Linked coupled-cluster equations

Show that the linked coupled-cluster equations can be expressed as... 

In this section we examine the fundamental relationship between many-body perturbation theory (MBPT) and coupled cluster theory. As originally pointed out by Bartlett, this connection allows one to construct finite-order perturbation theory energies and wavefunctions via iterations of the coupled cluster equations. The essential aspects of MBPT have been discussed in Volume 5 of Reviews in Computational Chemistry,as well as in numerous other texts. We therefore only summarize the main points of MBPT and focus on its intimate link to coupled cluster theory, as well as how MBPT can be used to construct energy corrections for higher order cluster operators such as the popular (T) correction for connected triple excitations. 

Coupled electron pair and cluster expansions. - The linked diagram theorem of many-body perturbation theory and the connected cluster structure of the exact wave function was first established by Hubbard211 in 1958 and exploited in the context of the nuclear correlation problem by Coester212 and by Coester and Kummel.213 Cizek214-216 described the first systematic application to molecular systems and Paldus et al.217 described the first ab initio application. The analysis of the coupled cluster equations in terms of the many-body perturbation theory for closed-shell molecular systems is well understood and has been described by a number of authors.9-11,67,69,218-221 In 1992, Paldus221 summarized the situtation for open-shell systems one must nonetheless admit... 

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. 

Having examined the coupled-cluster energy and seen that it is no higher than quadratic in the cluster amplitudes, let us now turn our attention to the structure of the linked projected coupled-cluster equations (13.2.23) ... 

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. 

The use of a similarity-transformed Hamiltonian in linked coupled-cluster theory means that the energy and amplitude equations contain terms that consist either of the Hamiltonian itself or of nested commutators of the Hamiltonian with cluster operators. For a system containing two noninteracting subsystems A and B, these nested commutators separate additively into nested commutators, each involving a single subsystem, for example. 

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. 

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. 

One popular modification of the standard coupled-cluster model is the quadratic configuration-interaction (QCI) model, originally introduced as a size-extensive amendment of the Cl model [33]. We here discuss the QCI singles-and-doubles (QCISD) model within the framework of similarity-transformed (linked) coupled-cluster theory, from which it is obtained by omitting certain commutators in the CCSD equations. Expanding the remaining commutators, we then go on to express the QCISD equations in a form that illustrates its historical connection to CISD theory. 

In the linked form, the truncated coupled-cluster equations corresponding to the fill] equations (13.9.1)-(13.9.3) may now be written as... 

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... 

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

The coupled cluster (CC) method is actually related to both the perturbation (Section 5.4.2) and the Cl approaches (Section 5.4.3). Like perturbation theory, CC theory is connected to the linked cluster theorem (linked diagram theorem) [101], which proves that MP calculations are size-consistent (see below). Like standard Cl it expresses the correlated wavefunction as a sum of the HF ground state determinant and determinants representing the promotion of electrons from this into virtual MOs. As with the Mpller-Plesset equations, the derivation of the CC equations is complicated. The basic idea is to express the correlated wave-function Tasa sum of determinants by allowing a series of operators 7), 73,... to act on the HF wavefunction ... 

These two equations together are the basic equations of non-degenerate (singlereference) Brillouin-Wigner coupled cluster (Bwcc) theory. We emphasize that these equations are obtained directly from Brillouin-Wigner perturbation expansion. In particular, we have not used the linked cluster theorem and neither have we employed... 

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... 

The additive separability of each commutator leads to a formulation of coupled-cluster theory where each term (i.e. each expectation-value expression) in the energy or in the amplitude equations is separately size-extensive. The linked equations are therefore said to he termwise size-extensive. No terms that violate the size-extensivity arise and no cancellation of such terms ever occurs. 

Termwise size-extensivity makes the linked formulation of coupled-cluster theory particularly convenient for developing approximate but rigoously size-extensive models from the energy or amplitude equations, we may omit any commutator contribution without destroying size-extensivity. We shall see examples of this technique in our discussion of quadratic Cl theory in Section 13.8.2 and in the development of perturbation theory in Section 14.6. 

There exist equations which link explicitly the dependence of / on the bond coupling delay effects and on AEpp and the reader is referred to the original literature [4 b, 5 b, 8]. Of course, in the most general situation / will depend also on the curvature of ascent of the ground states toward the crossing point. Such an effect is related to overlap repulsion and is discussed later in the chapter by reference to the X3 clusters. Other effects like electrostatic and steric interactions should enter into the final / value, but are not going to be discussed in this chaper. 

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