# Molecular density functional theory based on a hard particle reference potential

- Molekulare Dichtefunktionaltheorie basierend auf einem Hartkörper-Referenzpotential
- Molecular density functional theory based on a hard-particle reference potential

Korden, Stephan; Leonhard, Kai (Thesis advisor); Roth, Roland (Thesis advisor)

*Aachen : Publikationsserver der RWTH Aachen University (2016)*

Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2015

Abstract

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.

Institutions

- Chair and Institute of Technical Thermodynamics [412110]

### Identifier

- URN: urn:nbn:de:hbz:82-rwth-2015-066903
- RWTH PUBLICATIONS: RWTH-2015-06690