Photoinitiated unimolecular decomposition of NO2: Rotational dependence of the dissociation rate

Article abstract of DOI:10.1063/1.479841

Photoinitiated unimolecular decomposition rate constants of rotationally excited NO₂ molecules have been measured near dissociation threshold (D₀) by employing a double resonance technique. Rotational selectivity has been achieved by using narrow-linewidth (0.015 cm^-1) infrared excitation to prepare specific rotational levels (N′ = 1,3,...,15, K′_a=0) of the (1,0,1) vibrational level. The picosecond-resolution pump-probe technique has then been used to photodissociate the molecules thus tagged and to monitor the appearance of the NO product. Data have been obtained for two progressions of average excess energies, 〈E〉-D₀: (i) 10cm^-1+E^rot₁₀₁ and (ii) 75cm^-1+E^rot₁₀₁, where 〈E〉 denotes an average over the pump laser linewidth and E^rot₁₀₁ is the rotational energy of the (1,0,1) X²A₁ intermediate vibrational level. The measured rate constants do not display any noticeable dependence on N′, which is a reflection of significant rovibronic interaction. Spin-rotation interaction, which has been implicated as the main source of rovibronic coupling for small values of N′, is not likely to yield such a result. A model is proposed to describe the influence of rotation on the dissociation rate. The experimental data are consistent with a Coriolis coupling mechanism causing transitions to occur between K_a levels.

Full text of DOI:10.1063/1.479841

Photoinitiated unimolecular decomposition of NO 2 : Rotational dependence of the
dissociation rate
I. Bezel, P. Ionov, and C. Wittig
Citation: The Journal of Chemical Physics 111, 9267 (1999); doi: 10.1063/1.479841
View online: http://dx.doi.org/10.1063/1.479841
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/111/20?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Nanosecond time-resolved IR emission from molecules excited in a supersonic jet: Intramolecular dynamics of N
O 2 near dissociation
J. Chem. Phys. 123, 154306 (2005); 10.1063/1.2049271
State-to-state rotational relaxation rate constants for CO + Ne from IR–IR double-resonance experiments:
Comparing theory to experiment
J. Chem. Phys. 120, 7483 (2004); 10.1063/1.1687314
Specific rate constants k (E,J) for the dissociation of NO 2 . II. Linewidths of rotationally selected NO 2 near to
the dissociation threshold
J. Chem. Phys. 115, 6531 (2001); 10.1063/1.1398306
Rotationally specific rates of vibration–vibration energy exchange in collisions of NO (X 2 Π 1/2 ,v=3) with NO (X
2 Π,v=0)
J. Chem. Phys. 111, 9296 (1999); 10.1063/1.479843
Direct measurement of eigenstate-resolved unimolecular dissociation rates of HOCI
J. Chem. Phys. 107, 10344 (1997); 10.1063/1.474227
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

JOURNAL OF CHEMICAL PHYSICS
VOLUME 111, NUMBER 20
22 NOVEMBER 1999
Photoinitiated unimolecular decomposition of NO2: Rotational dependence
of the dissociation rate
I. Bezel, P. Ionov,a) and C. Wittigb)
Department of Chemistry, University of Southern California, Los Angeles, California 90089
͑Received 21 June 1999; accepted 1 September 1999͒
Photoinitiated unimolecular decomposition rate constants of rotationally excited NO2 molecules
have been measured near dissociation threshold (D0) by employing a double resonance technique.
Rotational selectivity has been achieved by using narrow-linewidth ͑0.015 cmϪ1͒ infrared excitation
to prepare specific rotational levels (N ϭ1,3,...,15, KЈϭ0) of the ͑1,0,1͒ vibrational level. The
Ј
a
picosecond-resolution pump–probe technique has then been used to photodissociate the molecules
thus tagged and to monitor the appearance of the NO product. Data have been obtained for two
progressions of average excess energies, E ϪD : ͑i͒ 10 cmϪ1ϩEr1o0t1 and ͑ii͒ 75 cmϪ1ϩE1ro0t1
,
͗ ͘
0
rot
where ͗E͘ denotes an average over the pump laser linewidth and E is the rotational energy of the
101
2
˜
͑1,0,1͒ X A1 intermediate vibrational level. The measured rate constants do not display any
noticeable dependence on N , which is a reflection of significant rovibronic interaction. Spin–
Ј
rotation interaction, which has been implicated as the main source of rovibronic coupling for small
values of N , is not likely to yield such a result. A model is proposed to describe the influence of
Ј
rotation on the dissociation rate. The experimental data are consistent with a Coriolis coupling
mechanism causing transitions to occur between Ka levels. © 1999 American Institute of Physics.
͓S0021-9606͑99͒01744-4͔
I. INTRODUCTION
Much work has been done on the photodissociation
quantum yield at different temperatures2 and it has been
shown that dissociation can occur when the photon energy is
less than D0 as a consequence of molecular rotation.3 Thus,
rotational energy is available for dissociation. Also, the av-
erage dissociation rate constant as a function of photon en-
ergy has been measured for expansion cooled ͑i.e., almost
non-rotating͒ molecules4–8 and has been found to be smaller
than for room temperature samples.5,9,10 Though such experi-
ments demonstrate unequivocally that rotational energy, on
average, increases the dissociation rate constant, state-
specific information cannot be gleaned.
There are numerous scientific benchmark systems that
people return to—as pilgrims to Mecca—seeking knowledge
and testing their comprehension of the world around them.
They include hydrogen in quantum physics, drosophila ͑fruit
fly͒ DNA in genetics, and the value of ␲ in arithmetic. For
gas phase physical chemistry, NO2 could be such a system.
Its importance to fundamental chemical processes has
spawned extensive scrutiny for decades, and strong optical
absorption in a convenient wavelength region has enabled
spectroscopists to generate and analyze a vast amount of
data. It has a sparse enough density of vibronic states to
resolve them independently, yet sufficiently strong intramo-
lecular interactions to lead to chaotic dynamics. The large
amount of experimental data on NO2 makes it an ideal test
ground for examining theoretical methods and models. Since
it is relatively simple, precise computer simulations can be
carried out; yet it is too complex for these simulations to
succeed completely.
The near-threshold dissociation of NO2 occurs on the
2
˜
ground X A1 potential energy surface ͑PES͒ via a barrierless
pathway which leads to NO(X 2⌸1/2,3/2)ϩO(3P2,1,0).11–15 In
the energy range of interest in the present study ͑i.e., less
then 120 cmϪ1 above D0) only the ground vibrational state
of NO(X 2⌸1/2) and the lowest oxygen spin–orbit state
(3P2) are accessible.
Photoexcitation of NO2 is easy: strong absorption start-
In this paper, a specific issue concerning NO2 unimo-
ing at ϳ10 000 cmϪ1 is due to the low-lying A B2 electronic
2
˜
lecular decomposition near the dissociation threshold (D0
1
ϭ25,128.57Ϯ0.05 cmϪ1
)
is addressed, namely, how the
2
˜
state. Two other nearby electronic states ͑i.e., B B1 and
2
dissociation rate constant is affected by parent rotation. Ex-
citation energies near D0 are chosen because in this region
the rotational energy comprises a significant fraction of the
excess energy and therefore the effect of rotation is expected
to be prominent.
˜
C A2) are not optically accessible from the ground PES and
also have much smaller densities of states at the energies of
interest than does the ground PES.11,16–23 The X A1 and
2
˜
2
˜
A B2 electronic states are strongly coupled through the
asymmetric stretch vibration and are completely mixed
above 17 000 cmϪ1, giving rise to vibronic manifolds of A1
and B2 symmetries.16,23 These manifolds are coupled to one
another by rovibronic interactions, leading to rovibronic
a͒
Current address: Applied Materials, Inc., 3320 Scott Avenue, Santa Clara,
CA 95054.
b͒
Author to whom correspondence should be addressed. Electronic mail:
wittig@cheml.usc.edu
chaos above 23 000 cmϪ1 17
.
Above the dissociation thresh-
0021-9606/99/111(20)/9267/13/$15.00
9267
© 1999 American Institute of Physics
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9268
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Bezel, Ionov, and Wittig
old, however, the mixing of different zeroth-order rovibronic
states competes with decomposition. Consequently, the ex-
tent of rotationally induced energy transfer in this regime is
not a priori clear.
Rovibrational and rovibronic interactions in NO2 have
been studied from the bottom of the ground PES to excita-
tion energies near D0 .16,17,23–29 Delon et al. have estimated
the strengths of rovibronic interactions by using anticrossing
experiments with strong magnetic fields,25 as well as from
statistical analyses of absorption spectra for rotationally cold
molecules.17 In both cases it was deduced that second order
spin–orbit interaction is the main coupling mechanism. An-
other mechanism which is known to couple different rota-
tional states is perpendicular Coriolis interaction.27,28 Since
NO2 is a nearly prolate top, a symmetric top basis of rota-
tional eigenfunctions is generally adopted:17,24,25,30
FIG. 1. Schematic of the double resonance experiment. Rotational selectiv-
ity is achieved by using a narrow-linewidth nanosecond IR laser. Following
IR excitation, a picosecond pump pulse excites the ‘‘tagged’’ molecules to
energies above the dissociation threshold. After a set delay, a picosecond
probe pulse measures the amount of NO product via LIF. Only molecules
which have absorbed both the IR and picosecond pump photons have
enough energy to dissociate.
1
͉
N,K ,⌫ ϭ
͉
N,Ka
Ϯ
͉
N,ϪKa ),
͑1͒
͑
͘
͘
͘
a
&
where N is the rotational quantum number, Ka is the projec-
tion of the rotational angular momentum onto the molecular
a axis, and ⌫ is the symmetry of the rotational function.
Little is known about intramolecular coupling mecha-
nisms above dissociation threshold, and a study of the rota-
tional dependence of the dissociation rate constant can help
to establish their respective contributions. It is not the goal of
this paper to calculate absolute values of dissociation rate
constants of rotating NO2. What is pursued is how the dis-
sociation rate constants change with rotation and which
mechanisms participate. Thus, we are relieved of the neces-
sity to predict the rate itself—it has been measured—and we
can concentrate on the variation of the rate due to subtle
rovibrational dynamics.
Most, if not all, of the mechanisms responsible for rovi-
bronic couplings are known, though their respective impor-
tance in dissociating molecules is unclear. Moreover, experi-
ments have been limited to either small angular
momenta3–8,16,17,24,25,31–34 or high rotational temp-
eratures.2,5,9 In the latter, the data are averaged over a num-
ber of rotational states. In the former, only the first one or
two rotational states are observed, which makes extrapola-
tion of the measured dependencies to higher rotational states
ambiguous. In the present work, systematic measurements
have been carried out on a progression of rotational states
which spans approximately 100 cmϪ1 of rotational energy.
Other systems in which unimolecular decomposition has
been examined with rotational resolution are HFCO,35
HCO,36 CH3O,37 and CH2CO,38 and HOCl.39,40
the ͑1,0,1͒ intermediate vibrational level, and the brackets
denote an average over the linewidth of the picosecond pump
pulse. The first excitation progression, with the average pi-
cosecond pump photon energy set to excite the rotationless
system to energies just above D0 ͑i.e., 10 cmϪ1͒, was ex-
pected to be especially sensitive to the direction of energy
transfer from vibrational to rotational degrees of freedom.
The second excitation progression, set at the bottom of a
‘‘step’’ in the variation of the reaction rate constant versus
energy,5 was expected to be sensitive to the reverse direction
of energy transfer, i.e., from rotation to vibration. The N
Ј
quantum number was varied from 1 to 15, which corre-
sponds to ͑1,0,1͒ rotational energies up to 100 cmϪ1. The
dissociation rate constant was found to be unchanged. Were
100 cmϪ1 put into vibrational degrees of freedom, the rate
constant would have increased by a factor of approximately
two.5 Clearly, rotation affects the dissociation dynamics a
subtle way.
II. EXPERIMENT
The experimental arrangement is shown schematically in
Fig. 2. It is noteworthy that over 10 mJ of 3.4 ␮m radiation
was used in the first excitation step. This radiation derived
from an optical parametric oscillator/amplifier system ͑OPO/
OPA, STI Mirage 3000͒ pumped by an injection seeded
Nd:YAG laser ͑Continuum NY81C or Powerlite 9000͒.
Modifications to the Mirage 3000 enabled us to obtain 5–10
times more energy near 3.4 ␮m than is specified in the
manual. These modifications include ͑i͒ increasing the beam
size after the oscillator to match the aperture of the first
amplifier, ͑ii͒ using a single pass first amplifier stage ͑instead
of a resonant cavity͒ with corresponding adjustment of pump
beam delay, ͑iii͒ increasing the lengths of the KTP crystals in
the second amplifier stage from 15 to 20 mm, and ͑iv͒ care-
ful compensation of divergence and astigmatism of the pump
and seed beams. A color glass filter ͑Kopp 7-69͒ was added
between the oscillator and the first amplifier to eliminate 1.7
␮m radiation, which could cause interference in the first am-
In the present work, dissociation rate constants have
been measured by using the picosecond-resolution pump–
probe technique. The difference between this and earlier
work from our group5,8 is that in the present study single
rotational levels are selected by using double resonance, as
indicated in Fig. 1.
Measurements have been carried out for two progres-
sions of the average energies ͗E͘: D0ϩ10 cmϪ1ϩBN (N
Ј
Ј
ϩ1) and D0ϩ75 cmϪ1ϩBN (N ϩ1), where B is an NO
Ј
Ј
2
rotational constant, N is the rotational quantum number at
Ј
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9269
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Decomposition of NO2
DCM special laser dye with DTDCI saturable absorber and a
single-plate birefringent filter as a tuning element was used
in the dye laser to obtain a train of 2 nJϳ600 fs pulses at 678
nm. Pulses were selected at 10 Hz and amplified to ϳ2 mJ in
a three stage ‘‘red’’ dye amplifier. Concentrations of the dye
LDS 698 in MeOH in the different amplifier stages were
chosen to optimize beam quality and energy. A Schott glass
filter ͑RG780͒ was used as a solid state saturable absorber
between the first and second amplifier stages to reduce am-
plified spontaneous emission ͑ASE͒ and pulse broadening.
Approximately 40% of the 2 mJ 678 nm pulses was
tripled by using two 100 ␮m BBO crystals. This yielded a
probe pulse with ϳ2 ␮J of 226 nm radiation in a beam of ϳ3
mm diameter. The probe energy could be reduced by a half-
wave plate before the doubling stage.
To form the pump pulse the remaining 60% of the 2 mJ
678 nm pulses was focused into a cell containing D2O to
produce a continuum, out of which a ϳ30 cmϪ1 width was
selected by using a zero dispersion monochromator. A two
stage ‘‘blue’’ amplifier was used to bring the energy up to
100 ␮J with a beam diameter of ϳ6 mm. Coumarins 440 and
450 were used for 440 and 450 nm, respectively. To mini-
mize ASE and improve the beam quality, a telescope with a
pinhole in the focal plane and an interference filter were
placed between the first and the second amplifier stages. The
‘‘red’’ and ‘‘blue’’ amplifiers were pumped by ϳ35 mJ of
532 nm and ϳ45 mJ of 355 nm ͑ϳ80 ps pulses͒ from the
regenerative amplifier ͑Quantel International RGA60͒ seeded
with a fraction of the 1064 nm output of the Spectron SL903.
Pump and probe wavelengths and linewidths were mea-
sured with a double spectrometer ͑Spex 1401, 1800 grooves/
mm, 0.15 cmϪ1 resolution͒, calibrated against a wavemeter
͑Burleigh WA-4500, accuracy Ϯ1 cmϪ1͒ by using the tun-
able output of a distributed feedback dye laser set at 22 222.5
cmϪ1. This method enabled better absolute frequency cali-
bration than in our preliminary report.43,44 The uncertainty of
the average pump photon energy of Ϯ10 cmϪ1 is due to
calibration, measurement of the spectrum, and frequency
drift during the scan. The pump linewidth was ϳ30 cmϪ1
FWHM with a smooth close-to-Gaussian shape, while the
probe was ϳ50 cmϪ1 FWHM, which was enough to cover
the main band of the NO absorption spectrum.
After the probe beam encountered a computer controlled
delay stage ͑Aerotech ATS100͒ the probe and pump pulses
were combined collinearly by using a dichroic mirror. The
beams were then focused into a vacuum chamber. A 30 cm
focal length ͑f.l.͒ lens yielded the best signal/background ra-
tio ͑see below͒. The sizes of the pump and probe beams on
the lens were chosen to ensure that the probe was inside the
pump focal volume. The 3.4 ␮m pulse was counter-
propagating and arrived at the chamber ϳ50 ns before the
pump and probe pulses. The best signal was obtained with a
35 cm f.l. lens for the 3.4 ␮m pulse. All three beams inter-
sected ϳ1 mm from the nozzle orifice. LIF from NO was
collected, filtered, detected with a photomultiplier, preampli-
fied ͑Stanford Research fast preamplified SR445͒, digitized
͑Transiac 2001 and CAMAC 6001͒, averaged, and sent to a
computer. The integrated signal trace served as a measure of
the relative number of NO molecules.
FIG. 2. Diagram showing the experimental arrangement.
plifier. A frequency stabilization system ͑Laser Vision͒ was
used in some experiments to eliminate long term frequency
drift.
About 8% of the 3.4 ␮m beam traversed an opto-
acoustical cell containing ϳ1 Torr of room temperature
NO2. The opto-acoustical signal was monitored to ensure a
correct excitation frequency. The linewidth at 3.4 ␮m is es-
timated to be approximately 0.015 cmϪ1, based on resolution
of the rotational transitions 13 12, Kaϭ0 and 13 12,
Kaϭ1 of the (1,0,1) (0,0,0) band, which are 0.04 cmϪ1
apart.24 Table I lists the transitions used for the first excita-
tion step.
Much of the present technique for generating short pump
and probe pulses has been reported previously.5,41,42 The pri-
mary source of subpicosecond pulses is a dual-jet mode-
locked dye laser ͑Coherent 702-1͒ synchronously pumped by
2 W of second harmonic from an actively mode-locked
Nd:YAG laser ͑Spectron SL 903͒. The Nd:YAG laser was
driven at 76 MHz by a mode locked ͑CAMAC ML 4000͒.
TABLE I. Transitions used in the experiments; frequencies and assignments
from Ref. 24.
N
N
Ka h␯/cmϪ1
Other transitions within laser linewidth
Ј
Љ
1
3
5
7
9
2
0
0
0
0
0
0
0
0
2904.37 15 15, Kaϭ2
2
4
6
8
2908.54
2910.13
8
7, Kaϭ4
2911.68 10 9, Kaϭ4; 14 13, Kaϭ6; 20 19, Kaϭ8
2913.19 19 18, Kaϭ7; 14 13, Kaϭ5
2914.64 11 10, Kaϭ1
2916.06
2917.43 26 25, Kaϭ7
11 10
13 12
15 14
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9270
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Bezel, Ionov, and Wittig
The amount of NO signal was recorded as a function of
the pump–probe delay. The entire delay range was 33.2 ps
for all scans. Since some of the processes leading to NO
background were pump–probe delay dependent ͑see below͒,
data were recorded with the IR beam alternately on and off.
Precautions were taken to avoid long-term drift of the lasers:
after each 10 laser shots with the IR alternately on and off
the delay stage was moved to the next point, such that it took
only ϳ5 minutes to complete a scan over all delay points.
After that the direction of the delay stage travel was reversed
and the next scan was recorded. All IR-off scans were added
together and subtracted from the sum of the IR-on scans;
data handling details are given in Sec. III. A typical session
lasted 2–4 hours and consisted of 100–200 measurements
with the IR on and off for each pump–probe delay; the result
of the session will be referred to hereafter as a ‘‘scan.’’ Dur-
ing a ‘‘day’’ of measurements, 3–6 4 hour scans were re-
corded, each with a different IR transition. Approximately
eight scans were taken for each set of parameters. Typical
scans are shown in Fig. 3.
The cross correlation between the pump and the probe
pulses was recorded simultaneously with the LIF signal/
background scan. To measure the cross correlation, ϳ4% of
both the pump and the probe was split by a quartz window
and focused into a 100 ␮m BBO crystal tuned to generate the
difference frequency ͑DF͒. The DF wavelength was close to
that of the ASE of the ‘‘blue’’ amplifier and the pump beam
radiation, so the Spex 1401 spectrometer was used to select
only a narrow ͑ϳ0.5 cmϪ1͒ spectral region of the DF. When
recorded this way, the cross-correlation shape and position
did not depend on the exact frequency, which is probably the
consequence of negligible chirp in both the pump and the
probe pulses. The cross-correlation signal from the photo-
multiplier was amplified and measured by a gated integrator
and a boxcar averager.
NO2 samples were prepared as follows: 1.5–2 atm of O2
was sent through a bubbler containing NO2 ͑Matheson
99.5%͒ at 0 °C. The resulting mixture ͑80–85% O2 and 20–
15% NO2) was expanded into vacuum through a 0.2 mm
pulsed ͑200 ␮s duration͒ nozzle. The O2 pressure was opti-
mized to provide the best signal for each rotational level.
Precautions were taken to minimize NO2 decomposition on
metal surfaces: the delivery gas line was made of teflon and
was pumped constantly in order to provide a fresh mixture to
the nozzle orifice. These measures significantly reduced NO
background.
The presence of a strong LIF background from NO was
a serious obstacle. This background originated partially from
NO in the molecular beam. However, using oxygen as the
carrier gas and the nozzle modifications described above
minimized background to the point where signals could be
recorded. Another part of the background was due to photo-
induced processes: though one 450 nm photon is not ener-
getic enough to dissociate NO2 from the ground state, two
photon absorption brings NO2 far above the dissociation
threshold. In addition, 226 nm photons photodissociate NO2.
Another significant contribution to the background was due
to the pump pulse: two-photon excitation of NO or three-
photon excitation of NO2. A precise assignment has not been
FIG. 3. ͑a͒–͑c͒ Typical signal and cross correlation scans for different ex-
citation conditions. Circles: experimental product buildup traces ͑i.e., differ-
ence signals͒; thin lines: experimental cross correlations; thick lines: Eq. ͑8͒
is used to fit the buildup traces.
made, but it is a fact that some of the backgrounds are
pump–probe delay dependent and must be subtracted from
the signal. Figure 4 shows a typical background trace. Since
the processes contributing to the signal and to the back-
ground have different intensity dependencies, the intensities
of the pump and probe pulses were adjusted to optimize the
signal/background ratio. Even then, typical signal/
background ratios were often Ͻ3.
III. RESULTS
Data were obtained for two values of the average pump
photon energy h␯
: 22,233 and 22 298 cmϪ1. The cor-
͗
͘
pump
responding average excitation energies ͗E͘ are given by
E ϭEv1i0b1ϩErot ϩ h␯
,
͘
pump
͑2͒
͗ ͘
͗
101
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9271
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Decomposition of NO2
t Ϫt
i
Sfit t ͒ϭS
ϩA
baseline
0dt CC t ͒
͑
͑
Ј
Ј
͵
i
Ϫϱ
ϫ 1Ϫexp Ϫ t Ϫt Ϫt ͒/␶ ,
͔
͖
͑8͒
͓
͑
͕
Ј
i
0
where t0 is an offset for the delay between the signal and
cross correlation traces, A and ␶ are the amplitude and
buildup time parameters, respectively, CC(t ) is the experi-
Ј
mental cross correlation curve, and Sbaseline is the signal base-
line. The cross correlation baseline was set at the average of
the values measured at large negative and large positive de-
lays. The adjustable parameters are A, Sbaseline , t0 , and ␶.
Since product buildup was assigned a single exponential,
the use of a Levenberg–Marquardt fitting routine45 was fa-
cilitated by using analytical derivatives, i.e., ץ(␹2)/ץpi and
ץ2(␹2)/ץpiץpj , where pi and pj denote the fit parameters.
The experimental traces were fit and standard deviations for
the ␶ values were calculated. Confidence levels for individual
traces were used as weights for calculating the average ␶
values given in Table II.
FIG. 4. Example of the ‘‘background’’ signal. Squares show the background
recorded with the IR off. Other notations are the same as in Fig. 3. Note the
time dependence of the IR-off signal.
vib
101
rot
101
where E and E are the vibrational and rotational ener-
The ␶ values listed in Table II indicate clearly a depen-
gies, respectively, of the ͑1,0,1͒ intermediate vibrational
dence on h␯
. However, there is no significant depen-
͘
͗
pump
level. Using Ev1i0b1ϭ2906 cmϪ1, D0ϭ25 129 cmϪ1, and the
rot
dence on the IR photon frequency, i.e., on E . This is the
101
values of h␯
given above yields
͘
pump
͗
major experimental finding of this work.
E ϭD ϩ10 cmϪ1ϩE1ro0t1
͑3͒
͑4͒
͗ ͘
0
IV. DISCUSSION
and
A. The meaning of the experimental rates
E ϭD ϩ75 cmϪ1ϩE1ro0t1
.
͗ ͘
0
Considerable theoretical work has been devoted to uni-
molecular reactions of polyatomic molecules. For example, a
typical effective Hamiltonian approach treats a dissociating
molecule as a system of discrete levels coupled to product
continua, leading to resonance widths ប⌫.46 Isolated reso-
nances decay with single exponential rates ⌫i , whose aver-
Approximately eight scans were recorded for each set of
h␯
and Er1o0t1 values, and each scan yielded a parameter
͗
͘
pump
which we designated as ‘‘the rate constant.’’ These rate con-
stants were obtained as follows.
NO LIF signals were recorded with the IR beam alter-
nately off and on at different delay times, resulting in Soff(t)
and Son(t), respectively, where the bar signifies an average
over laser shots. Subtracting the former from the latter
yielded the difference signal function, S(t). Standard devia-
tions were calculated at each pump–probe delay, ti , by using
age value, ⌫ , is the conventional unimolecular decompo-
͗
͘
i
sition rate constant k. Unfortunately, in most cases, the
resonances overlap, making the situation more
complicated.47–52
The NO2 absorption spectrum does not display isolated
transitions even a few wave numbers above D0 .4,7,33,34
Though attempts have been made to extract dissociation rate
constants from the spectral features observed in the threshold
regime, there is no trust-worthy basis for doing so because of
severe spectral congestion,4,7,53,54 and therefore time resolved
␴ t ͒2ϭ␴ t ͒2ϩ␴ t ͒2,
͑5͒
͑
͑
͑
off i
i
on
i
where ␴(ti) is the standard deviation of the average at time
ti . ␴on(ti)2 is given by
2
Son t ͒2ϪS t ͒
͑
͑
i
i
on
␴
t ͒2ϭ
,
͑6͒
͑
on
i
NϪ1
TABLE II. Buildup times ͑␶͒ and standard deviations ͑␴͒ for different in-
termediate rotational levels ͑N ͒. The two values of h␯
correspond to
͘
Ј
͗
pump
where Son(ti)2 is the average of the squared IR-on signal at
ti ,Son(ti) 2 is the square of the average IR-on signal at ti ,
and N is the number of laser shots at ti ͑typically 100–200͒.
The expression for ␴off(ti)2 is also given by Eq. ͑6͒; i.e., the
word on is replaced by the word off. The experimental scan,
S(t), was fitted to a single exponential buildup convoluted
progressions beginning at average excess energies of 10 and 75 cmϪ1, as
indicated in the headings.
10 cmϪ1
75 cmϪ1
rot
101
/cmϪ1
N
␶/ps
␴/ps
␶/ps
␴/ps
Ј
E
1
3
5
7
9
11
13
15
1
5
5.8
6.5
6.3
6.2
6.1
6.5
5.7
6.0
0.2
0.3
0.4
0.5
0.3
0.3
0.2
0.3
5.1
5.0
4.6
4.9
4.9
4.7
4.7
4.1
0.3
0.4
0.2
0.2
0.3
0.3
0.2
0.3
2
with the measured cross correlation by minimizing the ␹
parameter:
13
24
38
56
77
101
M
2
S t ͒ϪS t ͒͒
͑ ͑
͑
i
fit
2
i
␹2ϭ
,
͑7͒
͚
␴ t ͒
͑
iϭ1
i
where M is the number of delay points and Sfit(ti) is given by
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9272
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Bezel, Ionov, and Wittig
measurements are needed for determining rate constants with
good accuracy. In the regime of overlapping resonances,
even the concept of resonances is somewhat abstract, since it
is no longer possible to observe them individually, no longer
possible to excite a single zeroth-order molecular eigenstate,
and decay is no longer described by a single exponential.
Moreover, in our experiments, NO2 is excited by a spectrally
broad laser pulse and therefore a number of zeroth-order vi-
bronic states with different decay rates are accessed. Hence,
the resulting kinetics are inherently nonexponential.
Abel et al. have monitored NO2 survival versus time fol-
lowing excitation with a broad linewidth femtosecond pulse
and have noted a nonexponential survival probability.4 This
was due in large part to the fact that a significant fraction of
the molecules were excited to energies below D0 . When
monitoring product buildup, neither the experiments of
Ionov et al.5 nor those discussed herein were able to discern
nonexponential character. It should be emphasized that since
NO product is monitored, the measurements are insensitive
to NO2 molecules which are excited to energies below D0 or
which dissociate very slowly.
FIG. 5. Fitting a two-exponential buildup with a single exponential. The
solid line is the function (1–0.5eϪ0.1t –0.5eϪ0.3t) convoluted with a 1.5 ps
FWHM Gaussian cross correlation ͑thin line͒; time ͑t͒ is in picoseconds.
The dashed line is the fit given by Eq. ͑8͒. The traces are barely distinguish-
able.
ϭ0.5, ␴ϭ1011 Ϫ1). The resulting trace was fit to a single
s
exponential buildup by using the same routine as with the
experimental traces, but with a 1.5 ps FWHM Gaussian as
CC(t) in Eq. ͑8͒, and with all ␴(ti)ϭ1 in Eq. ͑7͒. The best
To examine the sensitivity of our experiment to non-
exponential product buildup, let us assume that the signal
S(t), is a sum of contributions, each having a rate constant
ki :
fit was obtained with ␶Ϫ1ϭ1.8ϫ1011 Ϫ1, which is only 10%
s
smaller than the average decay rate constant. The traces are
barely distinguishable.
S t͒ϭ
͑
P 1Ϫexp Ϫk t͒ ,
͑9͒
͑
͕
͖
͚
i
i
The same procedure was used to obtain rate constants
for the different unimolecular decomposition models dis-
cussed below. Namely, product buildup was calculated and
convoluted with a 1.5 ps FWHM Gaussian cross correlation,
and the resulting trace was fit to a single exponential by
using Eq. ͑8͒ with a Gaussian CC(t).
i
where Pi is the probability of having the rate constant ki . To
examine modest ranges of ki , S(t) is expanded around the
average value of the ki , i.e.,
k ϭ
͗ ͘
P k .
͑10͒
The dissociation rate constants determined in our experi-
ments are close to the average rate constants given by Eq.
͑10͒. From random matrix analyses of the survival probabil-
ity, it has been argued that the decay of a system in the
regime of strongly overlapping resonances might be slower
than a single-exponential function with the parameter
͗k͘.47,49,55 However, in using the ͑calculated͒ data of Ref. 45
when product buildup is fit by using a single exponential the
same way as with our experimental data, the fit gives the
average rate constant to within ϳ5%.
͚
i
i
i
This yields
S t͒ϭ
P Ϫexp Ϫ k t
͗ ͘
P 1ϩ⌬k tϩ 1 ⌬k ͒2t2ϩ¯͒
͑
͑
͑
2
͕
͖
͚
͚
i
i
i
i
i
i
ϭ1Ϫ 1ϩ 1␴2t2ϩ¯͒exp Ϫ k t ,
͑11͒
͑
͕
͗ ͘
͖
2
where ⌬k ϭk Ϫ k and ␴2ϭ͚iPi(⌬ki)2 is the square of
͗ ͘
i
i
the variance.
To be observed, deviations from single exponential de-
Despite the fact that our measurements are not very sen-
sitive to the spread of the contributing rates, the average rate
constant can be determined with good accuracy. We estimate
that у30% variation of an average rate constant would be
detectable.
cay must exceed the noise level, which is ϳ10% in the
present case. This results in ␴ϳ1011 Ϫ1. In addition, the
s
signal due to product buildup is a convolution of the molecu-
lar response function with the pump-probe cross correlation,
whose temporal width is comparable to the dissociation
times. This couples the fit parameters. Thus, to see devia-
tions from single exponential buildup, a higher signal-to-
noise ratio ͑S/N͒ is required than could be achieved in the
present study. With the present S/N, we estimate ͑from nu-
merical simulations͒ that to observe nonexponential behavior
B. Influence of molecular rotation on the dissociation
rate
The microcanonical unimolecular decomposition rate
constant can be expressed as56,57
requires ␴у4ϫ1011 Ϫ1, which is larger than the average
s
rate constant itself.
W‡ E͒
͑
k E͒ϭ
͑
,
͑12͒
h␳ E͒
͑
Figure 5 illustrates some of the above considerations. A
trace was generated by convoluting a 1.5 ps FWHM Gauss-
ian cross correlation with two exponential buildups (k1
where ‡ denotes the transition state ͑TS͒, E is the total en-
ergy relative to the zero-point level of the parent, W‡(E) is
the number of open channels at the TS, ␳(E) is the reactant
ϭ1011 sϪ1 and k2ϭ3ϫ1011 Ϫ1) of equal weight (P1ϭP2
s
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9273
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Decomposition of NO2
density of states, and h is Planck’s constant. It is obvious, yet
significant, that uncertainties in k(E) accrue from uncertain-
ties in both W‡(E) and ␳(E). For nonrotating molecules on
a single PES, ␳(E) is the vibrational density of states, and
when two or more PES’s are strongly coupled, ␳(E) is the
corresponding vibronic density of states. When symmetry is
present, both W‡(E) and ␳(E) should be sorted according to
symmetry species.
At energies below D0 , it is possible to obtain ␳(E) spec-
troscopically, for example, by counting the number of levels
accessed in an absorption spectrum. However, above D0 ,
␳(E) cannot be obtained spectroscopically, since absorption
spectra are congested—even in the first few cmϪ1 above D0
with samples cooled to a few K.4,7,33,34 Consequently, ␳(E)
must be either calculated or obtained by extrapolation from
below D0 .12,14,16,22
The dramatic increase of ␳(E) observed near D0 ͑Refs.
1, 4, 7, 32, 58, and 62͒ casts doubt on the viability of ex-
trapolations in the immediate vicinity of D0 . In addition,
W‡(E) generally cannot be measured, necessitating accurate
electronic structure and dynamics calculations, and parent
rotation and nonzero electronic spin further complicate mat-
ters. Even state-of-the-art calculations of W‡(E) and ␳(E)
cannot reproduce near-threshold effects.12,13,59–61
FIG. 6. Dissociation rate constants of rotationally cold molecules. Squares
denote experimental values ͑Ref. 4͒. Horizontal error bars indicate the ex-
citation linewidths; vertical error bars are standard deviations. The two ar-
rows show the onsets of the double resonance progressions. The solid lines
are the linear interpolation used to obtain knonrot(E).
W‡ EϪE‡
͒
͑
rot
knomix E͒ϭ
ϭknonrot EϪE‡ ͒,
͑13͒
͑
͑
rot
h␳ E͒
͑
where E is the total energy including parent rotation, E‡rot is
the rotational energy of the molecule at the TS, and ␳(E) is
assumed to be constant throughout the modest range of E
values studied here. For NO2, both W‡(EϪEr‡ot) and ␳(E)
are symmetry-sorted according to vibronic species. Equation
͑13͒ implies that rotation and vibration can be separated at
the TS and that the number of open channels depends only
The above limitations notwithstanding, the energy de-
pendence of Eq. ͑12͒ can assist in the determination of how
the dissociation rate constant changes with parent rotation.
Namely, though W‡(E) and ␳(E) cannot be measured sepa-
rately, k(E) for rotationally cold NO2 ͓hereafter referred to
as knonrot(E), see Fig. 6͔ has been measured5 and will be
used as an empirical input to the model presented below; the
pathological region within the first few wave numbers near
D0 ͑Refs. 1, 4, 7, 32, 58, and 62͒ will be neglected.
For rotating polyatomic molecules, only the total angular
momentum and its projection on a laboratory-fixed axis have
good quantum numbers, i.e., J and MJ , respectively. Effects
due to nuclear spin can be neglected on the timescale of the
present studies. The rotational angular momentum and its
projection on a molecule-fixed axis ͓with the corresponding
͑prolate͒ symmetric top quantum number Ka͔ are not rigor-
ously conserved. That is, states with different N and Ka can
be coupled through rovibronic interactions.
Molecular rotation can influence the dissociation dynam-
ics via several mechanisms. Let us distinguish two types: ͑i͒
those that do not change the rotational state of the molecule,
such as centrifugal distortion of the potential and parallel
Coriolis interaction; and ͑ii͒ those that couple different rota-
tional states ͑while of course conserving total angular mo-
mentum, including its projection on a laboratory fixed axis͒
and destroy vibronic symmetry, such as perpendicular Cori-
olis and spin–rotation interactions.
on the excess energy EϪEr‡ot
.
E‡rot depends on the TS geometry through the rotational
constants A‡, B‡, and C‡, and the quantum numbers N, Ka
͑prolate limit͒, and Kc ͑oblate limit͒. For the present model-
ing, the TS is taken to be at Rϭ3 Å and ␪ϭ130° in Jacobi
coordinates.11,28,59 The influence of the TS geometry will be
discussed below. In the present case, B‡ and C‡ are close
enough to justify the symmetric top approximation:
E‡rot N,K ͒ϭB‡N Nϩ1͒ϩ A‡ϪB‡͒K2,
͑14͒
͑
͑
͑
a
a
which should be substituted into Eq. ͑13͒ to obtain
knomix(E). A linear interpolation to data measured at 6 K5 is
used to obtain knonrot(EϪE‡rot), as shown in Fig. 6.
Now consider the situation in which rotational states
with different Ka values are coupled by rovibronic interac-
tion. For the sake of simplicity electron spin is neglected in
this section, in which case N is a good quantum number.
Equation ͑12͒ is premised on an ergodic assumption: all
zeroth-order states are thoroughly mixed and the correspond-
ing classical motion is chaotic. Analyses of NO2 absorption
spectra in the range of a few thousands of wave numbers
below D0 show that the Ka quantum number breaks down,
leading to so-called rovibronic chaos.1,7,17,62 However, above
the reaction threshold, randomization of the Ka quantum
number competes with dissociation, and therefore the extent
of Ka mixing is not a priori clear.
Centrifugal distortion of the potential is too small to alter
the vibronic density of states significantly. However, the en-
ergy associated with centrifugal distortion may be of the
same order of magnitude as the energy available at the TS,
and therefore the number of open channels can be affected
significantly. If the rotational state is unchanged throughout
the course of reaction, Eq. ͑12͒ can be written56,57
Complete Ka mixing prior to dissociation gives
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9274
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Bezel, Ionov, and Wittig
N
W‡ EϪE‡ N,K ͒͒
͑
͑
a
rot
kfullmix N,E͒ϭ
͑
͚
Nϩ1͒h␳ E͒
͑
͑
K ϭ0
a
N
1
ϭ
knonrot EϪE‡ N,K ͒͒, ͑15͒
͑ ͑
a
rot
͚
Nϩ1 K ϭ0
a
where level countings for W‡ and ␳(E) are carried out over
vibronic levels. The Nϩ1 term accounts for the increase of
the density of available states due to Nϩ1 projections of the
rotational angular momentum on the molecular axis. Note
that Nϩ1 is used, rather than the 2Nϩ1 possible states that
arise from projecting the rotational angular momentum on
the molecular axis. This is due to the fact that most of the
vibronic levels in the vicinity of D0 have either A1 or B2
symmetry.16,22,23 Of the levels above D0 which have Ka
ϭ0, even N, and A1 total symmetry ͑following double-
resonance excitation͒, only half of the states can be mixed
efficiently, i.e., those with A1 or B2 rotational symmetry.
Those of B1 or A2 rotational symmetry require B1 or A2
vibronic wave functions, respectively, which can only arise
2
2
˜
˜
from B B1 and C A2 electronic states. However, since the
2
2
˜
˜
B B1 and C A2 vibronic manifolds have much smaller den-
sities of states, they are neglected.
To simulate the data, product buildup was calculated by
2
2
˜
˜
using the selection rules for the A B2 X A1 parallel tran-
sition. Namely, suppose the initially prepared rotational state
of the ͑1, 0, 1͒ vibrational level has quantum numbers N
Ј
and KaЈϭ0. The photoexcited dissociative levels then have
quantum numbers NϭN Ϯ1 and K ϭ0, with the photoex-
Ј
a
¨
citation probabilities proportional to the Honl–London fac-
tors:
2
2
N ϩ1͒
N
Ј
͑
Ј
and
͑16͒
N ϩ1͒ 2N ϩ1͒
N ϩ1͒ 2N ϩ1͒
͑
Ј Ј
͑
͑
͑
Ј
Ј
for the R and P branch transitions, respectively.63 The con-
tributions from the P and R branch transitions were averaged
accordingly.
FIG. 7. Comparison of simulated and experimental dissociation rate con-
Figure 7 shows the results of such simulations for two
extreme cases: ͑a͒ no Ka mixing, and ͑b͒ complete Ka mix-
ing. Obviously, these limiting cases are not able to reconcile
the data. With no Ka mixing, the rate constant increases with
N over the range 1рNр15, whereas for complete Ka mix-
ing, the rate constant decreases with N. Mixing of different N
via spin–rotation coupling does not change this. The above
dependencies are intuitive: if Kaϭ0 is preserved, the rate
constant increases because the energy available for reaction
increases with N. On the other hand, for complete Ka mix-
ing, the rate constant decreases because the high-Ka zeroth-
order rovibronic states accessed via energy transfer dissoci-
ate slowly ͑or even not at all͒ when Ka is sufficiently large.
At zero rotation the calculated rate constants do not co-
incide exactly with those obtained in the double resonance
experiments. However, this does not signify a discrepancy
between the double resonance results and the one-photon
measurements. The dissociation rate constants in the double
resonance experiments—extrapolated to zero rotation—are
stants. Circles and dashed lines are for E ϪD ϭ10 cmϪ1ϩE1ro0t1 ; squares
͗
͘
0
and solid lines are for E ϪD ϭ75 cmϪ1ϩE1ro0t1 . Open symbols indicate
͗
͘
0
experimental data; error bars indicate standard deviations. Lines indicate
simulations: ͑a͒ no Ka mixing; ͑b͒ complete Ka mixing. Thick lines indicate
a TS at Rϭ3 Å and ␪ϭ130° ͑Jacobi coordinates͒ with rotational constants
A‡ϭ3.02 cmϪ1 and B‡ϭ0.166 cmϪ1. Thin lines indicate a TS at Rϭ3.5 Å
and ␪ϭ120° with A‡ϭ2.3 cmϪ1 and B‡ϭ0.125 cmϪ1. These extreme cases
do not fit the data.
samples ͑where effects due to rotational excitation are mini-
mal͒ yields the same rate constants for E ϪD ϭ8 and 59
͗ ͘
0
cmϪ1, respectively, i.e., at approximately the same energies.5
The difference can be attributed to the linear interpolation
used for knonrot(E), which smoothes any structure in the
variation of the reaction rate constant with energy.
The only parameter in the model which is not deter-
mined experimentally is the TS geometry. The ground PES,
on which dissociation occurs, is barrierless, resulting in TS’s
at large separations for nonrotating molecules just above
threshold. Grebenshchikov et al. have studied TS positions
1.6ϫ1011 and 2.0ϫ1011 sϪ1 for E ϪD ϭ10 and 75 cmϪ1
,
͗ ͘
0
respectively. The one-photon results for expansion cooled
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9275
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Decomposition of NO2
Vsrϭ
͉
A
1ʈT11 vibronic͒
ʈ
B2
͉
͑18͒
by using classical trajectories. They found that as the excess
energy increases to just a few tens of cmϪ1 the TS moves
inward to separations of slightly more than 3 Å.59 At yet
higher energies, the variation of the TS position is modest.
For example, it moves further inward by only ϳ0.1 Å for the
next 1000 cmϪ1. Similar results can also be obtained by ana-
lyzing the PES calculated earlier by Katagiri and Kato.11
Namely, bending levels orthogonal to the reaction coordinate
͑
͗
͘
0
is 0.7Ϯ0.1 cmϪ1 near D0 .17 They pointed out that the qϭ0
term in Eq. ͑17͒ ͑i.e., corresponding to ⌬NϭϮ1, ⌬Kaϭ0)
can be neglected, since it couples the A1 and B2 vibronic
states to the relatively sparse B1 and A2 vibronic manifolds.
The surviving coupling terms have qϭϮ1, for which ⌬N
ϭ0,Ϯ1, ⌬KaϭϮ1. Referring to Eq. ͑17͒, for values of S
ϭ1/2, Ka ϭKa , and Ka ϭKaϩ1, the only surviving term in
result in barriers at ϳ3 Å for excess energies of ϳ80 cmϪ1
.
1
2
Klippenstein et al. also place the TS at ϳ3 Å.12 Furthermore,
NO2 rotation facilitates tightening of the TS near threshold,
since the binding energy at 3 Å is tens of cmϪ1, i.e., compa-
rable to the centrifugal barriers. Although the TS is thought
to be at much larger separations right at D0 , there is no
reliable model to describe quantitatively the TS tightening,
and for the present calculations the TS was set at 3 Å regard-
less of excitation energy and angular momentum. Different
TS geometries do not produce qualitatively different results,
as shown in Fig. 7.
From the comparison given in Fig. 7, it is concluded that
dissociation takes place in a regime of intermediate coupling.
Namely, rotational levels other than Kaϭ0 participate, but
the rotational dynamics are not fully ergodic on the dissocia-
tion time scale. To model such a regime, explicit values of
the coupling matrix elements are needed.
the summation is that with qϭ1, and Eq. ͑17͒ becomes
͉
␩ ,N ,K ,S,J Hsr ␩ ,N ,K ϩ1,S,J
͉
͉
͉
͘
͗
1
1
a
2
2
a
1
2
J
N1
3
2
ϭͱ
2N ϩ1͒ 2N ϩ1͒
͑
͑
ͯ
1
2
ͭ
ͮ
1
2
1
N2
N1
1
1
N2
ϫ
Vsr .
͑19͒
ͩ
ͪ
ͯ
0
ϪKa
Kaϩ1
This expression can be used to see how the Hsr matrix
elements scale with the rotational quantum numbers. In fact,
they are approximately constant with respect to N and Ka .
This is not obvious from the equation, but follows from the
explicit values of the 6-j and 3-j symbols.
C. Coriolis, spin–rotation, and inertial asymmetry
coupling mechanisms
2. Coriolis coupling
Coupling of different rotational states occurs via interac-
tions such as Coriolis, spin–rotation, and inertial asymmetry.
For NO2 at energies near D0 , spin–orbit and Coriolis inter-
actions are believed to be particularly important.17,27,28,30
Grebenshchikov et al. have used classical trajectory cal-
culations to demonstrate that the robustness of the Ka quan-
tum number decreases with increasing NO2 rotational exci-
tation due to perpendicular Coriolis and inertial asymmetry
couplings.28 The off-diagonal Coriolis matrix elements are
given by30
1. Spin–rotation coupling
It has been shown by Bird et al. that spin–rotation cou-
pling near the minimum of a potential well is dominated by
a second-order interaction which involves spin–orbit and
orbit–rotation terms.67 Moreover, Delon et al. have deduced
that for rotationally cold NO2 the mixing of the zeroth-order
rotational levels which occurs in the range 17 000–23 945
cmϪ1 is due primarily to this mechanism.17,25 An expression
for the relevant coupling matrix elements has been derived
by Hallin and Merer:30
␩ ,N,K
͉
HCor
͉
␩ ,N,K Ϯ1
͗
͘
1
a
2
a
ϭϪͱN Nϩ1͒ϪK
͑
K Ϯ1͒ ␩1ʈBLBϯCLC
ʈ
␩
2
, ͑20͒
͘
͑
͗
a
a
where A, B, and C are rotational constants and LB and LC are
components of the orbital angular momentum operator. The
S and J quantum numbers are conserved and therefore omit-
ted from the notation. Upon introducing the average magni-
tude of the reduced Coriolis matrix element:
␩ ,N ,K ,S,J͉Hsr͉␩ ,N ,K ,S,J
͘
2 2 a
2
͗
1
1
a
1
VC0 or
ϭ
͉
A
1ʈBLBϯCLC
ʈ
B2
͉,
͘
͑21͒
͗
N ϩSϩJ
2
ͱS Sϩ1͒ 2Sϩ1͒ 2N ϩ1͒ 2N ϩ1͒
͑
ϭ Ϫ1͒
͑
͑
͑
͑
1
2
where A1 and B2 denote the coupled vibronic species, the
average magnitude of the matrix element given in Eq. ͑20͒
can be written as
N1
1
N2
Ka
J
S
N1
S
N ϪK
1
a
1
ϫ
Ϫ1͒
͑
ͭ
ͮ
ͩ
ͪ
͚
ϪKa
1
N2
q
q
1
2
ϫ ␩1ʈT1q vibronic͒
ʈ
␩
,
͑17͒
͑
͗
͘
͉
␩ ,N,K ͉HCor͉␩ ,N,K Ϯ1 ͉
͘
1 a 2 a
͗
2
K Ϯ1͒VCor
͑
.
͑22͒
where J is the total angular momentum excluding nuclear
ϭͱN Nϩ1͒ϪK
͑
a
a
0
spin, S is the electronic spin, ␩1ʈT1q(vibronic)
ʈ
␩
͘
2
is the
͗
reduced matrix element for the vibronic part of the interac-
tion, which depends only on the vibronic states and
, and the curly brackets and big parentheses denote 6-j
and 3-j symbols, respectively. Delon et al. have found that
the average value of the magnitude of the reduced matrix
element:
The spin–rotation and Coriolis matrix elements have
͉
␩
very different dependencies on the rotational quantum num-
bers. Namely, the matrix elements given by Eq. ͑22͒ grow
almost linearly with increasing N ͑for low Ka), while the
matrix elements given by Eq. ͑19͒ stay approximately con-
stant with increasing N.
͘
1
͉
␩
͘
2
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9276
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Bezel, Ionov, and Wittig
3. Inertial asymmetry coupling
numbers. This mechanism couples rotational states according
to ⌬Nϭ0, ⌬KaϭϮ2. For a nearly prolate top, in the first
approximation:64
Off-diagonal matrix elements due to inertial asymmetry
also have a significant dependence on the rotational quantum
␩,N,K Ϯ2 ϭ 1 CϪB͒
¯ ¯
K Ϯ1͒͒ N Nϩ1͒Ϫ K Ϯ1͒ K Ϯ͒͒,
͑ ͑
͑23͒
ͱ N Nϩ1͒ϪK
͑ ͑
␩,N,K
͉
Hasym
͉
͑
͑
͑
͑
͗
͘
a
a
4
a
a
a
a
¯
¯
dPN,K E,t͒
where B and C are vibrationally averaged rotational con-
stants. For low Ka , the matrix elements grow approximately
quadratically with N.
͑
a
ϭϪkN,K PN,K E,t͒
͑
a
a
dt
˜
N
Because of the N dependencies of their off-diagonal ma-
trix elements, Coriolis and asymmetry couplings can both
take place to a significant extent in the present experiments,
even though their roles are believed to be secondary relative
to that of spin–rotation coupling at low N.17,25 For fast rota-
tion, asymmetry and/or Coriolis couplings are significant
even for modest values of their reduced matrix elements.
There is an important difference between these two
mechanisms. Inertial asymmetry coupling comes about be-
cause an asymmetric top Hamiltonian is approximated by a
symmetric top Hamiltonian. Thus, the perturbation does not
operate on the electronic and vibrational parts of the wave
function. The asymmetry couples different zeroth-order sym-
metric top rotational wave functions built on the same
electronic/vibrational state.65,66 On the other hand, Coriolis
coupling arises from motion in a rotating reference frame,
and therefore can couple different vibronic states.
˜ ˜
Ϫ
␥
P
E,t͒
͑
N,K
a
͑
͚ ͚
N,K ;N,K
a
a
˜ ˜
N
K ϭ0
a
˜ ˜
ϪPN,K E,t͒͒,
͑24͒
͑
a
where PN,K (E,t) is the population at energy E and time t of
a
˜ ˜
levels having N and Ka quantum numbers, ␥N,K
is the
a
;N,K
a
˜ ˜
mixing rate constant between (N,Ka) and (N,Ka), and N
˜
and N can assume the values JϮ1/2. The dissociation rate
constants kN,K are given by
a
kN,K ϭknonrot EϪE‡ N,K ͒͒
͑25͒
͑
͑
a
rot
a
with E‡rot(N,Ka) given by Eq. ͑14͒.
The set of Eqs. ͑24͒ was solved by using a fourth-order
Runge–Kutta routine with initial populations obtained by us-
¨
ing the optical selection rules and the Honl–London factors
The inertial asymmetry of NO2 is not large. At the bot-
tom of the ground PES the A, B, and C rotational constants
are 7.99, 0.43, and 0.41 cmϪ1, respectively.67 The B and C
values change slightly with vibration, whereas A is sensitive
to bending. Regardless, the molecule remains close to a sym-
metric top. For example, for the ͑1,0,1͒ vibrational level, A,
given by Eq. ͑16͒. Product buildup at energy E was calcu-
lated by using
N
Pprod E,t͒ϭ
P
E,0͒ϪP
E,t͒͒ ͑26͒
͑
N,K
a
͑
͑
͑
͚ ͚
N,K
a
N
K ϭ0
a
B, and C are 7.85, 0.43, and 0.405 cm
,
Ϫ1 24 while for ͑2,13,1͒,
Highly vibrationally
they are 16.0, 0.43, and 0.41 cmϪ1 27
.
excited molecules may become either more or less symmet-
ric than the ground state, depending on the shape of the PES.
Rotational constants near D0 have not been measured,
¯
but estimates from trajectory calculations suggest that ͉B
is only 0.013 cmϪ1 68
¯
ϪC͉ . With this amount of inertial
asymmetry, and given the large separations between Ka lev-
els, we conclude that the effects of asymmetry can be ne-
glected in the present study.
D. Ka mixing in competition with unimolecular
decomposition
A model is presented here in which vibronic levels hav-
ing particular N and Ka values dissociate, with rate constants
given by Eq. ͑13͒, in competition with Ka mixing ͑Fig. 8͒.
The following phenomenological equations are used to de-
scribe this competition:
FIG. 8. Schematic representation of the couplings. The stacks represent
zeroth-order rovibronic manifolds of A1 ͑dark͒ and B2 ͑light͒ vibronic
symmetry, sorted according to rotational quantum numbers. Each stack can
either dissociate with a rate given by Eq. ͑13͒ or undergo transitions to other
stacks with rates given by Eq. ͑28͒. Arrow thicknesses indicate qualitative
trends.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9277
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Decomposition of NO2
and Pprod(E,t) was then averaged over the 30 cmϪ1 FWHM
Gaussian model excitation line shape, g(E):
9͑a͔͒, as measured by Delon et al.,17,25 the simulated depen-
dence is almost indistinguishable from that of no Ka mixing
shown in Fig. 7͑a͒. This can be understood in terms of the Ka
changes which are needed for Ka mixing to cause measur-
able effects. For example, for high N, changing Ka from 0 to
1 or 2 has essentially no effect on the rate constant, as can be
seen from Eqs. ͑13͒ and ͑14͒.
ϱ
D
͐
dE g E͒P
E,t͒
͑
prod
͑
0
P t͒ϭ
͑
.
͑27͒
ϱ
D
͐
dE g E͒
͑
0
To obtain the total product buildup, P(t) traces for four tran-
To bring the high-N rate constants down, the coupling
matrix elements must be made an order of magnitude larger.
sitions ͑i.e., JϭNϮ1/2, NϭN Ϯ1, and ⌬K ϭ0) were
Ј
a
summed. The result was convoluted with the 1.5 ps FWHM
Gaussian model cross correlation and then it was fit to a
single exponential ͓Eq. ͑8͔͒ to obtain the parameter ␶, which
is compared to the experimental result.
In the model, the initially prepared Kaϭ0 levels couple
to other Ka levels via successive ⌬KaϭϮ1 couplings. Re-
call that our qualitative conclusion concerning the degree of
Ka mixing is that a significant fraction of the Ka phase space
is mixed prior to dissociation. Consequently, since the mea-
sured dissociation rates are equal to or exceed 1.6
˜ ˜
ϫ1011
s
Ϫ1, the rate constants ␥N,K
must be at least
a
;N,K
a
comparable to this.
In the strong coupling limit ͑i.e., V ӷ1, where V is the
␳
coupling matrix element͒ the Ka mixing rate constants are
given by Fermi’s golden rule:
2␲
V2␳.
͑28͒
˜ ˜
␥
aϭ
N,K ;N,K
a
ប
is proportional to V2 as long as ␳
˜ ˜
;N,K
a
In this regime, ␥N,K
a
is independent of rotation. For a given coupling mechanism,
i.e., one whose matrix elements scale as per Eqs. ͑19͒ or
˜ ˜
͑22͒, the ␥N,K
were varied to attain the best fit to the
;N,K
a
a
data.
As mentioned above, though the vibronic density of
states above D0 is not known, extrapolation from energies
below D0 ͑but avoiding the pathological region just a few
cmϪ1 below D0) gives ϳ0.38/cmϪ1 for each symmetry
species.16 In the following discussion, the relation between
the average coupling matrix elements and the transition rate
constant is assumed to be given by Eq. ͑28͒ with ␳
ϭ0.38/cmϪ1, even though V ӷ1 is not satisfied for small
␳
values of the matrix elements. A slightly better approxima-
tion is given by the picket fence model of Bixon and
Jortner:69
2
2
2
2
˜ ˜
ͱV ϩ ␲V ␳͒ .
͑
␥
aϭ
͑29͒
N,K ;N,K
a
ប
However, using Eq. ͑29͒ in place of Eq. ͑28͒, with V given
by any of Eq. ͑19͒ or ͑22͒, does not change the results. Equa-
tion ͑29͒ differs from Fermi’s golden rule for small V, i.e.,
small coupling rate constants. However, for small coupling
rate constants, contributions from Ka values other than zero
are negligible. Thus, using Eq. ͑28͒ only results in an under-
estimation of the effect of the smallest coupling matrix ele-
ments. Results that use Eq. ͑29͒ are given by thin lines in
Figs. 9 and 10. Traces corresponding to the same values of
coupling matrix elements are almost identical.
FIG. 9. Comparison of simulated and experimental dissociation rate con-
stants. The simulations assume spin–rotation coupling only; see Eq. ͑19͒.
Circles and dashed lines are for E ϪD ϭ10 cmϪ1ϩE1ro0t1 ; squares and
͗
͘
0
solid lines are for E ϪD ϭ75 cmϪ1ϩE1ro0t1 . Open symbols indicate ex-
͗
͘
0
perimental data; error bars indicate standard deviations. Thick lines are cal-
culated using Eq. ͑28͒, i.e., Fermi’s golden rule; thin lines are calculated
using Eq. ͑29͒, i.e., the picket fence model of Bixon and Jortner. ͑a͒ V0sr
Figure 9 shows rate constants which have been calcu-
lated by assuming that spin–rotation interaction is the sole
coupling mechanism. If Vs0r is taken to be 0.7 cmϪ1 ͓Fig.
ϭ0.7 cmϪ1; ͑b͒ Vs0rϭ10 cmϪ1
.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9278
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Bezel, Ionov, and Wittig
These values exceed the average rovibronic matrix element
of 0.28 cmϪ1 which was obtained by analyzing absorption
spectra for the same upper states in the region ϳ1500 cmϪ1
below D0 .17 If VC0 or were ϳ0.3 cmϪ1 at energies below D0 ,
it should have been detected: ͑i͒ spectroscopically in the re-
gion 16 000–19 362 cmϪ1, where Delon et al. report matrix
elements of ϳ0.2 cmϪ1 for both ͉0,1͘ and ͉0,3͘ rotational
levels;17 and ͑ii͒ in magnetic anticrossing experiments car-
ried out near 17 700 cmϪ1 25
.
This can be a manifestation of enhanced Coriolis cou-
pling at energies near D0 , due to large amplitude vibrations.
Alternatively, it can reflect an incorrect choice of the model
parameters. The value of the Coriolis reduced matrix element
obtained from the fit is a function of the TS geometry. For
example, variation of the Jacobi angle of the TS from 120° to
140° results in a twofold increase of the A‡ rotational con-
stant and a concomitant change of the Coriolis coupling
strength required to rationalize the data. It would be impru-
dent to include more than one fitting parameter in the above
model. However, when the magnitudes of the average cou-
pling matrix elements are known, such a model can be used
to predict dissociation rate constants of rotating molecules as
well as k(T), since the computational effort is minimal.
The constancy of the dissociation rate constant with in-
creasing rotational excitation is due to a balance between ͑i͒
increasing the total energy available for reaction and ͑ii͒ in-
cluding zeroth-order high-Ka states which are slowly disso-
ciating and/or nondissociating in the photoexcited ensemble.
The double resonance results are not in conflict with the
observed rate constant increase for room temperature
samples.5,9 In the work reported here, Ka was always ini-
tially zero. However, if high-Ka levels were excited initially,
Ka mixing would then increase the amount of energy avail-
able for dissociation and the rate constant would increase.
FIG. 10. Comparison of simulated and experimental dissociation rate con-
stants. The simulations assume Coriolis coupling only ͓VC0 orϭ0.3 cmϪ1, see
Eq. ͑22͔͒. Notations as in Fig. 8.
However, in this case, the rate constants at intermediate N
values drop, as shown in Fig. 9͑b͒. A broad range of cou-
pling strengths has been examined, and the conclusion is that
it is not possible to reconcile the data by using
N-independent Ka mixing rate constants.
A significantly better fit to the data is obtained when it is
assumed that Coriolis coupling dominates, as shown in Fig.
10. Good agreement is achieved by assuming that the aver-
age coupling rate constants between Ka and KaϮ1 are given
by
N Nϩ1͒ϪK K Ϯ1͒ ϫ4ϫ1010 sϪ1
͓ ͑
͑30͒
which corresponds to VC0 orϭ0.3 cmϪ1. Recall that the rel-
͑
͔
a
a
evant basis of comparison is the N variation of the rates,
rather than absolute values.
Ј
V. CONCLUSION
The N2 scaling of the coupling rate constant for ⌬Ka
ϭϮ1 enables the calculated rates to fit the data. As men-
tioned above, the first-order term of the spin–rotation inter-
action expansion does not work, since there is almost no N
dependence. Moreover, higher-order spin–rotation terms are
believed to be significantly smaller.17 Thus, we conclude that
Coriolis coupling can reconcile the experimental results.
Our verdict about the significance of Coriolis coupling is
supported by trajectory calculations which focus on Coriolis
and inertial asymmetry couplings, ignoring electron spin.59
This study shows that coupling between adjacent Ka levels
increases approximately linearly with N, with almost one
third of the available Ka phase space populated prior to dis-
sociation, regardless of the speed of rotation. Though it is
hard to extract the value of the average coupling matrix ele-
ment from the trajectory study, further insight can be gained
from spectroscopic studies which have been carried out be-
low D0 .
The variation of the NO2 unimolecular decomposition
rate constant with parent rotational excitation has been mea-
sured for two progressions of the average excess energies,
E ϪD : ͑i͒ 10 cmϪ1ϩE1ro0t1 ; and ͑ii͒ 75 cmϪ1ϩEr1o0t1
,
͗ ͘
where E
0
rot
101
is the rotational energy ͑with corresponding
quantum numbers N and KaЈ) of the ͑1,0,1͒ intermediate
Ј
vibrational level. The double resonance technique, combined
with the picosecond-resolution pump–probe technique, has
enabled N to be examined in the range 1рN р15 for KЈ
Ј
Ј
a
ϭ0, i.e., 0рEr1o0t1р100 cmϪ1. The addition of up to 100
cmϪ1 of rotational energy does not have a noticeable effect
on the rate constant. This is interpreted as evidence for com-
plex rovibrational dynamics.
A simple model has been used to calculate the dissocia-
tion rate constants for rotating molecules. Subject to the in-
herent limitations of the model, this exercise suggests that
coupling between different rotational states increases with
rotational excitation. A good fit is achieved when Ka couples
to KaϮ1 with a rate constant that varies as ϳN2. This cou-
pling is attributed to perpendicular Coriolis interaction, with
Values for
͉ 1,0͉HCor͉1,1 ͉ and ͉ 3,0͉HCor͉3,1 ͉ can be
͗ ͘ ͗ ͘
obtained by using Eq. ͑22͒ with VC0 orϭ0.3 cmϪ1. This yields
values of approximately 0.4 and 1.0 cmϪ1, respectively.
a reduced matrix element of VC0 orр0.3 cmϪ1
.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27

9279
J. Chem. Phys., Vol. 111, No. 20, 22 November 1999
Decomposition of NO2
38 A. Mellinger, M. V. Ashikhmin, and C. B. Moore, J. Chem. Phys. 108,
8944 ͑1998͒.
ACKNOWLEDGMENTS
39 R. J. Barnes, G. Dutton, and A. Sinha, J. Phys. Chem. A 101, 8374 ͑1997͒.
40 R. J. Barnes and A. Sinha, J. Chem. Phys. 107, 3730 ͑1997͒.
41 S. I. Ionov, G. A. Brucker, C. Jaques, L. Valachovic, and C. Wittig, J.
Chem. Phys. 97, 9486 ͑1992͒.
This research was supported by a grant from the Na-
tional Science Foundation. The authors have benefited
greatly from discussions with S. Ionov, S. Grebenshchikov,
K. Liu, and H. Reisler.
42 S. I. Ionov, G. A. Brucker, C. Jaques, L. Valachovic, and C. Wittig, J.
Chem. Phys. 99, 6553 ͑1993͒.
1
˚
43 P. I. Ionov, I. Bezel, S. I. Ionov, and C. Wittig, Chem. Phys. Lett. 272, 257
͑1997͒.
R. Jost, J. Nygard, A. Pasinski, and A. Delon, J. Chem. Phys. 105, 1287
͑1996͒.
2 C. M. Roehl, J. J. Orlando, G. S. Tyndall, R. E. Shetter, G. J. Vazquez, C.
A. Cantrell, and J. G. Calvert, J. Phys. Chem. 98, 7837 ͑1994͒.
3 U. Robra, H. Zacharias, and K. H. Welge, Z. Phys. D 16, 175 ͑1990͒.
4 B. Abel, H. H. Hamann, and N. Lange, Faraday Discuss. 102, 147 ͑1995͒.
5 S. I. Ionov, G. A. Brucker, C. Jaques, Y. Chen, and C. Wittig, J. Chem.
Phys. 99, 3420 ͑1993͒.
44 The different excitation energy from the value reported in Ref. 43 is due to
better calibration of the monochromator used to measure the excitation
pulse spectrum. Previously, it was calibrated by using the nearest strong
line of a mercury lamp ͑435–833 nm͒ which is ϳ700 cmϪ1 to the blue of
the region of interest. In the present work, it is calibrated against a wave
meter ͑Burleigh WA-4500͒ using the tunable output of a DFDL laser set at
6 G. A. Brucker, S. I. Ionov, Y. Chen, and C. Wittig, Chem. Phys. Lett. 194,
301 ͑1992͒.
22 222.5 cmϪ1
.
7 J. Miyawaki, K. Yamanouchi, and S. Tsuchiya, J. Chem. Phys. 99, 254
͑1993͒.
45 W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Nu-
merical Recipes ͑Cambridge University Press, Cambridge, UK, 1992͒.
46 C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon In-
teractions ͑Wiley, New York, 1992͒.
8 I. Bezel, D. Stolyarov, and C. Wittig, J. Phys. Chem. in press, ͑1999͒.
9 H. Gaedtke, H. Hippler, and J. Troe, Chem. Phys. Lett. 16, 177 ͑1972͒.
10 B. Abel, paper presented at ACS Meeting, Anaheim CA, 1999.
11 H. Katagiri and S. Kato, J. Chem. Phys. 99, 8805 ͑1993͒.
12 S. J. Klippenstein and T. Radivoyevitch, J. Chem. Phys. 99, 3644 ͑1993͒.
13 D. M. Wardlaw and R. A. Marcus, J. Chem. Phys. 83, 3462 ͑1985͒.
14 J. Troe, Chem. Phys. 190, 381 ͑1995͒.
47 U. Peskin, H. Reisler, and W. H. Miller, J. Chem. Phys. 101, 9672 ͑1994͒.
48 U. Peskin and W. H. Miller, J. Chem. Phys. 102, 8874 ͑1995͒.
49 M. Desouter-Lecomte and F. Culot, J. Chem. Phys. 98, 7819 ͑1993͒.
50 T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M.
Wong, Rev. Mod. Phys. 53, 385 ͑1981͒.
15 L. B. Harding, H. Stark, J. Troe, and V. G. Ushakov, Phys. Chem. Chem.
Phys. 1, 63 ͑1999͒.
51 V. A. Mandelshtam, T. P. Grozdanov, and H. S. Taylor, J. Chem. Phys.
103, 10074 ͑1995͒.
16 R. Georges, A. Delon, and R. Jost, J. Chem. Phys. 103, 1732 ͑1995͒.
17 A. Delon, R. Georges, and R. Jost, J. Chem. Phys. 103, 7740 ͑1995͒.
18 C. F. Jackels and E. R. Davidson, J. Chem. Phys. 64, 2908 ͑1976͒.
19 C. P. Blahous III, B. F. Yates, Y. Xie, and H. F. Schaefer III, J. Chem.
Phys. 93, 8105 ͑1990͒.
52 M. L. Mehta, Random Matrices ͑Academic, San Diego, 1991͒.
53 C. Wittig and S. I. Ionov, J. Chem. Phys. 100, 4714 ͑1994͒.
54 J. Miyawaki, K. Yamanouchi, and S. Tsuchiya, J. Chem. Phys. 100, 4716
͑1994͒.
55 L. Zhu and W. L. Hase, Chem. Phys. Lett. 175, 117 ͑1990͒.
56 T. Baer and W. L. Hase, Unimolecular Reaction Dynamics: Theory and
Experiments ͑Oxford University Press, New York, 1996͒.
57 R. G. Gilbert and S. C. Smith, Theory of Unimolecular and Recombina-
tion Reactions ͑Blackwell Scientific Publications, Oxford, 1990͒.
20 G. D. Gillispie and A. U. Khan, J. Chem. Phys. 65, 1624 ͑1976͒.
21 C. F. Jackels and E. R. Davidson, J. Chem. Phys. 65, 2941 ͑1976͒.
22 B. M. Toselli and J. R. Barker, J. Phys. Chem. 93, 6578 ͑1989͒.
23
¨
E. Haller, H. Koppel, and L. S. Cederbaum, J. Mol. Spectrosc. 111, 377
͑1985͒.
24 V. Dana and J. P. Maillard, J. Mol. Spectrosc. 71, 1 ͑1978͒.
25 A. Delon, P. Dupre, and R. Jost, J. Chem. Phys. 99, 9482 ͑1993͒.
26 A. Delon, R. Jost, and M. Lombardi, J. Chem. Phys. 95, 5701 ͑1991͒.
27 A. Perrin, C. Camy-Peyret, and J.-M. Flaud, J. Mol. Spectrosc. 88, 237
58 A. Delon, S. Heilliette, and R. Jost, Chem. Phys. 238, 465 ͑1998͒.
59
¨
S. Y. Grebenshchikov, C. Beck, H. Flothmann, R. Schinke, and S. Kato, J.
Chem. Phys. 111, 619 ͑1999͒.
60 R. F. Salzgeber, V. A. Mandelshtam, C. Schlier, and H. S. Taylor, J.
Chem. Phys. 110, 3756 ͑1999͒.
͑1981͒.
28
¨
61 R. F. Salzgeber, V. A. Mandelshtam, C. Schlier, and H. S. Taylor, J.
Chem. Phys. 109, 937 ͑1999͒.
S. Y. Grebenshchikov, H. Flothmann, R. Schinke, I. Bezel, C. Wittig, and
S. Kato, Chem. Phys. Lett. 285, 410 ͑1998͒.
29 S. L. Coy, K. K. Lehmann, and F. C. DeLucia, J. Chem. Phys. 85, 4297
͑1988͒.
62 S. I. Ionov, H. F. Davis, K. Mikhaylichenko, L. Valachovic, R. A. Beau-
det, and C. Wittig, J. Chem. Phys. 101, 4809 ͑1994͒.
63 G. Herzberg, Molecular Spectra and Molecular Structure ͑Van Nostrand-
Reinhold, New York, 1966͒.
30 K.-E. J. Hallin and A. J. Merer, J. Mol. Spectrosc. 65, 163 ͑1977͒.
31 A. Delon and R. Jost, J. Chem. Phys. 95, 5686 ͑1991͒.
32 J. Miyawaki, K. Yamanouchi, and S. Tsuchiya, J. Chem. Phys. 101, 4505
͑1994͒.
64 H. W. Kroto, Molecular Rotation Spectra ͑Dover, Mineola, 1992͒.
65 E. B. Wilson and B. Howard, J. Chem. Phys. 4, 260 ͑1936͒.
66 E. B. Wilson, J. Chem. Phys. 4, 313 ͑1936͒.
33 T. J. Butenhoff and E. Rohlfing, J. Chem. Phys. 98, 5460 ͑1993͒.
34 T. J. Butenhoff and E. Rohlfing, J. Chem. Phys. 98, 5469 ͑1993͒.
35 Y. Choi and C. B. Moore, J. Chem. Phys. 97, 1010 ͑1992͒.
36 J. D. Tobiason, J. R. Dunlop, and E. A. Rohlfing, J. Chem. Phys. 103,
1448 ͑1995͒.
67 G. R. Bird, J. C. Baird, A. W. Jache, J. A. Hodgeson, J. R. F. Curl, A. C.
Kunkle, J. W. Bansford, J. Rastrup-Andersen, and J. Rosenthal, J. Chem.
Phys. 40, 3378 ͑1964͒.
68 S. Y. Grebenshchikov ͑private communication͒.
37 S. Dertinger, A. Geers, J. Kappert, J. Wiebrecht, and F. Temps, Faraday
Discuss. 102, 31 ͑1995͒.
69 M. Bixon and J. Jortner, J. Chem. Phys. 48, 715 ͑1968͒.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.59.222.12 On: Sat, 29 Nov 2014 02:19:27