Free energy of particle insertion an exact analysis of the origin singularity for simple liquids

In free energy simulations involving the creation of a new particle in a condensed medium, there is a well-known ‘origin singularity’ when the particle is first introduced, unless the coupling of the particle with its surroundings is increased very slowly. This singularity is analysed theoretically for a liquid that interacts with the new particle through a soft-sphere or a Lennard-Jones potential, using a virial expansion of the free energy derivative. For a softsphere potential u(r)= λ^αAr^-n (where λ is a coupling constant), we show rigorously that the derivative of the free energy with respect to λ is infinite at the origin if and only if α < n/3. This confirms a previous, approximate, prediction based on scaled particle theory. For a Lennard-Jones potential u(r) = λ^αAr^-12 + λ^βBr^-6, such that α ≤ 2β, the derivative of the free energy is again infinite if and only if α < n/3. If the volume of the solute is proportional to the coupling constant, there is no singularity.