CHARGE : Atomic charges from difference density

Author: Nick Spadaccini, Computer Science Department, University of Western Australia, Nedlands, 6907 WA, Australia

CHARGE calculates a charge associated with an atom from the difference density. The difference density (\(\Delta \rho _{mol}\) ) is partitioned according to the Hirshfeld method.

Hirshfeld Partitioning

The Hirshfeld method apportions the electron density among the atoms by the appropriate weighting. The weights are related by the atomic contribution to the promolecular density,

(4.1) \[
    w_{A}(\Vector{r}) = \frac{\rho_{\rm{atm}}^A(\Vector{r})} { \rho_{\rm{pro}}(\Vector{r})}

The fragment of the deformation density apportioned to atom A is,

(4.2) \[
            \Delta\rho_{\rm{frag}}^A(\Vector{r}) = w_{A}(\Vector{r}) 

The net atomic charge, \(Q_{A}\), is derived from the integration of the difference density fragment,

(4.3) \[
            Q_{A} = \int_{\rm{input}} {\Delta\rho_{\rm{frag}}^{A}(\Vector{r})d\Vector{v} }

An alternative scheme is based on the atomic contributions to the total promolecular potential \(V_{pro}\) defined as the sum of the electronic and nuclear contributions.

Density And Potential Profiles

The promolecular density or potential is the sum of the atomic densities or potentials. These latter values are derived from the Clementi and Roetti atomic wavefunctions. Associated with each atom type are the parameters \(A_{k}\), \(n_{k}\) and \(z_{k}\) for k=1,...,m such that the density is,

(4.4) \[
        \rho_{\rm{atm}}(\Vector{r}) =\sum_{k=1}^{m}{ A_k r^{n_k}e^{-z_k r} }

and the potential is

(4.5) \[
        V_{\rm{atm}}(\Vector{r}) =4\pi \sum_{k=1}^{m}{ A_k \left(
                      \frac{\gamma(n_k+3,z_k r) }{r z_{k}^{n_k+3} }  +
                      \frac{(n+1)! }{  z_{k}^{n_k+2} }  -
                      \frac{\gamma(n_k+2,z_k r) }{ z_{k}^{n_k+2} } \right)
                        + \frac{Z_A}{r} }

The last term is the nuclear contribution and \(\gamma \) (n,x) is the Incomplete Gamma Function.

The density profiles (e/ \(bohr^{3}\) ) and potential profiles (e/bohr) are stored at 44 discrete values of r (bohrs) for the points,

\(r_{1}= 0. ; r_{k+1}= 1.15(r_{k}+ .01)\) ; 0. ≤ r ≤ 31.2

The divisions are chosen so that the density of points is greatest in the region of steepest gradient. The density or potential value at any general point is linearly interpolated from the profile.

Density, Distances And Errors

The difference density ( \(\Delta \rho _{mol}\) ) must be input from the FOURR file map. CHARGE partitions this density into atomic contributions and integrates over the input region to obtain a charge. If the input map is the asymmetric unit the values obtained are total atomic charges for atoms at general positions or fractions governed by the site symmetry for atoms at special positions. The user must determine the fraction of an atom present in the input map and calculate the total charge accordingly.

The user may specify the effective range of atom contributions in two ways. The border option in the program initiation line determines the region beyond the input map for which atom contributions are included. The default value of 6 implies that any atom within this distance of the input map edges is included. Also the distance beyond which an atom contribution is zero may be set by the contact option.

Estimates of \(\sigma \) (Q) are determined for spherical regions of various radii following the method of Davis and Maslen. The estimates are derived from the values of \(\sigma \) (F), the errors in the structure factor amplitudes used to derive the difference density. For each atom the program outputs a value of \(\sigma \) (Q) for a radius set within the program. The absolute radii are generated from relative radii, derived from the density profiles and stored in the database. These values are rescaled so that the total volume of the atoms in the cell equals the cell volume. The radii used are listed.

Since values of atomic radii are not unique the variation of \(\sigma \) (Q) with r is also output so that the user may determine \(\sigma \) (Q) at an alternative radius if desired. The scheme assumes a centrosymmetric structure so that phase errors are not included .

File Assignments

  • Reads lrcell: and symmetry data from the input archive bdf

  • Reads difference density from a mapfile

  • Reads profile data base from the file designated by the macro profile:



The weights here are determined from the promolecular density. The contributions of atoms up to 6 beyond the input map edges are included in the calculation. However, an atom will not contribute to the density if the point is greater than 6 away.

CHARGE poten bord 4 cont 4.5

Here the weights are determined from the promolecular potential. Atoms up to 4 from the input map edges are included in the calculation. The effective range of a atom is set to 4.5.


  • Clementi, E and Roetti, C. 1974. Atomic Data and Nuclear Data Tables 14, 177-478.

  • Davis, C.L. and Maslen, E.N. 1978. Acta Cryst. A34, 743-746.

  • Hirshfeld, F. 1977. Israel Journal of Chemistry 16, 198-201.