A large part of computational chemistry is devoted to studying electrons in their ground states, i.e. their most stable states with lowest possible energy, within a molecule or material. As the Schrödinger equation that describes the detailed interactions of electrons with each other and their environment is too complicated to solve exactly, many efficient computer algorithms and approximations have been developed to calculate accurate ground states. Little is presently known, however, about the risk that these methods fail to recover the ground state and instead mistakenly converge to a less stable state with higher energy. The present project aims to make progress by finding conditions under which it can be mathematically guaranteed that the ground state is found. The project also aims to analyze why available computational methods work so well in practice, despite the lack of mathematical guarantees.