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

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

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

Linear response theory: static 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!

Δ has disappeared because of derivative Stationary For sufficiently short t

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+

Bennett-Chandler approach Conditional average: given that we start on top of the barrier 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

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

Transition state theory The motion of the particle is ballistic Transition state theory Assumed transition state In the product or reactant state in can exchange energy

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

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

t→∞: θ=1 For both Low value of κ

Bennett Chandler Approach MD simulation: At t=0 q=q1 Determine the fraction at the product state weighted with the initial velocity Transmission coefficient MD simulation to correct the transition state result!

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

Diffusive barrier crossing Consider the following “membrane” TST: Flux = P(0)  <v>/ω Bennett Chandler Flux= P(0)  <v>/ω  κ Diffusive barrier crossing Flux= P(0)  Deff/ω2

Diffusive barrier crossing Diffusion coefficient on the barrier Green Kubo relation for the diffusion coefficient!

Reaction coordinate r t→∞: θ=1 For both q F(q) Low value of κ

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