Common issues: convergence criteria and step length

Since the PES is not an analytic function the optimization and the search of the local roots must be performed numerically in an iterative fashion. In the sections that follow a wide landscape of techniques will be described. But all of them have two points in common:

convergence criteria:

we start from an initial structure but since the mathematically exact minimum will never be reached some convergence criteria must be adopted.

step length:

all the techniques have an algorithm to predict a geometry step to get progressively closer to the stationary point. But while most strategies provide a displacement vector that points to a certain direction, few of them are able to predict with accuracy the length of the vector along this direction. In this case a step length must be determined.

The convergence criteria is not unique. The most common criterion is to consider that the search has converged to a point when the norm of the gradient is lower than a threshold (actually this is the only requirement for a stationary point) ^2.5. Sometimes the maximum component of the gradient vector is required to be under a threshold as well. Other convergence criteria are the change in energy between the previous and the current structure. Otherwise the RMS of the displacement predicted by the algorithm for the next step search.

The step length prediction is not unique either. The trust radius approximation considers a fixed step length to the direction of optimization. Line search technique is a very useful strategy where an interpolated one-dimensional polynomial function describes the profile in the displacement direction. In this case the length of the displacement is the length to reach the minimum in the polynomial function.

Line search usually needs additional energy evaluations for such interpolation, but some techniques exist that avoid this waste [12]. Other techniques, such as the Rational Function Optimization method used in this thesis, and outlined in section 1.3.4.2, have an implicit step size determination.