Wave and cluster operators

At the beginning we introduce a special Slater determinant, the reference determinant (called the vacuum state, it can be the Hartree-Fock determinant) bo and we 

34) represents a very ambitious task. It assumes that we will find an operator T such that the wave operator (e ), as with the touch of a wizard s wand, will make an ideal solution of the Schrodinger equation from the Hartree-Fock function. The formula with exp(T) is an Ansatz. The charming sounding word Ansatz can be translated as arrangement or order, but in mathematics it refers to the construction assumed. 

In literature we use the argument that the wave operator ensures the size consistency of the CC. According to this reasoning, for an infinite distance between molecules A and B, both and 4 o functions can be expressed in the form of the product of the wave functions for A and B. When the cluster operator is assumed to be of the form (obvious for infinitely separated systems) T=Ta + Tb, then the exponential form of the wave operator expfT - - Tb) ensures a desired form of the product of the wave function fexpCr -l- 7b)]Oo = exp Ta exp Tb q. If we took a finite Cl expansion (Ta + then we would not get the product but the sum which 

As a cluster operator f we assume a sum of the excitation operators (see Appendix U) 

In the CC method we want to obtain correct results with the assumption that /max of eg. (10.35) is relatively small (usually 2 -e- 5). If /max were equal to N, i.e. to the number of electrons, then the CC method would be identical to the full (usually unfeasible) Q method. 

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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... 

Figure 3. (a) The overlaps of the CCSD ( ), QECCSD (V), and ECCSD (A) wave functions with the full Cl wave function for the STO-3G (146) model ofN2-(b) The difference between the CCSD and ECCSD cluster operators T ( ) and the difference between the ECCSD cluster operators T and S (O), as defined by eq (61), for the STO-3G (146) model ofN2-... 

As shown in Figure 3 (b), the CCSD and ECCSD cluster operators T are very similar only for smaller values of R. This explains why the CCSD and ECCSD wave functions, Po ° and, respectively, and the corresponding... 

Using a cluster operator, T, and an exponential ansatz [60,61], the coupled cluster wave function is written as... 

Thus,although we may choose a cluster operator S as containing quasi—open and open operators only, a wave-operator of the form exp(S) or exp(S) will generate closed operators stemming from the expansion involving products of quasi-open operators. The intermediate normalization is thus incompatible with the above choice of S. 

Where H is the similarity-transformed Hamiltonian, eq (14), with respect to two independent cluster operators T and Z or, more precisely, with respect to the excitation operator T and the deexcitation operator Z The advantage of eq (36) over the expectation value of the Hamiltonian with the CC wave function, which can also improve the results for multiple bond breaking (28, 127), is the fact that EcC(z,j is a finite series in T and Z. Unfortunately, the power series expansions of (Z,7), eq (36), in terms of T and Z contain higher powers of... 

In the simplest case of a nondegenerate ground state of a closed shell system, one employs the HF wave function as a reference 0) and represents the /-body cluster operators I] as linear combinations of the /-fold excitation operators G( ... 

It should be emphasized that the absence of terms in the wave operators in the preceding equations does not. reflect further truncation [e.g., with respect to exp(7 + T2 + T3)] rather it is a consequence of the triangle inequalities involving the (irreducible) ranks of the Hamiltonian, the external space operators, and the cluster operators. More specifically, a matrix element vanishes identically unless it has an overall rank of 0 from... 

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