Entropy and the Reaction Barrier

Proceeding with my goal to read the collected works of Raphael Levine, I recently came across a 1977 paper that he wrote with Noam Agmon, entitled, "Energy, Entropy and the Reaction Coordinate: Thermodynamic-Like Relations in Chemical Kinetics".

I was blown away.

The paper explains a series of empirical conjectures, collectively called "Free Energy Relationships (FERs)". FERs relate the (kinetic) rate of a reaction to its (thermodynamic) driving force. As such, they are thermodynamic-like rules for kinetic behavior. Note that this is strange! Thermodynamics is usually formulated for systems at equilibrium. These are unchanging, by definition, hence there is no rate!

What REALLY impressed me about this work was that they manage to explain FERs without any quantum mechanics at all! Instead, they use a very novel and simple statistical/information-theoretic model.

Agmon and Levine describe a model of a reaction where the progress of the reaction is parameterized by a number n between 0 and 1. This is a point in a 1-dimensional convex set of numbers (i.e. a closed line segment). Because the set is convex, there is a probability interpretation. They define an energy function that has a linear component, and introduce a barrier by adding a term that is just the Shannon entropy of mixing times a generalized "temperature". The entropy creates a barrier, and they proceed to show that this barrier follows the rules expected of LFERs as well as rules relating transition state position to exo/endoergicity (versions of "Hammond's Postulate").

Note that there is nothing quantum here, just probabilities! The convex set was just numbers, not density matrices.

The main idea that I took away is that FER's are more general than one might think (if one was trained as a "traditional" quantum chemist). They don't depend, for example, on details like how a transition state is defined for a molecule (or a series of molecules with similar shape).

I have been contrasting this in my head with another paper by Sason Shaik (another Israeli chemist of who I am a fan) and AC Reddy. It is not clear to me just yet how to bring the two pictures together, since the Shaik model of crossing VB surfaces intrinsically invokes two energies at the ends of the reaction coordinate (one for each surface). The Shaik/Reddy picture ultimately produces two potential surfaces with a diabatic interpretation. Also, it is clearly wavefunction-based. The Agmon/Levine model cannot be wavefunction based because the Hilbert space is not a convex set. It could be generalized to density matrices, but then it is not immediately clear how to discuss the "twin state" invoked by Shaik & Reddy. Since the states in the Shaik/Reddy picture are VB states and may be non-orthogonal, the definition of an appropriate mixing entropy is also not clear (see, for example, a paper by Dieks and van Dijk).