The Free-Valence Index

One possible approach to the study of chemical reactivity is to determine the degree that the atoms in a molecule are bonded to adjacent atoms relative to their theoretical maximum bonding power. If particular atoms are not much bonded compared to the maximum, we could say that they have considerable free valence and especially reactive positions. Coulson defines a free valence index, atom i as follows  

At first Coulson chose 4,5 for the maximum possible bonding power (in bond order units) because this was the highest degree of bonding he encountered in the usual calculations however, the theoretical maximum is easily shown to be 4. 732. Consider the molecule trimethylenemethane  

The central atom of trimethylenemethane is bonded by three (T bonds and three ir bonds to its neighbors. Since the CHz 

Calculate the energy levels, DE, and bond orders for symmetrical trimethylenemethane. Verify that Spij for the central atom is 4. 732. Calculate an energy of conversion of trimethylenemethane into methylene-cyclopropane. 

From the 3 values we can argue that butadiene could well be more reactive to neutral nonpolar reagents, such as free radicals at the 1 and 4 carbons, than at the 2 and 3 carbons. Neutral nonpolar reagents are specified here so as to avoid commitments that might have to be modified later by consideration of charge distribution effects. 

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Both the bond character and the free valence index (but not the energy) depend in principle on the choice of the canonical configurations. In general, however, it is easy to make a proper choice on the basis of chemical intuition. In naphthalene the results for both quantities are as given below (see also Table 24 and p. 210). o.I0... 

We can also give a free valence number (quite different from the free valence index in V.B.) for atom i for this method of approximation (see also pp. 269 and 282) in the form Ii — C — 2 Pij, that is to say, by taking the sum of the bond i... 

The free valence index was proposed as a measime of the residual valency of the ith atom in Ji-electron molecular orbitals [Coulson, 1946] it is defined as ... 

The free-valence index is actually found to be of greatest importance in free-radical reactions. Early work by SwareRl° investigated the rate... 

Thus, we see that the free valence index of the ith position is meaningful when a radical reagent produces a perturbation of p.j as well when it is far away. 

The free valence indexes 3. are calculated by the following equation (cf. pp. 56-57) ... 

Fig. 1. — Calculations by molecular orbital method, Delocalization energies (DE)are given below each formula, the bend orders (p) are shown by figures near each bond and the free-valence indexes (F)for each position are placed at arrow points. The letter T denotes a predicted tripletground state. The free valence indexes given for XXIV—XXX are those calculated for the free radicals.

Another example of the molecular reactivity-structure relationship may be given using the notion of the free valence index introduced by Coulson. 

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