## 1 Introduction  
Recently, hydrogen-rich materials have attracted great attention as potential candidates for high-temperature superconductivity, which is referred to as the holy grail of condensed matter physics. Since hydride causes high-frequency vibrations of hydrogen atoms at high pressure, it leads to stronger electron-phonon coupling, making it a potential high-temperature superconducting material [1, 2]. Almost 90 years ago, theoretical physicists [PERSON] and [PERSON] predicted that hydrogen, the lightest element in nature, would be extremely likely to metalize in a high pressure environment [3]. An early study in the 1960s on metallic hydrogen found superconductivity at temperatures close to or even above room temperature, which suggests that \(T_{c}\) of molecular hydrogen under high pressure is about 100-240 K, and that \(T_{c}\) of atomic hydrogen can reach 300-350 K also under high pressure [4, 5]. However, it is currently still difficult to perform relevant experiments due to the stringent synthesis requirements and ultra-sensitivity of metallic hydrogen under high pressure [6, 7]. As a result of development of computational tools for predicting crystal structures and calculating properties of materials, ultra-high pressure generators have emerged. Nowadays, theoretical and experimental research on hydrogen-rich compounds covers binary hydride materials formed by almost all elements, which also includes the lanthanum-based high-temperature superconducting material LaH\({}_{16}\)[8], as well as CeH\({}_{9}\)[9, 10, 11], CeH\({}_{16}\), and YbH\({}_{10}\)[12, 13]. Research on these materials has found that binary lanthanide superconducting materials with high hydrogen content at high pressure are characterized by high symmetry and that they exhibit a space group structure of hexagonal and cubic crystal systems under 200-300 GPa. In addition to lanthanide superconducting materials, there are also hydrogen-rich superconducting materials with yttrium elements, which are chemically similar to lanthanides and often classified as rare earth metals YH\({}_{6}\), YH\({}_{9}\), and YH\({}_{10}\)[14, 15]. Among the transition elements, many actinide-related superconducting materials have also been found, such as AcH\({}_{10}\), AcH\({}_{12}\), AcH\({}_{16}\), and ThH\({}_{9-10}\)[16, 17, 18]. Furthermore, some common metal elements have been found to form hydrogen-rich materials with superconducting properties, which include CaH\({}_{6}\)[19, 20, 21, 22], MgH\({}_{6}\)[23], MgH\({}_{12}\)[24], to name a few. In addition to the hydrogen-rich superconducting materials containing metallic elements, hydrogen-rich materials with nonmetals have also attracted much attention. In recent research, [PERSON] reported the superconductivity of a C-S-H system with a maximal superconducting transition temperature of \(287.7\pm 1.2\) K (approximately 15 degrees Celsius) at \(267\pm 10\) GPa [25]. Besides, it was predicted that \(T_{c}\) values of PbH\({}_{8}\), SbH\({}_{4}\), and AsH\({}_{8}\) would be 178 K, 150 K, and 100 K above 200 Gpa, 350 GPa, and 150 GPa respectively, as covalent hydrogenates [26, 27]. It has been found that LaH\({}_{10}\) with face-centered cubic structure is a good metal, forming high electron-density energy bands at the Fermi level. The finite occupancy of the La-4\(f\) orbitals, which are successively pushed at the Fermi level as the external pressure increases [28], ensures a particularly high \(T_{c}\) in La-H systems. The localized La-4\(f\) orbitals become more stable compared to the La-6s and La-5\(d\) orbitals due to the pressure of the external environment, which leads to new prominent quantum states originating from strong electronic correlations [28]. Therefore, in order to more accurately describe the properties and physical laws of hydrogen-rich high-temperature superconductors at high pressures, the quantitative theoretical calculations were used in our research.