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Hubbard-$U$ corrected Hamiltonians for non-self-consistent random-phase approximation total-energy calculations: A study of ZnS, TiO$_2$, and NiO

In non-self-consistent calculations of the total energy within the random-phase approximation (RPA) for electronic correlation, it is necessary to choose a single-particle Hamiltonian whose solutions are used to construct the electronic density and non-interacting response function. Here we investigate the effect of including a Hubbard-$U$ term in this single-particle Hamiltonian, to better describe the on-site correlation of 3$d$ electrons in the transition metal compounds ZnS, TiO$_2$ and NiO. We find that the RPA lattice constants are essentially independent of $U$, despite large changes in the underlying electronic structure. We further demonstrate that the non-self-consistent RPA total energies of these materials have minima at nonzero $U$. Our RPA calculations find the rutile phase of TiO$_2$ to be more stable than anatase independent of $U$, a result which is consistent with experiments and qualitatively different to that found from calculations employing $U$-corrected (semi)local functionals. However we also find that the +$U$ term cannot be used to correct the RPA's poor description of the heat of formation of NiO.