Drude oscillators provide a harmonic description of charge fluctuations and are widely studied as a model system and for ab initio calculations. In the dipole approximation the Hamiltonian describing the interaction of Drudes is quadratic, so it can be diagonalized exactly, but the energy diverges at short range. Here we consider the quantum mechanics of Drude oscillators interacting through the full Coulombic Hamiltonian for which the interaction energy does not have this defect. This protypical model for interactions between matter includes electrostatics, induction, and dispersion. Potential energy curves for rare-gas dimers are very closely matched by Drude correlation energies plus a single exponential function. The exact and accurate results presented here help to delineate between the basic properties of the physical model and the effects that arise from the dipole approximation.