Coupled-cluster theory variational

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. 

First, we note that the determination of the exact many-particle operator U is equivalent to solving for the full interacting wavefunction ik. Consequently, some approximation must be made. The ansatz of Eq. (2) recalls perturbation theory, since (as contrasted with the most general variational approach) the target state is parameterized in terms of a reference iko- A perturbative construction of U is used in the effective valence shell Hamiltonian theory of Freed and the generalized Van Vleck theory of Kirtman. However, a more general way forward, which is not restricted to low order, is to determine U (and the associated amplitudes in A) directly. In our CT theory, we adopt the projection technique as used in coupled-cluster theory [17]. By projecting onto excited determinants, we obtain a set of nonlinear amplitude equations, namely,... 

The ostensible impracticality of a variational coupled cluster theory raises an important question regarding the physical reality of the coupled cluster energy as computed using projective, asymmetric techniques. Quantum mechanics dictates that physical observables (such as the energy) are expectation... 

W. Kutzelnigg, Mol. Phys., 94, 65 (1998). Almost Variational Coupled Cluster Theory. 

Certain Schrodinger equation based methods, such as coupled cluster theory, are not based on a variational principle. They fall outside schemes that use the energy expectation value as a optimization function for simulated annealing, although these methods could be implemented within a simulated annealing molecular dynamics scheme with alternative optimization function. 

By ab initio we refer to quantum chemical methods in which all the integrals of the theory, be it variational or perturbative, are exactly evaluated. The level of theory then refers to the type of theory employed. Common levels of theory would include Hartree-Fock, or molecular orbital theory, configuration interaction (Cl) theory, perturbation theory (PT), coupled-cluster theory (CC, or coupled-perturbed many-electron theory, CPMET), etc. - We will use the word model to designate approximations to the Hamiltonian. For example, the zero differential overlap models can be applied at any level of theory. The distinction between semiempirical and ab initio quantum chemistry is often not clean. Basis sets, for example, are empirical in nature, as are effective core potentials. The search for basis set parameters is not usually considered to render a model empirical, whereas the search for parameters in effective core potentials is so considered. 

Hence, if we are to make bE = 0, we can avoid these terms. But to do so we have to have E optimum with respect to the location of the atomic basis functions, t (R) the MO coefficients, c(R) and the Cl coefficients, C(R). The first cannot be satisfied unless the atomic orbital basis set is floated off the atomic centers to an optimum location [105], while the second requires optimum MO coefficients, and the third optimum Cl coefficients. In practice, we will introduce atomic orbital derivatives explicitly, so the AOs can follow their atoms. Now focusing only on the MO and Cl coefficients, in SCF we have optimum MOs and no Cl term. In MCSCF, both terms would vanish, whUe in Cl, the MO derivatives would remain, but the Cl coefficients contribution would vanish. In the non-variational coupled-cluster theory, neither will vanish and this means that CC theory forces us into some new considerations for analytical forces. 

Theory. Usually we do not solve the fundamental equations directly. We use a theory, for example, Har-tree-Fock theory [3], Moller-Plesset perturbation theory [4], coupled-cluster theory [5], Kohn s [6, 7], Newton s [8], or Schlessinger s [9] variational principle for scattering amplitudes, the quasiclassical trajectory method [10], the trajectory surface hopping method [11], classical S-matrix theory [12], the close-coupling approximation... 

Alternatively, there are perturbation methods to estimate Ecorreiation- Briefly, in these methods, you take the HF wavefunction and add a correction—a perturbation—that better mimics a multi-body problem. Moller-Plesset theory is a common perturbative approach. It is called MP2 when perturbations up to second order are considered, MP3 for third order, MP4, etc. MP2 calculations are commonly used. Like CISD, MP2 allows single and double excitations, but the effects of their inclusion are evaluated using second-order perturbation theory rather than variationally as in CISD. An even more accurate type of perturbation theory is called coupled-cluster theory. CCSD (coupled-cluster theory, singles and doubles) includes single and double excitations, but their effects are evaluated at a much higher level of perturbation theory than in an MP2 calculation. 

In the case of the variational methods, SCF, MCSCF and CI, o0 = l o) and we have the normal expectation value. For the non-variational methods such as Mpller-Plesset perturbation or coupled cluster theory, the energy is calculated as a transition expectation value, where I FoO =... 

Well-defined variational (Cl-type see Configuration Interaction), perturbational (MPn see M0ller-Plesset Perturbation Theory), and coupled cluster (CC see Coupled-cluster Theory) techniques have all been employed to determine anharmonic force fields. Important conclusions of these studies include (1) Near equilibrium, the correlation energy is a low-order function of the bond distances,even a linear approximation is meaningful(2) For open-shell species, spin contamination can significantly deteriorate results if a... 

Extended articles on the most common electron correlation methods such as limited configuration interaction (Cl see Configuration Interaction), M0ller-Plesset many-body perturbation theory (MBPT see M0ller Plesset Perturbation Theory), variation-perturbation methods (such as PCILO see Complete Active Space Self-consistent Field (CASSCF) Second-order Perturbation Theory (CASPT2) and Configuration Interaction), and coupled cluster theory (CC see Coupled-cluster Theory), as well as on explicitly ri2-dependent wave functions (see rxi Dependent Wave functions), can be found elsewhere. 

For reasons of computational practicality and efficiency, molecular electronic coupled-cluster energies are determined using a nonvariational projection technique. The chief deficiency of this approach is not so much the loss of boundedness (since the coupled-cluster energy is nevertheless rather accurate), but the difficulties that it creates for the calculation of properties as the conditions for the Hellmann-Feynman theorem are not satisfied - even in the limit of a eomplete one-electron basis. Fortunately, as discussed in Section 4.2.8, this situation may be remedied by the construction of a variational Lagrangian [14]. In this formulation, the conditions of the Hellmann-Feynman theorem are fulfilled and molecular properties may be calculated by a proeedure that is essentially the same as for variational wave functions. The Lagrangian formulation of the energy is also related to a variational treatment of coupled-cluster theory applicable to excited states, as discussed in Section 13.6. 

Equation (13.5.8), which represents the generalization of the Hellmann-Feynman theorem to coupled-cluster wave functions, is shown in Exercise 13.1 to give size-extensive first-order properties. For variational wave functions, the Hellmann-Feynman theorem contains the real average value of the operator V (4.2.51) rather than a transition expectation value as in (13.5.8). Likewise, to ensure teal properties, we may in coupled-cluster theory work in terms of the manifestly real expression... 

Of the 33 invited speakers and the seven who contributed talks, 17 accepted our invitation to contribute a chapter to this book. These chapters are complemented by three additional chapters from individuals to help develop a more cohesive book as well as an overview chapter. Approximately half of the chapters are focused on the development of ab initio electronic structure methods and consideration of specific challenging molecular systems using electronic structure theory. Some of these chapters document the dramatic developments in the range of applicability of the coupled-cluster method, including enhancements to coupled-cluster wavefunctions based on additional small multireference configuration interaction (MR-CISD) calculations, the method of moments, the similarity transformed equation of motion (STEOM) method, a state-specific multireference coupled-cluster method, and a computationally efficient approximation to variational coupled-cluster theory. The concentration on the coupled-cluster approach is balanced by an approximately equal number of chapters discussing other aspects of modem electronic stracture theory. In particular, other methods appropriate for the description of excited electronic states, such as multireference... 

The reasons for not invoking the variation principle in the optimization of the wave function are given in Chapter 13, which provides a detailed account of coupled-cluster theory. We here only note that the loss of the variational property characteristic of the exact wave function is unfortunate, but only mildly so. Thus, even though the coupled-cluster method does not provide an upper bound to the FCI energy, the energy is usually so accurate that the absence of an upper bound does not matter anyway. Also, because of the Lagrangian method of Section 4.2.8, the complications that arise in connection with the evaluation of molecular properties for the nonvariational coupled-cluster model are of little practical consequence. 

Chapter 13 discusses coupled-cluster theory. Important concepts such as connected and disconnected clusters, the exponential ansatz, and size-extensivity are discussed the Unked and unlinked equations of coupled-clustCT theory are compared and the optimization of the wave function is described. Brueckner theory and orbital-optimized coupled-cluster theory are also discussed, as are the coupled-cluster variational Lagrangian and the equation-of-motion coupled-cluster model. A large section is devoted to the coupled-cluster singles-and-doubles (CCSD) model, whose working equations are derived in detail. A discussion of a spin-restricted open-shell formalism concludes the chapter. 

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