Understanding the correlation function of the inhomogeneous hard-sphere fluid

We present a new functional for the correlation $g(r)$ at contact for an inhomogeneous distribution of hard spheres. This term is a key input into classical density functional theories developed using Statistical Associating Fluid Theory, a widely used approach for handling complex liquids in which hydrogen bonding plays an important role. We use a thermodynamic approach to derive an exact formula for the correlation expressed as a derivative of the free energy functional. We evaluate this approach using the approximate free energy of the ``White Bear'' version of Fundamental Measure Theory, and test our approach (and two previously published approximations) against correlation values found from Monte Carlo simulations.