Method for locating low-energy solutions withinDFT+U

The widely employed $\text{DFT}+U$ formalism is known to give rise to many self-consistent yet energetically distinct solutions in correlated systems, which can be highly problematic for reliably predicting the thermodynamic and physical properties of such materials. Here we study this phenomenon in the bulk materials ${\text{UO}}_{2}$, CoO, and NiO, and in a ${\text{CeO}}_{2}$ surface. We show that the following factors affect which self-consistent solution a $\text{DFT}+U$ calculation reaches: (i) the magnitude of $U$; (ii) initial correlated orbital occupations; (iii) lattice geometry; (iv) whether lattice symmetry is enforced on the charge density; and (v) even electronic mixing parameters. These various solutions may differ in total energy by hundreds of meV per atom, so identifying or approximating the ground state is critical in the $\text{DFT}+U$ scheme. We propose an efficient $U$-ramping method for locating low-energy solutions, which we validate in a range of test cases. We also suggest that this method may be applicable to hybrid functional calculations.