In this project in the Theoretical and Computational Chemistry Subprogram of the Physical Chemistry Program, Lovett will study integral equations which relate the density and the pair correlation function. The problems are important for the determination of the spatial variation of the density in interfacial regions such as liquid-vapor or liquid-solid interfaces, of the profile of fluids adsorbed onto walls or solids, and of the variation of ion concentrations at electrolyte-electrode boundaries. An abstract argument shows that the number of solutions to the integral equations is very sensitive to the approximations made for the pair correlation function kernels, with the most likely consequence being the creation of an integral equation which possesses no solution. All the numerical work to date had been undertaken without a clear understanding of how the number of solutions varies with the approximation. The goal of the present work is to show how the number of solutions varies explicitly with the approximations and to provide a concrete connection between the abstract picture and the calculations.
