The Pauli Exclusion Operator: example of Hooke's atom

TL;DR

This paper introduces the Pauli Exclusion Operator (PEO) as a means to incorporate the Pauli principle directly into the Hamiltonian of multi-electron systems, simplifying the treatment of electron exchange symmetry.

Contribution

The paper presents the formulation of PEO for multi-electron systems, including explicit forms for 2D two-electron states, and demonstrates its application to Hooke's atom with analytical and numerical methods.

Findings

PEO can be expressed in closed form for 2D two-electron states

Application of PEO yields accurate energies and states for Hooke's atom

PEO relates to standard variational methods with Slater determinants

Abstract

The Pauli Exclusion Operator (PEO) which ensures proper symmetry of the eigenstates of multi-electron systems with respect to exchange of each pair of electrons is introduced. Once PEO is added to the Hamiltonian, no additional constraints on multi-electron wave function due to the Pauli exclusion principle are needed. For two-electron states in two dimensions ($2D$) the PEO can be expressed in a closed form in terms of momentum operators, while in the position representation PEO is a non-local operator. Generalizations of PEO for multi-electron systems is introduced. Several approximations to PEO are discussed. Examples of analytical and numerical calculations of PEO are given for isotropic and anisotropic Hooke's atom in~$2D$. Application of approximate and kernel forms of PEO for calculations of energies and states in~$2D$ Hooke's atom are analyzed. Relation of PEO to standard variational calculations with the use of Slater determinant is discussed.