Minimisation methods

Murtagh B A and Sargent R W 1970 Computational experience with quadratically convergent minimisation methods Comput. J. 13 185... 

In order to use a derivative minimisation method it is obviously necessary to be able to calculate the derivatives of fhe energy wifh respecf to the variables (i.e. the Cartesian or interna] coordinates, as appropriate). Derivatives may be obtained either analytically or numerically. The use of analytical derivatives is preferable as fhey are exact, and because they can be calculated more quickly if only numerical derivatives are available then it may be more effective to use a non-derivative minimisation algorithm. The problems of calculating analytical derivatives with quantum mechanics and molecular mechanics were discussed in Sections 3.4.3 and 4.16, respectively. 

Quantum mechanical calculations are restricted to systems with relatively small numbers of atoms, and so storing the Hessian matrix is not a problem. As the energy calculation is often the most time-consuming part of the calculation, it is desirable that the minimisation method chosen takes as few steps as possible to reach the minimum. For many levels of quantum mechanics theory analytical first derivatives are available. However, analytical second derivatives are only available for a few levels of theory and can be expensive to compute. The quasi-Newton methods are thus particularly popular for quantum mechanical calculations. 

Following the recently developed energy minimisation method described by M.U. Schmidt [2], for the first time three dimensional crystal structure determinations were carried out with P.Y.13 and P.Y.14 [3]. The studies showed the molecules to be planar with the terminal phenyl rings twisted by 4-10° with respect to the acetoacetyl fragments, the degree of planarity of P.Y.14 found to be a little lower than that of P.Y.13. 

Determine thermodynamic relationships between these species by using the free energy minimisation method, i.e., the fact that the free energy is minimum at equilibrium. 

Fig 5 3 A schematic one-dimensional energy surface Minimisation methods move downhill to the nearest minimum The statistical weight of the narrow, deep minimum may he less than a broad minimum which is higher in energy... 

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