Cohomology of groups of diffeomorphisms related to the modules of differential operators on a smooth manifold

Let M be a manifold and T ^* M be the cotangent bundle. We introduce a 1-cocycle on the group of diffeomorphisms of M with values in the space of linear differential operators acting on C ^∞(T ^* M). When M is the n-dimensional sphere, S ⁿ, we use this 1-cocycle to compute the first-cohomology group of the group of diffeomorphisms of S ⁿ, with coefficients in the space of linear differential operators acting on contravariant tensor fields.