Coupled-cluster method concept

The concept of the coupled cluster method as an eigenvalue problem may be easily generalized to include excited states (in this case, states that are not the lowest in energy within a given symmetry). We may write the more general right-hand problem as... 

Recently Lindroth (30) has introduced the concept of complex coordinates into nonvariational methods. The idea of Lindroth is to produce a basis set of one-particle complex rotated functions as solutions of the one-particle complex-rotated problem, and then to apply this basis within a bound-state method for a many-particle system. In this way the complex rotation has been combined with the many-body perturbation theory (30) and with the coupled cluster method (31). Apart from the fact that the resulting energy is complex and its real and imaginary part can be interpreted as the position and width of a resonance under consideration, the use of complex coordinates has the advantage that singularities caused by the degeneracy on the real axis between the resonance and the adjacent continuum are not present in the complex energy plane. 

Parallel to these endeavors, work started in Germany on new concepts to account for electron correlation. The independent electron pair approach (lEPA) was developed by Ahlrichs and Kutzelnigg, followed a few years later by the CEPA (coupled electron pair approach).The relation of these methods to contemporary Moller-Plesset second order (MP2) and coupled cluster treatments is discussed in Ref. 60. Work on circular dichroism by Ruch and on the chemical shift by Voitlander showed the variety of ab initio problems treated. The special priority program of the DFG from 1966-1970 demonstrated the intended impact. 

Even though computers were an essential tool in quantum chemical calculations, the main challenge was the further development of methods and concepts to describe even more facets of chemistry and with higher accuracy. Methods that account for electron correlation were extended to be able to describe energy surfaces more reliably. Several variants of the CEPA Ansatz (CEPA-1, CEPA-2) were developed as well as the method of self-consistent electron pairs (SCEP). Formulations using canonical or localized orbitals (e.g., pair natural orbitals, PNO, as a kind of optimized virtual orbitals) were put forth. These methods were extensively used for two decades, primarily in Germany, until coupled cluster formulations became more popular. ... 

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 concept of extremal electron pairs is discussed in the context of coupled-cluster theory and the MP2-R12 method. Using extremal pairs the numerical stability of R12-methods is considerably improved, which is demonstrated for CCSD(T)-R12 calculations of the molecules F2, N2, and Be2-... 

We begin this paper by shortly recapitulating the concept of extremal pair functions (Sect. 2). Then we consider extremal pair functions in the context of Moller-Plesset perturbation theory (Sect. 3) and coupled-cluster theory (Sect. 4). We then come to the main topic of this paper, the use of extremal pairs in R12-methods. To this end we formulate a new access to R12-theory starting with two-electron systems (Sect. 5) and generalizing it to n-electron systems (Sect. 6). We show then how extremal pairs arise in a natural way in R12-methods (Sect. 7). We finish (Sect. 8) by giving numerical examples which demonstrate the gain in numerical stability by using extremal pairs in Recalculations. 

The failing of MBPT is that it is basically an order-by-order perturbation approach. For difficult correlation problems it is frequently necessary to go to high orders. This will be the case particularly when the single determinant reference function offers a poor approximation for the state of interest, as illustrated by the foregoing examples at 2.0 R. A practical solution to this problem is coupled-cluster (CC) theory. In fact, CC theory simplifies the whole concept of extensive methods and the linked-diagram theorem into one very simple statement the exponential wavefunction ansatz. 

As discussed in Section 4.3. a computational method is said to be size-extensive if a calculation on the compound system AB consisting of two noninteracting systems A and B yields a total energy equal to the sum of the energies obtained in separate calculations on the two subsystems. This property of the coupled-cluster model is demonstrated for the linked formulation in Section 13.3.1, leading to the concept of termwise size-extensiAdty in Section 13.3.2. In Section 13.3.3, we consider size-extensivity in the unlinked formulation of coupled-cluster theory, demonstrating how size-extensivity in this case arises from a cancellation of terms that individually violate size-extensivity. 

In the present chapter, which deals with theoretical concepts applied to vanadium and molybdenum oxide surfaces, we will restrict the discussion to binary oxide systems. So far, mixed metal oxide systems have not been studied by quantitative theory. Theoretical methods that have been used to study oxide surfaces can be classified according to the approximations made in the system geometry where two different concepts are applied at present, local cluster and repeated slab models. Local cluster models are based on the assumption that the physical/chemical behavior at selected surface sites can be described by finite sections cut out from the oxide surface. These sections (surface clusters) are treated as fictitious molecules with or without additional boundary conditions to take the effect of environmental coupling into account. Therefore, their electro-... 

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