the role of the thrid order term in the funcational expansion of free energy of an inhomogeneous fluid

the third order term in the density functional is required for study of an inhomogeneous fluid. this term can be chosen such that, the thermodynamic potential gives the correct bulk pressure. the density porofile of a hard sphere fluid near hard wall which is an inhomogeneous fluid is obtained and it is shown that, if the third order term is ignored, although an exact bulk direct correlation function which gives the correct bulk pressure is used, the thermodynamic potential does not yield the correct density profile.