# Use first-principles calculations to predict and investigate the physical properties of electrodes

By solving the Kohn-Sham equation, we can accurately predict the important characteristics of the electrochemical properties of the material. Among the various physical properties of secondary batteries, we mainly discuss the interlayer intercalation voltage, the structural stability of the electrode active material, and the diffusion of lithium.

①Battery voltage
When the positive electrode material changes from x=x1 to x2, the voltage to the metal lithium negative electrode can be given by equation (1). Here, ΔG is the Gibbs free energy change value of reaction (3).

The ΔG of the above reaction has three parts. The internal energy change represented by ΔE can be easily obtained by dividing the ground state energy of the electrode material obtained by first-principle calculations. In formula (2), after obtaining the ground state energy values ​​of Lix1MX, Li (metal) and Lix2MX, subtract the ground state energy value of Lix2MX from the sum of the ground state energy of Lix1MX and Li (metal) to calculate the internal energy change. P△V can be ignored in solid-phase reactions, because in this type of solid-phase reaction, the value of AE is 3~4eV per molecule, and P△V is 10-5eV. T△S is caused by thermal energy and is about 0.025 eV, which can also be ignored. The change value of Gibbs free energy is approximately equal to the change value of internal energy, so it can be obtained from first-principle calculations.

②Structural stability of electrode material
The phase change caused by the change of the lithium content in the electrode active material can be obtained by first-principles calculation. By comparing the thermodynamic energies of the substances produced by the phase change, we can predict the possible reactions and the products produced by the reactions. The thermodynamic energy is approximately equal to the ground state energy value obtained by first-principles calculations. However, if the structure of the by-product is unknown, predicting the phase transition requires comparison among the various possible ground-state configurations caused by changes in the lithium content, so new methods must be adopted to simplify the process. For example, experimental information on the possible structure of the lithium content is very useful for studying phase transitions. In LixMO2, x=0.5, the possible structures are spinel, layered structure and rock salt structure. By calculating the ground state energy of the transition metal M, it is easy to predict the corresponding phase transition of the slang transition metal oxide. We can also study the thermodynamic reaction energy and phase transition mechanism.

For cases other than x=0 and 0.5 to 1, it is recommended to use the group expansion method because the arrangement of lithium atoms in the structure becomes irregular. The arrangement of the rational position and the irrational position during the charging and discharging process causes the formation of a new phase, thereby affecting the voltage and stability of the battery. Through the group expansion method, the ground state energies of different structures can be obtained first, and then new phases can be predicted through geometric relations. This method can more effectively study the relationship between physical content and ground state energy.

③Lithium diffusion
The diffusion of lithium in the electrode active material can be predicted by first principles. Diffusion refers to the movement of atoms due to the gradient distribution of chemical potential in a non-equilibrium state. If this kind of movement is not far from the equilibrium state, it will also help maintain the equilibrium state. When the system is in equilibrium, considering the amplitude of the fluctuations, the changes in motion such as lithium diffusion coefficient can be obtained. Most of the lithium in the electrode active material is in the lattice position, and only has a very short time in the non-equilibrium state. The movement of lithium is considered to be a continuous movement from one lattice position to another lattice position. The transient theory can be used to calculate the frequency of equilibrium lithium ion transitions between lattice positions. This theory gives a statistical meaning Of various lithium orbitals. The transition frequency of lithium can be calculated by averaging the movement of lithium, as expressed by the following formula:

In the formula, υ* is the pre-vibration factor: △Eb is the activation energy of lithium diffusion. There are several methods for calculating the energy value obtained by first-principle calculations to calculate the dispersal path and activation energy of lithium. The first is the elastic band method, which predicts the intermediate state of lithium movement through the extrapolation of the initial state and the final state. The transient state connected by the energy acting as an elastic band is unstable, so the diffusion path and energy change can be accurately calculated. Figure 1 shows the diffusion of lithium in the layered structure and the corresponding activation energy

④Application
VASP (Vienna ab initio fitting data package) is a commercial software program based on in-function theory, which is widely used in first-principle calculations. The actual calculation starts with four input files (INCAR, POSCAR, POTCAR and KPOINTS).

INCAR file: INCAR is the core input file of VASP. Figure 2 shows the basic format of an INCAR file. Each line consists of an identifier, “=” and the corresponding value. Under “lonis Relaxation”, NSW=9 means that the number of ion steps is 9.

FPOSCAR file: The POSCAR file contains lattice geometry parameters and ion position information. Since accurate atomic position information can be obtained through first-principles calculations, it is more effective to use known atomic arrangements. Figure 3 shows a typical POSCAR file. The first row shows the calculated cubic BN structure, the second row provides the scale factor of the lattice constant, and the next three rows give the Bravais lattice vector.

As shown in Figure 3, these are three face-centered vectors, all 3.57Å. “1 1” in the sixth row is the number of each type of atom. If it is “121”, it means that there are two atoms of one kind, and one atom of each other two kinds. Lines 7-9 show the arrangement of these atoms. “Direct” means that the position of the atom is given by the Cartesian coordinate system. The last two lines give the three-dimensional coordinates of each atom.

KPOINTS file: The energy calculation in VASP is performed in a reciprocal lattice. KPOINTS determines the energy density information collected from the reciprocal lattice. Large lattice vectors use small KPOINTS values, and vice versa. Large KPOINTS involves complicated calculations due to the large amount of information, but the given structure is more accurate. KPOINTS should be set according to the purpose of calculation and the characteristics of the material.

POTCAR file: The POTCAR file determines the types of atoms in the POSCAR file, and also contains the pseudopotentials of each atom type used in the Hamiltonian operator. VASP gives the pseudopotentials of all atoms in the periodic table. The user can input the information that needs to be calculated into a file. This is easily accomplished by the cat’s Unix command.

After saving the four initial files in the same folder, you can type in the >vasp command to start the calculation: various output files will be generated in the same folder, including the more important OSZICAR, CHG, CHGCAR and DOSCAR. The OSZICAR file provides important information for calculating electrode voltage or lithium diffusion activation energy. Figure 4 is an example of an OSZICAR file.