Unconventional approach to orbital-free density functional theory derived from a model of extended electrons

TL;DR

This paper introduces a novel geometric algebra-based model of extended electrons to generalize an existing orbital-free density functional theory equation for fermionic systems, with implications for condensed matter physics.

Contribution

It develops a space-time model of electrons using geometric algebra, extending a bosonic equation to fermionic charge and spin distributions, and derives coupled equations for many-electron systems.

Findings

Model aligns with hydrogen atom results.

Wavefunctions satisfy Schrödinger equation.

Provides a many-electron wavefunction with fewer variables.

Abstract

An equation proposed by Levy, Perdew and Sahni in 1984 [PRA 30, 2745 (1984)] is an orbital--free formulation of density functional theory. However, this equation describes a bosonic system. Here, we analyze on a very fundamental level, how this equation could be extended to yield a formulation for a general fermionic distribution of charge and spin. This analysis starts at the level of single electrons and with the question, how spin actually comes into a charge distribution in a non-relativistic model. To this end we present a space-time model of extended electrons, which is formulated in terms of geometric algebra. Wave properties of the electron are referred to mass density oscillations. We provide a comprehensive and non-statistical interpretation of wavefunctions, referring them to mass density components and internal field components. It is shown that these wavefunctions comply with the Schr\"odinger equation, for the free electron as well as for the electron in electrostatic and vector potentials. Spin-properties of the electron are referred to intrinsic field components and it is established that a measurement of spin in an external field yields exactly two possible results. The model agrees with the results of standard theory concerning the hydrogen atom. Finally, we analyze many-electron systems and derive a set of coupled equations suitable to characterize the system without any reference to single electron states. The model is expected to have the greatest impact in condensed matter theory, where it allows to describe an $N$-electron system by a many-electron wavefunction $\Psi$ of four, instead of 3N variables. The many-body aspect of a system is in this case encoded in a bivector potential.