Linked cluster theorem

Consider some function G(x) having a formal Taylor series expansion 

To define the cumulanC GrM we represent G(-M) by M points contained in a box, and we draw all diagrams in which these points are grouped together in an arbitrary way (Fig. 5.15). A group of m connected points stands for the cumulant G Thus Fig. 5.15, for instance, stands for 

consider all contributions to G M consisting of nm,l m M m-poinf. cumulants. Clearly 

The summation over M eliminates the constraint ETnnnjn = M, and G(x) takes the form 

The application to the grand partition function is obvious. We in Eq. (5.2) formally multiply Z by xM. thus defining a function Z(x, [/ ]) reducing to Z[n for x — 1. The cumulants as defined here are identical to the cumulants 

Consider some function (jr(n-) having a formed Taylor series expansion 

Grand Canonical Description of Solutions at Finite Concentration. 

3j wh( jre diagrammatically two points aie connected if tlie corresponding cliains interact. The linked cluster theorem (5.7) follows. 

To dt rive tiie linked cluster theortirn for density correlations we dtjfine the generating functional 

---

The cluster expansions of the correlation functions and potentials of mean force can be found by studying the semi-invariant expansion of logg( n ) and by the use of the linked cluster theorem. The method is a straightforward extension of those given in the preceding section and details can be found in the paper by Allnatt and Cohen.3 The linked cluster method is simpler and... 

The organization of this chapter is as follows. In Sect. 5.1 we present the basic formalism and work out the Feynman rules for the grand canonical ensemble. Diagrammatic representations valid in the thermodynamic limit are derived for both thermodjmamic quantities and correlation functions. The proof of the Linked Cluster Theorem is given in Appendix A 5.1. Section 5.2... 

Ignoring the correction terms in Eq. (5.6), this argument is valid only if we switch off the interaction among different chains, i.e. it holds for an ideal solution . The generalization to mutually interacting chains follows by virtue of the Linked Cluster Theorem. In the present context this important theorem can be formulated as follows. 

We finally need the diagrammatic representation of the correlation functions. As shown in Appendix A 52, as a simple consequence of the general Linked Cluster Theorem all correlation functions or cnmulants defined here... 

To derive the linked cluster theorem for density correlations we define the generating functional... 

We introduced the variable x for bookkeeping in the application of the linked cluster theorem. 

We first consider the grand potential f7[/ip], which according to the Linked Cluster Theorem can be written as... 

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... 

For the analysis of the various formalisms, manipulation of the equations, generating normal product of terms via Wick s theorem, and particularly for indicating how the proofs of the several different linked cluster theorems are achieved, we shall make frequent use of diagrams. For the sake of uniformity, we shall mostly adhere to the Hugenholtz convention/1/. All the constituents of the diagrams will be operators in normal order with respect to suitable closed-shell determinant taken as the vacuum. We shall refer to the creation/annihilation operators with respect to this vacuum after the h-p transformation.The hamiltonian H will also be taken to be in normal order with respect to... 

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... 

---