Pairing theorem Alternant hydrocarbon

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. 

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... 

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. 

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... 

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. 

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. 

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... 

By an odd-alternant hydrocarbon is usually meant a free radical, such as the benzyl radical (Fig. 6-3). In this example there are seven carbon-atoms, hence seven atomic-orbitals and seven molecular-orbitals which must accommodate the seven jt-electrons. Of the seven energy-levels, six of them will occur quite naturally in pairs, according to the Theorem we have just proved, as shown schematically in Fig. 6-4. How can the seventh be... 

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. 

As a matter of fact, the hole and particle occupancies are identical for any bipartite networks treated within Jt-approximation, up to FCl/PPP. This is a simple corollary of the generalized pairing theorem of McLachlan [94] stating that the jr-electron charge density matrix of the alternant hydrocarbons is of the form... 

These conclusions can also be interpreted in terms of the m - n rule developed above and are a direct consequence of the properties of the Eq. (1) matrix. The Pairing Theorem does not completely define the spectrum of molecular orbitals in an alternant hydrocarbon, because (1) is consistent with the presence of an even number of molecular orbitals. Whilst this might generally mean that the number of non-bonding molecular orbitals is 0, there are instances when there are 2, 4, etc. For example, both butadiene and cyclo-butadiene are even alternant hydrocarbons, but the former has no non-bonding molecular orbitals and the latter 2 (see Fig. 13). 

In the case where the hydrocarbon is an even AH, the coefficients and will differ at most in sign (pairing theorem). Consequently, so both MOs are depressed equally in energy by the heteroatom. The lowest n- transition should therefore remain unchanged. The first n n absorption bands of even, alternant, heteroconjugated molecules should therefore appear in the same place as those of the isoconjugate AHs—and they do. The UV spectra of benzene and pyridine (Fig. 6.13) illustrate this well, the first absorption maxima being at 254 and 258 nm, respectively. 

Alternant hydrocarbons exhibit certain symmetry properties which are reflected by the special arrangement of their orbital energy levels and the specific form of their molecular orbitals. These special properties are often referred to as the pair-theorem or, using the terminology of physics, it is called the particle-hole symmetry. It holds under the first-neighbor approximation in the 7c-electron system. In the name of this symmetry, the term particle refers simply to the electrons of the molecule. We say that the occupied levels are filled with particles. 

Uncharged perimeters with 4N+2 jt electrons are alternant hydrocarbons and the pairing theorem (cf. Section 1.2.4) applies 0 and 0 are paired, as are and 0a> their coefficients are equal on all odd atoms and... 

Bolton, J.R., Fraenkel,G.E. Electron spin rescmance study of the pairing theorem for alternant hydrocarbons. J. Chem. Phys. 40, 3307 (1965). 

Charged alternant hydrocarbons occur as pairs of negatively and positively charged ions, and the pairing theorem relates the states of the former to the states of the latter. While at the standard PPP level of theory the absorption spectra of the members of a pair are predicted to be identical, and indeed experimentally almost exactly are, their MCD spectra are expected to be mirror images of each other and indeed nearly are (e.g., diphenylmethyl anion and cation). ... 

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