Difference electron density map ($\Delta \rho (\mathbf{r})$)

A difference electron density map is an electron density map calculated to reveal disagreement between measured diffraction amplitudes and those predicted by a crystal structure model. The common $F_o-F_c$ map combines observed-minus-calculated amplitude information with phases, usually derived from the model. In practice map coefficients include weighting rather than simply subtracting raw measured intensities.^[ref]

For one simplified convention, let $F_c(\mathbf{h})=|F_c(\mathbf{h})|e^{i{\varphi }_c(\mathbf{h})}$. A schematic unweighted difference map isThe crystallographic phases remain necessary even though the coefficients represent differences. Positive features can indicate density missing from the model; negative features can indicate excessive modeled density. Errors in positions, crystallographic site occupancies or atomic displacement parameters can all contribute.

An original one-dimensional idealization takes ${\rho }_o(x)=3+2\cos (2\pi x)$ and ${\rho }_c(x)=3+\cos (2\pi x)$, with matching phases. Their difference isIt is positive near0 and negative near $1/2$, although its integral over one period is zero. Positive and negative residuals therefore need not mean that the total electron count is wrong.

Experimentally the difference electron density map guides corrections to a fitted macromolecular structural model. Finite diffraction data, uncertain phases and model-derived coefficients affect its interpretation; a peak alone does not uniquely identify an atom or chemical species.