# The role of spin in the calculation of Hubbard $U$ and Hund's $J$ parameters from first principles

## TL;DR

This paper revises the calculation of Hubbard U and Hund's J parameters in DFT+U methods by explicitly incorporating spin effects, leading to more accurate predictions for strongly-correlated materials.

## Contribution

It introduces spin-aware definitions for Hubbard and Hund's parameters within the linear response framework, aligning them with the current DFT+U functional and enabling easier computation.

## Key findings

- Accurately reproduces experimental band gaps and magnetic moments in manganese oxide.
- Shows the necessity of oxygen Hubbard corrections for bond length preservation.
- Demonstrates numerical stability even for near-filled electronic subspaces.

## Abstract

The density functional theory (DFT)+$U$ 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 this linear response approach, demonstrating that the conventional Hubbard $U$ formula, unlike the conventional 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 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 [M(H$_2$O)$_6$]$^{n+}$ (for M = 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 beyond the reach of the standard DFT+$U$ approach. 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 towards avenues for further development of DFT+$U$-type methods.