## 3 Results
At a high \(T_{e}\), the electrons occupy high-energy delocalized states, the electron density becomes more homogeneous compared to that at zero \(T_{e}\), and the LDA potential, derived from the exchange-correlation functional of a homogeneous electron gas, becomes more accurate [30]. As a result, although the experimental and ab initio permittivities disagree for some metals and energy ranges at 300 K (after comparing Figure 1 and Figures 2 and 3), that does not invalidate the calculations at higher \(T_{e}\). The procedure to measure the optical constants is very sensitive to the experimental conditions, quality of the metal surface, and data extraction methods that may be at the origin of the disagreement at 300 K. Some other experimental data sets reported in the literature agree well with ab initio calculations [24]. Additionally, the ab initio permittivity describes an ideal case of a pure metal and perfect illumination conditions that are hard to achieve experimentally. Figure 1: Drude–Lorentz fit of experimental permittivity at 300 K reported in the _Handbook on Optical Constants of Metals_(Adachi, 2012) [29]. The real \(\epsilon_{1}\) and imaginary \(\epsilon_{2}\) parts of the permittivity are shown. An electron–phonon equilibrium is considered. Figure 2: Drude–Lorentz fit of permittivity calculated using ab initio methods for Cr, W, and Ti. The real \(\epsilon_{1}\) and imaginary \(\epsilon_{2}\) parts of permittivity are shown at different electron temperatures corresponding to different levels of electron–phonon nonequilibrium. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Coefficient**} & \multicolumn{3}{c}{**Cr**} & \multicolumn{3}{c}{**W**} & \multicolumn{3}{c}{**Ti**} \\ \cline{2-10}  & **300 K** & **10,000 K** & **25,000 K** & **300 K** & **10,000 K** & **25,000 K** & **300 K** & **10,000 K** & **25,000 K** \\ \hline \(f_{0}\) & 0.132 & 0.168 & 0.133 & 0.156 & 0.249 & 0.185 & 0.115 & 0.241 & 0.199 \\ \(\Gamma_{0}\) & 0.110 & 0.154 & 0.138 & 0.120 & 0.137 & 0.107 & 0.177 & 0.417 & 0.447 \\ \(f_{1}\) & 0.567 & 0.312 & 0.388 & 0.015 & 0.052 & 0.131 & 0.694 & 0.783 & 0.377 \\ \(\Gamma_{1}\) & 2.382 & 2.855 & 4.571 & 0.171 & 0.840 & 1.040 & 1.932 & 2.592 & 2.188 \\ \(\omega_{1}\) & 1.935 & 1.411 & 1.578 & 0.686 & 0.652 & 0.325 & 0.936 & 1.751 & 1.823 \\ \(f_{2}\) & 0.002 & 0.569 & 0.204 & 0.003 & 0.008 & 0.015 & 0.306 & 0.374 & 0.274 \\ \(\Gamma_{2}\) & 0.100 & 2.437 & 2.105 & 0.100 & 0.333 & 0.579 & 1.363 & 1.760 & 1.685 \\ \(\omega_{2}\) & 2.608 & 3.302 & 3.446 & 1.114 & 2.412 & 2.343 & 2.062 & 3.024 & 3.126 \\ \(f_{3}\) & 0.471 & 0.038 & 0.121 & 2.038 & 1.837 & 1.860 & 0.602 & 0.140 & 0.189 \\ \(\Gamma_{3}\) & 1.640 & 0.396 & 1.342 & 6.709 & 7.195 & 9.992 & 1.626 & 0.878 & 1.666 \\ \(\omega_{3}\) & 3.241 & 3.423 & 3.976 & 3.668 & 3.933 & 4.813 & 3.689 & 4.223 & 4.690 \\ \(f_{4}\) & 0.128 & 0.153 & 0.495 & 0.376 & 0.348 & 0.175 & 0.370 & 0.419 & 0.544 \\ \(\Gamma_{4}\) & 0.001 & 0.001 & 1.512 & 0.961 & 1.023 & 0.952 & 0.001 & 0.212 & 0.458 \\ \(\omega_{4}\) & 6.773 & 7.187 & 9.614 & 5.141 & 5.209 & 5.347 & 6.882 & 7.275 & 8.058 \\ \(\chi_{7}^{2}\) & 0.299 & 0.521 & 0.320 & 0.587 & 0.301 & 0.588 & 0.878 & 0.436 & 0.691 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Drude-Lorentz model fitting coefficients for the permittivity calculated using ab initio methods for Cr, W, and Ti. To understand other effects of a high electron temperature on permittivity, we analyzed the intraband and interband contributions separately. The contribution of intraband transitions to the permittivity generally increases with electron temperature, as shown in Figure 4. The high electron temperature also influences the relative strength of the interband transitions and shifts the interband transition energy, as can be seen from the evolution of the intraband transitions. The high electron temperature also influences the relative strength of the interband transitions and shifts the interband transition energy, as can be seen from the evolution of the intraband transitions.