⚗️Computational Chemistry Unit 14 – Transition State Theory & Reaction Dynamics

Key Concepts

• Transition state theory (TST) describes the rates of elementary chemical reactions based on the properties of the transition state
• Assumes a quasi-equilibrium between reactants and the activated complex (transition state)
• Reaction dynamics studies the atomic-level details of chemical reactions, including the motion of atoms during the reaction
• Potential energy surface (PES) represents the potential energy of a system as a function of its atomic coordinates
  • Stationary points on the PES correspond to reactants, products, and transition states
• Reaction coordinate describes the progress of a reaction from reactants to products
• Transition state is the highest-energy point along the reaction coordinate, representing the activated complex
• Reaction rate is determined by the free energy difference between the reactants and the transition state

Historical Context

• Transition state theory was developed in the 1930s by Eyring, Evans, and Polanyi
• Built upon earlier work by Arrhenius, who proposed the concept of activation energy
• Eyring equation relates the reaction rate to the free energy of activation
• Development of computational methods in the 1960s and 1970s enabled the application of TST to more complex systems
• Advances in quantum chemistry and molecular dynamics simulations have expanded the scope of reaction dynamics studies
• Nobel Prize in Chemistry awarded to Eyring (1966) and Karplus, Levitt, and Warshel (2013) for their contributions to the field

Theoretical Framework

• TST assumes that reactants and the activated complex are in quasi-equilibrium
• Reaction rate is determined by the concentration of the activated complex and its rate of crossing the transition state
• Eyring equation: $k = \frac{k_B T}{h} e^{-\Delta G^{\ddagger}/RT}$, where $k$ is the rate constant, $k_B$ is Boltzmann's constant, $h$ is Planck's constant, and $\Delta G^{\ddagger}$ is the free energy of activation
• Hammond's postulate states that the structure of the transition state resembles the nearest stable species (reactant or product)
• Marcus theory describes electron transfer reactions and the relationship between the reaction rate and the reorganization energy
• Transition state theory can be extended to include quantum mechanical effects, such as tunneling

Mathematical Foundations

• Potential energy surfaces are multi-dimensional functions that describe the potential energy of a system as a function of its atomic coordinates
• Stationary points on the PES are characterized by zero first derivatives (gradients) and classified by their second derivatives (Hessian matrix)
  • Minima have all positive eigenvalues of the Hessian matrix
  • Transition states have one negative eigenvalue corresponding to the reaction coordinate
• Intrinsic reaction coordinate (IRC) connects the transition state to the reactants and products
• Reaction rate constants can be calculated using statistical mechanics and partition functions
• Variational transition state theory (VTST) improves upon TST by optimizing the dividing surface between reactants and products
• Kinetic isotope effects (KIEs) can provide insights into the reaction mechanism and the nature of the transition state

Computational Methods

• Electronic structure methods, such as density functional theory (DFT) and ab initio methods, are used to calculate potential energy surfaces and locate transition states
• Geometry optimization techniques, such as the Newton-Raphson method and quasi-Newton methods, are employed to find stationary points on the PES
• Transition state optimization can be performed using methods like the nudged elastic band (NEB) and the dimer method
• Molecular dynamics simulations can be used to study the atomic-level details of chemical reactions and to sample reaction pathways
• Enhanced sampling techniques, such as umbrella sampling and metadynamics, can be used to overcome barriers and explore rare events
• Quantum dynamics methods, such as wave packet propagation and the multi-configuration time-dependent Hartree (MCTDH) method, can be used to study quantum effects in reaction dynamics

Applications in Chemistry

• Catalysis: TST and reaction dynamics studies help in understanding and designing more efficient catalysts
• Enzyme kinetics: TST can be applied to study the reaction mechanisms of enzyme-catalyzed reactions
• Atmospheric chemistry: Reaction dynamics studies are crucial for understanding the complex chemical processes in the atmosphere
• Combustion chemistry: TST and reaction dynamics are used to model and optimize combustion processes
• Photochemistry: Excited-state potential energy surfaces and non-adiabatic dynamics are studied using TST and reaction dynamics methods
• Drug design: Understanding the reaction mechanisms of drug-target interactions can aid in the development of new therapeutics

Limitations and Challenges

• TST assumes a quasi-equilibrium between reactants and the activated complex, which may not always hold true
• Accurate calculation of potential energy surfaces can be computationally demanding, especially for large systems
• Identifying the correct reaction coordinate and transition state can be challenging, particularly for complex reactions with multiple steps
• Inclusion of quantum effects, such as tunneling and non-adiabatic transitions, can be difficult and computationally expensive
• Sampling rare events and overcoming high barriers in molecular dynamics simulations remains a challenge
• Extending TST and reaction dynamics methods to condensed-phase systems, such as reactions in solution or at interfaces, introduces additional complexity

Advanced Topics and Future Directions

• Transition path sampling (TPS) methods can be used to study rare events and generate ensembles of reactive trajectories
• Markov state models (MSMs) can be constructed from molecular dynamics simulations to describe the kinetics of complex systems
• Machine learning techniques, such as neural networks and Gaussian process regression, can be used to construct accurate potential energy surfaces and accelerate reaction dynamics simulations
• Non-equilibrium transition state theory (NE-TST) extends TST to systems far from equilibrium, such as reactions driven by external forces or in the presence of a temperature gradient
• Quantum computing and quantum algorithms may provide new opportunities for simulating quantum dynamics and studying reaction mechanisms
• Multiscale modeling approaches, combining quantum mechanics and molecular mechanics (QM/MM), can be used to study reactions in complex environments, such as in enzymes or at solid-liquid interfaces