The ‘density’ of the interacting objects (planets and sun) becomes a sum of delta functions weighed by the mass of the planet/sun at that position.
The hamiltonian for one of the quasiplanets, i.e. planets that do not interact but reproduce the same density, now consists of its kinetic energy, a ‘gravitational hartree term’ which is the gravitational interaction with all other objects besides the object in question. The external potential could for example be the gravitational potential of the Milky way or of the sun if you treat it as external to the planets. Since planets are macroscopic and distinguishable there is no exchange correlation function