“Van der Waals” Wells are Important in Chemical Reactions

University of Florida, QTP Nov. 6, 2002

Acknowledgments:

Dunyou Wang (now at NASA/Ames), Tiao

Xie (Emory), David Manolopoulos (Oxford),

$$ from US Dept. of Energy

Cl + HD D+HCl, H+DCl reaction

• Importance of this reaction– It plays a central role in fundamental chemical kinetics, and

has served as a critical test case for bimolecular reaction rate theory, especially transition-state and kinetic isotope effect. And, the theory of isotope effects was derived from it.

– This reaction is also a prototype for a host of Cl reactions that are in atmospheric chemistry and photochemical air pollution.

– This reaction is the rate determining step in the mechanism of the Cl2 + H2 2HCl chain reaction.

Studies of the Cl + H2 reaction

• Experimental studies:– Rate constants for Cl + H2 and D2 reactions over the temperature

range 296-3000 K. – Branching ratio of Cl + HD reaction has been studied in crossed

molecular beam experiment.

• Theoretical studies:– Many potential energy surfaces have been constructed for this

reaction, among which, the G3 surface most successful one.– VTST have been used to calculate rate constants on these

surfaces, and compared with experimental data. Truhlar and co.– Quantum reactive scattering on G3 and a new pes

Manolopous, Werner and co-workers

The “G3” potential energy surface

• G3 surface was constructed by Truhlar et al. in 1996.• It’s based on the so-called GQQ surface, which has been

shown to give good agreement with experiment on Cl + H2 and D2 reactions.

• G3 surface improves on the GQQ surface in the region of Cl-H-H bending potential.

• Linear saddle point geometry:RHCl (Å) = 1.4011

RHH’ (Å) = 0.9896

RH’Cl (Å) = 2.3907

V (kcal/mol) = 7.88

G3 Success

Cl + H2Cl + D2

Failure of the G3 surface

Branching ratio determined in cross-beam experiment as a function of collision energy for HD(j=0).

K. Liu (1999)

Collision energy (kcal/mol)

Contour Plot of G3 Surface

Cl

H

H

R

r

Jacobi Coordinates

G3 surface and Bian-Werner surface

• BW and G3 surface are broadly similar– Barrier height: (kcal/mol)

7.88 (G3) 7.61 (BW)– Saddle point frequencies (cm-1)

bending: 581 (G3) 540 (BW)

stretching: 1358 (G3) 1360 (BW)

• Difference– Imaginary frequency (cm-1)

1520i (G3) 1294i (BW)

This indicates that G3 surface has a thinner barrier.– BW has a Van der Waals well with a depth of 0.5 kcal/mol at a

T-shape equilibrium geometry.

G3 surface and Bian-Werner surfaces

Theory and ExperimentManolopoulos Science (1999)

G3 and BW surfaces

Cl H D

H

D

Cl

Prob to form HCl reduced

On BW relative to G3

Conclusion

Van der Waals well (very shallow) in

Cl+HD has a significant effect on

branching ratio for Cl + HD(j=0) but not

on rate constant

The O(3P)+HCl Reaction

A challenging reaction, non-linearsaddle point, ‘heavy-light-heavy’ system.

H. Koizumi, G. C. Schatz, and M. S. Gordon, J . Chem. Phys. (1991).

W. H. Thompson and W.H. Miller, J . Chem. Phys. (1996).

O. I. Tolstkhin, K. Nobusada and H. Nakamura, J . Chem Phys. (1998)

F. J . Aoiz, L. Bañares, J . F. Castillo, M. Menèdez, and J . E.Verdasco, PCCP (1999).

F. Matzkies and U. Manthe, J . Chem. Phys. (2000).

Barrier height of KSG adjusted down by KSG to get agreement with exp on k(T). Those calculations werenot converged so later calcs showed disagreement withExperiment - barrier height too small. New surface ‘S4’by Ramuchandran, barrier height is higher than KSG, but ...

RATE CONSTANT FOR O(RATE CONSTANT FOR O(33P)+ HCl ON S4 P)+ HCl ON S4

S.Z.B.A.T.L.R.G.L JPC (2001)

1

10

100

1000

104

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Smith

Fontijn QM/JS - S4QM/JS - KSG

ICVT/ μ - 4OMT S

(k cm

3

/ - ) 10molec sec X

16

1000/ ( )T deg K

The exact expression for k(T)

k(T)=1

hQreactdEN(E)exp(−E/kBT)

0

∞∫

N(E) = (2J + 1) P

,i f

,J K

,i f

∑

K = − J

J

∑

J = 0

∑ ( )E

P

,i f

,J K

( )E = | S

i, f

J, K

(E) |

2

N(E) is the Cumulative Reaction Probability

NTST(E)= (2J+1)

J =0∑ θ

n=0∑

K=−J

J∑ (E-En,J,K

TS )

En,J,KTS = V0+Evib

TS+EJ,KTS

(Variational) Transition State Theory

‡

TST Derivation

k(T)=1

hQreactdEN(E)exp(−E/kBT)

0

∞∫

kTST(T)=kBT

hQTS

Qreactexp[−(V0+E0

TS)/kBT]

NTST(E)= (2J+1)

J =0∑ θ

n=0∑

K=−J

J∑ (E-En,J,K

TS )

En,J,KTS = V0+Evib

TS+EJ,KTS

POTENTIALS FOR O(POTENTIALS FOR O(33P)+ HCl REACTIONP)+ HCl REACTION

The O(3P)+HCl Reaction

Configuration (bohr and degrees) of the saddle point and the Van der

Waals minima in the appropriate set of Jacobi coordinates.

O-HCl Cl-OH

Saddle Point vdW Well Saddle Point vdW Well

R 4.56 6.22 4.50 4.21

r 2.66 2.46 2.42 1.90

γ 23.4 0. 0 26.3 74.8

The O(3P)+HCl Reaction

O H Cl

O HCl

O H

Cl

-1.6

9.8 kcal

-5.2

The O(3P)+HCl Reaction

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.3 0.4 0.5 0.6 0.7 0.8

KSG

S4

CRP (J=0)

E (eV)

S. Skokov, T. Tsuchida, S. Nanbu, J. M. Bowman, and S. K. Gray, J Chem. Phys(2000).

K. Nobusada, H. Nakamura, Y. Lin, B. Ramachandran, J. Chem. Phys. (2000)

CRP(J=0) =

Pi,fi,f∑ (E )

The O(3P)+HCl ReactionXie, Wang, Bowman, Manolopoulos (2002)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40

CRP (J = 0)

E (eV)

1

3

4

2

5

6

7

8

9

The O(3P)+HCl Reaction

0.000

0.005

0.010

0.015

0.020

0.2358480 0.2358485 0.2358490 0.2358495 0.2358500

Resonance 1

CRP

E (eV)

The O(3P)+HCl Reaction

Bound states

Quasi-bound states

Resonances and density of states

Resonances are therefore like bound states in some respects, or bound states are resonances with zero widths.

Eth

Resonances and lifetimes

Ψn(t)=ψ ne−iEnt / h =ψ ne

−i(Ern−iΓn2

)t/ h

=ψ ne−iErnt/ h

e−

Γn2

t / h

Pn(t)=Pn(0)e−Γnt/ h

The more conventional relationship is givenas follows:

This is unimolecular decay of an (isolated) resonance,with a decay rate equal to Γ / h

The (quasi) bound state approach

Resonances are quasibound eigenstates

with complex energy eigenvalues, Er,n-i n /2

HC = H - iU(R)

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

2.0 3.0 4.0 5.0 6.0 7.0 8.0

F00

(R)

R (bohr)

x 200

0

5

10

15

20

25

30

2.0 3.0 4.0 5.0 6.0

HN2 -> H+N

2

Vmin

(R)

R (bohr)

Quasibound State calculations

A primitive basis of twenty Legendre functions,

Eight vibrational functions of HCl for O+HCl

(range: 1.6 a0 to 3.3a0) and 8 OH vibrational functions

for Cl + OH (range 1.2a0 to 3.6a0) and

100 sine functions in R for each arrangement

Ranges of R are [3.4a0 ,10.2a0] for the O+HCl channel and

[3.2a0 , 8.0a0] for the C+lOH channel.

Length of the absorbing potential: 2.0a0

A contraction scheme was used to reduce the direct product

basis from to 16,000 to 4770.

400 of the real wavefunctions used to construct complex H-matrix.

The range of was 0.001 to 0.5 h, in

steps of 0.01 h.

Quasibound State calculations

O-HCl well Cl-OH well(vR, ν, νr) Ener (gy eV) (vR, ν, νr) Ener (gy eV)

(1,6, 0) 0.2 361 (5,0, 0) 0.1 939

(0,7, 0) 0.2 496 (2,1, 0) 0.2 040

(0, 8, 0) 0 . 2702 (6, 0, 0) 0 . 2124

(1, 8, 0) 0 . 2750 (3, 1, 0) 0 . 2246

(0, 9, 0) 0 . 2935 (0, 2, 0) 0 . 2355

(0, 10 , 0 ) 0 . 3194 (4, 1, 0) 0 . 2414

(1, 10 , 0 ) 0 . 3243 (1, 2, 0) 0 . 2580

(0, 11 , 0 ) 0 . 3787 (2, 2, 0) 0 . 2751

(0, 3, 0) 0 . 3110

Resonance Peak position Quasibound state energy

O-HCl well Cl-OH well

1 0.2359 0.2361 0.2355

2 0.2417 0.2414

3 0.2497 0.2496

4 0.2584 0.2580

5 0.2755 0.2750 0.2751

6 0.2923 0.2935

7 0.3113 0.3110

8 0.3252 0.3243

9 0.3761 0.3787

Comparison of resonance energies and quasiboundState energies of VdW wells (eV)

Comparison of resonance energies and quasiboundstate energies of VdW wells (eV)

Resonance Probability Width VdW Well Overlap

1 0.169E-01 0.001 O-HCl Cl-HO 1.2e-12 9.3e-5

2 0.677E-04 1.02 Cl-HO 1.6e-6

3 0.405E-06 11.3 O-HCl 7.5e-12

4 0.331E-02 0.306 Cl-HO 6.0e-6

5 0.613E-03 5.65 O-HCl Cl-HO 1.7e-11 1.2e-5

6 0.261E-04 66.5 O-HCl 7.0e-9

7 0.701E-01 0.677 Cl-HO 2.80e-4

8 0.220E-02 50.0 O-HCl 1.24e-8

9 0.377E-01 69.3 O-HCl 1.48e-6

Overlap = quasibound density in the saddle point region

Assignment of resonances

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40

CRP (J = 0)

E (eV)

O

Cl

Cl

Cl

O

Cl

O

OCl

Quasibound state wavefunctions

4 6 8 10

150

120

90

60

30

R (bohr)

gamma (deg)

4 6 8 10

3.0

2.5

2.0

R (bohr)

r (bohr)

O-HCl state at 0.2496 eV

Quasibound state wavefunctions

Cl-HO state at 0.2414 eV

4 5 6 7

150

120

90

60

30

R (bohr)

gamma (deg)

4 5 6 7

3.5

3.0

2.5

2.0

1.5

R (bohr)

r (bohr)

CONCLUSIONSCONCLUSIONS

Resonances in the tunneling region due toVan der Waals minima.

Important effect on k(T) - increasing, why?a) Resonances “prepare complexes”b) Non-adiabaticity?

Recall

Question bend zpe. Do wells destroy bending Adiabaticity?

En,J,KTS = V0+Evib

TS+EJ,KTS

Other examplesOther examples

OH+HNO3

Negative T-dependence

indicates fairly complex

and positive T-dependence

indicates a barrier, as usual.