A permutation invariant collective variable to track and drive vacancy dynamics in simulations of solids

Vacancy dynamics in oxides are vital for understanding redox reactions and resulting memristive effects or catalytic activity. We present a method to track and drive vacancies which we apply to metadynamics simulation of oxygen vacancies (V$_{\mathrm{O}}$) in rutile, demonstrating its effectiveness. Using the density functional based tight binding method, it is possible to explore the free energy hyperplane of oxygen vacancies in TiO$_{2}$. We show that the migration of V$_{\mathrm{O}}$ in TiO$_{2}$ is governed by the jump with the higherst number of realizations. Free energy profiles are consistent with minimum energy paths.