  In this work, we generalize the local spin analysis of Clark and Davidson [J.
Chem. Phys. 115(16), 7382 (2001)] for the partitioning of the expectation value
of the molecular spin square operator, $\langle S^2 \rangle$, into atomic
contributions, $\langle S_A \cdot S_B \rangle$, to the noncollinear spin case
in the framework of density functional theory (DFT). We derive the working
equations and we show applications to the analysis of the noncollinear spin
solutions of typical spin-frustrated systems and to the calculation of magnetic
exchange couplings. In the former case, we employ the triangular H$_3$He$_3$
test molecule and a Mn$_3$ complex to show that the local spin analysis
provides additional information that complements the standard one-particle spin
population analysis. For the calculation of magnetic exchange couplings,
$J_{AB}$, we employ the local spin partitioning to extract $\langle S_A \cdot
S_B \rangle$ as a function of the interatomic spin orientation given by the
angle $\theta$. This, combined with the dependence of the electronic energy
with $\theta$, provides a methodology to extract $J_{AB}$ from DFT calculations
that, in contrast to conventional energy differences based methods, does not
require the use of $ad-hoc$ $S_A$ and $S_B$ values.
