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We consider a molecule in the Born-Oppenheimer approximation interacting witha plate of infinite thickness, i.e, a half--space, which is perfectlyconducting or dielectric. It is well--known in the physics literature that inthis case the atom or molecule is attracted by the plate at sufficiently largedistances. This effect is analogous to the well--known van der Waalsinteraction between neutral atoms or molecules. We prove that the interactionenergy $W$ of the system is given by $W(r,v)= -C(v)r^{-3} +\mathcal{O}(r^{-4})$, where $r$ is the distance between the molecule and theplate and $v$ indicates their relative orientation. Moreover, $C(v)$ ispositive and continuous, thus the atom or molecule is always pulled towards theplate at sufficiently large distances, for all relative orientations $v$. Forsome specific systems we provide sharper estimates of $W(r,v)$. This asymptoticbehavior is well--known in the physics literature, however, we are not aware ofany previous rigorous results, even on existence of a ground state of thesystem. For pedagogical reasons, we often start with the case of a hydrogenatom and then we generalize the arguments to deal with a general molecule.

On the van der Waals interaction between a molecule and a half-infinite plate