Theoretical calculations

IK Ivan A. Kruglov AK Alexander G. Kvashnin AG Alexander F. Goncharov AO Artem R. Oganov SL Sergey S. Lobanov NH Nicholas Holtgrewe SJ Shuqing Jiang VP Vitali B. Prakapenka EG Eran Greenberg AY Alexey V. Yanilkin

This protocol is extracted from research article:

Uranium polyhydrides at moderate pressures: Prediction, synthesis, and expected superconductivity

**
Sci Adv**,
Oct 12, 2018;
DOI:
10.1126/sciadv.aat9776

Uranium polyhydrides at moderate pressures: Prediction, synthesis, and expected superconductivity

Procedure

The evolutionary algorithm USPEX (*14*–*16*) is a powerful tool for predicting thermodynamically stable compounds of given elements at a given pressure. We performed variable-composition searches in the U-H system at 0, 5, 25, 50, 100, 200, 300, 400, and 500 GPa. The first generation (120 structures) was created using a random symmetric generator, while all subsequent generations contained 20% of random structures, and 80% of structures created using heredity, soft mutation, and transmutation operators. Within USPEX runs, structure relaxations were performed at the generalized gradient approximation level [with the functional from ref. (*21*)] of density functional theory using the projector-augmented wave method (*22*) as implemented in the Vienna Ab initio Simulation Package (VASP) code (*23*–*25*). Plane-wave kinetic energy cutoff was set to 600 eV, and the Brillouin zone was sampled by the Γ-centered *k*-mesh with a resolution of 2π × 0.05 Å^{−1}.

To establish stability fields of the predicted phases, we recalculated their enthalpies with increased precision at various pressures with a smaller pressure increment (from 1 to 10 GPa), recalculating the thermodynamic convex hull at each pressure. The phases that were located on the convex hull are the ones stable at given pressure. Stable structures of elemental H and U were taken from USPEX calculations and from (*26*) and (*27*), respectively.

The superconducting properties were calculated using the QUANTUM ESPRESSO (QE) package (*28*). The phonon frequencies and EPC coefficients were computed using density-functional perturbation theory (*29*) using the plane-wave pseudopotential method and Perdew-Burke-Ernzerhof exchange correlation functional (*21*). Convergence tests showed that 60 rydberg (Ry) is a suitable kinetic energy cutoff for the plane-wave basis set. Electronic band structures of UH_{7} and UH_{8} were calculated using both VASP and QE and demonstrated good consistency. Comparison of the phonon densities of states calculated using the finite displacement method [VASP and PHONOPY (*30*)] and density-functional perturbation theory (QE) showed perfect agreement between these methods.

Critical temperature was calculated from the Eliashberg equation (*31*), which is based on the Fröhlich Hamiltonian *c*^{+} and *b*^{+} relate to creation operators of electrons and phonons, respectively. The matrix element of electron-phonon interaction *u*_{q} is the displacement of an atom with mass *M* in the phonon mode *q*,*j*. Within the framework of Gor’kov (*32*) and Migdal (*33*) approach, the correction to the electron Green’s function _{log}/*E*_{F}). As a result, it will lead to integral Eliashberg equations (*31*). These equations can be solved using an iterative self-consistent method for the real part of the order parameter Δ(*T*, ω) (superconducting gap) and the mass renormalization function *Z*(*T*, ω) (for more details, see the supplementary materials) (*34*).

In our calculations of the EPC parameter λ, the first Brillouin zone was sampled using a 2 by 2 by 2 *q*-points mesh and a denser 24 by 24 by 24 *k*-points mesh (with Gaussian smearing and σ = 0.03 Ry, which approximates the zero-width limits in the calculation of λ). The superconducting transition temperature *T*_{c} was also estimated using the Allen-Dynes–modified McMillan equation (*35*)._{log} is the logarithmic average frequency, and μ* is the Coulomb pseudopotential for which we used widely accepted lower and upper bound values of 0.10 and 0.15. The EPC constant λ and ω_{log} were calculated as

Note: The content above has been extracted from a research article, so it may not display correctly.

Q&A

Your question will be posted on the Bio-101 website. We will send your questions to the authors of this protocol and Bio-protocol community members who are experienced with this method. you will be informed using the email address associated with your Bio-protocol account.