Connected cluster theorem

The connected cluster theorem of Cizek5 states that... 

Needless to say, it was the second quantization formalism of quantum field theory, enabling the exploitation of Wick s theorem together with a representation via Feynman-like graphs or diagrams—the mathematical techniques relied upon by all the above authors [32-34]—that made it possible to carry out the general proof of the extensive nature of RSPT and to unscramble the general structure of MBPT wave functions and energies. The principal results of these efforts are usually referred to as the linked cluster and connected cluster theorems (see below). 

The connected cluster theorem, which is the basis of the CC theory as will be seen below, then fine-tunes the linked-cluster theorem for the wave function in the following... 

Connected Cluster Theorem. Define the cluster operator T that generates all the... 

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 ... 

For the sake of correctness, it is necessary to note that we disregarded such theoretically important concepts as the linked cluster (linked connected graph) theorem and the exclusion principle violating (EPV) diagrams. This is in accordance with our aim to maintain the practical nature of this review. The linked cluster theorem and EPV diagrams are of importance in the fourth and higher orders of the perturbation theory which, in our opinion, shall hardly be accessible to routine calculations in the foreseeable future. For detailed information on the linked cluster theorem and EPV diagrams see Refs.9,33,34,4a ... 

We may recall that the desirability of ensuring size-extensivity for a closed-shell state was one of the principal motivations behind the formulation of the MBPT for the closed-shells. The linked cluster theorem of Bruckner/25/, Goldstone/26/ and Hubbard/27/, proving that each term in the perturbation series for energy can be represented by a linked (connected) diagram directly reflects the size-extensivity of the theory. Hubbard/27/ and Coester/30/ even pointed out immediately after the inception of MBPT/25,26/, that the size-extensivity is intimately related to a cluster expansion structure of the associated wave-operator that is not just confined only to perturbative theory. The corresponding non-perturbative scheme for the closed-shells was first described by Coester and Kummel/30,31/ in nuclear physics and this was transcribed to quantum chemistry... 

The 1inked-cluster theorem for energy, from the above analysis, is a consequence of the connectivity of T, and the exponential structure for ft. Size-extensivity is thus seen as a consequence of cluster expansion of the wave function. Specfic realizations of the situation are provided by the Bruckner—Goldstone MBPT/25,26/, as indicated by Hubbard/27/, or in the non-perturbative CC theory as indicated by Coester/30,31/, Kummel/317, Cizek/32/, Paldus/33/, Bartlett/21(a)/ and others/30-38/. There are also the earlier approximate many-electron theories like CEPA/47/, Sinanoglu s Many Electron Theory/28/ or the Cl methods with cluster correction /467. 

The subscript c indicates that only connected diagrams have to be taken. The last equation follows after applying a linked cluster theorem. Furthermore the abbreviation <.. .. >=<4>qI I > been used. The... 

Using the connected cluster form of H defined above, as well as the techniques of Wick s theorem and normal ordering, we may derive a programmable form of the energy expression in the CCSD approximation. In accord with Eq. [50] and the normal-ordered Hamiltonian, the energy is given by... 

Sect, 4.3j wh( jre diagrammatically two points aie connected if tlie corresponding cliains interact. The linked cluster theorem (5.7) follows. 

Linked Cluster Theorem. Only linked (or connected) vacuum diagrams contribute to the energy, while all unlinked (or disconnected) vacuum diagrams mumally cancel out, so that... 

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... 

It should be mentioned that there is a difiiculty connected with the freezing out procedure described above. Namely it turns out that the linked cluster theorem is violated (Keiter 1968). This has no implications for the evaluation of diagrams for the coupled RE-ion and phonon or conduction-electron system as long as one uses the mean field or random-phase approximation (RPA). However the method can not be easily applied to more sophisticated approximation schemes. For more details we refer to Fulde and Peschel (1972). Since most of the following considerations are restricted to the above approximations the method can be applied and has considerable advantages because of its simplicity. 

A great simplification of eq. (4.38) is achieved by exploiting the linked cluster theorem [84]. This implies that in eq. (4.36) or eq. (4.38), we calculate only the terms represented by linked connected (LC) diagrams. Therefore, eq. (4.38) can be written as... 

In a truncated coupled cluster approach, tlie two vectors are not connected by the adjoint operation but without truncations a representation of the exact state situation is retrieved and one state is the adjoint of the other. The generalized Hellmann-Feynman theorem is proven to hold... 

Since we can make a graph taut, the theorem above implies that it is possible to achieve the lower bound in synchronization costs for elementary graphs. We note that imposing a cluster ordering in a graph will not affect the property of well-posedness. The reason is that, by definition, anchor clusters are not connected by any cycle in the constraint graph. Therefore, no cycles can be formed by serializing between anchors in different clusters. 

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