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converge faster than those of the molecules, which contain more electron pairs.

Although the calculations reported here have been carried out in a small basis, there is no reason to believe that our conclusions regarding the convergence of the coupled-cluster hierarchy would be different had the calculations been carried out in larger basis. In particular, we conclude that the CCSDT model is incapable of predicting AEs to within 1 kJ/mol.

3.2.The CCSD(T) Model

As shown in the previous section, the coupled-cluster hierarchy converges rapidly, the error in the total error being reduced by an order of magnitude at each new level of theory. Unfortunately, from section 2.3, we recall that the cost of the coupled-cluster calculations increases very rapidly with the inclusion of higher-order connected excitations. In practice, while it is possible to carry out CCSD calculations for fairly large systems and basis sets (more than 10 atoms at the cc-pCVQZ level), the full CCSDT model is presently too expensive for routine calculations. However, since we are anyway forced to neglect the connected quadruples (CCSDTQ) in our calculations, the overall quality of our calculations will not be adversely affected if we make an approximation in the treatment of the connected triples whose error is not larger than that incurred by neglecting the quadruples. In practice, therefore, any approximate treatment of the triples that gives an error of the order of 10 % or less would be welcome.

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 in CCSDT [see Eq. (2.5)] to in CCSD(T), the CCSD(T) method avoids completely the storage of the triples amplitudes.

In Table 1.3, we have listed the contributions from the single and connected double and triple excitations to the AEs of CO,

HF, and at the valence-electron CCSDT/cc-pV5Z and CCSD(T)/ccpV5Z levels [20]. The second column of Table 1.3 contains the CCSD singles and doubles contributions to the correlation energy, the third column the triples contributions as obtained in the CCSD(T) method, and the last column the difference between the triples contributions in CCSDT and CCSD(T) – that is, the energy contribution that originates from the full relaxation of the triples. The error incurred by employing

8

Chapter 1

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.

However, the success of the CCSD(T) model stems not only from the fact that it gives a good approximation to the full triples correction. From Table 1.3, we note that the CCSD(T) model usually overestimates the contributions from the triples, the only exception being This overestimation is particularly significant for where the CCSD(T) triples correction is 3 kJ/mol larger than the full triples correction. For comparison, it is seen from Table 1.2 that, in the cc-pVDZ basis, the connected quadruple excitations add 3.55 kJ/mol to the AE. The overestimation of the triples contribution by the CCSD(T) model will thus partly cancel the error incurred by ignoring the connected quadruples. In general, therefore, we may expect that the CCSD(T) AEs will not be improved by going to the full CCSDT method [20]. In this sense, the CCSD(T) model represents a very accurate method for the calculation of AEs, which may only be improved upon by simultaneously including more terms from the connected triples as well as contributions from the connected quadruples [20-22].

4.AN ILLUSTRATIVE EXAMPLE:

THE ATOMIZATION ENERGY OF CO

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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and the C and O atoms at 0 K. The experimental AE of CO is known to be 1071.8(5) kJ/mol [23, 24]; in Table 1.4, we have collected the various contributions to the theoretical AE of CO, as calculated at the CCSD(T) level in the limit of a complete one-electron basis. Note that the calculated AE of 1072.0 kJ/mol is within the experimental error bars, even though it constitutes less than 0.5 % of the total energy of the CO molecule.

4.1.Electronic and Nuclear Contributions

Let us discuss the various contributions to the calculated AE of CO. The first row of Table 1.4 contains the energies obtained from separate Hartree-Fock calculations on CO and its constituents. The large error of 32 % arises since the Hartree-Fock model is incapable of describing the complicated changes in the electronic structure that occur as electron pairs are broken. Although we might hope that the errors associated with the breaking of electron pairs to some extent cancel in the enthalpies of isogyric reactions (i.e., reactions that conserve the number of electron pairs), we clearly need to go beyond the one-determinant Hartree-Fock description for a satisfactory theoretical prediction of AEs.

10	Chapter 1
In the second row of Table 1.4,	we have listed the corrections to

the Hartree-Fock energies that are obtained from CCSD calculations. Clearly, we now have a better description of the atomization process, the error in the calculated AE being only -19.6 kJ/mol (2 %). Still, we are far away from the prescribed target accuracy of 1 kJ/mol.

To improve on the CCSD description, we go to the next level of coupled-cluster theory, including corrections from triple excitations – see the third row of Table 1.4, where we have listed the triples corrections to the energies as obtained at the CCSD(T) level. The triples corrections to the molecular and atomic energies are almost two orders of magnitude smaller than the singles and doubles corrections. However, for the triples, there is less cancellation between the corrections to the molecule and its atoms than for the doubles. The total triples correction to the AE is therefore only one order of magnitude smaller than the singles and doubles corrections.

With the triples correction added, the error relative to experiment is still as large as 15 kJ/mol. More importantly, we are now above experiment and it is reasonable to assume that the inclusion of higher-order excitations (in particular quadruples) would increase this discrepancy even further, perhaps by a few kJ/mol (judging from the differences between the doubles and triples corrections). Extending the coupledcluster expansion to infinite order, we would eventually reach the exact solution to the nonrelativistic clamped-nuclei electronic Schrödinger equation, with an error of a little more than 15 kJ/mol. Clearly, for agreement with experiment, we must also take into account the effects of nuclear motion and relativity.

From Table 1.4, we note that the zero-point vibrational energy (ZPVE) correction is large and negative, reducing the error at the CCSD(T) level to only 2.2 kJ/mol. A further inclusion of the firstorder relativistic correction brings the error down to only 0.2 kJ/mol, an excellent result. However, before we become too enthusiastic about this result, it should be pointed out that the error in the perturbative treatment of the triples correction in CCSD(T) is quite large (2 kJ/mol, see Table 1.2), partly cancelling the error that arises from the neglect of quadruple and higher excitations. In addition, there are unknown (but probably small) non-Born-Oppenheimer corrections. In conclusion, it seems possible to calculate AEs to an accuracy of 1 - 2 kJ/mol, but very difficult to reduce it further. We shall later present a statistical analysis (based on more molecules) that confirms this tentative conclusion.