Coulson-Rushbrooke theorem

Tyutyulkov analysed cross-conjugated systems like [14] theoretically, using the Coulson-Rushbrooke theorem and Wannier transformation of the Bloch MO. His results hold multiple significance. First, it was suggested, by taking into account the interaction between the one-dimensional chains of the quasi-one-dimensional polymer, that the critical temperature may reach 10 -10 K (see p. 226). Secondly, the radical centres of hydrocarbons... 

In simple 7t-electron theory the alternant hydrocarbons have some special features. In these planar unsaturated hydrocarbons each second carbon atom is labelled with a star ( ), resulting in a division of the atoms into two sets, the starred and the unstarred, with no two atoms of the same set neighbors. One feature is the so called Coulson-Rushbrooke theorem, or the pairing theorem the bonding (occupied) 7C-orbitals are given in the form,... 

Corrosion inhibitors, [1.2,4]triazino[4,3-ojbenzimidazoles, 59, 155 Coulson-Rushbrook theorem, 55, 273 Coumarins, see l-Benzopyran-2-ones Coupling reactions, trifluoromethyl iodide with aryl halides, 60, 12 Covalent hydration in 6-nitro-l 1,2,4]triazolo[ 1,5-a)-pyrimidines, 57, 107 of coordinated ligands, 58, 138 Creutz-Taube ion, 58, 124 Criss-cross cycloadditions, of... 

The Coulson Rushbrooke theorems have many important consequences that will lead us a long way towards a qualitative understanding of the electronic structure of conjugated molecules, particularly of their excited states. Dewar developed a simple perturbation method PMO theory) to evaluate the HOMO LUMO gap and the associated excitation energy for 7t-systems with an even number of conjugated atoms.284,285,288 Because the NBMO of odd AHs can be determined so easily, the system of interest is dissected into two odd AHs. Some examples are shown in Scheme 4.1. 

In the next chapter we proceed to a discussion of atomic charges, bond orders and free valences, none of which depends on taking any explicit empirical value for a or /J. In Chapters Five and Six we deal with the Hiickel Rule of Aromaticity and the Coulson-Rushbrooke Theorem on Alternant Hydrocarbons, both of which are also independent of any numerical values assumed for these basic parameters. 

An illustration of this aspect of the Coulson-Rushbrooke Theorem is again provided by the alternant hydrocarbon, butadiene. The LCAO-coefficients of the two pairs of complementary orbitals in the molecule display this alternation of sign, as examination of equations (2-67) confirms (atoms 1 and 3 of Fig. 2-6 may be considered, for this purpose, as the starred atoms). 

We now prove the second part of the Coulson-Rushbrooke Theorem that in pairs of complementary orbitals the LCAO-coefficients of orbitals centred on the starred atoms are the same, while those of orbitals centred on the unstarred atoms are the same in magnitude but opposite in sign. Let us consider the 7th orbital of energy e, (let it be, for argument s sake, a bonding orbital) and its associated set of LCAO-coefficients c/rh r = 1, 2,. .., n. These will satisfy a series of secular equations, the rth one of which, on the simple Hiickel assumptions, is... 

For our proof of part 3 of the Coulson-Rushbrooke Theorem we now focus attention on the matrix C (equation (6-27)). We are assuming that the LCAO-MO coefficients are a) normalised, i.e. 

Were r to be a starred atom, we know from part 2 of the Coulson-Rushbrooke Theorem ( 6-4) that... 

Hence, because of the result expressed in equation (6-37), the pairing of MO s and the consequent symmetry of LCAO-coefficients between pairs of complementary orbitals, the charge density on the rth carbon-atom of a neutral, even, alternant hydrocarbon in its ground state, is unity. This is the essence of the third part of the Coulson-Rushbrooke Theorem. Its proof depends on the fact that the square of a quantity is the same as the square of minus that quantity, and is thus seen to be a natural consequence of parts 1 and 2 of the Theorem. 

A more abstract, matrix-proof which, in essence, derives parts 1 and 2 of the Coulson-Rushbrooke Theorem in just one operation, is therefore presented, for the interested reader, in Appendix D. 

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 have just proved that any matrix which can be partitioned as the matrix M in equation (Dl) has a spectrum of eigenvalues which is symmetric about zero and that the eigenvectors corresponding to these complementary eigenvalues differ only in the sign of the last (n — m) components ( coefficients ). It is now a simple matter to show that this is the basis of parts 1 and 2 of the Coulson-Rushbrooke Theorem. 

We now see that parts 1 and 2 of the Coulson-Rushbrooke Theorem are really a special case of the matrix theorem proved in D1 in fact, this matrix theorem is, in its turn, only a special case of the famous Perron-Frobenius Theorem on matrices with non-negative elements (1907-1912)R4T. 

Alternant and Non-Alternant Hydrocarbons The Coulson-Rushbrooke Theorem... 

The remaining sections of this chapter are devoted to formal proofs of these three distinct parts of the Coulson-Rushbrooke Theorem. There are very many and varied (though, of course, equivalent) proofs of part l of the theorem the particular exposition detailed in the next section ( 6.3) will not necessarily be the one which the reader will find the most obvious or straightforward. It does, however, have a certain aesthetic charm which should become evident as the proof progresses ... 

An interesting corollary to part 2 of the Coulson-Rushbrooke Theorem (which does not require assumption c), equation (6-4)) concerns the nodes in the various LCAO-MO s discussed in our sample calculation on the alternant hydrocarbon, butadiene ( 2.10). As an example, we consider just the highest-bonding and lowest-anti-bonding orbitals of butadiene which, in 2.7, we called 4 2 and 4V These are, of course, complementary orbitals their nodal behaviour has been redrawn in Fig. 6-6 which is a simplified... 

Charge distribution in alternant hydrocarbons is the concern of the third (and main) part of the Coulson-Rushbrooke Theorem, to the proof of which we now turn. 

This is the crucial result which will enable us, within a few lines, to prove part 3 of the Coulson-Rushbrooke Theorem. Those readers who have preferred not to follow in detail equations (6-23)-(6-37) may, by accepting the validity of the latter equation, pick up the argument again from here. 

Finally, we note that, once more, we have not made any specific reference to P in proving part 3 of the Coulson-Rushbrooke Theorem hence, this part of the Theorem also does not depend on the numerical value assumed for P, nor does it require that the resonance integrals, / , should be the same for all bonds—i.e., assumption c) (equation (6-4)) is again not necessary. 

Assumptions on Which All Three Parts of the Coulson-Rushbrooke Theorem Depend... 

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... 

A Theorem in Matrix Algebra Embodying Parts 1 and 2 of the Coulson—Rushbrooke Theorem... 

When parts of 1 and 2 of the Coulson-Rushbrooke Theorem are proved as in this Appendix, the necessity of the condition stipulated by equation (6-2) (all Coulomb integrals, acr, to be the same (a), for r = 1, 2,..., n) is even more immediately evident than it was in the previous proof ( 6.3). If just one of the... 

For the ground states of alternant hydrocarbons, for which the Coulson-Rushbrooke theorem tells us that = q, I, it follows that... 

Huckel theory for the even alternant hydrocarbons leads to the Coulson-Rushbrooke theorem and some other characteristic results shown by McLach-lan to be valid also in the Pariser-Parr-Pople model. These are the well-known pairing relations between electronic states of alternant hydrocarbon cat-and anions. This particle-hole symmetry is analogous to the situation discussed in Chapter 4 for electrons and holes in atomic subshells. 

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