In the process of researching cathode materials, the allowable discharge voltage must be considered. The design of the cathode material is related to the working voltage and cut-off voltage of the battery, and by investigating the design of the cathode material, it is possible to develop a new cathode material. It can be seen from the energy levels shown in Figure 1 that the potential difference of the battery can be calculated from the Fermi energy level difference between lithium and transition metal ions. In addition, the conductivity of a positive electrode material can be obtained by the difference between the redox couple of transition metals and the 2p orbital energy band of oxygen. This energy band also provides more information about the possibility of synthesizing new materials.

Figure 2 shows the potential of metallic lithium and several representative cathode materials and the energy levels of redox couples.Because the 3 ^{+}/4 ^{+} redox couple of nickel ions partially overlaps the 2p energy band of oxygen, the charging process As nickel oxidizes, O^{2-} decomposes into O_{2} and releases electrons from the 2p band. Therefore, it is impossible to generate NiO_{2}. When LiNiO2 is charged, nickel ions are oxidized to 4^{+ }and react with the 2p orbital of oxygen to produce gas.

In order to develop a new material, the possibility of its synthesis and the potential of the material must be predicted, and the calculation of the electronic energy state must be given. The energy level of the surrounding electrons can be calculated by the wave function and current density. For example, the electronic configuration of lithium is (1s)2 (2s)1, and each electron can be derived from the Schrddinger wave equation based on the spherical symmetry field of an isolated atom Wave function.

The wave function of a molecule is a linear combination of the wave number of each atomic orbital. If we assume that the electron interaction between hydrogen atoms A and B in a hydrogen molecule is negligible, and in the spherical symmetric field of Is without considering the changes in the angle θ and φ of the atomic function X (r), the molecular orbital can be expressed as p(r)=CAXA(r) +CBXB (r): But for elements of the second period that contain more electrons, the atomic orbital becomes more complicated. For example, the atomic orbit of carbon monoxide contains 10 variable parameters, and its characteristic function and energy value need to be solved by a 10 x 10 matrix. A simple molecule requires complex calculations. In order to simplify the calculation process, experimental methods are needed to study the bonding mechanism and chemical reactivity. With the latest developments in technology, non-experimental methods such as first-principles calculations or ab initio calculations have been suggested to use atomic number and composition to calculate material properties. Figure 3 shows the expected results for the cathode material.

**Introduction to first-principles calculations**First-principles calculations use quantum mechanics based on the most basic information to derive the physical properties of materials. Unlike experimental methods, it can calculate physical properties only from atomic number and material composition without experimental data. Through the theorems of quantum mechanics, many properties such as energy and structure can be calculated.

The purpose of first-principles calculations is to obtain the wave function of a given system that contains all the information by solving the time-related Schrodinger equation. However, in the study of cathode materials for secondary batteries, it is sufficient to solve the time-independent Schrddinger equation. This-the time-independent Schridiger equation can be expressed as

Hψ=Eψ s

_{(1)}

In the formula, H is the Hamitoian operator; ψ is the wave function; E is the total energy of the system. In a system with multiple electrons and nuclei, it is almost impossible to solve equations for all particles. Therefore, a lot of assumptions must be made. According to the Bomopeheiner approximation, the state of electrons surrounding the nucleus is directly determined by the nuclear coordinates, because the electrons move much faster than the nucleus, and the position of the nucleus (→Rn) can be regarded as a number in the Hamiltonian operator. Parameters rather than variables. We can rewrite the Hamiltonian operator in more detail as follows:

In the formula, T is kinetic energy: Vee is the electrostatic interaction between electrons: the third term is the Coulomb interaction between electrons and atomic nuclei; the fourth term is Ewald energy or Meidelung energy, which is defined by the interaction between nuclei with charge number Z It is caused by the interaction of Coulomnb, which can be regarded as a constant and ignored when calculating the wave function.

Under the above assumptions, the Rayleigh-Ritz change theorem [71] is used to solve the Schrodinger equation. According to the theorem, the ground state energy value of the Hamiltonian operator can be calculated by formula (3):

In other words, when the square integrable function φ changes to minimize the energy, E[φ] becomes the ground state energy. The single electron wave function in the Slale determinant is used to represent the change in the median value, which is the so-called Haree-Focke self-consistent field method. This allows us to obtain the ground state energy and its corresponding wave function, which contains the physical properties of the system.

Another method to solve Schrodinger equation (1) is the density function theory with electron density as a variable. It is simpler than the 4N-dimensional (N: total number of electrons in the system) HartreeFock self-consistent field method when dealing with three-dimensional problems. Hohenberg-Kohn theory provides a theoretical basis for reducing dimensionality. According to this theory, the energy and physical properties of the ground state are determined by the electron density. Therefore, the ground state energy is given by:

In the formula, F[ρ] is an independent function of the system; v(→r) is the Coulomb potential energy between nuclei. Energy has changed from a function of wave function (E[φ]) to a function of electron density (E[p]). According to the density function theory, if the independent function F[p] is known, the ground state energy of the system is determined by each potential energy. Since it is difficult to calculate F[ρ] accurately, it must be approximated. Kohn and Sham used the following method to obtain approximate values:

F[p] is the sum of three functions. Among them, T.[p] is the kinetic energy of non-interacting electrons; J[ρ] is the traditional Coulomb energy, that is, Hartree potential energy; E.[ρ] is the exchange-correlation energy caused by quantum effects. This exchange correlation energy is the difference between the kinetic energy of the electron and the kinetic energy of the non-interacting electron in formula (5), which leads to different J[p] values through the interaction between the electrons. The kinetic energy Ts[p] can be derived from the Slate determinant of the independent electron orbital function, and the Hartree potential J[p] can be obtained by the traditional method of calculating the Coulomb energy.

By minimizing the system energy E[φ] through function theory, we can obtain the following single-electron equation, namely the Kohn-Sham equation. Because υeff varies with Q(r), the equation can be solved by self-iteration.

In the process of solving equations, the exchange correlation energy needs to be approximated. A widely used method is LDA (Local Density Approximation), which was first proposed by Kohn and Sham. Under this assumption, the exchange-correlation energy can be expressed as

Here, εxc. is the exchange-correlation energy of each electron in the distributed electron cloud. In order to solve the LDA problem of the inhomogeneous electron cloud in transition metal oxides, the GGA (Generalized Gradient Approximation) method, which shows the electron density and gradient, was introduced. However, since GGA is often not more accurate than LDA, it is necessary to choose an approximate method according to the characteristics of the system.

By applying the wave function obtained by the Hartree-Fock method or density function theory, we can obtain the ground state energy, which allows us to calculate various physical properties. In the next blog, we will have the physical properties calculated based on the ground state energy.