The Ozone Isotope Effect

Answers and Questions

The Ozone Isotope Effect

Answers and Questions

Dynamical studies of the ozone isotope effect:

A status report

Ann. Rev. Phys. Chem. 57, 625–661 (2006)

R. Schinke S.Yu. Grebenshchikov, M. V. Ivanov and P. Fleurat-Lessard

Some basic facts about O, O2 and O3

isotopes of oxygen:

16O (0.99763), 17O (3.7× 10−4), 18O (2.0× 10−3)

6 7 8

zero point energies (ZPE) of O2:

EZPE ≈ ω/2 ω ≈√

f/µ µ = m1m2/(m1 +m2)

66: 790.4 cm−1

68: -22.2 cm−1 (1 eV = 8066 cm−1)

88: -45.2 cm−1

(for comparison: kBT = 220 cm−1 at 300K)

forms of ozone:

ozone is predicted by theory to exist in two different forms:

D , cyclic O C , open O3h 2v3 3

R R1 2a

a

a

a

a = 60° a = 117°

However, only Open Ozone exists in the gas phase; the central atom is special:

687 ⇋ 6 + 87 or 68 + 7 EZPE(87) < EZPE(68)

687 ⇋ 67 + 8 is not possible

Some ‘historical’ remarks about O3 isotope effect

1981: Mauersberger measures the fractionation δ(50O3) ∼ 13%(heavy Ozone 668) in the stratosphere (balloon experiments)

1985: Thiemens measures the fractionation δ(49O3) ∼ 11% (667)in laboratory experiments

–1990: more laboratory experiments• Mauersberger et al. Adv. At. Mol.Opt. Phys. 50, 1–54 (2005)

δ(MO3) =

[

(MO3/48O3)meas.

(MO3/48O3)cal.− 1

]

× 100

• very large enrichments

• no apparent mass dependence

• δ(49O3) ≈ δ(50O3)

• “ozone isotope effect” or “ozoneanomaly”

Ozone recombination or formation rate constants

Ozone formation rate:

d[O3]dt

= krec(T ) [O] [O2] [M] [O] ≪ [O2] ≪ [M]

Mauersberger and coworkers measured (under controlled conditions in thelaboratory) krec for several [O,O2] combinations (relative to 666):

6 + 66: krec = 1.00 (normalization)

6 + 88: krec = 1.50 (largest ratio)

8 + 66: krec = 0.92 (smallest ratio)

6 + 68: krec = 1.45

etc.

The measured krec/k666 show a large variationwith no apparent systematic dependence ....

... until they were represented as function of the ZPE difference between thetwo possible diatomic channels:

∆ZPE = EZPE(products)− EZPE(reactants)

D

exothermic endothermic

8 + 66 866 86 + 6+ 23 cm

6 + 88 688 68 + 8- 23 cm-1 -1

Symmetric

666, 868 etc.

Janssen et al. (2001)

The symmetric moleculesbehave differently than thenon-symmetric ones!

• The fractionation constants follow from

the recombination rate constants krec

• Therefore, the krec are the focus of most

theoretical studies

Recombination vs. isotope exchange reaction

(1) O+ PQ→ (OPQ)∗ formation of highly excited complex

(2) (OPQ)∗ → O+ PQ inelastic process (e.g., vib. relaxation)

(OPQ)∗ → OP+Q isotope exchange

(3) (OPQ)∗ +M→ OPQ+M stabilization (energy transf. mechanism)

• relaxation, isotope exchange and recombination are intimately related:they proceed through the same O∗

3 complex.

• reactions (2) are well defined (bi-molecular collisions) and can be rigorouslytreated; they are independent of pressure p.

• stabilization step (3) involves many collisions with M and is extremelycomplicated to treat (for example, master equation); it shows a strong pdependence.

• at low pressures: isotope exchange is much faster than stabilization

O+O2 ⇋ O∗3 interaction potential

• first ‘reasonable’ potential energy surface (PES) calculatedby Siebert et al. in 2001 and 2002

• multi reference configuration interaction (MRCI)

• cc-pVQZ basis set

• global PES

• V (R1, R2, α)

R1 and R2 are the two O–O–O bond lengths, α is the angle

2D contour representations

2

3

4

5

2 3 4 5

R2

[a0]

R1 [a0]

E [e

V]

R1 [a0]

1.0

0.5

0.03 4 5 6

30

60

90

120

150α

[deg

.] — V (R1, R2, α)

— three equivalent wells + cyclic well

— accurate vibrational energies

— very small dissociation barrier(0.006 eV)

— narrow transition state

— quite ‘harmonic’, compact potential(correlation with excited products?)

cyclic O3

– better calculations increaseDe

and decrease the barrier!

– simple modification =⇒potential II

– artificial removal of barrier=⇒ potential III

0.80

0.90

1.00

1.10

3 4 5 6 7 8 9

E [eV

]R1 [a0]

I

III

II

+0.006 eV

−0.014 eV

0

5

10

15

σ [a

02 ]

I

0

5

10

15

σ [a

02 ]

II

0

10

20

30

40

0 1000 2000 3000

σ [a

02 ]

Ec [cm-1]

IIIj=0j=10j=20j=40

exchange reaction

O+O2(j )→ O2(j′) + O

classical trajectory calculations

initial state resolved crosssections for isotopic exchange

σexj (Ecoll.)

depend strongly on the transitionstate barrier!

Exchange reaction rate constant kex(T )

exp.

artificial PES

original PES

• poor agreement with experimental rate

– quantum effects — unlikely for three heavy O atoms– PES (transition state) — much better (i.e., more expensive) calc. do notchange the TS structure

– non-adiabatic effects, i.e. breakdown of BO approximation?

(II)

(III)

New (2004) ab initio calculations in the transition-state region‘at our computational limit’

g[d

eg]

R a[ ]0

Rr

-100

-200

+100

0

g

4 4.5 5 5.5 6 6.5 7 7.5 80

20

40

60

80

100

120

140

160

180

— AQCC, av6z basisset

— not a full PES(RO2 = r fixed)

[see also: Holka et

al., J.Phys.Chem. A36, 9927 (2010)]

The structure of a ‘narrow’ TS with the barrier below the asymptote is confirmed!

Non-adiabatic transitions between different electronic states all correlating with

O(3P ) + O2(X3Σ−

g ) (open shell system)

2003-06-04-S1-WK.CDR

E[c

m-1

]E

[cm

-1]

[a ]0R

S

T

Q

(a)

(b)

j=0

j=1

j=2

– 3 × (5 + 3 + 1) = 27different electronic statescorrelate with the groundstate asymptote.

– Thus, transitions due tonon-adiabatic, spin-orbit orRenner-Teller coupling arepossible!

spin-orbit splitting

Isotope dependence of exchange reaction

the ratio R8,6 =k8+66→86+6k6+88→68+8

has been measured (directly)

it is 1.27 at room temperature (∆ZPE = ±23 cm−1 ≪ 2kBT = 440 cm−1)

DZPE

DZPE

8 + 66 (866)* 86 + 6 6 + 88 (688)* 68 + 8

exothermic endothermic

• Quantum mechanics automatically includes ∆ZPE —classical mechanics, however, does not!

• simple trick: we add ∆ZPE to V (R1, R2, α) in the asymptotic channels(thereby making the PES mass-dependent).

4 5 6 7-800

-600

-400

-200

0

E

[cm

-1

]

R [a.u.]

O3 original PES

O3 PES + ∆ZPE

– the classical method (—) with mass-dependent PES works well; slightunderestimation of ratio R8,6

– another classical method (—) gives even better results; it is, however, muchmore “expensive” (about 95% of trajectories are not counted)

Recombination within the strong-collision model

– deactivation and activation of the excited complex in multiple collisions withM is very difficult to describe.

– strong-collision model: stabilization occurs in a single collision withfrequency ω, which is the sole parameter!

ω ∝ p and ω ∝ ∆E/collision

– for each trajectory (i) we define a stabilization probability

P(i)stab = 1− e−ωτi τi = survival time of complex

– low-pressure limit: P(i)stab ≈ ωτi linear p dependence

– high-pressure limit: P(i)stab ≈ 1 every complex-forming trajectory

is stabilized

pressure dependence of recombination rate krec

Hippler et al. Lin and Leu

-1

-2-3-4-5 10101010

-1

-3p [molec. cm ]

3

stab

k

(p)

[cm

s

]

222119 2018-15

-14

-13

-12

-11

10101010 1010

10

10

10

10

T=300K

[ps ]

p = 7× 1023 ω

[p] =molec./cm3

[ω] = ps−1

the high-p be-haviour is notunderstood!

temperature dependence of recombination rate krec

100 1000100 100010-35

10-34

10-33

10-32

CHAPERON

(b)

T [K]

ENERGY TRANSFER

(a)

kr / [A

r] [

cm6 m

olec

ule-2

s-1]

T [K]

• ET mechanism yields T dependence, which is too weak at lower T

temperature dependence of recombination rate krec

100 1000100 100010-35

10-34

10-33

10-32

CHAPERON

(b)

T [K]

ENERGY TRANSFER

(a)

kr / [A

r] [

cm6 m

olec

ule-2

s-1]

T [K]

• ET mechanism yields T dependence, which is too weak at lower T

• multiplication with f(T ) = kexexp(T )/kexcal(T ) yields very good agreement (?)

�

�

�

�

Recombination within the chaperon model

• The chaperon mechanism is a one-step process (J. Troe):

Ar · · ·O+O2 → O∗

3 +Ar

Ar · · ·O2 +O → O∗

3 +Ar ,

where Ar · · ·O and Ar · · ·O2 are weakly bound vdW dimers.

• kr,CH(T ) ≈ KArO(T ) kArO+O2→O3+Ar(T ) [M]

where KArO is the equilibrium constant of the Ar + O ⇌ Ar · · ·O system.

• Both, kArO+O2→O3+Ar and KArO strongly depend on T .

temperature dependence of recombination rate krec

100 1000100 100010-35

10-34

10-33

10-32

CHAPERON

(b)

T [K]

ENERGY TRANSFER

(a)

kr / [A

r] [

cm6 m

olec

ule-2

s-1]

T [K]

• Chaperon mechanism yields reasonable T dependence at lower T .

• However, is it really a one-step mechanism?

←− Troe et al.

Isotope dependence of recombination rate

at low pressures: krec ∝ ω 〈〈τ〉〉aver.

DZPE

DZPE

8 + 66 (866)* 86 + 6 6 + 88 (688)* 68 + 8

exothermic endothermic

smaller 〈〈τ〉〉aver. ⇒ smaller krec larger 〈〈τ〉〉aver. ⇒ larger krec

krec = 0.92 krec = 1.50

comparison of exp. and calculated recombination rate coefficients

D

exothermic endothermic

8 + 66 866 86 + 6+ 23 cm

6 + 88 688 68 + 8- 23 cm-1 -1

Symmetric

666, 868 etc.

norma.

– the overall dependence is well reproduced by the classical calculations —

comparison of exp. and calculated recombination rate coefficients

D

exothermic endothermic

8 + 66 866 86 + 6+ 23 cm

6 + 88 688 68 + 8- 23 cm-1 -1

Symmetric

666, 868 etc.

norma.

– the overall dependence is well reproduced by the classical calculations —when ∆ZPE is included!

– however, the rates for the symmetric molecules are too high by about 15%

∼ 15%

Classical vs. statistical (RRKM) calculations

• The classical results for the isotope dependence agree with the statistical(RRKM) results of Marcus et al. (1999–2002)

• They agree because in both approaches ∆ZPE is included.Otherwise, the two methods are quite different!

• Marcus et al. introduced a so-called non-statistical parameter

η ≈ 1.18

in order to (artificially) decrease the rates for the symmetric molecules.

• With η = 1 very poor results for measured fractionations (Marcus) !

• Up to now, there is no computational verification nor a realunderstanding of this rescaling!

Is the O+O2 ⇋ O∗3 statistical?

– low density of states near dissociation threshold (ρ ≈ 0.1 per cm−1)

– shape of wave functions, assignability even close to threshold

– slow intramolecular rotational-vibrational energy transfer (see below)

– molecular beam experiment at 0.32 eV collision energy for the O+O2 exchangereaction shows a clear forward–backward asymmetry (Van Wyngarden et al. J.Am. Chem. Soc. 129, 2866 (2007)

– exact quantum mechanical calculations for collision energies as low as 0.01–0.05 eV and j = 0 also show clear forward–backward asymmetry (Sun et al.

PNAS 107, 555 (2010))

0

10

20

30

40

σ [a

02 ]

I

j=0

j=20

ClassicalStatistical

0

20

40

60

80

100

0 200 400 600 800

σ [a

02 ]

Ec [cm-1]

III

j=0

j=20

Comparison between

classical and statistical

σ(Ecoll.,j)

— the state-specific statisticalcross sections are verydifferent from the classicalones!

— the dependence on Ec and jis very different

— however, the averaged rateconstants are similar — whatdoes that mean?

Need for quantum mechanical calculations

• classical (as well as statistical) calculations are questionable at very low energies

• the difference between symmetric and non-symmetric O3 strongly indicatesthat the symmetry of the quantum states is important

• in quantum mechanics (schematic):

Hsym =

(

hsym 0

0 hanti−sym

)

Hamiltonian block-diagonal

• wavefunctions are either symmetric or anti-symmetric, without any couplingbetween the two sets

• this may affect the energy flow in O∗3 and thus 〈〈τ〉〉aver. and/or ω ∝ ∆Ecoll

• symmetry is not included in classical mechanics nor in the

statistical approach

Quantum mechanical resonances

– resonances are thecontinuation of the truebound states into thecontinuum

– Eres = E0 − iΓ/2

– lifetime = τ = Γ−1

– S.Yu. Grebenshchikov, R.Schinke, and W.L. Hase InComprehensive Chemical

Kinetics, Vol. 39

original PES (2001)

10-3

10-2

10-1

100

101

102

0 200 400 600 800 1000

Γ [c

m-1

]

E - Ethres [cm-1]

(0,12,0)(8,0,0)

quantum mechanical resonances (J = 0)Babikov et al. (2003)

18

What are the very long-lived states between the two thresholds (shaded area)?

O3

• The long-lived resonances between thresholds are the vdW states in the ‘upper’channel 8 · · · 66.

• Decay only by coupling to the main O3 well and subsequently to the continuumof the other vdW well 6 · · · 86, i.e., they are almost real bound states.

• Do such delocalized vdW states contribute to the recombination???

S. Yu. Grebenshchikov

– most complete quantum mechanical calculations up to now

krec(T ) = Q−1r

∑

JK

(2J + 1)∑

n

Γn(JK)ωω + Γn(JK)

e−En(JK)/kbT

– resonance energies En(JK) and widths Γn(JK) for J ≤ 40 and K ≤ 10(several thousand!!)

– simplified PES: no vdW wells and only one (rather than three) O3 well

results presented in next talk!

Vibrational energy transfer in O∗3 +Ar collisions

• classical trajectory calculations

– problem: separation of vibrational and ‘active’ rotational (Ka) energy– maximum impact parameter; what is a ‘collision’?

• ‘infinite order sudden’ approximation

– quantum mechanical approximation, full PES– τcoll ≪ τrot

• ‘breathing sphere’ approximation

– drastic quantum mechanical approximation– average full 6D PES over Ar−O3 orientations =⇒ 4D PES– preserves symmetry!

-6000 -4000 -2000 010-4

10-3

10-2

10-1

100

BSA

IOSA

-∆E

[cm

-1]

E [cm-1]

Ivanov et al. Mol. Phys. 108, 259(2010)

black: 668 (non-symmetric)

red: 686 (symmetric)

• trajectory and IOS calculations agreewell

• no apparent difference betweensymmetric and non-symmetric O3

• ∆Evib ≈ 0.5–1 cm−1 near threshold

-6000 -4000 -2000 010-4

10-3

10-2

10-1

100

BSA

IOSA

-∆E

[cm

-1]

E [cm-1]

Ivanov et al. Mol. Phys. 108, 259(2010)

black: 668 (non-symmetric)

red: 686 (symmetric)

• trajectory and IOS calculations agreewell

• no apparent difference betweensymmetric and non-symmetric O3

• ∆Evib ≈ 0.5–1 cm−1 near threshold

∆Eexp ≈ 10–20 cm−1

Other approach to collisional energy transfer:

Ivanov and Babikov

(Tuesday afternoon)

Intramolecular vibrational–rotational energy flow

• classical trajectory calculations, Eint ≈ Ethreshold: higly excited ozone

• Eint = Erot(t) + Evib(t) = constant

– Erot(t) = AK2a +BK2

b + CK2c

– Kx projection of J on body-fixed x-axis– J = constant

• Evib ←→ Erot energy flow (Coriolis coupling)

• magnitude and direction depend strongly on Ka

• similar calculations (with similar results) by Kryvohuz and Marcus:J.Chem.Phys. 132, 224304 and 224305 (2010)

0 100 200 300 400-120

-100

-80

-60

-40

-20

0

20

40

60

∆ Tr =

-∆ E

v [c

m-1]

t [ps]

Ka(0)=2

Ka(0)=6

Ka(0)=10

Ka(0)=14

Ka(0)=18

– low Ka: flow from vibrationto rotation

– high Ka: flow from rotationto vibration

– possible mechanism ofstabilization:

1. flow of energy from vib. torot. during collisions with M

2. removal of rot. energy incollisions with M

low Ka

high Ka

0 100 200 300 400-120

-100

-80

-60

-40

-20

0

20

40

60

∆ Tr =

-∆ E

v [c

m-1]

t [ps]

Ka(0)=2

Ka(0)=6

Ka(0)=10

Ka(0)=14

Ka(0)=18

– low Ka: flow from vibrationto rotation

– high Ka: flow from rotationto vibration

– possible mechanism ofstabilization:

1. flow of energy from vib. torot. during collisions with M

2. removal of rot. energy incollisions with M

low Ka

high Ka

Quantum Mechanics ???

Open Questions

• magnitude and T dependence of kex? T dependence of krecom?

transition-state (‘reef’) structure of PES is essential

• dynamical-weighting state-averaged CASSCF orbitals

• up to 10 excited 1A states included

• ‘smooth’ change of orbitals through ‘reef’ region

Open Questions

• magnitude and T dependence of kex? T dependence of krecom?

transition-state (‘reef’) structure of PES is essential

• magnitude of energy transfer per collision with M (1 cm−1 vs. 10 cm−1)

quantum mechanical test of intramolecular V → R energy transfer

Open Questions

• magnitude and T dependence of kex? T dependence of krecom?

transition-state (‘reef’) structure of PES is essential

• magnitude of energy transfer per collision with M? (1 cm−1 vs. 10 cm−1)

quantum mechanical test of intramolecular V → R energy transfer

• why are symmetric and non-symmetric isotopomers formedwith different rates (η ≈ 1.15)?

different rates of intramolecular V −R energy transfer for sym. and non-sym.complexes?

Open Questions

• magnitude and T dependence of kex? T dependence of krecom?

transition-state (‘reef’) structure of PES is essential

• magnitude of energy transfer per collision with M? (1 cm−1 vs. 10 cm−1)

quantum mechanical test of intramolecular V → R energy transfer

• why are symmetric and non-symmetric isotopomers formedwith different rates (η ≈ 1.15)?

different rates of intramolecular V −R energy transfer for sym. and non-sym.complexes?

Calculations will be very, very demanding!!

.... or something else has been ignored:

presentation by

P. Reinhardt and F. Robert

(Tuesday afternoon)