Automated free-energy calculation from atomistic simulations

We devise automated workflows for the calculation of Helmholtz and Gibbs free energies and their temperature and pressure dependence and provide the corresponding computational tools. We employ nonequilibrium thermodynamics for evaluating the free energy of solid and liquid phases at a given temperature and reversible scaling for computing free energies over a wide range of temperatures, including the direct integration of $P\text{−}T$ coexistence lines. By changing the chemistry and the interatomic potential, alchemical and upscaling free energy calculations are possible. Several examples illustrate the accuracy and efficiency of our implementation.

Figure 1

(a) $G\left(\infty \right)-G\left(N\right)$ at 1000 K and zero pressure as a function of system size from 128 to 16 000 atoms and a switching time of 100 ps. (b) $G(N=6750)$ calculated for each switching time. $G\left(\infty \right)$ is shown in dashed red line. Switching times from 1 ps to 10 ns are used for 6750 atoms.

Figure 2

The pressure-temperature phase diagram for Ti using an embedded atom method (EAM) potential [14]. The regions of stability for bcc, hcp, and liquid are shown in red, green, and blue, respectively. The gradients in the color indicate decreasing free energy; darker color indicates decreasing free energy. Coexistence lines calculated using Algorithm 3 are shown in gray. The coexistence line is further verified by using Algorithm 1, the results of which are shown as gray circles. The three independent computations are in excellent agreement.

Figure 3

Pressure-temperature phase diagram of Si calculated using three different potentials. Stillinger-Weber (SW) potential [26], angular-dependent potential (ADP) [27], and spectral neighbor analysis potential (SNAP) [28] are shown.

Figure 4

$C_P$ of Cu calculated using an embedded atom method (EAM) potential [16]. Orange line: Algorithm 2 and Eq. (14), yellow circles: experiment [32], red circles: molecular dynamics (MD) calculations using Eq. (19).

Figure 5

Free energy of the CuZr (red) and ${\mathrm{CuZr}}_2$ (blue). Solid line: free energy calculated using Algorithm 2, circles: reference from Tang and Harrowell [33].

Figure 6

Illustration of the two routes by which the free energy for Cu within atomic cluster expansion (ACE) can be calculated. Starting from the reference system, it can be directly calculated using Algorithm 1. Alternatively, first, the free energy of the embedded atom method (EAM) potential is calculated using Algorithm 1, after which the system is upsampled to ACE using Algorithm 4.

Figure 7

Energy dissipation during the calculation of $G_{\mathrm{ACE}}$ using Algorithm 1 (red circles) and for switching between the embedded atom method (EAM) and atomic cluster expansion (ACE) potentials (blue circles).

Figure 8

(a) $\mathrm{\Delta }F$ as a function of $\lambda \left(t\right)$ along the alchemical integration path that switches a 48 at.% Zr B2 structure to 52 at.% Zr in the same lattice. (b) The free energy as a function of the Zr concentration calculated using Algorithm 4 (red line). Values reported by Tang and Harrowell [33] are shown as blue circles, while those calculated using Algorithm 1 are shown as red circles.