Treatment of the K-quantum number in unimolecular reaction theory: Insights from product correlations

Article abstract of DOI:10.1021/ja0273602

The connection between the K-quantum number and product correlations in the barrierless unimolecular dissociation of symmetric-top molecules is explored to establish a qualitative diagnostic for the treatment of the K-rotor dynamics in unimolecular reaction theory. We find that fragment scalar and vector correlations can provide guidance in this matter, and the photodissociation dynamics of thermal NCNO to form CN and NO at several dissociation wavelengths are presented to demonstrate the utility of this approach. The goodness of the K-quantum number can be related to the amount of energy in the conserved vibrational modes at the inner transition state. On the basis of measured correlated vibrational distributions, the K-quantum number is found to be approximately conserved at the inner transition state for the photodissociation of NCNO at 514, 520, and 526 nm. The methodology, involving a comparison of product distributions from the photodissociation of jet and thermal ensembles at identical wavelengths, is general and may be applied to previously studied systems that dissociate along barrierless potential energy surfaces, CF₃NO and CH₂CO. In addition, vector correlations serve as a means to probe the K-mixing at the outer transition state, and measured v-j correlations in the photodissociation of thermal NCNO are presented.

Full text of DOI:10.1021/ja0273602

Published on Web 11/09/2002
Treatment of the K-Quantum Number in Unimolecular
Reaction Theory: Insights from Product Correlations
W. Sean McGivern and Simon W. North*
Contribution from the Department of Chemistry, Texas A&M UniVersity,
3255 TAMU, College Station, Texas 77842
Received June 18, 2002
Abstract: The connection between the K-quantum number and product correlations in the barrierless
unimolecular dissociation of symmetric-top molecules is explored to establish a qualitative diagnostic for
the treatment of the K-rotor dynamics in unimolecular reaction theory. We find that fragment scalar and
vector correlations can provide guidance in this matter, and the photodissociation dynamics of thermal
NCNO to form CN and NO at several dissociation wavelengths are presented to demonstrate the utility of
this approach. The “goodness” of the K-quantum number can be related to the amount of energy in the
conserved vibrational modes at the inner transition state. On the basis of measured correlated vibrational
distributions, the K-quantum number is found to be approximately conserved at the inner transition state
for the photodissociation of NCNO at 514, 520, and 526 nm. The methodology, involving a comparison of
product distributions from the photodissociation of jet and thermal ensembles at identical wavelengths, is
general and may be applied to previously studied systems that dissociate along barrierless potential energy
surfaces, CF3NO and CH2CO. In addition, vector correlations serve as a means to probe the K-mixing at
the outer transition state, and measured v-j correlations in the photodissociation of thermal NCNO are
presented.
Introduction
niques have become more powerful, greater detail of the reactant
and transition-state structures and energetics required for eq 1
Unimolecular reactions occurring along barrierless potential
energy surfaces represent an important class of reactions that
are of fundamental importance in atmospheric, interstellar, and
combustion chemistry. The ability to accurately model these
reactions is necessary for studying systems that are intractable
to examine experimentally because of extreme conditions or
complexity. A number of statistical theories have been developed
to provide both rate constants and product distributions for this
type of reaction.1-3 These theories are derived from the tenet
that only information regarding the reactant and the transition
state is required to calculate rate constants and product quantum
state distributions. Although the derivation of these theories is
based on dynamical considerations, their implementation only
requires the specification of a small number of constants of the
motion.
has become available, allowing accurate rate constants and
product state distributions to be calculated. However, the
accuracy of the calculated molecular properties has increased
the need to correctly incorporate other quantities to refine the
calculated rate constants further. The angular momentum of the
K-rotor, described by the projection of the total angular
momentum vector, J, on the symmetry axis of symmetric top,
is such a quantity. Hase has recently noted that the proper
treatment of the K-rotor in both the reactant and the transition
state remains an important unsolved issue in unimolecular rate
theory.1,2,4-6
The treatment of the K-quantum number is particularly
important in the study of thermal association rate constants
where the reaction rates are a strong function of the pressure.5,6
In the high-pressure limit, the expression for the thermal rate
constant is independent of the nature of the K-rotor if the
reaction occurs over a barrier. However, on a barrierless
potential, where the location of the transition state in an RRKM
calculation is given by the point at which the sum of states of
the transition state is a minimum, the high-pressure limiting rate
The workhorse for predicting microcanonical rate constants,
Rice-Ramsperger-Kassel-Marcus (RRKM) theory, typically
specifies only the reaction energy, E, and the reactant rotational
angular momentum quantum number, J:
Nq(E-E0,J)
k(E,J) )
(1)
hF(E,J)
(1) Holbrook, K. A.; Pilling, M. A.; Robertson, S. H. Unimolecular Reactions,
2nd ed.; Wiley-Interscience, New York, 1996.
(2) Baer, T.; Hase, W. L. Unimolecular Reaction Dynamics; Oxford University
Press: New York, 1996.
(3) Truhlar, D. G.; Garrett, B. C.; Klippenstein, S. J. J. Phys. Chem. 1996,
100, 12771.
where k(E,J) is the microcanonical rate constant, E0 represents
the threshold energy, Nq(E-E0,J) is the sum of states of the
transition state, F(E,J) is the density of states of the reactant
molecule, and h is Planck’s constant. As computational tech-
(4) Hase, W. L. Acc. Chem. Res. 1998, 31, 659.
(5) Zhu, L.; Chen, W.; Hase, W. L.; Kaiser, E. W. J. Phys. Chem. 1993, 97,
311.
* Address correspondence to this author. E-mail: swnorth@tamu.edu.
(6) Zhu, L.; Hase. W. L. Chem. Phys. Lett. 1990, 175, 117.
9
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K-Quantum Number in Unimolecular Reaction Theory
A R T I C L E S
constant is dependent on whether the K-rotor is active or
adiabatic.6 In the low-pressure limit, the rate is found to be
strongly dependent on the treatment of the K-quantum number
irrespective of the type of potential energy surface. In general,
rate constants will be overestimated if a K-adiabatic system is
treated with an active K-rotor. Additionally, the physical
interpretation of the collisional efficiency derived from fitting
low-pressure falloff data is strongly dependent on the treatment
of the K-quantum number.4 The study of K-mixing is therefore
critical both to allow accurate modeling of unknown reaction
rates and to understand the physics of thermal association
reactions.
The “goodness” of the K-quantum number has been explored
in numerous studies on the spectroscopy of highly excited
molecules.7 One finds that the mixing of the K-levels is
substantial for excitation to states near the dissociation limits
for a number of small molecules.8,9 Above the threshold there
is a competition between the time scales of K-mixing and
unimolecular reaction. Few studies, however, have been able
to address the nature of the K-rotor above dissociation
thresholds.10-13 Several of these have involved optical-optical
double-resonance experiments in molecular beams which are
limited to low J and K and are fundamentally different from
thermal systems where higher-order Coriolis coupling terms can
be important. Although most rate calculations have assumed
that K is not a good quantum number (K-active model), it is
difficult to distinguish between treatments of K based on kinetics
data due to the uncertainty in several of the other inherent
assumptions.4
It is well-known that scalar and vector correlations can
provide significant insight into the dynamics of direct photo-
dissociation reactions that occur on excited-state potential energy
surfaces, which are often characterized by a localized release
of potential energy into product translational motion.14,15 More
recently, the study of scalar and vector correlations has also
been shown to be a powerful means to examine dissociation
reactions along barrierless potentials.16-22 We find that the study
of product scalar and vector properties from thermal systems
near vibrational thresholds provides a means to examine the
nature of the K-rotor angular momentum. Marcus has qualita-
tively described a physical mechanism by which energy is
partitioned among the vibrational and rotational energy levels
in a barrierless reaction.23 The conserved-mode vibrational
distribution is determined early in the reaction at the so-called
“inner” transition state and maps directly to the products. In a
thermal sample prior to optical excitation, the parent symmetric-
top molecules have a broad distribution of parent rotational
quantum states, J, but have restricted values of K. Coriolis
coupling of the rotational motion to the vibrations of the
molecule will result in a significant broadening of the K-
distribution. The coupling of vibrational motion into rotation
results in a significant change in the vibrational energy of the
conserved modes at the inner transition state, which yields a
different product vibrational distribution from that of a K-
adiabatic system. The transitional modes remain coupled until
the system reaches the “outer”, orbiting, transition state.24
Constraints may be manifested in the rotational distributions,
but vector correlations, specifically those between the relative
velocity, v, and the product rotational angular momentum vector,
j, have been shown to be a sensitive probe of the K-rotor activity
at the outer transition state.13,19 The activity of the K-rotor
strongly affects the sum of states at this transition state, which
is important for calculating rates for reactions just above
vibrational thresholds.25,26,29,30
We have recently examined the fragment correlations in the
photodissociation of room temperature nitrosyl cyanide (NCNO)
using transient frequency modulation Doppler spectroscopy at
520 nm.27 The present study represents an extension of that work
to include other wavelengths that span a range of dissociation
lifetimes. On the basis of these additional results, we believe
that this method for probing the K-rotor dynamics at the inner
transition state may be applied to other systems that undergo
barrierless unimolecular dissociation. To this end, we have
examined previous work by other groups on two prototypical
systems for barrierless unimolecular reactions, CF3NO and CH2-
CO. We find that both systems show evidence for approximate
K-mixing, which is qualitatively consistent with the rates of
dissociation relative to the rotational periods of the parent
molecules as well as their relative densities of reactant states.
Experiment
The transient frequency modulation (FM) Doppler spectroscopy
technique used in the NCNO studies has been described in detail
previously.27,28,56,57 Briefly, the output of a near-infrared diode laser
operating on the A-X electronic transitions of CN near 800 nm was
frequency modulated using an electrooptic phase modulator. The
frequency-modulated beam primarily consists of the fundamental and
two sideband frequencies, equally spaced above and below the
fundamental, that are 180° out of phase. Preferential absorption of one
of the sidebands causes amplitude modulation at the sideband spacing
frequency, 192 MHz in the present experiment. This spacing is
significantly less than the width of the absorption features of interest,
and scanning the output of the laser over a single rotational state
provides a spectrum that is a near derivative of the absorption feature.
The photolysis beam was generated from a Nd:YAG-pumped dye laser
operating with Coumarin 500 dye. The sample is introduced via a needle
valve into a long-path-length absorption cell at pressures of 10-30
mTorr, and the photolysis and probe beams propagate collinearly along
(7) Hamilton, C. E.; Kinsey, J. L.; Field, R. W. Annu. ReV. Phys. Chem. 1986,
37, 493.
(8) Abramson, E.; Field, R. W.; Imre, D.; Innes, K. K.; Kinsey, J. L. J. Chem.
Phys. 1985, 83, 453.
(9) Dai, H. L.; Korpa, C. L.; Kinsey, J. L.; Field, R. W. J. Chem. Phys. 1985,
82, 1688.
(10) Mellinger, A.; Ashikmin, M. V.; Moore, C. B. J. Chem. Phys. 1998, 108,
8944.
(11) Ionov, P. I.; Bezel, I.; Ionov, S. I.; Wittig, C. Chem. Phys. Lett. 1997, 272,
257.
(12) Bezel, I.; Ionov, P.; Wittig, C. J. Chem. Phys. 1999, 111, 9267.
(13) North, S. W.; Hall, G. E. Ber. Bunsen-Ges. Phys. Chem. 1997, 101, 496.
(14) Sato, H. Chem. ReV. 2001, 107, 2687.
(15) Hall, G. E.; Houston, P. L. Annu. ReV. Phys. Chem. 1989, 40, 375.
(16) Nadler, I.; Pfab, J.; Reisler, H.; Wittig, C. J. Chem. Phys. 1984, 81, 653.
(17) Qian, C. X. W.; Noble, M.; Nadler, I.; Reisler, H.; Wittig, C. J. Chem.
Phys. 1985, 83, 5573.
(23) Marcus, R. A. Chem. Phys. Lett. 1988, 144, 208.
(24) An early reference to inner and outer transition states in barrierless
dissociation can be found in Rai, S. N.; Truhlar, D. G. J. Chem. Phys.
1983, 79, 6046.
(18) Qian, C. X. W.; Ogai, A.; Reisler, H.; Wittig, C. J. Chem. Phys. 1989, 90,
209.
(19) North, S. W.; Hall, G. E. J. Chem. Phys. 1997, 106, 60.
(20) Costen, M. L.; Katayanagi, H.; Hall, G. E. J. Phys. Chem. A 2000, 104,
10247.
(21) Morgan, C. G.; Drabbels, M.; Wodtke, A. M. J. Chem. Phys. 1996, 104,
7460.
(22) Spasov, J. S.; Cline, J. I. J. Chem. Phys. 1999, 110, 9568.
(25) Miller, W. H. J. Chem. Phys. 1976, 65, 2216.
(26) Klippenstein, S. J. J. Chem. Phys. 1994, 101, 1996
(27) McGivern, W. S.; North, S. W. J. Chem. Phys. 2002, 116, 7027.
(28) North, S. W.; Hall, G. E. Annu. ReV. Phys. Chem. 2000, 51, 243.
(29) Chesnavich, W. J.; Bowers, M. T. J. Am. Chem. Soc. 1977, 99, 1705.
(30) Chesnavich, W. J.; Bowers, M. T. J. Chem. Phys. 1977, 66, 2306.
9
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A R T I C L E S
McGivern and North
its length. To ensure that Doppler spectra are acquired prior to
translational or rotational relaxation of the CN radicals, the signal is
sampled from the first 100 ns after the photolysis pulse. A fast
photodiode is used to measure the high-frequency intensity variation,
and standard phase-sensitive detection electronics are used to convert
the magnitude of the beat signal to a dc voltage for analysis and archival.
Analysis and Discussion
In NCNO, the incident photon excites the system to the first
singlet excited state, S1. A large barrier to dissociation (>5000
cm-1) exists on the S1 surface, and for wavelengths >450 nm
the system internally converts back to the ground state, where
dissociation occurs.31 A qualitatively similar dissociation mech-
anism is also present in CF3NO and CH2CO as discussed below.
Scalar Correlations and the K-Quantum Number. There
are several prescriptions for calculating product state distribu-
tions in barrierless reactions.1,2 The simplest theory, phase space
theory (PST), assumes that both the conserved and transitional
modes are strongly coupled throughout the dissociation, which
is equivalent to locating a variational RRKM transition state at
infinity.32,33 The resulting product state distributions are thus
simply determined by the degeneracies of the individual product
quantum states. While PST has been successful at modeling
rotational product state distributions at low energies in numerous
systems, it typically overestimates rate constants and has been
found to underestimate product vibrational excitation. A model
that overcomes these shortcomings of PST assumes that the
vibrational modes are strongly coupled only up to the inner
transition state, at which point the conserved modes become
adiabatic. The transitional modes, which evolve into fragment
rotations, remain strongly coupled until a second outer transition
state.34 This separation of coupling regimes is the principle
behind both the vibrationally adiabatic variational RRKM
treatment of Marcus23 and the separate statistical ensembles
(SSE) method proposed by Wittig and co-workers.35
An accurate variational RRKM treatment of dissociation to
vibrationally excited products requires a transition state to be
located for each conserved-mode vibrational channel. Evaluation
of the transition-state location at each ascending vibrational
channel is performed at decreasing available energies to account
for the energy in the individual product vibrations. The transition
states become looser for each successively higher vibrational
channel. This treatment provides a greater flux at each of the
increasing vibrational levels relative to that of the ground state
than would be expected from a PST calculation, where the
transition states are all located at infinity. This greater relative
flux for the vibrationally excited channels will therefore provide
more vibrationally excited products than would be predicted
using PST. The SSE model is an alternative to the RRKM
treatment for calculating product state distributions without
consideration of the rate constants. The SSE model utilizes the
density of states of the “unhindered” vibrational degrees of
freedom of the parent molecule to calculate the product
vibrational distribution. The SSE model is equivalent to a pure
PST model in which a third degree of freedom, in addition to
Figure 1. Schematic vibrational and rotational energies of the reactant and
transition state in a prolate symmetric top. The dark gray areas represent
the vibrational energy at each point along the reaction coordinate, and the
light gray areas represent the rotational energy. In a K-adiabatic system,
the change in the rotational energy between the reactant and transition state
is small. K-mixing removes energy from the vibrational reservoir in a prolate
top, causing less vibrational energy to be present at the inner transition
state.
those removed to account for the reaction coordinate and angular
momentum conservation, is removed.2 A theoretical connection
between the variational RRKM treatment and the SSE method
is currently undeveloped.36 We have chosen to use SSE in the
present calculations because of its relative simplicity of
implementation and previously demonstrated ability to describe
the global product vibrational distributions derived from NCNO
photodissociation.17,18,31
The principle of using scalar distributions to interrogate the
nature of the K-rotor at the inner transition state relies on this
model of the different coupling regimes associated with the
conserved and transitional modes. Strong coupling of the K-rotor
energy to the vibrational energy reservoir prior to the inner
transition state would result in a uniform K-distribution when
the vibrations become adiabatic, although the extent to which
the K-rotor fully samples phase space remains an open ques-
tion.37 The effect of K-mixing on the product vibrational
distributions is dependent on the geometry of the reactant
molecule.38 For a thermal sample of a prolate symmetric top,
the K-distribution is peaked at zero, and randomization would
be expected to remove energy from the vibrational energy
reservoir, yielding cold vibrational distributions. A schematic
diagram of the vibrational and rotational energies for the reactant
and transition states of a prolate top is shown in Figure 1. For
an oblate symmetric top, the K-distribution is peaked at the
parent rotational quantum number, (J, and a uniform K-
distribution would add energy to the vibrational energy reservoir
prior to the inner transition state. In the present system, NCNO
is a near-prolate symmetric top, so a K-active system would
produce colder fragment vibrational distributions than if the
system were K-adiabatic.
(31) Noble, M.; Nadler, I.; Reisler, H.; Wittig, C. J. Chem. Phys. 1984, 81,
4333.
(32) Pechukas, P.; Light, J. C. J. Chem. Phys. 1965, 42, 3281.
(33) Klots, C. E. J. Phys. Chem. 1971, 75, 1526.
(34) The applicability of the adiabatic treatment of the conserved-mode vibrations
is uncertain for larger systems that have low-frequency vibrational modes.
(35) Wittig, C.; Nadler, I.; Reisler, H.; Noble, M.; Catanzarite, J.; Radhakrishnan,
G. J. Chem. Phys 1985, 83, 5581.
(36) Green, W. H., Jr.; Moore, C. B.; Polik, W. F. Annu. ReV. Phys. Chem.
1992, 43, 591.
(37) Perry, D. S.; Bethardy, G. A.; Davis, M. J.; Go. J. Faraday Discuss. 1995,
102, 215.
(38) Aubanel, E. E.; Wardlaw, D. M.; Zhu, L.; Hase, W. L. Int. ReV. Phys.
Chem. 1991, 10, 249.
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K-Quantum Number in Unimolecular Reaction Theory
A R T I C L E S
Figure 2. Typical FM Doppler spectra. The upper panel shows an R-branch
for j ) 11.5 CN radicals derived from the photodissociation of NCNO at
526 nm, and the lower panel shows a Q-branch for j ) 11.5 CN from
photodissociation at 514 nm. The data in the lower panel have been
symmetrized to remove the effects of the R12 satellite transition. The circles
represent experimental data. The dashed lines represented simulated Doppler
profiles derived from PST rotational distributions for the coincident V ) 0
and V ) 1 NO fragments, and the solid lines are the best-fit weighted sums
of these basis functions.
Correlated vibrational branching ratios are determined ex-
perimentally by first simulating Doppler profiles for each
energetically available vibrational channel using PST. The
details of this procedure are provided elsewhere.27 The correlated
vibrational branching ratios are then determined by weighting
these basis functions to provide the best fit to the measured
Doppler profiles. Figure 2 shows typical FM Doppler spectra
at 514 and 526 nm and their corresponding fits. The circles
represent experimental data. The dashed lines represent the
simulated basis functions for the V ) 0 and V ) 1 channels,
and the solid lines represent the best-fit weighted sum of the
simulations.27 The differences between these two contributions
are significant, providing a robust method for determination of
the correlated vibrational branching ratios. Shown in Figure 3
are correlated vibrational branching ratios determined from
Doppler spectra for several CN rotational quantum states for
the photodissociation of NCNO at three wavelengths. The 520
nm results have been reported previously.27 The solid and dashed
lines are the results of SSE theory under K-adiabatic and
K-active conditions at the inner transition state, respectively.
All of the simulations include thermal averaging to account for
the internal energy distribution of the parent molecules at room
temperature. We find that the K-adiabatic SSE values are
significantly larger than the K-active values and provide the
best fit to the experimental data at all wavelengths.39 We
therefore conclude that the K-rotor is not strongly coupled into
the vibrational reservoir early in the reaction.
Figure 3. Correlated vibrational branching ratios at 526, 520, and 514 nm
as a function of CN rotational quantum number N. Open circles represent
experimental values. Solid and dashed lines are the simulations from SSE
theory, where the K-quantum number is treated as adiabatic and active,
respectively.
than the smaller differences in absolute measurements typically
observed in the global branching ratios. At a correlated level,
numerous independent measurements of the vibrational branch-
ing ratio may be made, corresponding to individual rovibrational
states of the detected fragment, and can provide additional
certainty in the measured values.
Numerous studies have identified SSE as an effective
description of the energy partitioning of a dissociating system
in a molecular beam. In the typical application of the SSE model,
K-adiabaticity is implicit because only the parent thermal
vibrational energy and the photon energy are used to determine
the vibrational branching.35 However, the use of an adiabatic
K-rotor in an SSE calculation has little effect on the calculated
vibrational branching of a sample in a molecular beam, since
the rotational cooling of the parent precludes a detailed analysis
of the K-rotor dynamics. The observation of significant changes
in the vibrational branching ratio requires large values of J, the
parent rotational quantum number, to ensure that the differences
between the K-adiabatic and K-active cases are sufficient to
permit unambiguous identification.
The photodissociation of ketene to give singlet CH2 and CO
is one of the most significant benchmarks for the study of
unimolecular dissociation on barrierless potentials.14 As with
NCNO, vibrational branching ratios may distinguish between
an active and adiabatic K-rotor. Global measurements of the
CO vibrational distribution derived from the 308 nm photodis-
sociation of room temperature ketene were made by Nesbitt et
al. using laser-induced fluorescence.40 Those authors measured
a value of 0.09 ( 0.05 for the (V ) 1)/(V ) 0) ratio of
populations, which were found to be consistent with the SSE
Although global vibrational branching ratios can provide
similar information in principle, the use of correlated branching
fractions has significant advantages. At excitation energies near
the threshold for vibrational excitation, the differences between
K-adiabatic and K-active global branching ratios are typically
small and may be difficult to distinguish. Correlated studies
examine larger differences between relative measurements rather
(39) The predictions of the K-active SSE model and PST are very similar for
all wavelengths studied.
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A R T I C L E S
McGivern and North
model, which treated K as adiabatic implicitly. However, more
recent correlated measurements have revealed that the global
vibrational branching ratio may be significantly lower than the
previously reported value.41 The correlated CO vibrational
branching ratio of jet-cooled CH2CO has been measured recently
by Costen et al.20 Interestingly, the measured correlated
vibrational branching ratios in the room temperature sample are
the same or lower than the corresponding jet-cooled branching
ratios for each measured CH2 transition common to both studies,
despite the additional vibrational energy that would be expected
from the internal energy of the parent molecule in the room
temperature sample. It is unlikely that changes in the global
CH2 rotational distribution could compensate for the factor of
3 increase in the correlated vibrational branching ratio predicted
by the K-adiabatic SSE model in going from jet-cooled to room
temperature parent molecules. We therefore believe that this
difference may be indicative of approximate K-mixing at the
inner transition state. Experiments are currently underway to
remeasure the global CO vibrational distributions.
The differences between global branching ratios in the bulb
and the molecular beam may still provide some insight into the
nature of the K-rotor in systems for which correlated measure-
ments of the vibrational branching have not been made. The
photodissociation of CF3NO to give CF3 and NO provides an
example in which the K-rotor appears to play an important role
in the overall dynamics of the room temperature system.
Although correlated measurements have been made by detecting
the NO fragment to identify the wavelength-dependent photo-
dissociation dynamics,22 no studies of the correlated NO
vibrational branching ratios have been undertaken. Like NCNO,
CF3NO is a near-prolate symmetric top, and dissociation occurs
along the S0 potential energy surface after internal conversion
from the first singlet excited state when following excitation at
wavelengths longer than 610 nm.42-45 Houston and co-work-
ers44,45 measured global vibrational and rotational distributions
for the photodissociation of both room temperature and jet-
cooled CF3NO at wavelengths ranging from 600 to 660 nm,
corresponding to 1172-2687 cm-1 of energy above the dis-
sociation threshold.46 At 625 nm, which corresponds to 111
cm-1 above the threshold for V ) 1 formation in the NO
fragment, a (V ) 1)/(V ) 0) ratio for NO of 0.022 ( 0.011 was
observed in a jet-cooled sample.45 At room temperature,
dissociation at the same wavelength resulted in a (V ) 1)/(V )
0) ratio for NO of 0.018 ( 0.009.42 Although the differences
are small and the estimated errors are relatively large, the room
temperature sample would be expected to have a significantly
larger vibrational branching ratio if the K-rotor were adiabatic
as a result of the significant parent thermal vibrational energy.
We suggest that this lower vibrational branching ratio in the
bulb is a signature of K-mixing at the inner transition state,
although clearly more experiments are necessary to confirm this
hypothesis. It should be noted that the appearance rates for
products from the photodissociation of CF3NO are dominated
by internal conversion to the ground state. The extent to which
K-mixing occurs on the ground or excited state is not revealed
by measurements of the product distributions, which show only
that K-mixing has occurred prior to reaching the inner transition
state.
The mixing of the K-quantum number in a symmetric top
requires the coupling of vibrational motion to rotational motion
about the symmetry axis through either Coriolis coupling or
Fermi interactions.47 It is difficult to develop a general prescrip-
tion for how to treat K in unimolecular dissociation since the
competition between K-mixing and dissociation time scales will
depend on numerous factors including the density of states, the
magnitudes of J and K, and the dissociation rates. Perry et al.
have examined the time scale of K-mixing in near-prolate
symmetric tops and found that the scrambling of K to states
near J requires several hundred rotations of the parent mol-
ecule.37 Treating the frequency of parent rotation, ω, as a
“conceptual limit” to the rate of Coriolis mixing as described
by Perry et al.,37 one would not expect significant K-mixing if
ω were less than or equal to the dissociation rate. The
dissociation rate of NCNO has been examined in detail both
experimentally48 and theoretically,49-51 though these studies did
not extend to the available energies of the present work.
Variational RRKM calculations that are consistent with the
previous studies at lower energies yield a unimolecular dis-
sociation rate constant of 5 × 1011 s-1 for 520 nm photolysis.
The average rate of rotation of NCNO, assuming it to be a
symmetric top with a rotational constant for rotation about the
axis perpendicular to the symmetry axis of 0.175 cm-1, is 1.4
× 1012 s-1 at 295 K. This difference is sufficient to remove
any significant laboratory frame anisotropy of the departing
fragments; however, it is unlikely to be sufficiently long to allow
extensive K-mixing to values near J in a thermal sample. As a
result, the observation that K is approximately adiabatic at the
inner transition state in NCNO dissociation is not unreasonable.
However, the unimolecular dissociation rates of CF3NO and
ketene are significantly slower and the state densities signifi-
cantly larger than those of NCNO at wavelengths corresponding
to a few hundred cm-1 above the threshold for formation of
vibrationally excited diatomic fragments. The dissociation rates
of CF3NO at the wavelengths of interest in the studies by
Houston and co-workers44,45 were on the order of 107-108 s-1
.
The average rotational frequency of CF3NO at 295 K is 1.1 ×
1012 s-1, several orders of magnitude higher than the dissociation
rate. K-mixing would be expected to be more extensive in CF3-
NO than in the faster NCNO dissociation, which is consistent
with the postulated K-mixing mechanism apparent from the
difference between the bulb and jet experiments. Similarly, the
rate constant for the dissociation of ketene on the singlet
potential energy surface was calculated to be 9 × 109 s-1 by
Klippenstein et al. for 308 nm photodissociation.52 At 295 K,
the average rotational frequency of ketene is 2 × 1012 s-1, still
significantly higher than the dissociation rate. As in CF3NO,
K-mixing would be expected to take place prior to the inner
(40) Nesbitt, D, J.; Petek, H.; Foltz, M. F.; Filseth, S. V.; Bamford, D. J.; Moore,
C. B. J. Chem. Phys. 1985, 83, 223.
(41) The analysis of the correlated distributions is unpublished, but a comparison
of the jet and thermal Doppler spectra can be found in ref 28.
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K-Quantum Number in Unimolecular Reaction Theory
A R T I C L E S
transition state, although whether the mixing takes place in the
ground- or excited-state surfaces is unknown. Some evidence
for incomplete K-mixing in the near-threshold photodissociation
of NO2 has been observed previously using optical double
resonance to measure rates from specific parent (J, K) states.
The lack of complete K-mixing was attributed to the fast
dissociation of NO2 relative to the parent rotational period. Later
classical trajectory calculations on this system confirmed this
hypothesis, finding that several molecular rotations are necessary
to obtain appreciable K-mixing.53
Vector Correlations and the K-Rotor. Vibrational distribu-
tions can provide no information on the dissociation dynamics
after the conserved modes are decoupled at the inner transition
state. However, vector correlations, namely, correlations among
µ, the molecular transition dipole moment vector, v, the relative
velocity vector, and j, the rotational angular momentum vector
of the detected fragment, can provide detailed information on
the long-range anisotropy of the potential and exit channel
couplings. These long-range effects do not influence the rate
constant except at energies close to vibrational thresholds but
play a significant role in the product state distributions.
Correlations involving µ may be probed by utilizing a polarized
laser for photolysis, which defines a preferred direction in the
laboratory frame. An anisotropic distribution of photofragments
in the laboratory frame of reference may be probed using several
techniques, including laser-induced fluorescence, numerous mass
spectrometric methods, and the transient FM method described
in the present work. However, in the case of NCNO, the
dissociation is sufficiently slow relative to the rotational period
of the parent molecule that the correlation between the transition
dipole moment of the parent molecule and the speed and rotation
of the product molecules has disappeared. The correlation
between v and j does not depend on the alignment in the
laboratory frame and remains important.
The relationship between the v-j correlations and K-mixing
in barrierless unimolecular dissociation reactions has been
discussed in detail by North and Hall.13 Briefly, the K-quantum
number may be related to the v-j correlation by transforming
the system into a basis in which the projections of the angular
momenta of the fragments along the relative velocity vector
are good quantum numbers. Since the relative velocity vector
is strictly perpendicular to the orbital angular momentum of
the fragments about the parent center of mass, these fragment
projections map directly to the K-quantum number. In a K-active
system, there is no restriction on the projection of the fragment
angular momentum on v, leading to small correlations between
v and j. K-adiabatic systems, however, are limited to only a
small number of such projections and lead to strong correlations
between v and j. Experimentally, these correlations may be
probed by examining the differences between Q- and R-branch
transitions with the same initial rotational state.54,55 We have
previously reported the v-j correlations in the 520 nm photo-
dissociation of NCNO.27 The v-j correlations were found to
be small and were well-represented by treating the system at
the outer transition state as K-active. The v-j correlations
measured at 514 and 526 nm were also found to be small for
all measured rotational states, consistent with K-scrambling at
the outer transition state.
The presence of a K-active outer transition state and a
K-adiabatic inner transition state is surprising, since the system
would be expected to pass between the inner and outer transition
states in less than a picosecond. This motion would require
strong Coriolis coupling to allow the K-quantum number to
partially scramble on this time scale. While this is not altogether
unreasonable, other factors may affect the magnitudes of the
v-j correlations. The strict relationship between the K-quantum
number and the total projection of the rotational angular
momentum vectors on the relative velocity axis is true only
within the axial recoil approximation for a true symmetric top
molecule. Significant rotation of the molecular complex as
dissociation occurs allows states with different total projections
along the relative velocity axis to mix, which would be expected
to degrade the observed v-j correlations. The dissociation axis
and the symmetry axis of the near-symmetric top do not strictly
coincide, which leads directly to an additional blurring of the
v-j correlation. Finally, the broadening of the observed Doppler
spectra due to parent translational motion increases the overall
uncertainty in the measured correlations. The measurement of
v-j correlations in a collimated effusive beam of room
temperature molecules would significantly decrease the trans-
lational blurring, allowing a more accurate determination of the
v-j correlations, while maintaining the thermal parent rotational
energy required to probe the nature of the K-rotor. These
experiments are currently underway in our laboratory.
Conclusion
The extent of coupling of the K-rotor motion in barrierless
unimolecular dissociation reactions has been examined by
studying fragment correlations in the photodissociation of
NCNO. We find that both product scalar and vector correlations
can provide a diagnostic on the nature of the K-rotor in
unimolecular dissociation, particularly when a comparison
between results from jet and thermal experiments is possible.
At three dissociation wavelengths, vibrational branching ratios
of the NO radical were determined for several coincident CN
rotational quantum states. The correlated vibrational branching
ratios are found to be well-described by a separate statistical
ensembles treatment in which the K-rotor is assumed to be
adiabatic at the transition state. The use of vibrational distribu-
tions as a probe of K-rotor activity has been applied more
generally to CF3NO and CH2CO, which were both found to be
approximately K-active at the transition state. As expected,
K-mixing appears to become more extensive as the dissociation
rate decreases relative to the parent rotational period. Vector
correlations, which serve as a probe of the K-rotor activity at
the outer transition state, were also measured for NCNO
photodissociation. The v-j correlations were found to be small
at both 514 and 526 nm, which is indicative of K-mixing at the
outer transition state.
Acknowledgment. We thank Dr. Gregory E. Hall, Dr.
Stephen J. Klippenstein, Dr. William L. Hase, and Dr. Robert
R. Lucchese for insightful discussions. The reviewers of this
manuscript are also acknowledged for their helpful comments.
This work was supported by a grant from the Robert A. Welch
Foundation (A-1405).
(53) Grebenshchikov, S. Y.; Flo¨thmann, H.; Schinke, R.; Bezel, I.; Wittig, C.;
Kato, S. Chem. Phys. Lett. 1998, 285, 410.
(54) Hall, G. E.; Houston, P. L. Annu. ReV. Phys. Chem. 1989, 40, 375.
(55) Gordon, R. J.; Hall, G. E. AdV. Chem. Phys. 1996, 96, 1.
(56) Li, R.; Derecskei-Kovacs, A.; North, S. W. Chem. Phys. 2000, 254, 309.
(57) Li, R.; McGivern, W. S.; North, S. W. Chem. Phys. Lett. 2001, 334, 47.
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