# Exploring the Effect of the Number of Hydrogen Atoms on the Properties of Lanthanide Hydrides by DMFT

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Discussion

## 3. Methods

**Structural predictions**. Crystal structure investigation by particle swarm optimization (CALYPSO) [30,31], based on PSO algorithm [32,33], provides stoichiometric compositions via Gibbs enthalpies for the equation of state and convex hull. The structure searches were carried out at 400 GPa with primitive cells of LaH${}_{10}$ and LaH${}_{18}$ for more than 600 structures.

**Ab initio calculations**. Structural optimization and computations of enthalpy were performed using VASP code [34]. Electronic structures were calculated by QUANTUM ESPRESSO (QE) [35] code.

**EPC calculations**. We performed the electron-phonon coupling (EPC) calculation using QE with a kinetic energy cutoff of 90 Ry. In order to perform reliable calculation of the electron-phonon coupling in metallic systems, we have employed k-meshes of $2\times {0.045}^{-1}$ for the electronic Brillouin zone integration and q-meshes of $2\times {0.09}^{-1}$ for LaH${}_{10}$ and LaH${}_{18}$ compounds.

**Methods**. We combined the use of QE and CASTEP [36,37] by using input file format conversion, pseudopotentials, and K-point grids. Core libraries for DMFT quantum embedding were used for the many-body corrections, which provided the total free energies, electronic densities, and Kohn–Sham levels occupancies [9,38].

**Post-processing**. For estimating the superconducting transition temperature, ${T}_{c}$ we used the Allen–Dynes modified McMillan equation [40]:

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**The structural parameters of LaH${}_{10}$ and LaH${}_{18}$ were calculated and the results are shown below.

Space Group | Lattice Parameters (Å) | Atoms | Atomic Coordinates (Fractional) | |||
---|---|---|---|---|---|---|

x | y | z | ||||

LaH${}_{10}$ (400 GPa) | Fm-3m | a = b = c = 4.59461 $\alpha $ = $\beta $ = $\gamma $ = 90° | H(32f) | 0.62021 | 0.62021 | 0.62021 |

H(8c) | 0.25000 | 0.25000 | 0.25000 | |||

La(4a) | 0.00000 | 0.00000 | 0.00000 | |||

LaH${}_{18}$ (400 GPa) | Fmmm | a =5.78715 b = 7.06134 c = 3.33380 $\alpha $ = $\beta $ = $\gamma $ = 90° | H(16o) | 0.32905 | 0.27446 | 0.00000 |

H(16l) | 0.08379 | 0.25000 | 0.25000 | |||

H(16k) | 0.25000 | 0.38271 | 0.25000 | |||

H(16o) | 0.33762 | 0.07756 | 0.00000 | |||

La(8h) | 0.00000 | 0.61864 | 0.00000 | |||

La(4a) | 0.00000 | 0.00000 | 0.00000 |

**Table A2.**Calculated variation of $\lambda $, ${\omega}_{log}$ and $N\left({E}_{\mathrm{f}}\right)$ for LaH${}_{10}$ and LaH${}_{18}$ compounds in Fm-3m and Fmmm phase at 400 GPa, respectively.

Method | $\mathit{\lambda}$ | ${\mathit{\omega}}_{log}$$\left(\mathbf{K}\right)$ | $\mathit{N}\left({\mathit{E}}_{\mathbf{f}}\right)$ (States/eV/f.u.) | |
---|---|---|---|---|

LaH${}_{10}$ | DFT | 1.35 | 1536 | 0.80 |

DMFT | 1.76 | 1405 | 1.23 | |

LaH${}_{18}$ | DFT | 1.65 | 1083 | 0.75 |

DMFT | 2.22 | 909 | 0.83 |

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**Figure 1.**Eliashberg function ${\alpha}^{2}F\left(\omega \right)$ for (

**a**) LaH${}_{10}$ and (

**b**) LaH${}_{18}$; the spectral weight at the Fermi level is calculated with different extents of approximation: (i) DFT and (ii) with the full-charge self-consistent formalism (DFT + DMFT + CSC).

**Figure 2.**The superconducting temperature ${T}_{c}$ obtained by the Allen and Dynes formalism for LaH${}_{10}$ and LaH${}_{18}$ at 400 GPa. We obtained a theoretical estimation for LaH${}_{10}$ of ${T}_{c}=155.98$ K by DFT and ${T}_{c}=184.33$ K by DMFT, and a theoretical estimation for LaH${}_{18}$ of ${T}_{c}=134.69$ K by DFT and ${T}_{c}=139.85$ K by DMFT. We have used values of $U=6$ eV and $J=0.6$ eV. All these calculations have been carried out using Fm3m-LaH${}_{10}$ and Fmmm-LaH${}_{18}$ at 400 GPa.

**Figure 3.**(

**a**) Density of states produced by DFT calculations. In (

**b**,

**c**), we show the density of states, obtained within the oneshot DFT + DMFT and within the full-charge self-consistent DFT + DMFT + CSC, respectively. $tDOS$ and $fDOS$ signify the density of states corresponding to the lattice and f impurity Green’s function, respectively. All calculations were carried out in the Fm3m phase of LaH${}_{10}$ at 400 GPa.

**Figure 4.**(

**a**) Density of states produced by DFT calculations. In (

**b**,

**c**), we show the density of states, obtained within the one-shot DFT + DMFT and within the full-charge self-consistent DFT + DMFT + CSC, respectively. $tDOS$ and $fDOS$ signify the density of states corresponding to the lattice and f impurity Green’s function, respectively. All calculations were carried out in the Fmmm phase of LaH${}_{18}$ at 400 GPa.

**Figure 5.**Schematic of the workflow for DFT + DMFT interfaced with Allen–Dynes. Outline of the theoretical platform’s core modules and their interrelationships. To begin with, structures are modeled by crystal structure analysis by particle swarm optimization (CALYPSO), and the pseudopotentials are generated with OPIUM. The main core engines are CASTEP and QE DFT software. We have used format conversion of input files, pseudo-potentials, and K-point grids to ensure interoperability between QE and CASTEP. We employed core libraries to provide many-body corrections using quantum embedding, which also provides updated values for the forces and energies. During the post-processing phase, we used the DMFT + a2F approach to obtain values for the Eliashberg function and superconducting temperature ${T}_{c}$. As a final step, data was archived for future usage using HDF5.

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**MDPI and ACS Style**

Wei, Y.; Chachkarova, E.; Plekhanov, E.; Bonini, N.; Weber, C.
Exploring the Effect of the Number of Hydrogen Atoms on the Properties of Lanthanide Hydrides by DMFT. *Appl. Sci.* **2022**, *12*, 3498.
https://doi.org/10.3390/app12073498

**AMA Style**

Wei Y, Chachkarova E, Plekhanov E, Bonini N, Weber C.
Exploring the Effect of the Number of Hydrogen Atoms on the Properties of Lanthanide Hydrides by DMFT. *Applied Sciences*. 2022; 12(7):3498.
https://doi.org/10.3390/app12073498

**Chicago/Turabian Style**

Wei, Yao, Elena Chachkarova, Evgeny Plekhanov, Nicola Bonini, and Cedric Weber.
2022. "Exploring the Effect of the Number of Hydrogen Atoms on the Properties of Lanthanide Hydrides by DMFT" *Applied Sciences* 12, no. 7: 3498.
https://doi.org/10.3390/app12073498