Steepest-descent techniques, potential energy

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. 

Here, we consider how we can approximately describe the size dependence of the potential energy in Fig. 23.3. As discussed above, molecular coordinates at the imaginary time t = 8/2 are distributed according to the exact ground state wavefunction, I o(/ )p. Each structure at t = /2 along the VPIMD trajectory is mapped onto a nearest local minimum structure in the configuration space. This can be realized by the steepest descent minimization technique ... 

In order to determine the reaction path one starts at the saddle point, determined by the Newton-Raphson technique described in the previous sections, and follows the steepest descent path to the reactant and product valleys. Along the steepest descent path one takes steps in the direction which leads to a maximal decrease in potential energy. Thus we wish to maximize... 

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