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The frequent practice of combining X-ray and neutron diffraction for distinguishing vibrational smearing from charge deformation due to chemical bonding is neither necessary nor completely satisfactory. The two effects occur principally in distinct regions of real and reciprocal space. They can be distinguished by X-ray data alone provided these extend to reciprocal radii d* > 2 A-1 and provided the refinement program allows explicitly for the bonding deformation. The success of this separation can be tested by comparison of the vibration ellipsoids of bonded atom pairs, which should have equal amplitudes in the bond direction. Application of this 'rigid-bond' test to four recent refinements using the charge deformation model shows the vibration parameters to be largely unbiased, as those from parallel spherical-atom refinements are not. Hydrogen vibration parameters cannot be derived from X-ray data because of large deformation densities at the nuclei; nor do they satisfy the rigid-bond postulate.

The electron-density distribution in a crystal is expressed in parametric form suitable for least-squares refinement against the X-ray intensities. The model rests on an expansion of the difference density in a basis of conveniently shaped deformation functions, both spherical and non-spherical, centred on the several atoms. The expansion coefficients are subject to refinement along with the atomic coordinates and atomic or molecular vibration parameters. As a test of the method, the difference density in fumaramic acid has been mapped by least-squares refinement, which reveals far more detail than could be obtained from the same data by conventional methods. No fully objective assessment of accuracy is available, but comparison with difference densities found in other structures by more standard procedures supports the validity of the model and its potential for extracting a maximum of information from the experimental data.

Abstract. Quantitative indices of a molecular charge distribution are provided by the values of volume integrals of the deformation density times a set of appropriate sampling functions. These integrals may, for example, represent the net charges in particular atomic or bond regions. A sharing function derived from the promolecule density defines a general partitioning of any molecule into bonded-atom fragments, whose net charges and multipole moments determine the electrostatic properties of the molecule. Expressions are available for estimating the experimental accuracy of such indices but they are not easy to evaluate nor very reliable.where p(r) is a chosen function of position and Bp(r) is the experimental deformation density at point r. To illustrate the generality of this formalism, we choose three representative examples:(1) Let PI(r) be a delta functionthat vanishes everywhere except at point ro. The corresponding quantity PI is then simply the deformation density 8p(ro) at point ro.(2)' Let pi,r) = 1 inside a sphere of radius R around atom a, zero outside. P2 then measures the net charge increment inside the sphere, a possible measure of the net atomic charge. The customary output of a charge-density study is a set of contour maps depicting the deformation density in selected planes through the molecule or other structural unit. Other modes of graphical representation have been devised to aid in the three-dimensional visualization of spatial features of the charge distribution (Smith et at, Chapter XVI). Such maps are appropriate for answering qualitative questions about the deformation density, e.g. (a) Where are the lone-pair maxima? (b) Is a particular bond straight or bent? (c) Is a given bond peak circular or elliptical in cross section? But often we wish to ask more quantitative questions: (d) What is the net charge on a particular atom? (e) How much excess charge has accumulated in a given covalent bond? (f) What is the Coulomb interaction energy of two neighboring molecules in the crystal?To make possible answers to such questions, we must present our results in a more quantitative form. Contour diagrams, however appealing to our esthetic sense, cannot be fed into a computer for producing numerical answers. In deciding how to quantify our charge-density results, we can adopt two approaches. We can ask what kind of information is potentially useful to whoever is likely to require it. Or we can ask what quantities are most reliably obtainable from our crystallographic study. In favorable cases we may hope for an appreciable overlap between the answers to these two questions.Let us take the rather general expressiondipole moment, referred to an origin at atom a. (We assume that Bp is defined for an isolated molecule; otherwise we must modify P3 so that it vanishes outside an appropriate volume assigned to the chosen molecule.) Our object is to choose a suitable set of sampling functions pm (r), leading to corresponding numerical indices Pm' that can serve to quantify the significant features of a...

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