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Rate Theory (overview)macroscopic view (phenomenological)
rate of reactionsexperimentsthermodynamics
Van ‘t Hoff & Arrhenius equation
microscopic view (atomistic)statistical mechanicstransition state theories
effect of environmentEyring theory
static: potential of mean force
dynamic: Kramer’s theory
computing reaction rateoptimizating transition states
simulating barrier crossing
practical
normal mode analysis
practical next weekThursday, 22 December 2011
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Chromophore in water
CASSCF(6,6)/3-21G//SPCE molecular dynamics
p-hydroxybenzylidene acetone (pck)
resonance structuresThursday, 22 December 2011
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Rate of photoisomerization of double bond
uni-molecular process, initiated by photon absorption
Thursday, 22 December 2011
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Rate of photoisomerization of double bond
uni-molecular process, initiated by photon absorption
Thursday, 22 December 2011
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measuring reaction rate
simulation & pump-probe fluorescence
Thursday, 22 December 2011
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kinetics & thermodynamicsapproaching equilibrium
[A] + [B] = [A]0
d[A]
dt= −k+[A] + k− ([A]0 − [A]) = − (k+ + k−) [A] + k−[A]0
[A] =k− + k+e−(k−+k−)t
k+ + k−[A]0
conservation law
unimolecular process
so that
solution of the differential equations
d[A]
dt= −k+[A] + k−[B]
d[B]
dt= +k+[A]− k−[B]
Ak+�k−
B
Thursday, 22 December 2011
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equilibrium constant & reaction free energy
limt→∞
[A] =k−
k+ + k−[A]0 lim
t→∞[B] = [A]0 − [A]∞ =
k+k+ + k−
[A]0
K =[B]∞[A]∞
=k+k−
= exp
�−∆G
RT
�
approaching equilibrium
eventually....
kinetics & thermodynamics
Thursday, 22 December 2011
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temperature dependence of reaction rates
Gibbs-Helmholtz relation
G = H − TS S =H −G
T
�∂G
∂T
�
p
= −S =G−H
T
�∂G
∂T
�
p
− G
T= −H
T
Thursday, 22 December 2011
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temperature dependence of reaction rates
Gibbs-Helmholtz relation
�∂G
∂T
�
p
= −S =G−H
T
�∂G
∂T
�
p
− G
T= −H
T
T
�∂
∂T
�G
T
��
p
= −H
T
G = H − TS S =H −G
T
Thursday, 22 December 2011
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temperature dependence of reaction rates
Gibbs-Helmholtz relation
�∂G
∂T
�
p
= −S =G−H
T
�∂G
∂T
�
p
− G
T= −H
T
T
�∂
∂T
�G
T
��
p
= −H
T
�∂
∂T
�G
T
��
p
= − H
T 2
G = H − TS S =H −G
T
Thursday, 22 December 2011
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temperature dependence of reaction rates
Gibbs-Helmholtz relation
�∂G
∂T
�
p
= −S =G−H
T
�∂G
∂T
�
p
− G
T= −H
T
T
�∂
∂T
�G
T
��
p
= −H
T
�∂
∂T
�G
T
��
p
= − H
T 2
G = H − TS S =H −G
T
Thursday, 22 December 2011
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temperature dependence of reaction ratesVan ‘t Hoff equation
lnK = −∆G
RT
d lnK
dT= − 1
R
d
dT
�∆G
T
�
p
=∆H
RT 2
equilibrium constant
Gibbs-Helmholtz predicts effect of temperature on equilibrium constant
d lnK
d1/T= −∆H
R
Thursday, 22 December 2011
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temperature dependence of reaction ratesVan ‘t Hof equation
lnK = −∆G
RT
d lnK
dT= − 1
R
d
dT
�∆G
T
�
p
=∆H
RT 2
K =k+k−
d
dTln k+ − d
dTln k− =
∆H
RT 2
equilibrium constant
Gibbs-Helmholtz predicts effect of temperature on equilibrium constant
relation between equilibrium and rate constant
d lnK
d1/T= −∆H
R
Thursday, 22 December 2011
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temperature dependence of reaction ratesVan ‘t Hof equation
lnK = −∆G
RT
d lnK
dT= − 1
R
d
dT
�∆G
T
�
p
=∆H
RT 2
d lnK
d1/T=
∆H
R
K =k+k−
d
dTln k+ − d
dTln k− =
∆H
RT 2
equilibrium constant
Gibbs-Helmholtz predicts effect of temperature on equilibrium constant
relation between equilibrium and rate constant
therefore
d
d1/Tln k = −E
R+ a
Thursday, 22 December 2011
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temperature dependence of reaction ratesArrhenius equation
activated state
K‡ =[A‡]
[A]
A � A‡ → B
Thursday, 22 December 2011
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temperature dependence of reaction ratesArrhenius equation
activated state
K‡ =[A‡]
[A]
d lnK‡
d1/T=
∆H‡
R
ln k = ln a− Ea
R
1
T
k = a exp
�− Ea
RT
�
A � A‡ → B
Thursday, 22 December 2011
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microscopic picturestatistical mechanics
partition function
β =1
kBTK =
pB
pA=
QB
QA=
�B exp[−βH]dpdq�A exp[−βH]dpdq
Thursday, 22 December 2011
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microscopic picturestatistical mechanics
partition function
H = T + V
β =1
kBT
H =�
i
p2i
2mi+ V (q1, q2, .., qn)
Hamiltonian
K =pB
pA=
QB
QA=
�B exp[−βH]dpdq�A exp[−βH]dpdq
Thursday, 22 December 2011
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microscopic picturestatistical mechanics
partition function
H = T + V
β =1
kBT
H =�
i
p2i
2mi+ V (q1, q2, .., qn)
K =
�B exp[−βV ]dq�A exp[−βV ]dq
Hamiltonian
integrate over momenta
equilibrium determined solely by potential energy surface
K =pB
pA=
QB
QA=
�B exp[−βH]dpdq�A exp[−βH]dpdq
Thursday, 22 December 2011
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microscopic picture
rare eventτrxn � τeq k = 1/τrxn
compute rates from simulations
Thursday, 22 December 2011
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microscopic picture
rare event
initial rate
basic assumptions
stationary conditions
τrxn � τeq k = 1/τrxn
dρ(p, q)
dt= 0
compute rates from simulations
Thursday, 22 December 2011
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microscopic picture
rare event
initial rate
basic assumptions
stationary conditions
τrxn � τeq k = 1/τrxn
dρ(p, q)
dt= 0
J = kcAcA = �Θ(x‡ − x)�
compute rates from simulations
flux
Thursday, 22 December 2011
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microscopic picture
rare event
initial rate
basic assumptions
stationary conditions
sampling problem...
τrxn � τeq k = 1/τrxn
dρ(p, q)
dt= 0
J = kcAcA = �Θ(x‡ − x)�
ρ(x‡) =
�exp[−βV (x)]δ(x− x‡)dx�
exp[−βV (x)]dx
compute rates from simulations
flux
Thursday, 22 December 2011
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Eyring theoryassumptions
classical dynamics
no recrossing
molecules at barrier in thermal equilibrium with molecules in reactant well
Thursday, 22 December 2011
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Eyring theoryobservations
barrier is flat: f(x‡) =dU
dx|x=x‡ = 0
δL
Thursday, 22 December 2011
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Eyring theoryobservations
barrier is flat: f(x‡) =dU
dx|x=x‡ = 0
δN δLthere are molecules in
δL
Thursday, 22 December 2011
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Eyring theoryobservations
barrier is flat: f(x‡) =dU
dx|x=x‡ = 0
δN δLthere are molecules in
v >δL
δtreaction if
δL
Thursday, 22 December 2011
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Eyring theoryobservations
barrier is flat: f(x‡) =dU
dx|x=x‡ = 0
N rxn = δNvdt
δL
δN δLthere are molecules in
v >δL
δtreaction if
dtthe number of molecules passing TST in
δL
Thursday, 22 December 2011
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Eyring theoryobservations
barrier is flat: f(x‡) =dU
dx|x=x‡ = 0
N rxn = δNvdt
δL
k+ =N rxn
Ndt=
δN
N
v
δL
δN δLthere are molecules in
v >δL
δtreaction if
dtthe number of molecules passing TST in
δL
reaction rate
Thursday, 22 December 2011
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Eyring theoryobservations
barrier is flat: f(x‡) =dU
dx|x=x‡ = 0
N rxn = δNvdt
δL
k+ =N rxn
Ndt=
δN
N
v
δL
δN
N=
q‡
qA
k+ =q‡
qA
v
δL
δN δLthere are molecules in
v >δL
δtreaction if
dtthe number of molecules passing TST in
partition function
δL
reaction rate
Thursday, 22 December 2011
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Eyring theory
partition function of TST
q‡ =1
hδL
� ∞
−∞exp[−β(
p2
2m+ V (x‡)]dp
q‡ =δL
h
� ∞
−∞exp[−β
p2
2m]dp exp[−βV (x‡)]
q‡ =δL
h
�2mkBTπ exp[−V (x‡)
kBT]
Thursday, 22 December 2011
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Eyring theory
partition function of TST
only positive velocities contribute
q‡ =1
hδL
� ∞
−∞exp[−β(
p2
2m+ V (x‡)]dp
q‡ =δL
h
� ∞
−∞exp[−β
p2
2m]dp exp[−βV (x‡)]
q‡ =δL
h
�2mkBTπ exp[−V (x‡)
kBT]
�v+� =�∞−∞ vΘ(v) exp[−β p2
2m ]dp�∞−∞ exp[−β p2
2m ]dp
�v+� =1m
122mkBT√2mkBTπ
=
�kBT
2πmThursday, 22 December 2011
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Eyring theorytaking together to express rate
k+ =δL
h
√2mkBTπ
δLqA
�kBT
2πmexp
�−V (x‡)
kBT
�
k+ =kBT
hqAexp
�−V (x‡)
kBT
�
Thursday, 22 December 2011
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Eyring theorytaking together to express rate
partition function of A
k+ =δL
h
√2mkBTπ
δLqA
�kBT
2πmexp
�−V (x‡)
kBT
�
k+ =kBT
hqAexp
�−V (x‡)
kBT
�
qA =1
h
� x‡
−∞exp
�−V (x)
kBT
�dx
� ∞
−∞exp
�−β
p2
2m
�dp
qA =1
h
�2πmkBT
� x‡
−∞exp
�−V (x)
kBT
�dx
Thursday, 22 December 2011
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Eyring theorytaking together to express rate
partition function of A
k+ =δL
h
√2mkBTπ
δLqA
�kBT
2πmexp
�−V (x‡)
kBT
�
k+ =kBT
hqAexp
�−V (x‡)
kBT
�
qA =1
h
� x‡
−∞exp
�−V (x)
kBT
�dx
� ∞
−∞exp
�−β
p2
2m
�dp
qA =1
h
�2πmkBT
� x‡
−∞exp
�−V (x)
kBT
�dx
Thursday, 22 December 2011
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Eyring theoryharmonic approximation
V (x) ≈ 1
2kf (x− xA)
2
V (x) ≈ 1
2mω2
A(x− xA)2
ωA =
�kfm
Thursday, 22 December 2011
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Eyring theoryharmonic approximation
V (x) ≈ 1
2kf (x− xA)
2
V (x) ≈ 1
2mω2
A(x− xA)2
qA =1
h
�2πmkBT
�2kBT
mω2A
√π
qA =1
h2πkBT
1
ωA
partition function
ωA =
�kfm
Thursday, 22 December 2011
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Eyring theoryharmonic approximation
Final result: Eyring equation
V (x) ≈ 1
2kf (x− xA)
2
V (x) ≈ 1
2mω2
A(x− xA)2
qA =1
h2πkBT
1
ωA
k+ =ωA
2πexp
�−V (x‡)
kBT
�
partition function
ωA =
�kfm
Thursday, 22 December 2011
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Eyring theoryharmonic approximation
Final result: Eyring equation
V (x) ≈ 1
2kf (x− xA)
2
V (x) ≈ 1
2mω2
A(x− xA)2
qA =1
h2πkBT
1
ωA
k+ =ωA
2πexp
�−V (x‡)
kBT
�
partition function
ωA =
�kfm
attempt frequency
probability to be at barrier (Boltzmann factor)Thursday, 22 December 2011
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static solvent effects: potential of mean forceoops
Thursday, 22 December 2011
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Dynamic solvent effects: Kramers Theory
friction due to interactions
thermal noise (Brownian motion): Langevin dynamics
d�v�dt
= − g
m�v� = −γ�v�
�v� = �v�0e−γt
dv
dt= −γv + FR(t)
�FR� = 0
�FR(t1)FR(t2)� = φ(t2 − t1) ≈ fδ(t2 − t1)
coupling between reaction coordinate and other coordinates
noise properties
solution
v = v0 exp [−γt] + exp [−γt]
� t
0exp [γt]FR(τ)dτ
Thursday, 22 December 2011
![Page 42: Rate Theory (overview) - Max Planck Society · Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van ‘t Hoff & Arrhenius equation](https://reader035.vdocuments.site/reader035/viewer/2022062602/5ee1c6d9ad6a402d666c88eb/html5/thumbnails/42.jpg)
Dynamic solvent effects: Kramers Theorycoupling between reaction coordinate and other coordinates
solution
v = v0 exp [−γt] + exp [−γt]
� t
0exp [γt]FR(τ)dτ
�v� = v0 exp [−γt]
limt→∞
�v� = 0
�v2� = v20 exp [−2γt] +f
2γ(1− exp [−2γt])
limt→∞
�v2� = f
2γ
1
2m�v2� = 1
2kBT
f = 2γkBT/m�FR(t)FR(0)� = δ(t)2γkBT/m
properties
equipartition theorem:
Thursday, 22 December 2011
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Dynamic solvent effects: Kramers TheoryFokker-Planck equation
∂P (r, v; t)
∂t= −v
∂P (r, v; t)
∂r+
1
M
∂U
∂r
∂P (r, v; t)
∂v+ γ
∂
∂v(vP (r, v; t)) +
γkBT
M
∂2P (r, v; t)
∂v2
probability
P (r, v; t)drdv
r, r + dr
v, v + dv
to find a particle at
with velocity
Thursday, 22 December 2011
![Page 44: Rate Theory (overview) - Max Planck Society · Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van ‘t Hoff & Arrhenius equation](https://reader035.vdocuments.site/reader035/viewer/2022062602/5ee1c6d9ad6a402d666c88eb/html5/thumbnails/44.jpg)
Dynamic solvent effects: Kramers TheoryFokker-Planck equation
∂P (r, v; t)
∂t= −v
∂P (r, v; t)
∂r+
1
M
∂U
∂r
∂P (r, v; t)
∂v+ γ
∂
∂v(vP (r, v; t)) +
γkBT
M
∂2P (r, v; t)
∂v2
stationary solution
∂P
∂t= 0
P (r, v) =1
Qexp
�−�mv2
2+ V (r)
�/kBT
�
Boltzmann distribution
Thursday, 22 December 2011
![Page 45: Rate Theory (overview) - Max Planck Society · Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van ‘t Hoff & Arrhenius equation](https://reader035.vdocuments.site/reader035/viewer/2022062602/5ee1c6d9ad6a402d666c88eb/html5/thumbnails/45.jpg)
Dynamic solvent effects: Kramers TheoryFokker-Planck equation
∂P (r, v; t)
∂t= −v
∂P (r, v; t)
∂r+
1
M
∂U
∂r
∂P (r, v; t)
∂v+ γ
∂
∂v(vP (r, v; t)) +
γkBT
M
∂2P (r, v; t)
∂v2
steady state solution
P (r, v) = Y (r, v)1
Qexp
�−�mv2
2+ V (r)
�/kBT
�
r ∼ rA ⇒ Y (r, v) = 1
r ∼ rC ⇒ Y (r, v) = 0
boundary conditions
Thursday, 22 December 2011
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Dynamic solvent effects: Kramers Theoryfree energy surface surface
U(r) = U(rB)−1
2mω2
B(r − rB)2
U(r) = U(rA) +1
2mω2
A(r − rA)2
Thursday, 22 December 2011
![Page 47: Rate Theory (overview) - Max Planck Society · Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van ‘t Hoff & Arrhenius equation](https://reader035.vdocuments.site/reader035/viewer/2022062602/5ee1c6d9ad6a402d666c88eb/html5/thumbnails/47.jpg)
Dynamic solvent effects: Kramers Theoryfree energy surface surface
U(r) = U(rB)−1
2mω2
B(r − rB)2
U(r) = U(rA) +1
2mω2
A(r − rA)2
k+ =ωA
2πωB
��γ2
4+ ω2
B − γ
2
�exp [− (U(rB)− U(rA)) /kBT ]
stationary solution to Fokker-Planck equation
Thursday, 22 December 2011
![Page 48: Rate Theory (overview) - Max Planck Society · Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van ‘t Hoff & Arrhenius equation](https://reader035.vdocuments.site/reader035/viewer/2022062602/5ee1c6d9ad6a402d666c88eb/html5/thumbnails/48.jpg)
Dynamic solvent effects: Kramers Theoryfree energy surface surface
U(r) = U(rB)−1
2mω2
B(r − rB)2
U(r) = U(rA) +1
2mω2
A(r − rA)2
k+ =ωA
2πωB
��γ2
4+ ω2
B − γ
2
�exp [− (U(rB)− U(rA)) /kBT ]
stationary solution to F-P equation
Thursday, 22 December 2011
![Page 49: Rate Theory (overview) - Max Planck Society · Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van ‘t Hoff & Arrhenius equation](https://reader035.vdocuments.site/reader035/viewer/2022062602/5ee1c6d9ad6a402d666c88eb/html5/thumbnails/49.jpg)
Dynamic solvent effects: Kramers Theoryfree energy surface surface
U(r) = U(rB)−1
2mω2
B(r − rB)2
U(r) = U(rA) +1
2mω2
A(r − rA)2
k+ =ωA
2πωB
��γ2
4+ ω2
B − γ
2
�exp [− (U(rB)− U(rA)) /kBT ]
limiting cases
γ/2 � ωB k+ =ωAωB
2πγexp
�−∆U ‡/kBT
�
γ/2 � ωB k+ =ωA
2πexp
�−∆U‡/kBT
�high friction
low friction
stationary solution to F-P equation
Thursday, 22 December 2011
![Page 50: Rate Theory (overview) - Max Planck Society · Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van ‘t Hoff & Arrhenius equation](https://reader035.vdocuments.site/reader035/viewer/2022062602/5ee1c6d9ad6a402d666c88eb/html5/thumbnails/50.jpg)
Dynamic solvent effects: Kramers Theoryfree energy surface surface
U(r) = U(rB)−1
2mω2
B(r − rB)2
U(r) = U(rA) +1
2mω2
A(r − rA)2
k+ =ωA
2πωB
��γ2
4+ ω2
B − γ
2
�exp [− (U(rB)− U(rA)) /kBT ]
limiting cases
γ/2 � ωB k+ =ωAωB
2πγexp
�−∆U ‡/kBT
�
γ/2 � ωB k+ =ωA
2πexp
�−∆U‡/kBT
�high friction
low friction
transmission coefficient
k+ = κkTST+
stationary solution to F-P equation
Thursday, 22 December 2011