Theory of electron structure The ground state of electrons: chemical bonding and electron density

The properties of matter are naturally divided into two main categories, determined by the ground state of the electron and the excited state of the electron. The difference between the two is very clear in the physical properties of materials, and therefore it also determines the framework for theoretical development in the field of electronic structure. The reason for this distinction is that materials consist of the combination of atomic nuclei and electrons.

First, since the energy scale of an electron is often much larger than the energy scale of a more massive nucleus, the lowest energy ground state of an electron determines the structure and low-energy motion of the nucleus. From diamond, the hardest known material, to soft graphite, or the many complex crystals and molecules formed from elements in the periodic table, the various forms of matter are to a large extent the manifestation of the ground state of electrons. At the same time, because atomic nuclei move through most materials on a much larger time scale than electrons, electrons can be thought of as always in their instantaneous ground state, This is also called an adiabatic approximation or Born-Oppenheimer approximation. Since the ground state of the electron is a very important part of the electronic structure, in order to accurately predict the theory, we need a very accurate method to distinguish the small energy difference between different phases of matter. So far, the most widely used "first principle" setting calculation method is density functional theory. That's what we're going to focus on. In addition, the most accurate method for multi-body calculations is quantum Monte Carlo, which is explicitly designed to calculate the properties of the ground state or thermal equilibrium state.

On the other hand, for the structure formed by a given nucleus, the excited state of an electron is the essence of those "electronic properties" - including conductance, optical properties, thermal excitation, phenomena of extrinsic electrons in semiconductors, and so on. These properties are determined by the excitation energy spectrum and the characteristics of the excited state. There are two main types of excitation of electrons: an increase or decrease of electrons, and an excitation with a constant number of electrons. Because excitation can often be roughly thought of as a perturbation of the ground state, perturbation theory methods are often the key to theoretical understanding and calculation of these properties. The excited states of electrons can also be coupled to the motion of the nucleus, which leads to other effects such as electron-phonon interactions. The potential effects on metals are large, such as phase transitions and even new states of matter (such as superconducting states). We may be exposed to the theory and calculation of electron-phonon interactions in the future, but we will not be exposed to the details of phase transitions and superconductivity.

The next few articles will start with the ground state.

The stable structure of a solid is usually classified according to the ground state of the electrons, because it determines the bonding of atoms, more precisely, the lowest energy electron state determines the spatial structure of the nucleus, or conversely, the spatial structure of the nucleus provides the potential field that can determine the Schrodinger equation of the electron. 

The five types of structures formed are as follows:

closed-shell systems are typically noble gases and molecular solids. For noble gases, the characteristic structure is a solid with a close-packed structure, and for non-spherical molecules, the structure is more complex. The electron states are not very different from those in atoms (or molecules) alone, and their binding is usually explained as a result of the van der Waals force's attraction and repulsion counterbalancing each other when overlapping, but there actually seem to be other important mechanisms at play.

Ionic crystals are composed of elements that differ greatly in electronegativity. They are characterized by the transfer of charge that forms ions with closed shells, resulting in the larger anions forming close-packed structures (hcp, fcc, or bcc), while the smaller cations are in the locations where the Coulomb attraction is greatest. However, experiments and calculations have shown that it is impossible to identify charges associated only with ions, so the key is that the ion crystal is an insulator with a band gap.

A metal system is a conductor and has no band gap for electrons to excite because its band is half full. At the same time, the metal energy band is also easy to accept other different numbers of electrons, so that metals with different valences can form alloys, but also make it easier for metals to form close-packed structures. Since the isotropic electron gas is a typical example, we will begin from this point to understand the properties of condensed matter, especially the sp-bonded metals, also known as "simple metals". Other metals, especially many transition metals (or metals dominated by D-bonds), also have important mechanical and magnetic properties, as well as some examples of many-body effects, which are also a theoretical challenge today.

covalent bonding involves a transition from an isolated atomic or ionic state to a bonding state in a solid where the electron band is just filling the band gap. Because of the directional nature of covalent bonds, covalent crystals have a loose structure, unlike the close packing of other types of structures. One of the most successful theories is that of semiconductors and their conversion under high pressure to more densely packed metal systems.

Hydrogen bonding is formed due to the special case of hydrogen, because it is the only active element without inner electrons, so there is no repulsive force of inner electrons when protons and electrons attract each other, as in other elements. The properties of water and ice are greatly affected by this property, because protons can be shared between different water molecules. In addition to this, hydrogen bonding is crucial in many molecules, especially in biological activities, which is a huge challenge in its application in complex materials. Of course, we will not focus too much on the complexity of hydrogen bonds, just know that the electronic structure method can also be used to describe the strength of hydrogen bonds in different situations.

Bonding in actual materials is usually a combination of several of the above mentioned. For example, in a metal there can also be directional covalent bonds or ionic bonds caused by charge transfer. Molecular crystals can contain strong covalent and intramolecular ionic bonds, as well as weaker intermolecular van der Waals bonds. The list goes on.

Now let's talk about another concept of the ground state, electron density.

It plays a fundamental role in the theory of electronic systems and can be measured by X-rays or diffraction of high-energy electrons. Since the total electron density of all but the lightest atoms is largely contributed by the core (generally the part of the atom other than the valence electrons), electron density can reveal the following properties:

The density of nuclei is essentially the density of atoms

debye-waller factor, which describes the effect of thermal motion and zero-point energy on the average density of diffraction (also dominated by nuclei)

Density change due to bonding or charge transfer