Implementation of Coupled Cluster Theory

Recently, it has been shown that equation-of-motion coupled cluster theory provides a unique perspective on the CCSD(T) 

It is perhaps not immediately clear how one may go about solving the Tj and T2 amplitude equations given in Eqs. [152] and [153] for the individual amplitudes, tf and Iff- . A simple rearrangement of the equations, however, provides a more palatable form of these expressions that leads to a simple iterative approach for determining the coupled cluster wavefunction amplitudes. For example, the first few terms of Eq. [152] may be written as 

To determine the values of the amplitudes, one must solve this set of coupled nonlinear equations iteratively. A simple starting approximation for tf and on the left-hand sides of the equations may be obtained by setting all the amplitudes on the right-hand side to zero. Hence, for the T amplitudes we have 

This initial guess may then be inserted on the right-hand sides of the equations and subsequently used to obtain new amplitudes. The process is continued until self-consistency is reached. For the special case in which canonical Hartree-Fock molecular orbitals are used, the Fock matrix is diagonal and the T2 amplitude approximation above is exactly the same as the first-order perturbed wave-function parameters derived from Moller-Plesset theory (cf. Eq. [212]). In that case, the Df and arrays contain the usual molecular orbital energies, and the initial guess for the T1 amplitudes vanishes. 

The form of Eqs. [152] and [153] is perhaps misleading in that many of the terms appear to be computationally more expensive than is necessary. For example, Eq. [153] contains the following term which is quadratic in the T2 amplitudes  

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A superior method for the calculation of excited-state PE surfaces is CC2, which is a simplified and computationally efficient variant of coupled-cluster theory with single and double excitations [22], CC2 can be considered as the equivalent of MP2 for excited electronic states. Efficient implementations of CC2 with density fitting [23] and analytic gradients [24] allow reaction path calculations for rather large systems. Being a singlereference method, CC2 fails in the vicinity of conical intersections of excited states with the electronic ground state. 

A second purpose of the present work is to assess the performance of the explicitly correlated coupled-cluster model CCSD(F12) that we have recently implemented in the TuR-BOMOLE program package [68, 69]. This model has the potential to yield electronic molecular energies at the level of coupled-cluster theory with single and double excitations (CCSD [37, 70]) at the limit of a complete one-particle basis set. In conjunction with corrections for higher excitations (connected triples and connected quadruples) it should be possible to compute the barrier height for the above reaction with an accuracy of about 1-2 kJ mol that is, with an error of about 0.5-1.0%. 

Most of the models described above have also been implemented at correlated levels of tlieory, including perturbation theory. Cl, and coupled-cluster theory (of course, the DFT SCRF process is correlated by construction of the functional). Unsurprisingly, if a molecule is subject to large correlation effects, so too is the electrostatic component of its solvation free energy. 

With increasing use of such models, methods are likely to become more concisely defined in the near future. At present, the models for which protocols and parameters have been most clearly defined and where a fair number of applications have appeared applying those models in a consistent fashion include the aheady noted AM1/TIP3P model (more generally AMl/OPLS when solvents other than water are employed in the MM region) and a similarly fashioned HF/3-21G/OPLS model (Freindorf and Gao 1996). Implementations carrying the QM level as far as coupled-cluster theory have been reported (Kongsted et al. 2003). 

Korona T, Przybytek M, Jeziorski B. Time-independent coupled cluster theory of the polarization propagator. Implementation and application of the singles and doubles model to dynamic polarizabilities and Van der Waals constants, 2006. Submitted to Mol. Phys 104 2302-2316... 

Evangelista, F. A. AUen, W. D. Schaefer m, H. F. Coupling term derivation and general implementation of state-specific multireference coupled cluster theories, J. Chem. Phys. 2007,127, 024102-024117. 

X. Li and J. Paldus,/. Chem. Phys., 101, 8812 (1994). Automation of the Implementation of Spin-Adapted Open-Shell Coupled-Cluster Theories Relying on the Unitary Group Formalism. 

In Volume 5 of this series, R. J. Bartlett and J. E Stanton authored a popular tutorial on applications of post-Hartree-Fock methods. Here in Chapter 2, Dr. T. Daniel Crawford and Professor Henry F. Schaefer III explore coupled cluster theory in great depth. Despite the depth, the treatment is brilliantly clear. Beginning with fundamental concepts of cluster expansion of the wavefunction, the authors provide the formal theory and the derivation of the coupled cluster equations. This is followed by thorough explanations of diagrammatic representations, the connection to many-bodied perturbation theory, and computer implementation of the method. Directions for future developments are laid out. 

Ab initio and DFT Calculations. - For small molecules consisting of light atoms, the ab initio correlated methods implemented by Gauss in coupled cluster theory are the most accurate. In this review period the chemical shifts in the (E) and (Z)- isomers of the penta-l,3-dienyl-2-cation were calculated at MP2/tzp... 

The thesis begins with Section 2, where a brief history about the explicitly correlated approaches is presented. This is followed by Section 3 with general remarks about standard and explicitly correlated coupled-cluster theories. In Section 4, the details about the CCSD(F12) model relevant to the implementation in TuRBOMOLE are presented. The usefulness of the developed tool is illustrated with the application to the problems that are of interest to general chemistry. A very accurate determination of the reactions barrier heights of two CH3+CH4 reactions has been carried out (Section 5) and the atomization energies of 106 medium-size and small molecules were computed and compared with available experimental thermochemical data (Section 6). The ionization potentials and electron affinities of the atoms H, C, N, O and F were obtained and an agreement with the experimental values of the order of a fraction of a meV was reached (Section 7). Within all applications, the CCSD(F12) calculation was only a part of the whole computational procedure. The contributions from various levels of theory were taken into account to provide the final result, that could be successfully compared to the experiment. 

Although a wide variety of theoretical methods is available to study weak noncovalent interactions such as hydrogen bonding or dispersion forces between molecules (and/or atoms), this chapter focuses on size consistent electronic structure techniques likely to be employed by researchers new to the field of computational chemistry. Not stuprisingly, the list of popular electronic structure techniques includes the self-consistent field (SCF) Hartree-Fock method as well as popular implementations of density functional theory (DFT). However, correlated wave function theory (WFT) methods are often required to obtain accmate structures and energetics for weakly bound clusters, and the most useful of these WFT techniques tend to be based on many-body perturbation theory (MBPT) (specifically, Moller-Plesset perturbation theory), quadratic configuration interaction (QCI) theory, and coupled-cluster (CC) theory. 

A time-dependent coupled cluster theory with unrestricted electron spins and full treatment of orbital rotation has been implemented to calculate the polarizabilities and dispersion coefficients. Illustration calculations on Li, Ar, HCl, CO, N2, O2, and H2O at the coupled cluster singles and doubles level have demonstrated the reliabihty of the method. Comparisons with HF and MP2 results have further shown the importance of high-order electron correlation effects whereas basis sets of the aug-cc-pVXZ family have been compared. 

The calculation of molecular properties can be carried out at three distinct levels (i) ab initio, (ii) semi-empirical, (iii) empirical. Ab initio methods have increased enormously in accuracy and efficiency in the last two decades and are the focus of our discussion here. Ab initio methods have developed in two directions first, the level of approximation has become increasingly sophisticated and, hence, accurate. The earliest ab initio calculations used the Hartree-Fock/self-consistent field (HF/SCF) methodology, which is the simplest to implement. Subsequently, such methods as Moller-Plesset perturbation theory, multi-configuration self-consistent field theory (MCSCF) and coupled-cluster theory have been developed and implemented. Relatively recently, density functional theory (DFT) has become very popular, since it yields an accuracy much greater than that of HF/SCF while requiring relatively little additional computational effort. 

Schutz M (2002) A new, fast, semi-direct implementation of linear scaling local coupled cluster theory. Phys Chem Chem Phys 4 3941-3947... 

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