Tables for
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. 142-143

Section Atom-centred multipole expansion functions

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

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

[Scheme scheme4]


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.Google Scholar
Hansen, N. K. & Coppens, P. (1978). Testing aspherical atom refinements on small-molecule data sets. Acta Cryst. A34, 909–921.Google Scholar

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