Unlinked coupled-cluster equations

Clearly, the fulfilment of conditions (13.3.26) and (13.3.27) for the noninteracting systems A and B implies the fulfilment of conditions (13.3.28)-(13.3.30) for the compound system - as expected for a size-extensive computational model. Note that, in our demonstration of size-extensivity in the unlinked formulation, no assumptions were made about the projection space. Thus, although the linked and unlinked coupled-cluster equations may give different solutions when the projection space is not closed under de-excitations, both solutions are size-extensive. 

In conclusion, in the unlinked formulation of coupled-cluster theory, the amplitude equations yield solutions that are size-extensive but not termwise so. In general, therefore, it is not trivial to make size-extensive approximations to the unlinked coupled-cluster equations and the linked equations are to be preferred for such purposes. 

In this exercise, we compare the linked and unlinked coupled-cluster equations with the FCI eigenvalue problem for the Hz molecule in a minimal basis, containing the gerade (g) and ungerade (u) MOs. 

Show that the linked and unlinked coupled-cluster equations are equivalent... 

Invoking the resolution of the identity, we may rewrite the unlinked coupled-cluster equations (13E.6.3) and (13E.6.4) in the form... 

Let us now compare the coupled-cluster equations in the linked and unlinked forms. We b n by reiterating that these two forms of the coupled-cluster equations are equivalent for the standard models in the sense that they have the same solutions. Moreover, applied at the important CCSD level of theory, neither form is superior to the other, requiring about the same number of floating-point operations. The energy-dependent unlinked form (13.2.19) exhibits mcMe closely the relationship with Cl theory, where the projected equations may be written in a similar form (13.1.18). On the other hand, the linked form (13.2.23) has some important advantages over the unlinked one (13.2.19), making it the preferred form in most situations. 

In the present subsection, we have established that the coupled-cluster nnodel is size-extensive - at least in the similarity-transformed, linked formulation of the theory. Note that nothing has been assumed about the nature of the cluster operators for the subsystems - size-exlensivity occurs irrespective of how we truncate these operators and what excitation operators are left out In Section 13.3.3, we shall see that the coupled-cluster state is also size-extensive in the unlinked formulation of the coupled-cluster equations. 

In Section 13.3.1, size-extensivity was demonstrated for the linked formulation of coupled-cluster theory. In this subsection, we consido- size-extensivity in the alternative, unlinked formulation of the theory, thereby establishing size-extensivity also in those (rare) cases where the linked and unlinked formulations differ. More important, the discussion in the present subsection illustrates that, in unlinked coupled-cluster theory, size-extensivity arises from a cancellation of contributions that individually violate size-extensivity. This behaviour is in contrast to that of linked coupled-cluster theory, where size-extensivity occurs separately for all commutators that contribute to the equations as discussed in Section 13.3.2. 

The unlinked coupled-cluster Schrodinger equation is given by... 

For the full coupled-cluster wave function, the equivalence of the equations (13.2.16) and (13.2.20) is trivial for truncated cluster expansions, on the other hand, the equivalence of the linked and unlinked forms of the amplitude equations is less obvious and requires special attention. First, the equivalence of the energy expressions (13.2.18) and (13.2.22) is easily established since for any choice of amplitudes... 

Chapter 13 discusses coupled-cluster theory. Important concepts such as connected and disconnected clusters, the exponential ansatz, and size-extensivity are discussed the Unked and unlinked equations of coupled-clustCT theory are compared and the optimization of the wave function is described. Brueckner theory and orbital-optimized coupled-cluster theory are also discussed, as are the coupled-cluster variational Lagrangian and the equation-of-motion coupled-cluster model. A large section is devoted to the coupled-cluster singles-and-doubles (CCSD) model, whose working equations are derived in detail. A discussion of a spin-restricted open-shell formalism concludes the chapter. 

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