We analyse the possible existence of non-perturbative contributions in heavy $\bar Q Q$ systems ($\bar Q$ and $Q$ need not have the same flavour) which cannot be expressed in terms of local condensates. Starting from QCD, with well defined approximations and splitting properly the fields into large and small momentum components, we derive an effective lagrangian where hard gluons (in the non-relativistic aproximation) have been integrated out. The large momentum contributions (which are dominant) are calculated using Coulomb type states. Besides the usual condensate corrections, we see the possibility of new non-perturbative contributions. We parametrize them in terms of two low momentum correlators with Coulomb bound state energy insertions $E_n$. We realize that the Heavy Quark Effective lagrangian can be used in these correlators. We calculate the corrections that they give rise to in the decay constant, the bound state energy and the matrix elements of bilinear currents at zero recoil. We study the cut-off dependence of the new contributions and we see that it matches perfectly with that of the large momentum contributions. We consider two situations in detail: i) $E_n>> ��_{QCD}$ ($M_Q \rightarrow \infty$) and ii) $E_n << ��_{QCD}$, and briefly discuss the expected size of the new contributions in $��$ , $J/��$ and $B_{c}^{\ast}$ systems.

28 pages, LaTeX. Minor changes, some comments and numerical results added. To be published in Phys. Rev. D