Coupled-cluster technique

The second step of the calculation involves the treatment of dynamic correlation effects, which can be approached by many-body perturbation theory (62) or configuration interaction (63). Multireference coupled-cluster techniques have been developed (64—66) but they are computationally far more demanding and still not established as standard methods. At this point, we will only focus on configuration interaction approaches. What is done in these approaches is to regard the entire zeroth-order wavefunc-tion Tj) or its constituent parts double excitations relative to these reference functions. This produces a set of excited CSFs ( Q) that are used as expansion space for the configuration interaction (Cl) procedure. The resulting wavefunction may be written as... 

J. Cizek, Theor. Chim. Acta, 80, 91 (1991). Origins of Coupled Cluster Technique for Atoms... 

Another recently created correlated method is the no(virtual)pair DF (DCB) coupled-cluster technique of E. Eliav, U. Kaldor and I. Ishikawa [31,32,44]. It is based on the DCB Hamiltonian Equation 3. Correlation effects are taken into account by action of the excitation operator... 

There are at least three types of cluster expansions, perhaps the most conventional simply being based on an ordinary MO-based SCF solution, on a full space entailing both covalent and ionic structures. Though the wave-function has delocalized orbitals, the expansion is profitably made in a localized framework, at least if treating one of the VB models or one of the Hubbard/PPP models near the VB limit -and really such is the point of the so-called Gutzwiller Ansatz [52], The problem of matrix element evaluation for extended systems turns out to be somewhat challenging with many different ideas for their treatment [53], and a neat systematic approach is via Cizek s [54] coupled-cluster technique, which now has been quite successfully used making use [55] of the localized representation for the excitations. 

Later work evaluated the two-dimensional potential energy surface using various correlation treatments including many-body perturbation theory and coupled cluster techniques Evaluation of the vibrational spectrum was explicitly anharmonic in nature, mak-... 

The exponential ansatz given in Eq. [31] is one of the central equations of coupled cluster theory. The exponentiated cluster operator, T, when applied to the reference determinant, produces a new wavefunction containing cluster functions, each of which correlates the motion of electrons within specific orbitals. If T includes contributions from all possible orbital groupings for the N-electron system (that is, T, T2, . , T ), then the exact wavefunction within the given one-electron basis may be obtained from the reference function. The cluster operators, T , are frequently referred to as excitation operators, since the determinants they produce from fl>o resemble excited states in Hartree-Fock theory. Truncation of the cluster operator at specific substi-tution/excitation levels leads to a hierarchy of coupled cluster techniques (e.g., T = Ti + f 2 CCSD T T + T2 + —> CCSDT, etc., where S, D, and... 

We emphasize that the present discussion focuses only on high-spin open-shell systems to which a single-determinant reference wavefunction is applicable. Coupled cluster techniques for low-spin cases, such as open-shell singlets, have been pursued in the literature for many years, however, and provide a fertile area of research (Refs. 158, 167-170). 

This becomes more complicated once the algorithm is parallelized to utilize modem computer architectures. Again one may therefore consider to utilize existing implementations of coupled cluster techniques to benefit from the work done in this area. The first step would be to derive a restricted algorithm that can be compared to non-relativistic closed shell algorithms. 

Non-coupled cluster techniques for generating MC wave functions exist and have been employed, for example, to study the bonding and structure of titanium hydrides and titanium organometallics " by Gordon and co-workers. These papers also highlight that within the realm of computational chemistry, a sophisticated MC treatment is needed, in many cases to qualitatively describe the bonding of an organometallic. 

CFOUR, Coupled-Cluster techniques for Computational Chemistry, a quantum-chemical program package by Stanton JF,... 

We should mention a special coupled-cluster technique, which has proved to be very valuable in four-component calculations. If the electronic structure of a molecule is not even qualitatively well described by a single electronic... 

Stanton IF, Gauss J, Harding ME, Szalay PG (2010) CFOUR, coupled-cluster techniques for computational chemistry (http // www.cfour.de)... 

The four C s in the name of the program stand for Coupled-Cluster techniques for Computational Chemistry. The predecessor of this program was known as the Mainz-Austin-Budapest version of ACES II which was replaced by CFOUR in April 2009 (see http //www.aces2. de/). At present, the development of the program is continued at the University of Mainz (J. Gauss), at the University of Texas at Austin (J.F. Stanton and M.E. Harding), and at Eotvos Lorand University (EG. Szalay). 

The two relativistic four-component methods most widely used in calculations of superheavy elements are the no-(virtual)pair DF (Coulomb-Breit) coupled cluster technique (RCC) of Eliav, Kaldor, and Ishikawa for atoms (equation 3), and the Dirac-Slater discrete variational method (DS/DVM) by Fricke for atoms and molecules. " Fricke s DS/DVM code uses the Dirac equation (3) approximated by a Slater exchange potential (DFS), numerical relativistic atomic DS wavefunctions, and finite extension of the nuclei. DFS calculations for the superheavy elements from Z = 100 to Z = 173 have been tabulated by Fricke and Soff. A review on various local density functional methods applied in superheavy chemistry has been given by Pershina. ... 

Applications to atoms are in most cases based on the publicly available programs using finite difference methods for integration in the solution of the (multi-configurational) Dirac-Hartree-Fock equations. The problem of introducing electron correlation in this framework is most successfully accomplished by employing complete active space (CAS) and restricted active space (RAS) techniques (see Ref. 84 for a recent application with further references to the literature) or coupled-cluster techniques. ... 

Clearly, we have at our disposal a systematic procedure for improving our description of the electronic system. As demonstrated in Section 5.8, this technique can be very powerfiil, providing, for the calculation of electron correlation energies, an attractive alternative to the Cl and coupled-cluster techniques of Chapters 11 and 13. 

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