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

Abstract

We consider a molecule in the Born–Oppenheimer approximation interacting with a plate of infinite thickness, i.e., a half-space, which is perfectly conducting or dielectric. It is well known in the physics literature that in this case the atom or molecule is attracted by the plate at sufficiently large distances. This effect is analogous to the well-known van der Waals interaction between neutral atoms or molecules. We prove that the interaction energy W of the system is given by W(r,v)=−C(v)r−3+O(r−4), where r is the distance between the molecule and the plate and v indicates their relative orientation. Moreover, C(v) is positive and continuous, thus the atom or molecule is always pulled toward the plate at sufficiently large distances, for all relative orientations v. For some specific systems, we provide sharper estimates of W(r, v). This asymptotic behavior is well known in the physics literature; however, we are not aware of any previous rigorous results, even on the existence of a ground state of the system. For pedagogical reasons, we often start with the case of a hydrogen atom and then we generalize the arguments to deal with a general molecule.