Efficient approach to compute melting properties fully from ab initio with application to Cu

Applying thermodynamic integration within an ab initio-based free-energy approach is a state-of-the-art method to calculate melting points of materials. However, the high computational cost and the reliance on a good reference system for calculating the liquid free energy have so far hindered a general application. To overcome these challenges, we propose the two-optimized references thermodynamic integration using Langevin dynamics (TOR-TILD) method in this work by extending the two-stage upsampled thermodynamic integration using Langevin dynamics (TU-TILD) method, which has been originally developed to obtain anharmonic free energies of solids, to the calculation of liquid free energies. The core idea of TOR-TILD is to fit two empirical potentials to the energies from density functional theory based molecular dynamics runs for the solid and the liquid phase and to use these potentials as reference systems for thermodynamic integration. Because the empirical potentials closely reproduce the ab initio system in the relevant part of the phase space the convergence of the thermodynamic integration is very rapid. Therefore, the proposed approach improves significantly the computational efficiency while preserving the required accuracy. As a test case, we apply TOR-TILD to fcc Cu computing not only the melting point but various other melting properties, such as the entropy and enthalpy of fusion and the volume change upon melting. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional and the local-density approximation (LDA) are used. Using both functionals gives a reliable ab initio confidence interval for the melting point, the enthalpy of fusion, and entropy of fusion.

Figure 1

Free energy approach for determining the melting temperature $T^{\mathrm{m}}$ at a constant pressure by the crossing point of the Gibbs energies of solid and liquid, $G^{\mathrm{solid}}$ and $G^{\mathrm{liquid}}$.

Figure 2

Schematic representation of the TOR-TILD method. Details on the different steps are given in the main text. Note that for visualization purposes the curvature along the volume axis has been made somewhat smoother in the shown free energy surfaces.

Figure 3

Single snapshot of a $10\times 10\times 20$ supercell with 8000 atoms for simulating the coexistence of solid and liquid.

Figure 4

Impact of finite size effects on the liquid free energy at 1400 K. The DFT calculations correspond to the LDA functional.

Figure 5

Example for the efficiency of the present TOR-TILD method. The figure represents results of a thermodynamic integration for the liquid phase at 1600 K from different reference potentials to DFT (low converged values and LDA functional). The “ref1” potential has been fitted to DFT solid energies, “ref2” to DFT liquid energies, and “lit” is a Cu potential available from literature [65]. (a) Standard deviation [Eq. (10)] versus the coupling parameter $\lambda$. (b) Convergence of the mean value of the free energy for $\lambda =1$, referenced to the converged value. (c) Total CPU time taken to converge the mean value to below 1 meV/atom on a single core.

Figure 6

Thermodynamic properties of Cu calculated with our methodology using the PBE and LDA functional.

Figure 7

Vacancy contribution to the (a) Gibbs energy and (b) entropy of fcc Cu for PBE (orange lines) and LDA (blue). The dotted lines indicate the respective melting points (black for experimental one).