Coulson Rushbrooke pairing theorems

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... 

Coulson bond order, see Bond order Coulson-Golebiewski method, 117-118 Coulson-Rushbrooke pairing theorem, 88-110, 159-166... 

MaUion, R.B., Rouvray, D.H. The golden jubilee of the Coulson-Rushbrooke pairing theorem. J. Math. Chem. 5(1), 399 (1990). See also J. Math. Chem. 8, 1991... 

Before leaving HMO, let us mention an important theorem about HMO energies for alternant systems (% electron systems having no odd rings) due to Coulson and Rushbrooke [90], known as the Coulson-Rushbrooke pairing theorem ... 

There are many remarkable theorems dealing with abstract and realistic models of bipartite lattices [94, 101-104]. The weU-known Coulson-Rushbrooke pairing... 

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,... 

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. 

The above result, usually called the Pairing Theorem, was first proved in 1940 by Coulson and Rushbrooke 20>. There are a number of proofs... 

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... 

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. 

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. 

Coulson and Rushbrooke showed that the 2n n MOs of such an even AH show certain striking regularities, comprised in what is now usually termed the pairing theorem. We assume first that the conjugated atoms have been divided as above into starred and unstarred sets. The pairing theorem can then be summarized by the following statements ... 

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