Validity of the Assumption of Two-Body Interactions in Molecular Physics

The treatment of three-body van der Waals forces between hydrogen or helium atoms, given in an earlier paper, is extended to the heavy rare gas atoms. Two of the three atoms are supposed to be close together (e.g., nearest neighbors in a crystal), whereas the third atom is considerably farther away. Expressions are given for the induced dipole-dipole and dipole-quadrupole components of the field. The electronic charges in the atoms are represented by Gaussian distribution functions. The results are applied to a determination of the most stable crystal structure of the heavy rare gases at absolute zero temperature (excluding zero-point energy); it is found that the face-centered cubic crystal is favored over the hexagonal close-packed by about one-tenth of one percent of the van der Waals cohesive energy, if the first and second shells of neighbors are excluded from the region of summation for the third atom. If this restriction is dropped, then the difference between the two structures vanishes, while the effect itself rises abruptly and very steeply. The effect of three-body interactions in general is to weaken the attractive forces in the crystalline state. Under the above restriction, the relative magnitude of these interactions is twenty percent of the van der Waals cohesive energy for xenon. The three-body forces vanish identically if the Gaussian goes over into a Dirac $\delta$ function.