Pair theorem

LUMO are paired orbitals according to a (pairing) theorem derived by McConnell and McLachlan (cf. Fig. 6.5). Adding an electron to or removing an electron from an aromatic hydrocarbon occurs with minimal changes in structure. 

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... 

Pairing theorem [46] Distribution of energy levels in benzenoid systems ... 

Pairing theorem again. And how is the characteristic polynomial of a bipartite graph look like Recalling the Sachs theorem one can easily write the expression for Pa (x) ... 

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... 

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