  To go beyond Gaussian approximation to the Hohenberg-Kohn free energy playing
the key role in the density functional theory (DFT), the density functional
\textit{integral} representation would be relevant, because field theoretical
approach to perturbative calculations becomes available. Then the present
letter first derives the associated Hamiltonian of density functional,
explicitly including logarithmic entropy term, from the grand partition
function expressed by configurational integrals. Moreover, two things are done
so that the efficiency of the obtained form may be revealed: to demonstrate
that this representation facilitates the field theoretical treatment of the
perturbative calculation, and further to compare our perturbative formulation
with that of the DFT.
