Cluster operator expansion

Eqs (46) and (49) are the basic equations of the ECC theory described in ref 124. The approximate ECC methods, such as ECCSD, are obtained by truncating the many-body expansions of cluster operators T and Z at some excitation level < N. so that T is replaced by eq (4), and Z is replaced by... 

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. 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

The only nonzero terms in the Hausdorff expansion are those in which the Hamiltonian, has at least one contraction with every cluster operator, T , on its right. 

This is often referred to as the connected cluster form of the similarity-transformed Hamiltonian. This expression makes the truncation of the Haus-dorff expansion even clearer since the Hamiltonian contains at most four annihilation and creation operators (in n) can connect to at most four cluster operators at once. Therefore, the Hausdorff expansion must truncate at the quartic terms. 

As discussed in detail in Refs. 77 and 82, for example, this expansion is not N-fold (where N is the number of electrons in the system) for the lower perturbational orders, but truncates to include only modest excitation levels. For example, the first-order wavefunction, which may be used to compute both the second- and third-order energies, contains contributions from doubly excited determinants only, whereas the second-order wavefunction, which contributes to the fourth- and fifth-order perturbed energies, contains contributions from singly, doubly, triply, and quadruply excited determinants. Furthermore, the sum of the zeroth- and first order energies is equal to the SCF energy. This determinantal expansion of the perturbed wavefunctions suggests that we may also decompose the cluster operators, T , by orders of perturbation theory ... 

The partitioning of the electronic Hamiltonian and the corresponding breakdown of the cluster operators leads to an expansion of the coupled cluster effective Hamiltonian, H, in orders of perturbation theory through the Haus-dorff expansion given in Eq. [122] ... 

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... 

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. 

Limiting ourselves to the HF reference case for convenience, by combining a cluster-type expansion of ft given in Eq. (26) and its order-byorder series, the low-order contributions to the wave operator may be expressed as... 

It is important to note, however, that there are fundamental differences between FSCC and SRCC with respect to the nature of their excitation operators. For a given truncation of the cluster operators beyond simple double excitations, the determinantal expansion space available in an FSCC calculation is smaller than those of SRCC calculations for the various model space determinants. A class of excitations called spectator triple excitations must be added to the FSCCSD method to achieve an expansion space that is in some sense equivalent to that of the SRCC. But even then, the FSCC amplitudes are restricted by the necessity to represent several ionized states simultaneously. Thus, we should not expect the FSCCSD to produce results identical to a single reference CCSD, nor should we expect triple excitation corrections to behave in the same way. The differences between FSCC and SRCC shown in Table I and others, below, should be interpreted as a manifestation of these differences. 

Equations (70) or (76) and (73) are the basic equations of the new EOMXCC theory. In order to solve an eigenvalue problem (70) we must decide about the source of information about the cluster operator T that defines H. We find the cluster amplitudes defining T by projecting Eq. (73) against the excited configurations included in the many-body expansion of... 

Both the lack of freezing and the inability of ions to enter the pores containing motionally restricted water can be explained by the existence of fragmented clusters such as monomors, dimers etc. Thus, the presence of these fragmented clusters prevent the necessary aggregation and co-operative expansion needed for an ice-like structure to exist, while at the same time they are less able to hydrate ions resulting in low solubilities and consequently low rejections in the desalination sense (1,2 ). This could be the microscopic mechanistic basis for the solution-diffusion model so... 

It is obvious that 0o cannot serve as a vacuum in the strict sense of the traditional hole-particle formalism, since the valence orbitals in are partially occupied. A straightforward cluster expansion in the occupation number representation from tpo would thus entail two problems (a) there is no natural choice of vacuum to effect a cluster expansion, and (b) the occupation number representation of cluster operators would refer to orbital excitations with respect to the entire oi thus necessitating the considerations of virtual functions which are by themselves combination of functions. If we want to formulate a many-body theory using if>o as the reference function, we need constructs where these cause no problems. 

If the cluster operator is connected, one can easily show that the dressed Hamiltonian and the matrix elements are also connected via multi-commutator expansion. Hence, the proof of the connectedness of the first term of Eq. (7) is quite... 

The coupled-cluster (CC) method is an attempt to find such an expansion of the wave function in terms of the Slater determinants, which would preserve size consistency. In this method, the wave function for the electronic ground state is obtained as a result of the operation of the wave operator exp (T) on the Hartree-Fock function (this ensures size consistency). The wave operator exp (T) contains the cluster operator T, which is defined as the sum of the operators for the Z-tuple excitations, Ti up to a certain maximum I = Zmax. Each 2) operator is the sum of the operators each responsible for a particular l-Uiple excitation multiplied by its amplitude t. The aim of the CC method is to find the t values since they determine the wave function and energy. The method... 

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