This post is being presented as a background explanation for the post above. If you have no interest in chemistry, ignore this post.
The Hammett equation is an empirical relationship that relates the effect of a distant substituent in a molecule with the reactive centre. Such a centre might react with something, or be in equilibrium with another form. An example of the latter could be an acid or an amine, which could be in equilibrium with its ionized form, thus
X – H ⇋ X- + H+
Now, further suppose X is part of a molecular structure where some distance away there is a substituent Y, in which case
Y—-X – H ⇋ Y—-X- + H+
and —- is the hydrocarbon structure separating X and Y. Now it is observed that different Y can alter the equilibrium position or rate of reaction of the function X, and the reason is, the substituent alters the potential energy of electrons around wherever it is attached, and that alters to a lesser degree the potential energy around the next carbon atom, and so on. Thus the effect attenuates with distance. There are two complicating factors.
The first is what is called (in my opinion, misleadingly) electron delocalization. It is well known that carbon-carbon double bonds can “delocalize”. What that means is, if you have a molecule with a structure C=C-C(+/-) , where (+/-) means there is either a positive or negative electric charge (or the start of another double bond) on the third carbon atom, then the molecule behaves as if it were C-C-C with two additional electrons that make the bonds effectively 1.5 bonds, and half the charge is at each end. That particular system is called allyl.
The Hammett equation
log(K/Ko) = ρσ
relates the effect of distant substituents on a reactive site. Here K is a rate or equilibrium constant, Ko is a reference constant (to make the bracketed part a number – you cannot have a logarithm of apples!) and this should give a straight line when the logarithm is plotted against the axis that measures σ. ρ is the slope of the line, which attenuates as the path between substituent and site increases provided we assume each intervening chemical bond is localized. If so, σ is a specific value for the substituent. The straight line results because the values of σ are assigned to the substituent to ensure you get a straight line provided the attenuation is dependent on the substituents always having the same values of their assigned σ. What you are doing is empirically relating the attenuation of the electric potential change caused by the substituent at one point by the time it reaches the reactive site. Of course, there is always scatter, but it should be random.
To understand why delocalization becomes relevant, we have to consider what is actually meant. In chemistry textbooks you will often see mechanisms postulated to explain what is going on, and you will see electron pairs moving about. Electrons do not hold hands and move about, and they are never “localised” in as much as they can be anywhere in the region of the molecule at any given instant. The erroneous concept comes from the Copenhagen Interpretation of Quantum Mechanics, whereby the intensity of the wave function gives the probability of finding charge. The chemical bond arises from the interference between two wave functions and the interference zone has two electrons associated with it, and if you agree with my Guidance Wave Interpretation, half the periodic time (because a wave needs a crest and a trough per period, and in the absence of a nodal surface, which is only generated in the so-called antibonds, you need two electrons to provide both in one cycle). What I consider to be localised is not the electrons but the wave interference zone. If you follow the Copenhagen Interpretation, such an interference zone represents regions of enhanced electron density. If the wave interference zone is restricted to a certain volume of space, that characteristic space in the molecule conveys characteristic properties to the molecule because there is enhanced electron density at a lower potential within that region.
Why does it become localised at all, after all the waves can go on forever. The simplest answer is that because of molecular structure, e.g. the carbon atom has four orbitals directed towards the corners of a tetrahedron because that is the optimal distribution to minimize electron repulsion between the four carbon electrons. Interference to create single bonds is “end-on”, in which case for the wave to proceed its axis has to turn a corner, and it cannot do that without a change of refractive index, which requires a change of total energy. However, the allyl system, and a number of others, can delocalize because the axes of the orbitals are normal to any change of direction, and the orbitals can interfere sideways (i.e. normal to the orbital axes) as opposed to the end-on interference in single bonds. So, to get delocalization, the bonding must involve sideways interference of atomic orbitals, while the single bonds are invariably end-on. The reason why cyclopropane was of interest is that if the atomic waves have axes directed towards a tetrahedron, and an angle of 109.4 degrees between them, and since the resultant structure of cyclopropane perforce has angles of 60 degrees between the inter-atomic axes, then either there is partial sideways interference, or the bonds are “bent”. The first should permit delocalization; The second is ambiguous.
If we now reconsider the Hammett equation, we see why it is a test for delocalization. First, if there is delocalization, the value of ρ increases because there is no attenuation over the delocalized zone (i.e. overall the distance has fewer links to attenuate in it). There is, of course, the base value of how much change a substituent can cause anyway. Now, in the cyclopropyl systems I discussed in the previous post, the cyclopropane ring gave a value of ρ that was about 30% higher than a C – C link. My argument was that this is expected if there is no delocalization because there are two routes around the ring, and the final effect should be the sum of the two routes, which is what was found.
The value of σ also changes with delocalization for a limited number of substituents, namely those that can delocalize and amplify a certain effect if demanded. An example is if the reactive site generates a demand for more electron charge, then a substituent such as methoxyl will supply extra by delocalizing its lone pairs on the oxygen, or alternatively, if the demand is to disperse negative charge, a nitro group will behave as if it takes on more. Thus the effect of a limited number of substituents can address the question of whether there is delocalization. The saddest part of the exercise outlined in the previous post is that the first time it was ever deployed to answer a proper question, those who used it on the whole did not seem to appreciate the subtleties available to them. For the ionization of the 2-phenylcyclopropane carboxylic acids, the results obtained in water were too erratic, thanks to solubility problems. The results of reactions in ethanol had an acceptable value of ρ to get a result, but the authors overlooked the effect of two routes, and did not bother to examine the values of σ.