rate theory (overview) - max planck society · rate theory (overview) macroscopic view...

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

Chromophore in water

CASSCF(6,6)/3-21G//SPCE molecular dynamics

p-hydroxybenzylidene acetone (pck)

resonance structuresThursday, 22 December 2011

Rate of photoisomerization of double bond

uni-molecular process, initiated by photon absorption

Thursday, 22 December 2011

measuring reaction rate

simulation & pump-probe fluorescence


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


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


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




�∂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




�∂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


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


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



relation between equilibrium and rate constant

d lnK

d1/T= −∆H

R


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



relation between equilibrium and rate constant

therefore

d

d1/Tln k = −E

R+ a


temperature dependence of reaction ratesArrhenius equation

activated state

K‡ =[A‡]

[A]

A � A‡ → B


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


microscopic picturestatistical mechanics

partition function

β =1

kBTK =

pB

pA=

QB

QA=

�B exp[−βH]dpdq�A exp[−βH]dpdq



partition function

H = T + V

β =1

kBT

H =�

i

p2i

2mi+ V (q1, q2, .., qn)

Hamiltonian

K =pB

pA=

QB

QA=




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=



microscopic picture

rare eventτrxn � τeq k = 1/τrxn

compute rates from simulations


microscopic picture

rare event

initial rate

basic assumptions

stationary conditions

τrxn � τeq k = 1/τrxn

dρ(p, q)

dt= 0



microscopic picture

rare event

initial rate

basic assumptions



dρ(p, q)

dt= 0

J = kcAcA = �Θ(x‡ − x)�


flux


microscopic picture

rare event

initial rate

basic assumptions


sampling problem...


dρ(p, q)

dt= 0

J = kcAcA = �Θ(x‡ − x)�

ρ(x‡) =

�exp[−βV (x)]δ(x− x‡)dx�

exp[−βV (x)]dx


flux


Eyring theoryassumptions

classical dynamics

no recrossing

molecules at barrier in thermal equilibrium with molecules in reactant well


Eyring theoryobservations

barrier is flat: f(x‡) =dU

dx|x=x‡ = 0

δL




dx|x=x‡ = 0

δN δLthere are molecules in

δL




dx|x=x‡ = 0


v >δL

δtreaction if

δL




dx|x=x‡ = 0

N rxn = δNvdt

δL


v >δL

δtreaction if

dtthe number of molecules passing TST in

δL




dx|x=x‡ = 0

N rxn = δNvdt

δL

k+ =N rxn

Ndt=

δN

N

v

δL


v >δL

δtreaction if


δL

reaction rate




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


v >δL

δtreaction if


partition function

δL

reaction rate


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]


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

Eyring theorytaking together to express rate

k+ =δL

h

√2mkBTπ

δLqA

�kBT

2πmexp

�−V (x‡)

kBT

�

k+ =kBT

hqAexp

�−V (x‡)

kBT

�


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


Eyring theoryharmonic approximation

V (x) ≈ 1

2kf (x− xA)

2

V (x) ≈ 1

2mω2

A(x− xA)2

ωA =

�kfm



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



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



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

static solvent effects: potential of mean forceoops


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


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:


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



∂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



∂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


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



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



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


stationary solution to F-P equation



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


limiting cases

γ/2 � ωB k+ =ωAωB

2πγexp

�−∆U ‡/kBT

�

γ/2 � ωB k+ =ωA

2πexp

�−∆U‡/kBT

�high friction

low friction




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


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+



rate theory (overview) - max planck society · rate theory (overview) macroscopic view...

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