  The one-body density matrix is derived within the Extended Thomas-Fermi
approximation. This has been done starting from the Wigner-Kirkwood
distribution function for a non-local single-particle potential. The links
between this new approach to the density matrix with former ones available in
the literature are widely discussed. The semiclassical Hartree-Fock energy at
Extended Thomas-Fermi level is also obtained in the case of a non-local
one-body Hamiltonian. Numerical applications are performed using the Gogny and
Brink-Boeker effective interactions. The semiclassical binding energies and
root mean square radii are compared with the fully quantal ones and with those
obtained using the Strutinsky averaged method.
