## 3 Results  
### MgO Surface Model  
MgO belongs to a cubic crystal system with a space group number of \(Fm3m\) (225), meaning it has a sodium chloride crystal structure. To verify the rationality of the calculation method and accuracy, the structure of the crystal cell was optimized when studying the crystal. The lattice constant after structure optimization was 4.26 A. In order to study the catalytic decomposition and catalytic reduction of N\({}_{2}\)O by MgO, it was necessary to establish a reasonable surface model to characterize the surface properties of MgO. Surface energy is a measure of the breaking of chemical bonds between molecules during surface formation, and density functional theory (DFT) is a theoretical calculation based on the principles of quantum physics, which is very suitable for measuring surface energy [22]. Therefore, we first studied the energy of different MgO surfaces to determine the stable surface of MgO and compared the surface energies of two low-index MgO surfaces (100, 110). These two surfaces are plate models of 3-7 layers of MgO (100) and MgO (110) in the optimized MgO crystal, wherein each layer contains 8 Mg ions and 8 O ions. To avoid mutual interference between the mirror surfaces, a 12 A vacuum layer was constructed perpendicular to the surface, and the surface energies of each surface were calculated as follows [23]:  
\[\sigma_{surf}=\frac{(E_{slab}-NE_{bulk})}{2A_{surf}} \tag{7}\]  
where \(\sigma_{surf}\) is the surface energies of each surface, \(E_{slab}\) is the total energy required to build the surface, \(E_{bulk}\) is the energy of the MgO cell after structure optimization, \(N\) is the number of MgO cells in the surface model, and \(A_{surf}\) is the surface area of the MgO (100) surface.  
The surface energy calculated from Equation (7) is shown in Figure 1. It is clear that the surface energy of the MgO (100) surface is lower than that of the MgO (110) surface for the 3-7 layers of MgO (100) and MgO (110) models, indicating that the MgO (100) surface is more stable than the MgO (110) surface. Further, for alkaline earth metal oxides, the MgO (100) surface is more stable and compact than MgO (110) surface. These results are consistent with the results of [PERSON] et al. [23]. When the number of surface layers gradually increases from three to seven, the surface energy of the MgO (100) surface changes by 1%. Considering the calculation and its accuracy, the MgO (100) five-layer surface model was used in subsequent calculations to study the performance of MgO catalytic decomposition and catalytic reduction of N\({}_{2}\)O.  
The constructed five-layer MgO (100) model is shown in Figure 2, wherein it can be seen that the highest occupied molecular orbital (HOMO) on the MgO (100) surface is mainly attributed to the oxygen sites of the edge mode, which indicates that the oxygen sites on the surface are more likely to participate in the catalytic reaction process of the surface than the magnesium sites on the surface. As shown in Figure 2c, the results of the partial density analysis (PDOS), the price band of the MgO (100) model is mainly contributed to by the 2p orbital of the surface oxygen ions, mainly distributed between the  
N\({}_{2}\)O and the surface after adsorption is the O vacancy on the surface, with a distance of 3.760 A. Note that the bond lengths of the N-O bond and N-N bond of the N\({}_{2}\)O molecule did not change significantly before and after adsorption. In addition, according to the Mulliken charge analysis, the charge transfers of all eight adsorption configurations were small, wherein the maximum charge transfer occurred when N\({}_{2}\)O was adsorbed at the Mg site at the N end of N\({}_{2}\)O. After adsorption, N\({}_{2}\)O transferred 0.012 \(e\) to the surface of MgO (100). According to the comprehensive adsorption energy, stable aggregate structure after adsorption, and charge transfer, N\({}_{2}\)O is in a state of physical adsorption when in this structure.