Exchange-correlation energy of a quasi-two-dimensional electron gas: A Hartree self-consistent approach

We present an approach to the exchange-correlation energy of an electron gas confined in a finite-width quantum well, which avoids the local-density approximation. We focus on densities large enough to have electrons in more than one subband. This imposes a self-consistent treatment of the Hartree interaction (i.e., $q=0$ excitation processes) while the q≠0 interaction can be treated perturbatively. By writing the Hamiltonian in terms of appropriate Hartree self-consistent creation operators, we can cancel most but not all $q=0$ Coulomb contributions; a few mixed ($q=0$, q≠0) terms do remain in the correlation energy. A numerical calculation of the exchange energy shows that the effect of the Hartree processes turns out to stay rather small in the whole two-subband filling domain.