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

2. Diffusion in porous material

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

4. 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

5. 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!

6. Δ has disappeared because of 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. 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

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

10. 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

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

13. 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

14. t→∞: θ=1
For both
Low value of κ

15. 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!

18. 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

19. 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

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

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

22. 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