Thermodynamic geometry of the Gaussian core model fluid

The three-dimensional Gaussian core model (GCM) for soft-matter systems has repulsive interparticle interaction potential $\phi (r) = \varepsilon\, {\rm exp}\left[ -(r/\sigma)^{2} \right]$, with $r$ the distance between a pair of atoms, and the positive constants $\varepsilon$ and $\sigma$ setting the energy and length scales, respectively. $\phi (r)$ is mostly soft in character, without the typical hard core present in fluid models. We work out the thermodynamic Ricci curvature scalar $R$ for the GCM, with particular attention to the sign of $R$, which, based on previous results, is expected to be positive/negative for microscopic interactions repulsive/attractive. Over most of the thermodynamic phase space, $R$ is found to be positive, with values of the order of $\sigma^3$. However, for low densities and temperatures, the GCM potential takes on the character of a hard-sphere repulsive system, and $R$ is found to have an anomalous negative sign. Such a sign was also found earlier in inverse power law potentials in the hard-sphere limit, and seems to be a persistent feature of hard-sphere models.