Long-Range TST for predicting rate constants of barrier-less reactions at low temperatures

Yuri Georgievskii and Stephen J. Klippenstein

2

• At large interfragment separations J ~= Jorb and the expression for NE,J is simplified

• LR interaction potential can be represented by a sum of a few 1/Rn terms

• When a single term in the potential dominates

an analytical expression for the capture rate constant is obtained

Where C1 is a unique constant corresponding to a given angular form f0

nnn

R

fVRV

)(),(

nR

fVRV

)(),( 00

nB

n TkVCk /22/1/20

2/11 )(

Long Range TSTLong Range TST

3

( ) 2 /k T q

1/ 2 1/ 2( ) 2k T qd T

2 / 31/ 2 1/ 61( )k T C qQ T

LR-TST: Analytic ResultsLR-TST: Analytic Results

Dipole-Dipole

Dipole-Quadrupole

1/ 2 2 / 3 1/ 61 1 2( ) ( )k T C d d T

1/ 2 1/ 21( )k T C dQ T

Ions Neutrals

Ion-Induced Dipole

– Langevin Result

Ion-Dipole

Ion-Quadrupole

4

O(3P) + H3+

-8

-6

-4

-2

0

2 2.5 3 3.5 4 4.5 5

O(3P) + H

3

+

x doubly occ.y doubly occ.z doubly occ.

V (

kca

l/mo

l)

ROH

(A)

3x10-10

4x10-10

5x10-10

6x10-10

7x10-10

8x10-109x10-101x10-9

2x10-9

10 100

TSTFehsenfeldMcEwanCollinsk

(cm

3 mo

lecu

le-1

s-1

)

Temperature (K)

5

CO + H3+

Jarrold, Bowers, DeFrees, McLean, Herbst, Astrophys. J, 303, 392 (1986)

6

Reaction T, K Expt D-D D-Q D-iD Disp Ratio

OH+isoprene 58 0.78 1.45 3.85 2.50 4.23 5.54 7.10

OH+Propene 58 0.57 2.13 2.97 2.23 3.84 4.8 8.42

OH+1-Butene 23 4.27 2.71 2.63 2.00 3.44 4.55 1.07

OH+Z-2-Butene 23 3.89 1.73 2.77 2.02 3.43 4.24 1.09

OH+E-2-Butene 23 4.52 0 2.76 2.03 3.44 4.18 0.92

CN+C2H2 25 4.60 0 4.08 1.38 3.94 4.98 1.08

CN+C2H4 25 4.35 0 2.87 1.44 4.04 4.49 1.03

CN+C2H6 25 1.13 0 1.34 1.43 4.09 4.24 3.75

CN+CH3CCH 15 3.8 3.26 3.43 1.34 3.76 5.04 1.33

CN+CH2CCH2 15 4.4 0 2.84 1.39 3.84 4.3 0.98

LR-TST: comparison with experimentLR-TST: comparison with experiment

7

• LR-TST agrees with Classical Trajectories to within 10%• A single term in the potential expansion is generally not

sufficient• At higher temperatures chemical bonding region important• In all cases LR-TST provides an upper bound for the

reaction constant• Situation in which LR-TST rate constant estimate is too high

even at low temperatures usually indicates the presence of an inner TS state which effects the rate constant.

• Alternatively, this effect may be related to the presence of multiple electronic surfaces

• Difference between LR-TST and experiment usually increases with temperature in accordance with the growing role of the inner TS region

Long range TST: conclusionsLong range TST: conclusions

8

R

VInner TS Outer TS

0

),(/1),(/1),(/1 ## JENJENJEN outerinnereff

“Two-transition-states” model“Two-transition-states” model

Inner transition state:– Direct VTST or RRHO TST

Outer transition state: – Long range TST

9

Minimum Energy Path

Inner Transition State CN + C2H6 Minimum energy path Inner Transition State CN + C2H6 Minimum energy path

Minimum Energy Path +

Zero Point Energy

10

CN + C2H6 2TS ResultsCN + C2H6 2TS Results

11

Ethene, Propene, 1,Z,E,Iso-butenes + O

X

A

J=2

J=1

J=0

M=3

M=2

M=1

Potential energy surface structure:

Alkenes + O(3P)Alkenes + O(3P)

12

Inner transition state:

o Parabolic barrier with tunneling, quantum harmonic normal modes, classical rotations

o Optimization & frequencies calculation: CAS(6e5o2s)PT2/ADZ

o Energies: CAS(6e5o2s)PT2(mix=2,shift=0.2)/ATZ Outer transition state:

o Long range TSTo Center-of-mass-to-center-of-mass reaction coordinateo Effective isotropic interaction chosen to fit the ab initio

minimized LR reactive fluxo Energies: CAS(4e3o3s)PT2

MethodsMethods

13

O(3P) + Alkenes: Rate ConstantsO(3P) + Alkenes: Rate Constants

2D Graph 1

T (K)

1 10 100 1000

k (c

m3

molecule

-1 s

-1)

10-12

10-11

10-10

iso- butene

trans- butenecis- butene

1- butene

propene

ethene

Barrier Heights (1/cm)

Cis-Butene -312Trans-Butene -386Iso-Butene -3371-Butene -168Propene -51Ethylene 324

14

• The “two transition state” feature seems to be common for many barrierless reactions at low temperatures

• There is a broad range of temperatures from tens to several

hundreds K in which it is important to take into account both effects from short range and long range interactions for correct prediction of the rate constant

• Because of the conservation of energy and angular momentum in the TS region the resultant 2TS rate constant is essential reduced in comparison with its value for either of the TS separately. It is

crucially important to use the E,J-resolved level of the theory for rate constant calculations

• LR-TST in conjunction with the 2TS model provides a way for a

quantitative prediction of the rate constant

ConclusionsConclusions

15

Direct configurational space sampling:

o Conserved modes relaxation: UMP2/6-31G*o SP energy calculation: CASPT2/CC-PVDZ

Minimum energy path:

o Constrained optimization: CASPT2/6-311G**o SP energy calculation: RQCISD(T)/CC-PVDZ,CC-PVTZ, CC-

PVQZ, AUG-CC-AVDZ, AUG-CC-AVTZ Zero-point energy:

o Constrained optimization: CASPT2/CC-PVTZo Frequency calculation:CASPT2/AUG-CC-PVDZ

Electronic structure methodsElectronic structure methods

16

• ZPE correction to the potential explains the KIE for CN + ethane reaction

• orientation-dependent ZPE correction is needed to provide quantitatively accurate agreement with experiment

• KIE increases as the temperature decreases. At low temperatures the outer TS starts to play bigger role and KIA should become smaller. It would be interesting to check this prediction

CN+C2H6/ C2D6 kinetic isotope effectCN+C2H6/ C2D6 kinetic isotope effect