Fig. 1. Workflow of the simulation of WDM with the SDFT and DPMD methods. (a) Stochastic orbitals are employed in SDFT to perform molecular dynamics simulations on smaller systems (32 atoms in a bulk B) and initial training data are collected that include atomic positions, energies, forces, and virial tensors. (b) The gathered training data are used to construct a DP model with the temperature-dependent DPMD model. The deep neural network contains both embedding and fitting networks. (c) The new model enables simulations to be performed on large systems (16 384 atoms) and at extremely high temperatures (350 eV). (d) Several physical quantities such as radial distribution functions, static structure factors, dynamic structure factors, and shear viscosities can be calculated, and the data are converged with large systems and long trajectories.

Fig. 2. Root-mean-square error (RMSE) of atomic forces arising from the SDFT calculations for B. The temperature is set to 350 eV and the density to 2.46 g/cm³. The RMSE is evaluated with respect to (a) the number of stochastic orbitals N_sto, (b) the number of k-points in the Brillouin zone N_k, and (c) the product of N_sto and N_k. For each data point, the RMSE is obtained via nine independent SDFT calculations with different sets of stochastic orbitals.

Fig. 3. Forces acting on each B atom as obtained from nine independent SDFT calculations with different stochastic orbitals. The B system has a density of 2.46 g/cm³, and the temperature is 17.23 eV. For each calculation, the force acting on each atom along the γ direction is denoted by F_sto, and their average is F_ave. N_KS and N_sto are the numbers of KS and stochastic orbitals, respectively. Two sets of Monkhorst–Pack k-points are utilized: a 1 × 1 × 1 k-point mesh (the Γ k-point) and a shifted 2 × 2 × 2 k-point mesh. The RMSE of forces between F_sto and F_ave is computed from Eq. (13). The RMSE is obtained via the mentioned nine independent SDFT calculations.

Fig. 4. (a) Comparison of forces acting on each B atom in a 32-atom cell for densities of 2.46 and 12.3 g/cm³ at a temperature of 17.23 eV. (b) Comparison of forces acting on each C atom for densities of 4.17 and 12.46 g/cm³ at a temperature of 21.54 eV. In the SDFT and KSDFT calculations, we denote the forces acting on each atom along the γ direction (γ ∈ x, y, z) by F_sto and F_KS, respectively. The RMSE of forces between F_sto and F_KS is also shown.

Fig. 5. DOS for the B and C systems as calculated by the SDFT and KSDFT methods. The densities are selected as (a) 2.46 g/cm³ and (b) 12.3 g/cm³ for the B system, and (c) 4.17 g/cm³ and (d) 12.64 g/cm³ for the C system. The Fermi energy is set to zero. We use two sets of k-point sampling in the KSDFT and SDFT calculations. In addition, we adopt two XC functionals, namely, PBE⁵⁹ and corrKSDT.⁶⁵

Fig. 6. Radial distribution functions g(r) of B systems with a density of 2.46 g/cm³ at temperatures of (a) 86 eV and (b) 350 eV. The g(r) obtained by Ext-FPMD with the local density approximation (LDA) functional comes from Blanchet et al.¹⁹ The SDFT calculations are performed with the PBE⁵⁹ and corrKSDT⁶⁵ XC functionals. The number of B atoms is set to 32 in the first-principles molecular dynamics simulations. DPMD denotes the model trained by the traditional DP method⁵⁰ and DPMD-T the model trained by the TDDP method.⁵¹N is the number of B atoms in a cell, which ranges from 32 to 16 384 in the deep-potential-based simulations.

Fig. 7. Static structure factors S(q) of B with a density of 2.46 g/cm³ at temperatures of (a) 86 eV and (b) 350 eV. The SDFT calculations are performed with the PBE⁵⁹ and corrKSDT⁶⁵ XC functionals. The number of B atoms is set to 32 in the first-principles molecular dynamics simulations. DPMD denotes the model trained by the traditional DP method⁵⁰ and DPMD-T the model trained by the TDDP method.⁵¹N is the number of B atoms in a cell and ranges from 32 to 16 384 in the deep-potential-based simulations.

Fig. 8. Intermediate scattering functions F(q, t) of B with a density of 2.46 g/cm³ as calculated from DPMD trajectories. Three system sizes (256, 2048, and 16 384 atoms) are adopted in DPMD simulations at two temperatures of (a) 86 eV and (b) 350 eV. Three wave vectors are chosen: q = 0.51, 1.02, and 2.50 Å⁻¹.

Fig. 9. Ion–ion dynamic structure factors S(q, ω) of B with a density of 2.46 g/cm³ as computed from DPMD trajectories. Three system sizes (256, 2048, and 16 384 atoms) are adopted. The wave vectors q are chosen to be (a) 0.51 Å⁻¹, (b) 1.02 Å⁻¹, and (c) 2.50 Å⁻¹.

Fig. 10. Stress autocorrelation functions [Eq. (21)] of warm dense B with a density of 2.46 g/cm³ at temperatures of (a) 86 eV and (b) 350 eV. (c) Shear viscosity of B. DPMD simulations are used, with cells containing 32, 108, 256, 864, 2048, 6912, and 16 384 atoms. Error bars represent standard deviations.