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

Diffusion in porous material

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

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

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

βF(q) q q*q* q Reaction coordinate

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

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

Linear response theory: static

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

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

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 of the barrier Probability to find q on top of the barrier Computational scheme: 1.Determine the probability from the free energy 2.Compute the conditional average from a MD simulation

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

Ideal gas particle and a hill q 1 is the estimated transition state q * is the true 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 x t 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 Shooting Shifting A r0r0 o r0r0 n B rTrT n rTrT o r0r0 o r0r0 n A B rtrt ptpt n ptpt o pp rTrT o rTrT n