The density functional theory $\left(\mathrm{DFT}\right)+U$ method is a pragmatic and effective approach for calculating the ground-state properties of strongly correlated systems, and linear-response calculations are widely used to determine the requisite Hubbard parameters from first principles. We provide a detailed treatment of spin within the linear-response framework, demonstrating that the conventional Hubbard $U$ formula, unlike the conventional $\mathrm{DFT}+U$ corrective functional, incorporates interactions that are off-diagonal in the spin indices and places greater weight on one spin channel over the other. We construct alternative definitions for Hubbard and Hund's parameters that are consistent with the contemporary $\mathrm{DFT}+U$ functional, expanding upon the minimum-tracking linear-response method. This approach allows Hund's $J$ and spin-dependent $U$ parameters to be calculated with the same ease as for the standard Hubbard $U$. Our methods accurately reproduce the experimental band gap, local magnetic moments, and the valence band edge character of manganese oxide, a canonical strongly correlated system. We also apply our approach to a complete series of transition-metal complexes ${\left[M{\left({\mathrm{H}}_2\mathrm{O}\right)}_6\right]}^{n+}$ (for $M=\text{Ti}$ to Zn), showing that Hubbard corrections on oxygen atoms are necessary for preserving bond lengths, and demonstrating that our methods are numerically well behaved even for near-filled subspaces such as in zinc. However, spectroscopic properties appear to be beyond the reach of standard $\mathrm{DFT}+U$. Collectively, these results shed new light on the role of spin in the calculation of the corrective parameters $U$ and $J$, and point the way toward avenues for further development of $\mathrm{DFT}+U$ and related methods.

• Received 25 February 2018
• Revised 19 November 2018

DOI:https://doi.org/10.1103/PhysRevB.98.235157