We show how two-point correlation functions recently derived within non-isotropic random wave models can be obtained in the appropriate limit in terms of the exact Green function of the quantum system. Since no statistical model is required for this derivation, this shows that taking the wavefunctions as Gaussian processes is the only assumption of those models. We also show how for clean systems the two-point correlation function based on an energy average defines a Gaussian theory which substantially reduces the spurious contributions coming from the normalization problem.