# Density Functional Theory Study on the Adsorption of Co(II) in Aqueous Solution by Graphene Oxide
## 2 Theoretical Calculation Method  
### Density Functional Theory  
Density functional theory (DFT) is a method of calculating energy as the density of particles in a system. Since the energy of a molecular system has a one-to-one correspondence with the electron density, and the electron density of a molecule is only a function of the position of each atom in space (x, y, z), the distribution of electron density in space can be obtained by solving the 3n degrees of freedom of n particles, and then the motion state of microscopic particles can be described qualitatively or quantitatively [2]. At present, density functional theory is one of the most widely used quantum mechanical methods, which is usually used to calculate the binding energy of molecules in chemistry and the band structure of solids in physics [3].  
The theoretical basis of DFT is the theory of heterogeneous electron gas proposed by [PERSON] and [PERSON], namely the Hohenberg-Kohn theorem [4], which mainly contains two parts. One is that the ground-state energy of identical Fermi subsystems regardless of spin is the only functional of the particle number density function. The second is that under the condition of a constant particle number, the energy functional reaches the minimum value in the correct particle number density function, which is equal to the ground-state energy of the system. The first part of quantification shows that the particle population density function is the basic variable to determine the ground-state physical properties of the multiparticle system. All the ground-state physical properties of the multiparticle system, such as energy, wave function and the expected values of all operators, are uniquely determined by the particle population density function. The second part shows that the minimum value of the energy functional can be determined by the ground-state particle population density function, and the minimum value is equal to the ground-state energy, so the variation in the energy functional to the particle population density is the way to determine the ground state of the system.  
[PERSON] and [PERSON] proposed that in order to find a concrete expression of the energy functional, the whole system of the interacting functional can be replaced by a separate, non-interacting functional, and all errors can be represented by a term called the exchange correlation functional \(\text{Ex}[\rho]\)[5]. Therefore, the choice of the exchange correlation functional directly determines the accuracy of the calculated result. However, in the current DFT research, there is no specific function that can accurately express \(\text{Ex}\). \(\text{Ex}\) is usually written as the sum of exchange energy \(\text{Ex}\) and correlation energy \(\text{E}\). Different processing methods of the two constitute different methods in DFT calculation. Common functionals include Generalized Gradient Approximation (GGA) functionals, Local Density Approximation (LDA) functionals and mixed functionals. The solution of the energy of different systems is the process of obtaining the atomic coordinates through the established structural model and then solving the appropriate functional relations.