Computer implementation of coupled cluster theory

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%. 

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. 

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. 

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. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

For molecules that require a multireference description, use of a single reference post-HF method can often fail since the dynamic correlation space is insufficient. Multireference post-HF methods are quite taxing in terms of computational resources and comprise a very active area of theoretical development." A method that has shown some recent promise is multireference coupled cluster (MRCC) theory, and the implementation proposed by Mukheijee and coworkers" (often labeled as MkCC or MkMRCC) has garnered much interest." ... 

With regard to the electronic structure methodology, major obstacles must be surmounted before improvements can be made. Calculations with Coupled-Cluster methods, an obvious next step, are far more computationally costly than the presently used MP2, or B3LYP methods. In fact, there are extremely few direct ab initio calculations of anharmonic vibrational spectroscopy at higher than MP2 or DPT levels, even for small polyatomics. From the point of view of ab initio anharmonic spectroscopy, the leap from MP2 to the Coupled-Cluster method seems a bottleneck. One can draw encouragement from faster Coupled-Cluster implementations, so far employed with the perturbation theory anharmonic analysis [116,117]. 

Erik Deumens work in computational chemistry has focused on two projects the development of a theory for the description of the dynamics of electrons and nuclei in molecular reactions as fully coupled degrees of freedom, and the implementation of this theory in computer software. The second project is the building of infrastructure for computation at QTP (Quantum Theory Project, Gainesville, FL), including building clusters, software libraries and courses. 

Computer simulations of excess proton conductivity in water have reached a powerful level [8,92,93,102]. Importantly, simulations extend to quantum-mechanical proton dynamic features, so that proton motion can be coupled to details of the molecular environmental dynamics. A recent feature article explored an analytical theory in order to rationalize these complex processes that involve interconversion of proton-bearing clusters and proton transfers [103]. With a simple two-state empirical valence bond model (see below for details), which implements in a classical way the above-mentioned idea of two limiting protonated structures, namely the 11502 and the H30 cluster, it was indeed observed that the two alternative sequences are equivalent with similar life times for both clusters, and that conversions between the two clusters are purely fluctuative. 

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