Direct correlation function for the square-well potential

An analytical expression of direct correlation function (DCF) for the square-well potential is developed. The development is based on the first-order mean spherical approximation and its extension to the functionality of the existing radial distribution function. The developed DCF is a combination of a special polynomial function introduced in this work. The combination is piecewise in four regions [0, lambda - 1], [lambda - 1,2 - lambda], [2 - lambda,1], and [1,lambda] for lambda < 1.5 or [0,2 - lambda], [2 - lambda, lambda - 1], [lambda - 1, 1], and [1, lambda] for lambda > 1.5. In addition, the DCF is continuous to second-order inside hard core and discontinuous at r=1 and r=lambda outside it. The behavior of DCF is analyzed by some detail calculations.