Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 8.7, pp. 713-734

Chapter 8.7. Analysis of charge and spin densities

P. Coppens,a Z. Sub and P. J. Beckerc

a732 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


Becker, P. (1990). Electrostatic properties from X-ray structure factors. Unpublished results.
Becker, P. & Coppens, P. (1985). About the simultaneous interpretation of charge and spin density data. Acta Cryst. A41, 177–182.
Becker, P. & Coppens, P. (1990). About the Coulombic potential in crystals. Acta Cryst. A46, 254–258.
Bentley, J. (1981). In Chemical Applications of Atomic and Molecular Electrostatic Potentials, edited by P. Politzer & D. G. Truhlar. New York/London: Plenum Press.
Bergevin, F. de & Brunel, M. (1981). Diffraction of X-rays by magnetic materials. I. General formulae and measurements on ferro- and ferrimagnetic compounds. Acta Cryst. A37, 314–324.
Bertaut, E. F. (1978). Electrostatic potentials, fields and gradients. J. Phys. Chem. Solids, 39, 97–102.
Blume, M. (1985). Magnetic scattering of X-rays. J. Appl. Phys. 57, 3615–3618.
Blume, M. & Gibbs, D. (1988). Polarization dependence of magnetic X-ray scattering. Phys. Rev. B, 37, 1779–1789.
Bonnet, M., Delapalme, A., Becker, P. & Fuess, H. (1976). Polarized neutron diffraction – a tool for testing extinction models: application to yttrium iron garnet. Acta Cryst. A32, 945–953.
Boucherle, J. X., Gillon, B., Maruani, J. & Schweizer, J. (1982). Spin densities in centrally unsymmetric structures. J. Phys. (Paris) Colloq. 7, 227–230.
Boyd, R. J. (1977). The radial density function for the neutral atoms from helium to xenon. Can. J. Phys. 55, 452–455.
Brown, P. J. (1986). Interpretation of magnetization density measurements in concentrated magnetic systems: exploitation of the crystal translational symmetry. Chem. Scr. 26, 433–439.
Brunel, M. & de Bergevin, F. (1981). Diffraction of X-rays by magnetic materials. II. Measurements on antiferromagnetic Fe2O3. Acta Cryst. A37, 324–331.
Buckingham, A. D. (1959). Molecular quadrupole moments. Q. Rev. Chem. Soc. 13, 183–214.
Buckingham, A. D. (1970). Physical chemistry. An advanced treatise, Vol. 4. Molecular properties, edited by D. Henderson, pp. 349–386. New York: Academic Press.
Condon, E. U. & Shortley, G. H. (1935). The theory of atomic spectra. Cambridge University Press.
Coppens, P. (2001). The structure factor. International tables for crystallography, Vol. B, edited by U. Shmueli, Chap. 1.2. Dordrecht: Kluwer Academic Publishers.
Coppens, P., Koritsanszky, T. & Becker, P. (1986). Transition metal complexes: what can we learn by combining experimental spin and charge densities? Chem. Scr. 26, 463–467.
Cromer, D. T., Larson, A. C. & Stewart, R. F. (1976). Crystal structure refinements with generalized scattering factors. J. Chem. Phys. 65, 336–349.
Cruickshank, D. W. (1949). The accuracy of electron density maps in X-ray analysis with special reference to dibenzyl. Acta Cryst. 2, 65–82.
Dahl, J. P. & Avery, J. (1984). Local density approximations in quantum chemistry and solid state physics. New York/London: Plenum.
Delley, B., Becker, P. & Gillon, B. (1984). Local spin-density theory of free radicals: nitroxides. J. Chem. Phys. 80, 4286–4289.
Desclaux, J. B. & Freeman, A. J. (1978). J. Magn. Magn. Mater. 8, 119–129.
Epstein, J. & Swanton, D. J. (1982). Electric field gradients in imidazole at 103 K from X-ray diffraction. J. Chem. Phys. 77, 1048–1060.
Fermi, E. (1928). Eine statitische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente. Z. Phys. 48, 73–79.
Forsyth, J. B. (1980). In Electron and magnetization densities in molecules and solids, edited by P. Becker. New York/London: Plenum.
Freeman, A. J. & Desclaux, J. P. (1972). Neutron magnetic form factor of gadolinium. Int. J. Magn. 3, 311–317.
Gillon, B., Becker, P. & Ellinger, Y. (1983). Theoretical spin density in nitroxides. The effect of alkyl substitutions. Mol. Phys. 48, 763–774.
Gordon, R. G. & Kim, Y. S. (1972). Theory for the forces between closed-shell atoms and molecules. J. Chem. Phys. 56, 3122–3133.
Hamilton, W. C. (1964). Statistical methods in physical sciences. New York: Ronald Press.
Hellner, E. (1977). A simple refinement of density distributions of bonding electrons. Acta Cryst. B33, 3813–3816.
Hirshfeld, F. L. (1984). Hellmann–Feynman constraint on charge densities, an experimental test. Acta Cryst. B40, 613–615.
Hirshfeld, F. L. & Rzotkiewicz, S. (1974). Electrostatic binding in the first-row AH and A2 diatomic molecules. Mol. Phys. 27, 1319–1343.
Hirshfelder, J. O., Curtis, C. F. & Bird, R. B. (1954). Molecular theory of gases and liquids. New York: John Wiley.
Hohenberg, P. & Kohn, W. (1964). Inhomogeneous electron gas. Phys. Rev. B, 136, 864–867.
Holladay, A., Leung, P. C. & Coppens, P. (1983). Generalized relation between d-orbital occupancies of transition-metal atoms and electron-density multipole population parameters from X-ray diffraction data. Acta Cryst. A39, 377–387.
International Tables for Crystallography (2001). Vol. B, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers.
International Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press.
Johnston, D. F. (1966). Theory of the electron contribution to the scattering of neutrons by magnetic ions in crystals. Proc. Phys. Soc. London, 88, 37–52.
Levine, I. L. (1983). Quantum chemistry, 3rd edition. Boston/London/Sydney: Allyn and Bacon Inc.
Lovesey, S. W. (1984). Theory of neutron scattering from condensed matter. Oxford: Clarendon Press.
Marshall, W. & Lovesey, S. W. (1971). Theory of thermal neutron scattering. Oxford University Press.
Massa, L., Goldberg, M., Frishberg, C., Boehme, R. F. & La Placa, S. J. (1985). Wave functions derived by quantum modeling of the electron density from coherent X-ray diffraction: beryllium metal. Phys. Rev. Lett. 55, 622–625.
Miller, K. J. & Krauss, M. (1967). Born inelastic differential cross sections in H2. J. Chem. Phys. 47, 3754–3762.
Mook, H. A. (1966). Magnetic moment distribution of Ni metal. Phys. Rev. 148, 495–501.
Moss, G. & Coppens, P. (1981). Pseudomolecular electrostatic potentials from X-ray diffraction data. In Molecular electrostatic potentials in chemistry and biochemistry, edited by P. Politzer & D. Truhlar. New York: Plenum.
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.
Rees, B. (1976). Variance and covariance in experimental electron density studies, and the use of chemical equivalence. Acta Cryst. A32, 483–488.
Rees, B. (1978). Errors in deformation-density and valence-density maps: the scale-factor contribution. Acta Cryst. A34, 254–256.
Schweizer, J. (1980). In Electron and magnetization densities in molecules and crystals, edited by P. Becker. New York: Plenum.
Spackman, M. A. (1992). Molecular electric moments from X-ray diffraction data. Chem. Rev. 92, 1769.
Stevens, E. D. & Coppens, P. (1976). A priori estimates of the errors in experimental electron densities. Acta Cryst. A32, 915–917.
Stevens, E. D., DeLucia, M. L. & Coppens, P. (1980). Experimental observation of the effect of crystal field splitting on the electron density distribution of iron pyrite. Inorg. Chem. 19, 813–820.
Stevens, E. D., Rees, B. & Coppens, P. (1977). Calculation of dynamic electron distributions from static molecular wave functions. Acta Cryst. A33, 333–338.
Stewart, R. F. (1977). One-electron density functions and many-centered finite multipole expansions. Isr. J. Chem. 16, 124–131.
Stewart, R. F. (1979) On the mapping of electrostatic properties from Bragg diffraction data. Chem. Phys. Lett. 65, 335–342.
Su, Z. & Coppens, P. (1992). On the mapping of electrostatic properties from the multipole description of the charge density. Acta Cryst. A48, 188–197.
Su, Z. & Coppens, P. (1994a). Rotation of real spherical harmonics. Acta Cryst. A50, 636–643.
Su, Z. & Coppens, P. (1994b). On the evaluation of integrals useful for calculating the electrostatic potential and its derivatives from pseudo-atoms. J. Appl. Cryst. 27, 89–91.
Thomas, L. H. (1926). Proc. Cambridge Philos. Soc. 23, 542–548.
Tofield, B. C. (1975). Structure and bonding, Vol. 21. Berlin: Springer-Verlag.