# Ab Initio Calculations of Transport and Optical Properties of Dense Zr Plasma Near Melting

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## Abstract

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## 1. Introduction

## 2. Computational Methods

#### 2.1. Problem Formulation

**k**-points grid size, number of bands, cutoff energy for plane waves, etc.) are chosen to provide the convergence of thermodynamic properties. Then the QMD simulation is carried out at a given temperature (the temperature is maintained by a thermostat) and the ionic trajectories are calculated. At each ionic step, the electronic structure is calculated in the framework of finite-temperature DFT. The ionic positions are determined using the Hellman–Feynman forces obtained from the electronic structure calculation in which the Born–Oppenheimer approximation is applied. After reaching equilibrium, a set of ionic configurations is selected for the next stages of the study. The number of configurations is chosen to ensure the convergence of the calculated properties, while the distance between neighboring configurations should be large enough to ensure the absence of correlations. In this work, 9 configurations were chosen for averaging after the system had reached the equilibrium state, and every 300th QMD step was chosen.

**k**-points grid, more number of bands) which can provide better accuracy compared with the calculations at the first stage. As a result of the second stage, we have the eigenstates and eigenvalues for electronic bands and corresponding Fermi-weights.

- the complex dielectric constant $\epsilon \left(\omega \right)={\epsilon}_{1}\left(\omega \right)+i{\epsilon}_{2}\left(\omega \right)$ as$${\epsilon}_{1}\left(\omega \right)=1-\frac{{\sigma}_{2}\left(\omega \right)}{\omega {\epsilon}_{0}},\phantom{\rule{2.em}{0ex}}{\epsilon}_{2}\left(\omega \right)=\frac{{\sigma}_{1}\left(\omega \right)}{\omega {\epsilon}_{0}},$$
- the complex refractive index $n\left(\omega \right)+ik\left(\omega \right)$ as$$n\left(\omega \right)=\sqrt{\frac{\left|\epsilon \left(\omega \right)\right|+{\epsilon}_{1}\left(\omega \right)}{2}},\phantom{\rule{2.em}{0ex}}k\left(\omega \right)=\sqrt{\frac{\left|\epsilon \left(\omega \right)\right|-{\epsilon}_{1}\left(\omega \right)}{2}};$$
- the normal spectral reflectivity as$$R\left(\omega \right)=\frac{{\left(1-n\left(\omega \right)\right)}^{2}+k{\left(\omega \right)}^{2}}{{\left(1+n\left(\omega \right)\right)}^{2}+k{\left(\omega \right)}^{2}};$$
- the absorption coefficient as$$\alpha \left(\omega \right)=2k\left(\omega \right)\frac{\omega}{c},$$
- the normal spectral emissivity as$$\mathcal{E}\left(\omega \right)=1-R\left(\omega \right).$$

#### 2.2. Kubo–Greenwood Formula

#### 2.3. Kramers–Kronig Transform

`C++`program was developed by the authors [42]. The function ${\sigma}_{1}\left(\omega \right)$ is given as a one-dimension array with different values ${\sigma}_{i}$ for different values ${\omega}_{i}$ with a step $\mathsf{\Delta}\omega $. As ${\sigma}_{1}\left(\omega \right)$ may have sawtooth-like behavior at low temperatures due to the small size of the system we use the trapezoidal rule for integration to avoid additional interpolation errors. In this program, we can set a different upper integration limit ${\omega}_{\mathrm{max}}$ and change the integration step $\mathsf{\Delta}\omega $. The cubic spline of the original function ${\sigma}_{1}\left(\omega \right)$ is used in the case of a small integration step $\mathsf{\Delta}\omega $ (implemented using

`GSL`library). A user can also increase the upper integration limit ${\omega}_{\mathrm{max}}$ beyond the maximum frequency value of the array. In this case, additional elements of the ${\sigma}_{1}\left(\omega \right)$ array are populated with the last value of the original array. Convergence according to the parameter $\eta $ was tested and achieved at $\eta \le {10}^{-6}$.

## 3. Results and Discussion

## 4. Conclusions

- We have presented the dynamic electrical conductivity of dense Zr plasma at $T=2250$ K and $\rho =6$ g/cm${}^{3}$ using the QMD approach and Kubo–Greenwood formula for the first time.
- We have analyzed the influence of simulation parameters of the numerical integration (integration range, integration step) in KKT and the choice of an exchange-correlation functional on the obtained results using normal spectral emissivity as an example.
- We have shown that the inner shell electrons give a limited contribution to the optical properties, so taking into account only the valence electrons provides a good estimate for transport and optical properties.
- We have demonstrated good agreement with the results of our calculations with experiments.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The real and imaginary part of the dynamic electrical conductivity of liquid Zr at $\rho =6$ g/cm${}^{3}$ and $T=2250$ K.

**Figure 2.**Electronic DOS of liquid Zr at $\rho =6$ g/cm${}^{3}$ and $T=2250$ K, calculated from QMD. The green, red, and violet areas represent the partial (projected) DOS for ideal bcc Zr at $\rho =6$ g/cm${}^{3}$. Energy is given relative to the Fermi level ${E}_{F}$. The Fermi–Dirac distribution at $T=2250$ K is shown by the orange dashed line.

**Figure 3.**The real part of the dynamic conductivity calculated with a different broadening $\mathsf{\Delta}E$ of the Gaussian used for the representation of $\delta $-function in Equation (1). The bottom subfigure shows the deviation from the reference value (with $\mathsf{\Delta}E=0.05$ eV) in percentages. The inset is a closer look at the region 0–1 eV.

**Figure 4.**The imaginary part ${\sigma}_{2}\left(\omega \right)$ of the conductivity calculated using (3) with different ${\omega}_{\mathrm{max}}$ (5–200 eV). The real part ${\sigma}_{1}\left(\omega \right)$ of the conductivity corresponds to Figure 1, at $\omega \ge 50$ eV a constant value ${\sigma}_{1}\left(50\phantom{\rule{3.33333pt}{0ex}}\mathrm{eV}\right)=\mathrm{27,748}$ Sm/m is used for the continuation of ${\sigma}_{1}\left(\omega \right)$. The integration step $\mathsf{\Delta}\omega =0.001$ eV.

**Figure 6.**The behaviour of ${\sigma}_{1}\left(\omega \right)$ (left axis) and normal spectral emissivity $\mathcal{E}$ at wavelength $\lambda =900$ nm (right axis) which are calculated using different ${\sigma}_{1}\left(\omega \right)$ integration range. The green, blue, and olive lines are the three different options for the continuation of ${\sigma}_{1}\left(\omega \right)$-curve.

**Figure 7.**The normal spectral emissivity $\mathcal{E}$ at $\lambda =900$ nm (left axis) which is calculated using ${\sigma}_{1}\left(\omega \right)$ with different integration steps $\mathsf{\Delta}\omega $; and the difference between $\mathcal{E}$ at $\lambda =900$ nm (right axis) which is calculated with a given $\mathsf{\Delta}\omega $ and with $\mathsf{\Delta}{\omega}_{\mathrm{min}}=5\times {10}^{-6}$ eV.

**Figure 8.**Normal spectral emissivity calculated with a different broadening $\mathsf{\Delta}E$ for the Gaussian used for the representation of $\delta $-function in the Kubo–Greenwood formula. The bottom subfigure shows the deviation from the reference value (when $\mathsf{\Delta}E=0.05$ eV) in percentages.

**Figure 9.**The real part of the dynamic electrical conductivity of liquid Zr calculated with different XC-functionals.

**Figure 10.**Normal spectral emissivity of liquid Zr: colored curves—data obtained in this work using different XC-functionals, the black circles—experiment [16], the black line is its linear approximation, the black square—experiment [13], the black up triangle—experiment [17], the black diamond—experiment [14], the black down triangles—experiment [47].

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**MDPI and ACS Style**

Fokin, V.; Minakov, D.; Levashov, P.
Ab Initio Calculations of Transport and Optical Properties of Dense Zr Plasma Near Melting. *Symmetry* **2023**, *15*, 48.
https://doi.org/10.3390/sym15010048

**AMA Style**

Fokin V, Minakov D, Levashov P.
Ab Initio Calculations of Transport and Optical Properties of Dense Zr Plasma Near Melting. *Symmetry*. 2023; 15(1):48.
https://doi.org/10.3390/sym15010048

**Chicago/Turabian Style**

Fokin, Vladimir, Dmitry Minakov, and Pavel Levashov.
2023. "Ab Initio Calculations of Transport and Optical Properties of Dense Zr Plasma Near Melting" *Symmetry* 15, no. 1: 48.
https://doi.org/10.3390/sym15010048