Coupled cluster energy

The additivity of E and the separability of the equations determining the Cj eoeffieients make the MPPT/MBPT energy size-extensive. This property ean also be demonstrated for the Coupled-Cluster energy (see the referenees given above in Chapter 19.1.4). However, size-extensive methods have at least one serious weakness their energies do not provide upper bounds to the true energies of the system (beeause their energy funetional is not of the expeetation-value form for whieh the upper bound property has been proven). 

Mebel, A. M., Morokuma, K., lin, M. C., 1995, Modification of the GAUSSIAN-2 Theoretical Model The Use of Coupled-Cluster Energies, Density Functional Geometries, and Frequencies , J. Chem. Phys., 103, 7414. 

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. 

S. Hirata, M. Nooijen, I. Grabowski, and R. J. Bartlett, Perturbative corrections to coupled-cluster and equation-of-motion coupled-cluster energies a determinantal analysis. J. Chem. Phys. 114, 3919 (2001). 

NONITERATIVE CORRECTIONS TO COUPLED-CLUSTER AND EQUATION-OF-MOTION COUPLED-CLUSTER ENERGIES DEFINING THE EXACT METHOD OF MOMENTS OF COUPLED-CLUSTER EQUATIONS... 

The computational problem, then, is determination of the cluster amplitudes t for aU of the operators included in tlie particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wave functions expressed as dctcnninants of the HF orbitals. This generates a set of coupled, nonlinear equations in the amplitudes which must be solved, usually by some iterative technique. With the amplitudes in hand, the coupled-cluster energy is computed as... 

P. Piecuch, J. Cizek, and J. Paldus, Int. ]. Quantum Chem., 42, 165 (1992). Behaviour of the Coupled Cluster Energy in the Strong Correlated Limit of the Cyclic Polyene Model. Comparison with the Exact Results. 

Koch H, Jdrgen H, Jensen A, Jorgensen P, Helgaker T, Scuseria GE, Schaefer III HF (1990) Coupled cluster energy derivatives. Analytic Hessian for the closed-shell coupled cluster singles and doubles wave function Theory and applications. J Chem Phys 92 4924-4940... 

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... 

Presently, we are able to determine the coupled cluster energy based on the variational Lagrangian and expectation values for real operators... 

This property of the coupled cluster energy is commonly known as size consistency. ... 

This is the natural truncation of the coupled cluster energy equation an analogous phenomenon occurs for the amplitude equation (Eq. [45]). This truncation depends only on the form of f and not on that of T or on the number of electrons. Equation [49] is correct even if T is truncated to a particular excitation level. 

Using the truncated Hausdorff expansion, we may obtain analytic expressions for the commutators in Eq. [52] and insert these into the coupled cluster energy and amplitude equations (Eqs. [50] and [51], respectively). However, this is only the first step in obtaining expressions that may be efficiently implemented on the computer. We must next choose a truncation of T and then derive expressions containing only one- and two-electron integrals and cluster amplitudes. This is a formidable task to which we will return in later sections. 

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... 

The task of determining the left-hand ground state eigenvector of H is thus reduced to determining the amplitudes 1% . The ground state coupled cluster energy may then be written as... 

The two-electron component, however, produces four equivalent fully contracted terms, and therefore contributes to the coupled cluster energy ... 

This equation is not restricted to the CCSD approximation, however. Since higher excitation cluster operators such as T3 and T4 cannot produce fully contracted terms with the Hamiltonian, their contribution to the coupled cluster energy expression is zero. Therefore, Eq. [134] also holds for more complicated methods such as CCSDT and CCSDTQ. Higher excitation cluster operators can contribute to the energy indirectly, however, through the equations used to determine the amplitudes, and t-h, which are needed in the energy equation above. 

The coupled cluster energy, on the other hand, does not suffer from this lack of size extensivity for two reasons (1) the amplitude equations in Eq. [50] are independent of the coupled cluster energy and (2) the Hausdorff expansion of the similarity-transformed Hamiltonian in Eq. [106], for example, guarantees that the only nonzero terms are those in which the Hamiltonian is con-... 

T. D. Crawford and. F. Stanton, Int. J. Quantum Chem. Symp, 70, 601 (1998). Investigation of an Asymmetric Triple-Excitation Correction for Coupled Cluster Energies. 

T. J. Lee and A. P. Rendell,/. Chem. Phys., 94,6229 (1991). Analytic Gradients for Coupled-Cluster Energies That Include Noniterative Connected Triple Excitations Application to cis-and tra s-HONO. 

H. Koch, H. J. Aa. Jensen, P. j0rgensen, T. Helgaker, G. E. Scuseria, and H. F. Schaefer, /. Chem. Phys., 92, 4924 (1990). Coupled-Cluster Energy Derivatives. Analytic Hessian for the Closed-Shell Coupled-Cluster Singles and Doubles Wave Functions Theory and Applications. 

The approach outlined above combines the calculation of response functions (i.e. of frequency-dependent properties) with the theory of analytic derivatives developed for static higher-order properties. In the limit of a static perturbation all equations above reduce to the usual equations for (unrelaxed) coupled cluster energy derivatives. This is an invaluable advantage for the implementation of frequency-dependent properties in quantum chemistry programs. 

Curtiss, L. A. Redfem, P. C. Raghavachari, K. Baboul, A. G. Pople, J. A. (1999b). Gaussian-3 theory using coupled cluster energies, Chemical Physics Letters, 314, pp. 101-107. 

For really accurate enzyme calculations, we argue that detailed QM/MM calculations are needed. These should include single-point calculations with 600-1000 atom QM systems and estimations of QM/MM free energies, as well as extrapolations to coupled-cluster energies, as in the thorough studies by Thiel and coworkers on XO and aldehyde oxidoreductase. We have started similar studies on DMSOR and... 

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]. 

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