Factorization of the direct correlation function. I. Basic results

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Abstract
We examine the possibility of applying the factorization technique to the Ornstein–Zernike equation in closed form for systems of molecular fluids in D‐dimensional space. We find a coordinate space transform which puts the equation in pseudo‐one‐dimensional form by reducing the convolution term to one involving parallel vectors. The resulting factorization equations greatly resemble the one‐dimensional ones, but contain the additional variables which are present for D≳1. The inverse transform is constructed and shown to exhibit a characteristic difference between odd and even dimensionality. For odd D a geometric interpretation can be given for all transforms, while even D requires an additional Hilbert transform. Expansion of the transforms in harmonics is carried out for D=2 and 3 to obtain the radial transforms. For D=3 agreement with known results is obtained. The transformation under a shift in the choice of molecular origin is considered. The transformation to the pseudo‐one‐dimensional form is invariant. The factorization itself depends on a certain surface associated with each molecular species. It is invariant if each surface is kept fixed with respect to the molecule rather than the shifted molecular origin. Distortion or displacement of the surface leads to a different factorization.