Cluster expansion fundamentals

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. 

As noted in Section 2, the fundamental bond function in cluster expansion theory is the Mayer / function defined in Eq. (8). Both si and g can be expressed very simply in terms of graphical series containing / bonds. 

Such a theory is presented in this chapter corroborated by various experimental data, the combination of microscopic quantum mechanics in the shape of density functional theory (DFT) with the statistical approach of the cluster expansion (CE) provides an approach for predicting the segregation on a fundamental level. In the following three sections, we describe experimental and theoretical ways to determine and predict the surface segregation properties of materials. Before coming to the probably most modem approach to predict surface in Section 11.3.3, we present some experimental methods in Section 11.3.1 and the basis for ab initio modehng in Section 11.3.2. 

It is important to note, however, that there are fundamental differences between FSCC and SRCC with respect to the nature of their excitation operators. For a given truncation of the cluster operators beyond simple double excitations, the determinantal expansion space available in an FSCC calculation is smaller than those of SRCC calculations for the various model space determinants. A class of excitations called spectator triple excitations must be added to the FSCCSD method to achieve an expansion space that is in some sense equivalent to that of the SRCC. But even then, the FSCC amplitudes are restricted by the necessity to represent several ionized states simultaneously. Thus, we should not expect the FSCCSD to produce results identical to a single reference CCSD, nor should we expect triple excitation corrections to behave in the same way. The differences between FSCC and SRCC shown in Table I and others, below, should be interpreted as a manifestation of these differences. 

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