Perturbative Expansion in Reciprocal Space: Bridging Microscopic and Mesoscopic Descriptions of Molecular Interactions

Determining the Fourier representation of various molecular interactions is important for constructing density-based field theories from a microscopic point of view, enabling a multiscale bridge between microscopic and mesoscopic descriptions. However, due to the strongly repulsive nature of short-ranged interactions, interparticle interactions cannot be formally defined in Fourier space, which renders coarse-grained approaches in $\textit{k}$-space somewhat ambiguous. In this paper, we address this issue by designing a perturbative expansion of pair interactions in reciprocal space. Our perturbation theory, starting from reciprocal space, elucidates the microscopic origins underlying zeroth-order (long-range attractions) and divergent repulsive interactions from higher-order contributions. We propose a systematic framework for constructing a faithful Fourier space representation of molecular interactions, capturing key structural correlations in various systems, including simple model systems and molecular coarse-grained models of liquids. Building upon the Ornstein-Zernike equation, our approach can be combined with appropriate closure relations, and to further improve the closure approximations, we develop a bottom-up parametrization strategy for inferring the bridge function from microscopic statistics. By incorporating the bridge function into the Fourier representation, our findings suggest a systematic, bottom-up approach to performing coarse-graining in reciprocal space, leading to the systematic construction of a bottom-up classical field theory of complex aqueous systems.