Alternant Hiickel Hydrocarbons

The effects of various structural factors of the molecular graph on the HOMO-LUMO separation, Ahl, of alternant conjugated hydrocarbons were investigated and some graph-theoretic formulas were obtained for Ahl- As can be derived from equation (77), when computed at the Hiickel level, the absolute hardness r] equals half the HOMO-LUMO separation Ahl as an important consequence of this fact, all theoretical relationships derived for Ahl are also valid for r). An approximation to Ahl for benzenoid hydrocarbons was derived by Cioslowski as a function of simple graph invariants, namely N, M, and KSC ... 

In the HMO or extended Hiickel approach, the individual ionization potentials should be set equal to orbital energies. The inadequacy of the HMO treatment is apparent with odd alternant hydrocarbons (e.g., allyl, benzyl), where a constant value is obtained, in disagreement with the experiment. Streitwieser and Nair (105) showed, however, that reasonable results can be obtained with the co technique. 

It should be noted, however, that planarity had not been excluded for corannulene until the X-ray analysis was performed. The molecule is a non-alternant hydrocarbon because it contains an odd-membered ring. Hence, the it-electron density distribution cannot be uniform in the ground state. One of such polar structures is the double Hiickel aromatic structure 66, c consisting of peripheral 14n and central 6ir systems. The contribution of resonance as shown in 6 c will be at its maximum when the whole molecule is planar like coronene (8). Notwithstanding, the planar corannulene will have huge angle strain. If we assume that all C-C bonds are 1.40 A in length and the... 

A typical failure of the Hiickel method in the case of non-alternant systems is its prediction that all the hydrocarbons XVII-XXIII should be aromatic of these only azulene (XXII) shows aromatic properties. 

Hiickel MOs of alternant hydrocarbons have some noteworthy properties, outlined below. 

The rate of success falls whenever these conditions are not met. For example, condition (1) is not always satisfied in aromatic substitution reactions. The FOs of polycyclic aromatic hydrocarbons are not well separated from the other MOs,66 so subjacent orbital control may intervene. Particular care should be taken with non-alternant hydrocarbons because they tend to react under charge control. This is even truer in heteroaromatic systems (cf. Exercise 15, p. 119, and Exercise 16, p. 121), where Hiickel... 

However, QMe (or B) may vary with type of radical, especially charge type, and also with the nature of the a-atom. Thus, methyl hyperfine splittings in alternant hydrocarbon anion- and cation-radicals, for which Hiickel MO theory predicts identical spin populations at the methylated positions, typically differ by a factor of at least two.15... 

In the case of alternant hydrocarbons it is possible to show that the finite changes in SCF charges, bond orders, and free valences due to changes in the parameters Wf and y (Greenwood and Hayward, 1960), have properties which are completely analogous to those of the corresponding quantities used in the Hiickel approach of Section IVB. The SCF results incorporate those of the Hiickel method as a special case, in which electron repulsion terms can be dropped from the non-linear equations without invalidating the derivations. The theoretical techniques used to obtain the analytical properties are essentially different from those described previously for Hiickel theory, but the result can be stated briefiy in similar terms, and this will sufiice for present purposes. 

We would also like to mention here another important point. Since the Hiickel and topological matrices are closely related for a particular conjugated molecule, all properties of a molecule (i.e. energy, MOs, bond orders, charge densities) which may be derived from the topological matrix by mathematical treatment must be dependent on the molecular topology 23>. This may be one reason why the predictive power of elementary Hiickel theory is in many cases (e.g. for alternant hydrocarbons) as good as that of any more elaborate approach 58>. 

According to Hiickel s rule, annulenes with 4n tt electrons are not aromatic. Cyclobutadiene and cyclooctatetraene are [4n]-annulenes, and their properties are more in accord with their classification as cyclic polyenes than as aromatic hydrocarbons. Among higher [4n]-annulenes, [16]-annulene has been prepared. [16]-Annulene is not planar and shows a pattern of alternating short (average 134 pm) and long (average 146 pm) bonds typical of a nonaromatic cyclic polyene. 

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. 

Consider the Hiickel Hamiltonian-matrix, HI, for an alternant hydrocarbon, constructed on the basis of the simple Huckel-approximations (equations (6-2)-(6-5)) as an example, the matrix, HI, for butadiene is shown in equation (2-54). If the (m) starred atoms are labelled from 1, 2,..., rn, and the unstarred ones from m + 1, m + 2,. .., n, then by an exactly similar argument to that used when discussing the corresponding secular-determinant for an alternant hydrocarbon in 6.3, the matrix HI may be partitioned as in equation (D7). 

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... 

In alternant hydrocarbons, however, the Hiickel bond orders vanish between pairs of atoms in the same set," so that the elements F , like the resonance integrals, vanish unless r and s are in different sets. The upshot of all this, as Pople showed, is that his equations, Eqs. 17, when applied to alternant hydrocarbons yield solutions which have all the pairing properties of the Hiickei molecular orbitals. Pople thus succeeded in showing that the Huckel scheme can be very easily adapted to include electron repulsion, at least under the assumption of zero differential overlap. 

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