  We present the implementation of the Hubbard ($U$) and Hund ($J$) corrected
Density Functional Theory (DFT+$U$+$J$) functionality in the Quickstep program,
which is part of the CP2K suite. The tensorial and L\"owdin subspace
representations are implemented and compared. Full analytical DFT+$U$+$J$
forces are implemented and benchmarked for the tensorial and L\"owdin
representations. We also present the implementation of the recently proposed
minimum-tracking linear-response method that enables the $U$ and $J$ parameters
to be calculated on first principles basis without reference to the Kohn-Sham
eigensystem. These implementations are benchmarked against recent results for
different materials properties including DFT+$U$ band gap opening in NiO, the
relative stability of various polaron distributions in TiO$_2$, the dependence
of the calculated TiO$_2$ band gap on +$J$ corrections, and, finally, the role
of the +$U$ and +$J$ corrections for the computed properties of a series of the
hexahydrated transition metals. Our implementation provides results consistent
with those already reported in the literature from comparable methods. We
conclude the contribution with tests on the influence of the L\"owdin
orthonormalization on the occupancies, calculated parameters, and derived
properties.
