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In non-self-consistent calculations of the total energy within therandom-phase approximation (RPA) for electronic correlation, it is necessary tochoose a single-particle Hamiltonian whose solutions are used to construct theelectronic density and non-interacting response function. Here we investigatethe 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 transitionmetal compounds ZnS, TiO$_2$ and NiO. We find that the RPA lattice constantsare essentially independent of $U$, despite large changes in the underlyingelectronic structure. We further demonstrate that the non-self-consistent RPAtotal energies of these materials have minima at nonzero $U$. Our RPAcalculations find the rutile phase of TiO$_2$ to be more stable than anataseindependent of $U$, a result which is consistent with experiments andqualitatively different to that found from calculations employing $U$-corrected(semi)local functionals. However we also find that the +$U$ term cannot be usedto correct the RPA's poor description of the heat of formation of NiO.

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