多通道多位井速率常数的计算Rate Constants for Multi-Channel, Multi-Well Reactions

张绍文

北京理工大学

化学反应速率理论

碰撞理论（经典碰撞理论，轨线法，量子散射理论）

过渡状态理论（传统过渡状态理论，变分过渡状态理论）

过渡状态理论的基本假设

玻恩 -奥本海默近似 反应物微观状态保持玻尔兹曼分布 不返回假定 运动分离假定

化学反应速率常数的计算 正则系综速率常数的计算

传统过渡态理论

正则变分过渡态理论

TkvR

B BeQ

Q

h

TkTk /0)(

),(min)(

)(),( /)(

sTkTk

eQ

sQ

h

TksTk

s

TksvR

B B

考虑到量子隧道效应时

)()()( TkTTk

微正则系综速率常数的计算 传统过渡态理论

)(

)()(

Eh

ENEk

QeETEf

dETEfEkTk

TkE B /)(),(

),()()(

/

0

微正则变分过渡态理论

)(

)},({min)(Eh

SENEk s

隧道穿透系数的计算

BWK近似

2

1

2/1))((22

)(

x

xdxExV

m

eE

dEe

dEeET

v

TkE

TkE

B

B

0

/0

/)()(

Master Equation Method

Why Use Master Equation

• Calculate pressure dependence of rate constants• Calculate branching ratios of multi-channel

reactions• High accuracy

Single Well Multi-channel Case

B + C

D + E

F + G

A

• Methodology

A B + C

E

E0

Eik(Ei)

Ej

Rij Rji

A

B+C

ni

nj

0

)(][][

jE

iijijijiji nEkdEnRnRM

dtdn

• ni is the population of reactant molecules at energy Ei.

• [M] is the concentration of bath gas.

• Rij is the rate of collision-induced excitation from Ej to Ei of the reactant molecule on collision with a bath gas molecule (Energy transfer coefficient).

• k(Ei) is the microcanonical rate constant of the reaction at energy Ei

1. Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and Recombination Reactions; Blackwell: London, 1990.

2. Klippenstein, S. J., Harding, L. B. J. Phys. Chem. 1999, 103, 9388.

3. Diau, E. W. G, Lin M. C. J. Phys. Chem. 1995, 99, 6589.

4. Robertson, S. H., Pilling, M. J., Baulch, D. L., Green, N. J. B. J. Phys. Chem. 1995, 99, 13452.

j

iiijijiji nknRnREMtn )(][

dd

j

iiijijiji nknPnPEZ

tn )(

dd

Z: collision number per unit time, collision frequency, (time-1)

Pi(E,E’): probability of energy transferred per collision, (energy-1)

ij

jiiiiijij PEZkJjiEPZJ

dt

;,

d Jnn

gJg unik kuni is pressure dependent thermal rate constants

gBg unik

1SJSB jiBfB ijiii ,0;/1

Energy Transfer Rate Coefficient

Pij=c(Ej)exp[-(Ej-Ei)/], Ei < Ej

Exponential down model

Pji f(Ei) = Pij f(Ej)

f(Ei) = [(Ei) exp(- Ei/kBT)]/Q

c(E) is normalization coefficient; is energy transfer constant; f(E) is distribution function; (E) is density of state; Q is partition function.

E

Ei

Ej

Rij Rji

1d),(0

iji EEEP

Microcanonical Rate Constant

)(

)},({min)(Eh

SENEk

R

GTSs

NGTS(E,S) is the sum of states of the Generalized Transition State (GTS).

R(E) is the density of states of the reactant.

1. Garrett, B. C.; Truhlar, D. G.; Grev, R. S.; Magnuson, A. W. J. Phys. Chem. 1980, 84, 1370.2. Hase, W. L. Acc. Chem. Res. 1998, 31, 659.3. Forst, W. Theory of Unimolecular Reactions; Academic: London, 1973. 4. Baer, T.; Hase, W. L. Unimolecular Reaction Dynamics. Theory and Experiment; Oxford: New York, 1996.5. Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and Recombination Reactions; Blackwell: London,

1990.

Rate Constants for Multi-channel Reaction

B + C

D + E

F + G

A

j

iiijijiji nknRnREMtn )(][

dd

i

iij

ijijiji knnRnREMtn )(][

dd

Sum over channels

Rate Constants for Multi-Well Multi-channel Reaction

),...,()()()()()()(

)()()()()(d)(),(d

d0

'''

MIiEnEknnEfEkKEnEk

EnEkEnEkEZnEEnEEPZt

En

ipimRidiRiidi

M

ijjij

M

ijijiiE ii

i

i

1. J. A. Miller, S. J. Klippenstein, S. H. Robertson, J. Phys. Chem. A 2000, 104, 7525-7536

2. S. J. Klippenstein, J. A. Miller, J. Phys. Chem. A 2002, 106, 9267-9277

Z: collision number per unit time, collision frequency, (time-1)

Pi(E,E’): probability of energy transferred per collision, (energy-1)

Pi(E’,E) fi(E) = Pi(E,E’) fi(E’)

kij(E)j(E)=kji(E)i(E)

M

IiE idiRimR

M

IiE idi

R

ii

dEEfEkKnndEEnEkt

n00

)()()()(d

d

RmB nnn

wt

wG

d

d

N

jjj

t wggetw j

0

)0()(

Solution to the Master Equation

• Finding the eigenvalue and eigenvectors ?

• Solving the stiff ordinary differential equations ?

0.001

0.01

0.1

1

0.1 1 10 100 1000 10000

Pressure (Torr)

p

Branching fraction of product (stiff ODE results)

C2H5+O2=C2H4+HO2 的产物产率

实验 理论

Master Equation Study of HMX decomposition

H

O

H

N

H

O

N

H

OO

N

NN

N

H

O

H

N

O

HH

NO

O

H H

H HN

N

O O

O O

N N

H H

N N

O O

N

H H

H

H

H

H

H

O

H

N

N

N

N

O

O

N

N

H

N

H

O

O

O

O

N

O

O

O

N+I

II

P

0

10

20

30

40

50

Reaction Coordinate

E (

kcal

/mol

)

NO2 Fission (44.17 kcal/mol)

HONO Elimination (47.29 kcal/mol)

Potential energy profile of the HONO elimination and NO2 fission chnnels

-6

-4

-2

0

2

4

6

8

10

-3 -2 -1 0 1 2 3 4

log{P/Torr}

log{k(T,P)/s

-1}

500 K800 K1000 K1500 K

-2

0

2

4

6

8

10

-3 -2 -1 0 1 2 3 4

log{P/Torr}

log{k(T,P)/s

-1}

500 K800 K1000 K1500 K

Pressure dependent rate constants

a b

NO2 fission HONO elimination

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-3 -2 -1 0 1 2 3 4

log{P/Torr}

log{kNO2 (T)/k

HONO (T)} 500 K

800 K1000 K1500 K

Branching ratios vs pressure

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

400 800 1200 1600

T/K

log{k

NO2(T)/k

HONO(T)}

0.005 Torr0.01 Torr0.05 Torr1 Torr10 Torr1000 TorrHigh pressure lim it

Branching ratios vs temperature