  While some of the most elegant applications of topological insulators, such
as quantum anomalous Hall effect, require the preservation of Dirac surface
states in the presence of time-reversal symmetry breaking, other phenomena such
as spin-charge conversion rather rely on the ability for these surface states
to imprint their spin texture on adjacent magnetic layers. In this work, we
investigate the spin-momentum locking of the surface states of a wide range of
monolayer transition metals (3$d$-TM) deposited on top of Bi$_{2}$Se$_{3}$
topological insulators using first principles calculations. We find an
anticorrelation between the magnetic moment of the 3$d$-TM and the magnitude of
the spin-momentum locking {\em induced} by the Dirac surface states. While the
magnetic moment is large in the first half of the 3$d$ series, following Hund's
rule, the spin-momentum locking is maximum in the second half of the series. We
explain this trend as arising from a compromise between intra-atomic magnetic
exchange and covalent bonding between the 3$d$-TM overlayer and the Dirac
surface states. As a result, while Cr and Mn overlayers can be used
successfully for the observation of quantum anomalous Hall effect or the
realization of axion insulators, Co and Ni are substantially more efficient for
spin-charge conversion effects, e.g. spin-orbit torque and charge pumping.
