  Beginning from the semiclassical Hamiltonian, the Fermi pressure and Bohm
potential for the quantum hydrodynamics application (QHD) at finite temperature
are consistently derived in the framework of the local density approximation
with the first order density gradient correction. Previously known results are
revised and improved with a clear description of the underlying approximations.
A fully non-local Bohm potential, which goes beyond of all previous results and
is linked to the electron polarization function in the random phase
approximation, for the QHD model is presented. The dynamic QHD exchange
correlation potential is introduced in the framework of local field
corrections, and considered for the case of the relaxation time approximation.
Finally, the range of applicability of the QHD is discussed.
