Detailed resource on the method of electronic structure crystallography for revealing the experimental electronic structure and structure-property relationships of functional materials
Electronic Structure Crystallography and Functional Motifs of Materials describes electronic structure crystallography and functional motifs of materials, two of the most challenging topics to realize the rational design of high-performance functional materials, emphasizing the physical properties and structure-property relationships of functional materials using nonlinear optical materials as examples.
The text clearly illustrates how to extract experimental electronic structure information and relevant physicochemical properties of materials based on the theories and methods in X-ray crystallography and quantum chemistry. Practical skills of charge density studies using experimental X-ray sources are also covered, which are particularly important for the future popularization and development of electron structure crystallography.
This book also introduces the related theories and refinement techniques involved in using scattering methods (mainly X-ray single-crystal diffraction, as well as polarized neutron scattering and Compton scattering) to determine experimental electronic structures, including the experimental electron density, experimental electron wavefunction, and experimental electron density matrix of crystalline materials.
Electronic Structure Crystallography and Functional Motifs of Materials includes information on: - Basic framework and assumptions of the first-principle calculations, density matrix and density function, and Hartree-Fock (HF) and Kohn-Sham (KS) methods - Analysis of topological atoms in molecules, chemical interaction analysis, coarse graining and energy partition of the density matrix, and restricted space partition - Principles of electronic structure measurement, including thermal vibration analysis, scattering experiments, and refinement algorithm for experimental electronic structure - Independent atom model, multipole model, X-ray constrained wavefunction model, and other electron density models
Electronic Structure Crystallography and Functional Motifs of Materials is an ideal textbook or reference book for graduate students and researchers in chemistry, physics, and material sciences for studying the structures and properties of functional crystalline materials.
Table of Contents
About the Authors xi
Foreword 1 xii
Foreword 2 xiv
Preface xvi
Abbreviations xxi
Introduction xxiii
1 Overview of Electronic Structure Crystallography 1
1.1 Introduction 1
1.1.1 History of Electronic Structure Crystallography 4
1.1.2 The Beginnings of X-ray Crystallography and Quantum Mechanics 4
1.1.3 The Nascent Period of Experimental Electronic Structure Research 5
1.1.4 Developments of Pseudo-atom Models 5
1.1.5 Developments of Experimental Electron-density Matrix Models 9
1.1.6 Developments of Experimental Electron Wavefunction Models 12
1.1.7 Developments in Electron Diffraction-Based Studies of Electronic Structures 14
1.2 Basic Descriptors of Electronic Structure 15
1.2.1 Electron Density 15
1.2.2 Residual Density 16
1.2.3 Deformation Density 17
1.2.4 Electron Wavefunction and Density Matrix 19
1.3 Experimental Characterization of Electronic Structure 21
1.3.1 Experimental Electronic Structure Measurement with X-ray Single-crystal Diffractometer 22
1.3.1.1 X-ray Source 22
1.3.1.2 Goniometer 23
1.3.1.3 X-ray Detector 23
1.3.1.4 Cryogenic Systems 23
1.3.2 Key Aspects of Experimental Electronic Structure Measurement 24
1.3.2.1 Single-crystal Samples 24
1.3.2.2 Measurement Process 25
1.3.2.3 Data Correction 25
1.3.2.4 Examination of the Quality of Electronic Structure Refinement 26
References 26
2 First-Principles Calculations of the Electron Density Functions 35
2.1 Introduction 35
2.2 Basic Framework and Assumptions of the First-Principles Calculations 36
2.3 Density Matrix and Density Function 38
2.3.1 Basic Definition 38
2.3.2 Electron Density 39
2.3.3 Momentum Density 40
2.4 Hartree-Fock (HF) and Kohn-Sham (KS) Methods 42
2.4.1 Basic Theoretical Framework 42
2.4.2 Periodic Solutions of Hartree-Fock (HF) and Kohn-Sham (KS) Equations 44
2.4.3 Calculation of Crystal Density Matrix and Density Function 45
2.4.4 Pseudopotentials 46
2.4.5 Basis Set 47
References 48
3 Topological Indices and Properties of Electronic Structures 49
3.1 Introduction 49
3.2 Analysis of Topological Atoms in Molecules 50
3.2.1 Topological Description of the Electron Density 50
3.2.2 Gradient Vector Field and Topological Atoms 53
3.2.3 Bond Path and Molecular Topological Graph 54
3.2.4 Laplacian 54
3.2.5 Topological Properties of Chemical Bonds 55
3.2.5.1 Electron Density at Bond Critical Points 55
3.2.5.2 Bond Radius and Bond Path Length 55
3.2.5.3 Laplacian of Electron Density at the Bond Critical Points 56
3.2.5.4 Ellipticity 56
3.2.5.5 Energy Density of Bond Critical Points 56
3.2.5.6 Delocalization Index and Bond Order 57
3.2.6 Topological Atomic Properties 59
3.2.6.1 Atomic Charges 59
3.2.6.2 Atomic Volume 59
3.2.6.3 Atomic Kinetic Energy 59
3.2.6.4 Laplacian 60
3.2.6.5 Total Atomic Energy 60
3.2.6.6 Atomic Dipole Moment 62
3.2.6.7 Atomic Quadrupole Moment 62
3.2.6.8 Atomic Information Entropy 63
3.3 Chemical Interaction Analysis 63
3.3.1 Source Function 63
3.3.2 Electron Localization Function 65
3.3.3 Reduced Density Gradient 68
3.4 Coarse Graining and Energy Partition of the Density Matrix 69
3.4.1 Partition of the Density Matrix in Real Space 69
3.4.2 Energy Partition 72
3.4.3 Electron Population Statistics 75
3.5 Restricted Space Partition 77
3.5.1 ω-Restricted Partition 77
3.5.2 Restricted Electron Population Analysis 80
3.5.3 Quasi-continuous Distribution 81
3.5.4 Electron Localization Indicators (ELI) 82
3.5.4.1 Same-spin Electron Pairs 83
3.5.4.2 Singlet and Triplet Electron Pairs 84
3.5.4.3 ELI in Momentum Space 85
3.6 Intermolecular Interaction Energy 86
3.6.1 Interaction Energy of Experimental Electron Density 86
3.6.2 Pseudoatomic Representation of Electrostatic Interactions 88
3.6.2.1 Multipole Expansion Approximation 88
3.6.2.2 Exact Potential and Multipole Moment (EPMM) Model 89
3.6.2.3 Promolecular Approximation 89
3.6.3 Non-electrostatic Interactions 90
3.6.4 Lattice Energy 91
3.6.5 Interaction Energies Obtained from Experimental Charge Analysis 93
References 94
4 Principles of Electronic Structure Measurement 97
4.1 Introduction 97
4.2 Thermal Vibration Analysis 100
4.2.1 Lattice Dynamics 101
4.2.2 Atomic Displacement Parameters 103
4.2.3 Rigid Fragment Analysis 106
4.2.4 Neutron Diffraction-assisted Analysis 108
4.2.4.1 Temperature 108
4.2.4.2 Absorption 109
4.2.4.3 Extinction 109
4.2.4.4 Thermal Diffuse Scattering 109
4.2.4.5 Multiple Scattering 109
4.3 Scattering Experiments 110
4.3.1 X-ray Diffraction 110
4.3.2 Polarized Neutron Diffraction 111
4.3.3 Compton Scattering 112
4.4 Refinement Algorithm for Experimental Electronic Structure 113
4.4.1 Least-square Method 113
4.4.1.1 Mathematical 113
4.4.1.2 Least-square Refinement of Structure Factors 114
4.4.1.3 Parameter-estimated variance and covariance 116
4.4.2 Maximum Entropy Method 117
References 120
5 Pseudo-atom Models 123
5.1 Introduction 123
5.2 Independent Atom Model 124
5.3 Kappa Model 125
5.4 Multipole Model 126
5.4.1 Multipole Spherical Harmonics 126
5.4.2 Real Spherical Harmonic Density Function 127
5.4.3 Radial Distribution Functions 128
5.4.4 Multipole Model Framework 129
5.4.5 Aspheric Atomic Scattering Factors 130
5.4.6 Multipolar Model of Core Electron Expansion 131
5.5 Spin Density Model 132
5.5.1 Pure Spin Contribution 132
5.5.1.1 Atomic Orbital Model of Spin Density 132
5.5.1.2 Multipole Refinement of Spin Density 134
5.5.2 Spin and Orbital Contributions 135
5.5.3 Non-collinear Magnetism 136
5.5.4 Combinatorial Refinement of Electron Density and Spin Density 136
5.6 Other Electron Density Models 137
5.6.1 The X-ray Atomic Orbital (XAO) Model 137
5.6.1.1 Atomic Single-electron Orbitals in a Crystal Field 137
5.6.1.2 Electron Density and Structure Factor 141
5.6.2 X-ray Molecular Orbital Model (XMO) 143
5.6.2.1 Molecular Orbital and Electron Density 143
5.6.2.2 Structure Factors for Monocentric and Bicentric Terms 144
5.6.2.3 Processing of Temperature Factors 147
5.6.3 Molecular Orbitals with Variable Occupation Numbers Model (moon) 149
References 150
6 Density Matrix Model 153
6.1 Introduction 153
6.2 Density Matrix Model 154
6.2.1 Definition of the Density Matrix 154
6.2.2 Localized Model of the Density Matrix 154
6.3 Correlation of Density Matrix to Scattering Experiments 156
6.3.1 Dynamic Scattering Factor 156
6.3.2 Static Structure Factor 157
6.3.3 Elastic Scattering 157
6.3.4 Inelastic Scattering 158
6.4 Reconstruction and Refinement of the Density Matrix 160
6.4.1 Bayesian Method 160
6.4.2 Combined Refinement of Different Types of Data 161
6.4.3 Refinement of the One-electron Reduced Density Matrix (1-RDM) 163
6.4.4 Combinatorial Refinement of Structure Factor and Compton Profile Data 165
6.4.5 Spin-resolved One-order Reduced Density Matrix (1-SRDM) Refinement 165
6.4.5.1 Basic Framework 165
6.4.5.2 Molecular Modeling 166
6.4.5.3 Magnetic Structure Factor and Magnetic Compton Profile 167
6.4.5.4 Variation of the Basis Functions 167
6.4.5.5 Variation of Spin Population Matrices 168
References 169
7 Electron Wavefunction Models 171
7.1 Introduction 171
7.2 X-ray Constrained Wavefunction (XCW) Model 173
7.2.1 Mathematical Framework 173
7.2.2 Hirshfeld Atom Refinement 174
7.2.2.1 Selection of Wavefunction 174
7.2.2.2 Electron Density 175
7.2.2.3 Hirshfeld Atomic Partitioning Method 176
7.2.2.4 Calculation of the Structure Factor 176
7.2.3 X-ray Constrained Wavefunction Refinement 178
7.2.3.1 Special Treatment for Thermal Vibrations 178
7.2.3.2 Density Matrix Representation of Structure Factor 179
7.2.3.3 Experimental Constrained Wavefunction Refinement 179
7.2.4 Open Shell System Method 180
7.2.5 Treatment of Relativistic Effects 182
7.3 The X-ray-Constrained Extremely Localized Molecular Orbital Method 183
7.3.1 Theoretical Extremely Localized Molecular Orbitals 183
7.3.2 Refinement of the Experimentally Constrained Extremely Localized Molecular Orbitals 185
References 187
8 Functional Electronic Structures and Functional Motif of Materials 189
8.1 Introduction 189
8.2 Material Functional Motif 190
8.2.1 Crystal Structure 191
8.2.2 Electronic Structure 193
8.2.3 Magnetic Structure 195
8.2.4 Modulated Defects 197
8.2.5 Statistical Defects 199
8.2.6 Local Defects 200
8.3 Functional Electronic Structures 201
References 206
Index 209
