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:
16
O (0.99763),
17
O (3.7 × 10−4),
6
18
O (2.0 × 10−3)
7
8
zero point energies (ZPE) of O2:
EZPE ≈ ω/2
66:
790.4 cm−1
68:
-22.2 cm−1
88:
-45.2 cm−1
ω≈
p
f /µ
µ = m1m2/(m1 + m2)
(1 eV = 8066 cm−1)
(for comparison: kBT = 220 cm−1 at 300 K)
forms of ozone:
ozone is predicted by theory to exist in two different forms:
a
a
R1
a
R2
a
D 3h , cyclic O 3
C 2v , open O 3
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
• Mauersberger et al. Adv. At. Mol.
–1990: more laboratory experiments
Opt. Phys. 50, 1–54 (2005)
M
48
( O3/ O3)meas.
− 1 × 100
δ( O3) =
(MO3/48O3)cal.
M
• very large enrichments
• no apparent mass dependence
• δ(49O3) ≈ δ(50O3)
• “ozone isotope effect” or “ozone
anomaly”
Ozone recombination or formation rate constants
Ozone formation rate:
d[O3]
= krec(T ) [O] [O2] [M]
dt
[O] ≪ [O2] ≪ [M]
Mauersberger and coworkers measured (under controlled conditions in the
laboratory) 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 variation
with no apparent systematic dependence ....
... until they were represented as function of the ZPE difference between the
two possible diatomic channels:
∆ZPE = EZPE(products) − EZPE(reactants)
exothermic
endothermic
8 + 66
6 + 88
866 86 + 6
+ 23 cm -1
688 68 + 8
- 23 cm -1
Janssen et al. (2001)
The symmetric molecules
behave differently than the
non-symmetric ones!
Symmetric
666, 868 etc.
D
• 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
(3) (OPQ)∗ + M → OPQ + M
isotope exchange
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 rigorously
treated; they are independent of pressure p.
• stabilization step (3) involves many collisions with M and is extremely
complicated to treat (for example, master equation); it shows a strong p
dependence.
• at low pressures: isotope exchange is much faster than stabilization
O + O2 ⇋ O∗3 interaction potential
• first ‘reasonable’ potential energy surface (PES) calculated
by 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
120
— V (R1, R2, α)
90
— three equivalent wells + cyclic well
60
30
— accurate vibrational energies
cyclic O3
— very small
(0.006 eV)
5
dissociation
barrier
1.0
E [eV]
R2 [a0]
α [deg.]
150
— narrow transition state
0.5
4
— quite ‘harmonic’, compact potential
(correlation with excited products?)
0.0
3
4
5
6
R1 [a0]
3
2
2
3
4
R1 [a0]
5
II −0.014 eV
1.10
– simple
modification
potential II
=⇒
E [eV]
– better calculations increase De
and decrease the barrier!
III
+0.006 eV
I
1.00
0.90
– artificial removal of barrier
=⇒ potential III
0.80
3
4
5
6
R1 [a0]
7
8
9
15
I
2
σ [a0]
10
5
exchange reaction
O + O2(j ) → O2(j ′) + O
0
15
II
10
2
σ [a0]
classical trajectory calculations
5
initial state resolved cross
sections for isotopic exchange
0
40
j=0
j=10
j=20
j=40
depend strongly on the transition
state barrier!
2
σ [a0]
30
σjex(Ecoll.)
III
20
10
0
0
1000
2000
-1
Ec [cm ]
3000
Exchange reaction rate constant k ex(T )
exp.
artificial PES
original PES
(III)
(II)
• poor agreement with experimental rate
– quantum effects — unlikely for three heavy O atoms
– PES (transition state) — much better (i.e., more expensive) calc. do not
change the TS structure
– non-adiabatic effects, i.e. breakdown of BO approximation?
New (2004) ab initio calculations in the transition-state region
‘at our computational limit’
180
+100
0
160
— AQCC, av6z basis
set
140
g [deg]
120
100
r
— not a full PES
(RO2 = r fixed)
R
g
80
60
-200
40
[see also: Holka et
al., J.Phys.Chem. A
36, 9927 (2010)]
-100
20
0
4
4.5
5
5.5
6
6.5
7
7.5
8
R [a 0]
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(X 3Σ−
g ) (open shell system)
E [cm-1 ]
(a)
S
T
Q
(b)
E [cm-1 ]
j=0
j=1
j=2
spin-orbit splitting
R [a0 ]
– 3 × (5 + 3 + 1) = 27
different electronic states
correlate with the ground
state asymptote.
– Thus, transitions due to
non-adiabatic, spin-orbit or
Renner-Teller coupling are
possible!
Isotope dependence of exchange reaction
the ratio
k8+66→86+6
R8,6 = k
6+88→68+8
has been measured (directly)
it is 1.27 at room temperature (∆ZPE = ±23 cm−1 ≪ 2kBT = 440 cm−1)
8 + 66
(866)*
86 + 6
6 + 88
(688)*
68 + 8
D ZPE
D ZPE
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).
0
-1
E [cm ]
-200
-400
-600
O3 original PES
O3 PES + ∆ZPE
-800
4
5
R [a.u.]
6
7
– the classical method (—) with mass-dependent PES works well; slight
underestimation of ratio R8,6
– another classical method (—) gives even better results; it is, however, much
more “expensive” (about 95% of trajectories are not counted)
Recombination within the strong-collision model
– deactivation and activation of the excited complex in multiple collisions with
M is very difficult to describe.
– strong-collision model: stabilization occurs in a single collision with
frequency ω, which is the sole parameter!
ω∝p
and
ω ∝ ∆E/collision
– for each trajectory (i) we define a stabilization probability
(i)
Pstab
= 1 − e−ωτi
(i)
– low-pressure limit: Pstab ≈ ωτi
(i)
– high-pressure limit: Pstab ≈ 1
τi = survival time of complex
linear p dependence
every complex-forming trajectory
is stabilized
pressure dependence of recombination rate krec
-1
[ps
10
-11
-5
10
-4
]
-3
10
-2
10
10
10
p = 7 × 1023 ω
-12
-1
]
T=300K
k stab (p) [cm
3
s
[p] = molec./cm3
[ω] = ps−1
-13
10
-14
Hippler et al.
10
Lin and Leu
-15
10
18
10
19
10
10
20
10
p [molec. cm
21
-3
]
10
22
the high-p behaviour is not
understood!
temperature dependence of recombination rate krec
10
(a)
-32
(b)
CHAPERON
-2
-1
k / [Ar] [cm molecule s ]
ENERGY TRANSFER
-33
10
-34
10
-35
r
6
10
100
T [K]
1000
100
T [K]
1000
• ET mechanism yields T dependence, which is too weak at lower T
temperature dependence of recombination rate krec
10
(a)
-32
(b)
CHAPERON
-1
k / [Ar] [cm molecule s ]
ENERGY TRANSFER
-2
10
-33
6
-34
10
-35
r
10
100
T [K]
1000
100
T [K]
1000
• ET mechanism yields T dependence, which is too weak at lower T
ex
ex
• multiplication with f (T ) = kexp
(T )/kcal
(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.
•
k r,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
10
(a)
-32
(b)
CHAPERON
←− Troe et al.
-2
-1
k / [Ar] [cm molecule s ]
ENERGY TRANSFER
-33
10
-34
10
-35
r
6
10
100
T [K]
1000
100
T [K]
1000
• Chaperon mechanism yields reasonable T dependence at lower T .
• However, is it really a one-step mechanism?
Isotope dependence of recombination rate
at low pressures:
8 + 66
(866)*
86 + 6
krec ∝ ω hhτ iiaver.
6 + 88
(688)*
68 + 8
D ZPE
D ZPE
exothermic
smaller hhτ iiaver. ⇒ smaller krec
krec = 0.92
endothermic
larger hhτ iiaver. ⇒ larger krec
krec = 1.50
comparison of exp. and calculated recombination rate coefficients
exothermic
endothermic
8 + 66
6 + 88
866 86 + 6
+ 23 cm -1
688 68 + 8
- 23 cm -1
Symmetric
666, 868 etc.
norma.
D
– the overall dependence is well reproduced by the classical calculations —
comparison of exp. and calculated recombination rate coefficients
exothermic
endothermic
8 + 66
6 + 88
866 86 + 6
+ 23 cm -1
688 68 + 8
- 23 cm -1
∼ 15%
Symmetric
666, 868 etc.
norma.
D
– 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%
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 real
understanding 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 exchange
reaction 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))
Comparison between
classical and statistical
σ(Ecoll.,j)
40
I
Classical
Statistical
σ [a20]
30
20
— the state-specific statistical
cross sections are very
different from the classical
ones!
j=0
10
j=20
0
100
III
j=0
80
σ [a20]
— the dependence on Ec and j
is very different
60
— however, the averaged rate
constants are similar — what
does that mean?
40
20
j=20
0
0
200
400
600
-1
Ec [cm ]
800
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 indicates
that the symmetry of the quantum states is important
• in quantum mechanics (schematic):
Ĥsym =
ĥsym
0
ĥanti−sym
0
!
Hamiltonian block-diagonal
• wavefunctions are either symmetric or anti-symmetric, without any coupling
between the two sets
• this may affect the energy flow in O∗3 and thus hhτ iiaver. and/or ω ∝ ∆Ecoll
• symmetry is not included in classical mechanics nor in the
statistical approach
Quantum mechanical resonances
– resonances
are
the
continuation of the true
bound states into the
continuum
– lifetime = τ = Γ−1
– S.Yu. Grebenshchikov, R.
Schinke, and W.L. Hase In
Comprehensive Chemical
Kinetics, Vol. 39
101
Γ [cm-1]
– Eres = E0 − iΓ/2
10
original PES (2001)
2
100
10
(8,0,0)
-1
(0,12,0)
10-2
10-3
0
200
400
600
E - Ethres [cm-1]
800
1000
quantum mechanical resonances (J = 0)
Babikov et al. (2003)
18
What are the very long-lived states between the two thresholds (shaded area)?
S. Yu. Grebenshchikov
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 continuum
of the other vdW well 6 · · · 86, i.e., they are almost real bound states.
• Do such delocalized vdW states contribute to the recombination???
– most complete quantum mechanical calculations up to now
krec(T ) =
Q−1
r
P Γn(JK)ω −En(JK)/k T
b
e
(2J + 1)
n ω + Γn (JK)
JK
P
– 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!
0
10
Ivanov et al. Mol. Phys. 108, 259
(2010)
-1
10
black: 668 (non-symmetric)
-1
-∆E [cm ]
IOSA
red: 686 (symmetric)
-2
10
BSA
• trajectory and IOS calculations agree
well
-3
10
• no apparent difference between
symmetric and non-symmetric O3
• ∆Evib ≈ 0.5–1 cm−1 near threshold
-4
10-6000
-4000
-2000
-1
E [cm ]
0
0
10
Ivanov et al. Mol. Phys. 108, 259
(2010)
-1
10
black: 668 (non-symmetric)
-1
-∆E [cm ]
IOSA
red: 686 (symmetric)
-2
10
BSA
• trajectory and IOS calculations agree
well
-3
10
• no apparent difference between
symmetric and non-symmetric O3
• ∆Evib ≈ 0.5–1 cm−1 near threshold
-4
10-6000
-4000
-2000
-1
E [cm ]
0
∆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) = AKa2 + BKb2 + CKc2
– 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)
60
low Ka
40
20
-1
∆Tr = -∆Ev [cm ]
– low Ka: flow from vibration
to rotation
– high Ka: flow from rotation
to vibration
0
-20
-40
Ka(0)=2
Ka(0)=6
Ka(0)=10
Ka(0)=14
Ka(0)=18
-60
-80
-100
-120
0
100
high Ka
200
t [ps]
300
– possible
mechanism
stabilization:
of
1. flow of energy from vib. to
rot. during collisions with M
2. removal of rot. energy in
400
collisions with M
60
low Ka
40
20
-1
∆Tr = -∆Ev [cm ]
– low Ka: flow from vibration
to rotation
– high Ka: flow from rotation
to vibration
0
-20
-40
Ka(0)=2
Ka(0)=6
Ka(0)=10
Ka(0)=14
Ka(0)=18
-60
-80
-100
-120
0
100
high Ka
200
300
– possible
mechanism
stabilization:
of
1. flow of energy from vib. to
rot. during collisions with M
2. removal of rot. energy in
400
collisions with M
t [ps]
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 formed
with 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 formed
with 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)