Pauli Principle, anti-symmetry requirements

However, this is not a satisfactory description, especially, for electronic configurations in which there are unpaired electron spins, since this, so-called Hartree product form, does not comply with the anti-symmetry requirement of the Pauli Principle. In general, many-electron wave functions are written as Slater Determinants, which do exhibit the necessary anti-symmetry properties for electron exchange. ... 

The Hartree product wave function, equation 5.2, for helium does not comply with the anti-symmetry requirement of the Pauli Principle (42,47) that electronic wave functions must change sign on exchange of the coordinates for a pair of electrons. Fock identified this defect in the overestimation of the electron-electron repulsion term, which occurs for Hartree product wave functions, while Slater showed how to overcome this problem by writing the product wave function in the form of a determinant (6,7,42,47,64). 

If there would be no electron spin, the Pauli principle would only allow the antisymmetric solution with repulsion, and no homo-polar binding would exist. The existence of homo-polar chemistry appears to rest entirely on the presence of spin in connection with the Pauli principle. This yields an additional degree of freedom (which is not of importance for the energetic behaviour) and the anti-symmetry requirement of the Pauli principle can be satisfied with the spin coordinates. As a result of this, the eigenfunction symmetric in the coordinates of the centre-of-mass with a homo-polar binding is allowed it is only necessary to suppose the spins of the electrons different and anti-symmetrically related. 

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