Origin of structural trends a moments theorem

What is the origin of the structural trends that are observed in Figs. 4.5 and 

6 as a function of the number of valence electrons or electron count N1 For example, why do four-atom molecules with a = show the structural trend from tetrahedron - rhombus - linear chain - square as N runs from 0 to 8 Moreover, why do the curves for the square and the linear chain cross each other twice whereas those for the tetrahedron and the linear chain cross only once To answer these questions we now show that the moments of the eigenspectrum can be related directly to the topology of the molecule as was first done within a framework by Cyrot-Lackmann in 1967. We will then state a moments theorem that allows us to understand the origin of the structural trends in Figs. 4.5 and 4.6. 

we can write the pth moment as the trace over Hp, since 

However, as is well known, the trace is invariant with respect to choice of basis functions that are related by a unitary transformation. Thus, rather than working with the basis of eigenfunctions we may, following eqn (4.31), work with respect to the basis of atomic orbitals, to write 

We see, therefore, that the pth moment of the eigenspectrum is given by the sum over all bonding paths of length, p, that start and finish on the same atom within the molecule. 

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