The independent functions from an orbital product

We saw in Chapter 4 that the number of independent functions is reduced to one if two of the three electrons are in the same orbital. A similar reduction occurs in general. In our five-electron example, if b is set equal to a and c f d, there are only two linearly independent functions, illustrating a specific case of the general result that the number of linearly independent functions arising from any orbital product is determined only by the orbitals outside the doubly occupied set. This is an important point, for which now we take up the general rules. 

Assume we have a set of m linearly independent orbitals. In orderto do a calculation we must have m nl2 + S, where n is the number of electrons. Any fewer than this would require at least some triple occupancy of some of the orbitals, and any such product, S, would yield zero when operated on by utt,. This is the minimal number ordinarily there will be more. Any particular product can be characterized by an occupation vector, Y = [Y Y2 Ym] where y, = 0, 1, or, 2, and 

Clearly, the number of 2 s among the y, cannot be greater than n/2 — S. 

The general result states that the number of linearly independent functions from the set UTT, S(y) i = 1. / is the number of standard tableaux with repeated elements that can be constructed from the labels in the H product. As a general principle, this is not so easy to prove as some of the demonstrations of linear independence we have given above. The interested reader might, however, examine the case of two-column tableaux with which we are concerned. Examining the nature of the tt, for this class of tableau, it is easy to deduce the result using ffVff. This is all that is needed, of course. The number of linearly independent functions cannot depend upon the representation. 

The method for putting together a Cl wave function is now clear. After choosing the y s to be included, one obtains 

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