Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.2, pp. 11-12   | 1 | 2 |

Section 1.2.6. Effect of bonding on the atomic electron density within the spherical-atom approximation: the kappa formalism

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260–3000, USA
Correspondence e-mail:

1.2.6. Effect of bonding on the atomic electron density within the spherical-atom approximation: the kappa formalism

| top | pdf |

A first improvement beyond the isolated-atom formalism is to allow for changes in the radial dependence of the atomic electron distribution.

Such changes may be due to electronegativity differences which lead to the transfer of electrons between the valence shells of different atoms. The electron transfer introduces a change in the screening of the nuclear charge by the electrons and therefore affects the radial dependence of the atomic electron distribution (Coulson, 1961[link]). A change in radial dependence of the density may also occur in a purely covalent bond, as, for example, in the H2 molecule (Ruedenberg, 1962[link]). It can be expressed as [{\rho'_{\rm valence}} (r) = \kappa^{3} \rho_{\rm valence} (\kappa r) \eqno(](Coppens et al., 1979[link]), where ρ′ is the modified density and κ is an expansion/contraction parameter, which is > 1 for valence-shell contraction and < 1 for expansion. The [\kappa^{3}] factor results from the normalization requirement.

The valence density is usually defined as the outer electron shell from which charge transfer occurs. The inner or core electrons are much less affected by the change in occupancy of the outer shell and, in a reasonable approximation, retain their radial dependence.

The corresponding structure-factor expression is [\eqalignno{F({\bf H}) &= {\textstyle\sum\limits_{j}} [\{P_{j, \, {\rm core}}\,f_{j, \, {\rm core}} (H) + P_{j, \, {\rm valence}}\,f_{j, \, {\rm valence}} (H/\kappa)\}\cr &\quad \times \exp (2\pi i{\bf H}\cdot {\bf r}_j)], &(}]where [P_{j, \, {\rm core}}] and [P_{j, \, {\rm valence}}] are the number of electrons (not necessarily integral) in the core and valence shell, respectively, and the atomic scattering factors [f_{j, \, {\rm core}}] and [f_{j, \, {\rm valence}}] are normalized to one electron. Here and in the following sections, the resonant-scattering contributions are incorporated in the core scattering.


Coppens, P., Guru Row, T. N., Leung, P., Stevens, E. D., Becker, P. J. & Yang, Y. W. (1979). Net atomic charges and molecular dipole moments from spherical-atom X-ray refinements, and the relation between atomic charges and shape. Acta Cryst. A35, 63–72.
Coulson, C. A. (1961). Valence. Oxford University Press.
Ruedenberg, K. (1962). The nature of the chemical bond. Phys. Rev. 34, 326–376.

to end of page
to top of page