International
Tables for
Crystallography
Volume G
Definition and exchange of crystallographic data
Edited by S. R. Hall and B. McMahon

International Tables for Crystallography (2006). Vol. G, ch. 3.5, pp. 141-143

Section 3.5.3. Classification of data definitions

P. R. Mallinsona* and I. D. Brownb

aDepartment of Chemistry, University of Glasgow, Glasgow G12 8QQ, Scotland, and bBrockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario, Canada L8S 4M1
Correspondence e-mail:  paul@chem.gla.ac.uk

3.5.3. Classification of data definitions

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The rhoCIF dictionary is intended to be used in combination with the core CIF dictionary, or, after adaptation to DDL2, the mmCIF dictionary. The two dictionaries can be combined to form a composite dictionary by concatenation. Alternatively, the merging procedures described in Section 3.1.9[link] can be used to create a virtual dictionary at run time. Because the dictionary is not self-contained, its structure can be kept very simple, comprising only two categories: one to define a local axis system around each atom and one to contain the various parameters of the Hansen & Coppens (1978[link]) multipole function.

The two categories in the electron density CIF dictionary lie within the single ATOM category group and relate to the structural model. They are as follows:

Specification of local axes at each atom (§3.5.3.1[link])
 ATOM_LOCAL_AXES
Atom-centred multipole expansion functions (§3.5.3.2[link])
 ATOM_RHO_MULTIPOLE

3.5.3.1. Specification of local axes at each atom

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Data items in this category are as follows:

ATOM_LOCAL_AXES [Scheme scheme1] The bullet (•) indicates a category key. The arrow [(\rightarrow)] is a reference to a parent data item.

The ATOM_LOCAL_AXES category is used to define a local Cartesian axis system for each atom. Data items in this category are looped, with the category key ( _atom_local_axes_atom_label) being a pointer to the _atom_site_label defined in the ATOM_SITE category of the core CIF dictionary. The orientation of the Cartesian basis is defined in terms of vectors between atoms, and any arbitrary orientation can be described by including dummy atoms if necessary. The spherical coordinate system used in the description of the multipoles is defined in terms of the local Cartesian basis: z is the polar axis from which the angle [\theta] is measured, [\varphi] is measured from the x axis in the xy plane with the y axis having a value of [\varphi = +90 ^\circ].

Example 3.5.3.1[link] shows how the category is used in part of an electron density CIF relating to a charge density study of the complex of 1,8-bis(dimethylamino)naphthalene with 1,2-dichloromaleic acid (a `proton sponge') (Mallinson et al., 1997[link]).

Example 3.5.3.1. Definition of the local Cartesian axis system for the atoms in a proton sponge complex.

[Scheme scheme2]

3.5.3.2. Atom-centred multipole expansion functions

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Data items in this category are as follows:

ATOM_RHO_MULTIPOLE [Scheme scheme3] The bullet • indicates a category key. The arrow [(\rightarrow)] is a reference to a parent data item.

Data items in this category are also looped. The category key ( _atom_rho_multipole_atom_label) is also a child of _atom_site_label in the ATOM_SITE category, thereby linking the electron density multipole coefficients to a specific atom site in the list of atom coordinates. This category contains all the parameters of the Hansen & Coppens (1978[link]) function[\eqalign{\rho({\bf r}) &= P_c\rho_{\rm core}({\bf r}) + P_v\kappa^3 \rho_{\rm valence}(\kappa r)\cr &\quad + \textstyle\sum\limits_{0 \le l \le l_{\rm max}} \{\kappa'(l)^3 R(\kappa'(l),l,{\bf r})\}\textstyle\sum\limits_{-l \le m \le l} \{P(l,m) d(l,m,\theta,\varphi)\}, }]where r is a position vector with respect to the atomic nucleus, and [\rho_{\rm core}(\bf r)] and [\rho_{\rm valence}(\kappa\, r)] are the spherical core and valence electron densities, respectively. They are obtained from atomic orbital analytic wavefunctions such as those tabulated by Clementi & Roetti (1974[link]). They are also the Fourier transforms of the X-ray scattering factors given in _atom_rho_multipole_scat_core and _atom_rho_multipole_scat_valence.

[P_c] is the weight applied to the free-atom core (_atom_rho_multipole_coeff_Pc), [P_v] is the weight applied to the free-atom valence shell (_atom_rho_multipole_coeff_Pv), [P(0,0)] is the number of remaining electrons (_atom_rho_multipole_coeff_P00) and [P_c + P_v + P(0,0) = Z] (the atomic number) for a neutral atom. [\kappa] is the valence electron expansion factor (_atom_rho_multipole_kappa), [R(\kappa'(l),l,{\bf r})] is the radial function (Slater or equivalent) (_atom_rho_multipole_radial_*), [\kappa'(l)] is the multipole function expansion factor (_atom_rho_multipole_kappa_prime[l]), [P(l,m)] are the spherical harmonic coefficients (_atom_rho_multiple_coeff_P[lm]) and [d(l,m,\theta,\varphi)] is the spherical harmonic of order [l,m] at the angle [(\theta, \varphi)]. The summations are performed over the index ranges [0 \le l \le l_{\rm max}], [-l \le m \le l], where [l_{\rm max}] is the highest order of multipole applied.

Example 3.5.3.2[link] demonstrates how the category is used in the proton sponge complex of Example 3.5.3.1[link]. Only the first atom is shown in the example.

Example 3.5.3.2. Multipole expansion for an atom in the proton sponge complex of Example 3.5.3.1[link].

[Scheme scheme4]

References

Clementi, E. & Roetti, C. (1974). Roothan–Hartree–Fock atomic wavefunctions. Basis functions and their coefficients for ground and certain excited states of neutral and ionized atoms. At. Data Nucl. Data Tables, 14, 177–478.
Hansen, N. K. & Coppens, P. (1978). Testing aspherical atom refinements on small-molecule data sets. Acta Cryst. A34, 909–921.
Mallinson, P. R., Wozniak, K., Smith, G. T. & McCormack, K. L. (1997). A charge density analysis of cationic and anionic hydrogen bonds in a `proton sponge' complex. J. Am. Chem. Soc. 119, 11502–11509.








































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