(456d) Bridge Functionals Representing Higher-Order Terms in the Perturbative Classical Density Functional Theory of Solid-Fluid Transitions

Classical density functional theory (DFT) is a useful tool for predicting the solid-fluid phase transition based on a pairwise interaction potential model; it is more accurate than perturbation theory and less expensive than simulation.  The key element in DFT is an accurate and computationally feasible expression for the excess Helmholtz free energy as a functional of the density.  Weighted DFTs model this excess free energy as a volume integral over an appropriately weighted free energy density; these DFTs are highly accurate for hard spheres but rather cumbersome to apply to more complex potentials.  Perturbative DFTs use a formally exact functional expansion of the solid free energy about the liquid reference state, but in practice are typically limited to second order due to difficulty in computing the higher order terms.  We have been exploring a closure-based DFT that represents the sum of these higher order terms as a bridge function B, which may be represented as a function in real space B(r) or as a functional of the second order term B[gamma(r)] or the density B[rho(r)].  We present what we have learned about this bridge function for a variety of interaction potentials (hard sphere, power law repulsion, Lennard-Jones) using an inverse DFT approach.  There is a remarkable universality in the shape of the bridge function across different potentials, and only a portion of the bridge function seems to be relevant to the thermodynamic properties at coexistence.  We also present the formal graph theory expression for B[rho(r)] and comment on the prospects for calculating the values of the graphs.