in an atomic liquid [@chern17].

The atomic displacements $\mathbf u_i$ in the finite-temperature MD simulations are governed by the Newton equation of motion with the Langevin thermostat [@allen89; @rapaport11] $$\begin{aligned}
    \label{eq:langevin}
    m \ddot{\mathbf u}_i = \mathbf f^l_{i} + \mathbf f^e_{i} -\gamma \dot{\mathbf u}_i + \bm \xi_i(t),\end{aligned}$$ where $\gamma$ is the damping coeficient, $\bm\xi_i(t)$ are stochastic forces with statistical properties $\langle \xi^a_i \rangle = 0$ and $\langle \xi^a_i(t) \xi^b_j(t') \rangle = 2 \gamma k_B T \delta_{ij} \delta_{ab} \delta(t - t')$, $a, b = x, y, z$ denote the cartesean components, and $T$ is the temperature of the reservoir. Standard velocity-Verlet algorithm [@allen89; @rapaport11] is used to integrate the above Langevin equation in our simulations. The atomic forces within the BO-MD framework is computed using the finite temperature generalization of the Hellmann-Feynman theorem $\mathbf f_{i} = - {\rm Tr}(\rho_G \, \partial \mathcal{H} / \partial \mathbf u_i)$. There are two contributions to the atomic forces. The elastic restoring force is $$\begin{aligned}
    \mathbf f^l_i = -K_0 \mathbf u_i -K_1\,\sum_j \!^{'} \, \hat{\mathbf e}_{ij}  \cdot  (\mathbf u_i - \mathbf u_j),\end{aligned}$$ where the summation is restricted to the nearest neighbors. The electronic force originates from the $\mathbf u$-dependence of the transfer integral $t_{ij}$, and is given 