A Trigonometric Minimum Model for Refinement

Refinement of model parameters against observed diffraction data is a widely used technique across many specializations of crystallography.In most cases gradient-driven minimization is employed, usually in combination with exact or approximate second derivatives.Ultimately, such methods are variations of Newton's method for iterative root finding and assume a quadratic model around the minimum.The parameter adjustments d in each iteration are determined as the ratios of 1 st and 2 nd derivatives: d = -f'/f", where f is the function to be minimized.While the success of this approach is evident through the innumerable results it has produced, it is obviously handicapped by singularities if f" approaches zero.In such cases naïve use of Newton's method leads to overestimated, unfeasible parameter adjustments.Consequently, all practical minimization algorithms include shift damping, line search, or trust region methods to achieve numerical stability.A systematic inspection of crystallographic target functions commonly used in the refinement of atomic coordinates reveals that near-zero or negative 2 nd derivatives occur systematically, well within the convergence radius.We observe that in this context a trigonometric (sine or cosine) function is always a better fit to the shape of the minimum, compared to the quadratic model underlying Newton's method.The trigonometric minimum model leads to the formula d = -w/π arctan(π/w f' / f") for the estimate of the parameter adjustment.This function is free of singularities; computer implementations use atan2().By design the maximum shift length is w, which is the halfwidth of the minimum.The trigonometric minimum model allows us to achieve numerical stability by injecting easily obtainable prior knowledge into the minimization procedure, through the parameter w.We will report results of systematically comparing conventional minimizations with minimizations using the trigonometric minimum model.