Residual electron density

Definition

The residual density is defined as

[math] \Delta\rho(\textbf x)= \frac{1}{V}\sum_{\mathbf{h}} (|F_o(\textbf{h})|-|F_c(\textbf{h})|)\exp(i\phi_c(\textbf{h})\exp (-2\pi\mathrm i\mathbf{h}\cdot\mathbf{x}) [/math]

Here the subscripts o respectively c refer to the observed and calculated structure factors

Note

In the classical i.e. kinematical theory approach, the electron density of the unit cell content of a crystal is calculated from the observed structure factors (related to the diffracted intensities) and the corresponding phases derived from the electron density model. The calculated electron density is based on spherical electron densities of individual and independent atoms. This model is a good approximation for standard crystal structure calculations. Once an optimal structure model is obtained by iterative refinements, the two models can be compared by calculating the residual densities, which gives the deviation of the real structure from the spherical model approximation. This residual density usually reveals all aspects of the structures which depart from the spherical atomic densities. We can observe e.g. [math]\pi[/math]-bonds on aromatic C-C bonds but also electron densities related to pairs of electrons in the vicinity of metal atoms.