On the Quantum Theory of the Third Virial Coefficient

The quantum theory of the third virial coefficient $C$ is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of $C_{\mathrm{BE}}$ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for ${\mathrm{He}}^4$ and ${\mathrm{He}}^3$, by means of a comparison of the corresponding expansion for the second virial coefficient $B$ with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to $B$. Then the leading term of $C$ is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for $C$ is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.