Current Methods and Optimization Algorithms for the Refinement of X-Ray Crystal Structures

Abstract

Optimization algorithms and other techniques used for the crystallographic refinement of both small molecule and macromolecular X-ray crystal structures are reviewed in this work. Emphasis is made on the advantages and disadvantages of every method in its application to the refinement process of different problems, as well as on their actual implementation in current program packages. A description of the most basic (first-level) algorithms based on physical grounds, such as maximum-likelihood, least-squares, molecular dynamics, maximum-entropy, Fourier and graphical methods, and holographic methods, is followed by a second-level approach, including Newton methods, simulated annealing, and gradient methods, among others, and finally by a third-level study which includes matrix decompositions and iterative methods for the solution of the resulting linear systems of equations. In addition, by descending to the most basic mathematical and physical level, connections with other disciplines that use almost the same tools are easily made, showing not only the interdisciplinarity of crystallography but also the flow of information between different sciences.