Experimental evidence for the existence of non-nuclear maxima in charge densities is questioned. It is shown that the non-nuclear maxima reported for silicon are artifacts of the maximum entropy method that was used to analyze the x-ray diffraction data. This method can be improved by the use of appropriate prior information. We report systematic tests of the improved method leading to the absence of non-nuclear maxima in Si. Likewise, the non-nuclear maxima reported earlier in beryllium are not substantiated. [S0031-9007(96) [4] recently reported that the presence of the non-nuclear maxima in the Na clusters appears to be a basis-set or method-dependent effect. Unfortunately, no experimental methods exist to confirm or invalidate these theoretical results.We now turn to crystals. Periodicity allows accurate studies of the EDD by x-ray diffraction. In reality, perturbation of the periodicity by the presence either of a surface or of impurities cause oscillations in the EDD, the so-called Friedel oscillations. In the present paper, however, we refer to non-nuclear maxima with the periodicity of the lattice, such as found by Mei et al. [5] in the bcc lattices of lithium and sodium, using the HartreeFock program CRYSTAL [6]. In 1990 Sakata and Sato [7] analyzed the highly accurate x-ray Pendellösung data on Si measured by Saka and Kato [8] with the maximum entropy method (MEM). They found maxima in the electron density in the Si-Si bond. Recently, non-nuclear maxima were also found in Be by Iversen et al. [9], applying the MEM to structure factors that were measured by Larsen and Hansen [10]. These two findings are the first and only experimental support for the rather elusive non-nuclear maxima. They constitute a considerable challenge to theoretical solid state physics, since the many present wave-function calculations on Si, including our own [11], do not corroborate the result. Since we do not doubt the quality of the primary data in the MEM analysis, we decided to reassess the MEM procedure itself and test its ability to bring to light subtle features in electron density maps. In the course of the work we discovered the essential role of the prior assumptions in the analysis, a role that has not yet received the attention it deserves.Structure factors, obtained from a single crystal x-ray experiment, are directly related to the EDD by means of a Fourier transform. The problems one is facing when performing the inverse Fourier transformation are that (i) the experiment provides us with a limited number of structure factors, (ii) all structure factors are determined within a certain experimental error, and (iii) the phase of the structure factors is unknown. In case the crystal is centrosymmetric the phases can usually be derived without any ambiguity and only the first two problems remain.The traditional way to extract the EDD from a limited set of noisy data is to fit the structure factors by a model. The drawback of this method is that features which are not allowed by the model will never show up in the ED...

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The quality of the extraction of electron density distributions by means of a multipole refinement method is investigated. Structure factors of the urea crystal have been obtained from an electron density distribution (EDD) resulting from a density function calculation with the CRYSTAL95 package. To account for the thermal motion of the atoms, the stockholder-partioned densities of the atoms have been convoluted with thermal smearing functions, which were obtained from a neutron diffraction experiment. A POP multipole refinement yielded a good fit, R = 0.6%. This disagreement factor is based on magnitudes only. Comparison with the original structure factors gave a disagreement of 0.8% owing to differences in magnitude and phase. The fitted EDD still showed all the characteristics of the interaction density. After random errors corresponding to the experimental situation were added to the structure factors, the refinement was repeated. The fit was R = 1.1%. This time the resulting interaction density was heavily deformed. Repetition with another set of random errors from the same distribution yielded a widely different interaction density distribution. The conclusion is that interaction densities cannot be obtained from X-ray diffraction data on non-centrosymmetric crystals.

Abstract:In 1990 Sakata and Sato applied the maximum entropy method (MEM) to a set of structure factors measured earlier by Saka and Kato with the Pendellosung method. They found the presence of non-nuclear attractors, i.e., maxima in the density between two bonded atoms. We applied the MEM to a limited set of Fourier data calculated from a known electron density distribution (EDD) of silicon. The EDD of silicon was calculated with the program ADF-BAND. This program performs electronic structure calculations, including periodicity, based on the density functional theory of Hohenberg and Kohn. No non-nuclear attractor between two bonded silicon atoms was observed in this density. Structure factors were calculated from this density and the same set of structure factors that was measured by Saka and Kato was used in the MEM analysis. The EDD obtained with the MEM shows the same non-nuclear attractors that were later obtained by Sakata and Sato. This means that the non-nuclear attractors in silicon are really an artefact of the MEM.

The maximum-entropy method (MEM) has been applied for the deconvolution of the point-spread function (PSF) of two-dimensional X-ray detectors. The method is robust, model and image independent, and only depends on the correct description of the twodimensional point-spread function and gain factor of the detector. A signi®cant enhancement of both the spatial resolution and the contrast ratio has been obtained for two phase-contrast images recorded with an ultra-highresolution X-ray imaging detector. The method has also been applied to a Laue diffraction image of a protein crystal, showing an important improvement in both the peak separation of closely spaced diffraction peaks and the signal-to-noise ratio of medium and weak peaks. The principle of the method is explained and examples of its application are presented.