Crystallographic structure refinement electron density, molecular

Crystallographers share the fruits of their work in the form of lists of atomic coordinates, which can be used to display and study the molecule with molecular graphics programs (Chapter 11). Less commonly, because fewer people have the resources to use them, crystallographers share the final structure factors, from which electron-density maps can be computed. The audience for structure factors includes other crystallographers developing new techniques of data handling, refinement, or map interpretation. 

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

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