Abstract: We compute free energy functionals as power series in infinite dimensional spaces. We provide a rigorous framework to prove validity of density functionals for inhomogeneous, non-translation invariant systems with applications in classical density functional theory, liquid crystals, molecules with various shapes or other degrees of freedom. We use cluster expansions and compare them to large deviations techniques. A key technical tool is a combinatorial identity for trees which allows us to obtain convergence estimates in situations  where Banach inversion fails. This is joint work with Sabine Jansen and Tobias Kuna.