# High-Temperature Superconductivity in the Lanthanide Hydrides at Extreme Pressures
## 2 Discussion  
The idea that hydrogen-rich compounds could be potential high-\(T_{c}\) superconductors dates back to the turn of the millennium, when chemical pre-compression was proposed as a viable way to reduce the metallisation pressure of hydrogen in the presence of other elements, leading to observed \(T_{c}\) exceeding 150 K in the LaH\({}_{16}\) system. This indicates compressed hydrogen-rich compounds as potential room-temperature superconductors. The genesis of superconductivity in these hydrides is known to originate from electron-phonon interactions. The characteristic phonon frequency, the electron-phonon coupling strength, the density of states at the Fermi level, and the Coulomb pseudopotential \(\mu^{*}\) are the four factors that define \(T_{c}\) according to BCS theory. Density functional theory, using conventional pseudo-potentials, such as the PBE, is widely acknowledged to provide an accurate explanation of lattice dynamics. However, for compounds with weakly hybridised and localised \(f\) electrons, such as La and Ce, where many-body corrections are required, DFT is known to have difficulties in dealing with strong electronic correlations. We applied a density functional theory technique combined with dynamical mean-field theory (DMFT) in this study. The charge and spin local fluctuations, which are important for the local paramagnetic moment of lanthanide elements, are readily corrected by DMFT. DMFT is used to account for changes in the orbital character at the Fermi surface caused by spectral weight transfer related to Hubbard \(f\) band splitting. This influences low-energy electron-electron scattering processes via phonon momentum transfer, as stated in the Methods Section, according to the Allen-Dynes formalism. P6/mmmm-LaH\({}_{16}\) and P6/mmm-CeH\({}_{16}\) show very little dependence on pressure at 100 GPa. The phonon DOS of P6/mmm-LaH\({}_{16}\) and P6/mmm-CeH\({}_{16}\) computed at 250 GPa is shown in Figure 1, Panels (a) and (b), and the structural parameters are shown in Table 1. Then, we investigated the impact of various DMFT electronic charge self-consistency schemes (see Figure 1c,d). We found that correlation effects have a significant impact on \(\alpha^{2}F(\omega)\). In particular, we compared: (i) PBE density functional theory; (ii) DFT+DMFT with the full-charge self-consistent formalism (DFT+DMFT+CSC). We used the Koster-Slater interaction vertex for La and Ce correlated manyfold, with typical values for \(U=6\) eV and \(J=0.6\) eV. Interestingly, the full-charge self-consistent DMFT provides a large increase of the superconducting temperature (see Panels (e) and (f) in Figure 1). This confirms that many-body effects have a sizeable contribution to the prediction of the superconducting temperatures in lanthanide hydrides. Note that we used \(U=6\) eV and \(J=0.6\) eV throughout the rest of the paper. We focused on the LaH\({}_{16}\) and CeH\({}_{16}\) P6/mmm systems at 250 GPa. Although DMFT readily provides important corrections to the electronic structure, it is worth investigating how DMFT affects the structural qualities. Calculating forces with limited atomic displacement is not tractable due to the high computing overhead of performing many-body adjustments. Recently, we developed a method for the calculation of forces within DMFT, allowing for ultra-soft and norm-conserving pseudopotentials in the underlying DFT [22]. This opens up new possibilities for systems with heavy components that are not well-suited to all-electron computations. The structure relaxation at 250 GPa is seen in Figure 2. Typically, we obtained corrections for the bond lengths of the order of 5%. The many-body effects tend to slightly shorten the La-H and Ce-H bonds, somehow increasing the La-H and Ce-H covalency. At the same time, this La-H and Ce-H bond length reduction is accompanied by a moderate increase in the H-H distance. This behaviour is opposite what we previously found in CeH\({}_{9}\), where both the Ce-H and H-H bonds were longer within the DMFT treatment [21].