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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. 723
Section 8.7.3.7.1. General considerationsa 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 |
In the Born–Oppenheimer approximation, the electrons rearrange instantaneously to the minimum-energy state for each nuclear configuration. This approximation is generally valid, except when very low lying excited electronic states exist. The thermally smeared electron density is then given by![[\left \langle \rho ({\bf r}) \right \rangle =\textstyle\int \rho ({\bf r},{\bf R}) P({\bf R}) {\,\rm d} ({\bf R}), \eqno(8.7.3.79)]](/teximages/cbch8o7/cbch8o7fd110.svg)
where R represents the 3N nuclear space coordinates and P(R) is the probability of the configuration R. Evaluation of (8.7.3.79)![[link]](/graphics/greenarr.gif)
is possible if the vibrational spectrum is known, but requires a large number of quantum-mechanical calculations at points along the vibrational path. A further approximation is the convolution approximation, which assumes that the charge density near each nucleus can be convoluted with the vibrational motion of that nucleus, ![[ \left \langle \rho({\bf r}) \right \rangle =\textstyle\sum \limits _n\textstyle\int \rho _n ({\bf r}-{\bf u}-{\bf R}_n) P_n ({\bf u}) {\,{\rm d}}{\bf u}, \eqno (8.7.3.80)]](/teximages/cbch8o7/cbch8o7fd111.svg)
where ρn stands for the density of the nth pseudo-atom. The convolution approximation thus requires decomposition of the density into atomic fragments. It is related to the thermal-motion formalisms commonly used, and requires that two-centre terms in the theoretical electron density be either projected into the atom-centred density functions, or assigned the thermal motion of a point between the two centres. In the LCAO approximation (8.7.3.9), the two-centre terms are represented by ![[ \rho _{\mu \nu }({\bf r}) =P_{\mu \nu }\chi _\mu ({\bf r}-{\bf R}_\mu) \chi _\nu ({\bf r}-{\bf R}_\nu ), \eqno (8.7.3.81)]](/teximages/cbch8o7/cbch8o7fd112.svg)
where χμ and χν are basis functions centred at Rμ and Rν, respectively. As the motion of a point between the two vibrating atoms depends on their relative phase, further assumptions must be made. The simplest is to assume a gradual variation of the thermal motion along the bond, which gives at a point ri on the internuclear vector of length Rμν ![[ U_{{i}j}({\bf r}_{i}) =\left [U_{{i}j} ({\bf R}_\mu) | {\bf R}_\mu -{\bf r}_{i} | +U_{{i}j} ({\bf R}_\nu) | {\bf R}_\nu -{\bf r}_{i}| \right] /R_{\mu \nu }. \eqno (8.7.3.82)]](/teximages/cbch8o7/cbch8o7fd113.svg)
This expression may be used to assign thermal parameters to a bond-centred function.
