A functional integration approach – whose main ingredient is the Hubbard-Stratonovich transformation – for the quantum nonrelativistic many-fermion problem is investigated. With this method, the ground state energy correponds to a systematic expansion in powers of a small parameter related to the number of fermions. It is a functional of a potential determined by a self-consistent equation. The semiclassical Hartree energy is obtained at lowest order of the expansion, the exchange energy at first order, and the correlation energy at second order. This approach is applied to large neutral atoms, for which the correlation energy is computed. This approach is also applied to many-electron quantum dots with harmonic confinement. The self-consistent equation is solved as a function of a small parameter depending on the confinement strength. The Hartree and exchange energies are computed in powers of this parameter, and the correlation energy is computed at lowest order. The energy oscillations, arising from the Hartree energy, are also evaluated; they are related to the periodic orbits of the classical dynamics of the self-consistent potential.