Derivative, energy minimisation

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. 

In connection with the derivation of molecular symmetries by force field calculations we would like first to make a few cautionary remarks and then show how symmetry properties of energy minimisations may in certain cases facilitate the evaluation of conformational transition states (59). 

H - H interactions, both for all-(S) and (R,R,S,S) tetramers, even after conformational energy minimisation. Overall, most of the >40 kcal mol-1 enthalpic advantage of the [dimer + dimer to tetramer] process is lost on going from methylzinc to isopropylzinc-derived structures, because of the increased steric strain. The remaining [Zn-Oh square-based tetramers are all more strained in the isopropyl than in the Me series. As an extreme, the latter is less stable than two isolated monomers. What stands out is that the square-capped macrocycle is only modestly more strained in the isopropyl series, and lacks severe H - H interactions (only two H - H contacts are below 2.3 A). Forming the barrel isomer from this by making two additional Zn - N... 

One important point that we should bear in mind as we undertake a deeper analysis of molecular mechanics is that force fields are empirical-, there is no correct form for a force field. Of course, if one functional form is shown to perform better than another it is likely that form will be favoured. Most of the force fields in common use do have a very similar fqrm, and it is tempting to assume that this must therefore be the optimal functional form Certainly such models tend to conform to a useful picture of the interactions present in a system, but it should always be borne in mind that there may be better forms, particularly when developing a force field for new classes of molecule. The functional forms employed in molecular mechanics force fields are often a compromise between accuracy and computational efficiency the most accurate functional form may often be unsatisfactory for efficient computation. As the performance of computers increases so it becomes pcKsible to incorporate more sophisticated models. An additional consideration is that in order to use techniques such as energy minimisation and molecular dynamics, it is usually desirable to be able to calculate the first and second derivatives of the energy with respect to the atomic coordinates. 

Fig. 2 The types of molecules studied in the first decade of crystal structure prediction, as derived from a survey of published lattice energy minimisation based studies. Within each category, the molecules are generally the smaller and more rigid molecules whose published crystal structures are in common space groups with Z = l.

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