van der Waals Dispersion Force Between Three Atoms or Molecules

Abstract

Working within the electric dipole approximation, the leading non-additive retarded three-body dispersion potential is evaluated. Again two virtual photons are exchanged between each coupled pair. In order to avoid the use of sixth-order perturbation theory, necessitated when the interaction Hamiltonian is linear in the electric displacement field, a canonical transformation is performed on the $- \varepsilon_{0}^{ - 1} \vec{\mu } \cdot \vec{d}^{ \bot }$ form to yield an effective two-photon coupling Hamiltonian that is quadratic in the displacement field. Third-order perturbation theory is then used to obtain the dispersion energy shift, which holds for a scalene triangle configuration. At short inter-particle separation distance , the Axilrod-Teller-Muto result originally derived via semi-classical theory is reproduced. Explicit expressions for interaction energies corresponding to equilateral and right-angled triangle, and collinear geometries are also presented, along with their asymptotically limiting forms. Whether the dispersion force is attractive or repulsive depends on the spatial arrangement of the three objects.