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Transition State Theory & Rare Event Simulations: A Comprehensive Approach

Delve into the complex realm of rare event simulations, from Bennett-Chandler approach to diffusion in porous materials. Understand the microscopic and macroscopic phenomena, theories, and reactions behind transition states. Explore the power of linear response theory and computational schemes in analyzing chemical reactions.

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Transition State Theory & Rare Event Simulations: A Comprehensive Approach

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  1. Rare Event Simulations Theory 16.1 Transition state theory 16.1-16.2 Bennett-Chandler Approach 16.2 Diffusive Barrier crossings 16.3 Transition path ensemble 16.4

  2. Diffusion in porous material

  3. Chemical reaction Theory: macroscopic phenomenological Total number of molecules Make a small perturbation Equilibrium:

  4. Heaviside θ-function q q* βF(q) q Microscopic description of the reaction Theory: microscopic linear response theory Reaction coordinate Reaction coordinate Reactant A: Product B: Lowers the potential energy in A Increases the concentration of A Perturbation: Probability to be in state A

  5. Very small perturbation: linear response theory Linear response theory: static Outside the barrier gA =0 or 1: gA (x)gA (x)=gA (x) Switch of the perturbation: dynamic linear response Holds for sufficiently long times!

  6. Δ has disappeared because of derivative Derivative Stationary For sufficiently short t

  7. Eyring’s transition state theory Only products contribute to the average At t=0 particles are at the top of the barrier Let us consider the limit: t →0+

  8. Transition state theory • One has to know the free energy accurately • Gives an upper bound to the reaction rate • Assumptions underlying transition theory should hold: no recrossings

  9. Conditional average: given that we start on top of the barrier Bennett-Chandler approach Probability to find q on top of the barrier Computational scheme: • Determine the probability from the free energy • Compute the conditional average from a MD simulation

  10. cage window cage cage window cage q* q* βF(q) βF(q) q q Reaction coordinate

  11. Ideal gas particle and a hill q* is the true transition state q1 is the estimated transition state

  12. Bennett-Chandler approach • Results are independent of the precise location of the estimate of the transition state, but the accuracy does. • If the transmission coefficient is very low • Poor estimate of the reaction coordinate • Diffuse barrier crossing

  13. Transition path sampling xt is fully determined by the initial condition Path that starts at A and is in time t in B: importance sampling in these paths

  14. dp pt pt o n rT r0 rT rT rT r0 rt o o n n o o r0 r0 n n B A B A Walking in the Ensemble Shooting Shifting

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