The coupled-cluster model

As noted in Section 5.6, the Cl hierarchy of wave functions suffers from two serious shortcomings the lack of size-extensivity and the slow convergence towards the FCI limit. We shall now see how these problems are overcome through the introduction of the coupled-cluster model, which represents a nonlinear but manifestly separable parametrization of the correlated electronic state. The coupled-cluster model constitutes a particularly successful approach to electronic-structure theory, providing for many purposes - often in combination with perturbation theory - the most efficient strategy for the accurate calculation of electronic energies and wave functions. 

The lack of size-extensivity of the truncated d wave function arises from the use of the linear parametrization (5.6.1). To impose size-extensivity on our description, we recast the linear FCI expansion (5.6.5) in the form of a product wave function  

This coupled-cluster wave function is manifestly separable (see Section 4.3.4) and differs from the Cl wave function (5.6.5) by the presence of terms that are nonlinear in the excitation operators. Since the excitation operators such as (5.6.6) and (5.6.7) commute with one another, there is no problem with the wder of the operators in the product wave function (5.7.1). 

It should be noted that the Cl and coupled-cluster wave functions (5.6.5) and (5.7.1) are entirely equivalent provided all excitations are included in the expressions, differing only in 

The transition from a linear model for the wave function to a product model shifts the emphasis away from excitation levels and excited determinants towards excitations and excitation processes. Thus, applied to some electronic state or configuration 0 , each operator in (5.7.1) produces a superposition of the original state and a correction term that represents an excitation from the original state  

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An implementation of the cubic response function Eq. (30) for the coupled cluster model hierachy CCS, CC2 and CCSD was reported in Ref. [24]. 

The Cauchy moments are derived and implemented for the approximate triples model CC3 with the proper N scaling (where N denotes the number of basis functions). The Cauchy moments are calculated for the Ne, Ar, and Kr atoms using the hierarchy of the coupled-cluster models CCS, CC2, CCSD, CC3 and a large correlation-consistent basis sets augmented with diffuse functions. A detailed investigation of the one- and A-electron errors shows that the CC3 results have the accuracy comparable to the experimental results. 

A. Kohn and C. Hattig, Analytic gradients for excited states in the coupled-cluster model CC2 employing the resolution-of-the-identity approximation, J. Chem. Phys., 119 (2003) 5021-5036. 

A different study provides some further information coneeming the effects of electron correlation upon calculated intensities. In addition to SCF and MPn type methods, these authors considered also the coupled-cluster model which is an infinite-order generalization of MR Coupled-cluster, limited to single and double excitations (CCSD), was considered, as well as CCSD + T which includes the effects of connected triple excitations. The doubleharmonic approximation was made, consistent with most calculations of H-bonded systems. The [5s3pld/3slp] basis set is of the polarized double- variety, within reach of most workers. 

We would like to stress that this chapter is a review of coupled cluster theory. It is not primarily intended to provide an analysis of the numerical performance of the coupled cluster model, and we direct readers in search of such information to several recent publications. " Instead, we offer a detailed explanation of the most important aspects of coupled cluster theory at a level appropriate for the general computational chemistry community. Although many of the topics described here have been discussed by other au-thors, ° this chapter is unique in that it attempts to provide a concise, practical introduction to the mathematical techniques of coupled cluster theory (both algebraic and diagrammatic), as well as a discussion of the efficient... 

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. 

States in the Coupled-Cluster Model CC2 Employing the Resolution-of-the-Identity Approximation. 

For a laige system with many degrees of freedom, such a task is an expensive undertaking. A similar situation arises for any other nonvariational computational model. In the coupled-cluster model, for example, we would need to calculate the partial derivatives of the cluster amplitudes and the orbital-rotation parameters with respect to all perturbations of interest. Clearly, to make the calculation of gradients practical for such wavefunctions, we must come up with a better scheme for the evaluation of molecular gradients. 

The purpose of the present section is to introduce the coupled-cluster model. First, in Section 13.1.1, we consider the description of virtual excitation processes and correlated electronic states by means of pair clusters. Next, in Section 13.1.2, we introduce the coupled-cluster model as a generalization of the concept of pair clusters. After a discussion of connected and disconnected clusters in Section 13.1.3, we consider the conditions for the optimized coupled-cluster state in Section 13.1.4. 

Since the excitation operators (13.1.2) commute among one another, there are no problems with the order of the operators in this expression. The resulting wave function (13.1.5) corresponds to a particularly simple realization of the coupled-cluster model, in which only double excitations are allowed the coupled-cluster doubles (CCD) wave function. For a complete specification of this model, we must also describe the method by which the amplitudes are determined. We shall... 

Several points should be noted about the form of this wave function. First, the coupled-cluster state is manifestly in a product form, leading to a size-extensive treatment of the electronic system as discussed in Sections 4.3 and 13.3. In this respect, the coupled-cluster model differs fundamentally from the linear Cl model, for which the corresponding wave function is not in a product form ... 

By contrast, the nonlinear parametrization of the coupled-cluster model (13.1.7) means that the derivatives of the coupled-cluster state become complicated functions of the amplitudes... 

As discussed in Section 4.3. a computational method is said to be size-extensive if a calculation on the compound system AB consisting of two noninteracting systems A and B yields a total energy equal to the sum of the energies obtained in separate calculations on the two subsystems. This property of the coupled-cluster model is demonstrated for the linked formulation in Section 13.3.1, leading to the concept of termwise size-extensiAdty in Section 13.3.2. In Section 13.3.3, we consider size-extensivity in the unlinked formulation of coupled-cluster theory, demonstrating how size-extensivity in this case arises from a cancellation of terms that individually violate size-extensivity. 

In Section 14.6, we examine the relationship between coupled-cluster theory and perturbation theory. After a perturbation analysis of the coupled-cluster models, we consider various hybrid methods, in which perturbational corrections are applied (iteratively and noniteratively) within the framework of coupled-cluster theory. In particular, we apply a triples correction to the CCSD energy, arriving at the highly successful CX SD(T) approximation to the FCl electronic energy. 

Table 14.5 The order in the fluctuation potential to which the coupled-cluster model is correct. In general, the wave function of order N is correct to order N — I and the energy to order 3N/2]...

It should be emphasized that an order analysis of the coupled-eluster model can serve only as a crude indication of the overall quality of the coupled-cluster wave funetion and energy. If anything, we would expeet the coupled-cluster model to be more accurate and more flexible than indicated by a strict perturbation analysis. This expectation is based on the observation that, for example. 

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. 

We now turn our attention to the first standard model that incorporates the effects of dynamical correlation the configuration-interaction (Cl) model. This model, which arises naturally in the MO picture as the superposition of determinants, has been quite successful in molecular electronic applications but has more recently been superseded by the coupled-cluster model as a computational tcx)l of quantum chemistry, at least in the more common areas of application. 

The coupled-cluster model represents a significant improvement on the truncated Cl model in that it provides a description of the electronic structure that is both size-extensive and noore compact. On the other hand, it has proved difficult to extend the application of coupled-cluster theory to... 

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