Least squares optimization procedure

A starting set of force constant values, from which are calculated a set of normal coordinates, Qa, and normal frequencies, v, needs to be chosen. These are refined to observed data by a least-squares optimization procedure. We note that a normal mode can be visualized in terms of the local, usually symmetry (S, ), coordinates since... 

As formulated, the charge flow model provides a possibihty for simultaneous calculation of infiared band intensities associated with fundamental and binary overtone and combination transitions. A least squares optimization procedure for diese calculations has been applied [161]. The parameters and dC /dR are determined from the intensities of fundamental modes. Initial guess values of the highm- overtone terms are iteratively optimized to fit the observed intmisities of the binary overtone and combination bands. Reliable experimental data for overtone and combination band absorption intensities can be determined for very small molecules. 

Once a force-field model has been chosen, we need a mechanism whereby a set of force constants can be selected that gives optimal prediction of a (usually larger) set of observed frequencies. This could be done by a manual trial-and-error adjustment, but a least-squares fitting procedure is more satisfactory (for discussions of these methods see Duncan, 1975 Califano, 1976 Gans, 1977 Zerbi, 1977). In the present discussion we assume that we have reasonable starting force constants... 

There are situations where a net atomic charge model does not give the desired accuracy of fit to the electric potential. Since the least-squares fitting procedure is a curve-fitting process, it is expected that addition of new variables of appropriate mathematical form will improve the fit. The addition of a new fixed site adds implicit variables, x, y, z, of the site location and an explicit variable, q, the site charge. The new site can be treated as a dummy atom with net charge q, and this charge can be optimized. 

The Linked-Atom Least-Squares (LALS) procedure (44) was used to generate molecular models of cellotetraose. The glucose residues were kept in the standard Cj conformation all bond angles cmd bond lengths were fixed at standard values (43). The constrained model of the crystal structure was optimized against both X-ray data and non-covalent interatomic interactions, as described by Smith and Arnott (44). 

Equation 2.87 can be used to represent the CLD of polyolefins made with the combination of two or more metallocenes very well [35, 38]. This is a reasonably easy case, since the individual metallocenes can be tested separately to obtain the values of Flory s T parameter. For multiple-site catalysts, such as heterogeneous Ziegler-Natta and Philips catalysts, the procedure for obtaining r values for each site type is more elaborate and involves the deconvolution of the MWD into several Flory s distributions. This subject will not be covered in this chapter the reader is directed to references 39 and 40 at the end of the chapter for more information on this subject. Suffice to say that MWD deconvolution involves the use of a non-linear least-squares optimization routine to... 

Firstly, it has been found that the estimation of all of the amplitudes of the LI spectrum cannot be made with a standard least-squares based fitting scheme for this ill-conditioned problem. One of the solutions to this problem is a numerical procedure called regularization [55]. In this method, the optimization criterion includes the misfit plus an extra term. Specifically in our implementation, the quantity to be minimized can be expressed as follows [53] ... 

Most of the force fields described in the literature and of interest for us involve potential constants derived more or less by trial-and-error techniques. Starting values for the constants were taken from various sources vibrational spectra, structural data of strain-free compounds (for reference parameters), microwave spectra (32) (rotational barriers), thermodynamic measurements (rotational barriers (33), nonbonded interactions (1)). As a consequence of the incomplete adjustment of force field parameters by trial-and-error methods, a multitude of force fields has emerged whose virtues and shortcomings are difficult to assess, and which depend on the demands of the various authors. In view of this, we shall not discuss numerical values of potential constants derived by trial-and-error methods but rather describe in some detail a least-squares procedure for the systematic optimisation of potential constants which has been developed by Lifson and Warshel some time ago (7 7). Other authors (34, 35) have used least-squares techniques for the optimisation of the parameters of nonbonded interactions from crystal data. Overend and Scherer had previously applied procedures of this kind for determining optimal force constants from vibrational spectroscopic data (36). 

Chapter 2 summarizes the characteristics of process models and explains how to build one. Special attention is focused on developing mathematical models, particularly empirical ones, by fitting empirical data using least squares, which itself is an optimization procedure. 

Method III. The weighted nonlinear least squares parameter optimization procedure (34, 35) was applied to the entire set of points shown in Figure 5 for 0.1 M KNO-j. The value of C was fixed and the optimal values of and aHz were obtained. The model... 

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