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

Abstract

One of the classical problems in the application of the maximum-entropy method (MEM) to electron-density reconstructions is the uneven distribution of the normalized residuals of the structure factors jF obs Hj À jF calc Hja'H of the resulting electron density. This distribution does not correspond to the expected Gaussian distribution and it leads to erroneous features in the MEM reconstructions. It is shown that the classical 1 2 constraint is only one of many possible constraints, and that it is too weak to restrict the resulting distribution to the expected Gaussian shape. It is proposed that constraints should be used that are based on the higher-order central moments of the distribution of the structure-factor residuals. In this work, the in¯uence of different constraints on the quality of the MEM reconstruction is investigated. It is proposed that the use of a combined constraint on more than one central moment simultaneously would lead to again improved results. Oxalic acid dihydrate was chosen as model structure, from which several data sets with different resolutions and different levels of noise were calculated and subsequently used in the MEM. The results clearly show that the use of different constraints leads to signi®cantly improved results.

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

“…Therefore, electron-or nuclear-density distributions in real space smooth down with increasing order, x (de Vries et al, 1994;Palatinus and van Smaalen, 2002). Weighting of d j 4 is reported to be the best in the case of single-crystal X-ray diffraction (de Vries et al, 1994).…”

“…Dysnomia further adopts a linear combination of generalized F constraints (Palatinus and van Smaalen, 2002):…”

“…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).…”

“…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). This problem can be partly solved either by changing a weighting scheme for the F constraint (de Vries et al, 1994;Palatinus and van Smaalen, 2002) or by imposing a constraint based on higher-order moments of normalized residuals of structure factors, ΔF j = |F obs (h j ) − F(h j )|/σ(h j ), rather than their second-order moment (Palatinus and van Smaalen, 2002).…”

The maximum-entropy method in superspace

“…In a previous publication (Palatinus & van Smaalen, 2002), we have investigated these problems and it was demonstrated that the MEM with a uniform prior leads to artifacts that are larger than the differences between the true electron density and the electron density of a procrystal that is based on spherical atoms. A new constraint was proposed that is based on the higher-order central moments of the distribution of F MEM H À F obs Ha'H.…”