Coupled cluster method perturbation expansion

MBPT starts with the partition of the Hamiltonian into H = H0 + V. The basic idea is to use the known eigenstates of H0 as the starting point to find the eigenstates of H. The most advanced solutions to this problem, such as the coupled-cluster method, are iterative well-defined classes of contributions are iterated until convergence, meaning that the perturbation is treated to all orders. Iterative MBPT methods have many advantages. First, they are economical and still capable of high accuracy. Only a few selected states are treated and the size of a calculation scales thus modestly with the basis set used to carry out the perturbation expansion. Radial basis sets that are complete in some discretized space can be used [112, 120, 121], and the basis... 

Couple cluster methods differ from perturbation theory in that they include specific corrections to the wavefunction for a particular type to an infinite order. Couple cluster theory therefore must be truncated. The exponential series of functions that operate on the wavefunction can be written in terms of single, double and triple excited states in the determinantl " . The lowest level of truncation is usually at double excitations since the single excitations do not extend the HF solution. The addition of singles along with doubles improves the solution (CCSD). Expansion out to the quadruple excitations has been performed but only for very small systems. Couple cluster theory can improve the accuracy for thermochemical calculations to within 1 kcal/mol. They scale, however, with increases in the number of basis functions (or electrons) as N . This makes calculations on anything over 10 atoms or transition-metal clusters prohibitive. 

Compared with density functional methods, Hartree-Fock-based approaches play a less important role in electronic structure calculations of solids and surfaces, although the exact treatment of the exchange terms is conceptually very appealing. The inclusion of electronic correlation effects in the form of perturbation theory, coupled cluster methods, or configuration interaction expansion is very well developed for the calculation of molecular properties. However, in most cases these approaches are not suited for solid state systems. 

The coupled cluster theory may be derived from the many-body perturbation theory which we have presented above. Each coupled cluster approximation can be obtained by summing certain well-defined types of diagrammatic terms through all orders of the perturbation expansion. We shall not present here the details of the relation between coupled cluster and many-body perturbation theories. For a detailed discussion, the reader is referred to the review by Paldus and Li [81], published in 1999, entitled A critical assessment of coupled cluster method in quantum chemistry and the chapter on coupled cluster theory by Paldus [82] in the Handbook of Molecular Physics and Quantum Chemistry. 

A further difficulty arises from the fact that neither the WF s nor the wave functions obtained from a localization inside a molecule are solutions of the canonical Hartree-Fock equations. Therefore, any method treating correlation (perturbation theory, coupled cluster expansion, etc.) has to be reformulated for this case. This work has been done in the case of perturbation theory for the localization inside a molecule /27/ and it is in progress for the coupled cluster method /28/. 

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... 

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. 

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