Coupled-Cluster Wave Functions

Perturbation methods add all types of corrections (S, D, T, Q etc.) to the reference wave function to a given order (2, 3, 4 etc.). The idea in Coupled Cluster (CC) methods is to include all corrections of a given type to infinite order. The (intermediate normalized) coupled cluster wave function is written as... 

With the coupled cluster wave function (4.46) the Schrddinger equation becomes... 

The use of Cl methods has been declining in recent years, to the profit of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl or MCSCF based methods have been the prefen ed methods here. More recently propagator or equation of motion (Section 10.9) methods have been developed for coupled cluster wave functions, which allows calculation of exited state properties. 

For our purposes, CC theory and its finite order MBPT approximations offer a convenient, compact description of the correlation problem and give rapid convergence to the basis set (i.e. full Cl) limit with different categories of correlation operators (see Fig. 15.1). The coupled-cluster wave function is... 

In section c above, two different ways of writing the wave-function have been described by (4.12) and (4.13), which in principle can be extended to the exact wave-function. There is a third alternative which has many advantages and is connected with one of the presently dominating ways of solving the Schiodinger equation, and this is the coupled cluster wave-function. To define this wave-function it is convenient to define certain excitation operators. A general n-tuple excitation operator T is defined as... 

The corresponding coupled cluster wave-function is instead... 

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

The determination of a coupled cluster wave function does not follow the conventional variational procedure but a non-variational procedure where the excitation amplitudes are determined by a projection technique. We have that the coupled cluster energy for a molecule in vacuum is given by... 

For the optimization of the coupled cluster wave function in the presence of the classical subsystem we write the CC/MM Lagrangian as [24]... 

Turning now to the specific case of a coupled cluster wave function we write the time-dependent CC wave function as... 

The second general approach to correlation theory, also based on perturbation theory, is the coupled-cluster method, which can be thought of as an infinite-order perturbation method. The coupled-cluster wave function T cc is expressed as a power series,... 

V. Derivatives from Coupled-Cluster Wave Functions... 

We assume that we have carried out a Hartree-Fock calculation at the reference geometry X0 and used the orbitals to determine a coupled-cluster wave function... 

Kallay M, Surjan PR (2000) Computing coupled-cluster wave functions with arbitrary excitations. J Chem Phys 113 1359-1365. 

MR-MBPT methods, however, is not straightforward. The main problem here is similar With the coupled cluster wave function (4.46) the Schrodinger equation becomes... 

J. F. Stanton, On the extent of spin contamination in open-shell coupled-cluster wave functions, J. Chem. Phys. 101, 371-374 1994. 

Open-Shell Coupled-Cluster Wave Functions. 

R. A. Chiles and C. E. Dykstra,/. Chem. Phys., 74,4544 (1981). An Electron Pair Operator Approach to Coupled-Cluster Wave Functions. Application to He2, Bea, and Mg2 and Comparison with CEPA Methods. 

Conceptually related and numerically superior to 2/zp-TDA are the EOMIP-CC [264] and related EOMEA-CC [265] methods.42 In these approaches, the HF-SCF and coupled-cluster wave functions are obtained for the closed-shell state,43 and the similarity transformed Hamiltonian H (see Section II) parameterized by the closed-shell T amplitudes rather than the bare Hamiltonian... 

This defines an independent electron pair approximation in terms of extremal pairs, which can be regarded as a generalization of the independent electron pair approximation (IEPA) [4, 8] in terms of pairs (ij) constructed form (preferably) localized orbitals. As in the discussion in Paper I for MP2 [5], one can show that the extremal pairs defined in this section are related to approximate natural geminals corresponding to the coupled-cluster wave function. 

Chiles RA, Dykstra CE. An electron pair operator approach to coupled cluster wave functions. Applications to He2, Be2 and Mg2 and comparison with CEPA methods. J Chem Phys 1981 74 4544 1556. 

Such a basis set combines well with coupled-cluster wave functions to tend to converge in a consistent and predictable manner towards limits of the basis set and the theory. Calculation of the rotational g tensor and magnetizability involved use of rotational London orbitals [10]. Optimization, first order in derivatives of energy with respect to internuclear distances, yielded all reported geometric stmctures of... 

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