Electrostatic potential outside a charge distribution

Coppens, P.; Su, Z.; Becker, P. J.

doi:10.1107/97809553602060000615

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 720

Section 8.7.3.4.2.2. Electrostatic potential outside a charge distribution

P. Coppens,^a Z. Su^b and P. J. Becker^c

^a 732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,^bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and ^cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France

8.7.3.4.2.2. Electrostatic potential outside a charge distribution

| top | pdf |

Hirshfelder, Curtis & Bird (1954) and Buckingham (1959) have given an expression for the potential at a point $[{\bf r}_{i}]$ outside a charge distribution: $[\eqalignno{ \Phi ({\bf r}_{i}) &= {q\over r_{i}}+ {\mu _\alpha r_\alpha \over r_{i}^3}+ \textstyle{1\over2}\left [3r_\alpha r_\beta -r^2\delta _{\alpha \beta }\right] \displaystyle {\Theta _{\alpha \beta }\over r_{{i}}^5} \cr &\quad +\left [5r_\alpha r_\beta r_\gamma -r^2\left (r_\alpha \delta _{\beta \gamma }+r_\beta \delta _{\gamma \alpha }+r_\gamma \delta _{\alpha \beta }\right) \right] {\Omega_{\alpha \beta \gamma } \over 5r_{i}^7}+\ldots, \cr& & (8.7.3.61)}]$ where summation over repeated indices is implied. The outer moments q, μ_α, Φ_αβ and Ω_αβγ in (8.7.3.61) must include the nuclear contributions, but, for a point outside the distribution, the spherical neutral-atom densities and the nuclear contributions cancel, so that the potential outside the charge distribution can be calculated from the deformation density.

The summation in (8.7.3.61) is slowly converging if the charge distribution is represented by a single set of moments. When dealing with experimental charge densities, a multicentre expansion is available from the analysis, and (8.7.3.61) can be replaced by a summation over the distributed moments centred at the nuclear positions, in which case $[r_{i}]$ measures the distance from a centre of the expansion to the field point. The result is equivalent to more general expressions given by Su & Coppens (1992), which, for very large values of $[r_{i}]$ , reduce to the sum over atomic terms, each expressed as (8.7.3.61). The interaction between two charge distributions, A and B, is given by $[ E_{AB}=\textstyle\int \Phi _A({\bf r}) \rho _B({\bf r}) {\,{\rm d}}{\bf r},]$ where ρ_B includes the nuclear charge distribution.

References

Buckingham, A. D. (1959). Molecular quadrupole moments. Q. Rev. Chem. Soc. 13, 183–214.Google Scholar

Hirshfelder, J. O., Curtis, C. F. & Bird, R. B. (1954). Molecular theory of gases and liquids. New York: John Wiley.Google Scholar

Su, Z. & Coppens, P. (1992). On the mapping of electrostatic properties from the multipole description of the charge density. Acta Cryst. A48, 188–197.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 720