  Many positive electrode materials in lithium ion batteries include transition
metals which are difficult to describe by electronic structure methods like
density functional theory (DFT) due to the presence of multiple oxidation
states. A prominent example is the lithium manganese oxide spinel
Li$_x$Mn$_2$O$_4$ with $0\leq x\leq2$. While DFT, employing the local hybrid
functional PBE0r, provides a reliable description, the need for extended
computer simulations of large structural models remains a significant
challenge. Here, we close this gap by constructing a DFT-based high-dimensional
neural network potential (HDNNP) providing accurate energies and forces at a
fraction of the computational costs. As different oxidation states and the
resulting Jahn-Teller distortions represent a new level of complexity for
HDNNPs, the potential is carefully validated by performing X-ray diffraction
experiments. We demonstrate that the HDNNP provides atomic level details and is
able to predict a series of properties like the lattice parameters and
expansion with increasing Li content or temperature, the orthorhombic to cubic
transition, the lithium diffusion barrier, and the phonon frequencies. We show
that for understanding these properties access to large time and length scales
as enabled by the HDNNP is essential to close the gap between theory and
experiment.
