Transition State Theory

We derive the Eyring equation from the quasi-equilibrium assumption and statistical mechanics. Consider the bimolecular reaction:

Step 1: Quasi-Equilibrium

By the quasi-equilibrium hypothesis, the activated complex $AB^{\ddagger}$ is in thermodynamic equilibrium with the reactants. We can therefore define an equilibrium constant$K^{\ddagger}$ for this pseudo-equilibrium:

This gives us the concentration of the activated complex:

Step 2: Rate of Barrier Crossing

The rate of reaction equals the concentration of the activated complex multiplied by the frequency $\nu^{\ddagger}$ at which it crosses the barrier:

To find $\nu^{\ddagger}$, we treat the motion along the reaction coordinate as a one-dimensional classical translation. The partition function for this single translational degree of freedom over a small distance $\delta$ at the barrier top is:

The average velocity of crossing in the forward direction is:

The frequency of crossing is the velocity divided by the distance:

We can factor $K^{\ddagger}$ into two parts: the partition function for the reaction coordinate $q_{\text{rc}}$ and the remaining equilibrium constant $K^{\ddagger\prime}$that excludes this degree of freedom:

Step 3: Combining the Expressions

Multiplying $\nu^{\ddagger}$ by $K^{\ddagger}$:

The $\delta$ and $\sqrt{2\pi m^{\ddagger}}$ terms cancel beautifully, leaving:

Since the rate constant $k$ is defined by $\text{Rate} = k[A][B]$, we obtain:

where we now write $K^{\ddagger}$ to mean the equilibrium constant with the reaction coordinate factored out (dropping the prime for notational convenience).

Step 4: Thermodynamic Form

From thermodynamics, the equilibrium constant is related to the standard Gibbs energy of activation:

Substituting into the Eyring equation:

Since $\Delta G^{\ddagger} = \Delta H^{\ddagger} - T\Delta S^{\ddagger}$, we can write:

This is the complete Eyring equation. It separates the rate constant into an entropic factor $\exp(\Delta S^{\ddagger}/R)$ and an enthalpic (energetic) factor $\exp(-\Delta H^{\ddagger}/RT)$, providing deep physical insight into what controls reaction rates.

The potential energy surface (PES) provides the conceptual and computational foundation for transition state theory. For a chemical reaction, the PES is a multidimensional function that gives the electronic energy as a function of all nuclear coordinates. We illustrate the key concepts using the simplest nontrivial case: a collinear atom-transfer reaction.

The Collinear A + BC Reaction

Consider the reaction $A + BC \to AB + C$ constrained to collinear geometry. The system has two independent internal coordinates: the bond distances $r_{AB}$and $r_{BC}$. The potential energy $V(r_{AB}, r_{BC})$ defines a two-dimensional surface with the following key features:

• • Reactant valley: Located at large $r_{AB}$ and small $r_{BC}$ (A is far from BC, which is at its equilibrium distance).
• • Product valley: Located at small $r_{AB}$ and large $r_{BC}$ (AB is formed, C moves away).
• • Saddle point: The point of highest energy along the minimum energy path connecting the two valleys. This is the transition state.

The Minimum Energy Path and Reaction Coordinate

The minimum energy path (MEP), also called the intrinsic reaction coordinate (IRC), is the path of steepest descent from the saddle point to both the reactant and product valleys. It represents the most favorable pathway for the reaction. The arc length along this path defines the reaction coordinate $s$.

At the saddle point, the Hessian matrix of second derivatives of the potential energy has a special structure. In mass-weighted coordinates:

The eigenvalues of the Hessian at the saddle point are:

• • One negative eigenvalue $\lambda_1 < 0$: This corresponds to the reaction coordinate direction. The curvature is negative because the saddle point is a maximum along this direction. The associated imaginary frequency is $\nu^{\ddagger} = \frac{1}{2\pi}\sqrt{|\lambda_1|/\mu}$.
• • All other eigenvalues positive $\lambda_i > 0$: These correspond to the vibrational modes of the activated complex perpendicular to the reaction coordinate. The frequencies are $\nu_i = \frac{1}{2\pi}\sqrt{\lambda_i/\mu_i}$.

Curvature and Vibrational Frequencies

The vibrational frequencies of the transition state are directly related to the curvature of the PES. For a nonlinear transition state with $N$ atoms, there are$3N - 7$ real vibrational frequencies (one fewer than the usual $3N - 6$because the reaction coordinate replaces one vibration). These frequencies enter the partition function of the activated complex:

The imaginary frequency along the reaction coordinate determines the width of the barrier and is crucial for estimating quantum mechanical tunneling corrections:

where $s$ is the displacement along the reaction coordinate from the saddle point. A larger $|\lambda_1|$ means a sharper, narrower barrier, which enhances tunneling.

The LEPS Potential

A commonly used analytic form for the PES of atom-transfer reactions is the London–Eyring–Polanyi–Sato (LEPS) potential. For the collinear $A + BC \to AB + C$ reaction, the LEPS potential is:

where $Q_{ij}$ and $J_{ij}$ are the Coulomb and exchange integrals for each diatomic pair, expressed in terms of Morse potential parameters and Sato parameters that control the barrier height. The Coulomb integral for pair $ij$ is:

and the exchange integral is:

where $E_{\text{singlet}} = D_{ij}(e^{-2\beta_{ij}(r-r_e)} - 2e^{-\beta_{ij}(r-r_e)})$is the Morse potential and $E_{\text{triplet}} = \tfrac{1}{2}D_{ij}(e^{-2\beta_{ij}(r-r_e)} + 2e^{-\beta_{ij}(r-r_e)})$is the anti-Morse (triplet repulsion) curve.

Enzyme Catalysis

Enzymes are nature's catalysts, accelerating reactions by factors of $10^{6}$ to$10^{17}$. TST explains this enormous rate enhancement through the concept of transition state stabilization. An enzyme lowers$\Delta G^{\ddagger}$ by binding the transition state more tightly than the ground state substrate. Pauling first proposed this idea in 1946, and it has been confirmed by the success of transition state analog inhibitors — stable molecules that mimic the geometry of the transition state and bind tightly to the enzyme active site. Many clinically important drugs (protease inhibitors for HIV, statins for cholesterol) are designed as transition state analogs.

Atmospheric Reactions

TST is essential for predicting rate constants of atmospheric reactions where direct measurement may be difficult. The reactions governing ozone depletion, smog formation, and climate chemistry involve radical species and unusual transition states. For example, the reaction $\text{OH} + \text{CH}_4 \to \text{H}_2\text{O} + \text{CH}_3$ controls the atmospheric lifetime of methane. Computational TST with tunneling corrections provides rate constants in excellent agreement with experiment over wide temperature ranges.

Drug Design

Modern computational drug design leverages TST through free energy perturbation calculations. By computing$\Delta G^{\ddagger}$ for enzymatic reactions with different substrate modifications, medicinal chemists can predict how structural changes affect reaction rates and binding affinities. The concept of the transition state is central to understanding enzyme mechanism and designing effective inhibitors.

Catalytic Converters

Heterogeneous catalysis in automotive catalytic converters involves reactions on metal surfaces (Pt, Pd, Rh). TST applied to surface reactions explains how the catalyst lowers$\Delta G^{\ddagger}$ for the oxidation of CO and hydrocarbons and the reduction of NO$_x$. The Eyring equation, adapted for surface reactions using two-dimensional gas models for adsorbed species, successfully predicts turnover frequencies and selectivity patterns observed in experiments.