Rare Event Simulations Theory 16 1 Transition state

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

Theory: microscopic linear response theory Microscopic description of the reaction Reaction coordinate Reactant A:

Very small perturbation: linear response theory Linear response theory: static Outside the barrier g.

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

Eyring’s transition state theory At t=0 particles are at the top of the barrier

Transition state theory • One has to know the free energy accurately • Gives

Bennett-Chandler approach Conditional average: given that we start on top theofbarrier Probability to find

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

Ideal gas particle and a hill q* is the true transition state q 1

Bennett-Chandler approach • Results are independent of the precise location of the estimate of

Transition path sampling xt is fully determined by the initial condition Path that starts

Walking in the Ensemble dp Shooting n pt rt r 0 o r 0

Slides: 19

Download presentation

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

Diffusion in porous material

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

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

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

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

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

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

Bennett-Chandler approach Conditional average: given that we start on top theofbarrier Probability to find q onoftop the barrier Computational scheme: 1. Determine the probability from the free energy 2. Compute the conditional average from a MD simulation

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

Ideal gas particle and a hill q* is the true transition state q 1 is the estimated transition state

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

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

Walking in the Ensemble dp Shooting n pt rt r 0 o r 0 n Shifting pto r. Tn B A r. To r 0 n A B r. Tn