**4. Which minimization method should 1 use?**

The choice of Minimization algorithm is determined by a number of components, including the storage and computational requirements, the relative speeds with which the various parts of the calculation can be performed, the availability of analytical derivatives and the robustness of the method. Thus, any method that requires the Hessian matrix to be stored (let alone its inverse calculated) may present memory problems when applied to systems containing thousands of atoms. Calculations on systems of this size are invariably performed using molecular mechanics, and so the steepest descents and the conjugate gradients methods are very popular here. For molecular mechanics calculations on small molecules, the Newton–Raphson method may be used, although this algorithm can have problems with structures that are far from a minimum. For this reason it is usual to perform a few steps of Minimization using a more robust method such as the simplex or steepest descents before applying the Newton–Raphson algorithm Analytical expressions for both first and second derivatives are available for most of the terms found in common force fields. The steepest descent method can actually be superior to conjugate gradients when the starting structure is some way from the minimum. However, conjugate gradients are much better once the initial strain has been removed. 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 Minimization 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.

When using internal coordinates in a quantum mechanical Minimization it can be important to use an appropriate Z-matrix as input. For many systems the Z-matrix can often be written in many different ways as there are many combinations of internal coordinates. There should be no strong coupling between the coordinates. Dummy atoms can often help in the construction of an appropriate Z-matrix. A dummy atom is used solely to define the geometry and has no nuclear charge and no basis functions. Strong coupling between coordinates can give long 'valleys' in the energy surface, which may also present problems. Care must be taken when defining the Z-matrix for cyclic systems in particular. The natural way to define a cyclic compound would be to number the atoms sequentially around the ring. However, this would then mean that the ring closure bond will be very strongly coupled to all of the other bonds, angles and torsion angles. Some quantum mechanics programs are able to convert the input coordinates (be they Cartesian or internal) into the most efficient set for Minimization so removing from the user the problems of trying to decide what is an appropriate set of internal coordinates. For energy Minimizations redundant internal coordinates have been shown to give significant improvements in efficiency compared with Cartesian coordinates or non-redundant internal coordinates, especially for flexible and polycyclic systems [21]. The redundant internal coordinates employed generally com- comprise the bond lengths, angles and torsion angles in the system. These methods obviously also require the means to inter-convert between the internal coordinate representation and the Cartesian coordinates that are often used as input and desired as output. Of particular importance is the need to transform energy derivatives and the Hessian matrices (if appropriate) [22].
