Rushbrooke equality

A consequence of these findings is that the jr-electron charges are all exactly equal to 1, for the ground state of every alternant hydrocarbon. (Coulson and Rushbrooke, 1940). 

Note that metric collapse requires that stability inequalities that hold elsewhere in Ms must become strict equalities at the critical limit [e.g., (11.120) and (11.140)]. This in turn implies that critical exponent inequalities inferred from such stability conditions [see, e.g., G. S. Rushbrooke. J. Chem. Phys. 39, 842 (1963)] are necessarily equalities, if indeed the critical exponent assumption is valid at all. Such equality in critical-point relationships seems to be supported by all available experimental data, and its justification is straightforward in the metric framework, independent of subsidiary scaling hypotheses (cf. Sidebar 10.4). 

The reader may, however, object that there are a number of molecules, the alternant hydrocarbons which we discussed in great detail in Chapter Six, in which the charge densities, q at all carbon atoms, r, (r = 1,2,..., ) are identically unity, by part 3 of the Coulson-Rushbrooke theorem. Examining the 7r-electron charge at the various sites in such a molecule is, therefore, no longer a way of distinguishing one position from another. For example, is the a-position in naphthalene more reactive than the -position, and, if so, why Such a distinction cannot depend upon the qr, for, as we have just observed, they are all equal. In that case we shall just have to make appeal to the next-highest-order differential—a procedure which introduces a new set of properties, called polarisabilities, which have proved quite important in the study of this kind of system. The word polarisability is rather an unfortunate one, but we shall use it and deal here with so-called atom-atom polarisabilities . 

We see therefore that no part of the Coulson-Rushbrooke Theorem on alternant hydrocarbons depends on having all non-zero Hamiltonian matrix-elements, Hrs, equal. In order for the reader to be quite clear which assumptions, in the context of the simple Hiickel-method, are necessary for the Theorem to hold, we summarise them again below. We require... 

An important corollary to this theorem for alternant n-systems is the Coulson-Rushbrooke theorem, which states that the electronic charge on all atoms in an alternant system is equal to - e. This means that all atoms are uncharged, since one electron was originally removed from each atom to form the i-system. Consider the coefficient matrix... 

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