-pVTZ

Valence states, aug-cc-pVTZ

Rydberg states, aug-cc-pVTZ

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Figure 3: The error of the excitation energies (in electron volts) with respect to the CCSDT results is plotted against the CCSDT excitation energies (in eV) for the approximate triples Coupled-Cluster methods (CCSD(T), CCSD(T)(a)*, CCSDT-3, CC3). Left panels represent the results for valence states, while the right panels represent the Rydberg ones. Top panels stand for the cc-pVTZ results and the bottom panels stand for the aug-cc-pVTZ results. (Note here, that it only in the aug-cc-pVTZ basis sets is possible to describe the Rydbergstates properly.) Valence states, cc-pVTZ

Rydberg states, cc-pVTZ

Valence states, aug-cc-pVTZ

Rydberg states, aug-cc-pVTZ

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22

cc-pVTZ

cc-pVDZ

TZVP

Basis set

SD

mean

max. (-)

max. (+)

SD

mean

max. (-)

max. (+)

SD

mean

all valence Rydberg all valence Rydberg

all valence Rydberg all valence Rydberg all valence Rydberg all valence Rydberg


Value

0.06 0.20 -0.03 0.27 0.14 0.29

0.02 0.20 -0.10 0.27 0.16 0.26 0.51 0.51 0.48 -0.78 -0.05 -0.78

0.02 0.18 -0.08 0.27 0.13 0.29 0.45 0.41 0.45 -0.78 -0.06 -0.78

CIS(D)

0.01 0.14 -0.07 0.24 0.10 0.26

0.06 0.15 0.00 0.20 0.07 0.23 0.45 0.28 0.45 -0.54 0.02 -0.54

-0.01 0.13 -0.08 0.23 0.09 0.25 0.39 0.30 0.39 -0.55 -0.03 -0.55

CC2

0.09 0.04 0.12 0.12 0.11 0.11

-0.02 -0.06 0.00 0.12 0.11 0.12 0.30 0.20 0.30 -0.24 -0.24 -0.21

0.04 -0.002 0.06 0.11 0.12 0.09 0.27 0.27 0.24 -0.21 -0.21 -0.09

CCSD(2)

0.13 0.12 0.14 0.08 0.06 0.09

0.12 0.10 0.13 0.08 0.07 0.09 0.37 0.27 0.37 -0.01 0.03 -0.01

0.12 0.11 0.13 0.06 0.07 0.06 0.28 0.28 0.27 0.03 0.04 0.03

CCSD

0.04 0.04 0.04 0.06 0.05 0.06

0.09 0.08 0.10 0.05 0.05 0.05 0.27 0.19 0.27 0.02 0.02 0.04

0.07 0.07 0.07 0.04 0.05 0.03 0.17 0.17 0.14 0.000 0.00 0.03

CCSD(T)

0.02 0.02 0.02 0.05 0.03 0.06

0.02 0.02 0.03 0.04 0.03 0.04 0.17 0.10 0.17 -0.03 -0.01 -0.03

0.02 0.02 0.01 0.04 0.04 0.03 0.14 0.14 0.08 -0.09 -0.01 -0.09

CCSD(T)(a)*

0.04 0.02 0.05 0.03 0.02 0.03

0.04 0.02 0.05 0.03 0.02 0.04 0.15 0.08 0.15 -0.01 -0.01 0.00

0.04 0.02 0.04 0.03 0.02 0.03 0.10 0.09 0.10 -0.01 -0.01 0.00

CCSDT-3

0.01 0.00 0.01 0.02 0.02 0.01

0.01 0.01 0.01 0.01 0.02 0.01 0.05 0.03 0.05 -0.02 -0.02 -0.01

0.00 0.00 0.01 0.02 0.02 0.02 0.04 0.03 0.04 -0.03 -0.03 -0.03

CC3

Table 1: Statistical analysis on the error of excitation energies (in electron volts) calculated at different levels of CC theory in various basis sets, with respect to the CCSDT values for a total of 46 excited states: 16 singlet valence and 30 singlet Rydberg states.

Journal of Chemical Theory and Computation Page 22 of 24

aug-cc-pVTZ

aug-cc-pVDZ


23

max. (-)

max. (+)

SD

mean

max. (-)

max. (+)

SD

mean

max. (-)

max. (+)



all valence Rydberg all valence Rydberg

Value

-0.13 0.08 -0.23 0.29 0.22 0.26 0.27 0.27 0.20 -0.77 -0.68 -0.77

-0.21 0.04 -0.34 0.30 0.23 0.25 0.31 0.31 0.15 -0.76 -0.76 -0.72

0.54 0.45 0.54 -0.68 -0.05 -0.68

CIS(D)

-0.14 0.14 -0.29 0.30 0.12 0.25 0.46 0.46 0.12 -0.81 -0.08 -0.81

-0.16 0.12 -0.30 0.28 0.12 0.23 0.52 0.52 0.11 -0.72 -0.03 -0.72

0.49 0.36 0.49 -0.59 -0.07 -0.59

CC2

0.08 0.03 0.11 0.10 0.09 0.10 0.38 0.23 0.38 -0.13 -0.13 -0.03

-0.05 -0.06 -0.04 0.08 0.08 0.08 0.13 0.12 0.13 -0.20 -0.20 -0.16

0.40 0.30 0.40 -0.16 -0.16 -0.10

CCSD(2)

0.09 0.10 0.09 0.07 0.05 0.07 0.39 0.19 0.39 -0.06 -0.04 -0.06

0.06 0.09 0.05 0.05 0.04 0.05 0.20 0.17 0.20 -0.01 0.04 -0.01

0.35 0.28 0.35 -0.11 0.03 -0.11

CCSD

0.02 0.03 0.02 0.05 0.06 0.04 0.12 0.12 0.09 -0.17 -0.15 -0.17

0.05 0.07 0.05 0.04 0.04 0.03 0.16 0.16 0.11 0.00 0.00 0.02

0.14 0.13 0.14 -0.20 -0.02 -0.20

CCSD(T)

0.01 0.01 0.01 0.03 0.05 0.03 0.10 0.08 0.10 -0.14 -0.15 -0.02

0.01 0.01 0.01 0.03 0.03 0.03 0.09 0.06 0.09 -0.03 -0.02 -0.03

0.11 0.09 0.11 -0.20 -0.01 -0.20

CCSD(T)(a)*

0.02 0.02 0.02 0.04 0.02 0.04 0.09 0.07 0.09 -0.16 -0.01 -0.16

0.03 0.02 0.04 0.03 0.02 0.03 0.09 0.06 0.09 -0.01 -0.01 -0.01

0.11 0.08 0.11 -0.01 -0.01 -0.01

CCSDT-3

0.00 0.00 0.00 0.02 0.02 0.02 0.03 0.03 0.02 -0.05 -0.04 -0.05

0.01 0.00 0.01 0.02 0.02 0.01 0.03 0.03 0.03 -0.03 -0.03 -0.03

0.04 0.03 0.04 -0.03 -0.03 -0.02

CC3

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Basis set

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Er r orofRy dbe r g e xc i t a t i one ne r gi e s


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Accuracy of Coupled Cluster Excitation Energies in ... - ACS Publications

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