The semiclassical model of quantum field theory on curved spacetimes is considered as an intermediate step in the direction of some quantum theory of gravitation, which however already yields remarkable effects like the creation of particles by strong gravitational fields most prominently predicted by Hawking's evaporation of black holes and the Unruh effect. This can be seen as an indication for the severe fact, that the quantum field lacks a distinguished notion of vacuum and hence particle interpretation of states for general spacetimes even expressible via certain no-go-theorems by now. In order to reduce the huge amount of possible states and since we consider free quantum fields it is natural to restrict to quasifree states, which are completely determined by some scalar product on the space of solutions of the homogeneous field equation. A further physical motivated constraint is the Hadamard condition generalizing the energy condition of Minkowski quantum field theory and moreover allow for a reasonable renormalization of the energy momentum tensor. In the talk I present the construction of global symmetric and positive bisolutions for wave operators on sections in vector bundles over arbitrary globally hyperbolic spacetimes via Hadamard series', which then determine quasifree states satisfying the Hadamard condition.