Coupled cluster formalisms

Emrich K (1981) An extension of the coupled-cluster formalism to excited states (II) Approximations and tests. Nucl Phys A 351 397 138. 

K. Emrich, Nucl. Phys. A, 351, 379 (1981). An Extension of the Coupled Cluster Formalism to Excited States (l). 

Various approximations of Eq. (6), the exact wave function in the coupled-cluster formalism, have been discussed in the chemical literature. In particular, Cizek s coupled-pair many-electron theory (CPMET),5 also referred to as coupled-cluster doubles (CCD) by Bartlett,8 has re-... 

U.S. Mahapatra, B. Datta, D. Mukherjee, A state-specific multi-reference coupled cluster formalism with molecular applications. Mol. Phys. 94 (1998) 157. 

The EOM method has recently been developed in conjunction with the coupled cluster formalism to provide a powerful, correlated, approach.83,84,85... 

Landau, A., Khistyaev, K., Dolgikh, S., and Krylov, A. I. (2010). Frozen natural orbitals for ionized states within equation-of-motion coupled-cluster formalism, . Chem. Phys. 132, p. 014109, doi 10.1063/1.3276630. Langlet, J., Caillet, J., Berges, J., and Reinhardt, P. (2003). Comparison of two ways to decompose intermolecular interactions for hydrogen-bonded dimer systems, J. Chem. Phys. 118, pp. 6157-6166, doi 10.1063/l. 1558473. 

The coupled-cluster electronic state is uniquely defined by the set of the cluster amplitudes and these amplitudes are used to obtain the coupled-cluster energy from Eq. (33). Due to the fact that the Ansatz of the coupled-cluster wave function has the exponential parametrization [Eq. (28)] the energy is size-extensive. This is an obvious advantage of the coupled-cluster formalism compared to some other techniques (e.g. configuration interaction). For a general discussion of coupled-cluster theory and the coupled-cluster equations see Refs. [5, 36]. 

Let us now return to the Rayleigh-Schrodinger formalism which is used in developing the standard coupled cluster formalism. The single-reference coupled cluster expansion can be developed by first of all writing the Schrodinger equation for the non-degenerate case as... 

The application of the Brillouin-Wigner coupled cluster theory to the multireference function electron correlation problem yields two distinct approaches (i) the multi-root formalism which was discussed in Section 4.2.2 and (ii) the single-root formalism described in the previous subsections of this section. Section 4.2.3. The multiroot multi-reference Brillouin-Wigner coupled cluster formalism reveals insights into other formulations of the multi-reference coupled cluster problem which often suffer from the intruder state problem which, and in practice, may lead to spurious... 

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative methods, although the matrix diagonalization involved in CISD usually is so large that it has to be done iteratively. Solution of the coupled cluster equations must be done by an iterative technique since the parameters enter in a non-linear fashion. In terms of the most expensive step in each of the methods they may be classified according to how they formally scale in the large system limit, as shown in Table 4.5. 

The Method of Moments of Coupled-Cluster Equations An Overview of the Ground-State Formalism... 

The vertical IPs of CO deserve special attention because carbon monoxide is a reference compound for the application of photoelectron spectroscopy (PES) to the study of adsorption of gases on metallic surfaces. Hence, the IP of free CO is well-known and has been very accurately measured [62]. A number of very efficient theoretical methods specially devoted to the calculation of ionization energies can be found in the literature. Most of these are related to the so-called random phase approximation (RPA) [63]. The most common formulations result in the equation-of-motion coupled-cluster (EOM-CC) equations [59] and the one-particle Green s function equations [64,65] or similar formalisms [65,66]. These are powerful ways of dealing with IP calculations because the ionization energies are directly obtained as roots of the equations, and the repolarization or relaxation of the MOs upon ionization is implicitly taken into account [59]. In the present work we remain close to the Cl procedures so that a separate calculation is required for each state of the cation and of the ground state of the neutral to obtain the IP values. 

Note that in contrast to a general similarity transformation (e.g., as found in the usual coupled-cluster theory) the canonical transformation produces a Hermitian effective Hamiltonian, which is computationally very convenient. When U is expressed in exponential form, the effective Hamiltonian can be constructed termwise via the formally infinite Baker-Campbell-Hausdorff (BCH) expansion,... 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

Generalization of the method of moments of coupled-cluster equations to excited electronic states Exact formalism... 

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