## 2 Materials and Methods
### Chapman-Enskog Approach Using Effective Ion-Ion Potential  
Using the data for the diffusion coefficient form the QMD simulations, we tested the accuracy of the Chapman-Enskog approach using the effective ion-ion potential [24; 25; 26; 27; 28; 29; 30]. The idea of this approach is to describe ion-ion collisions using the effective field \(\Phi_{\text{eff}}(r)\) defined by the two particle radial distribution function \(g(r)\) (RDF):  
\[\Phi_{\text{eff}}(r)=-k_{B}T\ln g(r). \tag{5}\]  
The effective potential (5) is used to compute the classical center-of-mass scattering angle of two colliding particles:  
\[\theta_{c}(b,v)=\pi-2b\int_{r_{0}}^{\infty}\frac{dr}{r^{2}}\bigg{(}1-\frac{2 \Phi_{\text{eff}}(r)}{\mu v^{2}}-\frac{b^{2}}{r^{2}}\bigg{)}^{-1/2}, \tag{6}\]  
here \(\mu=m_{i}/2\) is the reduced mass of an ion.  
In Equation (6), \(r_{0}\) is the distance of the closest approach for a given impact parameter \(b\) defined as the solution of the following equation:  
\[1-\frac{2\Phi_{\text{eff}}(r_{0})}{\mu v^{2}}-\frac{b^{2}}{r_{0}^{2}}=0. \tag{7}\]  
Next, the scattering angle \(\theta_{c}(b,v)\) is used to compute the Coulomb logarithm:  
\[\ln\Lambda=\frac{1}{b_{\perp}^{2}}\int_{0}^{\infty}\sin^{2}\bigg{(}\frac{ \theta_{c}}{2}\bigg{)}bdb, \tag{8}\]  
where \(b_{\perp}=Z^{2}/(\mu v^{2})\).  
The Coulomb logarithm defines the first-order collision integral [34]:  
\[\Omega^{(1,1)}=\frac{\sqrt{\pi}Z^{4}e^{4}}{2\sqrt{2\mu T^{3}}}\ln\Lambda, \tag{9}\]  
where, in our case for Be, we set the charge to the number of valence electrons per atom \(Z=2\).  
Using the collision integral \(\Omega^{(1,1)}\), the diffusion coefficient is computed as  
\[D=\frac{3T}{8 nm_{i}\Omega^{(1,1)}}, \tag{10}\]  
with \(n\) being the number density of ions and \(m_{i}\) being the ion mass.  
In practice, \(g(r)\) is approximated or supplied as an input from other methods, such as the hypernetted chain approximation. Here, for benchmarking purposes, we use \(g(r)\) from QMD to test if the Chapman-Enskog method can reproduce the diffusion coefficient from QMD. In Figure 1, we show the Coulomb logarithm computed using the RDF \(g(r)\) from QMD calculations and provide comparison with the Coulomb logarithm computed using classical Debye plasma model [26; 28; 34; 53; 54] and using a screened potential with the screening length computed according to the Thomas-Fermi approximation [35]. The effective potential of ion-ion interaction derived using the RDF from the QMD simulations are shown in Figure 2, where we also compare the QMD data with the Thomas-Fermi approximation For illustration, from Figure 1, we see that the QMD result for the Coulomb logarithm of warm dense matter can significantly differ from the results of the traditional Debye plasma model. Additionally, as shown in Figure 1, the Thomas-Fermi approximation aligns much better with the QMC results than the Debye plasma model does. However, as we will demonstrate shortly, there is still a significant discrepancy between the Thomas-Fermi approximation and the QMC results when it comes to the diffusion coefficient.  
Figure 1: The generalized Coulomb logarithm obtained using the RDF from the QMD simulations. The QMD results are compared with the data obtained using the classical Debye potential and quantum screening model based on the Thomas–Fermi model (see Ref. [35]).  
Figure 2: Effective ion–ion interaction potential (5) using the radial distribution function from the quantum molecular dynamics (QMD) simulations at different temperatures. Additionally, we provide a comparison with the screened potential calculated using the Thomas–Fermi screening length.