Method summator identities

The method of the summator identities (the method of approximating an integral identity). A solution of the problem... 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

The 1/r solution is in fact just an Euler s method approximation to the integral for the PFTR, in which one approximates the integral as a summation. The calculation is not very accurate because we used a 0.2 moles/liter step size to keep the spreadsheet small, but it illustrates the method and the identity between Euler s method and a spreadsheet solution. 

The exact solutions are not valid if any of the model inputs differ from the distribution type that is the basis for the method. For example, the summation of lognormal distributions is not identically normal, and the product of normal distributions is not identically lognormal. However, the Central Limit Theorem implies that the summation of many independent distributions, each of which contributes only a small amount to the variance of the sum, will asymptotically approach normality. Similarly, the product of many independent distributions, each of which has a small variance relative to that of the product, asymptotically approaches lognormality. 

Using atomic densities calculated from tabulated atomic wave functions, the summation was found [214] to produce results equivalent to the most elaborate molecular Hartree-Fock calculations for a series of small molecules, at a fraction of the computing expense. Surface areas and volumes computed by the two methods were found virtually identical. The promolecule calculation therefore has an obvious advantage in the exploration of surface electron densities, surface areas and molecular volumes of macromolecules for the analysis of molecular recognition. 

For random correlations (7i—54) = 0 and the scattering is independent of This corresponds to a scattering pattern which is cylindrically symmetrical about the incident beam. For non-random correlations (7i—S4) is finite which leads to a pattern which has four-fold symmetry in as is often experimentally observed (Fig. 27). Thus in addition to the usual correlation function 7i(r), the coefficient T4.—S4) plays the role of another correlation function which characterises the shape of the correlated region. The evaluation of these correlation functions for perfect two-dimensional spherulites has been carried out and their substitution into eqn. (84) has been shown to lead to a scattering pattern which is identical with that which is directly calculated by the amplitude summation method. Thus, this formulation enables one in principle to describe scattering from systems ranging from random to highly ordered. 

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