The CCSD T Model

Among the various approximate methods for including the connected triple excitations, the CCSD(T) method is the most popular [19]. In this approach, the CCSD calculation is followed by the calculation of a perturbational estimate of the triple excitations. In addition to reducing the overall scaling with respect to the number of atoms K from K8 in CCSDT [see Eq. (2.5)] to K7 in CCSD(T), the CCSD(T) method avoids completely the storage of the triples amplitudes. 

CCSD(T) instead of CCSDT amounts to no more than 10 % of the total triples correction and 1 % of the total correlation energy, thus fulfilling our requirement for an acceptable approximate triples theory. 

To illustrate the difficulties associated with the accurate calculation of thermochemical data, we here consider the calculation of the AE of CO - that is, the difference in total energy between the CO molecule 

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The calculated AEs are very accurate, with typical errors of about 1 kJ/mol and errors larger than 2.3 kJ/mol occurring only for 03 (-10.7 kJ/mol) and HOF (-12.0 kJ/mol). For 03, the difference probably arises from an error in the calculation as the high accuracy of the CCSD(T) model does not extend to systems that are poorly represented by a single determinant. For the single-determinant HOF molecule, the discrepancy is most likely caused by an error in the tabulated value derived from experimental data. 

The prerequisites for high accuracy are coupled-cluster calculations with the inclusion of connected triples [e.g., CCSD(T)], either in conjunction with R12 theory or with correlation-consistent basis sets of at least quadruple-zeta quality followed by extrapolation. In addition, harmonic vibrational corrections must always be included. For small molecules, such as those contained in Table 1.11, such calculations have errors of the order of a few kJ/mol. To reduce the error below 1 kJ/mol, connected quadruples must be taken into account, together with anhar-monic vibrational and first-order relativistic corrections. In practice, the approximate treatment of connected triples in the CCSD(T) model introduces an error (relative to CCSDT) that often tends to cancel the... 

Of all the methods currently used in molecular electronic structure theory, the CCSD(T) model is probably the most successful, highly accurate level, at least for closed-shell molecular systems. For many properties of interest to chemists such as molecular structure, atomization energies, and vibrational frequencies, it provides numerical data of consistently high quality, sometimes surpassing that of experiment. Nevertheless, it does fail in certain cases, in particular for systems characterized by several important Slater determinants and also for certain properties such as indirect nuclear spin-spin couplings of magnetic resonance spectroscopy. 

For small molecules it is possible to perform accurate computations with post-Hartre—Fock approaches, and in this respect harmonic frequencies computed at the CCSD(T) (coupled clusters with single, double, and perturbative inclusion of triple excitation [27]) level, with basis sets of at least triple- quality reach an overall accuracy of 15—20cm for closed-shell systems [e.g., 28, 29]. For radicals, the situation is not so well assessed, but some recent investigations confirm that analogous accuracy can be reached [e.g., 30-34]. However, the unfavorable scaling of the CCSD(T) model with the number of active electrons limits its applicability to very small systems only. In addition, a simple reduction of the computational cost by combining correlated QM methods with small basis sets is not to be recommended due to the quite unpredictable accuracy of the results. Thus, the extension of computational studies to large systems requires cheaper and at the same time reliable electronic structure models. 

At the cc-pVTZ level, two models stand out from the others the MP2 model with a mean error of —0.12 pm and the CCSD(T) model with a mean error of -t-0.01 pm. Again, there is a certain element of cancellation of errors in the calculations since, at the cc-pVQZ level, the MP2... 

The performance of the CCSD model is likewise disappointing, being intermediate between MP2 and MP3. Clearly, the CCSD model is not well suited to the calculation of bond distances - only with the inclusion of the triples at the CCSDfT) level does the coupled-cluster model yield satisfactory results. Indeed, at the cc-pVTZ and cc-pVQZ levels, the CCSD(T) model performs excellently, with sharply peaked distributions close to the origin. 

At the cc-pVQZ level, the CCSD(T) bond distances differ from the experimental ones by just a few tenths of a picometer. To see if the errors can be reduced further, we here consider calculations where, in the cc-pVQZ basis, the correlation treatment is extended to the CCSDT model and where, for the CCSD(T) model, the basis is extended to cc-pV5Z [4]. Since these calculations are demanding, they have been carried out for only six molecules - see Table 15.5. We first note that, even though the bond lengths in general shorten from cc-pVQZ to cc-pV5Z, there are exceptions to this rule - in the HF molecule, for example, the bond distance lengthens. 

The CCSDT and CCSD(T) bond lengths are very similar - the largest difference being 0.06 pm for N2 in the cc-pVQZ basis. Except for the small cc-pVDZ basis, the CCSD(T) bonds are always longer than the CCSDT bonds. The effect of the triples is thus overestimated by the CCSD(T) model - by as little as 0.4% in HF and as much as 9.6% in N2. In general, however, the CCSD(T) model provides a good approximate description of the effect of triple excitations on bond distances, at least for molecules containing only first-row atoms. 

In Table 15.7, we have collected the equilibrium bond distances of the molecules in Table 15.1, calculated using the core-valence cc-pCVQZ basis set, correlating all the electrons in the system. For the CCSD(T) model, a comparison with experiment shows that, for 22 of the 29 bond lengths, the difference is less than or equal to 0.1 pm. This error is less than the intrinsic error of the CCSD(T) model and arises from a cancellation of errors. 

These similarities notwithstanding, some differences exist between bond distances and bond angles. Thus, for the bond angles, the differences among the correlated models are smaller than for bond distances. The CCSD and MP3 plots, for instance, are now almost indistinguishable. More important, for bond angles, the MP2 model appears to perform as well as the CCSD(T) model. However, since the experimental uncertainties are typically 0.3". their performances cannot be distinguished. 

At the cc-pVQZ level, the largest CCSD(T) error occurs for the O3 bond angle, which is overestimated by 0.4°. Although this error is not much larger than the experimental uncertainty (0.3°), it is probably genuine and related to the multiconfigurational electronic structure of O3 -indeed, a priori, we would expect the CCSD(T) model to exhibit the largest error for O3. The fact that this does not happen for cc-pVTZ may perhaps be taken as an indication that the CCSD(T) calculations are not fully balanced at the cc-pVTZ level. 

From Figure 15.8, we see that the calculated dipole moments are more sensitive to the choice of the A -electron model than to the choice of cardinal number for the basis set. Even in the smallest basis, the CCSD(T) model is considerably more accurate than the CCSD model. Thus, for the CCSD(T) model, the smallest basis (aug-cc-pVDZ) gives a mean error of —0.006 D and the largest basis (aug-cc-pVQZ) a mean error of -1-0.006 D, the corresponding mean absolute errors being 0.02 and 0.008 D. In comparison, for the CCSD model, the mean errors are 0.02 and 0.03 D at the aug-cc-pVDZ and aug-cc-pVQZ levels, respectively, and the mean absolute errors 0.04 D in both cases. Whereas, at the CCSD and CCSD(T) levels, the dipole moment increases with the cardinal number, at the Hartree-Fock level, the change is in the opposite direction. The performance of the MP2 model appears to be less systematic. 

In Table 15.17, we have listed the valence-electron CCSDT energies for the six selected molecules. The main point of interest is the difference between the CCSDT and CCSD(T) models. In most cases, the CCSDT model changes the energy only very little relative to the CCSD(T) model. The largest change occurs for the CH2 molecule, where the CCSDT model reduces the energy by about 0.8 mEi, and where the CCSD(T) triples correction constitutes only 88% of the total CCSDT triples correction. 

Having discussed the CCSD(T) model at some length, let us briefly consider the simpler MP2 and CCSD models. In relative terms, these models do not behave badly at all. As seen from Figure 15.11 and Table 15.21, the basis-set dependence is much like that of CCSDfT). except that MP2 overestimates the atomization energy by 3% while CCSD underestimates it by about 5%. Unfortunately, in absolute terms, these errors translate to 30-40 kJ/mol, making the calculations rather useless except in preliminary or exploratory investigations. As a curiosity, we note that, at the cc-pCVTZ level, the MP2 model is very accurate in the mean, with an emra of only —4.7 kJ/mol. However, this small error arises from a rather unsystematic cancellation of errors in the one- and A-electron treatments, as evidenced by the standard deviation of 30 kJ/mol and the mean absolute error of 23 kJ/mol. For the same reason, in the cc-pCVTZ basis, the largest MP2 error occurs for CO2 rather than fOT O3 as in the CCSD and CCSD(T) calculations. 

For large basis sets, the CCSD(T) model gives atomization energies close to the experimental values, particularly when extrapolation is carried out for the correlation part - see Table 15.23. The cc-pCV(56)Z extrapolation, for example, gives a mean error and a standard deviation of —0.9 and 2.8 kJ/mol, respectively. It is thus of some interest to examine what happens when the triples treatment is extended to the full CCSDT model. Since such calculations are demanding, they have been performed only for the six molecules in Table 15.27. For details of the calculations, we refer to Section 15.6.4. 

The CCSDT and CCSD(T) atomization energies are similar, the largest difference being 2.7 kJ/mol for the cc-pCVQZ calculation on N2. At the CCSD(T) level, the triples correction is underestimated by 11% for CH2 but overestimated for the other molecules, by as little as 3.2% for F2 and as much as 7.1% for N2. In general, therefore, the CCSD(T) model provides a useful approximate treatment of the connected triples for atomization eneigies. The difference between the CCSDT and CCSD(T) atomization energies is quite stable with respect to the cardinal number... 

For chemical accuracy - that is, for errors smaller than 5 kJ/mol - we must take into account the effects of the connected triples. As can he seen in Figure 15.15, the inclusion of connected triple excitations at the CCSD(T) level reduces the exothermicity of the reactions relative to the MP2 and CCSD levels. However, for quantitative agreement with experiment, basis sets of at least quadruple-zeta quality are needed. Thus, at the all-electron cc-pCVQZ level, the CCSD(T) model gives a mean error and a standard deviation of 2.7 and 3.2 kJ/mol, respectively, and the relative errors are 1 -2%. As the basis increases, the exothermicity increases and, at the cc-pcV6Z level, the mean error and the standard deviation are —0.7 and 3.1 kJ/mol, respectively. 

Although the cc-pCV(DT)Z enthalpies are considerably more accurate than the cc-pCVTZ enthalpies, beyond the quadruple-zeta level, there is little or no improvement upon extrapolation. This occurs since the errors in the description of the short-range correlation of the closed-shell products and reactants cancel even for rather small basis sets. Thus, for cardinal numbers greater than four, the main errors in the correlation treatment arise no longer from the basis set but from the CCSD(T) model itself. We note that the intrinsic error of the CCSD(T) model is similar for... 

However, to achieve this goal, we shall probably have to find different solutions to the basis-set problem so as to reduce the size of the basis needed for convergence to the basis-set limit. The obvious candidates here are the ejq>licitly correlated methods, elements of which must be introduced into the standard models in a black-box manner, to enable the user to approach the basis-set limit of, for example, the CCSD(T) model at a cost smaller than presently possible. Again, there is enough work going on in this direction for the prospects to appear bright and exciting. 

For OH distances shorter than 3.5oo the CCSD(T) model works well, giving about 90% of the full CCSDT triples correction. However, the model breaks down at longer distances. Also, the unrestricted CCSD(T) model does not provide a unifr m description of the dissociation process, performing not much better than the simpler unrestricted CCSD model in the intermediate region. Hybrid methods are considered in Chapter 14, following a general discussion of perturbation theory. 

When the Hartree-Fock description is reasonably accurate - as for the stable water molecule -the restricted CCSD model appears to provide a satisfactory representation of the FCI wave function. Although less accurate than CCSDT, it represents a useful compromise between accuracy and cost and can be routinely applied to relatively large molecular systems in a black-box manner. The excellent performance of the CCSDT model is quite remarkable but its cost is so high that it is applicable only to small systems. For most purposes, however, the CCSDT model may be replaced by the CCSD(T) model at little or no loss of accuracy, making it possible to carry out highly accurate calculations on systems containing up to ten atoms. 

Fig. 8.24. The interaction energy of the neon dimer (in pEh) plotted as a function of the intemuclear separation (in oq). On the left, we have plotted the interaction energies calculated at the Hartree-Fock level in a complete basis (lull line) and using the d-aug-cc-pVXZ sets with X < 4 (with the number of consecutive dots indicating the cardinal number). No counterpoise correction has been applied to any of the Hartree-Fock curves. On the right, we have plotted the (counterpoise-corrected) interaction energies for different ab initio models at the frozen-core d-aug-cc-pV5Z level the Hartree-Fock model (full line), the MP2 model (longer dashes), the CCSD model (shorter dashes) and the CCSD(T) model (dots). The thick grey line represents the potential-energy curve extracted from experiment [29].

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