Three electron bond system, spin states

Slater s method which has been described in Chapter i6 for the problem of three electrons may be applied to systems with any number of electrons. Each electron may exist in the field of any of the nuclei, and resonance among the different states, representing different electronic distributions, will occur. With four atoms having four valency electrons, 4 = 24 different arrangements of the electrons between the nuclei are possible, with 2 = 16 different spin states. In all there will therefore be 24 X 16 = 384 different complete wave functions. But as we have already seen, states with different values of the spin quantum number do not interact with one another. This permits a considerable reduction in the number of states to be considered. We are only concerned with stable configurations in which all the electrons in pairs neutralize their spin by the formation of a covalent bond, i.e. S—0 and S =0. 

It would appear that for a description of the four electron system it would be necessary to consider the superposition of the three given structures with different localizations of the valency bonds. However, it is still possible to make a further simplification. In the problem of three electrons, three structures also were possible with the bond between a and b and with a free atom c (spin function aj8a—jSaa) with a bond between atoms b and c and a free atom a (spin function aajS—ajSa) with the bond between atoms a and c and a free atom b (spin function aajS—jSaa). But we pointed out that these three functions were not independent, the third being a linear combination of the other two. Let us write down the appropriate functions for the four electron problems describing the states /, II and IIL The space coordinate part of any wave function of four electrons will have the form... 

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