First-principles Hubbard <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math> and Hund's <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math> corrected approximate density functional theory predicts an accurate fundamental gap in rutile and anatase <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>TiO</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>
Okan K. Orhan, David D. O’Regan
Abstract
Titanium dioxide (${\mathrm{TiO}}_{2}$) presents a long-standing challenge for approximate Kohn-Sham density functional theory (KS-DFT), as well as to its Hubbard-corrected extension, $\text{DFT+}\mathit{\text{U}}$. We find that a previously proposed extension of first-principles $\text{DFT+}\mathit{\text{U}}$ to incorporate a Hund's $J$ correction, termed $\text{DFT+}\mathit{\text{U}}\text{+}\mathit{\text{J}}$, in combination with parameters calculated using a recently proposed linear-response theory, predicts fundamental band gaps that are accurate to well within the experimental uncertainty in rutile and anatase ${\mathrm{TiO}}_{2}$. Our approach builds upon established findings that Hubbard correction of both the titanium $3d$ and oxygen $2p$ subspaces in ${\mathrm{TiO}}_{2}$, symbolically giving $\text{DFT+}{\mathit{\text{U}}}^{d,p}$, is necessary to achieve acceptable band gaps using $\text{DFT+}\mathit{\text{U}}$. This requirement remains when the first-principles Hund's $J$ is included. We also find that the calculated gap depends on the correlated subspace definition even when using subspace-specific first-principles $U$ and $J$ parameters. Using the simplest reasonable correlated subspace definition and underlying functional, the local density approximation, we show that high accuracy results from using a relatively uncomplicated form of the $\text{DFT+}\mathit{\text{U}}\text{+}\mathit{\text{J}}$ functional. For closed-shell systems such as ${\mathrm{TiO}}_{2}$, we describe how various $\text{DFT+}\mathit{\text{U}}\text{+}\mathit{\text{J}}$ functionals reduce to $\text{DFT+}\mathit{\text{U}}$ with suitably modified parameters, so that reliable band gaps can be calculated for rutile and anatase with no modifications to a conventional $\text{DFT+}\mathit{\text{U}}$ code.