Minimum false

False minima may entrap the unwary a structure may be mistaken for the ground state that does not represent the most stable conformer. If so, the calculated 

The stored output of a stochastic search, STOCHASTIC.MEM, for the various local minima of n-pentane using Program MM3 is accessed by the following four lines 

File 5-3 Output From a Stochastic Search of the n-Pentane Molecule in MM3. The input structure was subjected to 50 kicks.  

The energy differences among conforiiiers relative to the ground state are 0.0, 0.85, 1.62, and 3.32 kcal mol . The relative populations of the states, judged by the number of times they were found in a random search or 50 trials, are 0.16, 0.21, 0.15, and 0.08 when degeneracy is taken into account. In the limit of ver y many runs, a Boltzmann distr ibution would lead us to expect a ground state that is much more populous than the output indicates, but this sample is much too small for a statistical law to be valid. 

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Damp the diagonal of the least squares matrix. This stops the calculation from locating false minima by moving parameters too far in a given cycle of refinement. 

In this chapter we consider in detail how one goes about performing molecular mechanics calculations. There are a number of important considerations such as having an adequate starting model, choosing an appropriate energy minimization method, and the probability that there are many possible energy minima. There are also pitfalls that need to be avoided such as false minima, saddle points and unstable refinements. 

The use of k > n+1 points ensures that the complex does not collapse into a subspace. False minima are normally eliminated through the procedure of reflecting a point about the centroid, and because the set of points is distributed over the whole interval. If a false minimum presents particular difficulties, it can usually be eliminated by repeating the optimization with different sets of trial points. 

Choose a reasonable set of trial values for the adjustable parameters of the model if nonlinear fitting is to be done. A good choice of a° is very important to reduce computational time and to avoid false minima (see Goodness of Fit, toward the end of the discussion of tests). 

FIGURE 2 Example of a convergence to false minima of the least mean squares fit routine, (a) A fit produced by the computer of three sets of doublets (constrained to equal line widths and equal intensities of the components of each pair) to an experimental spectrum (b) a comparison between the superposition of the six lines calculated from (a) and the experimental data illustrates a substantial misfit especially in the positive velocity region of the spectrum (c) an alternate fit to the data (constrained to two doublets and a singlet of different width), giving a much better agreement between the calculated and experimental values. It should be noted, however, that the agreement obtained in (c) is not unique hence, additional experimental information must be exploited to arrive at a reliable estimate of the spectroscopic parameters. 

This procedure is called global fitting. The origin of tp is also taken with a parameter, but for simplicity this parameter is omitted here. Since f is a nonlinear function of k, values, it can have many false minima. In order to avoid such false minima and to choose the true minimum, we need to obtain f that is closest to zero then Eq. (5) will be satisfied. Substituting Eq. (5) into the right-hand side of Eq. (4), and equating the coefficients of hftp) of both sides, we obtain... 

Some problems with the DFT method such as false minima in the PSDs for pore sizes vrith a diameter of 2 and 3 times the molecular diameter of the adsorbate have been pointed out [25] and are open to much discussion [26]. As a contribution to overcome these limitations, a round robin inter-laboratoiy trial of PSD analysis using N2 adsorption at 77 K in conjunction with the DFT theory, launched by G. Newcombe and L.R. Radovic, is currently underway [27]. 

Certain structural features arise from a cursory analysis of any of the data sets. Each of the 3 above data sets yields maps similar to those in Figures 2 and 3 that clearly show an area of optimal chain rotation. (Because of the overlapped reflections, there is a false minimum at about 140°, 140° that aligns the chains along the shorter x axis. Chains rotated to the position of the false minima have short interatomic distances.) With similar clarity, the gg 06 position is eliminated by all of the above data sets, as exemplified in Table X, below. 

Automatic fitting of EPR spectra is possible, although most techniques have problems with false minima and some degree of manual steering is necessary. One of the frequently encountered problems is that the least-squares difference is rarely an adequate measure of goodness of fit. In practice it is usually necessary to obtain a reasonable fit manually before automatic fitting is capable of further optimizing the spin Hamiltonian parameters. 

Analogous to the second graphical solution shown in Figure 6.5, the optimization routine is modified to accommodate q = 2 and 0.025 < t/R. Note that the results of R = 0.0294 and t = 0.0002 are not unique and any combination along the curve from point A1 to A2 will satisfy the constraints. Furthermore, we must continually check that the weight is indeed minimized. For example, the optimization routine will often find false minima along the constraint line that satisfies all the constraints, but is not a global minimum of W = 0.2 N. 

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