  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.
