Effect of Many-Body van der Waals Forces on the Lattice Dynamics of Rare-Gas Crystals

The contribution of the many-body part of the van der Waals forces to the lattice-vibration Hamiltonian of rare-gas crystals is investigated, and the result is compared with the usual two-body lattice potential (a Lennard-Jones potential, for instance). While the static lattice energy is modified only by a few percent, the harmonic term and the successive anharmonicities are so much influenced by collective dispersion forces that their perturbational developments in terms of pair, triplet, etc., contributions are inadequate. In this paper we show that, by a simple renormalization procedure, these developments can be summed so as to obtain the exact energies in closed forms. In addition to revealing the existence of a theoretical singularity as a function of the crystal "refractivity" $\frac{4}{3}\pi$ $\mathrm{N}\alpha$, these closed forms are simple enough to be used in numerical computations of various crystal properties. No such calculation is attempted here.