  Analytic energy gradients with respect to nuclear motion are derived for
non-singlet compounds in the natural orbital functional theory. We exploit the
formulation for multiplets in order to obtain a simple formula valid for any
many-electron system in its ground mixed state with a total spin S and all
possible spin projection Sz values. We demonstrate that the analytic gradients
can be obtained without resorting to linear response theory or involving
iterative procedures. It is required a single evaluation, so integral
derivatives can be computed on-the-fly along the calculation and thus improve
the effectiveness of screening by the Schwarz inequality. Results for small and
medium size molecules with many spin multiplicities are shown. Our results are
compared with experimental data and accurate theoretical equilibrium
geometries.
