Standard coupled-cluster theory

The Hartree-Fock wave function is obtained by variational minimization of the energ functional with respect to orthonormal transformations of the orbitals 

The energy functional involves the Slater determinant, that is an antisymmetrized product of occupied spin orbitals. As the solution, the optimal wave function and the associated energy are obtained. 

In a given orbital basis, the Hartree-Fock description divides the orbital space into a set of occupied and virtual spin orbitals. From the Slater determinant any other determinant may be generated by replacing an occupied orbital by a virtual. Formally such operation is performed within the second quantization formalism by using the excitation operators 

An operator annihilates an electron in spin orbital i, while creates an electron in spin orbital a. The creation and annihilation operators satisfy the usual algebra [a, aj]+ = 6pg. Introducing the generic notation for excitation operators and noting that [r, iv] = 0, we may express the coupled-cluster wave function in the standard exponential form [35] 

To determine the cluster amplitudes and the electronic energy we multiply this equation from the left by the ground state (Hartree-Fock state (HF ) and all possible excited states, called projection manifolds 

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Comparison of the Biillouin-Wigner coupled cluster theory wth the standard coupled cluster theory. Cancellation of disconnected terms in the Brillouin Wigner coupled cluster theory Czechoslovak Chemical Communications 62, 829 (1997)... 

C2H6, respectively, using explicitly correlated coupled-cluster theory with singles and doubles combined with standard coupled-cluster theory with up to connected quadruple excitations. Transition-state theory has been used to compute the respective reaction rate constants in the temperature interval 250-1500 K. The computed rates for the reaction to ethane are orders of magnitude slower than those used in the mechanism of Norinaga and Deutschmann (Ind. Eng. Chem. Res. 46 (2007) 3547) for the modeling of the chemical vapor deposition of pyrolytic carbon. 

In standard coupled-cluster theory, the single-excitation manifold is parametrized in terms of the exponential operator... 

In standard coupled-cluster theory, we use the Hartree-Fock orbitals and then determine a set of nonzero single-excitation amplitudes together with higher-excitation amplitudes. Alternatively, we may use cxp Ti) to generate an orbital transformation to a basis in which the single-excitation amplitudes vanish. Since (13.8.1) and (13.8.2) generate the same state to first order, we may use exp(— ) rather than exp(ri) to generate this orbital transformation, as done in orbital-optimized coupled-cluster (OCC) theory [29,301. The OCC ansatz for the wave function is... 

In OCC and BCC theories, the wave function has the form of a standard coupled-cluster wave function with vanishing singles amplitudes. However, unlike standard coupled-cluster theory, where the orbitals are determined in a separate optimization of the reference state HF), the OCC and BCC orbitals are determined simultaneously with the optimization of the cluster amplitudes, making them more suitable for the description of correlation, in a manner reminiscent of MCSCF theory. In practice, the differences between the standard coupled-cluster wave functions and the BCC and OCC wave functions are small, except in systems characterized by Hartree-Fock singlet instabilities such as the allyl radical in Section 10.10.6. In such cases, the Harlree—Fock instability makes the standard approach unsuitable - the BCC and OCC models, by contrast, suffer from no such instabilities. 

The field of quantum chemistry has seen tremendous development over the last thirty years. Thanks to high-accuracy models such as coupled-cluster theory and standardized, widely available program packages such as Gaussian 98, what was once merely an esoteric tool of a few specialists has evolved into an indispensable source of knowledge for both the prediction and the interpretation of chemical phenomena. With the development of reduced scaling algorithms for coupled cluster... 

The exact FCI (frill configuration interaction) solution of the PPP or Hubbard model is possible for molecules with up to about 16 atoms in the pi system. Any of the standard methods for performing approximate ab initio calculations, such as limited configuration interaction, Moeller-Plesset perturbation theory, or coupled cluster theory, may be applied to these models as well. All are expected to be very accurate at low order when U is small, but all will have to be pushed to higher order as U increases. 

In standard time-independent coupled cluster theory the wavefunction is parameterized as... 

One of the most dramatic changes in the standard theoretical model used most widely in quantum chemistry occurred in the early 1990s. Until then, ab initio quantum chemical applications [1] typically used a Hartree-Fock (HF) starting point, followed in many cases by second-order Moller-Plesset perturbation theory. For small molecules requiring more accuracy, additional calculations were performed with coupled-cluster theory, quadratic configuration interaction, or related methods. While these techniques are still used widely, a substantial majority of the papers being published today are based on applications of density functional theory (DFT) [2]. Almost universally, the researchers use a functional due to Becke, whose papers in 1992 and 1993 contributed to this remarkable transformation that changed the entire landscape of quantum chemistry. 

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. 

More recent investigations with the RPH employ standard correlation-corrected methods such as M0ller-Plesset (MP) perturbation theory (see M0ller-Plesset Perturbation Theory) at second or fourth order (MP2, MP4) or coupled cluster (CC) methods (see Coupled-cluster Theory) in connection with DZP or TZP basis sets. The repertoire of methods has recently been extended by applying density functional theory (DFT) (see Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field) and some convincing results have been published (see Section 3). 

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