Dual Characterization of the Ornstein-Zernike Equation in Moment Space

TL;DR

This paper develops a minimal basis set approach for the molecular density functional theory of fluids, enabling efficient computation of solvation energies and molecular structure analysis through a moment space framework.

Contribution

It introduces a novel moment-based Hilbert space formulation of the Ornstein-Zernike equation, with two new basis expansions for flexible and rigid molecules, enhancing computational efficiency.

Findings

Provides a minimal, basis-independent Hilbert space formulation.

Introduces two basis expansions emphasizing multipolar and rotational properties.

Demonstrates applications in stability analysis and solvation energy calculations.

Abstract

The molecular density functional theory of fluids provides an exact theory for computing solvation free energies in implicit solvents. One of the reasons it has not received nearly as much attention as quantum density functional theory for implicit electron densities is the paucity of basis set expansions for this theory. This work constructs a minimal Hilbert space version of the Ornstein-Zernike theory over the complete spatial, rotational, and internal conformational space that leaves the choice of basis open. The basis is minimal in the sense that it is isomorphic to a choice of molecular property space (i.e. moments of the distribution), and does not require auxiliary grids. This can be exploited, since there are usually only a few `important' properties for determining the structure and energetics of a molecular fluid. Two novel basis expansions are provided which emphasize either the multipolar expansion (most useful for flexible molecules) or the rotational distribution (most useful for rigid bodies described using quaternions). The perspective of this work shows that radial truncation of the Bessel series over translation space determines an analytical extrapolation of these functions to the origin in reciprocal space. We provide a new density functional theory that naturally fits the moment-based, matrix approach. Three diverse applications are presented: relating the present approach to traditional rotational invariants, demonstrating the stability of convex optimization on the density functional, and finding analytical expression for dispersion contributions to the solvation free energies of point polarizable dipoles.