The population analysis techniques described in Section
may also be used in combination with spin
dependent DFT to investigate the distribution of electronic spin
within a system. In order to achieve this, the spin up and spin down
electronic populations are analysed separately and their sums and
differences yield the charges and net spins respectively. This
approach must be used because if any states within the system are
independent of spin this will yield an ill-conditioned matrix of
overlaps between the projected states 
and
attempting to invert this matrix will cause numerical problems (See
Appendix ).
This section demonstrates the use of spin population analysis
techniques in molecular and bulk systems. Subsection
examines the results for three diatomic molecules, including 
which is paramagnetic. The results for ferromagnetic and
antiferromagnetic crystals are compared in Subsection .