Experimental evidence for the existence of non-nuclear maxima in the electron-density distribution of metallic beryllium. A comparative study of the maximum entropy method and the multipole refinement method

Iversen, Bo B.; Larsen, Finn; Souhassou, M.; Takata, Masaki

doi:10.1107/s0108768194010360

Cited by 95 publications

(64 citation statements)

References 16 publications

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“…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.…”

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confidence: 62%

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

1996

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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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confidence: 62%

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

1996

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“…Wang & Klein, 1981;Yin & Cohen, 1983) and experimental (Spackman, 1986) valence densities. In contrast, the MEM density of Be based on single-crystal data (Larsen & Hansen, 1984) is quite smooth and such fine features do not appear (Takata, Sakata, Kumazawa, Larsen & Iversen, 1994;Iversen, Larsen, Souhassou & Takata, 1995). In order to gain an insight into the cause of the difference in the quality of the MEM densities, the observed structure factors, Fob s, are listed in Table 1 together with the calculated ones, FME M, corresponding to Fig.…”

Section: The Mem Charge Density Of Si Obtained Previouslymentioning

confidence: 99%

The Influence of the Completeness of the Data Set on the Charge Density Obtained with the Maximum-Entropy Method. A Re-examination of the Electron-Density Distribution in Si

Takata¹,

Sakata²

1996

Acta Cryst Sect A

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The charge densities derived with the maximum-entropy method (MEM) may be influenced to some extent by the completeness of the data set. In order to examine the effects of the incompleteness, structure-factor data of Si measured by the PendellOsung method [Saka & Kato (1986). Acta Cryst. A42, 469-478] were re-analysed by the MEM. This data set is incomplete: it contains all space-group-allowed reflections with sin 0/2 = 0.86i -I, and in addition 844 and 880 with sin0/2 = 1.04,~, -1. Results of a MEM analysis of the complete subset of data are compared with those from the full but incomplete set published previously [Sakata & Sato (1990). Acta Cryst. A46, 263-270]. The smaller but complete set was found to give a smooth charge-density distribution that is consistent with previous theoretical work. It is found that the sharp peak maximum at the bond midpoint reported previously is exaggerated owing to the highest-order reflection 880. The completeness of the data set appears to be one of the key factors for obtaining reliable charge densities with MEM. The incompleteness of the data set may cause non-physical fine features of the MEM density distribution.

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“…The field of its application to experimental (X-ray diffraction) charge densities is rapidly developing now (e.g. Kapphahn, Tsirelson & Ozerov, 1988;Craven & Stewart, 1990;Gatti, Bianchi, Destro & Merati, 1992;Destro & Merati, 1995;Iversen, Larsen, Souhassou & Takata, 1995;Tsirelson & Ozerov, 1996) owing to the possibility of analytically representing experimental p by various multipole techniques (Hirshfeld, 1971;Stewart, 1976;Hansen & Coppens, 1978), allowing analytical evaluation of the Laplacian, V2p, and gradient vector field of p. Although it seems that there is no fundamental restriction on the applicability of topological analysis to experimental p (Tsirelson, 1996), the numerical results in this case are affected by the thermal atomic motions in crystals (Tsirelson, 1996;Gatti, Bianchi, Destro & Merati, 1992) and in fact refer to the mean thermal nuclear distribution. In this situation, the only way to evaluate indirectly to what extent quantitative values of topological characteristics depend on the atomic thermal motion effects may be a comparative topological analysis of theoretical and experimental charge densities simultaneously.…”

Section: Introductionmentioning

confidence: 99%

A Topological Analysis of Charge Densities in Diamond, Silicon and Germanium Crystals

Abramov¹,

Okamura²

1997

Acta Cryst Sect A

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The Hansen-Coppens multipole model of charge density has been fitted to highly accurate published experimental and theoretical structure factors for diamond, silicon and germanium crystals. Analysis of both model experimental and theoretical charge densities using the resulting model parameters was performed in terms of Bader's topological theory. The general topology of the charge density appeared to be identical for all crystals, containing the four possible types of critical points of rank three, and no non-nuclear attractors between neighboring atoms were found within achieved accuracy. Theoretical and experimental values of charge density and its Laplacian show quantitative and semiquantitative agreement, respectively, at the critical points of model charge densities. For Ge crystals, such agreement is worse at the ring critical point. These results suggest the possibility of semiquantitative (within 10-30%) study of the topological characteristics of highly accurate X-ray charge densities of crystals displaying shared interatomic interactions. Comparative topological analysis of the chemical bond in this series of crystals is discussed in terms of the quantum topological theory.

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Experimental evidence for the existence of non-nuclear maxima in the electron-density distribution of metallic beryllium. A comparative study of the maximum entropy method and the multipole refinement method

Cited by 95 publications

References 16 publications

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

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

The Influence of the Completeness of the Data Set on the Charge Density Obtained with the Maximum-Entropy Method. A Re-examination of the Electron-Density Distribution in Si

A Topological Analysis of Charge Densities in Diamond, Silicon and Germanium Crystals

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