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
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Rare Event Simulations
Theory 16.1Transition state theory 16.1-16.2Bennett-Chandler Approach 16.2Diffusive Barrier crossings 16.3Transition path ensemble 16.4
Diffusion in porous material
Theory:macroscopic
phenomenological
A BChemical reaction
d
dtA
A B A B A B
c tk c t k c t
d
dtB
A B A B A B
c tk c t k c t
Total number of molecules
d0
dtA Bc t c t
Equilibrium:
0A Bc t c t
A B A
B A B
c k
c k
Make a small perturbation
A A Ac t c c t
d
dtA
A B A B A A
c tk c t k c t
B B Ac t c c t
0 expA A A B B Ac t c k k t 0 expAc t
1
A B B Ak k
11 1 BA B A B
A B
ck c c
k
Theory:microscopic
linear response theory
Microscopic description of the reaction
Reaction coordinate
*q qReactant A:
Product B: *q q
Perturbation: 0 *AH H g q q
Lowers the potential energy in AIncreases the concentration of A
* 1 * *Ag q q q q q q
Heaviside θ-function
0 * 0*
1 * 0
q qq q
q q
0A A Ac c c
0A A Ac g g
Probability to be in state A
βF
(q)
q
q*
q
Reaction coordinate
2 2
00
AA A
cg g
Very small perturbation: linear response theory
0A A Ac g g
0 *AH H g q q
Outside the barrier gA =0 or 1:gA (x) gA (x) =gA (x)
0 01A Ag g
0 0 0 01A A A Bc c c c
Switch of the perturbation: dynamic linear response
00
A A
A AA B
g g tc t c
c c
0 expAc t
Holds for sufficiently long times!
Linear response theory: static
0H H B
0A A A
0
00
d exp
d exp
A HA
H
0
0
d exp
d exp
A H BA
H B
0 0
2
0
0 0
2
0
d exp d exp
d exp
d exp d exp
d exp
AB H B H BA
H B
A H B B H B
H B
0 0 0
AAB A B
0exp
A A
A B
g g tt
c c
For sufficiently short t
01exp
A A
A B
g g tt
c c
Derivative
Δ has disappeared because of derivative
0A A
A B
g g t
c c
0A A
A BA
g g tk t
c
* ** A B
A
g q q g q qg q q q q
q q
0 *0 B
B
A BA
g q qq g t
qk t
c
' 0d
A t B t tdt
' ' 0A t B t t A t B t t
0 ' 0 'A B t A B t
Stationary
Eyring’s transition state theory
0 *0 B
B
A BA
g q qq g t
qk t
c
0 0 * *
*
q q q q t q
q q
At t=0 particles are at the top of the barrier
Only products contribute to the average
Let us consider the limit: t →0+
0lim *
tq t q q t
0 0 *
*TSTA B
q q q qk t
q q
* 1 * *B Ag q q g q q q q
**Bg q q
q qq
* 1 * *Ag q q q q q 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
Bennett-Chandler approach
0 0 * *
*A B
q q q q t qk t
q q
0 0 * * 0 *
*0 *A B
q q q q t q q qk t
q qq q
Conditional average:given that we start on top of the barrier
0 *q q t q Probability to find q on top of
the barrier
Computational scheme:
1. Determine the probability from the free energy2. Compute the conditional average from a MD simulation
cage window cage
βF
(q)
q
q*
cage window cage
βF
(q)
q
q*
Reaction coordinate
Ideal gas particle and a hill
q1 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
0exp
A A
A B
g g tt
c c
01 exp
A B t
BA
h x h xC t h t
h
1 in
0 elsewhereA
Ah
0 0 0
0 0 0
d
d
A B t
A
x N x h x h xC t
x N x h x
xt is fully determined by the initial condition
,B t A t
C t h x 0 0path AP h x N x
Path that starts at A and is in time t in B: importance sampling in these paths
Walking in the Ensemble
Shooting
Shifting
A
r0o r0
n
B
rTn
rTo
r0o
r0n
AB
rt
ptn pt
o
p
rTo
rTn
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