Quantal-Density Functional Theory in the Presence of a Magnetostatic Field

Abstract

Quantal density functional theory (Q–DFT) of electrons in an external electrostatic field is generalized to the added presence of a magnetostatic field. This Q–DFT constitutes the mapping from the interacting system of electrons in an external electrostatic and magnetostatic field in any state as described by Schrödinger theory to one of noninteracting fermions with the same density, physical current density, electron number, and canonical orbital and spin angular momentum. To formulate this Q–DFT, Schrödinger theory from the perspective of the individual electron via the corresponding ‘Quantal Newtonian’ first law is developed. It is shown that in addition to the external fields, each electron experiences an internal field which is comprised of components representative of electron correlations due to the Pauli exclusion principle and Coulomb interaction, the density, the kinetic effects, and a contribution due to the external magnetic field. These fields are derived from quantal sources that are expectations of Hermitian operators taken with respect to the system wave function. As such the intrinsic self-consistent nature of the Schrödinger equation is demonstrated. With the Schrödinger equation written in self-consistent form, the magnetic field, (in addition to the vector potential of the field component of the momentum), now appears explicitly in it. The ’Quantal Newtonian’ first law for the model system of noninteracting fermions is derived. It is shown that if the model fermions are subject to the same external potentials, then the only electron correlations that must be accounted for in the Q–DFT mapping are those of the Pauli principle, Coulomb repulsion and Correlation-Kinetic effects. The resulting local electron-interaction potential within Q–DFT is the work done in an effective field that is the sum of fields representative of these correlations. The corresponding many-body components of the total energy can be expressed in integral virial form in terms of the separate fields. To explicate this Q–DFT, it is applied to a quantum dot as represented by the exactly solvable two-dimensional Hooke’s atom in a magnetic field. A key observation is that as a result of the reduction in dimensionality due to the presence of the magnetic field, Correlation-Kinetic effects are significant.