Cluster operator diagrams

The cluster operator diagrams are particularly simple to interpret as matrix elements the diagrams always involve the reference determinant on the right (because they contain no lines below the interaction line) and an excited determinant on the left, for example,... 

Since the two cluster operators act on the reference determinant to produce a total excitation level of +2, we require the same Hamiltonian -2 diagram fragment used in Eq. [164]. Also, because the cluster operators act before the Hamiltonian operator in the matrix element, they are placed at the bottom of the diagram. Furthermore, because the operators commute, their vertical... 

Thus far we have only considered one (boson) vector field, namely, the direct product field R Xn of creation and annihilation operators. The coefficients of the creation and annihilation operator pairs in fact also constitute vector fields this can be shown rigorously by construction, but the result can also be inferred. Consider that the Hamiltonian and the cluster operators are index free or scalar operators then the excitation operators, which form part of the said operators, must be contracted, in the sense of tensors, by the coefficients. But then we have the result that the coefficients themselves behave like tensors. This conclusion is not of immediate use, but will be important in the manipulation of the final equations (i.e., after the diagrams have contracted the excitation operators). Also, the sense of the words rank and irreducible rank as they have been used to describe components of the Hamiltonian is now clear they refer to the excitation operator (or, equivalently, the coefficient) part of the operator. 

Fig. 4. Diagrams for (a) single, (b) double, and (c) triple antisymmetrized cluster operators.

Foundations to the CC methods were laid by Coester and Kuemmel,1 Cizek,2 Hubbard,3 Sinanoglu,4 and Primas,5 while Cizek2 first presented the CC equations in explicit form. Also Hubbard3 called attention to the equivalence of CC methods and infinite-order many-body perturbation theory (MBPT) methods. From this latter viewpoint, the CC method is a device to sum to infinity certain classes of MBPT diagrams or all possible MBPT diagrams when the full set of coupled-cluster equations is solved. The latter possibility would require solving a series of coupled equations involving up to IV-fold excitations for N electrons. Practical applications require the truncation of the cluster operators to low N values. 

Another way of introducing cluster operators is to define the operator 7) to sum only connected /-fold excitation diagrams in P mbpt, and by virtue of defining Cl = exp(T) the disconnected but linked mbpt diagrams are summed as the quadratic and higher terms in the exp(T) expansion. This is the essential relationship of MBPT to coupled-cluster theory. 

Diagrammatically, these cluster operators S,- represent the connected /-body terms, i.e. those diagrams which cannot be separated into topologically unconnected parts. Typically, the wave and cluster operators are related to each other due to the exponential ansatz... 

When multiple properties are to be considered simultaneously, other approaches may be used. For example, the cluster ternary diagram may be used as a graphical tool for the design of property-based recycle networks. SheUey and El-FIalwagi [23] introduced the concept of property clusters as the basis for property-based component-free design. Based on the property mixing expressions (e.g., Eq. [4.1]), the property operators maybe normalized via the division by a reference value ... 

Along this line, in a recent paper [37] we introduced the so-called quasiparticle-based MR CC method (QMRCC). The mathmatical structure of QMRCC is more or less the same as that of the well-known SR CC theory, i.e., the reference function is a determinant, commuting cluster operators are applied, normal-ordering and diagram techniques can be used, the method is extensive, etc. The point where the MR description appears is the application of quasiparticle slates instead of the ordinary molecular orbitals. These quasiparticles are second-quantized many-particle objects introduced by a unitary transformation which allows us to represent the reference CAS function in a determinant-like form. As it is shown in the cited paper, on one hand the QMRCC method has some advantages with respect to the closely related SR-based MR CC theory [22, 31, 34] (more... 

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 ... 

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... 

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