Coupled-cluster method definition

However, for non-variational wavefunctions, as for example in the case of Moller-Plesset perturbation theory and the coupled cluster methods, the density matrices here are not consistent with the definition in Eq. (2.20) and thus Eqs. (9.107) and (9.108). The density matrices defined in Eq. (12.1) are for those methods therefore... 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

Recent developments include exact [12-14, 44, 90, 91] and approximate [14, 90, 92-94] iterative schemes to determine Hg, the intermediate Hamiltonian method [21, 24, 95], the use of incomplete model spaces [43, 44] and some multireference open-shell coupled-cluster (CC) formalisms [16-20, 96, 97]. Only some eigenvalues of the intermediate Hamiltonian H, are also eigenvalues of H. The corresponding model eigenvectors of H, are related to their true counterparts as in Bloch s theory. Provided effective operators a are restricted to act solely between these model eigenvectors, the possible a definitions from Bloch s formalism (see Section VI.A) can be used. 

The theoretical treatment of the Van der Waals interaction, on the other hand, definitely requires the application of more sophisticated, correlated methods such as perturbation theory (performed mostly in the form of Moller-Plesset perturbation theory, MP), coiffiguration interaction (Cl) or coupled cluster (CC) approaches (see below). 

Abstract This chapter introduces to the basic definitions of the PCM model for a molecular solute. The basic electrostatic problem for the determination of the solute-solvent interaction is described within the Integral Equation Formalism (lEF-PCM), and the QM problem associated to the effective Hamiltonian of the molecular solute is formulated in terms of a basic energy functional which has the thermodynamic status of a free-energy for the entire solute-solvent system. The QM problems for the molecular solute is exemplified at the Hartree-Fock and at the coupled-cluster level methods. 

Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141 134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented. 

In this chapter we wiU finally follow the third approach, which means that we abandon the perturbation-theory approach all together and go back to the definitions of the properties as derivatives of the energy in the presence of the perturbation. We will illustrate with a few examples how this approach can be appfied to approximate expressions for the energy in the presence of both static as well as time-dependent perturbations. However, the presentation will be very brief and restricted to Mpller-Plesset perturbation theory and coupled cluster energies as nothing new is obtained for variational methods compared to the response theory approaches in Chapters 10 and 11. 

Frequency-dependent response functions can only be computed within approximate electronic structure models that allow definition of the time-dependent expectation value. Hence, frequency-dependent response functions are not defined for approximate methods that provide an energy but no wave function. Such methods include MoUer-Plesset (MP) perturbation theory, multiconfigurational second-order perturbation theory (CASPT2), and coupled cluster singles and doubles with non-iterative perturbative triples [CCSD(T)]. As we shall see later, it is possible to derive static response functions for such methods. 

Within the variation (supramolecular) approach, definitely the method of choice for interaction energies would be the coupled cluster CCSD(T) method (in which the single and double excitations are evaluated iteratively while the triple excitations are included in a non-iterative way). The CCSD(T) method yields a significant portion of the correlation energy. The MP2 method, including the double electron excitations at the second order of perturbation theory, overestimates the correlation interaction energy for stacking, as noted above. 

Consideration of the effect of electron correlation is needed to arrive at predicted intensities comparable in quantitative terms with experimental values. Since the number of molecules treated in calculations accounting for large proportion of correlation energy is limited, definite conclusions as to what approach is best for quantitative IR intensity predictions are still to come. Analytical derivative methods for higher order perturbation frieory proaches, configuration interaction treatment and, especially, coupled cluster theory, tq>pear to be the best hopes. Whether such calculations would become a routine exercise is yet to be seen. Fortunately, the studies carried out show that die double harmonic approximation works quite well as far as ab initio intensity predictimis are concerned. 

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