Alternant hydrocarbons Coulson-Rushbrooke theorem

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,... 

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). 

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. 

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 . 

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

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. 

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... 

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. 

Symmetry has taken us to a point where still quintic, quartic, and quadratic secular equations must be solved. However, a closer look at this equations shows that they can easily be solved. Apparently, a further symmetry principle is present, which leads to simple analytical solutions of the secular equations. Triphenylmethyl is an alternant hydrocarbon. In an alternant, atoms can be given two different colors in such a way that all bonds are between atoms of different colors hence, no atoms of the same color are adjacent. A graph with this property is bipartite, and its eigenvalue spectrum obeys the celebrated Coulson-Rushbrooke theorem [16]. 

Some reference must now be made to the properties of the basic set of equations for the special case of alternant hydrocarbons. The secular polynomial A e) acquires important analytical properties when all a s and all jS s take common values ao and o> and when, in addition, the molecule is alternant in the sense of the Coulson-Rushbrooke (1940) theorem. The following results are well known ... 

Based on this conclusion one can introduce the concept of the partial electron density and draw its contour map in the plane just above and below, say one Bohr radius, the molecular plane [24, 29, 34]. Again for XVI and XII the results of g3 are given in Fig. 5, which is the contribution of the highest three occupied Huckel MO s. Note that due to the pairing theorem proved by Coulson and Rushbrooke [35] the n-electron densities on all the component carbon 2pn orbitals are the same and the contour map of the conventional electron density cannot differentiate any of the local aromaticity of alternant hydrocarbon molecules. 

Fig. 2-8). We note from Fig. (2-8) that, in the ground state, v, = v2 = 2 and vj = v4 = 0 hence, ail we require from equations (2-67) are cu, r = 1, 2,..., 4 and c2r, r = 1, 2,..., 4. The application of equation (4-4) to these data is then summarised in Table 4-1. Thus we find that the n-electron charge-densities on all four carbon atoms of butadiene are unity. This is by no means fortuitous and is always the case for a certain class of molecules (called alternant hydrocarbons and dealt with in Chapter Six) to which butadiene belongs. The charge distributions in excited-state species will also be discussed in detail in the context of the Coulson-Rushbrooke Pairing-Theorem in 6.5. 

We may also note in passing that the Coulson-Rushbrooke Pairing -Theorem for the energy levels of alternant hydrocarbons (much discussed in Chapter Six and in Appendix D) is nicely illustrated by Fig. B2 (c/also... 

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