  This work seeks to advance the understanding of the smooth structure of the
moduli space of self-dual contact instantons (SDCI) on Sasakian 7-manifolds M.
A neighborhood of a smooth point of M is locally modeled on the first
cohomological group of an elliptic complex (1.4). There is a cohomological
obstruction to the smoothness for the moduli space, in terms of a second basic
cohomological group, in this paper we study conditions under which this
obstruction disappears, by computing a Weitzenb\"ock formula and using a
Bochner-type method to obtain a vanishing theorem. Given an SDCI on a Sasakian
bundle E, we find sufficient conditions for the vanishing of the obstruction in
the positivity of a couple of operators R and F depending on the curvatures of
the connection and the Riemann curvature of the Sasakian metric g. In
particular, we find that if M is transversely Ricci positive and F positive,
the moduli space of SDCI must be smooth. However, in general, the operator F is
not positive definite and we describe bundles over the Stiefel manifold for
which it is the case. Finally, we show that when the energy of the curvature is
less than the first non-zero eigenvalue of RicT the obstruction vanishes.
