Density Functional Theory for Hard Particles in N Dimensions

Recently it has been shown that the heuristic Rosenfeld functional derives from the virial expansion for particles which overlap in one center. Here, we generalize this approach to any number of intersections. Starting from the virial expansion in Ree-Hoover diagrams, it is shown in the first part that each intersection pattern defines exactly one infinite class of diagrams. Determining their automorphism groups, we sum over all its elements and derive a generic functional. The second part proves that this functional factorizes into a convolute of integral kernels for each intersection center. We derive this kernel for N dimensional particles in the N dimensional, flat Euclidean space. The third part focuses on three dimensions and determines the functionals for up to four intersection centers, comparing the leading order to Rosenfeld's result. We close by proving a generalized form of the Blaschke, Santalo, Chern equation of integral geometry.Comment: 2 figure

A Short Proof of the Reducibility of Hard-Particle Cluster Integrals

The current article considers Mayer cluster integrals of n-dimensional hard particles in the n>1 dimensional flat Euclidean space. Extending results from Wertheim and Rosenfeld, we proof that the graphs are completely reducible into 1- and 2-point measures, with algebraic rules similar to Feynman diagrams in quantum field theory. The hard-particle partition function reduces then to a perturbative solvable problem.Comment: 4 pages, 1 figure, revtex4-

Deriving the Rosenfeld Functional from the Virial Expansion

In this article we replace the semi-heuristic derivation of the Rosenfeld functional of hard convex particles with the systematic calculation of Mayer clusters. It is shown that each cluster integral further decomposes into diagrams of intersection patterns that we classify by their loop number. This extends the virial expansion of the free-energy by an expansion in the loop order, with the Rosenfeld functional as its leading contribution. Rosenfeld's weight functions follow then from the derivation of the intersection probability by generalizing the equation of Blaschke, Santalo, and Chern. It is found that the 0-loop order can be derived exactly and reproduces the Rosenfeld functional. We further discuss the influence of particle dimensions, topologies, and geometries on the mathematical structure of the calculation.Comment: Replaced by version accepted by Phys. Rev. E. Section II.A and Appendix B improve

Molecular density functional theory based on a hard particle reference potential

The aim of the current doctorate thesis is the development of a density functional theory (DFT) for classical intermolecular interactions. In the first part, we begin with an analysis of the structure of the grand-canonical potential and demonstrate that for pair potentials only two independent representations exists: the direct-correlation functional and its Legendre transformation with respect to the pair potential. Using far-reaching assumptions, this dual grand-canonical potential reduces to the free-energy functions of Flory-Huggins, Staverman-Guggenheim, and Guggenheim, from which again derive the lattice-models UNIQUAC, UNIFAC, and COSMO-RS. We conclude this first part by discussing possible generalizations of this approach to a continuum formulation. As is well known from quantum mechanics, the central problem of the DFT approach is the derivation of the functionals, which is further complicated for intermolecular interactions by their strongly repulsive potential. A well established approach is the separation of the potential into a flat but long-ranged contribution and the approximation of the repulsive part using the geometry of hard particles. In the second part of this work, we develop the necessary methods for the non-perturbative derivation of their corresponding hard-particle functionals.We first begin with a discussion of the fundamental measure theory and interpret the semi-heuristic Rosenfeld functional as the leading order of an expansion in the number of intersection centers of the particles. For the generalization of the approach we demonstrate the equivalence between intersection configurations and classes of Ree-Hoover diagrams, whose sum defines a generic functional decoupling into a convolute of intersection kernels. Each such kernel determines the local intersection probability of a set of particles under the group of translations and rotations. For the case of two particles this result has been first derived by Blaschke, Santalo, and Chern. Here, we generalize their approach to an arbitrary set of particles and obtain a closed expression for the free-energy functional and the n-particle densities for any dimension. As examples, we derive the functional of the free energy for up to four intersection centers, whose leading order agrees with Rosenfeld's result. We then calculate for the 2-particle density an upper limit of the contact probability for hard spheres, which is in excellent agreement with the result of Carnahan and Starling. Comparing the same level of approximation with Kirkwood's superposition ansatz for correlation functions of higher orders, shows that the contact probability of spheres is significantly overestimated by the superposition approximation. Finally, we derive the leading perturbative corrections for long-range interactions.With the methods developed in the current work, the hard-particle interaction is now the only known example, whose density functionals can be derived systematically to any order of precision. We conclude our work with a discussion of possible applications in biology and chemistry