Multipolar expansion for the non-additive third-order interaction energy of three atoms

This paper gives a multipolar expansion for the non-additive third-order interaction energy of three well-separated atoms, in spherically symmetric states. Each term in the expansion can be written as the product of a geometrical factor W depending only on the relative positions of the three atomic nuclei, and an interaction constant Z which depends solely on the atomic species involved in the interaction. Explicit formulae are presented for the first few W factors, namely those involved in the dipole-dipole, dipole-dipole- quadrupole, dipole-quadrupole-quadrupole and quadrupole- quadrupole-quadrupole terms. The formulae for the dipole-dipole- dipole case is equivalent to Axilrod and Teller's expression for the three-body van de Waals energy. In the dipole-dipole- quadrupole case, however, the result is in disagreement with an earlier calculation by Ayres and Tredgold. Formulae for the other two W factors have not appeared previously in the literature. Approximate values of the Z constants are also presented, in the four cases mentioned, for triplets of rare-gas atoms.