Fig. 5: Free energy of vacancy formation in BCC iron.
a Transformations used to determine the Gibbs free energy of the perfect crystal and the crystal with a defect. The alchemical pathway used here transforms the perfect crystal into the crystal with a defect and a single atom attached to a spring to avoid diffusion. ΔGFL and ΔGAL represent the Gibbs free energy changes calculated via the Frenkel–Ladd and alchemical pathways, respectively. b The intermediate state parameterized by λ for the alchemical pathway in (a). The atom to be removed is assigned an alchemical weight of 1 − λ, and the energy of the harmonic oscillator is scaled by λ. c The free energy of vacancy (Eq. (12)) computed by the Frenkel–Ladd path and alchemical path. d Statistical efficiency for the Frenkel–Ladd paths and alchemical path at 100 K against the switching time. Upper panel shows the deviation of Gibbs free energy from the reference value at the longest switching time (60 ps), and the lower panel shows average dissipated energies (Eq. (10)). For panels c and d, each data point represents the average of four statistically independent simulations, with standard deviations shown as error bars. Source data are provided as a Source Data file.
