Antisymmetry and the Slater Method

In discussing the helium atom (Section 1.2) the antisymmetry requirement on the electronic wavefunction was easily satisfied for with only two electrons the function would be written as a product of space and spin factors, one of which had to be antisymmetric, the other symmetric, lliis is possible even for an exact eigenfunction of the Hamiltonian (1.2.1), as well as for an orbital product. The construction of an antisymmetric many-electron function is less easy. We have seen in Section 1.2 that for a general permutation (involving both space and spin variables) an antisymmetric function has the property 

From the assumption of completeness of the set of all products of N spin-orbitals, each factor being drawn from an orthonormal complete set of spin-orbitals, we may write any wavefunction 

Now consider the coefficient of the product in which the spin-orbitals 

Tlie coelBicients of all spin-orbital products involving the same selection of spin-orbitals (differing only in their order), can therefore differ only in sign. If we use Cab...x for the coefficient of the products in which the in-orbitals occur in a standard order (e.g. dictionary order) and abbreviate this particular ordered set to jc, it follows that 

It will be convenient to include a normalizing factor Af in the function 

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