  The Rosenfeld functional provides excellent results for the prediction of the
fluid phase of hard convex particle systems but fails beyond the freezing
point. The reason for this limitation is the neglect of orientational and
distance correlations beyond the particle diameter. In the current article we
resolve this restriction and generalize the fundamental measure theory to an
expansion in intersection centers. It is shown that the intersection
probability of particle systems is described by an algebra, represented by
Rosenfeld's weight functions. For subdiagrams of intersection networks we
derive vertex functions that provide the building blocks for the free energy
functional. Their application is illustrated by deriving the Rosenfeld
functional and its leading correction which is exact in the third virial order.
Furthermore, the methods are used to derive an approximate functional for the
infinite sum over Mayer ring diagrams. Comparing this result to the White Bear
mark II functional, we find general agreement between both results.
