Coupled-Cluster Hierarchy

From these expressions, it is evident that the steep increase of the computational cost with X (the basis-set hierarchy) and m (the coupled-cluster hierarchy) severely restricts the levels of theory that can be routinely used for large systems or even explored for small systems. Nevertheless, we shall see that, with current computers, it is possible to arrange the calculations in such a manner that chemical accuracy (of the order of 1 kcal/mol 4 kJ/mol) can be achieved - at least for molecules containing not more than ten first-row atoms. 

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For small basis sets and molecules, it is possible to calculate the full set of energies in the coupled-cluster hierarchy, from the Hartree-Fock to the full configuration-interaction (FCI) energy [18]. Although such... 

Table 1.2 Convergence of the coupled-cluster hierarchy for the valence-shell electronic energies (Ee) and equilibrium AEs of N2 and HF in the cc-pVDZ basis. The energies are given in kJ/mol and are relative to the FCI values.

Let us make two more observations about the convergence of the coupled-cluster hierarchy. First, it converges faster for HF than for N2, reflecting the more complicated electronic structure of the multiple bond in N2. Second, the cancellation of errors in the calculated AEs becomes less pronounced as we move up in the coupled-cluster hierarchy. The cancellation diminishes since the coupled-cluster expansions of the atoms... 

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. 

The CCS, CC2, CCSD, CC3 hierarchy has been designed specially for the calculation of frequency-dependent properties. In this hierarchy, a systematic improvement in the description of the dynamic electron correlation is obtained at each level. For example, comparing CCS, CC2, CCSD, CC3 with FCI singlet and triplet excitation energies showed that the errors decreased by about a factor 3 at each level in the coupled cluster hierarchy [18]. The CC3 error was as small as 0.016 eV and the accuracy of the CC3 excitation energies was comparable to the one of the CCSDT model [18]. 

In the usual coupled-cluster hierarchy, MP2 energy functional, while linearized coupled-cluster single-doubles (L-... 

The hierarchy of coupled-cluster models provides a clear route towards the exact solution of the Schrodinger equation, but the slow basis-set convergence limits the accuracy sometimes even for small molecules. The way to overcome this problem is to combine the coupled-cluster model with the explicitly correlated approach. It can be done, in principle, for any model within the coupled-cluster hierarchy. The main task of this work is, however, the implementation of explicitly correlated CCSD model, hence the discussion will be focused on this particular model. 

We now have at our disposal two different approaches to the evaluation of electronic wave functions and energies, both developed within the framework of the exponential cluster ansatz the coupled-cluster hierarchy of Chapter 13 and the Mpller-Plesset hierarchy presented in this chapter. In... 

The calculations presented in this section have shown that, whereas improvements in the correlation treatment in general increase the bond length, improvements in the one-electron treatment reduce it. The coupled-cluster hierarchy converges smoothly, with a reduction in the error by a factor of 3-5 at each level. Thus, whereas the intrinsic error (based on A and Aabs in Figure 15.3) at the Hartree-Fock level is about —3 pm, it is reduced to about —0.8 pm at the CCSD level and to about —0.2 pm at the CCSD(T) level. In contrast, the Mpller-Plesset hierarchy oscillates, with typical errors of about 0.5 pm at the MP2 level, 1.2 pm at the MP3 level and 0.4 pm at the MP4 level. Even for the small molecules in this study, the CISD model is inadequate, with an intrinsic error of —1.7 pm. Based on these observations, we may state that, in terms of computational accuracy and cost, the only successful models are Hartree-Fock, MP2, CCSD and CCSD(T). 

Let us examine the convergence of the coupled-cluster hierarchy towards the FCI wave function. In Table 5.11, we have — for the water molecule in the cc-pVDZ basis — listed the differences between the (spin- and space-restricted) CCSD, CCSDT and CCSDTQ energies and the FCI eneigy [3]. Comparing with the Cl energies in Table 5.9, we find that the coupled-cluster hierarchy converges faster and more uniformly towards the FCI limit. In particular, there is a significant reduction in the error at each level of truncation - not just at even-order levels as in Cl theory. 

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