The Two-Channel Maximum-Entropy Method Applied to the Charge Density of a Molecular Crystal: α-Glycine

Abstract

A two-channel maximum-entropy method (MEM), first used to enhance magnetization densities from phased polarized neutron data by Papoular & Gillon [(1990). Europhys. Lett. 13,[429][430][431][432][433][434], has been applied to the electron deformation density. The resulting entropic densities are compared with standard deformation densities and with dynamic and static deformation maps obtained from multipole ref'mements. The procedure is illustrated with simulated and real single-crystal X-ray data sets on the molecular crystal of ot-glycine. Both a uniform prior and a prior equal to the MEMenhanced dynamic model deformation density are used in the MEM procedure, the result of which does not depend on the starting density. The method is judged by the appearance of the resulting maps and the values of the molecular dipole moment before and after the MEM. Compared with the conventional deformation density, the MEM procedure sharpens the peaks in the bond but flattens the weaker features, especially when a uniform prior is used. The dipole-moment criterion shows the non-uniform prior to be preferable to the uniform prior in reproducing electrostatic properties. The usefulness of the MEM in charge-density analysis remains open to discussion.

Accurate Charge-Density Studies as an Extension of Bayesian Crystal Structure Determination

“…from Papoular, Vekhter & Coppens, who worked on observed and simulated data sets for -glycine (Papoular et al, 1996). In the latter study, when all electrons were redistributed with a single-channel approach, the density of the H atoms was clearly¯at-tened and features below 2 e A Ê À3 were in general deemed to be scarcely signi®cant because the large dynamic range of the total density reduced the sensitivity level.…”

“…From the calculations redistributing L-and M-shell electrons, it is already evident that even the use of a nonuniform prior-prejudice distribution cannot prevent errors as high as 0.2 e A Ê À3 from appearing in the reconstructed total density. All-electron runs, in which no fragment is used, are therefore expected to perform even worse; we stress that all the MaxEnt distributions obtained so far in the literature, with the exception of the two-channel deformation densities of Papoular et al (1996), are precisely all-electron reconstructions. In particular, we reproduced here with calculation E the total density calculation with a NUP described by ; calculation F, using the UP, is equivalent to the ones described in Sakata & Sato (1990), Takata & Sakata (1996) and de Vries, Briels, Feil, te Velde & Baerends (1996) (compare maps in these papers with the one in Fig.…”

“…The appeal of the method is evident when counting the increasing number of applications to charge-density studies that have appeared in the crystallographic literature in the last ten years: see among the most recent ones Restori & Schwarzenbach (1995), Papoular et al (1996), Takata & Sakata (1996), and Yamamoto et al (1996), and the works cited in relevant sections of reviews on charge-density studies (Spackman & Brown, 1994) and on maximum-entropy methods in crystallography (Gilmore, 1996). In principle, MaxEnt maps are not tied to any particular multipolar expansion or radial deformation function and can mirror any degree of angular and radial deformation that is present in the observations.…”

“…Recourse to a two-channel formalism, which redistributes`positive-' and`negative-density' scatterers, ®tting a set of difference Fourier coef®cients, has also been made (Papoular et al, 1996) but there is no consensus on what the de®nition of entropy should be in a two-channel situation (Bricogne, 1988a;Sakata et al, 1993;Papoular et al, 1996); moreover, the shapes and numbers of positive and negative scatterers may need to differ in a way that is dif®cult to specify.…”

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

“…Further lowering of the artifacts could probably be achieved with the two-channel MEM (Papoular et al, 1996) or with the valenceonly MEM proposed by Roversi et al (1998). The latter method uses the re®ned structure parameters to create a core electron-density fragment, which is then considered to be known and is not included in the MaxEnt optimization.…”