The Pauli Exclusion Operator: example of Hooke's atom

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.