## 3 Results
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Coefficient**} & \multicolumn{2}{c}{**Fe**} & \multicolumn{2}{c}{**Au**} & \multicolumn{2}{c}{**Ni**} \\ \cline{2-10}  & **300 K** & **15,000 K** & **30,000 K** & **300 K** & **10,000 K** & **25,000 K** & **300 K** & **10,000 K** & **25,000 K** \\ \hline \(f_{0}\) & 0.234 & 0.134 & 0.110 & 0.099 & 0.184 & 0.235 & 0.165 & 0.149 & 0.191 \\ \(\Gamma_{0}\) & 0.254 & 0.141 & 0.148 & 0.060 & 0.146 & 0.150 & 0.069 & 0.077 & 0.100 \\ \(f_{1}\) & 0.639 & 0.664 & 0.088 & 0.045 & 0.006 & 0.753 & 0.499 & 0.037 & 0.762 \\ \(\Gamma_{1}\) & 1.895 & 4.302 & 1.167 & 0.461 & 0.618 & 7.133 & 2.009 & 0.707 & 8.704 \\ \(\omega_{1}\) & 1.280 & 1.325 & 0.770 & 2.530 & 2.457 & 2.069 & 0.628 & 0.638 & 2.754 \\ \(f_{2}\) & 0.280 & 0.211 & 0.020 & 0.152 & 0.651 & 0.013 & 0.103 & 0.278 & 0.001 \\ \(\Gamma_{2}\) & 0.904 & 1.399 & 0.587 & 1.263 & 6.075 & 0.491 & 1.820 & 2.648 & 0.037 \\ \(\omega_{2}\) & 2.512 & 2.821 & 1.401 & 3.545 & 3.738 & 3.207 & 2.797 & 1.967 & 4.502 \\ \(f_{3}\) & 0.026 & 0.013 & 0.414 & 0.146 & 0.001 & 0.006 & 0.180 & 0.046 & 0.030 \\ \(\Gamma_{3}\) & 0.754 & 0.006 & 2.677 & 2.107 & 0.072 & 0.300 & 0.532 & 0.404 & 0.513 \\ \(\omega_{3}\) & 6.131 & 6.352 & 2.970 & 5.261 & 4.016 & 6.251 & 5.277 & 5.250 & 5.021 \\ \(f_{4}\) & 0.721 & 1.570 & 1.107 & 0.512 & 0.368 & 0.657 & 0.313 & 0.686 & 0.255 \\ \(\Gamma_{4}\) & 2.311 & 7.397 & 8.961 & 1.493 & 1.194 & 2.158 & 1.458 & 7.070 & 8.779 \\ \(\omega_{4}\) & 7.492 & 12.540 & 10.025 & 7.431 & 7.470 & 10.654 & 7.297 & 6.737 & 7.250 \\ \(\chi^{2}_{r}\) & 1.621 & 0.688 & 0.405 & 3.486 & 4.930 & 0.146 & 0.392 & 0.919 & 0.565 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Drude-Lorentz model fitting coefficients for the permittivity calculated using ab initio methods for Fe, Au and Ni. Figure 3: Drude–Lorentz fit of permittivity calculated using ab initio methods for Fe, Au, and Ni. The real \(e_{1}\) and imaginary \(e_{2}\) parts of permittivity are shown at different electron temperatures corresponding to different levels of the electron–phonon nonequilibrium. of the optimization parameters \(f_{j},\omega_{j},\Gamma_{j}\) with temperatures from Tables 3 and 4. This is also seen from Figure 5, where the contributions to the imaginary part of permittivity from different oscillators (order 0 to 4) are plotted separately. Different terms can thus be easily identified. The intraband term (the oscillator of 0 th order) is high in the infrared range below 1 eV. While the maximum is at zero photon energy limit, the intraband contribution decreases quickly at higher photon energies. The interband terms have a Lorentzian shape with different heights, widths, and locations. These features are modeled by \(f_{j}\), \(\omega_{j}\), and \(\Gamma_{j}\) values with higher order corresponding to a higher central frequency \(\omega_{j}\). For some cases, the frequency of the fourth-order oscillator is higher than the chosen range and thus does not appear on the graph. Note that for some cases, \(\Gamma_{j}\) reaches the set constraint of 0.001 and thus does not provide any physical meaning. It happens, however, either for the fourth oscillator or for other insignificant oscillators (such as a tiny second oscillator for Cr). Additionally, the first-order oscillator is sometimes similar in behavior to the zero-order oscillator, as obtained by the optimization procedure (for example, for W, at 25,000 K) with both zero and first orders exhibiting a continuous decrease with \(\hbar\omega\), which is characteristic of an intraband trend. In this case, the intraband contribution was considered as a sum of the zero- and first-order oscillators. It is generally believed that for a strong laser excitation, equivalent to a high electron temperature in our case, the interband oscillators change their relative strength \(f_{j}\) (becoming stronger or weaker) and width \(\Gamma_{j}\), whereas the central frequency \(\omega_{j}\) remains constant [16].