Variational methods for the solution of the Ornstein-Zernicke equation in inhomogeneous systems

We show that the Ornstein-Zernicke equation and other equations of similar form obey a variational principle that can be used to derive approximate solutions. This method requires the use of an initial trial solution where the variational solution possesses a stationary "point" with respect to the trial solution when the latter is equal to the exact solution. We show that with even a very simple form of the trial solution the results are quite reasonable. Furthermore, we have demonstrated that by combining the variational method with an iterative expansion of the Ornstein-Zernicke equation it is possible to develop a self-consistent method of writing the direct correlation function.