Modelling of the charge density

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. 714-715

Section 8.7.3.2. Modelling of the charge density

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

8.7.3.2. Modelling of the charge density

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The electron density ρ(r) in the structure-factor expression $[ F_{\rm calc}\left ({\bf h}\right)\textstyle \int\limits _{\rm unit\ cell} \rho \left ({\bf r}\right) \exp \left (2\pi i{\bf h\cdot r}\right) {\,{\rm d}}{\bf r} \eqno (8.7.3.4a) ]$ can be approximated by a sum of non-normalized density functions $[g_{i}({\bf r})]$ with scattering factor $[f_{i}({\bf h})]$ centred at $[{\bf r}_{i},]$ $[ \rho({\bf r}) =\textstyle\sum\limits_{i}g_{i}({\bf r}) *\delta ({\bf r}-{\bf r}_{i}). \eqno (8.7.3.5)]$ Substitution in (8.7.3.4a) gives $[ F({\bf h}) =\textstyle\sum\limits_{i}\, f_{i}({\bf h}) \exp(2\pi i{\bf h\cdot r}_{i}). \eqno (8.7.3.4b)]$

When $[g_{i}({\bf r})]$ is the spherically averaged, free-atom density, (8.7.3.4b) represents the free-atom model. A distinction is often made between atom-centred models, in which all functions g(r) are centred at the nuclear positions, and models in which additional functions are centred at other locations, such as in bonds or lone-pair regions.

A simple, atom-centred model with spherical functions g(r) is defined by $[ \rho _{{\rm atom}}({\bf r})=P_{{\rm core}}\rho _{{\rm core}}({\bf r})+\kappa ^3P_{{\rm valence}}\rho _{{\rm valence}}(\kappa {\bf r}).\eqno (8.7.3.6)]$ This `kappa model' allows for charge transfer between atomic valence shells through the population parameter $[P_{{\rm valence}}]$ , and for a change in nuclear screening with electron population, through the parameter κ, which represents an expansion $[(\kappa \, \lt \, 1)]$ , or a contraction $[(\kappa \, \gt \, 1)]$ of the radial density distribution.

The atom-centred, spherical harmonic expansion of the electronic part of the charge distribution is defined by $[\eqalignno{ \rho _{{\rm atom}}({\bf r}) &=P_{{\rm c}}\rho _{{\rm core}}({\bf r})+P_{{\rm v}}\kappa ^3\rho _{{\rm valence}}(\kappa r) \cr &\quad +\textstyle\sum \limits _{l=0}^{l(\max)}\kappa ^{\prime\,3}R_l(\kappa ^{\prime}\zeta r)\textstyle\sum \limits _{m=0}^l \, \textstyle\sum\limits _pP_{lmp}d_{lmp}(\theta, \varphi), & (8.7.3.7)}]$ where p = ± when m is larger than 0, and $[R_l(\kappa ^{\prime }\zeta r) ]$ is a radial function.

The real spherical harmonic functions $[{d}_{lmp}]$ and their Fourier transforms have been described in International Tables for Crystallography, Volume B, Chapter 1.2 (Coppens, 2001). They differ from the functions $[{y}_{lmp}]$ by the normalization condition, defined as $[\int | d_{lmp} | {\,{\rm d}}\Omega=2-\delta _{l0}]$ . The real spherical harmonic functions are often referred to as multipoles, since each represents the component of the charge distribution ρ(r) that gives a non-zero contribution to the integral for the electrostatic multipole moment $[{q}_{lmp},]$ $[ {q}_{lmp}=-\textstyle\int \rho _{{\rm atom}}({\bf r}) {r}^l{c}_{lmp}{\,{\rm d}}{\bf r},\eqno (8.7.3.8)]$ where the functions $[{c}_{lmp}]$ are the Cartesian representations of the real spherical harmonics (Coppens, 2001).

More general models include non-atom centred functions. If the wavefunction ψ in (8.7.2.4) is an antisymmetrized product of molecular orbitals ψ_i, expressed in terms of a linear combination of atomic orbitals $[\chi _{\mu},\ \psi _{i}=\sum _\mu {c}_{{i}\mu }\chi _\mu ]$ (LCAO formalism), the integration (8.7.2.4) leads to $[ \rho ({\bf r}) =\textstyle\sum \limits _\mu \textstyle\sum \limits _\nu P_{\mu \nu }\chi _\mu ({\bf r}-{\bf R}_\mu) \chi _\nu ({\bf r}-{\bf R}_\nu), \eqno (8.7.3.9)]$ with R_μ and R_ν defining the centres of $[\chi _\mu]$ and $[\chi _\nu]$ , respectively, $[P_{\mu \nu }=\sum _{i}n_{i}{c}_{i\mu }{c}_{{i}\nu}]$ , and the sum is over all molecular orbitals with occupancy $[n_{i}]$ . Expression (8.7.3.9) contains products of atomic orbitals, which may have significant values for orbitals centred on adjacent atoms. In the `charge-cloud' model (Hellner, 1977), these products are approximated by bond-centred, Gaussian-shaped density functions. Such functions can often be projected efficiently into the one-centre terms of the spherical harmonic multipole model, so that large correlations occur if both spherical harmonics and bond-centred functions are adjusted independently in a least-squares refinement.

According to (8.7.2.4) and (8.7.3.9), the population of the two-centre terms is related to the one-centre occupancies. A molecular-orbital based model, which implicitly incorporates such relations, has been used to describe local bonding between transition-metal and ligand atoms (Becker & Coppens, 1985).

References

Becker, P. & Coppens, P. (1985). About the simultaneous interpretation of charge and spin density data. Acta Cryst. A41, 177–182.Google Scholar

Coppens, P. (2001). The structure factor. International tables for crystallography, Vol. B, edited by U. Shmueli, Chap. 1.2. Dordrecht: Kluwer Academic Publishers.Google Scholar

Hellner, E. (1977). A simple refinement of density distributions of bonding electrons. Acta Cryst. B33, 3813–3816.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 714-715