Probing the nature of the K-rotor in unimolecular reactions: Scalar and vector correlations in the photodissociation of NCNO

Article abstract of DOI:10.1063/1.1462581

The photodissociation of NCNO at 520 and 532 nm was examined using transient frequency modulation Doppler spectroscopy in order to study vector and scalar correlations in the dissociation. A small correlation between v and J was observed corresponding to Β₀⁰(22) bipolar moments ranging from 0.00±0.02 to -0.03±0.02 at 532 nm and 0.00±0.02 to -0.01±0.02 at 520 nm. These were well described by a helicity unrestricted PST formulation at both wavelengths.

Full text of DOI:10.1063/1.1462581

Probing the nature of the K -rotor in unimolecular reactions: Scalar and vector
correlations in the photodissociation of NCNO
W. Sean McGivern and Simon W. North
Citation: The Journal of Chemical Physics 116, 7027 (2002); doi: 10.1063/1.1462581
View online: http://dx.doi.org/10.1063/1.1462581
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 116, NUMBER 16
22 APRIL 2002
Probing the nature of the K-rotor in unimolecular reactions: Scalar
and vector correlations in the photodissociation of NCNO
W. Sean McGivern and Simon W. Northa)
Department of Chemistry, Texas A&M University, 3255 TAMU, College Station, Texas 77842
͑Received 13 November 2001; accepted 29 January 2002͒
The photodissociation dynamics of thermal NCNO at 520 and 532 nm have been examined using
transient frequency modulation Doppler spectroscopy to measure state-selected CN scalar and
vector correlations. Previous work has suggested that the global vibrational and rotational
distributions may be described using separate statistical ensembles/phase space theory ͑SSE/PST͒.
We find that the correlated vibrational and rotational distributions are well described by SSE at 520
nm if the K-rotor is considered inactive. At both wavelengths studied, the correlation between the
velocity and the rotational angular momentum vector of the CN product is found to be described by
phase space theory with no restriction of the projection of the rotational angular momentum vectors
along the relative velocity axis. This is indicative of approximate K-scrambling at the transition
state, and a discussion of these results in light of the evolution of the K-quantum number is
provided. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1462581͔
I. INTRODUCTION
et al. using laser induced fluorescence ͑LIF͒ to probe the CN
fragments and were well-described by SSE.8 However, the
large linewidth of the LIF probe laser relative to the width of
the fragment Doppler profiles limited their overall resolution.
Recent studies have demonstrated that the correlation
between the fragment velocity vector, v, and the rotational
angular momentum vector, J, can provide insight into the
torques on the departing fragments during dissociation. Al-
though these were initially applied to the investigation of
direct dissociation processes, even in a purely statistical dis-
sociation that is well-described by PST, one should expect
v–J correlations as a consequence of energy and angular
momentum conservation. Deviation from this trivial case can
be used as a probe of dynamical constraints. Of particular
interest is whether the v–J correlation can provide insight
into the role of the K-rotor energy, a subject of considerable
importance in unimolecular reaction theory.15 It is currently
unclear whether the K-rotor should be treated as adiabatic,
requiring its angular momentum to be conserved throughout
the reaction, or whether it should be allowed to couple into
the reaction coordinate. In the case of a prolate symmetric
top, rotation about the symmetry axis, which can be related
to the relative velocity axis in the axial recoil limit, is small
even though the overall angular momentum is large.
K-conservation restricts the available phase space by remov-
ing states in which the fragments have a large projection of
the angular momentum on the relative velocity axis ͑helicop-
tering motion͒. Since these states contribute to a positive
v–J correlation, the overall correlation becomes more
strongly negative. In a K-scrambled system, one recovers the
less polarized PST v–J correlation. Though we consider the
correlation between v and N, the nuclear rotational angular
momentum vector, we neglect the difference between N and
the spin-coupled angular momentum J, and refer to the cor-
relation as a ‘‘v–J correlation’’ to retain common usage.
In recent years, there has been a steady exodus from the
NCNO has become a benchmark molecule for the study
of barrierless reactions, and the visible photodissociation of
this molecule has been the subject of numerous
experimental1–9 and theoretical10–12 studies. Absorption of
light with wavelengths shorter than 585.3 nm corresponds to
a transition from ground electronic state NCNO to the first
excited singlet state (S1). For photolysis wavelengths longer
than 450 nm,7 the system internally converts back to the S0
state, where dissociation occurs along a barrierless potential
to form products in their electronic ground states.3 Previous
experimental work has focused primarily on the measure-
ment of global vibrational and rotational distributions for
both the CN and NO fragments at wavelengths correspond-
ing to 0–5000 cmϪ1 above the dissociation threshold. The
global rotational distributions were found to be well-
described by phase space theory13 ͑PST͒, but the measured
vibrational branching ratios were hotter than predicted by
PST.
To explain these observations, Wittig and co-workers
have proposed a theory called separate statistical ensembles
͑SSE͒, in which vibrational excitation of the products was
derived strictly from the vibrational degrees of freedom of
the parent and was decoupled from the parent rotational de-
grees of freedom.14 Physically, such a system can be realized
if the conserved modes randomize energy only until the sys-
tem reaches the inner transition state and the transitional
modes exchange energy until the outer transition state is
reached. The SSE/PST predictions were found to reproduce
the global vibrational and rotational distributions of CN and
NO products over a range of wavelengths.5,7 Correlated
speed distributions, which provide a much more stringent
test of the vibrational partitioning, were measured by Qian
a͒
Electronic mail: swnorth@tamu.edu
0021-9606/2002/116(16)/7027/8/$19.00
7027
© 2002 American Institute of Physics
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7028
J. Chem. Phys., Vol. 116, No. 16, 22 April 2002
W. S. McGivern and S. W. North
highly averaged environment of the room temperature bulb
towards the rarefied environment of the molecular beam. The
molecular beam is ideally suited to photodissociation experi-
ments in many ways, significantly reducing thermal averag-
ing effects, and, in some instances permitting state-to-state
studies. However, the use of a jet-cooled ensemble charac-
terized by low rotational temperatures precludes the study of
certain facets of photodissociation processes. The study of
the nature of the K-rotor requires a broad population of the
parent rotational quantum states in order to sample a large
region of phase space effectively. In a thermal system, the
additional angular momentum restrictions imposed by con-
servation of the K-rotor energy severely limit the available
phase space. The differences in the observed dynamics cor-
responding to these two cases are marked, providing a robust
method for determining the extent of K-scrambling. In a jet-
cooled sample, the available phase space is significantly lim-
ited by the parent rotational distribution irrespective of the
nature of the K-rotor, and differences between the predicted
v–J correlations in the K-scrambled and K-unscrambled
cases are minimal. The parent rotational energy also plays a
role in testing the applicability of SSE to specific systems. If
the K-rotor is active the initially well-defined vibrational en-
ergy distribution will blur as energy is exchanged between
the rotational and vibrational reservoirs, resulting in a mark-
edly different product vibrational energy distribution. This
difference should be particularly significant in the correlated
vibrational branching ratios. The study of room temperature
samples also serves a more practical function. A correct de-
scription of energy partitioning to systems relevant in atmo-
spheric, interstellar, and combustion chemistry necessarily
requires the application of statistical theories to thermal sys-
tems. Detailed tests of these theories on thermal samples are
thus desirable to enable their widespread application.
In the present work, we have examined the 520 and 532
nm photodissociation of NCNO at 295 K using transient fre-
quency modulation ͑FM͒ Doppler spectroscopy. As shown in
previous studies, this technique is ideal for studying both
scalar and vector correlations in barrierless dissociation
processes.16–18 A single mode diode laser ͑Ͻ1 MHz͒ is used
to probe the Doppler line shapes, providing velocity resolu-
tion superior to pulsed ultraviolet techniques. The transient
FM technique allows a detailed study of correlated vibra-
tional branching ratios for different CN states, providing a
stringent test of the applicability of SSE. In addition, we
have measured v–J correlations at both 520 and 532 nm for
several CN rotational states. Both measurements provide a
means to probe the evolution of the K-rotor energy during
the dissociation.
FIG. 1. Fragment internal temperature as a function of delay time for the
532 nm photodissociation of NCNO at 900 mTorr. The circles represent the
temperatures of a translational Gaussian that best fit the data for a 150 ns
gate as a function of time. The solid line is an exponential fit and is prima-
rily intended to guide the eye. The system appears to come to room tem-
perature ͑295 K͒ within several microseconds following the photolysis
pulse.
820 nm wavelength range. The 520 and 532 nm photolysis
beams were the output of a Nd:YAG-pumped dye laser op-
erating with Coumarin 500 dye. Under the conditions of the
present experiment, the modulated near-infrared probe beam
may be approximately represented as a combination of three
distinct frequencies, separated by ␻m . Preferential absorp-
tion of one of these frequencies induces a beat signal on the
overall beam intensity, which is detected with a fast photo-
diode. The beat signal is then amplified and separated using
standard phase-sensitive RF electronics into in-phase (I) and
quadrature (Q) components with respect to the mixing oscil-
lator. The resulting time-dependent decays are stored as a
function of laser wavelength, providing a two-dimensional
data set in both time and frequency. In order to ensure that
only nascent fragments are contributing to the observed
spectra, the Doppler line shapes are generated using signal
from only the first 100 ns following photolysis.
In order to quantify the initial parent temperature, time-
dependent Doppler spectra were measured for NCNO buff-
ered to ϳ900 mTorr with argon and fit to thermal Gaussian
distributions as a function of time. Shown in Fig. 1 are the
temperatures of the best-fit Gaussian distributions for Dop-
pler profiles measured at each time using a gate width of 150
ns. The solid line is a best-fit exponential function that is
intended to guide the eye. The system returns to room tem-
perature within several microseconds, ensuring that the par-
ent molecules are completely thermalized between laser
shots at 10 Hz even at the low sample pressures of the
present experiment.
II. EXPERIMENT
The transient frequency modulation absorption spec-
trometer has been described in detail previously,18,19 and we
present only the salient features of the experiment. CN radi-
cals arising from the photodissociation of NCNO were
probed using the output of a tunable diode laser that was
frequency modulated at ␻mϭ192 MHz by a resonant phase
modulator. The CN radicals were detected on Q1- and
R1-branch lines of the A–X electronic transition in the 780–
The NCNO sample was synthesized by condensing
NOCl on AgCN and collecting the gas above the mixture
after approximately 10 min.20 The NOCl reagent was pre-
pared by dripping a concentrated solution of NaNO2 into
concentrated HCl and collecting the evolved gas.21 NOCl
and NO2 impurities were removed from the NCNO sample
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J. Chem. Phys., Vol. 116, No. 16, 22 April 2002
The K-rotor in unimolecular reactions
7029
by trap-to-trap distillation from a dry ice/acetone bath.
Samples were prepared at the beginning of each day and
were held at Ϫ78 °C during the experiment. NCNO was in-
troduced neat into the reaction bulb through a needle valve
adjusted to provide total pressures of 20–30 mTorr.
III. RESULTS AND ANALYSIS
The absolute phase of the beat signal with respect to the
local oscillator used for mixing is unknown, and the resulting
I and Q components are linear combinations of the FM ab-
sorption and dispersion signals. These I and Q signals were
corrected using a recursive method described previously22 to
provide pure absorption and dispersion line shapes. FM ab-
sorption line shapes were used for the analysis for all rota-
tional states studied.
A. Vector correlations
The determination of v–J correlations from the differ-
ences between Q- and R-branch absorption lines has been
discussed previously16 and only a brief overview will be
given here. Using the bipolar moment theory of Dixon,23 the
Doppler line shape in the absence of any laboratory align-
ment is given by16
ϱ
1
0
0
͑2͒
D w͒ϭ
1Ϫh
␤
22͒ P w/ ͒ 2 f ͒d ,
v ͔v ͑v
͑
͓
͑
͑
2
v
͵
v
2
v
͉w͉
FIG. 2. FM Doppler absorption profiles for Nϭ11 and Nϭ24 at 532 nm.
The circles represent the Q-branch absorption data, and the lines represent
R-branch absorption data.
where P2(x) is the second Legendre polynomial, ␤00(22)v is
the velocity dependent v–J correlation given by P (v"J) ,
͗
͘
2
f( ) is the laboratory speed distribution, h(2) is ϩ1 for a
v2
v
Q-branch transition, and ϪJ/(2Jϩ3) for a R-branch line,
and w is the projection of the laboratory velocity, , on the
v
probe laser axis. For v perpendicular to J, ␤00(22)vϭϪ0.5,
and ␤00(22)vϭ1.0 for v parallel to J. Previous studies on
similar systems using the transient FM technique typically
used data that had been inverted to pure absorption space for
the determination of the v–J correlation.16,18,24 Shown in
Fig. 2 are overlapped Q- and R-branch FM absorption spec-
tra for Nϭ11 and 24. The differences between the Q- and
R-branches are extremely small, and errors in the inversion
due to slight noise in the FM data lead to errors in the mea-
sured v–J correlation. We have therefore chosen to fit the Q-
and R-branches in FM space. The R- and Q-branch lines
were simultaneously fit using a simplex minimization of the
the present work. Similar treatments have been utilized in
previous studies and have provided good fits to observed
spectra,16–18,26 though the neutral time-of-flight apparatus
utilized by Wodtke and co-workers had sufficient resolution
to differentiate a velocity dependence to the v–J parameter
of CO from the 308 nm photolysis of ketene.27 To account
for slight variations in step size of the laser wavelength, we
have examined the difference between the R- and Q-branch
lines at late times and adjusted the widths to account for
minor differences.28 The solid and dashed lines in Fig. 2
show absorption data for R- and Q-branches, respectively,
for Nϭ11 and 24. The R- and Q-branch spectra are very
similar, especially for Nϭ11, which is indicative of a v–J
correlation near zero. The experimentally determined v–J
correlations for the measured lines are shown in Fig. 3 as
open circles and are found to be small and to vary little with
N.
overall ␹ with three parameters,25 two of which are used for
2
a translational energy distribution of the form,
a
b
v
v
1Ϫ
,
ͩ ͪ ͩ
ͪ
vmax
vmax
with a third parameter representing the v–J correlation.
There is no a priori reason to expect that a single value
of ␤00(22) can describe the angular momentum polarization
effectively. We have examined the effects of the velocity
dependence on the overall fits by using a linear model for
␤00(22)v . Though this provided an improvement by approxi-
B. Scalar correlations
Previous studies by Pfab et al.1 and Wittig and
co-workers2,3 have determined global vibrational, rotational,
and electronic energy distributions for both CN and NO for
room temperature NCNO at several wavelengths throughout
the visible absorption region. The distributions were found to
be in good agreement with PST results for low values of N at
532 nm and 549 nm. PST was found to underestimate the
2
mately a factor of 2 in the total ␹ values, the average values
of ␤00(22) from the linear model were within Ϯ0.005 to the
best-fit constant values for all lines studied. Therefore, we
have chosen to report only the velocity independent values in
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7030
J. Chem. Phys., Vol. 116, No. 16, 22 April 2002
W. S. McGivern and S. W. North
FIG. 3. Measured and calculated values for ␤00(22) for several detected CN
rotational states derived from NCNO photodissociation at 532 nm ͑upper
panel͒ and 520 nm ͑lower panel͒. The open circles represent the experimen-
tally determined v–J correlations. The solid and dashed lines are ␤00(22)
values calculated using quantum PST for unrestricted ⌳ and ͉⌳͉ restricted to
values less than or equal to 5, respectively.
FIG. 4. R-branch Doppler line shapes for Nϭ1, 4, and 11 from the photo-
dissociation of NCNO at 520 nm. The circles represent the experimental
data. The solid line is the total fit, and the dashed lines represent contribu-
population of the higher N states at both wavelengths, likely
due to dissociation from vibrationally excited parent mol-
ecules.
tions from coincident fragments in ϭ0 and ϭ1 vibrational states.
v
v
For each of the CN rotational lines measured in the
present work, PST was used to generate the coincident NO
fragment rotational distributions. Vibrational averaging was
accomplished using a Beyer–Swinehart direct state count al-
gorithm to determine the vibrational energy distribution of
the parent molecule at 295 K.29 A Boltzmann average over
the parent thermal rotational energy was explicitly included,
fragments.31 From each calculated PST rotational distribu-
tion, a velocity distribution was calculated and used to gen-
erate the correct Doppler absorption line shape, which was
subsequently converted to FM space for comparison with the
data.22
and the average energy of
͉
K
͉
was included in the available
The relative weightings of the Doppler line shapes cor-
responding to the coincident vibrational components were
adjusted to provide the best fits to the measured spectra.
Figure 4 shows typical R-branch line shapes for three differ-
ent CN rotational states derived from NCNO at 520 nm. The
solid line is the overall fit to the data that minimizes the ␹2,
using the experimentally determined v–J correlation and the
dashed lines are the individual Doppler line shapes that rep-
energy. A full Boltzmann average for the K-rotor energy was
tested, but no significant deviation was observed from the
line shapes simulated by changing only the available energy.
In order to simplify the analysis, NCNO was considered to
be a prolate symmetric top with rotational constants of 0.175
cmϪ1 and 2.71 cmϪ1 as noted below. For all of the simula-
tions, the population of the NO(2 1/2) and NO(2
͟ ͟3/2) spin–
orbit states were assumed to be equal. Qian et al. have ob-
served systematic deviations from this limit for high N states
at approximately 515 nm,8 but these deviations were minor
for NϽ30.5. A centrifugal barrier was included in the PST
calculations in which the maximum value of the orbital an-
gular momentum, lmax , was given by
resent the ϭ0 and ϭ1 components. The shapes of the
v v
individual lineshapes are significantly different for different
values of N, indicative of a strongly changing vibrational
branching ratio. As is evident from the figure, the differences
between the simulated line shapes for the individual vibra-
tional components are large, providing a robust method for
determination of the coincident vibrational branching frac-
tions for each CN rotational state.
1/3
2/3
lmax l ,
maxϩ1͒ប2ϭ6␮C E /2͒
͑ ͑
T
6
where C6 is the attractive potential term from a Lennard-
Jones potential,30 ␮ is the reduced mass of NCNO, and ET is
the relative translational energy of the CN and NO
Figure 5 shows the vibrational branching fractions deter-
mined from the CN lines measured in this study at 520 nm.
For the purposes of comparison, correlated vibrational
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J. Chem. Phys., Vol. 116, No. 16, 22 April 2002
The K-rotor in unimolecular reactions
7031
parent molecule and transition state. However, Hase has
noted recently that the nature of the K-rotor remains a par-
ticularly vexing problem: Can K be considered a good quan-
tum number throughout the dissociation, or does Coriolis
coupling result in partial or complete mixing?15 When K re-
mains a good quantum number, the angular momentum in
the K-rotor must be conserved. The additional conserved
quantity must therefore be incorporated into statistical theo-
ries to correctly describe the reaction rates. A further com-
plication arises from the potential disparate natures of the
K-rotors in the reactant and transition state, which may be
individually K-conserved or K-scrambled.29
Statistical v–J correlations may be predicted effectively
using PST in which the angular momentum states are ex-
panded into a helicity basis.33 In this basis, the quantum
states are indexed by helicity quantum numbers representing
the projection of the fragment angular momenta on the rela-
tive velocity axis. If the molecule dissociates in the axial
recoil limit and the dissociation axis coincides with the long
symmetric top axis, the total helicity, ⌳, may be associated
with the K quantum number.34 Since the individual helicity
quantum numbers are bound by the fragment rotational
FIG. 5. Correlated vibrational fractions of coincident NO fragments in vϭ1
for the 520 nm photodissociation of NCNO. Circles represent the best-fit
values to the experimental data. The solid line represents the PST prediction,
the dashed line is the K-conserved SSE prediction, and the dotted–dashed
line is the K-scrambled SSE prediction.
quantum numbers,
tional quantum number, J0 . Although
͉
⌳
͉
is strictly bound by the parent rota-
is also bound by
branching fractions were calculated using PST as a function
of the detected CN rotational state. The ‘‘SSE, K-conserved’’
predictions are the PST predictions scaled by the ratio of the
global SSE vibrational branching fraction to the PST global
fraction when K is conserved at the inner transition state. The
details of the K-rotor treatment within the SSE model are
given below. The SSE predictions are in very good agree-
ment with the experimental vibrational branching fractions.
For most of the lines examined, both Q- and R-branch line
shapes were fit independently, and the branching fractions
typically differed by Ͻ0.02 between the two lines. The
Q-branch lines for Nϭ4 and Nϭ1 are strongly overlapped
with one another and their corresponding satellite
transitions,32 and no attempt was made to fit these branches.
The Nϭ30 line is near the threshold for formation of prod-
ucts from cold NCNO molecules at 520 nm and has contri-
butions from dissociation of highly vibrationally and rota-
tionally excited parent molecules. Although no evidence of a
͉K͉
J0 , the nature of the K-distribution imposes an additional
constraint. For a prolate symmetric top, the distribution of K
quantum numbers is Gaussian and peaked around Kϭ0. As
such, for moderately large values of J0 , the total helicity will
be constrained by K to values well below J0 for situations in
which K is a good quantum number. We have chosen to treat
NCNO as a prolate symmetric top with Aϭ2.71 cmϪ1 and
Bϭ(BϩC)/2ϭ0.175 cmϪ1, providing an average value for
˜
͉K͉
of 5 at 295 K.35 The calculation of the v–J correlations
when ͉⌳͉ is both unrestricted and when it is limited to a small
value by the parent K distribution provides two distinct simu-
lations that, when compared to measured v–J correlations,
can differentiate between K-conserved and K-scrambled
models.
We have performed phase space theory calculations in
the helicity basis ͑quantum PST͒ to determine the values of
␤00(22) for the range of CN rotational states of interest using
both ⌳-unrestricted and ⌳-restricted ͉͑⌳͉р5͒ models.33 The
thermal averaging used was similar to that used to determine
the vibrational branching ratios. Application of quantum PST
provides center-of-mass ͑COM͒ v–J correlations, which are
related directly to the K-distribution at the transition state in
the axial recoil limit. However, the thermal translational and
rotational motion of the parent molecules degrades these val-
ues beyond the transition state, and this degradation must be
incorporated into the simulations for comparison to the ex-
perimental values. Thermal translational and rotational mo-
tion reorients the center-of-mass velocity vector, blurring the
center-of-mass v–J correlations. The details of the thermal
translational depolarization calculations may be found in
Ref. 33. The rotational depolarization mechanism has not
been treated previously, and we have developed a simple
model to describe its effects, which is shown in the Appen-
dix. The results of the fully degraded laboratory-frame quan-
tum PST calculations are shown as the solid ͉͑⌳͉-
coincident ϭ1 contribution was observed, the high N line
v
shapes were slightly wider than the simulated spectra. These
states are extremely sensitive to subtle deviations of the par-
ent vibrational distribution and may contain a larger contri-
bution from vibrationally excited parent molecules than ex-
pected from a pure Boltzmann distribution.
IV. DISCUSSION
A. The nature of the K-rotor at the outer transition
state
A fundamental tenet of statistical theories of unimolecu-
lar dissociation is the ergodic hypothesis, where only a hand-
ful of constants of motion are required to fully describe
the dissociation rates. Rice–Ramsperger–Kassel–Marcus
͑RRKM͒ theory, which provides a powerful means to calcu-
late microcanonical rate constants, generally specifies only
the excess energy and the total angular momentum of the
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7032
J. Chem. Phys., Vol. 116, No. 16, 22 April 2002
W. S. McGivern and S. W. North
unrestricted͒ and dashed ͉͑⌳͉р5͒ lines in Fig. 3. The
thermally degraded v–J correlations initially become stron-
ger with increasing values of N, consistent with the behavior
of the COM v–J correlations. However, the effect of the
thermal degrading becomes more significant at lower speeds,
causing the values of ␤00(22) to approach zero for the highest
detected rotational states. As shown in Fig. 3, the
⌳-restricted values of ␤00(22) provide a poor fit to the ob-
served v–J correlations at both 520 and 532 nm. However,
the fully averaged PST values of ␤00(22) reproduce the ob-
served values well at both wavelengths. The ability of the
⌳-unrestricted simulations to reproduce the experimental
v–J correlations is strongly indicative of approximate
K-scrambling at the transition state.
Several factors may contribute to the slight deviations of
the observed ␤00(22) from the predicted ⌳-unconstrained
values. In NCNO, the dissociation axis does not coincide
with the symmetric top axis used to calculate the rotational
constants, which lies along the NC and NO centers-of-mass.
As a result, the correspondence between the total helicity and
the K quantum number is not necessarily strict. In addition,
the approximate semiclassical treatment of the rotational de-
polarization factor likely fails to capture some of the more
subtle depolarization effects. If the fragment rotational states
are ‘‘frozen-out’’ earlier along the reaction coordinate than
considered in our calculation, the rotational depolarization
would be more severe. The predicted correlations therefore
may be expected to slightly overestimate the strength of the
measured v–J correlations. Additional experiments, already
underway, using a pulsed slit-jet should reduce the uncer-
tainty in the measured v–J correlations.
nonrandom.16,37 Correlated studies provide highly stringent
tests of such proposed dissociation models. In the present
work, we have examined the partitioning of the vibrational
energy at the correlated level in a room temperature sample.
When compared to different statistical predictions, these dis-
tributions provide insight into not only the evolution of the
parent internal energy but also the physical processes that
dictate the final product state distributions.
Of particular interest with respect to NCNO is the valid-
ity of SSE theory, originally proposed by Wittig and co-
workers, to describe the correlated state distributions.14 In
SSE theory, the vibrational product distribution is assumed to
be determined early in the reaction, well before the ‘‘loose’’
outer transition state. The rotational energy is then allowed to
couple at longer range, resulting in PST rotational distribu-
tions in each vibrational coincident channel. SSE was found
to effectively reproduce measured global CN and NO rota-
tional distributions derived from the photodissociation of jet-
cooled NCNO at several wavelengths. A later study utilized
laser induced fluorescence ͑LIF͒ to measure nascent Doppler
profiles of CN radicals from NCNO in order to test SSE at a
correlated level.8 SSE was found to fit the resulting Doppler
profiles well for several rotational states at dissociation en-
ergies ranging from 411 to 2348 cmϪ1. However, the line-
width of the probe laser used in these experiments was esti-
mated to be ϳ0.07 cmϪ1, and the typical CN linewidth was
ϳ0.1 cmϪ1, effectively limiting the overall velocity resolu-
tion of the LIF experiments. Although the authors used the
trough between the F1 and F2 spin components to increase
their resolution, subtle differences between the shapes of the
partial contributions corresponding to the distinct vibrational
channels would have been difficult to discern from changes
in the correlated vibrational branching ratios. In addition, the
lines were measured in the B–X electronic transition, for
which no Q-branch is present. This precluded measurement
of the v–J correlations, which are expected to be large in a
jet-cooled sample and would tend to bias the R-branch fits
B. The nature of the K-rotor at the inner transition
state
Both photoexcitation and chemical activation processes
can produce rotationally and vibrationally excited product
molecules. The partitioning of this excess energy can have a
significant effect on the subsequent reactivities of the product
molecules, and an understanding of the relevant energy par-
titioning is therefore necessary for modeling the complex
reaction pathways common in atmospheric, combustion, and
interstellar chemistry. In a purely statistical dissociation the
product state distributions are determined by the degenera-
cies of the product molecules and the available energy. This
behavior is precisely that described by PST, and is usually
realized for energies very close to the dissociation threshold,
where the number of available states is small. Deviations
from this behavior are indicative of dynamical constraints on
the dissociating system. The determination of these dynami-
cal constraints is important in determining the overall energy
partitioning, which, in turn, leads to a fuller understanding of
processes that produce excited radicals.
toward apparently larger ϭ1 vibrational branching ratios.33
v
The use of a thermal sample provides an additional im-
portant test of the treatment of the K-rotor within the SSE
model. Optical excitation of a thermal sample prepares mol-
ecules with a fixed vibrational energy and a thermal distribu-
tion of J and K. If the K-rotor is active in the thermal
samples, the NCNO vibrational reservoir is allowed to ex-
change energy with the K-rotor until the K distribution is
uniform. In the case of a prolate symmetric top, this results
in a net decrease in the total vibrational energy and thus a
lower ϭ1 probability. However, an inactive K-rotor does
v
not couple with the vibrational reservoir, and the vibrational
energy remains constant throughout the reaction, irrespective
of the initial parent rotational motion. Therefore, the corre-
lated measurements may be able to distinguish between the
predictions of a K-conserved and K-scrambled model in a
thermal sample. Previous studies which have applied the
SSE model to NCNO photodissociation have focused on jet-
cooled samples, where the parent rotational energy is small
and the difference between the K-conserved and
K-scrambled models is minor. We have explicitly incorpo-
rated the rotational energy into the phase space calculation of
State-specific photodissociation studies, typically prob-
ing global vibrational and rotational distributions of one or
more photofragments, have provided significant insight into
the relevant constraints.36 However, systems that show be-
havior purely consistent with a statistical theory at a global
level may, upon more detailed examination at the correlated
level, be governed by physical processes that are distinctly
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J. Chem. Phys., Vol. 116, No. 16, 22 April 2002
The K-rotor in unimolecular reactions
7033
ACKNOWLEDGMENTS
the basis functions as noted above. Figure 5 shows two SSE
curves: one where the K-rotor is considered conserved, re-
sulting in no change in the vibrational energy reservoir, and
one allowing K-scrambling, which results in changes in the
vibrational reservoir. The latter model involves a weighted
average of the J, K states, and both models include contri-
butions from the parent vibrational energy, which was
weighted using a thermally averaged Beyer–Swinehart algo-
rithm.
The authors would like to thank Dr. H. Reisler and Dr.
C. X. W. Qian for information regarding the NCNO synthe-
sis. The authors also acknowledge Dr. G. E. Hall, Dr. S. J.
Klippenstein, and Dr. R. R. Lucchese for helpful discussions.
This research was supported by a grant from the Robert A.
Welch Foundation ͑Grant No. A-1386͒.
APPENDIX: ROTATIONAL DEPOLARIZATION
The agreement between the measured vibrational
branching fractions and the predicted K-conserved SSE val-
ues is very good but is poor for both the PST and
K-scrambled SSE predictions. Physically the SSE model de-
scribes the exchange of energy between conserved modes
within a vibrational reservoir up to the inner transition state.
The observation that a K-conserved model fits our measure-
ments suggests that K is a good quantum number until the
conserved modes cease to exchange energy. This suggests
that implementation of RRKM theory to calculate dissocia-
tion rates at these energies should treat the K-rotor as inac-
tive in the reactant. The high resolution of the Doppler spec-
tra and the ability to accurately fit a thermal sample weighted
by a simple distribution derived from a Beyer–Swinehart
algorithm gives us confidence that SSE theory describes the
dynamics at 520 nm well. We have avoided a detailed corre-
lated vibrational analysis on the 532 nm Doppler spectra,
since a 532 nm photon is below the threshold for formation
The rotational depolarization correction to the center-of-
mass ͑COM͒ correlations (u"J) may be estimated by apply-
ing the azimuthally averaged addition ͑AAA͒ theorem38 to
the COM correlation,
P
v"J͒ ϭ P u"v͒ P u"J͒ ,
͑ ͑
͘ ͗ ͗͘ ͘
2 2
͑
͗
2
where u and v are the relative velocity axes before and after
rotation, respectively, J is the fragment rotational angular
momentum vector, and P2(x) is the second Legendre poly-
nomial. The depolarization factor, P (u"V) , is calculated
͗
͘
2
semiclassically from the initial relative velocity and the tan-
gential velocity of the center-of-mass of the detected frag-
ment. The tangential velocity is calculated from the total
parent rotational quantum number, J0 , taking into account
rotation about the K-axis, which is given by the total frag-
ment helicity, ⌳. The depolarization factor is calculated for
each possible set of parent vibrational and rotational quan-
tum numbers and weighted accordingly using a Boltzmann
distribution of rotations and a Boltzmann-weighted vibra-
tional distribution calculated using a Beyer–Swinehart algo-
rithm.
of ϭ1 NO from cold parent molecules. Any ϭ1 NO mol-
v
v
ecules are exclusively derived from only hot bands, and the
resulting Doppler lineshapes are therefore exquisitely sensi-
tive to the initial parent distribution of energies.
Shown in Fig. 6 are calculated depolarization factors for
the dissociation of room-temperature NCNO at 532 nm when
probing Nϭ17 CN radicals. The rotational temperature used
to calculate the Boltzmann distributions was 295 K, and
these factors were calculated using the ground vibrational
state for the parent molecule. For low coincident NO rota-
V. CONCLUSION
The photodissociation of NCNO at 520 and 532 nm has
been examined using transient frequency modulation Dop-
pler spectroscopy in order to study vector and scalar corre-
lations in the dissociation. A small correlation between v and
J was observed corresponding to ␤00(22) bipolar moments
ranging from 0.00Ϯ0.02 to Ϫ0.03Ϯ0.02 at 532 nm and 0.00
Ϯ0.02 to Ϫ0.01Ϯ0.02 at 520 nm. These were well described
by a helicity unrestricted PST formulation at both wave-
lengths. Restrictions on the total helicity produced signifi-
cantly more negative v–J correlations, providing evidence
for approximate K-scrambling beyond the outer transition
state. The correlated vibrational branching ratios were also
measured using PST rotational distributions to generate the
correlated basis sets. The measured branching ratios are well
described by a K-conserved SSE model at 520 nm, implying
efficient K-mixing between the inner and outer transition
states. Future experiments, currently underway, will study
these various facets of NCNO photodissociation and will in-
clude molecular beam experiments to reduce the blurring
arising from the initial parent motion. These experiments
will further probe the validity of PST for the correlated ro-
tational distributions and SSE for the correlated vibrational
distributions in a low-temperature sample.
FIG. 6. Rotational depolarization factors as a function of rotational angular
momentum about the short symmetry top axis, J ͑units of ប͒. These depo-
Ј
larization factors describe the dissociation of NCNO at 532 nm by probing
the CN radicals with Nϭ17 at 295 K.
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7034
J. Chem. Phys., Vol. 116, No. 16, 22 April 2002
W. S. McGivern and S. W. North
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FIG. 7. Calculated values of the translationally and rotationally depolarized
v–J correlation for the 355 nm photodissociation of NO2 at 295 K. The
solid line represents the v–J correlations with both the translational and
rotational corrections, and the dashed line represents the v–J correlations
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increasing values of J0 . However, as the coincident frag-
ments become higher in energy, less translational energy is
available, and the rotational depolarization becomes more
important. When the coincident rotational quantum number,
NNO , exceeds 30, products may only be observed from ro-
tationally excited parent molecules. For these coincident
states, the depolarization factor begins at Ϫ0.50 at threshold
and increases smoothly for each coincident NO state. In
practice, this correction is applied to the calculated PST v–J
correlations, and the translational depolarization correction is
subsequently applied to these values.
As a test of the applicability of this depolarization
method, we have calculated rotational depolarization factors
for the photodissociation of NO2 at 355 nm in a room tem-
perature bulb. Since NO2 is triatomic, a limiting v–J corre-
lation of Ϫ0.5 is expected for all J states in the absence of
parent translational and rotational motion. This provides an
ideal method for directly measuring the depolarization fac-
tors. Baker et al. have measured v–J correlations for NO2
photodissociation under these conditions using laser induced
fluorescence ͑LIF͒ detection of the NO products.39 For jNO
ϭ14.5, the authors measured values for ␤00(22) of Ϫ0.31
Ϯ0.12 and Ϫ0.25Ϯ0.12 for the F1 and F2 spin states, re-
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tors as a function of the detected NO rotational state. The
solid line includes both translational and rotational contribu-
tions, while the dashed line shows the effect of the transla-
tional depolarization alone. The depolarized value of the v–J
correlation at jNOϭ14.5 was calculated to be Ϫ0.27, in good
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