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Photolysis of Nitrous Oxide Isotopomers
Studied by Time-Dependent Hermite Propagation
Matthew S. Johnson, Gert Due Billing, Alytis Gruodis, and Maurice H. M. Janssen
J. Phys. Chem. A, 2001, 105 (38), 8672-8680 • DOI: 10.1021/jp011449x
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The Journal of Physical Chemistry A is published by the American Chemical Society.
1155 Sixteenth Street N.W., Washington, DC 20036
8672
J. Phys. Chem. A 2001, 105, 8672-8680
Photolysis of Nitrous Oxide Isotopomers Studied by Time-Dependent Hermite Propagation
Matthew S. Johnson,*,† Gert Due Billing,† Alytis Gruodis,‡ and Maurice H. M. Janssen§
Department of Chemistry, UniVersity of Copenhagen, UniVersitetsparken 5, DK-2100 Copenhagen Ø,
Denmark, Department of General Physics and Spectroscopy, UniVersity of Vilnius, Sauletekio 9, b. 3,
2040 Vilnius, Lithuania, and Laser Centre and Department of Chemistry, Vrije UniVersiteit,
De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
ReceiVed: April 18, 2001; In Final Form: June 25, 2001
Nitrous oxide (N2O) plays an important role in greenhouse warming and ozone depletion. Yung and Miller’s
zero point energy (ZPE) model for the photolysis of N2O isotopomers was able to explain atmospheric
isotopomer distributions without invoking in situ chemical sources. Subsequent experiments showed enrichment
factors twice those predicted by the ZPE model. In this article we calculate the UV spectrum of the key N2O
isotopomers to quantify the influence of factors not included in the ZPE model, namely, the transition dipole
surface, bending vibrational excitation, dynamics on the excited state potential surface, and factors related to
isotopic substitution itself. The relative cross-sections are calculated as the Fourier transform of the correlation
function of the initial vibrational wave function and the time-propagated wave function, using a Hermite
expansion of the time evolution operator. The model uses the electronic structure data recently published by
Balint-Kurti and co-workers and makes several predictions. (a) The absolute values of the enrichment factors
decrease with increasing temperature. (b) Photolysis of N2O will produce “mass-independent” enrichment in
the remaining sample. (c) Much of the enrichment is due to decreased heavy isotopomer cross-section over
the entire absorption band, in contrast to the wavelength shift predicted by the ZPE model. Consequently, to
within the error of the calculation, we predict only minor enrichments at λ < 182 nm. The smaller bending
excursion of heavy isotopomers combines with the transition dipole surface to produce a smaller integrated
cross-section. This effect is partially countered by the larger fraction of heavy isotopomers in excited bending
states; the first three bending states have an integrated intensity ratio of ca. 1:3:6. The model agrees with
available experimental enrichment factors and stratospheric balloon infrared remote sensing data to within
the estimated error.
Introduction
[1]
Nitrous oxide (N2O) is the second most abundant nitrogen
species in the atmosphere, produced mainly by nitrifying and
denitrifying bacteria in the soil and ocean. The preindustrial
mixing ratio, measured in gas bubbles trapped in glacial ice,
was 276 ppbv.1 It is anticipated that the present-day mixing
ratio of 316 ppbv will increase to ca. 400 ppbv by the end of
the century. A third of the N2O source is influenced by human
activity, primarily nitrogen fertilizer, biomass burning, and
animal husbandry, with minor contributions from nylon production and motor vehicle emissions.2 N2O is a greenhouse gas3
with a per-molecule global warming potential 200-300 times
that of CO2; accordingly the Kyoto treaty addresses N2O
emissions. Unfortunately there are uncertainties of about 50%
in source and sink fluxes, making regulation difficult.2 Isotopic
analysis has been used to understand and/or improve the budgets
of atmospheric gases such as O3, CH4, CO2, and CO.4 Isotopic
distributions in the main N2O sources have been characterized;5
in this paper we will consider the main sink reaction.
The atmosphere attenuates incoming radiation. A gap between
the Shumann-Runge bands of molecular oxygen and the
Hartley band of ozone (and the Herzberg continuum of O2)
allows energetic solar radiation to reach the lower stratosphere.
At 20 km, the “UV window” extends from 197 to 214 nm.6,7
The UV solar radiation photolyses several gases central to
stratospheric chemistry, e.g., CFCs, OCS8, and N2O:
* To whom correspondence should be addressed. Phone: +45 3532 0303.
Fax: +45 3535 0609. E-mail: msj@kiku.dk.
† University of Copenhagen.
‡ University of Vilnius.
§ Vrije Universiteit.
N2O + hν f N2 + O(1D)
Reaction 1 (unit quantum yield, D0 ) 351 kJ/mol)9 removes
about 90% of atmospheric N2O. The remaining sink is reaction
with electronically excited oxygen atoms:
[2a]
N2O + O(1D) f 2NO
(6%)
[2b]
N2O + O(1D) f N2 + O2
(4%)10
Reactions [1] and [2] lead to an instantaneous lifetime of 123
years.7 Reaction [2a] is the dominant source of reactive odd
nitrogen to the stratosphere.11 The odd nitrogen reaction cycle
is now known to be the most important process removing
nonpolar ozone.12
It is instructive to review the UV spectroscopy, electronic
structure, and photodissociation dynamics of N2O in order to
understand isotope effects in reaction 1. Several authors have
described the increase in absorption cross-section with temperature, especially on the red side of the absorption maximum.13
The JPL advisory panel6 recommends the parametrization given
by Selwyn et al. for use in stratospheric modeling.13a The
absorption feature is a peak centered at 182 nm, composed of
a smooth-continuum absorption with weak vibronic structure
to high energies, suggesting a repulsive or predissociative upper
state. The absorption is dominated by the parallel transition
21A′(1∆) r X11A′(1Σ+), with additional involvement of the
perpendicular transition 11A′′(1Σ-) r X11A′(1Σ+) (the symmetry
species of the bent (linear) moiety is indicated).14 While the 1∆
r 1Σ+ transition is electronic-dipole-forbidden, the transition
becomes allowed as the molecule bends. Using molecular orbital
10.1021/jp011449x CCC: $20.00 © 2001 American Chemical Society
Published on Web 09/05/2001
Photolysis of Nitrous Oxide Isotopomers
notation, the parallel transition is 2π33π r 2π4; the 3π orbital
is in the plane of the bend for the 21A′(1∆) state.15 Normal
coordinate analysis of the vibronic structure indicates an upperstate bond angle of 115°,16 indicating that a bent ground-state
geometry will lead to a larger Franck-Condon overlap. Thus
the temperature dependence may arise from participation by
bending hot bands through vibronic interaction and the FranckCondon factors. Selwyn and Johnston recorded the UV spectrum
of several individual N2O isotopomers; unfortunately, their
original data is no longer available.16
Hopper mapped the electronic structure of N2O in the early
1980s at the MCSCF/CI level.14 For the linear C∞ν geometry
there are two low-lying electronic states, 1Σ- and 1∆. The 1∆
state splits into the 21A′ and 21A′′ Renner-Teller components
upon bending (Cs), while the 1Σ- state becomes 11A′′. The 21A′(1∆) and 11A′′(1Σ-) states are similar topologically, and the
21A′′(1∆) state increases in energy upon bending. Hopper found
evidence for a bent excited-state geometry with bond angle of
ca. 130°. More recently Janssen and co-workers performed
CASSCF calculations to understand the photodissociation
dynamics of the system.17 Recently, Balint-Kurti and co-workers
published the results of CASSCF calculations on the 1A′, 21A′,
and 11A′′ potential energy surfaces, along with transition dipole
moment surfaces.18
While the UV photodissociation of N2O is used as a clean
source of O(1D) for kinetics studies, molecular beam studies
show that the process is not simple. About 60% of the available
energy is present within the N2(1Σ+) fragment, the remainder
being translational.19 The dissociation dynamics favor vibrationally cold and rotationally hot molecular nitrogen. The N2
rotational distribution is peaked at J ) 74,19b confirming the
ab initio and vibronic structure analysis that the geometry of
the upper state favors a bent molecule.
In 1993 Kim and Craig proposed that the N2O isotopomer
concentrations in the troposphere arise from a combination of
light N2O produced by soil, N2O from the oceans, and a back
flux of isotopically heavy N2O from the stratosphere.20 Photochemistry (reactions [1] and [2]) was identified as the source
of the stratospheric enrichment. However, a study of the
photolysis of N2O at 185 nm indicated no significant enrichment,16,21 and it was proposed that the accepted stratospheric
chemistry be modified to incorporate new N2O formation
reactions involving radicals and/or molecules in excited states
and molecular nitrogen.22 Mass-independent enrichment in the
three-isotope plot of atmospheric samples was taken to support
the additional reactions.23 At about the same time, Rahn and
Wahlen showed that stratospheric air was enriched in 15N and
18O relative to tropospheric air in a pattern consistent with the
standard photochemical sinks.24
The state of knowledge was improved considerably by Yung
and Miller’s zero point energy (ZPE) model, which provides a
wavelength-dependent explanation for isotopic enrichment
through the stratospheric photolysis of N2O.25 The natural
abundances of N and O isotopes are shown in Table 1 along
with the abundances of the N2O isotopomers. For simplicity,
the isotopomer 14N14N16O will be written as 446, 14N14N18O as
448, and so on. Yung and Miller observed that an isotopically
substituted molecule will have a smaller ZPE than 446 and
proposed that this results in a greater separation from the upper
state and a blue-shifted spectrum (cf. Table 2). Since photolysis
in the stratosphere takes place on the red shoulder of the peak,
heavy isotopomers will have a lower cross-section, leading to
their enrichment in the remaining sample. The model assumes
that the potential energy surfaces are not affected by the mass
J. Phys. Chem. A, Vol. 105, No. 38, 2001 8673
TABLE 1: Natural Abundance (i.e., No Enrichment) of the
Six Most Abundant N2O Isotopomers40 a
isotope
abundance/%
isotopomer
abundance/%
14
N
15N
99.634
0.366
16O
99.762
0.038
0.200
446
447
448
456
546
556
sum (446, 447, 448, 456,
546, 556):
99.033
0.038
0.198
0.364
0.364
0.001
17
O
18O
a 14
99.998
N14N16O is abbreviated 446, and so on.
TABLE 2: Band Centers of the Fundamental Vibrational
Frequencies of N2O, Together with the Zero Point Energy
(ZPE), and the Difference of the ZPE from the ZPE of the
446 Isotopomer (∆ZPE)a
ijk
ν1
ν2
ν3
ZPE
∆ZPE
446
447
448
456
546
556
1284.903b
1264.704b
1246.885b
1280.354b
1269.892b
1265.334b
588.768c
586.362e
584.225e
575.434c
585.312c
571.894e
2223.757d
2220.074d
2216.711d
2177.657d
2201.605d
2154.726d
2343.10
2328.75
2316.02
2304.44
2321.06
2281.92
0.00
14.35
27.08
38.66
22.04
61.17
a Frequencies and energies are given in units of cm-1. b Reference
41. c Reference 42. d Reference 43. e Reference 44.
of the nuclei. Further, the upper state is assumed to be repulsive,
so that isotopic substitution shifts the energy levels on the lower
bound surface but not the continuum levels of the upper surface.
The ZPE model was able to explain the observed distribution
of N2O isotopomers without the new source reactions. The more
general theory of photoinduced isotopic fractionation (PHIFE)
has been extended in recent work and promises to be an
important tool for investigating planetary atmospheres.26
Enrichment factors obtained from subsequent experiments
were in qualitative agreement with the ZPE model but a factor
of 2 higher than predicted (cf. Figure 5). Questions were raised
about the role played by the transition dipole surface, upperstate dynamics and bending excitation, which are not included
in the ZPE model. The goal of this article is to dissect the UV
spectrum of N2O and determine the contribution of the additional
factors. The structure of the paper is as follows. The relative
photolysis cross-sections of the most important isotopomers are
calculated using a Hermite time propagation operator27 and
potential energy surfaces from the literature18 and our own work.
The isotopomer-specific UV absorption spectra contain contributions from the three lowest-bending vibrational states. The
wavelength- and temperature-dependent isotopic enrichment
factors calculated by the model are compared to laboratory
experiments, stratospheric remote-sensing data, and the ZPE
model. The results of the present model are found to be in good
agreement with laboratory experiments and the stratospheric
observations.
Theory
For the NN + O system we assume the following model
Hamiltonian:28
H)
p2 1
1
1 ∂
∂
p2 ∂ 2
+
+ V(R,θ) (1)
sin θ
2
2µ ∂R
2 µR2 mr 2 sin θ∂θ
∂θ
e
(
)(
)
where R is the distance from the oxygen atom to the center of
mass of the nitrogen molecule and θ is the angle between the
8674 J. Phys. Chem. A, Vol. 105, No. 38, 2001
Johnson et al.
Figure 1. (a) Cuts through the ground X11A′(1Σ+) and excited 21A′(1∆) potential energy surfaces are shown.18 A vertical transition from
the minimum of the ground-state surface leads to a location where the
maximum gradient of the A′ surface is in the bending dimension instead
of the bond-breaking dimension. (b) The excited-state potential energy
surface (the units of the Z-axis are 100 kJ/mol).
NN and the R axis. The reduced masses µ and m are defined as
µ)
m3(m1 + m2)
m1 + m2 + m3
(2)
m 1m 2
m1 + m2
(3)
m)
where the masses are numbered from left to right in the NNO
molecule. The interatomic N-N distance re is kept fixed in the
calculations. The potential V(R,θ) is that of the upper electronic
state. To calculate the absorption spectra, we use the relation29
σ(ω) ) Cω
∫-∞∞ dt exp(iEt/p)〈φ0| exp(- pi Ht)|φ0〉
(4)
where φ0 is the initial wave function. The absorption spectrum
is obtained as a Fourier transform with respect to the total energy
{E ) pω + Ei, where Ei ) pω1(n1 + 1/2) + 2pω2(n2 + 1/2) +
pω3(n3 + 1/2)} of the correlation function 〈φ(t0)|φ(t)〉. Here φ(t0)
) φ0, and φ(t) is the time-propagated wave function:
i
φ(t) ) exp - Ht φ(t0)
p
(
)
(5)
To propagate the wave function, we use the so-called Hermite
Figure 2. (a) A comparison of the calculated absorption spectrum for
446 N2O at 300 K with the experimental absorption band.6,13a The
calculated absorption is shifted to the red by ca. 1.5% in energy (or
<3 nm) relative to experiment. The 446 spectrum contains contributions
from the n2 ) 0, 1, and 2 vibrational states as shown (labeled 000,
010, and 020, respectively). The cross-sections shown for these states
reflect the partition function at 300 K and the degeneracy of the bending
states. (b) The same, but at 233 K, a temperature typical for the
photolysis of N2O in the stratosphere. (c) The enrichment is a sensitive
function of the absorption cross-section. To ensure that the calculation
was converged, a 40 000-term Hermite expansion was used. Starting
with the widest function, the curves show the state of the correlation
function for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, and 40 thousand terms.
The last three overlap on this figure. Extension of the polynomial to
80 000 terms showed no significant difference from the result at 40 000
terms.
propagator,27 which is based on the expression:
∞
(-i)m m
exp(-iHt/p) )
r exp(-τ2)Hm(H/Εmax) (6)
m!
m)0
∑
where τ ) Emaxt/2p and Hm is a Hermite polynomial. The
Hermite propagator (HP) method improves on the well-known
Chebyshev propagator, and is a computationally and numerically
efficient formulation of the spectral filter problem, resulting in
Photolysis of Nitrous Oxide Isotopomers
J. Phys. Chem. A, Vol. 105, No. 38, 2001 8675
Figure 3. (a) The components of the enrichment of 456 N2O are
displayed. The bending ground state alone shows more enrichment than
the excited bending states, and the total enrichment is less than would
be apparent only from the consideration of the vibrational ground state.
(b) The fraction of the absorption coming from the ground vibrational
state is shown as a function of wavelength for the 446 and 456
isotopomers. (c) The HP model predicts that the enrichment arises from
a shift in the intensity of the entire isotopomer band, in addition to a
slight wavelength shift. The cross-section of the 446 and 456 isotopomers is compared at 295 K.
an exponentially convergent propagation scheme. Emax is a
scaling energy, which for convenience is set to be the largest
energy of the system. The wave function is propagated from t0
) 0. The relation 〈φ0|φ(-t)〉 ) 〈φ0|φ(t)*〉 is used for negative
values of t. The time integral from -∞ to +∞ can then be
performed analytically, leading to
∞
σ(ω) )
1
Hm(H/Emax)|φ0 >
∑ Fm(ω) < φ0|
m)0
m
x2 m!
(7)
Figure 4. (a) The enrichment (defined in the text) of the most common
N2O isotopomers plotted vs wavelength, using the isotopomer crosssections at 233 K. (b) 295 K. It is seen that the absolute value of the
enrichment decreases with increasing temperature as explained in the
text. (c) The results of the ZPE model, which are independent of
temperature. The ZPE model predicts positive enrichments at wavelengths shorter than 182 nm, whereas the HP model predicts zero or
slightly negative enrichments.
and y ) E/Emax. The summation is carried out using the
recursion formula for Hermite polynomials, Hm(x) ) 2xHm-1
- 2(m - 1)Hm-2. A Morse wave function in the N-O bond
(r′) times a harmonic oscillator function in the bending angle γ
was used to calculate the initial wave function on the ground
electronic state. A simple “local mode” approach for the
vibrational wave function was chosen so that the effect of
isotopic substitution would be direct and independent of the
model. Thus we have
φ0(r′,γ) ) φm(r′)φΗ(γ)
(9)
where
where for the bending ground state
Hm(y)
pω
2xπ exp(-y2)
Fm(ω) ) C
Emax
xm!2m
(8)
1
ψΗ(γ) ) (2βh/a)0.25 exp - bhγ2/a
2
(
)
(10)
8676 J. Phys. Chem. A, Vol. 105, No. 38, 2001
Johnson et al.
TABLE 3: Intensity Factors fijk,m of the Isotopomers in the
Ground, First, and Second Bending Vibrational Statesa
isotopomer ijk
fijk,0
fijk,1
fijk,2
446
447
448
456
546
556
0.013574
0.013516
0.013465
0.013309
0.013513
0.013250
0.034648
0.034464
0.034300
0.033722
0.034392
0.033475
0.063470
0.063151
0.062865
0.061784
0.063010
0.061334
a The factor accounts for the product of the isotopomer-dependent
vibrational wave function with the transition dipole surface.
The Morse parameters were obtained from the ground-state
potential energy surfaces and the bending force constant taken
as 0.49 × 105 dyn/cm.30 To initialize the values of R and θ for
the wave function, we used the following relations connecting
the coordinates and angles:
r′ ) xR2 + (ǫre)2 - 2ǫreR cos θ
(14)
where ǫ ) m1/(m1 + m2). The calculation was carried out using
γ ) cos-1((ǫre - R cos θ)/re′)
Figure 5. (a) A comparison of the calculated enrichment (solid line)
with experiment and with the zero point energy model (dashed line)
for 448 N2O, 295 K. Experimental data is shown at the following
wavelengths: 185 nm,21 193 nm (upper),45 193 nm (middle),46 193 nm
(lower),47 207.6 nm (upper),45 207.6 nm (lower),46 and 211.5 nm.46 (b)
Comparison for 456 N2O, 295 K. Experimental data is shown at the
following wavelengths: 193 nm (upper),46 193 nm (lower),47 207.6
nm,46 211.5 nm,46 and 212.8 nm.48 (c) Comparison for 546 N2O, 295
K. Experimental data is shown at the following wavelengths: 193 nm
(upper),47 193 nm (lower),46 205.5 nm,34 207.6 nm,46 211.5 nm,46 and
212.8.48 The uncertainty in the HP enrichment factors is estimated to
be ca. 10 per mil.
Here βh and a are defined as
βh ) ωh/p
(11)
a ) ((re + re′)2/m2 + (re)2/m3 + (re′)2/m1)/(rere′)2 (12)
ωh ) kγrere′xa
(13)
(15)
a 128 × 128 grid with R in the range of 1-6 Å and θ between
-π/2 and +π/2. Mass enters the calculation of the cross-section
through the following terms: the reduced mass of the Morse
oscillator for the N-O bond, the ZPE of the N2O molecule Ei,
the parameter a in the bending wave function, and the reduced
mass of the relative motion µ and of the N2 molecule m. In
addition, the mass-dependent bending vibrational frequency
changes the partition function and Boltzmann factors used to
weigh the contribution from the various vibrational states. As
shown below, the net effect is “mass-independent” fractionation,
as defined by the mass spectroscopy community.
The absorption cross-section was calculated for the isotopomers 446, 447, 448, 456, 546, and 556. As seen in eq 16
below, σijk(T) depends on the result of the model calculation
σijk,m, the factor gm representing the number of (pseudo)degenerate angular momentum states of the degenerate bending
mode (gm ) m + 1; e.g., for m ) 2, there are three components,
02(20 and 0200), an intensity factor representing the square of
the integral of the vibrational wave function and transition dipole
moment surface (fijk,m), the Boltzmann population factor, the
partition function, and an arbitrary scaling factor C. The nuclei
are represented by the labels i, j, and k. The ground, first, and
second excited bending vibrational states were considered;
values of n2 > 2 are not important at the temperatures being
considered because of the Boltzmann factor. To sum:
C
σijk(T) )
(
∑ σijk,mgmfijk,m exp
ijk,m
kT
( ) (
-νijkbend
1 + 2 exp
kT
)
- mνijkbend
)
-2νijkbend
+ 3 exp
kT
(16)
The intensity factors fijk,m are presented in Table 3.
A calculation of the potential energy surface of N2O in the
ground state was performed using Gaussian 98.31 The potential
minimum was found using the Moeler-Plesset MP2 method
in the 6-311++G(3df,3pd) basis set.32 The potential energy
surface was created using a grid of geometries. The NN distance
was fixed to the equilibrium value of 1.154388 Å, and the N-O
distance varied from 0.96 to 11.17 Å. The result of our
Photolysis of Nitrous Oxide Isotopomers
calculation of the ground-state potential energy surface is not
significantly different from that of Balint-Kurti and co-workers18
but is useful because a larger range of geometries is explored.
Results and Discussion
1. The Temperature-Dependent UV Cross-Section. Figure
1a shows cuts through the X11A′(1Σ+) and 21A′(1∆) potential
energy surfaces, and Figure 1b shows the 21A′ surface itself.
The ground state is at a minimum for the linear molecule, and
the excited-state surface slopes downward as the molecule bends.
Combined with a small barrier to direct dissociation for the
linear species, this leads to a large rotational excitation of the
N2 photofragment. State-resolved product imaging experiments
show 1.4 eV of rotational excitation, an amount of energy
roughly equal to the translational energy of the photofragments.19,33 Several workers have noted the importance of the
bending vibration in the temperature dependence of the
spectrum.13,16,19b Clearly, bending in the ground state will lead
to a more favorable Franck-Condon factor with the upper state.
In addition, the transition is orbitally forbidden for a linear
molecule and vibronically allowed for the bent system. The
transition dipole surface of Balint-Kurti and co-workers18
displays a minimum for the linear system and increases with
increasing bending angle. The maximum of the absorption curve,
182 nm, corresponds to an energy of 6.8 eV, which is not large
enough to reach the minimum of the potential for a vertical
transition to the linear excited state.
The results of the calculation for the main isotopomer are
shown at 300 K (Figure 2a) and 233 K (Figure 2b). 446 NNO
accounts for 99% of the molecules in a sample at natural
abundance (Table 1), and so is representative of the UV
absorption spectrum of the compound. The calculated relative
cross-sections were scaled by a factor of 1.7 × 10-18 to match
the experimental cross-section, also shown in Figure 2. The
width of the absorption curve matches the experimental
observation very well. There is a small shift in energy of about
1.5% (or 3 nm), perhaps due to remaining uncertainties in the
calculation of the potential energy surfaces.
The absorption spectrum is composed of contributions from
the ground, first, and second bending vibration states, with a
higher “hot band” contribution at higher temperatures. These
absorptions appear on the red shoulder of the curve, in the region
of the stratospheric UV window. At room temperature, the
fraction of molecules in the first excited bending mode is 10%
(including degeneracy); even so, this accounts for a majority
of the transition intensity at wavelengths longer than 200 nm.
The model results indicate that the perpendicular 11A′′ r X1A′
transition does not make a significant contribution to the
observed spectrum. The small transition dipole moment18 leads
to an intensity ca. 2.5% of that of the 21A′ r X1A′ transition,
in agreement with the results of Janssen and co-workers.17
Considering the transition dipole surface (and ignoring the
Boltzmann factor), the transitions from the n2 ) 0, 1, and 2
states have an intensity ratio of ca. 1:3:6. Compared to
experimental data, Figure 2b shows that the model results have
a temperature dependence that is slightly too large, implying
that the model may overestimate the contribution of the excited
bending states. This may be due to the transition dipole surface,
or the approximation used in obtaining the vibrational wave
functions.
To demonstrate convergence, the results of the calculation
for 446 in the n2 ) 0 vibrational state is shown in Figure 2c.
For a 40 000-term Hermite expansion (cf. eq 7), the calculation
was converged. Continuing the calculation to 80 000 terms did
not significantly change the result.
J. Phys. Chem. A, Vol. 105, No. 38, 2001 8677
2. Enrichment Factors. It is interesting to consider the
lifetime of the dissociating system. The ZPE model assumes a
repulsive excited state and direct dissociation; the isotope effect
arises from the effect of mass on the ground-state energy levels.
At the other extreme, as noted by Umemoto,34 large isotope
effects have been described for several predissociative systems.35
As the lifetime of the excited-state wave function increases, we
can expect the effect of isotopic substitution on the wave
function on the upper state to play a larger role in determining
the final enrichment. N2O seems to be an intermediate case.
Some fine structure is observed on the blue side of the
absorption curve, corresponding to a bond angle of 115°, ν1 of
1372 cm-1, and ν2 of 1761 cm-1.16 A Franck-Condon model36
indicates that isotopic substitution has a significant effect on
the wave function on the upper state, and thus the enrichment,
and prompted us to initiate the present work to consider the
effects in greater detail. A separate dynamics calculation
indicates that after 1 ps, only 6% of the wave packet has left
the potential energy grid of the calculation.36 The long lifetime
of the wave packet explains the structured absorption at
wavelengths shorter than 180 nm.13,16 (The experimental absorption cross-section shown in Figures 2a and 2b is based on a
polynomial expression and does not have enough terms to show
the fine structure). For a vertical transition from the minimum
on the ground-state surface, the maximum gradient in the upper
state corresponds to bending.
The origin of the enrichment is illustrated in Figure 3 for the
456 isotopomer. Enrichment is defined as ǫ ) R - 1, where R
is the ratio of the isotopomer to the reference reaction or
photolysis rate, in this case R ) σ456/σ446. Because it is
independent of the extent of reaction, the enrichment is a
characteristic property of a given process. We have displayed ǫ
for each of the vibrational states, and it is seen that the
enrichment is largest for n2 ) 0 and smallest for n2 ) 2. A
consideration of the vibrational energy of the systems involved
using the ZPE model would predict the opposite trend, so the
effect must arise from the upper state. A curious feature is that
the total enrichment is less than would be predicted from simply
considering the enrichment due to each vibrational component.
Heavy systems have relatively greater populations in the n2 )
1 and n2 ) 2 states (Figure 3b), and as shown in Figure 2 and
Table 3, these states have disproportionately large cross-sections.
Since the enrichment arises because heavy isotopomers have a
relatively smaller absorption cross-section on the red shoulder
of the absorption, this effect decreases the observed enrichment.
It follows that greater absolute enrichments will be observed at
lower temperatures; for example, our model predicts greater
enrichments in stratospheric samples than for room-temperature
photolysis experiments. Figure 3c shows that the enrichment
arises largely from a shift in the intensity (cf. Table 3) of the
absorption, in contrast to the ZPE model, in which the
enrichment arises from a wavelength shift of the spectrum.
The enrichments of the most common isotopomers are shown
at 233 and 295 K, respectively, in Figures 4a and 4b. 233 K
was chosen because it is characteristic of the temperature of
the stratosphere, where maximum photolysis occurs.6,7 295 K
was chosen to represent the temperature of the laboratories in
which photolysis experiments were conducted. Figure 4c shows
the enrichments predicted by the ZPE model. This model does
not consider excited bending vibrational states and is independent of temperature. The Hermite polynomial (HP) and ZPE
models make the same general predictions regarding the ordering
of isotopomer enrichments.
8678 J. Phys. Chem. A, Vol. 105, No. 38, 2001
3. Comparison to Experimental and Stratospheric Enrichment Data. Figure 5a compares the results of several experiments, the ZPE model, and our model for the 448 isotopomer.
The enrichment curve was calculated at 295 K and shifted by
3 nm to reflect the shift between the calculated UV spectrum
and the experimental (Figure 2). Figure 5b presents the results
for 456, and 5c presents those for 546. The agreement for the
448 and 456 isotopomers is satisfactory, given the approximations in the HP model. The uncertainty in the enrichments
derived from the HP model are ca. 10 per mil, due to remaining
uncertainties in the potential energy and dipole moment surfaces,
errors in the local mode vibrational wave function, and uncertainties in the calculation itself. Due to the propensity of the
excited state to bend, it is possible that the stimulated/resonant
Raman cross-section for vibrational bending excitation in pulsed
laser photolysis experiments may be high enough to transfer
population into the excited bending modes and bias the results.
A bump of ca. 5 enrichment units is apparent at 207 nm on the
results of the HP model for the 546 isotopomer (see Figure 5c).
This is due to a transition from a region where n2 ) 1
enrichment dominates to one where n2 ) 0 dominates. Closer
inspection reveals this modulation in the enrichment plots of
all the isotopomers (cf. Figures 4a and 4b).
Figures 6a to 6c show a comparison of the models with
enrichment factors derived from data concerning the altitude
profiles of N2O isotopomers in the stratosphere.49 The profiles
were calculated based on infrared spectra measured using a
balloon-based interferometer and used to derive enrichment
factors for stratospheric photolysis. The HP model predicts larger
absolute enrichment factors at low stratospheric temperatures
due to the decreased population of the excited bending modes.
The radiative transfer model of Minschwaner, Salawitch, and
McElroy indicates that the maximum photolysis rate for N2O
occurs at 200 nm and an altitude of 29 km, for equinox at 5°
latitude, local noon.7 The diurnal average photolysis rate has a
maximum in the region of 0-15° latitude and at an altitude of
ca. 31 km. The authors state that the photolysis rate for N2O is
largest at wavelengths between 195 and 205 nm, over an altitude
range of 25-35 km. Work by Minshwaner indicates that the
maximum actinic flux at a solar zenith angle of 30° occurs at
205 nm.6 Following Yung and Miller,25 in Figure 6 we take
the wavelength of 205 nm as characteristic for the photolysis
of N2O in the stratosphere. While the temperature of N2O will
vary depending on season, altitude, and latitude, we have
assumed a temperature of 233 K to be representative of the lowlatitude stratosphere at an altitude of 30 km. The results of
Griffith et al. for the isotopic enrichment of stratospheric N2O
are presented in Figure 649 and compared to the ZPE model
and our model. Because of the decreased temperature, the model
predicts a higher enrichment than at room temperature. With
the possible exception of the 546 isotopomer, the HP model is
in excellent agreement with the available stratospheric data.
As noted by Minschwaner,7 there are significant contributions
to the photolysis rate of N2O associated with holes in the
Shuman-Runge spectrum of molecular oxygen, between 188
and 195 nm. Based on the ZPE model (cf. Figure 4c), Rahn et
al. have proposed that the positive enrichments in this region
may help to explain why stratospheric enrichments are less than
the experimental.45 The HP model provides another explanation
for the discrepancy. Since the bending vibrational distribution
of a sample is important for determining its enrichment, lower
enrichments are expected for photolysis at stratospheric temperatures compared to ambient laboratory experiments. As noted,
Johnson et al.
Figure 6. (a) Comparison with observed enrichment factors in the
earth’s stratosphere with model results for 233 K, 448 isotopomer. We
take 205 nm as a representative wavelength for N2O photolysis in the
stratosphere. (b) The 456 isotopomer. (c) The 546 isotopomer.
Stratospheric data from Griffith et al.49
the HP model predicts moderately to strongly negative enrichments at all wavelengths.
There is excess 17O in atmospheric N2O, relative to the
isotope’s nominal concentration in the environment, and further,
the 17O is enriched in a mass-independent manner.37 A massdependent enrichment arises from processes such as diffusion
and evaporation, whose rate depends on the mass of the
molecule. The composition of a given sample is defined by the
value δ ) Ri/Rstd - 1, where R is the ratio of the concentration
of the isotopically labeled species (e.g., 456) to the reference
species (e.g., 446) for the isotopically enriched sample i and
the reference standard. For a sink process with an isotopic
signature ǫ, the composition can be approximated as δf ) δ0 +
ǫ ln(f), where f is the fraction of the original material remaining.
For δ0 ) 0, δf ) ǫ ln(f). The traditional theory states that if we
consider the three-isotope factor F ) 17δ/18δ in N2O, F ) 0.51
for a mass-dependent process. Work by Thiemens et al. has
indicated that F > 0.51 for stratospheric N2O; i.e., there is excess
17O.37 Figure 7 shows the results of the model; “massindependent” enrichment is predicted through photolysis in the
stratosphere. Moreover the enrichment grows as temperature is
decreased from ambient to stratospheric. The straight-dashed
Photolysis of Nitrous Oxide Isotopomers
Figure 7. The cross sections of the three-isotope system 446, 447,
and 448 indicate that “mass-independent” enrichment, as defined by
the mass spectroscopy community, is predicted for photolysis throughout the range of wavelengths shown in the figure. The mass-dependent
line is given by 0.51, as explained in the text. The figure displays the
three-isotope factor for the HP model at 295 K, the HP model at 233
K, and the ZPE model.
line is characteristic of a mass-dependent enrichment. The threeisotope factor predicted by the ZPE model is shown for
comparison. Both of these models indicate that the UV
photolysis of N2O is not a “mass-dependent” process.
Conclusion
This paper has considered the behavior of the UV spectrum
of N2O on the red shoulder of the absorption peak, in the region
where absorption by the bending hot bands is important. It is
interesting to compare the transition intensity factors fijk,m for
the various isotopomers, as shown in Table 3. A decrease in
intensity factor with decreasing ZPE (isotopic substitution) is
observed. This can be understood as follows. The amplitude of
the bending motion is greater for a light isotopomer. Since for
N2O the transition dipole moment and absorption cross-section
are a sensitive function of bending angle, this results in an
intensity change at all excitation wavelengths. In contrast, in
the ZPE model the area under the absorption curve is constant
for each isotopomer; only the position of the curve is changed.
Thus the ZPE model predicts positive enrichments on the blue
side of the peak, whereas the HP model predicts enrichments
that are near zero or slightly negative. An experimental
determination of the enrichment factors for N2O at wavelengths
between 165 and 185 nm would be useful, given a suitable light
source. Another approach is to choose a system where readily
available photolysis sources lie on the blue side of the absorption
maximum. We have begun such studies on the molecule
carbonyl sulfide and will report the results of our experiments
and a comparison to the HP and ZPE models in a future
publication.8 It is interesting to note that experiments are
underway on the vibrational state-selected photodissociation of
N2O isotopomers.17c
The HP model can correctly account for the observed
distribution of 456, 546, and 448 isotopomers. The results of
this calculation indicate that the standard model for the chemistry
of the stratosphere6 is complete without additional N2O source
mechanisms. In addition, recent experimental studies have cast
doubt on the mechanisms.38
Toyoda and co-workers have recently demonstrated that 15N
and 18O are enriched by about twice as much in the high
stratosphere (z > 24.1 km) compared to the low stratosphere (z
< 24.1 km).39 This would initially appear to contradict our
conclusion that smaller absolute enrichments are expected at
higher temperatures, since temperature increases with altitude
in the stratosphere. However, Figures 4a and 4b show that
J. Phys. Chem. A, Vol. 105, No. 38, 2001 8679
photolysis wavelength has a larger effect on isotopic enrichment
than temperature. With increasing altitude, one passes through
the Chapman layer, and long-wavelength absorption by O3
decreases more rapidly than short-wavelength absorption by
O2.6,7 Thus, at higher altitudes there is expected to be more
photolysis at longer wavelengths where stronger enrichment
factors pertain.
The HP method has allowed an examination of the effect of
isotopic substitution, bending vibrational excitation, and transition dipole moment surface on isotopic enrichment through the
photolysis of N2O. The computation of the cross-sections was
run on a PC. However the method requires high-quality potential
energy and dipole moment surfaces of the ground and excited
states. In contrast, the ZPE method uses readily available
spectroscopic data. Using the Hermite propagator method, we
have sought to extend the utility of predictions made using the
ZPE model. For systems such as N2O, where the sloping
transition dipole surface makes the absorption cross-section
dependent on bending motion, low-lying vibrational states play
an important role. Further, the lifetime of the system on the
upper state indicates it is not purely dissociative. These factors
lead the ZPE model to underestimate the enrichment factors
for N2O. However, this should not limit its use for estimating
isotope effects in systems for which those constraints do not
apply. In conclusion, we anticipate that a great deal of
experimental and theoretical activity will continue to be
associated with isotopic fractionation effects in planetary
atmospheres. Miller and Yung’s photoinduced isotopic fractionation (PHIFE) theory is independent of a specific model
for the mechanism of the enrichment.26
Acknowledgment. We wish to thank the Danish Natural
Sciences Research Council, the Nordic Network for Chemical
Kinetics sponsored by the Nordic Academy for Advanced Study,
and the Lund University Supercomputing Center. In addition,
we wish to acknowledge valuable conversations and/or correspondence with Y. L. Yung, C. E. Miller, G. A. Blake, P.
Wine, N. W. Larsen, G. C. Groenenboom and O. J. Nielsen.
Blake, Wine, and Miller graciously shared material prior to
publication.
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