Modern Charge-Density Analysis

A concise summary is provided on the basic aspects of charge density (CD) analysis and an overview of the charge density research and developments over the last 10 years. A glimpse is given to the issues which are treated in more details in the remaining chapters of this book and to those few that, although of some importance, could not be covered. Advances in experimental methodologies and in the charge density model refinements, along with progresses in quantum mechanical methods and in the interpretation/understanding of chemical bonding and interactions are briefly summarized. The increasingly stronger connection between charge density research and challenging questions of relevance to chemistry and to materials and life science is overviewed.

We present the scientist embarking on an experimental charge-density study with some background knowledge of atomic and molecular motion in crystals, and provide an introduction to the way atomic motion is often modeled or estimated in charge-density studies, as well as examples of how these models can be validated, visualized and analyzed.

The extent to which the electron momentum density and spin density can be studied through the Compton effect is outlined. The isolation of a measurable spin-dependent effect in ferromagnets with helically-polarized high energy synchrotron radiation is described within the impulse approximation. The interpretation of directionally dependent data and the reconstruction of the full three dimensional spin density is illustrated by reference to recent results. The Compton method is compared and contrasted with the study of spin and orbital magnetization in non-resonant x-ray diffraction experiments.

Basic theoretical and some practical aspects of the interpretation of X-ray scattering experiments are described. Our focus is on model building and refinement associated with retrieving information related to electron density matrices from the measured data. The ill-posed nature of this inverse problem is emphasised and the physical significance, reliability and reproducibility of the properties obtained by data fitting are discussed through representative examples taken from recent studies. A special attention is devoted to the pseudoatom formalism widely used to interpret high-resolution single-crystal X-ray diffraction data to map the static electron distribution in solids.

There is considerable scope for using quantum mechanical wavefunctions to extract more information from diffraction experiments, particularly X-ray diffraction experiments on molecular crystals. This chapter examines the ideas behind this approach, and reviews some recent developments. For example, the possibility to optimise atomic positional information using ab initio wavefunction is discussed. This important issue is a prerequisite to obtaining accurate electronic properties from the X-ray diffraction experiments because the two sets of information are often highly correlated. The constrained wavefunction approach for obtaining electronic properties is described. The method is validated by comparing deformation density plots, dipole moments, and some response properties (such as the in-crystal polarisability and the bulk refractive indices) with independent experimental data or higher level calculations which do not incorporate experimental data as input. Future developments and directions are mentioned throughout the article and in the conclusion.

Wave-functions based on local models can be built for the computation of electronic properties, with a flexible degree of accuracy. In particular, x-ray or electron scattering related properties (diffraction structure factors and Compton scattering spectra) are of particular interest because of their acute sensitivity to the nature of the chemical bond. Local models are thus relevant in at least two cases: the computations of Compton spectra for disordered solids and, in their simplest form, refinement of chemical parameters, such as atomic orbital extension and orbitals mixing, against experimental scattering data set.

The aim of this chapter is to point out the relevance of magnetization density studies by means of polarized neutron diffraction (PND) for understanding the magnetic behaviour of materials. Recent advances in the PND technique are overviewed. Major improvements in PND data analysis concern magnetization density reconstruction using the Maximum of Entropy Method (MEM) and the new method based on the local susceptibility tensor approach for non collinear moments in strongly anisotropic materials. The well established model refinement methods are based on the atomic orbital (AO) model and the multipole model. The interest of magnetization density studies for studying novel molecule-based magnetic materials is demonstrated on the basis of selected examples.

The analysis and treatment of density matrices provides the decisive key for the understanding of molecular and solid state systems. The density matrices not only reflect the energetic state of the system as the whole, but also open a possibility for a decomposition of a system into parts accessible to our imagination and experience. The interplay between the density matrices and the space partitioning with the focus on chemically relevant decomposition of molecular systems from the viewpoint of energy as well as the topology and the creation of new functionals is an important subject on the long journey to the comprehension of quantum chemistry.

The relationship between charge density distributions and physical properties in solids is highly complex and usually not obvious. It is therefore the aim of this Chapter to outline concepts how to explore and to analyze the interplay of real space properties (e.g. the charge density distribution or its Laplacian) and reciprocal space properties in solids (e.g. electronic conductivity, superconductivity) by means of charge density analyses. In our case study, we will focus on quasi-one dimensional organometallic carbides, which are textbook examples of extended systems displaying pronounced orbital interactions and anisotropic physical properties in real and reciprocal space. We therefore investigated the electronic structures of the complex carbides Sc₃ TC₄ (T=Fe (1), Co (2), Ni (3)) by combined theoretical and experimental charge density studies. The structures of these organometallic carbides are closely related and display one-dimensional infinite TC₄ ribbons embedded in a scandium matrix. Our study highlights that despite the structural similarities of 1–3 even tiny differences in the electronic band structure are faithfully recovered in the properties of the Laplacian of the electron density. In our case, the shift of the Fermi level to higher energies for the Co(d ⁹) and Ni(d ¹⁰) carbides 2 and 3 relative to the Fe(d ⁸) analogue 1 is reflected in the charge density picture by a significant change in the polarization pattern displayed by the valence shell charge concentrations (VSCC) of the individual transition metal centers in the TC₄ units. Hence, precise high-resolution X-ray diffraction data provide a reliable tool to discriminate and analyze the local electronic structures of isotypic solids even in the presence of a severe coloring problem (Z(Fe)/Z(Co)/Z(Ni)=26/27/28). We further demonstrate that the presence of an axial VSCC at the iron atom is due to localized d _z ² states near the Fermi energy and reflected by a high electronic heat capacity at low temperatures (Sommerfeld coefficient γ=17mJ/K²mol in 1). On contrast, the lack of a narrow conduction band (and axial VSCCs at the transition metal) could be correlated in 2 and 3 with their smaller Sommerfeld coefficients (γ=5.7 and 7.7mJ/K²mol, respectively). Finally, we demonstrate that also the cobalt carbide 2 can be discriminated from its isotypic nickel congener 3 on the basis of its electronic properties. Indeed, only 2 is superconducting below 4.5K and displays a structural phase transition around 70K. Hence, this Chapter should help filling the gap between the various chemical and physical viewpoints on the interplay of chemical bonding and physical properties in solids.

The estimation of intermolecular interaction energies from experimental charge densities, obtained by high resolution X-ray crystallographic measurements, is overviewed. Two main approaches are explored: one based on the conventional theory of intermolecular forces and using the charge density functions directly to deduce electrostatic and possibly other contributions of the total interaction energy, and another involving the characterization of interaction strengths from local properties determined at critical points in the total electron density.

This chapter focuses on the use of a broad spectrum of QTAIM indicators in the context of interpreting the nature of chemical bonding and chemical reactivity in main group and transition metal compounds. These indicators include the classical bond critical point indicators ρ _b and ∇² ρ _b, as well as ones which are less dependent on local properties, such as the Valence Shell Charge Concentrations (VSCCs) on individual atoms, the profiles of bond ellipticities ε along the bond path, the delocalisation indices δ between atomic basins and the source function. In the main, the chosen examples come from very recent work and have been selected to illustrate the use of the QTAIM indicators in providing a full picture of the chemical bonding and, where appropriate the reactivity.

During the last decade charge density studies have matured and more and more studies target specific chemical, physical or biological issues rather than method development. Indeed a very wide range of information can be retrieved from analysis of charge densities. Here we review recent applications of charge density analysis in materials science with emphasis on thermoelectric, magnetic and porous materials.

This paper presents the present status of force fields, their need in nanosecond simulations of systems of current chemical and biological interest, and finally the important steps in charge density research that are paving the way to generic force fields more tightly related to the real physics of chemical intra- and intermolecular interactions. The transferability of force fields is discussed, emphasizing that “fitting” procedures will be progressively replaced by “learning” procedures of how a molecule react to an embedding environment. The various components of interactions between molecules are also discussed.

Experimental charge density methodologies have been extended to macromolecular structures and biocrystallography. Ultra-high resolution diffraction data can now be collected at third-generation synchrotron sources for well-ordered protein crystals. The molecular structure can then be refined using multipolar expansion of the atomic electron density, which is a more sophisticated model than the independent atom model (neutral spherical atoms). Several databases describing average electron densities in terms of multipolar atoms were built by different research groups. These library charge density parameters have been transferred, in the literature, to several small molecules and a few biomacromolecules. The construction of the molecular electron densities through database transfer yields a better crystallographic refinement, notably when the X-ray diffraction data are measured at atomic resolution (0.6–1 Å). The use of an aspherical atom model for the electron density in atomic resolution protein structures allows for the accurate description of their electrostatic properties. The applications to several protein structures including syntenin PDZ2 domain, influenza neuraminidase, human aldose reductase (at 0.66 Å resolution) and a DING phosphate binding protein have been reported.

This chapter discusses the ways in which the present techniques and tools of charge density analysis can contribute to the burgeoning field of crystal engineering. The principal focus is on use of theoretical and experimental charge densities, Hirshfeld surface analysis, as well as the calculation of energies of intermolecular interactions and their application to halogen bonded systems.

This chapter focuses on the electron density in solids under external stress. The modifications of electronic configurations and chemical bonding are studied through analysis of direct space indicators. Examples are given of elemental solids, ionic compounds and molecular crystals, including many experimental evidences.

Recent computational developments on the application of the Electron Localization Function in the solid state allow to perform a rich characterization of chemical changes along phase transitions induced by thermodynamic variables in crystals. Chemical entities, in the sense of the Lewis theory, can be idengified and classified according to the role they play in these processes. Covalent (SiO₂), ionic (BeO), molecular (CO₂, O₂), and metallic (Na, K) systems have been selected to illustrate the ability of ELF to gain insight into the global understanding of the transformations. Detailed topological analysis of the bonding reconstruction process clearly distinguishes transitions where the bonding nature of the solid is not altered, and just a reorganization takes place, to those where the chemical pattern suffers a dramatic change. We have highlighted the close relationship between energy, structure and bonding across several transition pathways and how ELF can be of help to anticipate pressure induced emerging structures and to discard among competitive transition mechanism

This chapter deals with the variation in the electron density distribution of crystalline solids that may occur as a function of temperature (T) changes. The main focus is on experimental electron densities studied by single-crystal X-ray diffraction. In the first part of the chapter, the requirement of temperatures as low as possible for accurate and precise investigations is discussed. Then T-driven physical and chemical changes in the solid state are closely examined. In particular, some test-cases are presented to show how temperature can be used as a variable parameter to explore the correlations between the structural and electronic degrees of freedom in a crystal. Cases of structural phase transitions and electronic spin transitions in molecular crystals are also discussed.

We review recent progress in femtosecond x-ray diffraction for mapping structural dynamics on atomic length and time scales. The chapter combines an introduction to the experimental techniques based on laser-driven x-ray sources with a discussion of recent prototype results allowing for the determination of transient charge density maps in molecular materials.

Conceptual density-functional theory (DFT) provides a mathematical framework for using changes of the electron density to understand chemical reactions and chemical reactivity. The key idea is that by studying the response of a molecule or materials to perturbations, one can decipher its reactivity preferences. If a system reacts favorably to a perturbation, then this indicates that the system will react favorably with a certain class of reagents. Differentials of the energy may thus be interpreted as reactivity indicators. Because of the key role of energy differentials, the mathematical framework of conceptual DFT is similar to classical thermodynamics, with state functions, variational principles, and Legendre transforms. In this chapter we use this thermodynamic simile to present the mathematical underpinnings of conceptual DFT. Applications to systems of interest to organic, inorganic, and biological chemists are used to demonstrate how these abstract concepts may be applied to concrete chemical problems.