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 ful®lling 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 ®xed `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  -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 maximumlikelihood multipolar re®nement.