On the Non-Pauli Electronic States of Atoms and Molecules

Schr\"odinger's equation for atoms and molecules supports solutions that are not totally antisymmetric under electron coordinate permutations. These non-Pauli eigenstates are generally regarded as unphysical, with interest in them centered largely on their role as possible ``contaminants'' in physical solutions constructed by methods that provide only approximate antisymmetry, such as exchange perturbation theories, many-body diagrammatic approaches, and variational methods in the absence of precise prior enforcement of basis-state antisymmetry. Here we report atomic and molecular non-Pauli Schr\"odinger solutions employing largely pedestrian methods as an alternative to the more complicated Wigner-Weyl approach based on theory of the symmetric group. Using the non-relativistic Hamiltonian operator and spin-orbital product representations in variational calculations, we show that every antisymmetric Schr\"odinger eigenstate of an $n$ electron atom or molecule is accompanied by 2$^{n}$-1 degenerate non-Pauli ``ghost'' solutions. As a consequence of this degeneracy, admixtures of non-Pauli states are always present in Pauli solutions having only approximate antisymmetry. These can significantly affect calculated expectation values, even in the face of precise energy predictions.