Steepest descent energy minimisation

Fig. 9.14 Mean neighbouring ring size, m ), versus (where n is the central ring size) for configurations generated under a number of conditions (all for A 2.57 A ). The magenta line corresponds to the most rapid quench (a steepest descent energy minimisation) whilst the black, red and green lines correspond to successively slower cooling rates. The cyan and violet lines show the effect of removing the defect four- and two-coordinate sites as described in the text. The dashed line shows the prediction of the Aboav-Weaire Law...

We shall not discuss all the numerous energy minimisation procedures which have been worked out and described in the literature but choose only the two most important techniques for detailed discussion the steepest descent process and the Newton-Raphson procedure. A combination of these two techniques gives satisfactory results in almost all cases of practical interest. Other procedures are described elsewhere (1, 2). For energy minimisation the use of Cartesian atomic coordinates is more favourable than that of internal coordinates, since for an arbitrary molecule it is much more convenient to derive all independent and dependent internal coordinates (on which the potential energy depends) from an easily obtainable set of independent Cartesian coordinates, than to evaluate the dependent internal coordinates from a set of independent ones. Furthermore for our purposes the use of Cartesian coordinates is also advantageous for the calculation of vibrational frequencies (Section 3.3.). The disadvantage, that the potential energy is related to Cartesian coordinates in a more complex fashion than to internals, is less serious. 

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