Modeling Mean-Field Interacting Electrons Self-Consistently

Abstract

We model mutually interacting electrons in a one-dimensional quantum well by using the lowest mean-field approximation available, the Hartree Approximation (HA). In the HA an electron interacts with the total charge density of the electron system. We consider spin-less electrons and as they obey Fermi-Dirac statistics they fulfill the Pauli principle, that only one electron can occupy each quantum state. The interaction potential seen by an electron in the system depends on the total density and thus the resulting eigenvalue problem is nonlinear with respect to the wavefunctions of the electrons, and needs to be solved self-consistently by iterations. In order to put the HA in context with higher order perturbation theory for the electron gas, the nonlinear problem in each iteration is solved by linear methods requiring the eigenstates to be orthonormal. The many-electron wavefunctions in the HA are simple linear products of the wavefunctions for each occupied state. The system does thus not satisfy the antisymmetry of the total wavefunction, and clearly the HA includes nonphysical self-interactions, that can be avoided by the Hartree-Fock Approximation (HFA) at a much higher computational cost. The HA is still a challenging project to learn from when studying computational methods applied to physics, and truly it is important for systems with a high electron density, especially when they contain a positive homogeneous charge background representing the ions, which partially cancels the self-interaction.