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International Journal of Quantum Chemistry

Proof of the linear independence of properly selected projected spin eigenfunctions

Z. Gershgorn

Published May 1, 1968• 12 citations
Publisher

Abstract

Abstract An inductive proof is given of Löwdin's theorem about the linear independence and completeness of a properly selected set of projected spin eigenfunctions.
DOI
10.1002/qua.560020304

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Abstract It is shown that the exclusion principle merely requires the electrons to be divided into two distinct groups, then, according to our assumption of the distinguishability of electrons, its mathematical representation can be the spatial double antisymmetrization or the spatial‐spin antisymmetrization; in the latter case an auxiliary spin double symmetrization is also required in order to distinguish the two types of electrons. The second requirement is used to recognize the fact that, although all electrons can be assumed to be indistinguishable under certain condition, their natural characteristic of two distinct groups must still be preserved.It is further shown that the double antisymmetrization (or double symmetrization) determines the spin quantum number M, whereas the Löwdin spin projector for nonorthogonal eigenfunctions and the Wigner matric basis for orthogonal eigenfunctions will decide the spin quantum number S, Both the spatial double antisymmetrizer and the spin double symmetrizer will reject all the unnecessary eigenfunctions and project out only one unique set of linearly independent eigenfunctions.