Analytic Properties of the Quantum Corrections to the Second Virial Coefficient

The usefulness of the perturbation expansion and the Wigner-Kirkwood expansion of the quantum-mechanical partition function is discussed for various interaction potentials. It is shown that, contrary to what is expected from the Wigner-Kirkwood expansion, quantum-mechanical diffraction corrections at high temperature to the classical partition function may involve nonanalytic forms of ℏ2. This occurs when the second-order perturbation term is finite in the classical limit, and the interaction potential has a cusp or singularity in any derivative. The second-order perturbation term is evaluated exactly for the exponential, screened Coulomb, and square barrier potentials, and the nonanalytic form (ℏ2)½ is found. For potentials more singular than 1/r at the origin, the diffraction corrections are analytic in ℏ2. A new method of deriving the Wigner-Kirkwood expansion from the perturbation expansion is given. The method allows one to subtract off any order of the perturbation expansion which may be evaluated separately, and is particularly useful for the screened Coulomb potential. The classical second virial coefficient and the O(ℏ2) and O(ℏ4) diffraction corrections are evaluated for the singular potential, u(r) = (gp/rp)e−r/ro, by using the Mellin transform of e−βu.