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Titanium dioxide (TiO$_2$) presents a long-standing challenge for approximateKohn-Sham density-functional theory (KS-DFT), as well as to itsHubbard-corrected extension, DFT+$U$. We find that a previously proposedextension of first-principles DFT+$U$ to incorporate a Hund's $J$ correction,termed DFT+$U$+$J$, in combination with parameters calculated using a recentlyproposed linear-response theory, predicts fundamental band-gaps accurate towell within the experimental uncertainty in rutile and anatase TiO$_2$. Ourapproach builds upon established findings that Hubbard correction to bothtitanium $3d$ and oxygen $2p$ subspaces in TiO$_2$, symbolically givingDFT+$U^{d,p}$, is necessary to achieve acceptable band-gaps using DFT+$U$. Thisrequirement remains when the first-principles Hund's $J$ is included. We alsofind that the calculated gap depends on the correlated subspace definition evenwhen using subspace-specific first-principles $U$ and $J$ parameters. Using thesimplest reasonable correlated subspace definition and underlying functional,the local density approximation, we show that high accuracy results from usinga relatively uncomplicated form of the DFT+$U$+$J$ functional. For closed-shellsystems such as TiO$_2$, we describe how various DFT+$U$+$J$ functionals reduceto DFT+$U$ with suitably modified parameters, so that reliable band gaps can becalculated for rutile and anatase with no modifications to a conventionalDFT+$U$ code.

physical review. b./physical review. b

First-principles Hubbard U and Hund's J corrected approximate  density-functional theory predicts an accurate fundamental gap in rutile and  anatase TiO2