Preface

Electronic structure is a central research topic in material science. According to the widely accepted density functional theory, the intrinsic properties and functions of a material are mainly determined by its electronic structure; thus, obtaining an accurate electronic structure is the key to studying the structure-property relationship. Electrons in materials are de Broglie (probability) waves in nature, satisfying the Pauli exclusion principle, with a density distribution in both the position space (referred to as electron density) and momentum space (referred to as momentum density). Electron density and momentum density are complementary and together determine the properties of a material. The electronic structure of a material usually refers to the electron and momentum densities, as well as the associated physical quantities that determine the collective behavior of electrons. Both the electron wavefunction and density matrix are fundamental physical quantities that describe the electronic structure of a material and are equivalent to pure states. Therefore, the electronic structure in this book contains information on three main aspects in real space and momentum space: electron density, density matrix, and wavefunction, where the terms electron density and charge density are used interchangeably in this book.

To date, the electronic structures of materials have been mainly obtained by first-principle calculations, and these electronic structures are usually a simplified and approximated representation of the true electronic structures, as the calculations involve many assumptions and approximations, ignore the working conditions of the materials, and rarely consider material-related experimental data. Compared to theoretical computations, the experimental electronic structures obtained by experimental means can be considered a more realistic representation of the actual electronic structures of materials.

X-ray, neutron, and electron scattering experiments are currently the main methods of probing the structure of matter at the (sub)atomic level, where scattering includes both coherent scattering (diffraction) and incoherent scattering (Compton scattering). Because electrons in materials can scatter X-rays and unpaired electrons can scatter neutrons, the scattering signal should contain information regarding the electronic structure of the material, suggesting that it is theoretically feasible to extract the electronic structure information of a material from the scattering signals, such as X-ray diffraction signals, through certain mathematical processes (e.g. electronic structure refinement). In fact, this has long attracted the attention of scientists. For example, the idea of using scattering experiments (mainly X-ray single-crystal diffraction) to determine the electron density distribution of materials was introduced by Nobel laureates Debye and Bragg and crystallographer Hirshfeld in 1915, 1922, and 1992, respectively.

"It seems that the experimental study of scattered radiation, in particular from light atoms, should get more attention, since along this way it should be possible to determine the arrangement of the electrons in the atoms."

- P. Debye (1915)

"Measurement of the net charges was as yet impossible, but not beyond reach. It seems that crystal analysis must be pushed to a far greater degree of refinement before it can settle the point."

- W. L. Bragg (1922)

"An accurate set of nuclear coordinates and a detailed map of the electron density can be obtained, by X-ray diffraction, only jointly and simultaneously, never separately or independently."

- F. L. Hirshfeld (1992)

Since the late twentieth century, with the rapid development of computer technology and automated equipment, an increasing amount of research has been conducted on the use of scattering techniques to determine the electronic structures of materials and become a research frontier of crystallography. Among the scattering techniques currently used to determine the electronic structures of materials, X-ray single-crystal diffraction is the most dominant method, followed by X-ray Compton scattering, polarized neutron diffraction, magnetic X-ray diffraction, and magnetic X-ray Compton scattering.

The mathematical processes used for experimental data to determine the microstructures (atomic or subatomic scales) of materials, whether crystal or electronic structures, are similar, as they all involve structure models with unknown parameters to fit the scattering experimental data, selecting the model that best fits the experimental data as the optimal structure model. Because the Fourier transform of electron density (also known as the structure factor, ignoring the phase for now) can be determined by the diffraction intensity, the structure factor bridges the structure model and the experimental data. Therefore, the best structure can be determined by minimizing the difference between the modeled value and the experimental value of the structure factor (or structure factor square) through least-squares iteration (or other minimization algorithms). Compared to crystal structures, experimental means for determining electronic structures differ significantly in terms of the structure model, refinement algorithm, data quality requirements, and structural analysis. ? Crystal structure determination involves independent-atom approximation and does not consider interatomic chemical bonding, whereas electronic structure models need to consider chemical bonding, as well as electron correlation effects, to some extent. ? Crystal structure can be viewed as a geometric structure of the arrangement of points with maximum electron densities. Given that the structure factor can be determined from diffraction experiments, the electron density and crystal structure can be fully determined by refining diffraction data. However, the refinement of diffraction data alone is not sufficient for wavefunctions and density matrices, on which it is necessary to impose constraints such as energy minimization, N-representability, and basis function orthogonality. Therefore, the refinement process must minimize the differences between the modeled and experimental values of the structure factor, as well as minimize the Lagrangian of the constraints. Alternatively, the electron density matrix can be obtained by joint refinement of the structure factor and the directional Compton-profile data, where the structure factor determines the diagonal elements of the density matrix in position space, while the Compton-profile data determine the off-diagonal elements of the density matrix in position space. ? Crystal structure refinement typically requires only low-angle diffraction data (structure resolution ?/2sin???>?~0.83?Å), whereas electronic structure refinement uses both low-angle and high-angle diffraction data (?/2sin???>?~0.4?Å) to accurately determine the temperature factor to obtain a reliable electronic structure, because the electronic structure parameters are strongly correlated with atomic temperature factors and the high-angle diffraction data are mainly contributed by the temperature factor with a small contribution of the valence electrons. It is of course possible to use neutron diffraction to determine the atomic temperature factor, but this method requires joint refinement of the X-ray diffraction data and neutron diffraction data to determine the electronic structure, which is more technically challenging than only refining the X-ray diffraction data. ? Generally, small errors in the intensity of the X-ray diffraction data can have a significant impact on the final refined electronic structure; thus, the quality of diffraction data for electronic structure analysis (e.g. consistency factor Rint or Rmerge should typically be less than 0.03) is higher than that for crystal structure refinement. ? Experimentally determined crystal structures (e.g. unit cell parameters, atomic coordinates, and temperature factors) can be used directly for structural analysis and presentation, whereas experimental electronic structures (e.g. three-dimensional scalar data of electron densities) are not in data formats that chemists are familiar with. This makes it necessary to first obtain some intuitive characteristic quantities through topological analysis (e.g. Bader's quantum theory of atoms in molecules) before subsequent electronic structure analysis and structure-property relationship studies.

Of note, the electron wavefunction is generally not considered an experimentally observable physical quantity; therefore, it is not a rigorous statement to say that electron wavefunctions are determined exclusively by experimental methods. Rather, the experimental electron wavefunctions in this book were computed using refinement techniques with the experimental data and contain certain experimental information. Compared to an Nth-order reduced density matrix, a first-order reduced density matrix can be greatly simplified while still describing most of the material's physical and chemical properties of interest. Therefore, the density matrices mentioned in this book refer to first-order reduced density matrices unless otherwise specified.

Except for electronic structure in the real (or position) space, energy levels and band structures...