Operators cluster

Cluster operator (general, single, double,. .. excitations)... 

The Brueckner-reference method discussed in Section 5.2 and the cc-pvqz basis set without g functions were applied to the vertical ionization energies of ozone [27]. Errors in the results of Table IV lie between 0.07 and 0.17 eV pole strengths (P) displayed beside the ionization energies are approximately equal to 0.9. Examination of cluster amplitudes amd elements of U vectors for each ionization energy reveals the reasons for the success of the present calculations. The cluster operator amplitude for the double excitation to 2bj from la is approximately 0.19. For each final state, the most important operator pertains to an occupied spin-orbital in the reference determinant, but there are significant coefficients for 2h-p operators. For the A2 case, a balanced description of ground state correlation requires inclusion of a 2p-h operator as well. The 2bi orbital s creation or annihilation operator is present in each of the 2h-p and 2p-h operators listed in Table IV. Pole strengths are approximately equal to the square of the principal h operator coefiScient and contributions by other h operators are relatively small. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

A standard method of improving on the Hartree-Fock description is the coupled-cluster approach [12, 13]. In this approach, the wavefunction CC) is written as an exponential of a cluster operator T working on the Hartree-Fock state HF), generating a linear combination of all possible determinants that may be constructed in a given one-electron basis,... 

The cluster operator T creates excitations out of the Hartree-Fock determinant and may be written as... 

The computational complexity of the coupled-cluster method truncated after a given excitation level m - for example, m = 2 for CCSD - may be discussed in terms of the number of amplitudes (Nam) in the coupled-cluster operator and the number of operations (Nop) required for optimization of the wavefunction. Considering K atoms, each with Nbas basis functions, we have the following scaling relations ... 

The method of moments of coupled-cluster equations (MMCC) is extended to potential energy surfaces involving multiple bond breaking by developing the quasi-variational (QV) and quadratic (Q) variants of the MMCC theory. The QVMMCC and QMMCC methods are related to the extended CC (ECC) theory, in which products involving cluster operators and their deexcitation counterparts mimic the effects of higher-order clusters. The test calculations for N2 show that the QMMCC and ECC methods can provide spectacular improvements in the description of multiple bond breaking by the standard CC approaches. 

The simplest way of introducing the approximate clusters into the correction (QVMMCC) is obtained by considering the following form of the cluster operator Z in eq (29) ... 

Eq (46) alone is not sufficient to determine two different cluster operators T and Z- Thus, in addition to the right eigenvalue problem involving H, eq (46), we consider the corresponding left eigenvalue problem. 

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... 

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... 

In this section, we derive basic equations for the monoexcited and biexcited cluster amplitudes at the CCSD level of approximation, i.e. with the cluster operators 7 being approximated by their singly and doubly excited cluster components... 

The implementation of the CC method begins much as in the MPPT/MBPT case one selects a reference CSF that is used in the SCF process to generate a set of spin-orbitals to be used in the subsequent correlated calculation. The set of working equations of the CC technique given above in Chapter 19.1.4 can be written explicitly by introducing the form of the so-called cluster operator T,... 

On the one hand, we can strive for a single cluster operator T, defining the valence universal wave operator U, U = exp(T), which will transform all the model space states ]< > ) into some linear combinations of the exact states jfl i), f = 1,2, , M, which in turn span the target space M, i.e.. 

Alternatively, we can define different cluster operators T i), one for each reference i), so that... 

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