The Pairing Theorem

In our simple MO approach, the 7t MOs of an even AH are written as linear combinations of the 2p AOs (/ , of the contributing carbon atoms. 

As in the case of methane (Section 1.10), the In AOs of the In carbon atoms represent a volume of orbital space capable of holding In pairs of electrons the MOs derived by scrambling them [equation (3.1)] should therefore between them also be able to hold In pairs of electrons. Consequently, our 2n AOs should give rise to 2n MOs. 

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  

Pairing Theorem for Even AHs. (1) The 71 MOs of an even AH appear in pairs S and O , with energies a -h and a — ol being the Coulomb integral for carbon. 

The MOs of AHs thus appear in pairs, spaced equally from the central level, a, in energy (Fig. 3.2). If one of the MOs of a pair is expanded in terms of the AOs 0 of starred atoms and (j)j of unstarred atoms. 

When Oc = Oi = a, the pairing theorem which has been widely used in discussion of alternant conjugated organic molecules [15-19] is applicable to the system above because the magnitude of each eleinent in R of Eq. (2) does not affect the generalisation of this 

The octahedral MLg is an example where the approximation is perfect. The resonance integrals, (Lm H (()in,), can be obtained by using the following equations  

The overlap integrals between the L s and atomic orbitals of central atom in ML octahedral structure are  

It is noteworthy that the stabilisation energies are always NP /Degeneracy where N = Number of ligand atoms. When the assumption Po(s) = Pa(p) = p ,(d) has been made, the orbital interaction diagram for this idealized situation (Oj = Op = aj = oi) is shown in Fig. 2. 

An alternant hydrocarbon can be labeled with asterisks to demonstrate the existence of two sets of carbon centers in the molecule such that no two atoms in the same set are nearest neighbors (see Section 8-9). The following discussion pertains to alternant systems. 

The simultaneous equations leading to Hiickel energies and coefficients are of the form 

If we have already found a value of x and a set of coefficients satisfying the simultaneous equations (A5-4), it is easy to show that these equations will also be satisfied if we insert —x and also reverse the signs of the coefficients for one set of centers or the other. If we reverse the coefficient signs for the set j, k, /, we obtain, on the left-hand side 

This proves that each root of an alternant hydrocarbon at x( 0) has a mate at —x and that their associated coefficients differ only in sign between one or the other sets of atoms. Note that, if x = 0, the c/ term vanishes, leaving coefficients for only one set of centers. Reversing all signs in this case corresponds to multiplying the entire MO by — 1, which does not generate a new (linearly independent) function. Thus, it is possible for an alternant system to have a single, unpaired root at x = 0. It is necessary for odd alternants to have such a root. An even alternant may have a root at x = 0, but, if it has one such root, it must have another since, in the end, there must be an even number of roots. 

A5-2 Demonstration That Electron Densities Are Unity in Ground States of Neutral Alternant Hydrocarbons 

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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 pairing theorem requires that the HOMOs and LUMOs of the alternant molecules anthracene and naphthalene are localized on the same atoms this is not the case for the nonalternant azulene. For the detailed calculations concerning the colordetermining interelectronic repulsion, see the paper by Michl and Thulstrup.8... 

Structure 12 is alternant, so we can apply the pairing theorem to evaluate the LUMO. The two FO are symmetrical about the energy a. The starred atoms have the same coefficients in both orbitals and the non-starred coefficients are equal but opposite in sign. 

One such regularity has already been mentioned, namely the Dewar — Longuet-Higgins formula, Eq. (1). Bearing in mind Eqs. (2) and (3) as well as the pairing theorem... 

Now compare the conclusions deduced to what was said in Chapter 2 and you will see that what has just been presented is nothing else but the pairing theorem formulated in terms of the graph theory. 

To illustrate the effect of solvation on temporary anions we will consider the naphthalene molecule. This molecule is particularly interesting because it is an alternant hydrocarbon (14), and for such molecules, the pairing theorem (15) predicts that the anion and cation spectra should be identical. This theorem is valid for both Huckel and PPP model Hamiltonians, but is not valid for ab initio or CNDO calculations. It has been found (1 ) to be true to a good approximation ( /0.1 eV) in organic glasses (16). The ETS spectra allows an examination of the validity of this... 

Figure 4. Anion excitation energies in eV for naphthalene. (ETS) Energies derived from electron transmission measurements in the gas phase (Soln) optical absorption studies in anions in MTHF glass (PPI, Cl) theoretical energies (PT) values derived from the cation spectrum by applications of the Pairing Theorem...

Due to the pairing theorem, the absorption spectra and transition polarization directions of two mutually paired alternant systems should be identical, as shown in Figure 2.25 for the radical anion and the radical cation of tetra-cene. Under the same conditions (i.e., = 0 if and v are nonneighbors),... 

From this mirror-image theorem it follows that MCD spectra of uncharged alternant systems that are paired with themselves should be zero. This is only true within the confines of the above approximations, in which the y." contributions to the MCD are neglected. The y contributions are in fact zero for uncharged alternant hydrocarbons since from the pairing theorem it follows that AHOMO = ALUMO. 

Graph Theory can also be applied directly to quantum chemistry a good illustration of this is the graph theoretical derivation of the Pairing Theorem, derived earlier by Coulson and Rushbrooke20) in a different way. 

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

Condition (83) is automatically fulfilled for AHs because of the Pairing Theorem. Moreover, condition (83) also holds for the majority of NAHs (but, of course, not for all). Since... 

Excellent correlation was also found between first ionization potential Ix v and the HOMO energies obtained by a slightly modified Hiickel calculation for a large series of benzenoid hydrocarbons (Koopmans theorem, Section 4.7). The ionization potentials /lv are also strongly correlated with the energies of the La bands, as would be expected on the basis of the pairing theorem (Section 4.6). A few outliers were noted, but a reinvestigation... 

If the transition considered is the HOMO LUMO transition of an alternant hydrocarbon, then first-order theory predicts that inductive perturbation will have no effect at all, because for = fo as a consequence of the pairing theorem. Small red shifts are in fact observed that can be attributed to hyper conjugation with the pseudo-7t MO of the saturated alkyl chain.290 On the other hand, alkyl substitution gives rise to large shifts in the absorption spectra of radical ions of alternant hydrocarbons whose charge distribution is equal to the square of the coefficients of the MO from which an electron was removed (radical cations) or to which an electron was added (radical anions), and these shifts are accurately predicted by HMO theory.291... 

This latter assumption is not actually necessary for the Pairing Theorem to hold but for simplicity we shall make the assumption here and examine it later.)... 

One of the most interesting and important of these theorems is known as the Pairing Theorem. In his study of the polyenes, and annulenes, Huckel demonstrated [149] that thdr energy levels... 

Using the Pairing Theorem and assuming that the HMO coulomb integral, a, is zero, EXCj can be expressed in terms of HMO energies, and e., as follows ... 

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