First Principles Derivation of Effective Ginzburg-Landau Free Energy models for Crystalline Systems

The expression of the free energy density of a classical crystalline system as a gradient expansion in terms of a set of order parameters is developed using classical density functional theory. The goal here is to extend and complete an earlier derivation by L{\"o}wen et al (Europhys. Lett.9, 791, 1989). The limitations of the resulting expressions are also discussed including the boundary conditions needed for finite systems and the fact that the results cannot, at present, be used to take into account elastic relaxation.