Critical Analysis of Non-Nuclear Electron-Density Maxima and the Maximum Entropy Method

Abstract

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...

Dysnomia, a computer program for maximum-entropy method (MEM) analysis and its performance in the MEM-based pattern fitting

“…Since representation of disordered crystal structures by split-atom models in Rietveld analysis is far from perfect, such spherical distributions of electrons for the disordered sites are ascribable to a bias toward procrystal prior densities. Even though MEM can reconstruct peaks that do not exist in prior densities (de Vries et al, 1996), false characteristics of procrystal prior ones can hardly be erased in powder diffraction because F obs (h j )'s for overlapped reflections are estimated on the basis of the same structural model in the preceding Rietveld analysis (Rietveld, 1969).…”

“…One of the errors is caused by a seriestermination effect arising from the availability of only a limited number of reflections (Jauch, 1994;de Vries et al, 1996;Palatinus and van Smaalen, 2002), although this effect is much less significant in MEM than in Fourier synthesis. On the analysis of X-ray diffraction data, MEM also tends to give large residual errors for some low-Q reflections that contribute most significantly to information entropy, S. Such a tendency of MEM often generates artifacts in calculated electron densities even if accurate standard uncertainties, σ(h j ), of F obs (h j )'s are obtainable (de Vries et al, 1994;Palatinus and van Smaalen, 2002).…”

The maximum-entropy method in superspace

“…Furthermore, the MEM has been used as an alternative method to multipole re®nements, with the purpose to compute accurate electron densities that reveal the bonding electrons. After the ®rst promising applications in this ®eld (Collins, 1982;Sakata & Sato, 1990), several warnings concerning the reliability and possible pathologies of the method appeared (Jauch, 1994;de Vries et al, 1996).…”

The generalizedFconstraint in the maximum-entropy method – a study on simulated data

“…For comparison, the computations using the ad hoc weighting (de Vries et al, 1996, referred to as static weighting hereafter) were performed on the noisy data sets. The F constraint with additional static weighting is de®ned as…”

“…In crystallography, one particular application is the investigation of the electron density in the crystal structure. After the ®rst promising applications in this ®eld (Collins, 1982;Sakata & Sato, 1990), several warnings concerning the reliability and possible pathologies of the method appeared (Jauch, 1994;de Vries et al, 1996). One of the obvious problems was that the distribution of the normalized residuals of the structure factors ÁFHa'H jF obs Hj À jF calc Hja'H strongly deviated from the expected Gaussian distribution.…”