Force-constant computations. Part II. Some model calculations to test the Fletcher–Powell method and to analyse the III-conditioned force-constant refinement

Abstract

The basic cause of non-convergence in refinements of on-diagonal symmetry force-constants is an unacceptable force-field, but such refinements have properties which permit an acceptable force-field to be generated. The oscillatory nature of some refinements employing the Gauss–Newton–Raphson method is shown to stem from the Gauss linear hypothesis whereby the Taylor series expansion of the eigenvalues is cut off after the linear term, by model calculations in which further terms are included. The Fletcher–Powell method, which takes these terms implicitly into account, is shown to refine smoothly to the best possible solution in a variety of examples involving three force-constants.