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, pp. 723-724
Section 8.7.3.7.2. Reciprocal-space averaging over external vibrationsa 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 |
Thermal averaging of the electron density is considerably simplified for modes in which adjacent atoms move in phase. In molecular crystals, such modes correspond to rigid-body vibrations and librations of the molecule as a whole. Their frequencies are low because of the weakness of intermolecular interactions. Rigid-body motions therefore tend to dominate thermal motion, in particular at temperatures for which kT (k = 0.7 cm−1) is large compared with the spacing of the vibrational energy levels of the external modes (internal modes are typically not excited to any extent at or below room temperature).
For a translational displacement (u), the dynamic density is given by with ρ(r) defined by (8.7.3.81) (Stevens, Rees & Coppens, 1977). In the harmonic approximation, P(u) is a normalized three-dimensional Gaussian probability function, the exponents of which may be obtained by rigid-body analysis of the experimental data. In general, for a translational displacement (u) and a librational oscillation (ω), If correlation between u and ω can be ignored (neglect of the screw tensor S), P(u, ω) = P(u)P(ω), and both types of modes can be treated independently. For the translations where F−1 is the inverse Fourier transform operator, and Ttr(h) is the translational temperature factor.
If R is an orthogonal rotation matrix corresponding to a rotation ω, we obtain for the librations in which f(h) has been averaged over the distribution of orientations of h with respect to the molecule;
Evaluation of (8.7.3.85) and (8.7.3.86) is most readily performed if the basis functions ψ have a Gaussian-type radial dependence, or are expressed as a linear combination of Gaussian radial functions.
For Gaussian products of s orbitals, the molecular scattering factor of the product ψμψν = Nμexp[−αμ(r − rA)2] × Nνexp [−αν (r − rB)2], where Nμ and Nν are the normalization factors of the orbitals μ and ν centred on atoms A and B, is given by where the centre of density is defined by rc = (αμ rA + αν rB)/(αμ + αν).
For the translational modes, the temperature-factor exponent is simply added to the Gaussian exponent in (8.7.3.88) to give For librations, we may write As , for a function centred at r, which shows that for ss orbital products the librational temperature factor can be factored out, or Expressions for P(ω) are described elsewhere (Pawley & Willis, 1970).
For general Cartesian Gaussian basis functions of the type the scattering factors are more complicated (Miller & Krauss, 1967; Stevens, Rees & Coppens, 1977), and the librational temperature factor can no longer be factored out. However, it may be shown that, to a first approximation, (8.7.3.90) can again be used. This `pseudotranslation' approximation corresponds to a neglect of the change in orientation (but not of position) of the two-centre density function and is adequate for moderate vibrational amplitudes.
Thermally smeared density functions are obtained from the averaged reciprocal-space function by performing the inverse Fourier transform with phase factors depending on the position coordinates of each orbital product where the orbital product χμχν is centred at . If the summation is truncated at the experimental limit of (sin θ)/λ, both thermal vibrations and truncation effects are properly introduced in the theoretical densities.
References
Miller, K. J. & Krauss, M. (1967). Born inelastic differential cross sections in H2. J. Chem. Phys. 47, 3754–3762.Google ScholarPawley, G. S. & Willis, B. T. M. (1970). Temperature factor of an atom in a rigid vibrating molecule. II. Anisotropic thermal motion. Acta Cryst. A36, 260–262.Google Scholar
Stevens, E. D., Rees, B. & Coppens, P. (1977). Calculation of dynamic electron distributions from static molecular wave functions. Acta Cryst. A33, 333–338.Google Scholar