X-Ray Charge Densities and Chemical Bonding

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ISBN: 9780195098235 | 0195098234
Cover: Hardcover
Copyright: 5/8/1997

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This book deals with the electron density distribution in molecules and solids as obtained experimentally by X-ray diffraction. It is a comprehensive treatment of the methods involved, and the interpretation of the experimental results in terms of chemical bonding and intermolecular interactions. Inorganic and organic solids, as well as metals, are covered in the chapters dealing with specific systems. As a whole, this monograph is especially appealing because of its broad interface with numerous disciplines. Accurate X-ray diffraction intensities contain fundamental information on the charge distribution in crystals, which can be compared directly with theoretical results, and used to derive other physical properties, such as electrostatic moments, the electrostatic potential and lattice energies, which are accessible by spectroscopic and thermodynamic measurements. Consequently, the work will be of great interest to a broad range of crystallographers and physical scientists.

Chapter 1 Scattering of X-rays and Neutrons

(19)

1.1 Outline of this chapter

(1)

1.2 Introduction to the theory of X-ray scattering

(8)

1.2.1 Classical treatment of X-ray scattering

(2)

1.2.2 Quantum-mechanical treatment: the first Born approximation

(2)

1.2.3 Scattering by a periodic crystal

(2)

1.2.4 The structure factor formalism in terms of atomic densities

(2)

1.3 Resonance scattering of X-rays

(7)

1.3.1 Classical treatment

(7)

1.3.2 Quantum-mechanical treatment: the second Born approximation

(2)

1.3.3 The power series expansion of the scattering operator

(1)

1.3.4 The optical theorem and the relation between f" and f'

(2)

1.4 Neutron scattering

(4)

1.4.1 Properties of neutrons

(1)

1.4.2 The neutron scattering length

(3)

Chapter 2 The Effect of Thermal Vibrations on the Intensities of the Diffracted Beams

(27)

2.1 The normal modes of a crystal

(4)

2.1.1 Phonons, internal and external modes

(1)

2.1.2 The frequency of the normal modes

(4)

2.2 The effect of thermal vibrations on the Bragg intensities

(13)

2.2.1 The Born-Oppenheimer approximation

(1)

2.2.2 The harmonic temperature factor

(3)

2.2.3 Beyond the harmonic approximation

(5)

2.2.4 Comparison of the anharmonic formalisms

(1)

2.2.5 Quantum-statistical treatments

(3)

2.3 The relation between the atomic temperature factors and lattice dynamics

(9)

2.3.1 General expression

(1)

2.3.2 The Debye approximation

(1)

2.3.3 The rigid-body model for molecular crystals

(6)

2.3.4 The rigid-bond test

(1)

Chapter 3 Chemical Bonding and the X-ray Scattering Formalism

(23)

3.1 The breakdown of the independent-atom model

(6)

3.1.1 Qualitative considerations

(2)

3.1.2 The electron density and the LCAO formalism

(4)

3.2 Improved scattering models

(17)

3.2.1 The spherical atom kappa formalism

(1)

3.2.2 Modified spherical scattering factor for the hydrogen atom

(1)

3.2.3 Examples of results obtained with the k-formalism

(2)

3.2.4 The multipole description of the charge density of aspherical atoms

(8)

3.2.5 Aspherical atom scattering factors

(3)

3.2.6 The aspherical density functions of the Hirshfeld formalism

(2)

Chapter 4 Least-Squares Methods and Their Use in Charge Density Analysis

(18)

4.1 Least-squares equations

(7)

4.1.1 Background

(1)

4.1.2 General formalisms for linear least-squares

(2)

4.1.3 Explicit expressions for structure factor least-squares

(2)

4.1.4 Variances and covariances of the least-squares parameter estimates

(3)

4.1.5 Uncorrelated linear combinations of variables

(1)

4.2 The least-squares parameters in charge density analysis

(4)

4.2.1 The paramaters in a charge density refinement

(1)

4.2.2 Parameter restrictions imposed by site and local symmetry and chemical equivalence

(1)

4.2.3 The scale factor

(2)

4.3 Physical constraints of the electron density

(3)

4.3.1 The electroneutrality constraint

(2)

4.3.2 The Hellmann-Feynman constraint

(1)

4.4 Joint refinement of X-ray and neutron data

(4)

4.4.1 The use of complementary information

(1)

4.4.2 Differences in temperature parameters

(1)

4.4.3 Relative weighting of the X-ray and neutron data

(1)

4.4.4 Estimate of the goodness of fit

(2)

Chapter 5 Fourier Methods and Maximum Entropy Enhancement

(31)

5.1 General expressions

(4)

5.1.1 The total density

(2)

5.1.2 The residual density

(1)

5.1.3 Least-squares minimization and the residual density

(1)

5.2 Deformation densities

(21)

5.2.1 Definition of the deformation density

(1)

5.2.2 The choice of a model for the reference state

(6)

5.2.3 The choice of structural parameters in the calculation of the deformation density

101

(4)

5.2.4 Combining Fourier and least-squares methods: the model deformation density

105

(4)

5.2.5 Phase contributions to the deformation density

109

(2)

5.2.6 The variances and covariances of the experimental density

111

(4)

5.3 Maximum entropy enhancement of electron densities

115

(6)

5.3.1 The mathematics of entropy maximization

115

(2)

5.3.2 Application of MEN to single-crystal data on silicon

117

(1)

5.3.3 The two-channel maximum entropy method

118

(3)

Chapter 6 Space Partitioning and Topological Analysis of the Total Charge Density

121

(21)

6.1 Methods of space partitioning

122

(8)

6.1.1 The stockholder partitioning concept

122

(2)

6.1.2 Space partitioning based on the atom-centered multipole expansion

124

(1)

6.1.3 Atomic fragments defined by discrete boundaries

124

(6)

6.2 Space partitioning based on the topology of the total electron density

130

(4)

6.2.1 The definition of critical points

130

(2)

6.2.2 The surface of zero flux

132

(2)

6.3 Chemical bonding and the topology of the total electron density distribution

134

(8)

6.3.1 The Laplacian of the electron density

134

(2)

6.3.2 Classification of bonds based on the topology of the electron density

136

(3)

6.3.3 Topological analysis of experimental densities

139

(3)

Chapter 7 The Electrostatic Moments of a Charge Distribution

142

(23)

7.1 Moments of charge distribution

143

(4)

7.1.1 Definitions

143

(2)

7.1.2 The origin dependence of the electrostatic moments

145

(2)

7.2 Electrostatic moments from diffraction data

147

(8)

7.2.1 Atomic electrostatic moments in terms of the parameters of the multipole formalism

147

(2)

7.2.2 Molecular moments as a sum over the pseudoatom moments

149

(1)

7.2.3 The electrostatic moments of the deformation density

149

(2)

7.2.4 Electrostatic moments of a subvolume of space by Fourier summation

151

(3)

7.2.5 Error analysis of diffraction moments

154

(1)

7.3 Discussion of experimental results

155

(10)

7.3.1 A compilation of X-ray electrostatic moments

155

(5)

7.3.2 Dipole moments

160

(3)

7.3.3 Second moments and quadrupole moments

163

(2)

Chapter 8 X-ray Diffraction and the Electrostatic Potential

165

(27)

8.1 Definitions and units

166

(2)

8.1.1 Definition of the electrostatic potential

166

(1)

8.1.2 the electric field

167

(1)

8.1.3 The electric field gradient

167

(1)

8.1.4 Units

168

(1)

8.2 Evaluation of the electrostatic potential and its derivatives in reciprocal space

168

(6)

8.2.1 the electrostatic potential by Fourier summation of the structure factors

168

(2)

8.2.2 The (0) term in the potential summation

170

(2)

8.2.3 The electric field and the electric field gradient by direct Fourier summation of the structure factors

172

(2)

8.2.4 Application of the F summation to the electrostatic potential in Lalanine

174

(1)

8.3 Evaluation of the electrostatic functions in direct space

174

(7)

8.3.1 Basic expressions

174

(3)

8.3.2 Analytical evaluation of the electrostatic properties due to the spherical components of the charge density

177

(1)

8.3.3 Electrostatic properties and the multipole expansion

177

(4)

8.4 Comparison of diffraction results with theory and other experimental values

181

(5)

8.4.1 The comparison of experimental and theoretical potentials

181

(2)

8.4.2 The relation between the electrostatic potential at the nuclear position, the 1s binding energy and the net atomic charge

183

(1)

8.4.3 The electric field gradient at the nuclear position

184

(2)

8.5 The electrostatic potential outside a charge distribution

186

(6)

8.5.1 The electrostatic potential outside a charge distribution in terms of the multipole moments

186

(1)

8.5.2 Net atomic charges reproducing the electrostatic potential

186

(6)

Chapter 9 The Electron Density and the Lattice Energy of Crystals

192

(19)

9.1 The total energy of a system

192

(1)

9.2 Density functional expressions for the energy

192

(3)

9.2.1 The Hohenberg Kohn theorem

192

(1)

9.2.2 Density functional expressions

193

(1)

9.2.3 The total energy as a function of the electrostatic potential at the nuclear position

194

(1)

9.3 The cohesive energy of ionic crystals

195

(8)

9.3.1 The point-charge model

195

(1)

9.3.2 Fourier series for the total electrostatic energy

196

(1)

9.3.3 The accelerated convergence method and the electrostatic potential in a point-charge crystal

196

(2)

9.3.4 The lattice energy of a crystal consisting of spherical ions

198

(2)

9.3.5 Application to the lattice energy of alkali halides

200

(3)

9.4 The cohesive energy of molecular crystals

203

(8)

9.4.1 Molecular interactions

203

(3)

9.4.2 Interactions between molecules in crystals

206

(1)

9.4.3 Expressions for the evaluation of the electrostatic contribution to the lattice energy of molecular crystals

206

(2)

9.4.4 Calculated lattice energies of molecular crystals

208

(3)

Chapter 10 Charge Density Studies of Transition Metal Compounds

211

(36)

10.1 The study of transition metal complexes

211

(1)

10.2 On the electronic structure of transition metal atoms

212

(8)

10.2.1 Crystal field splitting of d orbitals

212

(2)

10.2.2 Effects of covalency on the orbital populations

214

(2)

10.2.3 The relation between on the occupancies of transition-metal valence orbitals and the multipole population parameters

216

(4)

10.3 The electric field gradient at the nucleus of a transition metal atom

220

(7)

10.3.1 Electric field gradient expressions for transition metal elements

220

(1)

10.3.2 Comparison with results from Mossbauer spectroscopy

221

(6)

10.4 Electron density studies of octahedral and distorted octahedral complexes

227

(3)

10.4.1 Complexes with CO, CN, and NH, ligands

227

(1)

10.4.2 First-row transition metal sulfides

228

(2)

10.5 Electron density studies of iron (II) porphyrins

230

(8)

10.5.1 The electronic structure of iron (II) tetraphenyl porphyrins

230

(1)

10.5.2 Experimental results for iron porphyrins and comparison with theory

230

(8)

10.6 The electron density in metal-metal bonds of transition metal complexes

238

(3)

10.6.1 Metal-metal bonding

238

(1)

10.6.2 Bonding between chromium atoms

238

(2)

106.3 Mn--Mn and Fe--Fe bonding

240

(1)

10.7 Anharmonic thermal motion and the bonding anisotropy of transition metal complexes

241

(6)

10.7.1 Atomic asphericity and anharmonic thermal motion

241

(1)

10.7.2 A model study of theoretical example: anharmonic refinement of bis(pyridine) (meso-tetraphenylporphinato)iron(II)

244

(3)

Chapter 11 The Charge Density in Extended Solids

247

(24)

11.1 Covalently bonded extended solids

248

(9)

11.1.1 Silicon

248

(5)

11.1.2 Charge density studies of silicates

253

(4)

11.2 Metallic solids

257

(11)

11.2.1 Specific aspects of metals

257

(1)

11.2.2 The charge density in beryllium metal

258

(3)

11.2.3 The charge density in vanadium and chromium

261

(3)

11.2.4 The charge density in copper

264

(1)

11.2.5 Intermetallic compounds and the critical-voltage electron diffraction method

265

(3)

11.3 Ionic solids

268

(3)

11.3.1 Do completely ionic solids exist?

268

(3)

Chapter 12 Electron Density Studies of Molecular Crystals

271

(17)

12.1 Why molecular crystals?

271

(2)

12.1.1 The importance of molecular crystals

271

(1)

12.1.2 The suitability factor

272

(1)

12.1.3 The oxalic acid project

273

(1)

12.2 Transferability of charge density parameters among related atoms

273

(4)

12.3 Selected studies of molecular crystals

277

(10)

12.3.1 Bent bonds

277

(5)

12.3.2 Molecular crystals with nonlinear optical properties

282

(1)

12.3.3 The effect of intermolecular interactions on the charge density

283

(4)

12.4 Concluding remarks

287

(1)

Appendix A Tensor Notation

288

(2)

Appendix B Symmetry and Symmetry Restrictions

290

(5)

Appendix C The 50% Probability Ellipsoid

295

(2)

Appendix D Spherical Harmonic Functions

297

(10)

Appendix E Products of Spherical Harmonic Functions

307

(4)

Appendix F Energy-Optimized Single Slater Values for Subshells of Isolated Atoms

311

(2)

Appendix G Fourier-Bessel Transforms

313

(2)

Appendix H Evaluation of the Integrals AN, l1,l2,k (Z, R) Occurring in the Expression for the Peripheral Contribution to the Electrostatic Properties

315

(2)

Appendix I The Matrix M 1 Relating d-Orbital Occupancies Pij to Multipole Populations Plmp

317

(1)

Appendix J The Interaction Between Two Nonoverlapping Charge Distributions

318

(2)

Appendix K Conversion Factors

320

(1)

Appendix L Selected Exercises

320

(8)

References

328

(19)

Index

347

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X-Ray Charge Densities and Chemical Bonding

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X-Ray Charge Densities and Chemical Bonding

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