Charge Density Studies

Charge density analyses can provide experimental information on the concentration of electron density around atoms and in intra- and intermolecular bonds, including the location of lone pairs. Transition metal d-orbital populations can be estimated from the asphericity of the charge distribution around such metal centers. A number of physical properties that depend upon the electron density distribution can also be calculated. These include atomic charges, dipole and higher moments, electric field gradients, electrostatic potentials and interaction 

Diffraction is the key experimental technique in crystal engineering. It provides the means of accurately characterising the product of a crystal synthesis endeavor. However, the applications of diffraction are much wider than simply structure determination. An overview of the contribution of crystallography to many areas of chemistry is provided in a recent issue of the journal Chemical Society Reviews dedicated to crystallography [49]. The purpose of this chapter has been to take a broader view of diffraction studies and their present and future potential to play an important role in the continued development of all aspects of crystal engineering. 

1 For example, see W. Clegg, A. J. Blake, R. O. Gould, P. Main, Crystal Structure Analysis Principles and Practice, Oxford University Press, Oxford, 2002. 

3 Presently at http //journals.iucr.org/ services/df/checking/checkfull. html 

Larsen in The Application of Charge Density Research to Chemistry and Drug Design, G. A. Jeffrey, J. F. Piniella (Eds.), NATO AST.Physics, Vol. 250, p.187, Plenum, New York 1991. 

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Maximum Entropy charge density studies Bayesian viewpoint and test applications... 

The Maximum Entropy (abbreviated MaxEnt) method has been used in the field of accurate charge density studies for some time now (see Section 2.2) it has the potential to overcome some of the limitations of traditional multipolar modelling, but great care must be taken not to apply it outside the range of validity of its own foundations. 

Model bias in multipolar charge density studies... 

The main sources of error in charge density studies based on high-resolution X-ray diffraction data are of an experimental nature when special care is taken to minimise them, charge density studies can achieve an accuracy better than 1% in the values of the structure factor amplitudes of the simplest structures [1, 2]. The errors for small molecular crystals, although more difficult to assess, are reckoned to be of the same order of magnitude. 

A second approach which is not subject to the limitations imposed by the choice of a parametrised model of the density, is the MaxEnt method. 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 [17-20], and the works cited in relevant sections of reviews... 

Since 1993, a number of studies have been devoted to assessing the limitations of the MaxEnt method when applied to charge density studies, especially in conjunction with uniform prior-prejudice distributions. We summarise here the main points that have arisen from these model studies. 

None of the studies mentioned in Section 2.2 has explicitly addressed the main issue of the redistribution of core electron densities under MaxEnt requirements in the absence of high-resolution observations. This is indeed the key to explaining the unsatisfactory features encountered so far in the applications of the method to charge density studies. 

If these structural features are not well represented by a mild redistribution of random independent constituents from an initially given prior prejudice, and arise instead from some degree of correlation between the scatterers, they cannot be expected to be satisfactorily dealt with by the method. For these reasons, substructures which scatter well beyond the experimental resolution should be left out of the subset of scatterers distributed at random. The data sets commonly collected for charge density studies do not as a rule extend beyond 0.4 A resolution, but scattering from the atomic core does extend well beyond this limit.2... 

Most of the relevant features of the charge density distribution can be elegantly elucidated by means of the topological analysis of the total electron density [43] nevertheless, electron density deformation maps are still a very effective tool in charge density studies. This is especially true for all densities that are not specified via a multipole model and whose topological analysis has to be performed from numerical values on a grid. 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

Under general hypotheses, the optimisation of the Bayesian score under the constraints of MaxEnt will require numerical integration of (29), in that no analytical solution exists for the integral. A Taylor expansion of Ao(R) around the maximum of the P(R) function could be used to compute an analytical expression for A and its first and second order derivatives, provided the spread of the A distribution is significantly larger than the one of the P(R) function, as measured by a 2. Unfortunately, for accurate charge density studies this requirement is not always fulfilled for many reflexions the structure factor variance Z2 appearing in Ao is comparable to or even smaller than the experimental error variance o2, because the deformation effects and the associated uncertainty are at the level of the noise. 

Both the determination of the effective number of scatterers and the associated rescaling of variances are still in progress within BUSTER. The value of n at the moment is fixed by the user at input preparation time for charge density studies, variances are also kept fixed and set equal to the observational c2. An approximate optimal n can be determined empirically by means of several test runs on synthetic data, monitoring the rms deviation of the final density from the reference model density (see below). This is of course only feasible when using synthetic data, for which the perfect answer is known. We plan to overcome this limitation in the future by means of cross-validation methods. 

The core and valence monopole populations used for the MaxEnt calculation were the ones of the reference density (electrons in the asymmetric unit iw = 12.44 and nvalence = 35.56). The phases and amplitudes for this spherical-atom structure, union of the core fragment and the NUP, are already very close to those of the full multipolar model density to estimate the initial phase error, we computed the phase statistics recently described in a multipolar charge density study on 0.5 A noise-free data [56],... 

We have described in this paper the first implementation of this Bayesian approach to charge density studies, making joint use of structural models for the atomic cores substructure, and MaxEnt distributions of scatterers for the valence part. Used in this way, the MaxEnt method is safe and can usefully complement the traditional modelling based on finite multipolar expansions. This supports our initial proposal that accurate charge density studies should be viewed as the late stages of the structure determination process. 

Flensburg, C., Larsen, S. and Stewart, R.F. (1995) Experimental charge density study ofmethylam-monium hydrogen succinate monohydrate. A salt with a very short O-H-O hydrogen bond,. /. Phys. Chem., 99, 10130-10141. 

Stewart, R.F. (1977) A charge-density study of crystalline beryllium, Acta Cryst., A33, 33-38. 

Jauch, W. and Palmer, A. (1993) The maximum-entropy method in charge-density studies aspects of reliability, Acta Cryst., A49, 590-591. 

Roversi, P., Irwin, J.J. and Bricogne, G. (1998) Accurate charge density studies as an extension of bayesian crystal structure determination, A54(6(2)), 971-996. 

Of46,135 reflections measured (29,973 with I > 2a(T)), only 156 reflections were missing to sin 9/A= 1.34 A-1 5102 reflections were unique of which 2681 had been measured more than nine times (symmetry equivalents plus multiple measurements). The merging R values were R1 = 0.037 and R2 = 0.024 for 4809 accepted means. Examination of the reflection statistics (Table 2) with respect to F2/charge density study. 

The charge density study of benzoylacetone [8] revealed that the Laplacian at the bond critical points between the enol hydrogen and the oxygens has a negative value. This means that the bonds between that hydrogen and both the oxygens have covalent character. Furthermore the populations of the spherical valence parts of the multipole... 

Y. Kubota, M.Takata, M. Sakata, T. Ohba, K. Kifune, and T.Takati, A Charge Density Study of the Intermetallic Compound MgCu2 by the Maximum Entropy Method, J. Phys. Condensed Matter, 12,1259 (2000). 

The aspherical density formalism of Hirshfeld is a deformation model with angular functions which are a sum over spherical harmonics. It will be described in more detail in section 3.2.6. All three models have been applied extensively in charge density studies (for a comparison, see Lecomte 1991). 

D from the multipole population parameters, and a solution value of 11.6 D. With a nonuniform prior, a more acceptable, but still low, MEM value of 7.8 D is obtained. While this physical criterion shows the nonuniform prior to be preferable, the validity of the MEM enhancement in charge density studies remains to be assessed. 

Charge Density Studies of Transition Metal Compounds... 

Charge Density Studies of Transition Metal Compounds 221 where Qr is the expectation value of r-3 defined as... 

Silicates comprise more than 95% by weight of the earth s crust and mantle, and are widely used in glasses, ceramics, sieves, catalysts, and electronic devices. Crystals of silicates are often hard, and may show considerable extinction in their diffraction pattern, which means not only that small samples must be used, but also that ambient temperatures may be adequate for charge density studies.2... 

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