## 2 Discussion  
Electron-phonon interactions are thought to be the origin of superconductivity in lanthanide hydrides. According to the Migdal-Eliashberg theory, there are four factors to define \(T_{c}\): the characteristic phonon frequency \(\omega_{\text{log}}\), the electron-phonon coupling strength \(\lambda\), the density of states at the Fermi level \(N(\epsilon_{F})\), and the Coulomb pseudopotential, \(\mu^{\star}\). Density functional theory (DFT), which uses standard pseudo-potentials and exchange correlation functionals, e.g., PBE, to explain lattice dynamics, is universally considered to be correct. It is generally acknowledged that DFT struggles to describe systems possessing strong electronic correlations with strongly localized \(d\) and \(f\) shells, and the use of many-body corrections are necessary. In the following analysis, we used DFT combined with DMFT. In order to capture the local paramagnetic moment in lanthanide elements, one has to correct for local charge and spin fluctuations captured by DMFT. Changes in the orbital character at the Fermi surface produced by spectral weight transfer linked to Hubbard \(f\) band splitting are accounted for using DMFT. In accordance to the Allen-Dynes formalism, this affects low-energy electron-electron scattering processes via transfer of phonon momentum, which will be indicated in the following section. In Table 11 are given the structural parameters of Fm3m-LaH\({}_{10}\) and Fmmm-LaH\({}_{18}\), which show hardly any correlation at 400 GPa. In addition, quantities involved in the \(T_{c}\) calculations, namely, \(\lambda,\omega_{\text{log}}\) and \(N(E_{\text{f}})\), are summarized in Table 12. \(\mu^{\star}\) was taken to be 0.10. We compared the effect of different DMFT electronic charge self-consistency schemes. A comparison is shown in Figure 1 between (i) PBE DFT and (ii) DFT + DMFT with full charge self-consistent formalism (DFT + DMFT + CSC). According to our research, we found that the Eliashberg function, \(\alpha^{2}F(\omega)\), was significantly influenced by the correlation effects, which is defined in Equation (2). We utilized the Koster-Slater interaction vertex for the La correlated manifold, with values of \(U=6\) eV and \(J=0.6\) eV. From Figure 2 we can conclude that the DFT + DMFT + CSC method enables an increase in the superconducting temperature, which confirms the large contribution of the many-body effect to the prediction of the superconducting temperature of lanthanide hydrides, and justifies the need for more detailed correction models. We noted that for the higher composition of hydrogen, LaH\({}_{18}\), the correction to the critical temperature is less significant compared to the lower hydrogen composition case. Figure 1 shows that the latter has a few well determined peaks that are enhanced by the DMFT + CSC correction, whilst in the LaH\({}_{18}\) case, the Eliashberg function has a more evenly spread character with lower height of the peaks. Consequently, for the many-body corrections, we obtained an approximately 18% correction to LaH\({}_{10}\) critical temperature compared to just under 4% for LaH\({}_{18}\), as can be seen from Figure 2. Note that \(U=6\) eV and \(J=0.6\) eV are used in the following section of this paper. As shown in Figure 3, the La compound is represented by a two-band system in the absence of a long-range magnetic order in DFT calculation. DFT formalism uses a single Slater determinant model and consequently fails to capture any paramagnetic effects, with an associated magnetic multiplet (fluctuating magnetic moment). These effects usually cause a splitting of spectral features into satellites, as seen in Figure 3,\(c\), with a resulting enforcement of the \(f\) character at the Fermi level.