Functional Counterpart of Lagrangian Theorem and PerturbativeDensity Functional Theory: a Forgotten Idea

In this letter, we truncates the functional expansion of non-uniform first-order direct correlation function (DCF) around the bulk density at the lowest order. But the truncation is performed formally and exactly by making use of functional counterpart of the Lagrangian theorem of differential calculus. Consequently the expansion coefficient, i.e., the uniform second-order DCF, is replaced by its non-uniform counterpart whose density argument is appropriate mixture of calculated density distribution and the bulk density with a mixing parameter determined by a hard-wall sum rule. The non-uniform second-order DCF is then approximated by the uniform second-order DCF with an appropriate weighted density as its density argument. The present formally exact truncated functional expansion predicts the density distribution in good agreement with simulation data for hard sphere and Lennard-Jones fluid exerted by an external field.