Perfect pairing and resonance

The whole of VB theory rests upon one simple formula, which gives the value of the scalar product (0, i) for any two spin-paired functions. We first present this formula and then apply it, first in the 

We continue to use spin-paired functions (rather than the branching-diagram functions used in Section 6.7) in order to derive the standard matrix-element rules employed in simple VB theory. 

Such patterns may not always occur in the matching form , in which all arrows come head-to-head and tail-to-tail but, since any arrow —j 

In the pattern we distinguish islands, each formed by a closed sequence of arrows and chains, each formed by an open sequence the latter are of two types O-chains containing an odd number of centres, and E-chains containing an even number. The required result is then (Cooper and McWeeny, 1966a,b) for normalized spin eigenfunctions 

We note that Pij k = 6 is simply another structure in which the ends of any arrows attached to points i and j have been interchanged, and we may therefore use (7.3.1) to obtain fiy = (0 1 P, 0k) = (0,r x) from the superposition pattern of 0 and 0 i. For interchanges 1 2 and 3 4 the result is trivial, for P,y0. will differ from 0 only by a factor -1 due to one arrow reversal hence 1112 = 34 = —1. A more interesting case is the interchange 23, for which we obtain 

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