# Assessing Density-Functional Theory for Equation-Of-State

^{*}

## Abstract

**:**

## 1. Introduction

_{ZP}= (9/8)(γ/V)Rθ

_{D}.

_{D}is the volume-dependent Debye temperature. For each element considered, we use the reference values V

_{0}, γ(V

_{0}) and θ

_{D}(V

_{0}) from Ref. [8] to compute P

_{ZP}for each volume in the isotherm. This is then subtracted from the T = 0 K isothermal pressures for each volume. This correction is significant in some cases at lower pressures but very small for all solids approaching 100 GPa. Below we compare isotherms and their assumed uncertainty from Ref. [8] (corrected with the P

_{ZP}) with carefully performed zero-temperature density-functional-theory calculations up to 100 GPa for the 64 elements indicated in Figure 1. The scope of the study, as outlined in Figure 1, is chosen so that we are avoiding gases and liquids as well as the 3d transition metal manganese that has a very complex 58-atom crystal structure with many internal parameters. We also chose to not consider the divalent rare-earth elements Eu and Yb because their ambient-pressure phases are (incorrectly) predicted by DFT to be trivalent. Lastly, some elemental solids (Po-Ac) have not been measured up to 100 GPa and these we exclude as well.

## 2. Computational Method

_{max}= 8) inside non-overlapping (muffin-tin) spheres surrounding each atom and in Fourier series in the region between these muffin-tin spheres. There is a choice how to define the muffin-tin sphere radius and here it is chosen as 0.8 of the radius of a sphere with a volume equal the atomic volume (Wigner-Seitz radius). For some crystal structures, where atoms are very close, a smaller value is used to avoid overlapping muffin-tin spheres. The radial part of the basis functions inside the muffin-tin spheres are calculated from a wave equation for the l = 0 component of the potential that include all relativistic corrections including spin-orbit coupling for d and f states but not for the p states, following the comprehensive discussion of the spin-orbit interaction in [11].

^{2}is negative and raging from −3 to −0.2 Ry. For elements with Z < 14 we define 10 basis functions with an s semi-core state in addition to the valence states while for elements 19 < Z < 71 we add a semi-core p state for a total of 12 basis functions. For the 5d transition metals and up to Z = 83 we also add the 4f states as semi-core states (total of 14 basis functions). The actinide metals Th, Pa and U have been shown to be very well described with 6s and 6p semi-core states and 7s, 7p, 5f and 6d valence states [4] and we are replicating that setup here.

^{3}, is used for each calculated pressure). No structural relaxation is attempted for these small volume variations. The DFT total energies do not include the zero-point motion contribution.

## 3. Results

_{2}22) [18] but not the 24-atom distorted fcc (dfcc) or the suggested monoclinic (C2/m) phase [19]. The 24-atom dfcc has very similar pressure dependence as the hcp phase and the C2/m is excluded due to its high DFT total energy (the P6

_{2}22 phase has considerably lower energy).

## 4. Summary and Conclusions

_{0}~ 110 Å

^{3}and B

_{0}~ 2 GPa to very stiff osmium with V

_{0}~ 14 Å

^{3}and B

_{0}~ 400 GPa, or others with few (lithium; 3) or many (uranium; 92) electrons. We also include in this broad population the series of rare-earth metals (La-Lu) that have localized 4f electrons that are troublesome to describe within a DFT approach [22,24].

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Söderlind, P.; Young, D.A.
Assessing Density-Functional Theory for Equation-Of-State. *Computation* **2018**, *6*, 13.
https://doi.org/10.3390/computation6010013

**AMA Style**

Söderlind P, Young DA.
Assessing Density-Functional Theory for Equation-Of-State. *Computation*. 2018; 6(1):13.
https://doi.org/10.3390/computation6010013

**Chicago/Turabian Style**

Söderlind, Per, and David A. Young.
2018. "Assessing Density-Functional Theory for Equation-Of-State" *Computation* 6, no. 1: 13.
https://doi.org/10.3390/computation6010013