##################################################################### # # # Data dictionary for the multipole model of aspherical atomic # # density distributions # # # # This dictionary defines two new categories: # # atom_local_axes # # atom_rho_multipole # # and additional items to the definition of _atom_site_label # # # ##################################################################### data_on_this_dictionary _dictionary_name cif_rho.dic _dictionary_version 1.0.1 _dictionary_update 2005-06-14 _dictionary_history ; 1999-07-07 Created as rhoCIF dictionary by P.R. Mallinson. 2000-10-13 Simplified CIF structure into two loops, corresponding to newly-defined categories atom_rho and atom_local. Introduced reference to multipole formalism, more rigorous definition of local axis systems, and rationalised definition of dummy atoms. Clarified example of use of dummy atom. 2000-10-16 Additions and changes made by I.D.Brown to bring the dictionary into better conformance with the other CIF dictionaries. Category definitions added. Category names changed to atom_local_axes and atom_rho_multipole. Items ordered alphabetically. 2000-10-18 Removed ambiguities in description of _ATOM_LOCAL_AXES. Individual items in this category defined separately. Definition of core population Pc added. Substituted new example, with literature reference. 2000-10-20 Version 0.5. I.D.Brown. Corrected datanames and some spelling. added the equation for the electron density to the definition. 2000-10-23 Further clarification of _ATOM_LOCAL_AXES definition. 2002-03-04 Version 0.61. I.D.Brown. Addition of parent links and enumeration lists. 2002-10-18 Version 0.62. I.D.Brown. Further additions to the atom_rho_multipole category based on input from Paul Mallinson. Tightening up of definitions. _atom_local_axes_label changed to _atom_local_axes_atom_label to conform to CIF style. 2002-10-31 Version 0.63. P.R. Mallinson and I.D. Brown. Amended descriptions of rho_core(r) and rho_valence(kr) in definitions which refer to them. 2002-11-20 Version 0.64. P.R. Mallinson and I.D. Brown. Changed names _atom_rho_multipole_scat_*_source to _atom_rho_multipole_*_source. 2003-06-04 Version 0.65. P.R. Mallinson. Changed kappa', kappa" nomenclature to kappa, kappa'. 2003-06-14 Version 0.66. B. McMahon. Fixed a few typos; added a _definition for _atom_site_label explaining its extension to the core definition; added _definition to the category overviews; tidied up layout and other stylistic edits. 2003-07-02 Version 0.67. P.R. Mallinson. Expanded category overview definitions and _atom_rho_multipole_*_source definitions. Specified summation ranges in expressions used in _atom_rho_multipole_* definitions. 2003-07-11 Version 0.68. B. McMahon. Implemented Paul's fix for index ranges -l <= m <= l, and moved the example in the *_source items as suggested by IDB. 2003-08-19 Version 0.69. I.D.Brown. Made minor corrections suggested during final COMCIFS approval which was received on this date. 2003-08-19 Release version 1.0. IUCr. 2005-01-20 NJ Ashcroft: minor corrections to hyphenation, spelling and punctuation. 2005-06-14 NJ Ashcroft: category overview added for ATOM_SITE category. ; ######################################################## # # # Addition to category ATOM_SITE # # # ######################################################## data_atom_site_[rho] _name '_atom_site_[rho]' _category category_overview _type null _definition ; Data items in the ATOM_SITE category record details about the atom sites in a crystal structure, such as the positional coordinates, atomic displacement parameters, magnetic moments and directions. ; data_atom_site_label_rho _name '_atom_site_label' _category atom_site loop_ _list_link_child '_atom_local_axes_atom0' '_atom_local_axes_atom1' '_atom_local_axes_atom2' '_atom_local_axes_atom_label' '_atom_rho_multipole_atom_label' # added from the entry in the core dictionary to make a self-contained # unit (BM 2003-06-12) _type char _list yes _list_mandatory yes _definition ; The _atom_site_label is a unique identifier for a particular site in the crystal, and is fully defined in the core CIF dictionary. The child data names itemized here are in addition to those in the core dictionary. ; ######################################################## # # # Category ATOM_LOCAL_AXES # # # ######################################################## data_atom_local_axes_[rho] _name '_atom_local_axes_[rho]' _category category_overview _type null loop_ _example_detail _example # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ; Example 1 - This example shows how the local axes can be defined around each atom in terms of vectors between neighbouring atoms. If necessary, dummy atoms can be introduced into the atom_site list for this purpose. ; ; loop_ _atom_local_axes_atom_label _atom_local_axes_atom0 _atom_local_axes_ax1 _atom_local_axes_atom1 _atom_local_axes_atom2 _atom_local_axes_ax2 Ni2+(1) DUM0 Z Ni2+(1) N(1) X loop_ _atom_site_label _atom_site_fract_x _atom_site_fract_y _atom_site_fract_z _atom_site_occupancy DUM0 0.80000 0.80000 0.80000 0.0 ; # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - _definition ; This category allows the definition of local axes around each atom in terms of vectors between neighbouring atoms. High-resolution X-ray diffraction methods enable the determination of the electron density distribution in crystal lattices and molecules, which in turn allows for a characterization of chemical interactions (Coppens, 1997; Koritsanszky & Coppens, 2001). This is accomplished by the construction of a mathematical model of the charge density in a crystal and then by fitting the parameters of such a model to the experimental pattern of diffracted X-rays. The model on which this dictionary is based is the so-called multipole formalism proposed by Hansen & Coppens (1978). In this model, the electron density in a crystal is described by a sum of aspherical "pseudoatoms" where the pseudoatom density has the form defined in the _atom_rho_multipole_* items. Each pseudoatom density consists of terms representing the core density, the spherical part of the valence density and the deviation of the valence density from sphericity. The continuous electron density in the crystal is then modelled as a sum of atom-centred charge distributions. Once the experimental electron density has been established, the "atoms in molecules" theory of Bader (1990) provides tools for the interpretation of the density distribution in terms of its topological properties. Ref: Bader, R. F. W. (1990). Atoms in molecules: a quantum theory. Oxford University Press. Coppens, P. (1997). X-ray charge densities and chemical bonding. Oxford University Press. Hansen, N. K. & Coppens, P. (1978). Acta Cryst. A34, 909-921. Koritsanszky, T. S. & Coppens, P. (2001). Chem. Rev. 101, 1583-1621. ; data_atom_local_axes_atom_label _name '_atom_local_axes_atom_label' _category atom_local_axes _type char _list yes _list_link_parent '_atom_site_label' _list_mandatory yes _definition ; This item is used to identify an atom for which a local axis system is to be defined. Its value must be identical to one of the values given in the _atom_site_label list. ; data_atom_local_axes_atom0 _name '_atom_local_axes_atom0' _category atom_local_axes _type char _list yes _list_link_parent '_atom_site_label' _list_reference '_atom_local_axes_atom_label' _definition ; Specifies 'atom0' in the definition of a local axis frame. The definition employs three atom-site labels, 'atom0', 'atom1' and 'atom2', and two axis labels, 'ax1' and 'ax2', having values '+/-X', '+/-Y' or '+/-Z'. For the atom defined by '_atom_local_axes_atom_label', whose nuclear position is taken as the origin, local axis 'ax1' is the vector from the origin to atom0, axis 'ax2' is perpendicular to 'ax1' and lies in the plane of 'ax1' and a vector passing through the origin parallel to the vector atom1 -> atom2 (its positive direction making an acute angle with the vector parallel to atom1 -> atom2), and a right-handed orthonormal vector triplet is formed from the vector product of these two vectors. In most cases, atom1 will be the same as the atom specified by _atom_local_axes_atom_label. One or more 'dummy' atoms (with arbitrary labels) may be used in the vector definitions, specified with zero occupancy in the _atom_site_ description. The values of *_atom0, *_atom1 and *_atom2 must be identical to values given in the _atom_site_label list. ; data_atom_local_axes_atom1 _name '_atom_local_axes_atom1' _category atom_local_axes _type char _list yes _list_link_parent '_atom_site_label' _list_reference '_atom_local_axes_atom_label' _definition ; Specifies 'atom1' in the definition of a local axis frame. The definition employs three atom-site labels, 'atom0', 'atom1' and 'atom2', and two axis labels, 'ax1' and 'ax2', having values '+/-X', '+/-Y' or '+/-Z'. For the atom defined by '_atom_local_axes_atom_label', whose nuclear position is taken as the origin, local axis 'ax1' is the vector from the origin to atom0, axis 'ax2' is perpendicular to 'ax1' and lies in the plane of 'ax1' and a vector passing through the origin parallel to the vector atom1 -> atom2 (its positive direction making an acute angle with the vector parallel to atom1 -> atom2), and a right-handed orthonormal vector triplet is formed from the vector product of these two vectors. In most cases, atom1 will be the same as the atom specified by _atom_local_axes_atom_label. One or more 'dummy' atoms (with arbitrary labels) may be used in the vector definitions, specified with zero occupancy in the _atom_site_ description. The values of *_atom0, *_atom1 and *_atom2 must be identical to values given in the _atom_site_label list. ; data_atom_local_axes_atom2 _name '_atom_local_axes_atom2' _category atom_local_axes _type char _list yes _list_link_parent '_atom_site_label' _list_reference '_atom_local_axes_atom_label' _definition ; Specifies 'atom2' in the definition of a local axis frame. The definition employs three atom-site labels, 'atom0', 'atom1' and 'atom2', and two axis labels, 'ax1' and 'ax2', having values '+/-X', '+/-Y' or '+/-Z'. For the atom defined by '_atom_local_axes_atom_label', whose nuclear position is taken as the origin, local axis 'ax1' is the vector from the origin to atom0, axis 'ax2' is perpendicular to 'ax1' and lies in the plane of 'ax1' and a vector passing through the origin parallel to the vector atom1 -> atom2 (its positive direction making an acute angle with the vector parallel to atom1 -> atom2), and a right-handed orthonormal vector triplet is formed from the vector product of these two vectors. In most cases, atom1 will be the same as the atom specified by _atom_local_axes_atom_label. One or more 'dummy' atoms (with arbitrary labels) may be used in the vector definitions, specified with zero occupancy in the _atom_site_ description. The values of *_atom0, *_atom1 and *_atom2 must be identical to values given in the _atom_site_label list. ; data_atom_local_axes_ax1 _name '_atom_local_axes_ax1' _category atom_local_axes _type char _list yes _list_reference '_atom_local_axes_atom_label' loop_ _enumeration x X y Y z Z +x +X +y +Y +z +Z -x -X -y -Y -z -Z _definition ; Specifies 'ax1' in the definition of a local axis frame. The definition employs three atom-site labels, 'atom0', 'atom1' and 'atom2', and two axis labels, 'ax1' and 'ax2', having values '+/-X', '+/-Y' or '+/-Z'. For the atom defined by '_atom_local_axes_atom_label', whose nuclear position is taken as the origin, local axis 'ax1' is the vector from the origin to atom0, axis 'ax2' is perpendicular to 'ax1' and lies in the plane of 'ax1' and a vector passing through the origin parallel to the vector atom1 -> atom2 (its positive direction making an acute angle with the vector parallel to atom1 -> atom2), and a right-handed orthonormal vector triplet is formed from the vector product of these two vectors. In most cases, atom1 will be the same as the atom specified by _atom_local_axes_atom_label. One or more 'dummy' atoms (with arbitrary labels) may be used in the vector definitions, specified with zero occupancy in the _atom_site_ description. The values of *_atom0, *_atom1 and *_atom2 must be identical to values given in the _atom_site_label list. ; data_atom_local_axes_ax2 _name '_atom_local_axes_ax2' _category atom_local_axes _type char _list yes _list_reference '_atom_local_axes_atom_label' loop_ _enumeration x X y Y z Z +x +X +y +Y +z +Z -x -X -y -Y -z -Z _definition ; Specifies 'ax2' in the definition of a local axis frame. The definition employs three atom-site labels, 'atom0', 'atom1' and 'atom2', and two axis labels, 'ax1' and 'ax2', having values '+/-X', '+/-Y' or '+/-Z'. For the atom defined by '_atom_local_axes_atom_label', whose nuclear position is taken as the origin, local axis 'ax1' is the vector from the origin to atom0, axis 'ax2' is perpendicular to 'ax1' and lies in the plane of 'ax1' and a vector passing through the origin parallel to the vector atom1 -> atom2 (its positive direction making an acute angle with the vector parallel to atom1 -> atom2), and a right-handed orthonormal vector triplet is formed from the vector product of these two vectors. In most cases, atom1 will be the same as the atom specified by _atom_local_axes_atom_label. One or more 'dummy' atoms (with arbitrary labels) may be used in the vector definitions, specified with zero occupancy in the _atom_site_ description. The values of *_atom0, *_atom1 and *_atom2 must be identical to values given in the _atom_site_label list. ; ########################################### # # # category ATOM_RHO_MULTIPOLE # # # ########################################### data_atom_rho_multipole_[rho] _name '_atom_rho_multipole_[rho]' _category category_overview _type null loop_ _example_detail _example # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ; Example 1 - Multipole coefficients for the nickel ion in [Ni(H3L)][NO3][PF6], [H3L = N,N',N''-tris(2-hydroxy-3-methylbutyl)-1,4,7-triazacyclononane] [G.T. Smith et al. (1997). J. Am. Chem. Soc. 119, 5028-5034]. ; ; loop_ _atom_rho_multipole_atom_label _atom_rho_multipole_coeff_Pv _atom_rho_multipole_coeff_P00 _atom_rho_multipole_coeff_P11 _atom_rho_multipole_coeff_P1-1 _atom_rho_multipole_coeff_P10 _atom_rho_multipole_coeff_P20 _atom_rho_multipole_coeff_P21 _atom_rho_multipole_coeff_P2-1 _atom_rho_multipole_coeff_P22 _atom_rho_multipole_coeff_P2-2 _atom_rho_multipole_coeff_P30 _atom_rho_multipole_coeff_P31 _atom_rho_multipole_coeff_P3-1 _atom_rho_multipole_coeff_P32 _atom_rho_multipole_coeff_P3-2 _atom_rho_multipole_coeff_P33 _atom_rho_multipole_coeff_P3-3 _atom_rho_multipole_coeff_P40 _atom_rho_multipole_coeff_P41 _atom_rho_multipole_coeff_P4-1 _atom_rho_multipole_coeff_P42 _atom_rho_multipole_coeff_P4-2 _atom_rho_multipole_coeff_P43 _atom_rho_multipole_coeff_P4-3 _atom_rho_multipole_coeff_P44 _atom_rho_multipole_coeff_P4-4 _atom_rho_multipole_kappa _atom_rho_multipole_kappa_prime0 _atom_rho_multipole_kappa_prime1 _atom_rho_multipole_kappa_prime2 _atom_rho_multipole_kappa_prime3 _atom_rho_multipole_kappa_prime4 Ni2+(1) 2.38(4) 0.32(4) 0.00 0.00 -0.02(1) 0.00(2) 0.00 0.00 0.00 0.00 -0.08(1) 0.00 0.00 0.00 0.00 0.06(1) -0.04(1) 0.05(1) 0.00 0.00 0.00 0.00 -0.20(1) 0.08(1) 0.00 0.00 1.04(1) 0.44(1) 0.44 1.15(4) 0.44 1.15 ; _definition ; This category contains information about the multipole coefficients used to describe the electron density. High-resolution X-ray diffraction methods enable the determination of the electron density distribution in crystal lattices and molecules, which in turn allows for a characterization of chemical interactions (Coppens, 1997; Koritsanszky & Coppens, 2001). This is accomplished by the construction of a mathematical model of the charge density in a crystal and then by fitting the parameters of such a model to the experimental pattern of diffracted X-rays. The model on which this dictionary is based is the so-called multipole formalism proposed by Hansen & Coppens (1978). In this model, the electron density in a crystal is described by a sum of aspherical "pseudoatoms" where the pseudoatom density has the form defined in the _atom_rho_multipole_* items. Each pseudoatom density consists of terms representing the core density, the spherical part of the valence density and the deviation of the valence density from sphericity. The continuous electron density in the crystal is then modelled as a sum of atom-centred charge distributions. Once the experimental electron density has been established, the "atoms in molecules" theory of Bader (1990) provides tools for the interpretation of the density distribution in terms of its topological properties. Ref: Bader, R. F. W. (1990). Atoms in molecules: a quantum theory. Oxford University Press. Coppens, P. (1997). X-ray charge densities and chemical bonding. Oxford University Press. Hansen, N. K. & Coppens, P. (1978). Acta Cryst. A34, 909-921. Koritsanszky, T. S. & Coppens, P. (2001). Chem. Rev. 101, 1583-1621. ; data_atom_rho_multipole_atom_label _name '_atom_rho_multipole_atom_label' _category atom_rho_multipole _type char _list yes _list_link_parent '_atom_site_label' _list_mandatory yes _definition ; This item is used to identify the atom whose electron density is described with an atom in the ATOM_SITE category. Its value must be identical to one of the values in the _atom_site_label list. ; data_atom_rho_multipole_coeff_ loop_ _name '_atom_rho_multipole_coeff_Pc' '_atom_rho_multipole_coeff_Pv' '_atom_rho_multipole_coeff_P00' '_atom_rho_multipole_coeff_P10' '_atom_rho_multipole_coeff_P11' '_atom_rho_multipole_coeff_P1-1' '_atom_rho_multipole_coeff_P20' '_atom_rho_multipole_coeff_P21' '_atom_rho_multipole_coeff_P2-1' '_atom_rho_multipole_coeff_P22' '_atom_rho_multipole_coeff_P2-2' '_atom_rho_multipole_coeff_P30' '_atom_rho_multipole_coeff_P31' '_atom_rho_multipole_coeff_P3-1' '_atom_rho_multipole_coeff_P32' '_atom_rho_multipole_coeff_P3-2' '_atom_rho_multipole_coeff_P33' '_atom_rho_multipole_coeff_P3-3' '_atom_rho_multipole_coeff_P40' '_atom_rho_multipole_coeff_P41' '_atom_rho_multipole_coeff_P4-1' '_atom_rho_multipole_coeff_P42' '_atom_rho_multipole_coeff_P4-2' '_atom_rho_multipole_coeff_P43' '_atom_rho_multipole_coeff_P4-3' '_atom_rho_multipole_coeff_P44' '_atom_rho_multipole_coeff_P4-4' _category atom_rho_multipole _type numb _type_conditions esd _list yes _list_reference '_atom_rho_multipole_atom_label' _definition ; Specifies the multipole population coefficients P(l,m) for the atom identified in _atom_rho_multipole_atom_label. The multipoles are defined with respect to the local axes specified in the ATOM_LOCAL_AXES category. The coefficients refer to the multipole formalism described by Hansen & Coppens [1978, equation (2)] which gives the electron density at position vector r with respect to an atomic nucleus as rho(r) = Pc*rho_core(r) + Pv*k^3^*rho_valence(kappa*r) + sum{kappa'(l)^3^*R(kappa'(l),l,r)} *sum{P(l,m)*d(l,m,theta,phi)} where: Pc = _atom_rho_multipole_coeff_Pc Pv = _atom_rho_multipole_coeff_Pv P(0,0) = _atom_rho_multipole_coeff_P00 Pc + Pv + P(0,0) = Z (the atomic number) for a neutral atom kappa = _atom_rho_multipole_kappa, kappa'(l) = _atom_rho_multipole_kappa_prime[l], d(l,m,theta,phi) is the spherical harmonic of order l,m at the position (theta, phi) with respect to spherical coordinates centred on the atom. The summations are performed over the index ranges 0 <= l <= lmax, -l <= m <= l, respectively, where lmax is the highest order of multipole applied. The spherical coordinates are related to the local Cartesian axes defined in category ATOM_LOCAL_AXES, z is the polar axis from which the angle theta is measured, and the angle phi is measured from the x axis in the xy plane with the y axis having a value of phi = +90 degrees. R(kappa'(l),l,r) is defined in the _atom_rho_multipole_radial_* items. rho_core(r) and rho_valence(kappa*r) are the spherical core and valence densities, respectively. They are obtained from atomic orbital analytic wavefunctions such as those tabulated by Clementi & Roetti (1974). 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. Ref: Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. Hansen, N. K. & Coppens, P. (1978). Acta Cryst. A34, 909-921. ; data_atom_rho_multipole_configuration _name '_atom_rho_multipole_configuration' _category atom_rho_multipole _type char _list yes _list_reference '_atom_rho_multipole_atom_label' _definition ; This item defines the electronic configuration of the atom given in _atom_rho_multipole_atom_label as free text. ; data_atom_rho_multipole_core_source _name '_atom_rho_multipole_core_source' _category atom_rho_multipole _type char _list yes _list_reference '_atom_rho_multipole_atom_label' _example ; Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. ; _definition ; This item gives the source of the orbital exponents and expansion coefficients used to obtain the spherical core density of the atom defined in _atom_rho_multipole_atom_label. Alternatively, the core density may be obtained as described in the _atom_rho_multipole_scat_core item. Ref: Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. ; data_atom_rho_multipole_kappa_ loop_ _name '_atom_rho_multipole_kappa' '_atom_rho_multipole_kappa_prime0' '_atom_rho_multipole_kappa_prime1' '_atom_rho_multipole_kappa_prime2' '_atom_rho_multipole_kappa_prime3' '_atom_rho_multipole_kappa_prime4' _category atom_rho_multipole _type numb _type_conditions esd _list yes _list_reference '_atom_rho_multipole_atom_label' _definition ; Gives the radial function expansion-contraction coefficients (kappa = _atom_rho_multipole_kappa and kappa'(l) = _atom_rho_multipole_kappa_prime[l]) for the atom specified in _atom_rho_multipole_atom_label. The coefficients refer to the multipole formalism described by Hansen & Coppens [1978, equation (2)] which gives the electron density at position vector r with respect to an atomic nucleus as: rho(r) = Pc*rho_core(r) + Pv*kappa^3^*rho_valence(kappa*r) + sum{kappa'(l)^3^*R(kappa'(l),l,r)} *sum{P(l,m)*d(l,m,theta,phi)} where: Pc = _atom_rho_multipole_coeff_Pc Pv = _atom_rho_multipole_coeff_Pv P(0,0) = _atom_rho_multipole_coeff_P00 Pc + Pv + P(0,0) = Z (the atomic number) for a neutral atom P(l,m) = _atom_rho_multipole_coeff_P[lm], d(l,m,theta,phi) is the spherical harmonic of order l,m at the position (theta, phi) with respect to spherical coordinates centred on the atom. The spherical coordinates are related to the local Cartesian axes defined in category ATOM_LOCAL_AXES, z is the polar axis from which the angle theta is measured, and the angle phi is measured from the x axis in the xy plane with the y axis having a value of phi = +90 degrees. R(kappa'(l),l,r) is defined in the _atom_rho_multipole_radial_* items. rho_core(r) and rho_valence(kappa*r) are the spherical core and valence densities, respectively. They are obtained from atomic orbital analytic wavefunctions such as those tabulated by Clementi & Roetti (1974). 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. The order, l, of kappa' refers to the order of the multipole function, 0 <= l <= 4. The values of kappa' are normally constrained to be equal. Ref: Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. Hansen, N. K. & Coppens, P. (1978). Acta Cryst. A34, 909-921. ; data_atom_rho_multipole_radial_function_type _name '_atom_rho_multipole_radial_function_type' _category atom_rho_multipole _type char _list yes _list_reference '_atom_rho_multipole_atom_label' _definition ; Specifies the function R(kappa'(l),l,r) used for the radial dependence of the valence electron density in the multipole formalism described by Hansen & Coppens [1978, equation (2)] which gives the electron density at position vector r with respect to the nucleus of the atom specified in _atom_rho_multipole_atom_label as: rho(r) = Pc*rho_core(r) + Pv*k^3^*rho_valence(kappa*r) + sum{kappa'(l)^3^*R(kappa'(l),l,r)} *sum{P(l,m)*d(l,m,theta,phi)} where: Pc = _atom_rho_multipole_coeff_Pc Pv = _atom_rho_multipole_coeff_Pv P(0,0) = _atom_rho_multipole_coeff_P00 Pc + Pv + P(0,0) = Z (the atomic number) for a neutral atom kappa = _atom_rho_multipole_kappa, kappa'(l) = _atom_rho_multipole_kappa_prime[l], P(l,m) = _atom_rho_multipole_coeff_P[lm], d(l,m,theta,phi) is the spherical harmonic of order l,m at the position (theta, phi) with respect to spherical coordinates centred on the atom. The summations are performed over the index ranges 0 <= l <= lmax, -l <= m <= l respectively, where lmax is the highest order of multipole applied. The spherical coordinates are related to the local Cartesian axes defined in category ATOM_LOCAL_AXES, z is the polar axis from which the angle theta is measured and the angle phi is measured from the x axis in the xy plane with the y axis having a value of phi = +90 degrees. rho_core(r) and rho_valence(kappa*r) are the spherical core and valence densities, respectively. They are obtained from atomic orbital analytic wavefunctions such as those tabulated by Clementi & Roetti (1974). 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. This item need not be given if a Slater function is used. The parameters of the Slater function should be given using the _atom_rho_multipole_radial_slater_* items. Ref: Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. Hansen, N. K. & Coppens, P. (1978). Acta Cryst. A34, 909-921. ; data_atom_rho_multipole_radial_slater_ loop_ _name '_atom_rho_multipole_radial_slater_n0' '_atom_rho_multipole_radial_slater_zeta0' '_atom_rho_multipole_radial_slater_n1' '_atom_rho_multipole_radial_slater_zeta1' '_atom_rho_multipole_radial_slater_n2' '_atom_rho_multipole_radial_slater_zeta2' '_atom_rho_multipole_radial_slater_n3' '_atom_rho_multipole_radial_slater_zeta3' '_atom_rho_multipole_radial_slater_n4' '_atom_rho_multipole_radial_slater_zeta4' _category atom_rho_multipole _type numb _type_conditions esd _list yes _list_reference '_atom_rho_multipole_atom_label' _definition ; These items are used when the radial dependence of the valence electron density, R(kappa'(l),l,r), of the atom specified in _atom_rho_multipole_atom_label is expressed as a Slater-type function [Hansen & Coppens (1978), equation (3)]: R(kappa'(l),l,r) = [{zeta(l)}^{n(l)+3}^/{n(l)+2}!] *(kappa'(l)*r)^n(l)^ *exp(-kappa'(l)*zeta(l)*r) where: kappa'(l) = _atom_rho_multipole_kappa_prime[l] n(l) = _atom_rho_multipole_radial_slater_n[l] zeta(l) = _atom_rho_multipole_slater_zeta[l] R(kappa'(l),l,r) appears in the multipole formalism described by Hansen & Coppens [1978, equation (2)] which gives the electron density at position vector r with respect to an atomic nucleus as: rho(r) = Pc*rho_core(r) + Pv*kappa^3^*rho_valence(kappa*r) + sum{k'(l)^3^*R(kappa'(l),l,r)} *sum{P(l,m)*d(l,m,theta,phi)} where: Pc = _atom_rho_multipole_coeff_Pc Pv = _atom_rho_multipole_coeff_Pv P(0,0) = _atom_rho_multipole_coeff_P00 Pc + Pv + P(0,0) = Z (the atomic number) for a neutral atom kappa = _atom_rho_multipole_kappa, kappa'(l) = _atom_rho_multipole_kappa_prime[l], P(l,m) = _atom_rho_multipole_coeff_P[lm], d(l,m,theta,phi) is the spherical harmonic of order l,m at the position (theta, phi) with respect to spherical coordinates centred on the atom. The summations are performed over the index ranges 0 <= l <= lmax, -l <= m <= l respectively, where lmax is the highest order of multipole applied. The spherical coordinates are related to the local Cartesian axes defined in category ATOM_LOCAL_AXES, z is the polar axis from which the angle theta is measured, and the angle phi is measured from the x axis in the xy plane with the y axis having a value of phi = +90 degrees. rho_core(r) and rho_valence(kappa*r) are the spherical core and valence densities, respectively. They are obtained from atomic orbital analytic wavefunctions such as those tabulated by Clementi & Roetti (1974). 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. Ref: Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. Hansen, N. K. & Coppens, P. (1978). Acta Cryst. A34, 909-921. ; data_atom_rho_multipole_scat_core _name '_atom_rho_multipole_scat_core' _category atom_rho_multipole _type char _list yes _list_reference '_atom_rho_multipole_atom_label' _definition ; This item gives the scattering factor for the core electrons of the atom specified in _atom_rho_multipole_atom_label as a function of sin(theta)/lambda. The text should contain only a table of two columns, the first giving the value of sin(theta)/lambda, the second giving the X-ray scattering factor at this point in reciprocal space. The atomic core scattering factors are used in least-squares fitting of the items in _atom_rho_multipole_coeff_* and _atom_rho_multipole_kappa_* to experimental X-ray structure factors [see for example Coppens (1997)]. This item enables them to be supplied in the form of a numerical table. Normally they originate from atomic orbital analytic wavefunctions such as those tabulated by Clementi & Roetti (1974). Ref: Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. Coppens, P. (1997). X-ray charge densities and chemical bonding. Oxford University Press. ; data_atom_rho_multipole_scat_valence _name '_atom_rho_multipole_scat_valence' _category atom_rho_multipole _type char _list yes _list_reference '_atom_rho_multipole_atom_label' _definition ; This item gives the scattering factor for the valence electrons of the atom specified in _atom_rho_multipole_atom_label as a function of sin(theta)/lambda. The text should contain only a table of two columns, the first giving the value of sin(theta)/lambda, the second giving the X-ray scattering factor at this point in reciprocal space. The atomic valence scattering factors are used in least-squares fitting of the items in _atom_rho_multipole_coeff_* and _atom_rho_multipole_kappa_* to experimental X-ray structure factors [see for example Coppens (1997)]. This item enables them to be supplied in the form of a numerical table. Normally they originate from atomic orbital analytic wavefunctions such as those tabulated by Clementi & Roetti (1974). Ref: Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. Coppens, P. (1997). X-ray charge densities and chemical bonding. Oxford University Press. ; data_atom_rho_multipole_valence_source _name '_atom_rho_multipole_valence_source' _category atom_rho_multipole _type char _list yes _list_reference '_atom_rho_multipole_atom_label' _example ; Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. ; _definition ; This item gives the source of the orbital exponents and expansion coefficients used to obtain the spherical valence density of the atom defined in _atom_rho_multipole_atom_label. Alternatively the valence density may be obtained as described in the _atom_rho_multipole_scat_valence item. Ref: Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data Tables, 14, 177-478. ; # ------ EOF ------ EOF ------ EOF -------- EOF -------- EOF ------- EOF