Unit 10: Molecular Reaction Dynamics, Potential Energy Surfaces & Transition State Theory

Simple collision theory (Trautz and Lewis, 1916-1918) provided the earliest microscopic physical model for bimolecular reaction rate constants.

Fundamental Formulation

For an elementary bimolecular reaction $A + B \longrightarrow P$:

where:

1. $Z_{AB}$ is the collision density between species $A$ and $B$:

where $d_{AB} = \frac{d_A + d_B}{2}$ is the collision diameter, and $\mu = \frac{m_A m_B}{m_A + m_B}$ is the reduced mass.

1. $f_{\text{Boltzmann}} = \exp(-E_0 / R T)$ is the fraction of collisions possessing relative kinetic energy along the line of centers exceeding threshold $E_0$.
2. $P$ is the steric factor ($0 < P \le 1$).

The Bimolecular Rate Constant

Expressing in molar units ($k = r / [A][B]$):

Comparing with the Arrhenius equation $k = A \exp(-E_a / R T)$:

Physical Meaning of the Steric Factor $P$

For simple spherical atom-atom reactions (e.g., $K + Br_2$), $P \approx 1\text{--}4$ (the "harpoon mechanism" gives $P > 1$ due to long-range electron transfer). For polyatomic molecules with complex geometry (e.g., cyclization of hexatriene), $P$ drops to $10^{-4}\text{--}10^{-7}$, indicating that less than one in a million energetic collisions occurs with proper mutual orientation.

Polanyi Rules & Potential Energy Surface Topology Matrix

| Reaction Class | Canonical Chemical Reaction | $\Delta H_r^\circ$ ($\text{kJ/mol}$) | Barrier Location | Polanyi Classification | Optimum Energy for Reaction | Product Energy Disposal | |---|---|---|---|---|---|---| | Exothermic | $F + H_2 \longrightarrow HF + H$ | $-134.0$ | Entrance Valley | Early Barrier (Attractive PES) | Translational Energy ($E_{\text{trans}}$) | High Product Vibration ($v' = 2, 3$) | | Exothermic | $H + Cl_2 \longrightarrow HCl + Cl$ | $-188.0$ | Entrance Valley | Early Barrier (Attractive PES) | Translational Energy ($E_{\text{trans}}$) | High Product Vibration ($v' = 3 - 6$) | | Thermoneutral| $H + H_2 \longrightarrow H_2 + H$ | $0.0$ | Symmetric Saddle Point | Central Barrier | Both Translation & Vibration | Moderate Translation & Vibration | | Endothermic | $H + HF \longrightarrow H_2 + F$ | $+134.0$ | Exit Valley | Late Barrier (Repulsive PES) | Reactant Vibration ($E_{\text{vib}}$) | High Product Translation ($E_{\text{trans}}'$) | | Endothermic | $Cl + HCl \longrightarrow Cl_2 + H$| $+188.0$ | Exit Valley | Late Barrier (Repulsive PES) | Reactant Vibration ($E_{\text{vib}}$) | High Product Translation ($E_{\text{trans}}'$) | | Harpoon Reaction| $K + Br_2 \longrightarrow KBr + Br$ | $-175.0$ | Long-Range Curve Crossing | Harpoon Electron Jump ($R_c \approx 8\text{ Å}$)| Low Thermal Energy Suffices | Forward Stripping Rebound ($KBr$) |

In liquid solution, molecules are enclosed within solvent "cages", undergoing dozens of rapid repeated collisions (an encounter) before diffusing apart.

The Smoluchowski Equation (1917)

For an activationless reaction between neutral species where every encounter leads immediately to reaction ($E_a \approx 0$), the rate is strictly limited by the rate at which reactants diffuse toward one another. Solving Fick's second law for spherical diffusion to an absorbing sphere of radius $R_{AB} = r_A + r_B$:

where $D_A, D_B$ are diffusion coefficients.

Substituting the Stokes-Einstein relation $D_i = \frac{k_B T}{6 \pi \eta r_i}$: Assuming equal hydrodynamic radii $r_A = r_B = r$:

Remarkable Feature: The diffusion-controlled rate constant is independent of solute size and depends solely on temperature and solvent dynamic viscosity $\eta$! For water at $298.15\text{ K}$ ($\eta = 0.890\text{ cP}$):

The Debye Modification for Ionic Encounters (1942)

For charged ions ($z_A e$ and $z_B e$), Coulombic attraction or repulsion alters encounter flux:

Oppositely charged ions ($z_A z_B < 0$) accelerate diffusion ($k > k_{\text{diff}}$), while like-charged ions repel ($k < k_{\text{diff}}$).

Henry Eyring, Michael Polanyi, and Eugene Wigner (1935) established Transition State Theory (also known as Activated Complex Theory), which calculates absolute reaction rates from first-principles statistical mechanics without empirical parameters.

Postulates of Classical TST

1. Reactants are in quasi-thermal equilibrium with the activated complex:

1. The activated complex $[AB]^\ddagger$ is treated as an ordinary molecule, except that one vibrational mode along the reaction coordinate has transformed into a loose translational motion across the saddle point.
2. Every activated complex crossing the barrier in the forward direction proceeds irreversibly to product (transmission coefficient $\kappa_{\text{trans}} \approx 1$).

Partition Function Derivation

The pseudo-equilibrium constant is:

where $q$ are molecular partition functions per unit volume.

Factoring out the critical reaction-coordinate vibration:

In the classical limit ($h \nu \ll k_B T$):

The rate of barrier crossing is the vibrational frequency $\nu^\ddagger$:

The reaction rate is:

The unknown imaginary frequency $\nu^\ddagger$ cancels identically, yielding the foundational Eyring Equation:

The universal prefactor $\frac{k_B T}{h} \approx 6.21 \times 10^{12}\text{ s}^{-1}$ at $298.15\text{ K}$ represents the fundamental attempt frequency of chemical transformation.

Eyring recast Transition State Theory into macroscopic thermodynamic language by expressing the quasi-equilibrium constant in terms of standard activation free energy:

The Thermodynamic Eyring Equation

where $m$ is molecularity and $c^\circ = 1\text{ M}$ is standard state concentration.

Relation Between Arrhenius and Eyring Parameters

From the Arrhenius definition $E_a \equiv R T^2 \frac{d \ln k}{dT}$:

For ideal gas reactions of molecularity $m$:

• For unimolecular gas reactions ($m = 1$): $E_a = \Delta H^{\ddagger\circ} + R T$.
• For bimolecular gas reactions ($m = 2$): $E_a = \Delta H^{\ddagger\circ} + 2 R T$.

Physical Meaning of Activation Entropy ($\Delta S^{\ddagger\circ}$)

• $\Delta S^{\ddagger\circ} < 0$ (associative transition state): Two independent molecules combine to form a rigid, highly ordered transition state, losing translational and rotational degrees of freedom (typical bimolecular additions).
• $\Delta S^{\ddagger\circ} > 0$ (dissociative transition state): A molecule loosens bonds, releasing fragments or solvent molecules, increasing disorder.

The rate of ionic reactions in solution depends strongly on the ionic strength $I$ of the medium.

Brønsted-Bjerrum Formulation (1922-1924)

Consider an elementary reaction between ions $A^{z_A}$ and $B^{z_B}$ forming an activated complex $[AB]^{\ddagger (z_A + z_B)}$:

Thermodynamic equilibrium requires activities:

The observed rate is:

Thus, the rate constant $k$ is:

where $k_0$ is the rate constant at infinite dilution ($I \to 0$).

Incorporation of Debye-Hückel Limiting Law

At low ionic strength ($I < 0.05\text{ M}$), the activity coefficient is given by $\log_{10} \gamma_i = -A z_i^2 \sqrt{I}$:

For water at $298.15\text{ K}$, $A = 0.509\;(\text{mol/kg})^{-1/2}$:

Diagnostic Regimes of Primary Salt Effect

1. Like Charges ($z_A z_B > 0$): Slope $> 0$. Adding inert electrolyte shields repulsion, stabilizing the $[AB]^\ddagger$ complex and accelerating the reaction.

2. Opposite Charges ($z_A z_B < 0$): Slope $< 0$. Adding inert electrolyte stabilizes separated reactants more than the complex, decelerating the reaction.

3. Neutral Reactant ($z_A z_B = 0$): Slope $= 0$. Rate is virtually independent of ionic strength.

University Honors Research Monograph: Quantum Resonances & Feshbach Bound States in the F + H2 Reaction

The benchmark elementary bimolecular reaction $F(^2P) + H_2 \longrightarrow HF(v', j') + H$ is the gold standard of modern quantum reaction dynamics:

• Reactive Scattering Resonances:

In crossed molecular beam experiments with Velocity Map Imaging, Yuan T. Lee, K. Liu, and X. Yang detected sharp step-function peaks in the backward-scattered differential cross-section for producing $HF(v'=3)$ at precise collision energies ($E_{\text{coll}} \approx 0.040 - 0.052\text{ eV}$).

• The Feshbach Transition State Bound State:

Full quantum 3D wavepacket calculations on highly accurate ab initio potential energy surfaces (such as the FXZ and CBS surfaces) revealed that these resonance spikes arise from quasi-bound quantum states trapped inside a dynamic potential well located in the transition state region:

Because the entrance channel is vibrationally adiabatic, the colliding system is temporarily trapped in a metastable quantum state for several vibrational periods ($\sim 30 - 50\text{ fs}$) before tunneling out into the exit channel, proving that chemical reactions exhibit discrete quantum bound states at the transition state barrier.

While classical kinetics measures macroscopic ensemble thermal averages $k(T)$, molecular reaction dynamics probes single-collision events with defined quantum states, velocity vectors, and scattering angles.

Crossed Molecular Beam Experiments (Herschbach and Lee, Nobel Prize 1986)

Two collimated supersonic molecular beams collide at right angles in an ultra-high vacuum chamber ($P < 10^{-10}\text{ Torr}$):

1. Velocity Selection: Choppers select initial relative kinetic energy $E_{\text{coll}}$.

2. Rotatable Mass Spectrometer: Measures angular distribution $d\sigma / d\Omega$ and time-of-flight velocity distributions of scattered reaction products.

Dynamics Classification: Rebound vs. Stripping

1. Rebound Mechanism ($K + CH_3I \longrightarrow KI + CH_3$):

• Small impact parameter ($b < d$).
• Direct backward scattering ($\theta \approx 180^\circ$).
• Low product vibrational excitation; high translational energy release.

2. Stripping Mechanism ($K + Br_2 \longrightarrow KBr + Br$):

• Large impact parameter ($b > d$).
• Forward scattering ($\theta \approx 0^\circ$).
• Harpoon Model: At long distance ($R_c \approx 4\text{--}6\text{ Å}$), an electron jumps from potassium to bromine ($K + Br_2 \to K^+ + Br_2^-$), followed by strong Coulombic attraction that strips the halogen atom into high vibrational excitation.

The Potential Energy Surface (PES) is the hyper-dimensional function $V(\vec{R})$ expressing the electronic energy of a reacting system as a function of nuclear geometry under the Born-Oppenheimer approximation.

Collinear Triatomic Reactions ($A + B-C \longrightarrow A-B + C$)

For a collinear arrangement, the potential energy depends on only two internuclear distances: $R_{AB}$ and $R_{BC}$.

The London-Eyring-Polanyi-Sato (LEPS) Surface

Constructed from semi-empirical valence bond theory by summing Coulombic ($Q$) and exchange ($J$) integrals:

calibrated using experimental diatomic Morse potentials.

Topography of the PES

1. Reactant Valley: Large $R_{AB}$, equilibrium $R_{BC}$.

2. Product Valley: Equilibrium $R_{AB}$, large $R_{BC}$.

3. Transition State (Saddle Point $\ddagger$): A first-order saddle point where the gradient vanishes ($\nabla V = 0$), corresponding to a maximum along the reaction coordinate and a minimum along all perpendicular vibrational coordinates.

Polanyi's Rules for Barrier Location

John Polanyi (Nobel Prize 1986) related barrier location to energy partitioning:

1. Early Barrier (Attractive Surface, reactant valley):

Translational energy $E_{\text{trans}}$ is highly effective in promoting reaction; product energy appears primarily as vibration ($E_{\text{vib}}'$). Exothermic reactions generally possess early barriers.

2. Late Barrier (Repulsive Surface, product valley):

Vibrational energy $E_{\text{vib}}$ in the reactant bond is far more effective than translational energy in crossing the barrier. Endothermic reactions generally possess late barriers.

Observing chemical reactions at the most fundamental quantum mechanical level requires measuring 3D product velocity vectors using ion imaging and observing transient Transition States in real time using femtosecond pump-probe laser pulses.

1. Velocity Map Imaging (VMI) in Crossed Molecular Beams

Velocity Map Imaging, introduced by David Parker and André Eppink (1997), revolutionized experimental reaction dynamics by measuring full 3D differential cross-sections $\frac{d^2\sigma}{d\Omega dE}$ with 100% collection efficiency.

Principle of Electrostatic Immersion Lens Focusing

In a crossed molecular beam experiment:

1. Two collimated supersonic molecular beams (e.g., $F + H_2$ or $O(^1D) + CH_4$) intersect at $90^\circ$ inside an ultra-high vacuum scattering chamber ($P < 10^{-10}\text{ Torr}$).
2. Products formed in specific quantum states are selectively ionized by Resonance-Enhanced Multi-Photon Ionization (REMPI).
3. The resulting ion cloud expands according to product recoil velocities. An open electrostatic immersion lens system (repeller, extractor, and ground electrodes with smooth curved potentials) accelerates ions toward a 2D position-sensitive microchannel plate (MCP) detector backed by a phosphor screen and CCD camera.
4. Crucially, the inhomogeneous electric field configuration functions as an ion telescope: All ions having identical initial velocity vectors $(v_x, v_y, v_z)$ are mapped onto the exact same spatial coordinate $(X, Y)$ on the detector, regardless of where they were formed within the finite spatial laser ionization volume.

Image Reconstruction via Abel Inversion

The 2D CCD image represents a cylindrical projection of the true 3D cylindrically symmetric scattering distribution around the relative velocity vector $\vec{v}_{\text{rel}}$. Applying the mathematical Inverse Abel Transform:

reconstructs the true 3D velocity slice, revealing product scattering angle $\theta_{\text{c.m.}}$ (forward vs. backward rebound) and kinetic energy disposal $E_{\text{trans}}'$, proving the microscopic collision mechanism.

2. Femtosecond Transition State Spectroscopy (Ahmed Zewail)

Prior to femtosecond laser spectroscopy (Nobel Prize in Chemistry 1999 to Ahmed Zewail), the Transition State was regarded as an unobservable theoretical abstraction lasting only $\tau \approx 10 - 100\text{ fs}$ ($10^{-14} - 10^{-13}\text{ s}$).

The Pump-Probe Ultrafast Paradigm

1. Pump Pulse ($t = 0$): An ultra-short laser pulse ($\tau_{\text{pulse}} \approx 20 - 50\text{ fs}$) promotes a stable reactant molecule ($ICN$ or $NaI$) from its ground potential energy surface $V_0(R)$ onto an excited repulsive surface $V_1(R)$, launching a coherent nuclear wavepacket:

2. Dynamic Evolution: The wavepacket slides down the repulsive potential curve as the bond stretches ($R = R_0 \to R^\ddagger \to \infty$).

3. Probe Pulse ($t = \Delta t$): After an adjustable optical delay time $\Delta t$ (varied by moving a computer-controlled translation stage by $\Delta x = c \Delta t$, where $1\;\mu\text{m} \leftrightarrow 6.67\text{ fs}$):

• Tuning the probe laser to frequency $\lambda_{\text{free}}$ monitors the appearance of free dissociated products ($CN^\bullet$).
• Tuning the probe laser to a shifted frequency $\lambda_{\text{TS}}$ excites the transient complex $[I\cdots CN]^{\ddagger *}$ while the atoms are at intermediate separation $R^\ddagger$, emitting characteristic fluorescence.

By recording fluorescence as a function of optical delay $\Delta t$, the birth of a chemical molecule is watched in real time, clocking the transition state lifetime at precisely $200\text{ femtoseconds}$.