Computing energy barriers for rare events from hybrid quantum/classical simulations through the virtual work principle

Hybrid quantum/classical techniques can flexibly couple ab initio simulations to an empirical or elastic medium to model materials systems that cannot be contained in small periodic supercells. However, due to electronic nonlocality, a total energy cannot be defined, meaning energy barriers cannot be calculated. We provide a general solution using the principle of virtual work in a modified nudged elastic band algorithm. Our method enables ab initio calculations of the kink formation energy for $\langle 100\rangle$ edge dislocations in molybdenum and lattice trapping barriers to brittle fracture in silicon.

Figure 1

(Left) Convergence of DFT forces in a QM region of 12 perfect aluminium lattice ions with increasing buffer width. (Inset) Cartoon of the hybrid system. (Right) Comparison of NEB methods to calculate the migration barrier of a vacancy in fcc aluminum.

Figure 2

(a) Comparison of dipole and cylinder geometries to calculate the Peierls barrier of an $\langle 100\rangle \left(010\right)$ edge dislocation in MEAM molybdenum. (b) Localized work for the MEAM and hybrid systems as described in the main text, using Eq. (2).

Figure 3

Illustration of the buffer and QM regions used to treat $\langle 100\rangle \left(010\right)$ edge dislocations in this work. The supercell has periodic boundary conditions in the dislocation line direction, $\left[001\right]$.

Figure 4

(Top) Comparison of NEB methods to calculate the Peierls barrier of a $\langle 100\rangle \left(010\right)$ edge dislocation in molybdenum. (Bottom left) Line tension calculation using hybrid and MEAM force fields as described in the text. (Bottom right) Kinks formed in Frenkel-Kontorova models using the hybrid and MEAM values.

Figure 5

Minimum energy path for cleavage in the $\mathrm{Si}\left(110\right)\left[1\overline{1}0\right]$ fracture system, involving a blunt-sharp-blunt tip reconstruction with two bonds opening simultaneously. Inset images show near-tip region for the initial, transition, and final states following hybrid relaxation of the path, with undercoordinated atoms shown in green. Upper-right inset gives dependence of energy barrier on strain energy release rate, with fracture becoming easier as load increases.