# High-Temperature Superconductivity in the Lanthanide Hydrides at Extreme Pressures
## 3 Methods  
Figure 5 depicts our theoretical approach. We present a schematic overview of the theoretical platform's principal elements and their interrelationships. Our method establishes a modular framework for high-pressure material screening. Firstly, Crystal structure AnalXsis by Particle Swarm Optimization (CALYPSO) provided the stoichiometric compositions via Gibbs enthalpies for the equation of state and convex hull [23]. Inter-operability between Quantum Espresso (QE) [24; 25] was accomplished through input file format conversion, pseudopotentials, and \(\mathbf{k}\)-point grids. The many-body corrections were provided by core libraries via the DMFT quantum embedding, which resulted in corrected forces [22] and total free energies [26; 27]. The underlying structure relaxations were carried out using the QE and CASTEP packages in the framework of DFT and using the Perdew-Burke-Ernzerhof generalised gradient approximation (PBE-GGA) [28; 29]. Norm-conserving pseudopotentials were used to describe the core electrons and their effects on valence orbitals [30]. Valence electron configurations of \(5s^{2}5p^{6}5d^{1}6s^{2}\), \(5s^{2}5p^{6}4f^{1}5d^{1}6s^{2}\) (i.e., with explicitly included \(f\) electrons), and \(1s^{1}\) were used for the La, Ce, and H atoms, respectively. A plane-wave kinetic-energy cut-off of 1000 eV and dense Monkhorst-Pack \(\mathbf{k}\)-point grids with a reciprocal space resolution of \(12\times 12\times 12\) were employed in the calculation. Phonon frequencies and superconducting critical temperature were calculated using density functional perturbation theory as implemented in QE [31]. The \(\mathbf{k}\)-space integra  
Figure 5: DFT inter-operability for a consistent many-body platform. Schematic overview of the main modules of the theoretical platform and its interrelations. Firstly, structures are predicted by Crystal structure AnaLYSIS by Particle Swarm Optimization (CALYPSO), via Gibbs enthalpies for the equation of state and convex hull. The underlying core engines are the CASTEP and Quantum Espresso DFT software. Inter-operability between QE and CASTEP is achieved via format conversion of input files, pseudo-potentials, and \(\mathbf{k}\)-point grids. Core libraries are used to provide the many-body corrections, via quantum embedding, which in turn provides corrected forces and energies. In post-processing, the Eliashberg function and superconducting \(T_{c}\) are obtained with the DMFT+a2F approach. Finally, data are archived for future usage in hdf5. tion (electrons) was approximated by a summation over a \(20\times 20\times 12\) uniform grid in reciprocal space, with the Methfessel-Paxton smearing scheme, using a temperature of \(k_{\text{B}}\)T = 0.05 eV for self-consistent cycles and relaxations; the same grid (\(20\times 20\times 12\)) was used for evaluating DOS and coupling strength. Dynamical matrices and \(\lambda\) were calculated on a uniform \(5\times 5\times 3\) grid in \(\mathbf{q}\)-space for P6/mmm-LaH\({}_{16}\) and P6/mmm-CeH\({}_{16}\). In post-processing, the superconducting transition temperature \(T_{c}\) was estimated using the Allen-Dynes modified McMillan equation [32]:  
\[T_{c}=\frac{\omega_{\text{log}}}{1.2}\exp\biggl{[}\frac{-1.04(1+\lambda)}{ \lambda-\mu^{*}(1+0.62\lambda)}\biggr{]} \tag{1}\]  
where \(\mu^{*}\) is the Coulomb pseudopotential.