A High-Dimensional Neural Network Potential for Co$_3$O$_4$

The Co$_3$O$_4$ spinel is an important material in oxidation catalysis. Its properties under catalytic conditions, i.e., at finite temperatures, can be studied by molecular dynamics simulations, which critically depend on an accurate description of the atomic interactions. Due to the high complexity of Co$_3$O$_4$, which is related to the presence of multiple oxidation states of the cobalt ions, to date \textit{ab initio} methods have been essentially the only way to reliably capture the underlying potential energy surface, while more efficient atomistic potentials are very challenging to construct. Consequently, the accessible length and time scales of computer simulations of systems containing Co$_3$O$_4$ are still severely limited. Rapid advances in the development of modern machine learning potentials (MLPs) trained on electronic structure data now make it possible to bridge this gap. In this work, we employ a high-dimensional neural network potential (HDNNP) to construct a MLP for bulk Co$_3$O$_4$ spinel based on density functional theory calculations. After a careful validation of the potential, we compute various structural, vibrational, and dynamical properties of the Co$_3$O$_4$ spinel with a particular focus on its temperature-dependent behavior, including the thermal expansion coefficient.