Non-additive Intermolecular Potential in Gases I. van der Waals Interactions

The potential energy of van der Waals interactions between three spherically symmetric atoms at large separations has been investigated both by perturbation theory and by variation method. Without using any particular atomic models the van der Waals interaction is derived, from second and third-order perturbations, in the form {aligned} -μ₁₂{r₁₂}⁻⁶-μ₂₃{r₂₃}⁻⁶-μ₃₁{r₃₁}⁻⁶+ν(r₁₂r₂₃r₃₁)⁻³(3cosθ₁cosθ₂cosθ₃+1), {aligned} where r i j is the distance between i th and j th atoms, θ i 's are inner angles of a triangle formed by the three atoms. An approximate relation between µ and ν is found by variation method: {aligned} ν{=}{2R₁R₂R₃(R₁+R₂+R₃)}{(R₁+R₂)(R₂+R₃)(R₃+R₁)}, {aligned} in which {aligned} {1}{R₁}{=}{1}{μ₁₂α₃}+{1}{μ₃₁α₂}{1}{μ₁₂α₃}-{1}{μ₁₂α₃}, {aligned} α i being the polarizability of the i th atom. In case three atoms are identical the relation reduces to 4ν=3αµ. The accuracy of this simple relation is sufficient; for example the error is 1.8% for atomic hydrogen. The relation will be applied in a subsequent paper to the equation of state of rare gases.