# Why is the original Pauling's theory of resonance that uses superposition of wavefunctions not used today?

I have recently studied superposition of states (also the famous Schrodinger's cat), measurement problem, decoherence & so on. I then read the resonance concept from Atkins' Physical Chemistry where it was written as:

It is the superposition of wavefunctions representing different electron distributions in the same nuclear framework. [...] A better description of a molecule of $$\ce{HCl}$$ is $$\psi_\ce{HCl} = \psi_{\ce{H-Cl}} + \lambda\psi_{\ce{H^+} \ce{Cl^-}}$$.

Thus this is the general wavefunction which is superposition of all the states. $$\lambda$$ is a numerical coefficient (can be found out by variation theorem) whose square predicts the probability of being in the ionic state when we inspect the molecule. Just like typical wavefunction-collapse after measurement! This is the original Pauli's theory of resonance. Unfortunately Atkins halts down here; after all it is physical chem-book! Nowhere have I found this approach to resonance; no organic chem book I possess: Morrison, Boyd; Solomons; R.K. Gupta ... They all say it is just the weighted average of all Lewis structures. No, it is not correct! As is said here:

This essay contains information that I have never found in any organic chemistry textbook. This can mean two things: the information is too difficult or you don't really need to know it. Maybe both are true .

[...] When Linus Pauling developed resonance theory, he defined "resonance" as the superposition of wave functions. Unfortunately, this detail has been lost in the intervening decades, and modern chemists incorrectly treat "resonance" as the superposition of Lewis structures (resonance forms)

[...]Suppose you have a molecular AB that is a resonance hybrid:

and you want to calculate the electron density on atom A.

The usual (wrong) prediction procedure. The usual procedure says, "combine what you see in each Lewis structure." In other words, combine the electron density (ED) on A in form IV with the electron density on A in form V:

$$ED_\text{wrong} \text{(hybrid)} = 1/2 [ED(IV) + ED(V)]$$

The much-ignored (correct) prediction procedure. You may recall that electron density (ED) is given by the square of the wave function (WF). If we want to know the electron density on A, we must look at the square of the wave function near A:

$$ED\text{(hybrid)} = WF\text{(hybrid)} ^2 \implies ED\text{(hybrid)} = ED_\text{wrong} \text{(hybrid)} + \boxed{WF(IV)WF(V)}$$

The final formula shows the discrepancy between the right answer, ED(hybrid), and the usual (wrong) answer, EDwrong. The correct formula includes WF(IV)WF(V). This term is called the "interaction density" or "overlap density," and is a product of two different wave functions. The interaction density affects the electron density everywhere in the molecule, and its appearance tells us that the electron density obtained by the "usual" procedure is wrong.

Electron density is just one of many properties that interest chemists, but all property predictions -- electron density, bond type, atom charge, dipole moment, geometry, energy, etc. -- have something in common with electron density prediction: the prediction procedure must start with WF(hybrid), and their answers always include interaction terms.

This is one of the great reasons why the modern concept of resonance is wrong; they don't take the "interaction term"!

So, why does only one book (which is not organic) contain the correct interpretation? Why do people follow the wrong version? Resonance is not at all the weighted average of Lewis structures; rather it is just a superposition of wavefunctions? So, why do most of the sources (people & book) advocate the wrong theory? Can anyone tell me which organic chem book still follows Pauling's original theory of resonance?

Added: In my quest to find the resources which still define Pauling's theory of resonance of wavefunctions, Feynman's Lectures, which I have just completed reading, is really worthy to be gold to be noted. Feynman completely describes the benzene molecule as the superposition of base states; it can be found in the CALTECH site.

• If you look for the term "valence bond theory" you'll find more books that deal with the full/correct approach. I have a colleague who's specialty is applying V.B. theory in computational organic chemistry, so there are certainly people who understand and use this approach. However, there was a historical shift with the introduction of molecular orbitals theory where many people abandoned V.B. theory (M.O. calculations scale better for larger molecules). Unfortunately, this means many chemists only have a vague/qualitative understanding of V.B. theory. Mar 22 '16 at 19:02
• My two cents: when the concept of resonance is first introduced in general chem, or organic chemistry courses, describing it as a weighted average of resonance structures, is easier both for the instructors as well as for the students to gain a working intuition. Most curricula introduce quantum theory a little bit later. Mar 23 '16 at 5:41
• If I recall correctly VB calculations scale extremely poorly, and not really used with anything bigger then a benzene molecule. In other words, VB may sound like a good idea but full VB treatment makes no practical sense.
– Greg
Jan 7 '19 at 23:09

why does only one book (which is not organic) contain the correct interpretation?

I bet that there are many books with correct descriptions of it. Almost any computational chemistry book that makes mention to the subject.

Why do people follow the wrong version?

It depend of the individual. Many because ignorance. Others because they are lazy and prefer do not immerse in modern (¿1920-1930?) ideas of quantum mechanics. Others both of them.

Resonance is not at all the weighted average of Lewis structures; rather it is just a superposition of wavefunctions? So, why do most of the sources (people & book) advocate the wrong theory?

I am unsure about the exact meaning of sentence

It is the superposition of wavefunctions representing different electron distributions in the same nuclear framework.

In principle, any state (approximation to a solution of Schrödinger equation restricted to an antisymmetric subspace) is different to other state because the electron distributions are not the same.

Even more, eigenfunctions of the Hamiltonian operator are vectors that form a complete set. That implies that any state can be represented by a linear combination ("superposition") of eigenvectors (wavefunctions for example).

So "resonance" defined in such way is too vague. Arises so many question with this definition: When a superposition implies resonance? The wavefunctions used for the combination, correspond to a stationary state? Is Atkins speaking about a particular kind of difference in electron distribution?

There is an awesome amount of literature about general chemistry, inorganic and organic chemistry with very serious mistakes, conceptual mistakes. Almost every book I've read in the subject. I do not believe that they are wrong because of simplification in post of clarity, because I can realize in which sense is good clarity about something that is completely wrong and not useful. Speaking about in-existent thing that contradicts quantum mechanics principles and at the same time do not allow make any prediction is not useful imho.

Can anyone tell me which organic chem book still follows Pauling's original theory of resonance?

Not me, sorry, I am not familiar with organic textbooks. I've seen some titles like "Theoretical and Computational Models for Organic Chemistry" that suggest a good theoretical background, but I can not assure it. I recommend you do not pay too much attention to resonance.

• Ha! A nice answer; actually I was quite ignorant then; I had to learn that resonance is the weighted average of all the canonical structures. But in reality, as we all know (which, to me, was kind of revolutionary new idea then), resonance is the quantum superposition of wavefunctions describing different base states- this is greatly summed up in Feynman's Lectures & of course, in 'almost any computational chemistry book'. But really my main concern was the undergrad Organic texts which lack that description. Now, I'm pretty much more literate than the then me & I know quite many books ...
– user5764
Mar 22 '16 at 1:49
• now which bear this more accurate definition of resonance. Though till now, I've not found any undergrad Organic text that deals with this definition- may be due to keeping the facts straight rather than rigorously analysing. I was a bit lazy not to delete this question but, well, it was destined for your answer ;)
– user5764
Mar 22 '16 at 1:52
• Thanks. To my understanding those non rigorous ideas are useful to predict qualitatively with good probability of success. They allow to skip a rigorous treatment that can be very time consuming and sometimes technically impossible to afford up to date. It is not bad to use them, but it is convenient to be aware of the inaccuracies of this ideas/models. Mar 22 '16 at 16:00
• In my experience it is hard to find people that likes both approach, I suppose that this is the reason why there are not many good books in this sense (I do not know anyone). For the fast progress of chemistry the usage of both approach is a bless. :-) . But for "interdisciplinary" works, sometimes can be very difficult understand each other views, I certify that ;-). Mar 22 '16 at 16:04
• But what are the implications of seeing it as a "superposition of resonance structures" rather than a "superposition of wavefunctions"? Other than the fact that it is inconsistent with the original theory and that it provides a wrong estimate of electron density, as illustrated in the question post above? Oct 29 '19 at 0:27

# TL;DR:

It is a useful tool to use experimental values to skip math (and math that is tedious even without considering inter-molecular/environmental interactions like VDW and adsorption)

resonance concept from Atkins' Physical Chemistry where it was written as: It is the superposition of superposition of wavefunctions

$$\psi_\ce{HCl} = \psi_{\ce{H-Cl}} + \lambda\psi_{\ce{H^+} \ce{Cl^-}}$$.

That's a good interpretation, but a slightly incomplete one again. In here, if $$\ce{HCl}$$ is the hybrid form we want to study, it exists in a combination of:

• a "perfect" covalent bond: $$\ce{H-Cl}$$ : where the electron cloud is uniformly distributed.
• a "perfect" ionised form: $$\ce{H^+}$$ separated from $$\ce{Cl^-}$$ sufficiently for their local electron wave functions to not interact (yes, yes, WFs are infinite - and this separation distance should be infinite too - hence the quotes around "perfect")
• and a set of states where the electron cloud around the bond is shifted from ever-so-slightly to heavily polarised.

In effect, you're dealing with an integration problem over $$N$$ states where each state has a probability of $$\lambda i\,\, (1 \Leftarrow i \Leftarrow N)$$

incorrectly treat "resonance" as the superposition of Lewis structures (resonance forms)

Again, true, but this gives us three important advantages:

• It allows us to represent most bonds/moieties as discrete extreme states. Think of the simplistic representation of the benzene ring and the electron jumping around the circle alternating between a polar $$\ce{C^+ - C^-}$$ bond and a $$\ce{C=C}$$ double bond.

• Experimentally observed bond energies and polarisation constants give us a good relative approximation to allow us to represent the whole cohort of possible intermediate waveforms as a single:

$$\psi\textrm{(bond)} = \psi\textrm{(extreme-state 1)} + \lambda \psi\textrm{(extreme-state 2)} + \,\,\textrm{( more if the bond requires it)}$$

• And given the number of molecules being observed, we can assume that some of molecules must be in these extreme states - and describe chemical reactions in familiar terms. (Bonus: With the tendency of the molecule to revert to a stable superimposition state, we can further explain why reactions that rely on said marginal states can fully use up the reactant)