  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.
