Thermal smearing of theoretical densities

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

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International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 723-724

Section 8.7.3.7. Thermal smearing of theoretical densities

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

8.7.3.7. Thermal smearing of theoretical densities

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8.7.3.7.1. General considerations

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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 $[\left \langle \rho ({\bf r}) \right \rangle =\textstyle\int \rho ({\bf r},{\bf R}) P({\bf R}) {\,\rm d} ({\bf R}), \eqno(8.7.3.79)]$ where R represents the 3N nuclear space coordinates and P(R) is the probability of the configuration R. Evaluation of (8.7.3.79) 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, $[ \left \langle \rho({\bf r}) \right \rangle =\textstyle\sum \limits _n\textstyle\int \rho _n ({\bf r}-{\bf u}-{\bf R}_n) P_n ({\bf u}) {\,{\rm d}}{\bf u}, \eqno (8.7.3.80)]$ where ρ_n 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 $[ \rho _{\mu \nu }({\bf r}) =P_{\mu \nu }\chi _\mu ({\bf r}-{\bf R}_\mu) \chi _\nu ({\bf r}-{\bf R}_\nu ), \eqno (8.7.3.81)]$ where χ_μ and χ_ν are basis functions centred at R_μ and R_ν, 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 r_i on the internuclear vector of length R_μν $[ U_{{i}j}({\bf r}_{i}) =\left [U_{{i}j} ({\bf R}_\mu) | {\bf R}_\mu -{\bf r}_{i} | +U_{{i}j} ({\bf R}_\nu) | {\bf R}_\nu -{\bf r}_{i}| \right] /R_{\mu \nu }. \eqno (8.7.3.82)]$ This expression may be used to assign thermal parameters to a bond-centred function.

8.7.3.7.2. Reciprocal-space averaging over external vibrations

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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 $[ \rho _{\rm dyn} ({\bf r}) =\textstyle\int \rho ({\bf r}-{\bf u}) P({\bf u}) {\,{\rm d}}{\bf u}, \eqno (8.7.3.83)]$ 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), $[ \rho _{\rm dyn}({\bf r}) =\left (\textstyle\sum P_{\mu \nu }\chi _\mu \chi _\nu \right) *P({\bf u},\omega ). \eqno (8.7.3.84)]$ 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 $[\eqalignno{ \left \langle \rho \right \rangle _{{\rm trans}} &=\textstyle\sum P_{\mu \nu } \{ \chi _\mu ({\bf r}) \chi _\nu ({\bf r}) *P({\bf u}) \} \cr &=\textstyle\sum P_{\mu \nu }F^{-1} \{\, f_{\mu \nu }({\bf h}) \cdot T_{{\rm tr}}({\bf h})\}, & (8.7.3.85)}]$ where F⁻¹ is the inverse Fourier transform operator, and T_tr(h) is the translational temperature factor.

If R is an orthogonal rotation matrix corresponding to a rotation ω, we obtain for the librations $[\eqalignno{ \left \langle \rho \right \rangle _{{\rm libr}} &=\textstyle\sum P_{\mu \nu } \{ \chi _\mu({\bi R}{\bf r}) \chi _v({\bi R} {\bf r}) *P (\omega) \} \cr &=\textstyle\sum P_{\mu \nu }F^{-1}\left \{\left\langle \,f_{\mu \nu }({\bi R}{\bf h}) \right \rangle \right \}, & (8.7.3.86)}]$ in which f(h) has been averaged over the distribution of orientations of h with respect to the molecule; $[ \left \langle \,f_{\mu \nu }({\bf h}) \right \rangle =\textstyle\int f_{\mu \nu } ({\bi R}{\bf h}) P (\omega ) {\,{\rm d}}\omega. \eqno (8.7.3.87)]$

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 ψ_μψ_ν = N_μexp[−α_μ(r − r_A)²] × N_νexp [−α_ν (r − r_B)²], where N_μ and N_ν are the normalization factors of the orbitals μ and ν centred on atoms A and B, is given by $[\eqalignno{ f_{{\rm stat}}^{s,s}\left ({\bf h}\right) &=N_\mu N_\nu \exp \left (- {\alpha _\mu \alpha _\nu \over\alpha _\mu +\alpha _\nu }\left | {\bf r}_A-{\bf r}_B\right | ^2\right) \left ({\pi \over\alpha _\mu +\alpha _\nu }\right) ^{3/2} \cr &\quad \times \exp \left ({-\pi ^2\left | h\right | ^2 \over\alpha _\mu +\alpha _v}\right) \exp \left (2\pi i{\bf h}\cdot {\bf r}_c\right), & (8.7.3.88)}]$ where the centre of density $[{\bf r}_c]$ is defined by r_c = (α_μ r_A + α_ν r_B)/(α_μ + α_ν).

For the translational modes, the temperature-factor exponent $[-2\pi ^2\sum _{i}\sum _jU_{ij}h_{i}h_j]$ is simply added to the Gaussian exponent in (8.7.3.88) to give $[ \exp \left (-{\pi ^2 | h | ^2 \over \alpha _\mu +\alpha _\nu }\right) -2\pi ^2\sum \limits _{i}\sum \limits _jU_{{i}j}h_{{i}}h_j. ]$ For librations, we may write $[ {\bi R}{\bf r}={\bf r}+{\bf u}_{{\rm lib}} ]$ As $[({\bi R}{\bf h})\cdot {\bf r}={\bf h}\cdot{\bi R}^T{\bf r}={\bf h}\cdot {\bf r}-({\bi R}{\bf h})\cdot {\bf u}_{{\rm lib}}]$ , for a function centred at r, $[\eqalignno{ \left \langle \exp (2\pi {\bf h}\cdot {\bf r}_c) \right \rangle &=\textstyle\sum \exp [2\pi i({\bi R}{\bf h}) \cdot {\bf r}_c] P (\delta) {\,{\rm d}}\omega \cr &=\exp (2\pi {\bf h}\cdot {\bf r}_c) \textstyle\int \exp [-2\pi i({\bi R}{\bf h}) \cdot {\bf u}_{{\rm lib}}^c] P(\omega ) {\,{\rm d}}\omega, \cr&& (8.7.3.89)}]$ which shows that for ss orbital products the librational temperature factor can be factored out, or $[ f_{\,{\rm dyn}}^{s,s}=f_{\rm stat}^{s,s}\textstyle\int \exp [-2\pi i({\bi R}{\bf h}) \cdot{\bf u}_{\rm lib}^c] P(\omega ) \,{\rm d}\omega. \eqno (8.7.3.90)]$ Expressions for P(ω) are described elsewhere (Pawley & Willis, 1970).

For general Cartesian Gaussian basis functions of the type $[ \psi({\bf r}) = (x-x_A) ^m (y-y_A) ^n (z-z_A) ^p\exp (-\alpha | {\bf r}-{\bf r}_A| ^2),\eqno (8.7.3.91)]$ 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 $[\left \langle \rho \right \rangle = {1\over V}\sum P_{\mu \nu }\sum \limits _{{\bf h}}f_{\mu \nu }\left ({\bf h}\right) \exp \left [-2\pi i{\bf h}\cdot \left ({\bf r}-{\bf r}_c\right) \right] , \eqno (8.7.3.92)]$ where the orbital product χ_μχ_ν is centred at $[{\bf r}_c]$ . 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 Scholar

Pawley, 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

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 723-724