More on More O'Ferrall-Jencks Diagram

Transition State Theory

Arrhenius

In 1889, Svante Arrhenius proposed the first quantitative relationship between reaction rate and temperature.

His empirical equation introduced the concept of activation energy (\(E_a\)), a barrier molecules must surmount to react. However, Arrhenius’s model was macroscopic and provided no insight into the internal reconfiguration of bonds or the geometry of the species at the peak. While the pre-exponential factor (\(A\)) accounted for collision frequency and orientation, it lacked a derivation from first principles, a gap later addressed by Transition State Theory.

Eyring, Polanyi & Potential Energy Surface

Around 1930-31, Henry Eyring and Michael Polanyi applied QM to the hydrogen exchange reaction (shown below). By assuming a collinear geometry (the three atoms in a straight line), they reduced the system’s degrees of freedom to allow for a 3D visualization.

Using the London equation for the potential energy of a three-atom system and incorporating experimental data on vibrational frequencies and dissociation energies (a precursor to the LEPS (London-Eyring-Polanyi-Sato) method), they constructed the first potential-energy surface (PES). This was a three-dimensional diagram where the vertical axis represented potential energy, and the two horizontal axes represented the internuclear distances between the three hydrogen atoms.

This idea turned out to be revolutionary. For the first time, chemists could visualize a reaction not as a discrete jump between states, but as a continuous motion of a point mass rolling across an energy landscape. It is also highly intuitive, the reactants occupied a deep valley, the products another, and separating them was a "saddle"-shaped region.

Pelzer, Wigner & The Saddle Point

A year later, Hans Pelzer and Eugene Wigner refined this interpretation. They explicitly identified the “col” or “saddle point”. Along the Reaction Coordinate, transition state (TS) is the highest energy point on the minimum energy path and perpendicular to the reaction coordinate, TS is a minimum. It sits at the bottom of the valley perpendicular to the path. Knowing this will come handy as we discuss Parallel and Perpendicular effects in More O’Ferrall-Jencks Diagram.

Absolute Reaction Rate Theory

In 1935, Henry Eyring formalized Transition State Theory (TST) by introducing the quasi-equilibrium assumption. This postulate bridges the gap between kinetics and thermodynamics by treating the activated complex as if it were in a state of equilibrium with the reactants.

While a true equilibrium is impossible given the activated complex is constantly crossing the barrier to form products, TST assumes that the transition state species maintains a Boltzmann distribution of energy levels for all degrees of freedom except the reaction coordinate. This allows the concentration of the transition state to be expressed using an equilibrium constant, \(K^\ddagger\). By applying this assumption, Eyring replaced the empirical pre-exponential factor (\(A\)) from the Arrhenius equation with fundamental physical constants. He defined the rate constant \((k)\) as:

where \(T\) is temperature, \(k_B\) is the Boltzmann constant, \(h\) is Planck's constant, \(\kappa\) is the transmission coefficient, and \(K^{\ddagger}\) is the equilibrium constant for the formation of the transition state.

By expressing \(K^{\ddagger}\) in terms of thermodynamic quantities, Eyring linked the rate of reaction to the entropy (\(S^{\ddagger}\)) and enthalpy (\(H^{\ddagger}\)) of activation, making the relation between kinetics and thermodynamics explicit.

However, while TST can calculate a rate if the PES is known, it cannot predict how that surface will shift if the chemical structure is modified. This gap is filled by arguably the most famous heuristic in physical organic chemistry: the Hammond Postulate.

The Hammond Postulate

George Hammond, published his seminal work, "A Correlation of Reaction Rates" in 1955. He proposed that one could deduce the structure of the elusive transition state by looking at the stable species closest to it in energy. In his own words:

If two states, as, for example, a transition state and an unstable intermediate, occur consecutively during a reaction process and have nearly the same energy content, their interconversion will involve only a small reorganization of the molecular structures.

The consequences of this heuristic are obvious. In an exergonic reaction, the transition state is closer in energy to the reactants than to the products. Therefore, the transition state structurally resembles the reactants. This is termed an "early" transition state. Conversely, in endergonic reaction, the transition state is closer in energy to the products. Therefore, the transition state structurally resembles the products. This is termed a "late" transition state.

The Mountain Pass Analogy

The intuition is best captured by the analogy of crossing a mountain range. If one travels from a high plateau (high energy reactants) to a low valley (low energy products), the highest point of the pass is encountered early in the journey, near the plateau. Conversely, if one climbs from a low valley to a high plateau, the summit is reached late, just before stepping onto the plateau.

However, this intuition is not absolute. As illustrated in Modern Physical Organic Chemistry by Anslyn (Ch. 7, Q. 26), the postulate “fails” when the potential energy surface contains relatively flat sections. In the context of Marcus analysis (which I won’t cover here), this corresponds to parabolas of significantly different widths; specifically, an exergonic reaction can exhibit a “late” transition state if the energy surface near the reactants is broad and flat.

The Leffler Equation

It’s interesting to note that two years prior to Hammond’s publication, John E. Leffler proposed a mathematical formulation of this principle, essentially a quantitative version of the postulate. Leffler suggested that the properties of the transition state could be described as a linear combination of the properties of the reactants and products. This is encapsulated in the Leffler equation (often linked to the Brønsted catalysis law):

where, \(\Delta G^{\ddagger}\) is the change in activation free energy caused by a substituent. \(\Delta G^\circ\) is the change in the overall standard free energy of reaction and \(\alpha\) is the proportionality constant (\(0 \lt \alpha \lt 1\)).

Coefficient of \(\alpha\) serves as a coordinate for the transition state along the reaction path. For instance, \(\alpha \approx 0\) indicates the TS is reactant-like (Early), and \(\alpha \approx 1\) indicates the TS is product-like (Late).

This relationship, often visualized as a Linear Free Energy Relationship (LFER), provided the first axis for what would become the More O'Ferrall-Jencks diagram. It described movement along the reaction coordinate. However, it treated the reaction path as fixed. It could not describe what happens when the mechanism itself shifts sideways such as, when an \(S_N2\) reaction shifts towards more \(S_N1\) character.

More O'Ferrall, Jencks & Reaction Map

By the late 60s, the limitations of the single reaction coordinate were becoming clear. Mechanistic studies on \(\beta\)-eliminations and nucleophilic substitutions showed a continuum of behaviors. A reaction might be "mostly \(E_2\)" but with "some \(E_1cb\) character". To describe this, chemists needed a map that allowed for deviation from the central path.

In 1970, R.A. More O'Ferrall published a paper specifically addressing the relationship between \(E_2\) and \(E_1cb\) mechanisms. He proposed a two-dimensional plot where the axes represented the two bond orders changing during the elimination: the breaking of the C-H bond and the breaking of the C-LG (leaving group) bond.

This 2D representation converted the "reaction path" from a line into a vector on a plane. The transition state was no longer a fixed point on a fixed line; it was a saddle point that could drift across a surface defined by the two independent bond coordinates.

William P. Jencks, a giant in the field of catalysis, adopted this diagrammatic method to clarify the mechanisms of general acid-base catalysis and addition reactions to carbonyls. Jencks recognized that the ambiguity in these reactions, whether proton transfer precedes, accompanies, or follows nucleophilic attack, could be perfectly resolved by mapping the energy surface. This combined effort cemented the "More O'Ferrall-Jencks Diagram" (MOJ diagram) as the standard framework for analyzing concerted versus stepwise mechanisms.

The MOJ diagram is a contour plot of the potential energy surface (PES) projected onto a generic square defined by normalized bond orders.

The axes of the diagram define the "reaction space". Typically, these are bond orders, ranging from 0 (no bond) to 1 (full bond), or 1 to 0 in the case of bond breaking.

For a generic reaction such as the one above, X-Axis represents the progress of formation of the B-C bond (0 \(\rightarrow\) 1) and Y-Axis represents the progress of forming the A-B bond (0 \(\rightarrow\) 1). This creates a unit square. The “Z-axis”, projecting out of the page, represents the Gibbs Free Energy of the system.

We can consider the energy surface as defined by four distinct corners. The top-left (0,1) and bottom-right (1,0) represent stable energy minima for the Reactants and Products, respectively.

In contrast, the off-diagonal corners represent high-energy intermediates (note: they could be local minima). Here, the bottom-left (0,0) depicts a species where both bonds are broken (dissociative intermediate, e.g., a carbocation), while the top-right (1,1) represents a species with A-B-C bonds (associative intermediate, e.g., a pentacoordinate intermediate).

Knowing this, how can we actually infer mechanistic information ?

A straight line connecting Reactants to Products represents a perfectly synchronous mechanism. Along this path, the bond order of the breaking bond decreases at exactly the same rate as the forming bond increases. This is the ideal \(S_N2\) or \(E_2\) path.

If the system moves from Reactants to Intermediate with bond breaking first, and then to Products, it indicates dissociative mechanism (e.g., \(S_N1\), \(E_1\)). Conversely, when the system moves from Reactants to Intermediate with bond formation first, and then to Products, it indicates an associative mechanism

The energy surface over this square typically resembles a saddle. The Reactant and Product corners are valleys. The Intermediate corners are usually high peaks (assuming the intermediates are unstable). The transition state is the "pass" between the reactant and product valleys. However, the "pass" is not a single point but a region.

If the intermediates are relatively stable (low energy), the saddle region becomes broad and flat, allowing the transition state to "wander" significantly. If the intermediates are very high energy, the saddle is steep and narrow, confining the transition state to the central diagonal.

Exercise for the reader - How does the activation entropy \(\Delta S^{\ddagger}\) differ between a transition state confined to a steep, narrow saddle versus one that 'wanders' across a broad, flat saddle region? and How does this affect the rate of the reaction ?

Thornton's Rules

In 1967, Edward Thornton attempted an answer to how does the transition state move when we change a substituent, solvent, or leaving group? His insight was that the movement is governed by the curvature of the PES. These rules became essential for predicting "Hammond" and "Anti-Hammond" effects in MOJ diagrams.

Thornton modeled the transition state as a geometric saddle point, defined by two opposing types of curvature. The reaction coordinate mode represents motion along the reaction path; because the transition state is an energy maximum in this direction (like the top of a hill), the curvature is negative (yielding an imaginary frequency). Conversely, the perpendicular mode represents motion orthogonal to the path; because the transition state sits at the bottom of an energy valley in this direction, the curvature is positive (yielding a real vibrational frequency).

What does Real or Imaginary frequencies even mean ?

Using Hooke’s law, \(\nu \propto \sqrt{k}\), where \(k\) is the force constant representing the curvature of the potential energy surface. In the perpendicular mode, the transition state sits in an energy valley (minimum) where curvature is positive meaning \(k \gt 0\); the square root of a positive number yields a real frequency, representing stable oscillation with a restoring force. Conversely, along the reaction coordinate, the transition state is an energy maximum (hill) where curvature is negative \(k \lt 0\). Since the square root of a negative number is imaginary, this yields an imaginary frequency, physically signifying the instability that drives the system down the hill toward products rather than restoring it to the peak.

This difference in curvature i.e. negative parallel, positive perpendicular, dictates how the TS responds to energy changes.

The Hammond Effect (Parallel Effect)

A perturbation that raises the energy of the reactant relative to the transition state will shift the transition state along the reaction coordinate toward the reactants.

Imagine a ball balanced on the peak of a hill (the TS). If you raise the plateau on the reactant side (making reactants less stable, reaction more exergonic), the peak effectively shifts closer to the reactant side. This means making the reaction more exergonic leads to earlier TS and making it more endergonic shifts it to a later TS.

The Anti-Hammond Effect (Perpendicular Effect)

A perturbation that raises the energy of the intermediate perpendicular to the reaction path will cause the transition state to move away from that high-energy region.

Since the TS is a minimum in the perpendicular direction (it sits at the bottom of the saddle's cross-section), it behaves like a marble in a bowl. If you raise one side of the bowl (destabilize an intermediate), the marble rolls away from the rising side.

This means, if the carbocation intermediate is destabilized, the TS moves away from the carbocation corner, becoming more concerted (synchronous, \(S_N1 \rightarrow S_N2\) for instance). This effect shows change in TS perpendicular to the reaction diagonal.

Hereon, it’s simple vector addition.

There's a lot more for me to add here. Jencks' cross-interaction constant, using MOJ diagram with Hammett and other LFERs, nitroalkane anomaly, principle of non-perfect synchronization, effects of solvents on substitution reactions, mechanistic studies of enzymes and much much more. But I need to sleep and I think the reading list presented here does justice to these topics far better than I could.

Selected Bibliography