Perturbative density functional approximation in the view of weighted density concept and beyond

Functional expansion of non-uniform first-order direct correlation function (DCF) around bulk density is rewritten with the concept of weighted density, it is found that the resulting first-order expansion term is actually the first-order expansion term of the Taylor series expansion of the uniform first-order DCF with weighted density as its density argument, the higher-order terms are incorporated into the first-order expansion approximation by Lagrangian theorem. Based on the functional integration view, the present Letter devises a numerical procedure to predict excess Helmholtz free energy from the approximation for the non-uniform first-order DCF. The physical foundation of the present discussion also applies to electronic density functional theory.