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International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 720
Section 8.7.3.4.2.3. Evaluation of the electrostatic functions in direct spacea 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 |
The electrostatic properties of a well defined group of atoms can be derived directly from the multipole population coefficients. This method allows the `lifting' of a molecule out of the crystal, and therefore the examination of the electrostatic quantities at the periphery of the molecule, the region of interest for intermolecular interactions. The difficulty related to the origin term, encountered in the reciprocal-space methods, is absent in the direct-space analysis.
In order to express the functions as a sum over atomic contributions, we rewrite (8.7.3.51)![[link]](/graphics/greenarr.gif)
, (8.7.3.54) and (8.7.3.57) for the electrostatic properties at point P as a sum over atomic contributions. ![[\eqalignno{ \Phi ({\bf R}_p) &=\sum _{{{M}}\neq {P}}\displaystyle {Z_{{M}}\over | {\bf R}_{{MP}} | }- \sum_{{M}}\int \displaystyle {\rho _{{e,M}} ({\bf r}_{{M}}) \over | {\bf r}_p | }{\,{\rm d}}{\bf r}_{{M}}, &(8.7.3.62)\cr {\bf E} ({\bf R}_p) &=-\sum _{ {{M}}\neq {P}}\displaystyle {Z_{{MP}}{\bf R}_{{MP}}\over | {\bf R}_{{MP}} | ^3}+\sum_{{M}}\int \displaystyle {{\bf r}_p\rho _{{e,M}} ({\bf r}_{{M}}) \over | {\bf r}_p | ^3}{\,{\rm d}}{\bf r}_{{M}}, &(8.7.3.63)\cr \nabla {\bf E}_{\alpha \beta } ({\bf R}_p) &=-\sum_{{M}\neq {P}}\displaystyle {Z_{{M}} (3{R}_\alpha {R}_\beta -\delta _{\alpha \beta } | {R}_{{MP}} | ^2) \over | {R}_{{MP}} | ^5} \cr &\quad+\sum_{{M}}\int \displaystyle {\rho _{{e,M}} ({\bf r}_{{M}}) (3r_\alpha r_\beta -\delta _{\alpha \beta } | {\bf r}_p | ^2) \over | {\bf r}_p | ^5}{\,{\rm d}}{\bf r}_{{M}}, & (8.7.3.64)}%fd8.7.3.63fd8.7.3.64]](/teximages/cbch8o7/cbch8o7fd84.svg)
in which the exclusion of M = P only applies when the point P coincides with a nucleus, and therefore only occurs for the central contributions. ZM and RM are the nuclear charge and the position vector of atom M, respectively, while rP and rM are, respectively, the vectors from P and from the nucleus M to a point r, such that rP = r − RP, and rM = r − RM = rP + RP − RM = rP − RMP. The subscript M in the second, electronic part of the expressions refers to density functions centred on atom M.
Expressions for the evaluation of (8.7.3.62)–(8.7.3.64) from the charge-density parameters of the multipole expansion have been given by Su & Coppens (1992). They employ the Fourier convolution theorem, used by Epstein & Swanton (1982) to evaluate the electric field gradient at the atomic nuclei. A direct-space method based on the Laplace expansion of 1/|Rp − r| was reported by Bentley (1981).
References
Bentley, J. (1981). In Chemical Applications of Atomic and Molecular Electrostatic Potentials, edited by P. Politzer & D. G. Truhlar. New York/London: Plenum Press.Google Scholar
Epstein, J. & Swanton, D. J. (1982). Electric field gradients in imidazole at 103 K from X-ray diffraction. J. Chem. Phys. 77, 1048–1060.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
