Accurate charge-density studies as an extension of Bayesian crystal structure determination

Many hopes and much controversy have surrounded the application of the maximum-entropy (ME) method to accurate charge-density studies. This paper shows that viewing such studies as an extension of Bayesian crystal structure determination provides practical means of fulfilling many of the hopes invested in the ME method, while essentially eliminating its controversial aspects, the latter being explained in terms of a number of computational artefacts. The positional probability distribution of scatterers having maximum entropy relative to a given 'prior prejudice' is computed so as to reproduce a set of phased structure-factor amplitudes; core electrons can optionally be treated as a fixed 'fragment' and described using atomic core densities derived from ab initio wave functions. Fragment and prior-prejudice density distributions are computed by fast Fourier transforms and are thermally smeared by aliasing. These various algorithms have been implemented within the BUSTER computer program. Model studies on noise-free synthetic data sets for a-glycine, silicon and beryllium show that all-electron calculations give rise to artefacts when a uniform prior prejudice is used, while valence-only calculations using valence monopole priors are essentially free from artefacts. The maximum-entropy approach is thus optimally implemented by incorporating the prior knowledge of the existence of sharp atomic cores in the form of a fragment not subjected to entropy maximization. These results contribute to settling the debate about the putative existence of non-nuclear density maxima at special positions for crystalline silicon and beryllium, and prepare the ground for developing maximum-likelihood multipolar refinement.