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“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.