X-Ray Charge Densities and Chemical Bonding

Abstract

Chapter 1 Scattering of X-rays and Neutrons 3 1.1 Outline of this chapter 3 1.2 Introduction to the theory of X-ray scattering 3 1.2.1 Classical treatment of X-ray scattering 3 1.2.2 Quantum-mechanical treatment: the first Born approximation 5 1.2.3 Scattering by a periodic crystal 7 1.2.4 The structure factor formalism in terms of atomic densities 9 1.3 Resonance scattering of X-rays 11 1.3.1 Classical treatment 11 1.3.2-Quantum-mechanical treatment: the second Born approximation 13 1.3.3 The power series expansion of the scattering operator 15 1.3.4 The optical theorem and the relation between /" and /' 16 1.4 Neutron scattering 18 1.4.1 Properties of neutrons 18 1.4.2 The neutron scattering length 19 Chapter 2 The Effect of Thermal Vibrations on the Intensities of the Diffracted Beams 22 2.1 The normal modes of a crystal 23 2.1.1 Phonons, internal and external modes 23 2.1.2 The frequency of the normal modes 23 2.2 The effect of thermal vibrations on the Bragg intensities 27 2.2.1 The Born-Oppenheimer approximation 27 2.2.2 The harmonic temperature factor 28 viii Contents 2.2.3 Beyond the harmonic approximation 31 2.2.4 Comparison of the anharmonic formalisms 36 2.2.5 Quantum-statistical treatments 37 2.3 The relation between the atomic temperature factors and lattice dynamics 40 2.3.1 General expression 40 2.3.2 The Debye approximation 41 2.3.3 The rigid-body model for molecular crystals 42 2.3.4 The rigid-bond test 48 Chapter 3 Chemical Bonding and the X-ray Scattering Formalism 49 3.1 The breakdown of the independent-atom model 49 3.1.1 Qualitative considerations 49 3.1.2 The electron density and the LCAO formalism 51 3.2 Improved scattering models 55 3.2.1 The spherical atom kappa formalism 55 3.2.2 Modified spherical scattering factor for the hydrogen atom 56 3.2.3 Examples of results obtained with the K-formalism 57 3.2.4 The multipole description of the charge density of aspherical atoms 59 3.2.5 Aspherical atom scattering factors 67 3.2.6 The aspherical density functions of the Hirshfeld formalism 70 Chapter 4 Least-Squares Methods and Their Use in Charge Density Analysis 72 4.1 Least-squares equations 72 4.1.1 Background 72 4.1.2 General formalisms for linear least-squares 72 4.1.3 Explicit expressions for structure factor least-squares 74 4. L4 Variances and covariances of the least-squares parameter estimates 76 4.1.5 Uncorrelated linear combinations of variables 79 4.2 The least-squares parameters in charge density analysis 79 4.2.1 The parameters in a charge density refinement 79 4.2.2 Parameter restrictions imposed by site and local symmetry and chemical equivalence 80 4.2.3 The scale factor 81 4.3 Physical constraints of the electron density 83 4.3.1 The electroneutrality constraint 83 4.3.2 The Hellmann-Feynman constraint 85 4.4 Joint refinement of X-ray and neutron data 86 4.4.1 The use of complementary information 86 4.4.2 Differences in temperature parameters 86 4.4.3 Relative weighting of the X-ray and neutron data 87 4.4.4 Estimate of the goodness of fit 88 Contents xi 9.2.1 The Hohenberg-...

Charge density analysis of two polymorphs of antimony(iii) oxide

“…The rigid pseudoatom model for multipole refinement with X-ray diffraction data is well established, and a detailed description can be found in Coppens' recent book, 16 or Appendix A of the work by Flensburg et al 17 VALRAY98 18 was used for the multipole refinements as well as subsequent mapping of properties and topological analyses. The multipole expansion extended up to the octopole 19 level on both antimony and oxygen atoms, while expansion-contraction parameters of the valence monopoles on all atoms were refined; apart from satisfying the atomic site symmetries, no additional symmetry was imposed on any of the peudoatoms.…”

An Interacting Quantum Atoms Analysis of the Metal–Metal Bond in [M2(CO)8]n Systems

“…12 In principle, QTAIM is an unbiased method to analyze the chemical bond between two atoms, through a topological analysis of the electron density distribution, which is an observable and therefore it can be determined also experimentally. 13 A sufficient and necessary condition for a pair of atoms to be bonded is the existence of a bond path linking them, 11 implying the presence of a bond critical point (BCP) between the interconnected nuclei. However, although the BCP is an observable, the interpretation of the bond path in "chemical terms" may be not so immediate and the nature itself of a direct MM bond (if present) is not unambiguous.…”

Orbital ordering in LaMnO3: estimates of structure factors and comparison of measurement methods

“…For organic crystals with larger unit cells, where dislocation energies are prohibitively large and where bonding electrons constitute a greater fraction of the total, the situation is much more favourable for conventional X-ray diffraction (Coppens, 1997). Other factors in¯uencing this work include accurate determination of Debye±Waller factors, absorption corrections and polarization corrections.…”