= Difference electron density map
{title2=$\Delta\rho(\mathbf r)$}
A \b[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.<Electron Density Maps and Coefficient Files at RCSB PDB>[Introduction, difference maps and weighted amplitude/phase coefficient files]{ref}
For one simplified convention, let $F_c(\mathbf h)=|F_c(\mathbf h)|e^{i\phi_c(\mathbf h)}$. A schematic unweighted difference map is
$$
\Delta\rho(\mathbf r)=\frac1{V_c}\sum_{\mathbf h}
\bigl(|F_o(\mathbf h)|-|F_c(\mathbf h)|\bigr)
e^{i\phi_c(\mathbf h)}e^{-2\pi i\mathbf h\cdot\mathbf r}.
$$
The <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 is
$$
\Delta\rho(x)=\cos(2\pi x).
$$
It 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.
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