Transition state vibrational level thresholds foe the dissociation of triplet ketene

Transition state vibrational level thresholds for the dissociation of triplet ketene

Sang Kyu Kim , Edward R. Lovejoy, and C. Bradley Moore

Citation: The Journal of Chemical Physics 102 , 3202 (1995); doi: 10.1063/1.468631

View online: http://dx.doi.org/10.1063/1.468631

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Transition state vibrational level thresholds for the dissociation

of triplet ketene

Sang Kyu Kim,^a⁾Edward R. Lovejoy,^b⁾and C. Bradley Moore

Department of Chemistry, University of California at Berkeley

and Chemical Sciences Division of the Lawrence Berkeley Laboratory, Berkeley, California 94720

͑

Received 13 September 1994; accepted 14 November 1994͒

Rate constants for the unimolecular dissociation of ketene ͑CH CO͒ and deuterated ketene ͑CD CO͒

2

˜

3

˜ 3

1

have been measured at the threshold for the production of CH (X B ) or CD (X B ) and CO

X ⌺ ) by photofragmentation in a cold jet. The rate constant increases in a stepwise manner as

2

1

2

˜

1

ϩ

(

energy increases. This is in accord with the long-standing premise that the rate of a unimolecular

reaction is controlled by ﬂux through quantized transition-state thresholds at each energy level for

vibrational motion orthogonal to the reaction coordinate. The ﬁrst step in rate constant and/or

photofragment excitation ͑PHOFEX͒ spectrum gives accurate values for the barrier to dissociation

Ϫ1

above the zero-point energy of the products, 1281Ϯ15 cm for CH CO and 1071Ϯ40 cm for

2

CD CO. The measured rate constants are ﬁt by Rice–Ramsperger–Kassel–Marcus ͑RRKM͒ theory.

2

The vibrational frequencies at the transition state obtained from the ﬁts are compared with ab initio

results. Vibrational motions at the transition state orthogonal to the reaction coordinate are also

revealed in CO product rotational distributions. Calculations using an impulsive model which

includes vibrational motions at the transition state reproduce the experimental dependence of the

PHOFEX spectra on the CO J state quite well. The small dependence of rate constant on jet

temperature ͑4–30 K͒ indicates that the K quantum number for rotation about its symmetry axis is

a

I. INTRODUCTION

tures have also been calculated for HϩO by Leforestier and

2

8

Miller and in a more approximate way for OϩH ͑Ref. 9͒

2

Transition-state theory ͑TST͒ was developed in the

1

and HϩH2 ͑Ref. 10͒ by Bowman. In experimental rate stud-

ies, since the initial conditions for bimolecular reactions are

averaged over total angular momentum ͑impact parameter͒

and usually over thermal distributions of initial energy and

quantum state, these structures are predicted to be com-

pletely smoothed out.¹¹However, for unimolecular reactions,

much of this averaging may be eliminated experimentally

and the possibility exists to observe these quantized thresh-

olds at the transition state.

The Rice–Ramsperger–Kassel–Marcus ͑RRKM͒ theory

is a microcanonical ensemble version of transition state

theory for treating unimolecular reaction rates.¹²RRKM

theory is based on the additional assumptions that all vibra-

tional states in the excited molecule are equally probable and

that the vibrational energy ﬂows freely among the different

degrees of freedom at a rate much faster than the reaction

rate. In RRKM theory, the rate constant for total energy E

and total angular momentum J is given by

1

930’s and has been widely used in describing chemical

reaction rates. TST is a statistical theory based on the funda-

mental assumptions that there is a local equilibrium between

reactants and molecules crossing the transition state toward

products along the reaction coordinate and that the molecule

crossing through the transition state proceeds to products

without recrossing. Even though TST has been enormously

successful, the validity of these fundamental assumptions has

been debated for decades. Rate constants may be calculated

rigorously, given an exact potential energy surface ͑PES͒, by

solving the Schr o¨ dinger’s equation for the full quantum-

state-resolved scattering matrix, calculating the cumulative

reaction probability ͑CRP͒ as a function of total energy and

angular momentum, and averaging over the distributions of

2,3

initial energy and quantum state. Miller has derived a rig-

orous quantum mechanical formulation which allows the

CRP to be calculated directly from computations on a grid in

the region of the transition state without explicitly calculat-

ing the scattering matrix, a ‘‘transition-state theory’’ without

W͑E,J͒

4

–7

3

k͑E,J͒ϭ

,

͑1͒

approximations.

Truhlar and co-workers and Miller and

h␳͑E,J͒

6

co-workers reported steplike structures in such rigorously

where W(E,J) is the number of energetically accessible

states for vibration orthogonal to the reaction coordinate at

the transition state, ␳(E,J) is the density of vibrational states

of the reactant, and h is Planck’s constant. The rate constant

increases stepwise as the energy increases through each vi-

brational level or threshold at the transition state. This step

structure from W(E,J) is apparently not an artifact of treat-

ing motion along the reaction coordinate classically since the

calculated CRP’s for the HϩH reaction; this shows that the

CRP is controlled by quantized vibrational thresholds at the

transition state as in the not so rigorous TST. Similar struc-

2

^a^͒Present address: Division of Chemistry and Chemical Engineering, Cali-

fornia Institute of Technology, Pasadena, California 91125.

b͒

Permanent address: NOAA/ERL/Aeronomy, R/E/AL-2, 325 Broadway,

Boulder, Colorado 80303.

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3

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Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3203

quantum treatments of CRP²^–¹⁰exhibit quite similar struc-

tures. RRKM theory presumes that motion along the reaction

coordinate is decoupled from the bound vibrational motions

at the transition state. The passage through the transition

state is then vibrationally adiabatic, and the vibrational levels

are deﬁned over a sufﬁciently broad region near the transi-

tion state to give well-deﬁned reaction thresholds or quan-

tized channels connecting reactant to products.

The RRKM theory has been used widely for many de-

cades, and yet because of the lack of knowledge about po-

tential energy surfaces ͑PES͒ in the vicinity of the transition

state, vibrational frequencies at the transition state have been

guessed or simply ﬁt to explain thermally averaged experi-

mental results. Due to the difﬁculty of resolving spectra of

highly vibrationally excited reactant states, the density of

reactant states in Eq. ͑1͒ has usually been estimated from

data at much lower energies rather than measured directly.

Recent spectroscopic studies have shown that the actual den-

sity of states for a number of small molecules ͓HCCH ͑Ref.

1

3͒, HFCO ͑Ref. 14͒, D CO ͑Ref. 15͒, CH O ͑Ref. 16͒, NO

2 3 2

͑

Refs. 17 and 18͔͒ is three to ten times higher than predicted

FIG. 1. The three lowest potential energy surfaces of ketene along the

reaction coordinate. The ketene molecule is excited by a UV laser pulse to

the ﬁrst excited singlet state (S ), undergoes internal conversion to S and

by estimates based on levels at signiﬁcantly lower energy.

With the recent development of molecular beams and

lasers, it is now possible to measure rate constants for reac-

tants with well-deﬁned initial conditions. A pulsed jet expan-

sion is used to minimize the internal thermal energy and

angular momentum of the molecule in its ground state, while

high-resolution lasers are used to excite the molecule to well-

deﬁned reactant states and probe the fragments with

quantum-state resolution. In addition, recent high-level quan-

tum mechanical calculations now provide quantitative pre-

dictions for the properties of transition states of small

molecules. Therefore, unimolecular reaction rate theories

are being subjected to more stringent tests. Using a variety of

spectroscopic techniques such as overtone excitation, photon

excitation followed by internal conversion, and stimulated

emission pumping ͑SEP͒, kinetic studies of molecules pre-

1

0

1

intersystem crossing to T1 , and dissociates into CH2 ( a˜ A1)ϩCO ͑singlet

3

˜

3

channel͒ or CH (X B )ϩCO ͑triplet channel͒ fragments.

2

1

cm^Ϫ¹above threshold.¹⁷^,²⁶Thus the transition state must

Ϫ1

move from loose to tight within a few tens of cm

.

The three lowest electronic states of ketene are the

ground singlet state (S ), the ﬁrst excited singlet state (S ),

0

1

and the ﬁrst triplet state (T ). The ketene molecule in its

1

ground state is UV laser excited (S ←S ), internally con-

1

9

1

0

verts and/or intersystem crosses, and dissociates into meth-

22

ylene and CO products, Fig. 1. The ketene molecule disso-

ciates into singlet methylene and CO fragments ͑singlet

channel͒ on the barrierless ground singlet potential energy

Ϫ1 27–29

surface with a threshold energy of 30 116 cm

,

while it

pared in well-deﬁned quantum states have been carried out

3

dissociates into triplet methylene ͑ CH ͒ and CO fragments

_f_o_r_s_e_v_e_r_a_l_u_n_i_m_o_l_e_c_u_l_a_r_r_e_a_c_t_i_o_n_s_.20–22

2

on the triplet surface which has a small barrier about 1300

RRKM theory, Eq. ͑1͒, predicts that the rate constant

increases by steps with an amplitude equal to 1/[h␳(E,J)]

as the energy is increased through each vibrational level of

the transition state. Recently, steplike structures in a rate con-

stant have been reported for the unimolecular dissociation of

ketene on its triplet surface.²³Since there is a small barrier

for this reaction, the transition state is well deﬁned at the

saddle point of the PES along the reaction coordinate, and

the steps in rate constants have been attributed to vibrational

level thresholds of the transition state. Structures in rate con-

stants have also been reported for some barrierless unimo-

lecular reactions. Zewail and co-workers have noted non-

monotonic features in rate constants for NCNO

Ϫ1

cm above the triplet products and 2000 cm below the

30,31

singlet products.

Recently, unimolecular reaction of

ketene on singlet and/or triplet channels has been studied

extensively and provided a good example for testing a vari-

22,27–34

ety of statistical theories.

In this work, the rate constants for dissociation of rota-

3

tionally cold ͑4 K͒ ketene to CH ϩCO are measured as a

2

continuous function of energy. The quantized transition state

vibrational thresholds are revealed in clear steplike structures

observed in the unimolecular dissociation rate constants.

This is consistent with the fundamental postulates of transi-

tion state ͑RRKM͒ theory and provides a quantitative test of

unimolecular reaction rate theory.

2

4

dissociation. Wittig and co-workers reported steplike struc-

tures in rate constants for NO dissociation and ascribed

2

II. EXPERIMENT

those steps to vibrational levels of a variational transition

Ϫ1

state which tightens as energy increases just 100 cm above

Ketene ͑CH CO͒ was prepared by passing acetic anhy-

2

the reaction threshold.¹

8,25

For the same reaction, Tsuchiya

dride through a red-hot quartz tube, trapped at 77 K, and

distilled from 179 to 77 K twice before use. Deuterated

and co-workers report that the rate constant increases step-

wise when a new rotational product channel opens just 5

ketene ͑CD CO͒ was prepared by the same method from

2

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3204

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

deuterated acetic anhydride ͑Sigma, Ͼ99.0% assay͒. Ketene

was stored at 77 K. During experiments, the ketene vapor

pressure was kept at 50 Torr by a hexanes slush bath ͑179 K͒.

Helium carrier gas was bubbled through the sample and the

gas mixture was expanded through the 0.5 mm diameter ori-

ﬁce of a pulsed nozzle into the vacuum chamber. The back-

ing pressure of the carrier gas was 1.5 atm and the back-

vacuum chamber, scattered from a LiF diffuser, struck a solar

blind VUV PMT ͑EMR 542G-08-19, LiF window͒, and pro-

duced a signal to normalize for VUV intensity ﬂuctuations.

The linearity of normalization was carefully checked.³²

The nozzle valve driver ͑Newport BV100͒ and two

Nd:YAG lasers were triggered by an SRS DG535 pulse gen-

erator at a 10 Hz repetition rate. Scattered photolysis laser

light detected by a fast photodiode was used to trigger a

boxcar ͑SR250 from SRS͒. The gate output of the boxcar

was delayed by the desired amount with respect to the trigger

signal, ampliﬁed, and used to trigger the Q switch of the

other Nd:YAG laser. The molecular beam pulse was synchro-

nized with the laser pulses by the internal delay function of

the nozzle valve driver.

ground pressure in the vacuum chamber was maintained at

Ϫ4

10

Torr when the nozzle was on. Ketene was rotationally

cooled by the supersonic expansion and the rotational tem-

perature of ketene in the experiment was estimated to be

2

7

about 4 K. In some experiments the backing pressure of the

carrier gas was varied from 260 Torr to 2.2 atm to vary the

rotational temperature of ketene in the supersonic jet. A mix-

ture of 100 ppm of CO in He was expanded and ﬂuorescence

excitation spectroscopy used to estimate the temperature of

the molecular beam.

LIF of the CO product was detected by a solar-blind

VUV photomultiplier tube ͑PMT͒ ͑EMR 542G-09-19, 50

mm diam MgF window͒ 5 cm away from the interaction

2

The photolysis laser was a frequency-doubled dye laser

region ͑vide supra͒. A cultured quartz window in front of the

VUV PMT absorbed scattered VUV laser light. PMT signals

were preampliﬁed by 50 times ͑Hewlett-Packard, 8447D

OPT 001͒, sampled and integrated by a boxcar ͑SR250͒,

digitized by an A/D converter ͑DASH-8 Metrabyte͒, and

stored in a microcomputer ͑IBM PC/XT͒.³⁶Both photolysis

and probe laser frequencies were controlled by the same

computer.

͑

Lambda Physik FL2002E; 0.2 cm^Ϫ¹linewidth͒ pumped by

the second harmonic output of a Q-switched Nd:YAG laser

Spectra-Physics DCR-4͒. DCM, LDS698, and mixtures of

those dyes in methanol were used in the dye laser to generate

0–50 mJ/pulse of energy in the 680–720 nm range. The UV

energy was 6–10 mJ/pulse in the 340–360 nm region, with a

͑

3

Ϫ1

0

.4 cm linewidth and a 7 ns duration. The frequency of the

dye laser output was calibrated with an accuracy of Ϯ0.5

cm using the optogalvanic spectrum of Ne gas. During

Rate constants for photodissociation of ketene were de-

termined by measuring the appearance rates of CO frag-

ments. The VUV probe laser frequency was ﬁxed at a spe-

ciﬁc rovibronic transition of the CO product and the CO LIF

signal was monitored while scanning the delay time between

pump and probe lasers at a ﬁxed pump laser frequency. The

delay time was scanned using SR265 software to control the

delay between trigger signal and gate output of the boxcar

via an SR245 computer interface. Data acquisition was con-

trolled by the same software.

Ϫ1

scans the direction and intensity of the UV laser output were

actively maintained and maximized, respectively, by an Inrad

Autotracker II with a KDP crystal. The UV laser output was

almost completely separated from the fundamental visible

laser pulse ͑99.9% attenuated͒ through three reﬂections from

UV dichroic mirrors before it entered the vacuum chamber.

Tunable vacuum UV ͑VUV͒ was generated for the laser-

induced ﬂuorescence ͑LIF͒ probing of CO fragment in the

˜

1

˜ 1

ϩ

(

A ⌸–X ⌺ ) (vЈϭ3ϪvЉϭ0) band. The output of a dye

CO product rise curves were obtained at photolysis en-

Ϫ1

6

7

Ϫ1

laser ͑Lambda Physik FL3002E; linewidth Ϸ0.2 cm , pulse

length Ϸ7 ns; 15–20 mJ/pulse in the 430–450 nm range

with coumarin-440 dye in methanol͒ pumped by the third

harmonic output of another Q-switched Nd:YAG laser ͑Con-

tinuum YG682-10͒ was frequency tripled in Xe gas to gen-

ergies for which rate constants were in the 10 –5ϫ10 s

range. Proper alignment of photolysis and probe lasers is

critical for accurate rate measurements. The alignment was

validated by recording the CO product rise curve at a pho-

tolysis energy for which the CO rise is Ͻ50 ns. For a 20

2

erate VUV in the 143–150 nm range. The efﬁciency of tri-

mm UV laser beam cross section, the signal remained at its

Ϫ6 35

pling is about 10

.

The fundamental dye laser output was

maximum until 1.0 ␮s when the UV-excited sample began to

focused by a fused silica lens ͑f.l.ϭ7 cm͒ into the center of a

0 cm long tripling cell ﬁlled with 20–40 Torr of Xe gas.

move beyond the probing zone deﬁned by the cross section

2

1

of the VUV probe laser beam ͑7–12 mm ͒. For the measure-

6

Ϫ1

The VUV was collimated by a CaF lens ͑f.l.ϭ8 cm͒ located

ment of rates Ͻ5ϫ10 s , the cross section of the pump

laser beam was expanded by about four times to give uni-

form detection out to 2.0 ␮s, Fig. 2. Because of the decrease

in the photolysis laser energy density, the signal-to-noise

͑S/N͒ ratio decreased and yielded larger uncertainties. Rate

2

at the exit of the tripling cell.

The photolysis and probe lasers propagated colinearly

from opposite ends of the vacuum chamber and crossed the

molecular beam pulse at 90°. The interaction region, deﬁned

by the overlap of the lasers, was centered 3–4 cm from the

nozzle oriﬁce. The polarizations of both laser pulses were

parallel to the jet axis and perpendicular to the ﬂuorescence

detection axis ͑vide infra͒. A small portion of the photolysis

laser intensity was reﬂected from the entrance window of the

vacuum chamber, detected by a fast-response photodiode,

and used to normalize for photolysis laser intensity ﬂuctua-

tions. The VUV probe laser passed through the molecular

beam, was partially reﬂected from the exit window inside the

6

Ϫ1

constants as small as 1.0ϫ10 s were measured without

any correction to the raw experimental data.

Each CO product rise curve was averaged over 15–20

scans and measured three times at each photolysis energy.

Rate constant measurements were carried out as a function of

Ϫ1

photolysis energy at 2–4 cm intervals. To check the repro-

ducibility of the data, the three different measurements at

each photolysis energy were carried out on different days

with a new alignment of laser beams.

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Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3205

FIG. 2. Cross section of the VUV LIF detection geometry in the molecular

beam chamber. A cultured quartz window ͑C.Q.͒ is used in front of the VUV

PMT to reduce the scattered probe laser light. For measurements of rate

6

Ϫ1

constants less than 5ϫ10 s , the expanded photolysis laser is overlapped

with the VUV probe laser in the downstream portion of the molecular jet.

The distances of the PMT and the oriﬁce of the nozzle from the interaction

region are about 5 and 3 cm, respectively. The diameters of UV pump and

VUV probe lasers are about 10 and 4 mm, respectively.

FIG. 3. CO product rotational distributions from the dissociation of ketene.

The solid lines are calculated from ab initio data as in Ref. 30: ͑a͒ CH CO

2

Ϫ1

at 28 250 cm , 1 ␮s; ͑b͒ CD CO at 28 410 cm , 1 ␮s.

2

Photofragment excitation ͑PHOFEX͒ spectra were ob-

tained by monitoring the LIF signal for a single CO(v,J)

state while the photolysis laser frequency was scanned. The

delay time between the photolysis and probe lasers was ﬁxed

with a time jitter of less than 5 ns. PHOFEX curves were

obtained with 0.5–1 cm steps of the photolysis laser fre-

quency. Each data point was averaged for 10–20 laser shots

and PHOFEX curves were averaged for 3–5 scans.

transfer may occur. Figure 4 compares the CO rise curves for

Jϭ6 and 12. Relaxation from high to low J is clearly ob-

served beginning at 2.5 ␮s delay, signiﬁcantly after the vol-

ume of excited gas begins to move beyond the probe laser

beam.

Ϫ1

The CO product rise curves are least-squares ﬁt by a

single exponential. The pulse widths of pump and probe la-

sers ͑7 ns͒ are convoluted in the ﬁtting for rates faster than

7

Ϫ1

5

ϫ10 s as in Ref. 31. The standard deviation for the least-

III. RESULTS

A. CO product rotational distributions

CO product rotational distributions, obtained from rovi-

brationally resolved VUV LIF spectra for CO product by the

same method as described in Ref. 30, are shown for CH CO

2

Ϫ1

and CD CO at photolysis energies of 28 250 cm

and

2

Ϫ1

28 410 cm in Fig. 3. The rotational distributions are ap-

proximately Gaussian-shaped and peak near Jϭ12 for

CH CO and Jϭ11 for CD CO. The CO fragment acquires

2

angular momentum from the forces exerted as it passes

through the exit valley of the potential energy surface. An

impulsive force along the C–C bond at the geometry of the

transition state combined with the vibrational momenta at the

transition state model the observed distributions quite well.³⁰

B. Rate constants, k(E)

CO product rise curves are shown in Figs. 4 and 5. As

3

1

previously, the rate constants measured for different J

states of CO such as 13 and 6 are the same as those for

Jϭ12 within experimental error. Since the Q(12) transition

is one of the well isolated peaks in the LIF spectrum and

probes the most populated level of CO, it has been used for

all rate constant measurements.

FIG. 4. CO (vϭ0) product rise curves for CD CO dissociation at 28 900

2

cm ͑kϷ3ϫ10⁷

Ϫ1

s ͒. ͑a͒ Jϭ6; ͑b͒ Jϭ12; ͑c͒ ratio of the CO Jϭ6 and

Ϫ1

Jϭ12 signals. The LIF intensity for Jϭ6 is about 5 times less than that for

Jϭ12. The delay time at which rotational relaxation starts is marked by an

When the CO fragment is probed at a long delay time

after UV excitation, collisional rotation-to-translation energy

2

arrow on ͑c͒. The UV beam cross section was ϳ80 mm .

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Kim, Lovejoy, and Moore: Dissociation of triplet ketene

FIG. 5. Jϭ12 CO product rise curves with single exponential ﬁts. ͑a͒

Ϫ1

CH CO at 28 423 cm , and ͑b͒ CD CO at 28 500 cm

.

FIG. 7. Rate constant for CD CO dissociation as a function of the photolysis

2

energy. The CO͑vϭ0, Jϭ12͒ PHOFEX curve ͑⌬tϭ1.7 ␮s͒ is shown in

the reaction threshold region; its intensity is arbitrarily scaled. The solid line

is a RRKM ﬁt using the parameters in Table V.

squares ﬁt to an individual CO rise curve is much less than

the deviations between measurements made on each of three

different days and their mean. For CD CO, the rate constants

within 40 cm of threshold are too small to measure accu-

rately.

2

vibrational level thresholds are observed in the ﬁrst 200–300

cm region above the reaction thresholds for each isotope.

Ϫ1

Rate constants as a function of the photolysis energy,

C. Photofragment excitation spectra

k(E), are plotted as open circles in Figs. 6 and 7 for CH CO

2

and CD CO, respectively. Error bars shown on selected data

2

PHOFEX spectra probing the Q(12) transition of CO

points are two times the estimated standard deviation of the

mean of the three measurements from the true value. Distinct

steps in k(E) associated with the ﬁrst few transition state

from CH CO at 50 ns and from CD CO at 200 ns delay

2

times are shown by curve ͑a͒ in Figs. 8 and 9. The PHOFEX

signal can be expressed as

FIG. 6. Rate constant for CH CO dissociation as a function of the photolysis

FIG. 8. PHOFEX curves. ͑a͒ CO(Jϭ12), ⌬tϭ50 ns; ͑b͒ ͕1Ϫexp[Ϫk(E)

ϫ⌬t]͖ scaled for comparison to ͑a͒; ͑c͒ CO(Jϭ2), ⌬tϭ50 ns. The actual

2

energy. The error bars on selected data points represent twice the standard

deviation from three independent rate constant measurements. The solid line

is a RRKM ﬁt using parameters in Table V, but without tunneling.

ratio of the yields of CO͑vϭ0, Jϭ12͒ to CO͑vϭ0, Jϭ2͒ is about 20:1 at

Ϫ1

28 500 cm

.

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Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3207

FIG. 10. The CO͑vϭ0, Jϭ12͒ PHOFEX curves taken at 50 ns delay time

^a^t^͑^a^͒^s^o^lⁱ^d^lⁱⁿ^e^:^a^t²^.²^a^t^m^͑^Trot^ϭ⁴^K^͒^aⁿ^d^͑^b^͒^s^q^u^a^r^e^s^:^a^t²⁶⁰^T^o^r^r^͑^Trotϭ30

K͒ backing pressure of carrier gas.

FIG. 9. PHOFEX curves. ͑a͒ CO(Jϭ12), ⌬tϭ200 ns; ͑b͒ ͕1Ϫexp[Ϫk(E)

ϫ⌬t]͖ scaled for comparison to ͑a͒; ͑c͒ CO(Jϭ2), ⌬tϭ150 ns. The actual

ratio of the yields of CO͑vϭ0, Jϭ12͒ to CO͑vϭ0, Jϭ2͒ is about 20:1 at

Ϫ1

2

8 500 cm .

determined by the vibrational motion at the transition state.³⁰

Therefore, the comparison of PHOFEX curves for CO prod-

uct with different J_C_O’s gives a dynamically biased way to

observe thresholds for transition-state vibrations which

strongly inﬂuence CO rotation. The PHOFEX spectrum for

͒

͕ ͖

S͑E,v,J_C_O͒ϰ␴͑E͒P͑E,v,J_C_O1Ϫexp͓Ϫk͑E͒ϫ⌬t͔ , ͑2͒

where S is the LIF intensity when probing the CO(v,J)

product, ␴(E) is the absorption cross section of the parent

molecule for photon energy E, P(E,v,J_C_O͒ is the probability

for the CO fragment to be formed in the ͑v,J_C_O͒ quantum

state, k(E) is the ketene decay rate constant averaged over

the ensemble of initial excited states of ketene, and ⌬t is the

time delay between the pump and probe lasers.

CO͑vϭ0, Jϭ2͒ from CD CO, Fig. 9, shows steps and peaks

2

at higher photolysis energies where the structures in both

k(E) and the CO͑vϭ0, Jϭ12͒ PHOFEX curve become

washed out, thus revealing additional vibrational thresholds

at the transition state.

The absorption spectrum of CH CO at room temperature

CO͑vϭ0, Jϭ12͒ PHOFEX curves taken at 4.0Ϯ0.5 K

and 30Ϯ3 K ͑2.2 atm and 260 Torr backing pressure, respec-

tively͒ are almost identical, Fig. 10. Although the actual ro-

tational temperature of ketene in the molecular beam may be

different from that of the CO used for calibration, it is clear

that there is no dramatic effect of ketene rotational tempera-

ture on the rate constant in this range.

2

37

is a diffuse structureless band for the S ←S transition.

1

0

Presumably, this is the result of fast internal conversion and

intersystem crossing. The analysis of singlet CH PHOFEX

2

spectra shows that ␴(E) in the pulsed jet is nearly constant

Ϫ1

over ranges of several hundred cm

energy.

in photolysis

2

7,34

Curves b in Figs. 8 and 9 show ͕1Ϫexp[Ϫk(E)

ϫ⌬t]͖ using directly measured k(E) and the ⌬t of the

PHOFEX spectrum. If ␴(E) and P(E,v,J_C_O͒ are constant,

then curves ͑a͒ and ͑b͒ should have the same shape. In the

IV. ANALYSIS AND DISCUSSION

Ϫ1

ﬁrst 200–300 cm above the reaction threshold, the posi-

The observation of structures in k(E) is consistent with

the fundamental ideas of transition-state theory and RRKM

theory. The application of the RRKM model to ketene is

developed in the following sequence: ͑A͒ Coupling among

the three lowest electronic states (S ,S ,T ) for EϾ28 000

tions and amplitudes of steps in curves ͑a͒ and ͑b͒ are the

same. This conﬁrms the structures observed in k(E) and

shows that ␴(E)ϫP(E,v,J_C_O͒ is nearly constant over this

small range of energy just above threshold. At higher pho-

1

0

1

Ϫ1

tolysis energy, over 28 500 cm for CH CO, ͕1Ϫexp[k(E)

cm ; ͑B͒ Symmetry selection rules for the UV excitation

and nonradiative transitions; ͑C͒ Ab initio transition-state ge-

ometry and vibrational frequencies;³⁸͑D͒ RRKM equations;

and ͑E͒ RRKM ﬁts.

2

ϫ⌬t]͖ is still increasing with increasing energy while the

experimental curve starts to level off. This deviation indi-

cates a decrease in ␴(E)ϫP(E,v,J_C_O͒. A more striking dif-

ference in shape is exhibited by PHOFEX curve ͑c͒ in Fig. 8

for CO͑vϭ0, Jϭ2͒ which has a sharp peak near 28 500

A. Coupling of electronic states

Ϫ1

cm where the CO͑vϭ0, Jϭ12͒ PHOFEX curve starts to

Three electronic states of ketene are involved in the re-

level off. This peak and leveling off are consistent with a

signiﬁcant broadening of the product rotational distribution

at this energy. The width of the CO rotational distribution is

action dynamics, S , T , and S , Fig. 1. The ab initio cal-

1

0

3

8

culations of Allen and Schaefer give the zero-point levels

Ϫ1

of S and T at 21 195 and 19 150 cm above that for S₀

1

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129.170.65.78 On: Mon, 28 Oct 2013 05:59:40

3208

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

with an estimated uncertainty of 700 cm^Ϫ¹. The two excited

electronic states have nearly identical equilibrium

It appears far more likely that intersystem crossing is

faster than dissociation, especially in the threshold region. In

this case the wave functions for individual molecular eigen-

states are a statistical mixture of S , T , and S basis states

3

8

geometries with the CCO frame bent in plane at an angle of

29 deg. The ground state is very different with a linear CCO

frame that has considerably shorter bond lengths. At the

1

0

given by

Ϫ1

2

8 250 cm energy of the barrier to formation of triplet

products, the densities of vibrational states calculated using

͉

⌿_n

͘

ϭ

␣_n_i

͉

S₁_i

͘

ϩ

␤_n_j

͉

S₀_j

͘

ϩ

␥_n_k

͉

T₁_k

͘

,

͑3͒

^͚i

^͚j

͚

k

the Whitten–Rabinovitch approximation are ␳(S )Ϸ4,

1

4

Ϫ1 31

␳

(T )Ϸ26, and ␳(S )Ϸ1.4ϫ10 /cm

.

The sum of the

1

0

where the coefﬁcients ␣ , ␤ , and ␥_n_kof the individual

basis functions of each electronic state are related to the vi-

bronic level densities by

ni

nj

densities of states, ␳_t_o_t_a_l, is thus dominated by ␳(S ).

0

There are no direct measurements of rate constants or

matrix elements for couplings among these three electronic

states. However it is most likely that with respect to nano-

second time scale phenomena all three are strongly coupled

␳͑S1͒

͉

␣_n_i

͉

²Ϸ

,

^͚i

␳_t_o_t_a_l

2

7

together. For this to be so, it sufﬁces for two of the three

͑

4͒

couplings among the three states to be strong. The S ← S

␳ S

g ␳ T

1

0

͑

0͒

t ͑ 1͒

²Ϸ

͉

²Ϸ

.

3

9

͚_j^͉

^␤nj

͚

k

^␥nk

spectrum shows some possible vibrational structure with

^␳total

␳_t_o_t_a_l

Ϫ1

linewidths of about 200 cm suggesting an upper limit of

1

3

Ϫ1

4ϫ10

s

for the S decay rate. There is no rotational ﬁne

1

The eigenstates with the most S character are preferentially

1

structure or Q-branch feature observable; the latter would

excited by the UV laser pulse, since only S₁states carry

signiﬁcant oscillator strength. The photolysis pulses ͑0.4

Ϫ1

surely be seen unless the linewidth were at least 5 cm

.

12

Ϫ1

Thus S must decay with a rate of at least 10

s

. There-

1

cm linewidth; 7 ns duration͒ excite a set of about 6000

fore, the sum of the rates for internal conversion to S and

0

eigenstates with phase coherence among the S coefﬁcients

1

intersystem crossing to ^T1 must lie in the range

extending only over about ten adjacent states. Thus statistical

ﬂuctuations⁴⁰of the rate constants from eigenstate to eigen-

state should be completely averaged out. The dissociation

rate of ketene on the triplet surface is then given by the usual

transition state theory expression, Eq. ͑1͒, with

1

3

12

4

ϫ10 Ͼk ϩk Ͼ10 s^Ϫ¹. At least one coupling, rate of

ic isc

transfer between electronic states, is large. Could the other

two both be small?

If only k_i_s_cwere larger than 10¹²s^Ϫ¹, then the triplet

could be populated directly from S and decay without in-

␳(E,J)ϭ␳_t_o_t_a_l.

1

9

Ϫ1

volving S at a rate of at least 1/h␳(T )ϭ10 s at thresh-

0

1

B. Symmetry selection rules

old. Since the observed dissociation rate of the triplet at

6

Ϫ1

threshold is less than 4ϫ10 s , T and S must be coupled

1. Conserved quantities

1

0

9

Ϫ1

directly by kЈ or indirectly by k at greater than 10 s so

isc

ic

In the absence of external ﬁelds, total energy E_t_o_tand

that the rate of dissociation on the triplet surface at threshold

is controlled by the total density of vibronic levels, 1/h␳

total angular momentum (F,M ) are conserved throughout

F

.

the dissociation. Since the weak interaction is quite negli-

gible on the microsecond time scale, parity with respect to

inversion ͑⌸͒ is also conserved. On the time scale of photo-

total

12 Ϫ1

Thus if only k_i_s_cis larger than 10

s

, one or both of k and

ic

9

Ϫ1

kЈ must also be larger than 10 s

.

isc

If only k_i_cwere larger than 10 s^Ϫ¹, then T₁would

equilibrate with the two singlets at a rate given approxi-

mately by ͓␳(S )/g ␳(T )͔k ϩ kЈ where g is the num-

1

2

dissociation, the nuclear spin (I,M ) and nuclear spin sym-

I

27

metry ͑⌫ ͒ are known to be conserved. Therefore, the

ns

RRKM rate constant in Eq. ͑1͒ is more rigorously written as

1

t

1

isc

t

ber of triplet spin states strongly coupled. If the rate of dis-

sociation of a pure triplet state is faster than this equilibration

rate, then the rate of dissociation of ketene would be limited

by the intersystem crossing rate. The kinetic scheme, Fig. 1,

W͑E_t_o_t,J,I,⌸,⌫_n_s͒

k͑E_t_o_t,J,I,⌸,⌫ ͒ϭ

,

͑5͒

͑the sum of the UV photon

thermal

ns

^h^␳^͑^Etot ^,^J^,^I^,^⌸^,^⌫ns^͒

where FϭIϩJ, E ϭEϩE

and thermal excitation in the jet͒, ⌸ and ⌫_n_sare for the

ketene molecule following UV excitation.

tot

^gⁱ^v^e^s^a^r^a^t^e^c^oⁿ^s^t^aⁿ^t^o^f^kt

ϭ

͓(g ␳(T )/␳(S )͔kЈ

t 1 0

isc

ϩ ͓(␳(S )/␳(S )͔k for this case. The observed dissociation

1

0

isc

6

Ϫ1

rate, Fig. 6, increases strongly with energy from 4ϫ10 s

7

Ϫ1

2

. Angular momentum selection rules

near the threshold for triplet dissociation, to 2ϫ10 s just

2

50 cm higher in energy, to 3ϫ10 s 1866 cm higher²⁹

Ϫ1

8

Ϫ1

The photodissociation of ketene on the triplet surface

at the singlet threshold. Since S and T have similar geom-

etries and origins, the Franck–Condon factors for k_i_s_cand

therefore k_i_s_citself cannot depend strongly on energy. The

involves UV optical excitation (S ←S ), internal conversion

1

0

(S ↔S ), and intersystem crossing (T ↔S ,S ) as well as

0

1

0

1

dissociation. The symmetry species for ketene in any elec-

second term in the expression for k cannot be responsible

tronic state are classiﬁed in the C₂_vmolecular symmetry

t

for the data in Fig. 6. Since T and S have very different

͑MS͒ group which is isomorphic with the G CNPI group.

1

0

4

geometries, a strong dependence of kЈ on energy is likely.

This is a sensible approach even for the strongly bent

transition-state geometry since its torsional barrier is so

low³⁸that operationally the overall symmetry of the total

Hamiltonian is C . Ketene is a slightly asymmetric prolate

isc

However, at the high levels of vibrational excitation in these

two states, it is very unlikely indeed that the regular stepped

increases in rate in Fig. 6 result from vibronic structure in

2

v

kЈ .

top. The magnitude of the projection of rotational angular

isc

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129.170.65.78 On: Mon, 28 Oct 2013 05:59:40

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3209

momentum N on the ‘‘a’’ axis, ͉K͉, can be a nearly good

⌬

SϭϮ1; ⌬Jϭ0; ⌬Nϭ0,Ϯ1;

quantum number and thus selection or propensity rules for K

are given. The selection rules for angular momentum quan-

tum numbers and wave function symmetries for each process

follow.

͑

10͒

KϭKϪKЈϭ0,Ϯ1,Ϯ2; ϩ↔ϩ,Ϫ↔Ϫ.

These selection rules and corresponding symmetries ap-

ply equally for motion through the transition state. Since the

coupling mechanism͑s͒ for intersystem crossing is not

known, the number of spin states g paired with each vibra-

tional level at the transition state is not known ͑g ϭ1, 2, or

(

i) UV excitation (S ←S )

1

0

1

The optical transition from S ( A ) to S ( A ) is elec-

0

1

2

t

tronically forbidden but vibronically allowed for electric di-

t

pole radiation. From A vibronic symmetry in the S ground

1

0

3

͒.

state, A-type, parallel transition gives rise to a vibrational

2

states in S . B-type and C-type perpendicular transitions ex-

1

cite b and b vibrational states, respectively. The selection

1

2

3

. Nuclear spin statistics

rules for total angular momentum and parity are

Fϭ0͑FÞ0͒,Ϯ1; ϩ↔Ϫ.

With nuclear spin conserved, the selection rules become

Jϭ0͑JÞ0͒,Ϯ1; ⌬Iϭ0; ϩ↔Ϫ.

Nuclear spin statistics have important experimental con-

⌬

sequences for ketene. In CH CO, hydrogen atoms have a

nuclear spin of 1/2. Four nuclear spin wave functions gener-

ate the representation, ⌫ ϭ3A ꢀ B in C , where the three

A states correspond to Iϭ1 ͑ortho͒ nuclear spin states and

2

⌬

͑6͒

ns

1

2

2v

1

The selection rules for the K quantum number in the sym-

metric top limit are

the B state is the Iϭ0 ͑para͒ state. Since hydrogen nuclei

2

are fermions, symmetry operations which exchange them

must change the sign of the wave function. Thus the symme-

try class of the total wave function for the eigenstate, ⌫, must

be either B or B . Therefore, since the vibronic symmetry

⌬

KϭKЈϪKЉϭ0 ͑A-type, parallel͒;

͑7͒

KϭKЈϪKЉϭϮ1 ͑B-, C-types, perpendicular͒.

1

2

of CH CO in the ground state is A , only b ͑KЉ ϭ odd,

2

1

a

(

ii) Nonradiative transitions

KЉ ϭ odd͒ or b ͑KЉ ϭ odd, KЉ ϭ even͒ rotational states

c

2

a

c

The symmetry of the wave function and the conserved

quantities in Sec. IV B 1 are unchanged in nonradiative tran-

sitions. Thus internal conversion from A , B , and B vi-

bronic symmetries in S gives rise to a , b , and b vibra-

tional states in S with the selection rules

can combine with ortho (Iϭ1) states while a₁͑K_aЉ

ϭ even, KЉ ϭ even͒ or a ͑KЉ ϭ even, KЉ ϭ odd͒ rota-

c

2

a

c

1

2

1

tional states combine with para (Iϭ0) state.

1

2

1

For the case of CD CO, deuterons have a spin of 1 and

2

0

each deuteron can have three different nuclear spin wave

functions. Hence nine nuclear spin wave functions are gen-

⌬

Sϭ0; ⌬Jϭ0; ⌬Iϭ0;

erated for CD CO and give the representation,

͑8͒

2

KϭKϪKЈϭ0; ϩ↔ϩ,Ϫ↔Ϫ.

⌫_n_sϭ6A ꢀ 3B , where ﬁve A states correspond to the quin-

1

2

1

tet (Iϭ2) functions, one A state is the singlet (Iϭ0) func-

In the highly vibrationally excited S₀states of ketene

resulting from internal conversion, strong anharmonic and

Coriolis couplings are expected. In the case of strong

vibration–rotation coupling, K is not a good or nearly good

quantum number. In the extreme case, K may be completely

mixed thus giving molecular states with equal contributions

from basis states with K from ϪJ to J. This is the limit

usually assumed in applications of RRKM theory. The extent

of K mixing has not previously been established experimen-

tally for ketene; it will be seen below that K appears to be a

good, or nearly good, quantum number.

1

^tⁱ^oⁿ^,^aⁿ^d^t^h^r^e^e^B2 states are the triplet (Iϭ1) functions.

Since deuterons are bosons, ⌫ must be A or A . Therefore,

1

2

in the ground state CD CO ͑A vibronic symmetry͒, only a

2

1

͑

KЉ ϭ even, KЉ ϭ even͒ or a ͑KЉ ϭ even, KЉ ϭ odd͒

a c a c

2

rotational states combine with ortho (Iϭ2,0) states, while

^b1 ͑KЉ ϭ odd, KЉ ϭ ^o^d^d^͒^o^r^b2 ͑KЉ ϭ odd,

a

c

a

KЉ ϭ even͒ rotational states can combine with para (Iϭ1)

c

states. For CD CO, the ortho (Iϭ0,2) states combine with

2

KЉϭeven and the para (Iϭ1) states combine with KЉϭodd,

a

opposite to the case for CH CO. Nuclear spin statistical

2

weights for CH CO are 3 for ortho and 1 for para, while for

The three electron-spin wave functions for T each have

a different symmetry, ⌫ : A , B , and B . When ﬁrst order

2

1

CD CO they are 6 for ortho and 3 for para.

2

es

2

1

2

It should be noted that the nuclear spin wave function

and statistical weight are unchanged by UV excitation and/or

radiationless transition. Thus for states reached by perpen-

dicular dipole (b ,b ) transitions from the ground vibronic

spin–orbit coupling is the dominant mechanism in intersys-

tem crossing, the angular momentum selection rules are

given by, using the notation in Hund’s case ͑b͒, by

1

2

⌬

SϭϮ1; ⌬Jϭ0; ⌬Nϭ0,Ϯ1;

state, K even and odd are reversed ͑e.g., for CH CO S K

a

2

1

a

͑9͒

even corresponds to ortho͒. This is especially important near

the thresholds for rovibrational energy levels at the transition

state. Nuclear spin is not cooled in the jet expansion and so

the degeneracies, relative populations at room temperature,

control the weights of ortho and para species.

As the molecule passes through the transition state to

products, the symmetry of the wave function and the other

conserved quantities established during excitation are un-

KϭKϪKЈϭ0; ϩ↔ϩ,Ϫ↔Ϫ.

This case is possible for S ↔T (⌫ ϭA ) only. It is likely

0

1

es

2

that second or higher order intersystem crossing through

spin–orbit vibronic coupling or spin–orbit rotational cou-

pling is important. Here all three triplet spin functions may

be allowed and the corresponding selection rules are given

4

1

by

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129.170.65.78 On: Mon, 28 Oct 2013 05:59:40

3210

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

Ϫ1

II

TABLE I. The scaled ab initio vibrational frequencies for the C_stransition

H–C–C–O torsion, 252 cm for C–C–O bending, and 366

Ϫ1

states of CH CO and CD CO.

cm for CH wagging for the CH CO transition state. The

distinct steps in k(E) are observed in the ﬁrst 200–300 cm

above threshold, and are most likely to be associated with

these low-frequency vibrational modes.

2

Ϫ1

a

CD CO^b

CH CO

2

‡

1

‡

2

‡

3

‡

4

‡

5

‡

6

‡

7

‡

8

‡

9

␯

(aЈ) asym C–H͑D͒ stretch

(aЈ) sym C–H͑D͒ stretch

(aЈ) C–O stretch

3178

2997

2029

1183

472

523i

252

366

2380 ͑2573͒

2150 ͑2314͒

2029 ͑2153͒

874 ͑930͒

383 ͑406͒

514i (550i)

224 ͑232͒

(aЈ) CH͑D͒ scissor

2

2. Hindered internal rotation

(aЈ) CH͑D͒ rock

2

(aЈ) C–C stretch

(aЈ) C–C–O bend

The torsional degree of freedom cannot be treated as an

isolated harmonic vibration because the top of the barrier to

internal rotation about the C–C bond is of the same order as

the energy range of interest. The ab initio values for the

(aЉ) CH͑D͒ wag

276 ͑283͒

126 ͑127͒

2

(aЉ) H͑D͒–C–C–O torsion

154

a

barrier to internal rotation are 436 ͑CISD level͒ or 384

Taken from Ref. 38.

Frequencies in parentheses are unscaled vibrational frequencies calculated

by normal-mode analysis using ab initio force constants of CH CO. The

same scale factors as used for CH CO are multiplied to give the values

b

Ϫ1 38

with Davidson correction cm

.

Furthermore, the internal

2

rotation and the solid body rotation about the same axis are

signiﬁcantly coupled. Thus the Schr o¨ dinger’s equation for at

least these two coupled degrees of freedom must be treated

in detail.

2

without parentheses.

changed. Furthermore, some quantities not rigorously con-

served may be approximate constants of the motion or nearly

good quantum numbers.

The potential energy for the internal rotation is approxi-

mated as a cosine series⁴²

V͑␪͒ϭ͑V /2͓͒1Ϫcos͑2␪͔͒ϩ͑V /2͓͒1Ϫcos͑4␪͔͒ϩ••,

0

1

͑

11͒

C. Ab initio transition state structure

1

. Structure and vibrational frequencies

where V0 is the barrier height, V1 changes the shape of the

potential function, and ␪ is the dihedral angle between the

An ab initio calculation has been performed by Allen

II

3

8

II

3

CH and CCO planes. The ab initio C transition-state struc-

2

s

and Schaefer, III

for the (C ) in-plane bent AЉ

s

3

1

ϩ

I

s

ture is used for the evaluation of moments of inertia for both

CH2CO and CD2CO. The moment of inertia of the CH2

CH CO→CH ( B )ϩCO ͑ ⌺ ͒ and the (C ) out-of-plane

2

1

3

1 ϩ

bent AЈ CH CO→CH ( B )ϩCO ͑ ⌺ ͒ dissociation paths.

2

1

2

II

s

3

‘‘top’’ about the C–C bond, Itop , is 1.88 amu Å while that of

The C stationary point for AЉ ketene dissociation has been

2

I

s

the CO ‘‘frame’’, Iframe , is 16.9 amu Å for CH2CO. For this

found to be a true transition state. The C stationary point for

3

case, where IframeϾItop , it is convenient to use the principal

axes of the whole molecule as the coordinate system.⁴³For

an asymmetric top with a single twofold internal rotor, the

AЈ ketene dissociation is the potential energy maximum

along the torsional or internal rotational coordinate of the

C transition state. The barrier to internal rotation is calcu-

lated to be 384 cm and is expected to be signiﬁcantly

less.

The ab initio ͑DZP CISD, double-zeta basis with

polarization–conﬁguration interaction with singles and

doubles͒ values for transition-state vibrational frequencies,

which have been scaled in Ref. 38 by multiplying by the

ratio of experimental to ab initio values for ground state

ketene vibrational frequencies, are listed in Table I. Vibra-

II

s

44,45

Ϫ1

Hamiltonian can be written as

3

8

2

HϭH ϩF͑pϪP͒ ϩV͑␪͒.

͑12͒

rot

Here, H_r_o_tis the standard rigid-rotor asymmetric-top rota-

tional Hamiltonian and F is the effective rotational constant

for the internal rotation of the top about its symmetry axis

h²

␭gItop

Fϭ

;

rϭ1Ϫ

,

gϭ͑a,b,c͒.

͑13͒

2

͚

g

8

␲ rI

I_g

top

II

s

tional frequencies for the C transition state of CD CO are

2

calculated by normal-mode analysis using the ab initio force

Here, I_t_o_pis the moment of inertia of the top about the sym-

metry axis, (I ,I ,I ) are the principal moments of inertia of

constants of CH CO, scaled as for CH CO, and listed in

2

a

b

c

Table I. The corresponding zero-point energies are given in

the whole molecule, and (␭ ,␭ ,␭ ) are the direction co-

a b c

Table II. The scaled harmonic vibrational frequencies ob-

sines of the symmetry axis of the top to the principal axes.

Ϫ1

tained from quadratic force constants are 154 cm

for

Since there is a plane of symmetry, ␭ vanishes. The mo-

c

menta P and p are deﬁned as

_T_A_B_L_E_I_I_._T_h_e_z_e_r_o_-_p_o_i_n_t_v_i_b_r_a_t_i_o_n_a_l_e_n_e_r_g_i_e_s_o_f_k_e_t_e_n_e_a_n_d_m_e_t_h_y_l_e_n_e_.a

ץ

T

ץ

T

ˆ

Pϭ

P i; P ϭ

,

͑iϭa,b,c͒; pϭ

, ͑14͒

^͚i

i

˙

ץ␻

_i

ץ␪

CH CO

CD CO

2

1

b

S ( A )

6712

5295

3495

5556

4204

2609

where Tϭ(HϪV) and ␻ is the angular velocity about axis

0

1

i

c

TS

i. In the second term in Eq. ͑12͒, (pϪP) represents the

relative angular momentum of the top and the frame. The

Hamiltonian in Eq. ͑12͒ can be rewritten as

3

d

CH͑D͒ fragment

2

a

The unit is cm^Ϫ¹

.

b

c

Calculated from the values in Ref. 51.

Calculated from the scaled ab initio vibrational frequencies for ␯ Ϫ␯ ; ␯

2

‡Ј

2

‡

2

‡

1

‡

8

‡

9

‡Ј

HϭA P ϩB P ϩC P Ϫ2F͑␣P pϩ␤P p͒

a b c

a b

is from Table IV.

d

2

Calculated from the values in Ref. 56.

ϩF␣␤͑P P ϩP P ͒ϩFp ϩV͑␪͒,

͑15͒

a

b

a

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Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3211

TABLE III. Molecular parameters used in the calculation of hindered inter-

nal rotation energy levels of the transition state.

TABLE IV. The energy and nuclear spin of hindered internal rotor states for

different K^‡for the CH CO and CD CO transition states.

a

2

Parameters^a

CH CO

CD CO

CH CO

CD CO

2

‡

I_a

I_b

I_c

␭_a

␭_b

F

6.74

62.9

69.6

0.95

0.31

12.1

1.88

8.59

71.0

79.6

0.95

0.31

7.56

3.76

3.26

m

K ϭ0

K ϭ1

K ϭ2

K ϭ0

K ϭ1

K ϭ2

0

͑0͒

͑99͒

103

164

͑189͒

͑209͒

276

͑Ϫ1͒

Ϫ1

99

͑102͒

͑166͒

181

Ϫ2

͑Ϫ2͒

͑99͒

99

͑0͒

0

84

͑84͒

͑149͒

155

Ϫ1

͑Ϫ1͒

͑83͒

83

͑Ϫ3͒

Ϫ3

81

͑81͒

͑146͒

151

Ϫ1

ϩ1

Ϫ2

ϩ2

Ϫ3

ϩ3

Ϫ4

ϩ4

Ϫ5

ϩ5

Ϫ6

ϩ6

170

150

I_t_o_p

_A‡

͑172͒

͑231͒

236

͑151͒

͑195͒

201

Ј

3.35

218

190

188

_B‡

Ј

0.268

0.237

0.212

͑253͒

͑301͒

351

414

͑476͒

͑552͒

͑216͒

͑225͒

283

284

͑363͒

͑209͒

͑228͒

268

292

͑345͒

͑375͒

‡

C

0.242

278

321

244

͑383͒

514

͑327͒

͑439͒

447

͑254͒

͑313͒

327

a

2

Ϫ1

‡

The units are amu Å for I ͑xϭa,b,c, top͒ and cm for F, A , B , and C

x

͑

see Sec. IV C 2 for deﬁnitions͒. The ab initio ͑DZP CISD level calcula-

II

tion͒ C_stransition-state structure is used for both CH CO and CD CO ͑Ref.

2

514

583

400

3

8͒.

^aThe energy levels are in cm^Ϫ¹for V ϭ240, V ϭ20 cm for both CH CO

Ϫ1

0

1

2

and CD CO. The energy values in parentheses are ortho states and those

2

without parentheses are para states. The symmetric rotor energy term val-

ues for (N,K ) states are not included. The energy values listed in each

‡

where

column are those with respect to the corresponding zero-point energy levels

‡

Ј

‡

2

‡Ј

‡

2

‡

Ϫ1

A ϭA ϩF␣ , B ϭB ϩF␤ ,

for K ϭ0, 57 and 46 cm for CH CO and CD CO, respectively.

2 2

͑16͒

^␭aI_t_o_p

␭ I

b

top

␣

ϭ

,

␤ϭ

V ϭ240 cm and V ϭ20 cm^Ϫ¹. The hindered rotor en-

Ϫ1

I_a

I_b

0 1

‡

_a_n_d‡ denotes the transition-state geometry.

The Hamiltonian matrix can be set up using the

symmetric-prolate-top, free-rotor basis set. The nonvanishing

elements of the Hamiltonian matrix are given in Ref. 45. The

ergy levels coupled to each K are listed with respect to the

zero-point energy at K ϭ0. The shift of the zero-point ener-

gies with increasing K due to the coupling of the hindered

internal rotation to the overall rotation is less than 4 cm , as

‡

Ϫ1

‡

shown in Table IV.

matrix elements off-diagonal in K are very small for the

nearly symmetric-top ketene molecule at its transition state

‡

Ј

‡

Ϫ1

D. RRKM theory

͑

␤Ϸ0.0092, (B Ϫ C ) Ϸ 0.026 cm for CH CO͒, and

2

‡

K splitting is not expected to be observed with the resolu-

The RRKM rate constant for unimolecular reaction of

tion of features in this work. Thus the matrix elements off-

diagonal in K are neglected in the calculation. The matrix

ketene for the case of complete K mixing ͑KϭϪJ to J͒ in

the highly excited molecule can be calculated as follows:

‡

46

elements

͉m

͘

in

the

free-rotor

basis

set,

k͓E,J←͑JЉ,KЉ͒,⌫ ͔

_ϭ₍₂_␲₎Ϫ1/2 exp(Ϫim␪) are thus

ns

‡

N

‡

2

‡

͗

m

͉

H

͉

m

͘

ϭFm Ϫ2F␣K mϩ͑V /2͒ϩ͑V /2͒,

͑17͒

͑18͒

^gt^͚

K ϭϪN

‡

͚ P͑EϩEthermal^Ϫ^␧

n

‡

͒

n,N,K ,⌫

ns

0

1

ϭ

,

͑20͒

͑21͒

J

͚

^h^␳^͑^E^ϩ^Ethermal^Ϫ^WJ,K,⌫ns^͒

mϮ2

mϮ4

͘

ϭϪV /4,

KϭϪJ

0

ϭϪV /4.

͑19͒

‡

sϪ1

1

‡

^h^␯^‡i

␧

‡

ϭE ϩW

‡

ϩ

ͩ

ⁿi^ϩ

ͪ

,

Here, the diagonal matrix element term of (Ϫ2F␣K m)

n,N ,K ,⌫

th

N ,K

͚

ns

2

iϭ1

arises from the coupling of internal rotation and overall ro-

tation. The effective rotational constants about the principal

axes have been modiﬁed due to the effect of internal rotation.

The energy levels for hindered internal rotation for each

value of K are calculated by diagonalizing the above Hamil-

tonian matrix. The rovibrational energy levels are calculated

by adding the symmetric top approximation for the rotational

‡

1

2

‡

‡2

‡

‡2

W

‡

ϭ ͑B ϩC ͓͒N ͑N ϩ1͒ϪK ͔ϩA K ,

͑22͒

͑23͒

͑24͒

N ,K

1

2

W_J_,_K

ϭ

͑

͓ ͑

BϩC͒ J Jϩ1͒ϪK ϩAK ,

͔

‡

1

2

^Ethermalϭ ͑BϩC͓͒JЉ͑JЉϩ1͒ϪKЉ ͔ϩAKЉ .

Here rotational energies are approximated by those of a sym-

metric top with K replacing K for ketene. Since even and

‡

Ј

‡Ј

‡

energies using (A ,B ,C ) as rotational constants. A

0ϫ40 Hamiltonian matrix is diagonalized to give the eigen-

a

4

odd parity states are almost equally populated in reactants

and transition states,³⁴states with both parities are counted in

both the numerator and denominator of Eq. ͑5͒. In Eq. ͑20͒,

rovibronic states which are symmetry allowed with a nuclear

spin state represented by ⌫_n_sare counted in both numerator

and denominator of Eq. ͑5͒. E is the photolysis photon en-

ergy and E_t_h_e_r_m_a_lis the thermal rotational energy of the parent

molecule in the pulsed molecular jet for the (JЉ,KЉ) level. J

values. All parameters used in the calculation are listed in

Table III.

Energy levels are calculated numerically for a wide

range of V and V for comparison to the thresholds in the

0

1

measured rate constants. Table IV shows energy levels with

nuclear spin statistical weights for hindered rotor states for

‡

K ϭ0, 1, 2 for CH CO and CD CO transition states with

2

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3212

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

TABLE V. The parameters used in the RRKM ﬁts to experimental results.^a

Quantum mechanically, P(E ) can be calculated for one-

1

dimensional tunneling through the reaction barrier along the

CH CO

CD CO

2

reaction coordinate. Formulas for P(E ) ͑Ref. 46͒ for in-

1

48,49

‡

₂_.₅₀_c_mϪ1

₁_.₉₆_c_mϪ1

verted parabolic or Eckart barriers

give about the same

A

B

C

results.³¹The inverted parabolic function is used here.

The density of reactant states, ␳(E)ϭf␳_W_R(E), is taken

to be a constant f multiplied by the harmonic oscillator den-

sity calculated using the Whitten–Rabinovitch ͑WR͒

Ϫ1

0.268 cm

0.242 cm

4 K

0.237 cm

0.212 cm

4 K

Ϫ1

T_r_o_t

E_t_h

Ϫ1

28 250 ͑10͒ cm

28 310 ͑15͒ cm

b

₁₀₀₍₄₀₎_i_c_mϪ1

50

^␯im

•••

approximation

f/g_t^c

1.11 ͑0.11͒

1.19 ͑0.07͒

Ϫ1

V₀

240 ͑30͒ cm

20 ͑20͒ cm

Ϫ1

V

20 ͑20͒ cm

1

‡

sϪ1

͑

^E^ϩ^Ezp^͒

Ϫ1

d

Ϫ1

␯

͑C–C–O bend͒

250 ͑15͒ cm

185 ͑20͒ cm

240 ͑20͒ cm

7

‡

8

␳_W_R͑E,Jϭ0͒ϭ

.

͑26͒

s

Ϫ1

͑CH͑D͒ wag͒

290 ͑15͒ cm

͑sϪ1͒!⌸

iϭ1

␯_i

2

a

The values in parentheses are uncertainties obtained from the ﬁts.

b

The imaginary frequency ͑␯ ͒ is used only for the ﬁt to the ﬁrst step in

im

^H^e^r^e^Ezp is the zero-point energy of the ketene molecule and

k(E) for CH CO dissociation.

2

c

␳

(E)ϭ␳_W_R(E)ϫf; ␳(E) is used for the RRKM ﬁts. The ␳_W_R(E) is the

␯ is the ith fundamental vibrational frequency of S ketene

i

0

density of vibrational states of the reactant calculated from the Whitten–

taken from Ref. 51. The contributions of ␳(S ) and ␳(T ) to

4

Ϫ1

1

Rabinovitch approximation; ␳ (E)ϭ1.36ϫ10 ͑/cm ͒ for CH CO(S )

WR

2

0

4

Ϫ1

␳_t_o_t_a_lare less than 1% and are neglected. In Eq. ͑20͒, the

density of vibrational states for each nuclear spin state,

and 4.78ϫ10 ͑/cm ͒ for CD CO(S ) at Eϭ28 500 cm

.

2

0

d

Assignments are tentative ͑see the text͒.

␳

͑E,⌫ ͒, is approximated as half of the total vibrational

ns

states, ␳͑E,⌫ ͒ϭ[␳(E)/2] ͑Refs. 27 and 52͒ and the sum

ns

over K is made over all 2Jϩ1 values.

is the angular momentum of the excited state of ketene and

^WJ,K its rotational energy. N is the rotational angular mo-

An experimental CO (v,J) product rise curve is the sum

of CO product rise curves for different initial quantum states

of ground state ketene and the different optical transitions for

each. The measured rate constant, k_o_b_s, is obtained from a

single exponential ﬁt to the experimental CO product rise

curve which is in reality a multiexponential one. This is

equivalent to the mean reaction time rate constant,³¹and has

been found to give an accurate value for the true average rate

constant.⁵

‡

mentum of ketene at the transition state on the T surface

1

‡

and W

‡

^t^h^e^r^o^t^a^tⁱ^oⁿ^a^l^eⁿ^e^r^g^y^.^Eth is the reaction thresh-

N ,K

old, the difference between zero-point energies of the ground

state reactant (S ) and transition state on the T surface. The

0

1

number of vibrational degrees of freedom of the stable mol-

ecule, s, is 9 for ketene and n represents the set of eight

vibrational quantum numbers for the transition state. The

electron spin multiplicity is approximated by multiplying the

3,54

The population of a (J,KЈ) level in the S state is deter-

1

number of vibrational states of the transition state by g . This

¨

t

mined by the product of the Honl–London factor and the

initial population of (JЉ,KЉ) states of ketene in the ground

state prepared in the supersonic jet. Nuclear spin is found not

to be cooled down in the molecular jet.²⁷Initial (JЉ,KЉ)

states of ketene are otherwise populated according to a Bolt-

zmann distribution for a rotational temperature of 4 K. When

K is assumed to be completely mixed for highly vibra-

‡

approximation is made by setting N ϭJ rather than sum-

ming N over J Ϫ1, J , and J ϩ1. In the case where all

‡

three electron spin states are strongly coupled g ϭ3.

t

Ketene in the ground state is a near-prolate asymmetric

4

7

top ͑␬ϭϪ0.997͒. The symmetric top approximation is used

for the calculation of rotational energy levels for both reac-

‡

tants and transition state. The K and K in the symmetric top

tionally excited S , as Eq. ͑20͒ implies, the population of K

0

approximation are the projections of the rotational angular

momentum, N ͑JϭNϩS͒, on the molecule-ﬁxed a axis for

the molecule in the excited reactant state and the transition

state, respectively. The rotational constants of UV-excited

ketene molecules are assumed to be the same as those

states is determined by J. When K is assumed to be con-

served, its value at the transition state depends on which

couplings are responsible for intersystem crossing. Since the

coupling strengths are not known, calculations were done for

‡

one of the least restrictive possibilities, K ϭKЈ, KЈϮ2, and

‡

(

A,B,C) in the ground state. A , B , and C are the rota-

KЈϮ4 where KЈ is taken as KЉϮ1. Calculations made with

more restrictive selection rules were not sensibly different.

The angular momentum (J) distribution of the excited

reactant states from parallel ͑A-type͒ or perpendicular ͑B-,

C-type͒ transitions are identical within one unit of angular

momentum, and therefore RRKM ﬁts are not sensitive to the

type of transition. The line strengths for a perpendicular tran-

sition are used. The rate constants are calculated for each

angular momentum (J), and averaged over the distribution

of JϭJЉ, JЉϮ1 in the reactant states. The rate constants for

each nuclear spin state, thermally populated JЉ,KЉ state, and

allowed value of J are calculated separately and averaged

using the proper nuclear spin statistical weights as follows:

tional constants for the transition state as calculated from the

ab initio structure, Table V.³⁸These rotational constants for

the transition state are modiﬁed due to the strong coupling of

the hindered internal rotation to the overall rotation of the

molecule ͑See Sec. IV C 2͒.

P(E ) is the transmission probability for each quantized

1

reaction threshold of the transition state. Classically, the

^t^r^aⁿ^s^mⁱ^s^sⁱ^oⁿ^p^r^o^b^a^bⁱ^lⁱ^t^yⁱ^s^t^h^e^s^t^e^p^f^uⁿ^c^tⁱ^oⁿ^,^h⁽^E1⁾

h͑E ͒ϭ0, E Ͻ0,

1

͑25͒

ϭ1, E Ͼ0.

1

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129.170.65.78 On: Mon, 28 Oct 2013 05:59:40

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3213

FIG. 11. Rate constants ͑ﬁlled circles͒ and CO͑vϭ0, Jϭ12͒ PHOFEX

FIG. 12. The CO͑vϭ0, Jϭ12͒ PHOFEX curve taken at 1.7 ␮s reaction

curve taken at 50 ns delay time are shown with RRKM ﬁts for CH CO

2

time is shown with RRKM ﬁts for the CD CO dissociation in the reaction

2

dissociation in the reaction threshold region. The solid line is the RRKM ﬁt

threshold region. The solid line is the RRKM ﬁt when the threshold energy

Ϫ1

when the threshold energy is 28 250 cm and ␯ ϭ100i cm . The RRKM

Ϫ1

im

is 28 310 cm without tunneling correction. The dashed line is the RRKM

Ϫ1

ﬁts for ␯ ϭ60i cm ͑long dashes͒ and ␯ ϭ140i cm ͑short dashes͒ are

Ϫ1

im

ﬁt when ␯ ϭ60i cm .

im

shown for the same threshold energy. The PHOFEX intensity is arbitrarily

scaled for comparison with the rate constants.

Ϫ1 27

E ͑singlet͒ϭ30 116.2Ϯ0.4 cm

.

Therefore, the reaction

th

barrier relative to the products, E is determined by

b

k͑E͒ϭ

P͑⌫_n_s͒

_͚N͑JЉ,KЉ,⌫_n_s͒

͚

⌫

JЉ,KЉ

ns

͕

E ϭ͓⌬ Ϫ E ͑singlet͒

th

b

ST

Ϫ1

ϪE ͑triplet͖͔͒ϭ1281Ϯ15 cm

͑28͒

ϫ

P͓J←͑JЉ,KЉ͒,⌫ ͔k͓E,J←͑JЉ,KЉ͒,⌫ ͔.

th

͚

ns

J

or 3.66Ϯ0.05 kcal/mol. This is about 40% less than the ab

initio calculation of 5.8 kcal/mol.³⁸

͑

27͒

Here P͑⌫ ͒ is the statistical weight for ⌫ normalized to

unity, N(JЉ,KЉ,⌫ ͒ is the thermal rotational distribution of

ketene for ⌫_n_s͑zero for either even or odd KЉ͒, and

P[J←(JЉ,KЉ),⌫ ͔ is the probability for a transition to J

ns

For CD CO dissociation, k(E) could not be measured in

2

ns

the reaction threshold region ͑vide supra͒; however, the ﬁrst

distinct step in the PHOFEX spectrum for CO͑vϭ0, Jϭ12͒,

Fig. 12, gives the reaction threshold energy as 28 310Ϯ15

ns

from an initial state (JЉ,KЉ) for each ⌫ state.

Ϫ1

ns

cm . The solid line in Fig. 12 is the RRKM rate constant

calculated without tunneling. The zero-point energies of the

normal and deuterated ketene and triplet methylene mol-

E. RRKM ﬁts

ecules are calculated using the spectroscopic and theoretical

The RRKM formalism above may be used to extract

information on transition-state energy levels and structure

and on the density of reactant states from the steplike struc-

tures in the rate constants and PHOFEX spectra. Since there

are many more distinguishable features in the data than ﬁt-

ting parameters, the comparison also tests the validity of the

model.

results, respectively, and listed in Table II.⁵

1,56

The barrier

height relative to products, EЈ , is given by

b

3

EЈϭE ϩZPE͑ CH – CD )

b

2

Ϫ͓E ϩZPE͑CH CO–CD CO͒ϪEЈ ͔

th

2

th

ϭ1071Ϯ40 cm^Ϫ¹

.

͑29͒

1. Reaction threshold

3

For CH CO dissociation, the ﬁrst sharp step in k(E)

accurately deﬁnes the reaction threshold energy. It is repro-

Here ZPE͑ CH – CD ͒ is the difference of zero-point ener-

2 2

2

3

Ϫ1 56

gies between CH and CD fragments, 886 cm

,

and

2

duced well by RRKM calculations for a threshold energy,

ZPE͑CH CO–CD CO͒ is the difference of zero-point ener-

2

Ϫ1

E ϭ28 250Ϯ10 cm

and an imaginary frequency,

gies between the ground states of CH CO and CD CO, 1156

th

2 2

The reaction barrier is thus 1071Ϯ40 cm ͑3.06

␯_i_mϭ͑100Ϯ40͒i cm^Ϫ¹, Fig. 11. The singlet-triplet splitting of

cm

Ϫ1 51

.

Ϫ1

Ϫ1 55

CH , ⌬ , has been measured to be 3147Ϯ5 cm

.

The

Ϯ0.10 kcal/mol͒. The difference in ZPE’s between CH CO

2

ST

2

singlet threshold energy for CH CO dissociation has been

and CD CO at their T₁transition states is ͓E_t_h

Ϫ EЈ ϩZPE͑CH CO–CD CO)

͖

] ϭ 1096 Ϯ 20 cm , Fig.

th

2 2

2

1

Ϫ1

accurately measured from

CH₂PHOFEX spectra,

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129.170.65.78 On: Mon, 28 Oct 2013 05:59:40

3214

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

positions were not strongly sensitive to V₁in the range

Ϫ1

2

0Ϯ20 cm and no improved ﬁt could be found for values

outside of this range. The ﬁt to the position and amplitude of

higher levels, Fig. 6, indicates that the basic form of the

potential and the symmetries and degeneracies of the levels

are correct. Some caution must be exercised since the fea-

Ϫ1

tures observed at 28 390 and 28 450 cm correspond to en-

ergy levels calculated for mϭϩ2 and Ϫ3 at 28 407 and

Ϫ1

2

8 432 cm . The splitting between these states just at the

top of the barrier to internal rotation should be particularly

sensitive to the barrier shape and may therefore be incor-

rectly predicted. Since the nuclear spin state symmetry is

conserved, the threshold photon energies for para states of

CH CO at the transition state are those shown in Table IV

2

Ϫ1

and for ortho states are lower by Ϸ9.7 cm . For CD CO,

2

since ortho states (KЉϭ0) lie below para states (KЉϭ1) in

the ground state, threshold energies for para rovibrational

Ϫ1

states should be Ϸ5.0 cm ͑energy difference between 1₁₀

and 0₀₀states͒ lower than the energy term values listed in

Table IV, while those for ortho states are as listed. The ex-

cellent ﬁt to CD CO, Fig. 7, using the potential for CH CO

indicates that the vibration is correctly identiﬁed and that the

reduced mass of the motion is correctly formulated.

FIG. 13. Energy diagram for the dissociation of CH CO and CD CO on the

triplet surface. The energy differences ͑cm ͒ among the zero-point energy

levels are shown ͑uncertainties in text͒.

2

Ϫ1

2

13. The ab initio difference using Table IV for torsion and

harmonic, scaled values from Table I for the remaining seven

3. Bending vibrations of the transition state

Ϫ1

frequencies is 1091 cm

.

The ab initio frequencies of the three bending vibrations

which become free rotations in the products make them all

candidates for inﬂuencing k(E) in the range 200–400 cm

2. Hindered internal rotor levels at the transition state

Ϫ1

The step structures in the measured k(E) and J_C_Oϭ12

above threshold. Unfortunately, the features in k(E) are

much less distinct in this energy range than closer to thresh-

old. Fortunately, PHOFEX spectroscopy provides a means to

search for molecules coming through the transition state with

speciﬁc vibrations excited. This is because the product en-

ergy state distribution is determined by the positions and

momenta of each atom as it passes through the transition

state as well as by the potential in the exit valley.³⁰That

PHOFEX spectra exhibit the same shapes in the ﬁrst 200

Ϫ1

cm above threshold. The ab initio predictions of vibra-

tional frequencies, Table I, indicate that the torsion or inter-

nal rotation degree of freedom is responsible for the vibra-

tional structure closest to threshold. The barrier height V of

0

the potential, as deﬁned in Eq. ͑11͒ and shown in Fig. 14, is

chosen to ﬁt the ﬁrst energy spacing in CH CO. The level

2

P(E,vϭ0,J_C_O͒ peaks at J_C_Oϭ12 near threshold has con-

ﬁrmed the ab initio CCO angle of 116°.³

0,38

The ab initio

values for the geometry and vibrational frequencies at the

transition state reproduce the Gaussian shape of the observed

distributions quite well, Fig. 3 and Ref. 30. The C–C–O

bending mode is the largest contributor to the width of the

CO rotational distribution.³⁰The scaled ab initio frequency

Ϫ1

for C–C–O bending is 252 cm , and a noticeable change of

CO rotational distribution is expected to occur when the ﬁrst

quantum of C–C–O bending is energetically accessible at

the transition state. In the harmonic oscillator approximation,

the momentum distribution resulting from the node in the

ﬁrst excited state of the C–C–O bend is a bimodal Gaussian

function. The distribution of CO angular momentum trans-

ferred from C–C–O bending motion for vϭ1 is given by

2

͑J ͒²

͑⌬J_Ќ₁_͒2

Ќ

2

Ќ

P ͑J ͒ϭ

J

exp

ͫ

Ϫ

ͬ

,

͑30͒

1

Ќ

␲͑⌬J_Ќ₁_͒3

ͱ

FIG. 14. The model potential function for the hindered internal rotor,

Ϫ1

V(␪)ϭ(1/2)V ͑1Ϫcos 2␪͒ϩ(1/2)V ͑1Ϫcos 4␪͒, where V ϭ240 cm

0

1

0

where J is the CO angular momentum along the c axis and

J_Ќ₁is the width of the distribution calculated from the

Ϫ1

II

Ќ

and V ϭ0 ͑solid line͒ and Ϯ20 cm . The ab initio C_stransition-state

geometry is drawn. For C O is out of the plane with the CCO plane bisect-

1

⌬

I

s

classically allowed maximum linear momenta of C and O

ing the HCH angle. The calculated energy levels listed in Table IV are

shown for Kϭ0.

atoms for vϭ1. The P (J ) is convoluted with the CO an-

1

Ќ

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Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3215

FIG. 16. The ratio of the Jϭ2 and Jϭ12 CD CO PHOFEX curves in Fig.

2

9

. Curves as in Fig. 15 with the ratios for curves ͑a͒ and ͑b͒ being 2.13 ͑or

Ϫ1

FIG. 15. Ratio of the populations of CO͑vϭ0, Jϭ2͒ to CO͑vϭ0, Jϭ12͒,

1.62 if 28 700 cm is used͒ and 2.22, respectively.

P(E,vϭ0,J_C_Oϭ2͒/P(E,vϭ0,J_C_Oϭ12͒ for CH CO. ͑a͒ From the ratio of

2

the Jϭ2 and Jϭ12 PHOFEX curves in Fig. 8. ͑b͒ The calculated ratio of

the yields of CO(Jϭ2) to CO(Jϭ12) products using the model described

Ϫ1

in Sec. IV E 3 for a C–C–O bending frequency of 250 cm . The ratio of

the maximum and minimum points on curves ͑a͒ and ͑b͒ are 1.80 and 1.90,

respectively.

Ϫ1

added at 290 cm , most likely the CH wag calculated ab

2

Ϫ1

initio at 366 cm . For CD CO the ab initio CCO bending,

2

‡

7

Ϫ1 38

␯

, frequency is 224 cm

.

The ab initio isotope ratio for

‡

8

‡

8

the CH wag, ␯ (CH CO͒/␯ ͑CD CO) ϭ 1.33, combined

2

gular momentum transferred from the motions of other vi-

brations to give the spread in CO angular momentum for

dissociation via an individual vibrational level at the transi-

tion state. A distinct widening of the CO rotational distribu-

tion about the peak at J_C_Oϭ12 imposed by the forces in the

exit valley is expected when the molecule passes through the

ﬁrst excited C–C–O bending level of the transition state.

The peaks in Fig. 8, curve ͑c͒ and the change in slope of

curve ͑a͒ are a clear illustration.

‡

Ϫ1

with ␯ (CH CO)

ϭ

290 cm

gives an estimated

8

2

‡

8

Ϫ1

␯

(CD CO) ϭ 219 cm . This accidental resonance be-

2

‡

7

‡

8

tween ␯ and ␯ and the opportunity for coupling through

internal and overall rotation about the molecular axis sug-

gests the possibility of substantial perturbations. Inspection

of Fig. 7 shows that the ﬂattest region in the k(E) curve

Ϫ1

above 28 450 cm is 28 520–28 545 cm , 210–235 cm

‡

above threshold. Clearly neither ␯₇nor ␯₈can be in this

The overall CO rotational distribution for a given photon

energy, E, is calculated by summing the distributions for

each vibrational level at the transition state multiplied by the

corresponding statistical weight for each vibrational state.

All energetically accessible vibrational levels of the transi-

tion state are assumed to be equally probable. Table IV is

used for the vibrational frequencies of hindered internal ro-

tation. The yields of CO͑vϭ0, Jϭ2͒, P(E,vϭ0,J_C_Oϭ2͒,

and of CO͑vϭ0, Jϭ12͒, P(E,vϭ0,J_C_Oϭ12͒, are calcu-

lated as a function of the available energy and their ratios

compared to the PHOFEX data from Sec. III C in Fig. 15.

The overall shape of the ratio as a function of energy is

well matched by the calculation, and the sharp peaks at

predicted range.

For CD CO the ratio of CO͑vϭ0, Jϭ2, ⌬tϭ150 ns͒

2

and CO͑vϭ0, Jϭ12, ⌬tϭ200 ns͒ PHOFEX curves,

Fig. 16, is the product of ͓P(E,vϭ0,J_C_Oϭ2͒/

P(E,vϭ0,J_C_Oϭ12͔͒ and ͓͕1Ϫexp(Ϫk(E)ϫ1.5ϫ10 )͖/

Ϫ7

͕1Ϫexp(Ϫk(E)ϫ2.0ϫ10 )͖͔. The variation of the latter

with E is small and the structures in the curve are mainly due

to the change of relative CO(v,J) yields. A sharp peak is

observed at around 28 495 cm . As for CH CO, this sug-

gests that the vibrational frequency for C–C–O bending of

Ϫ1

2

CD CO at the transition state is Ϸ͑28 495–28 310ϭ185

2

Ϫ1

cm ͒, about 16% lower than the ab initio value. The peak is

Ϫ1

smaller than calculated as would be expected if the fre-

quency were pushed down by interaction with other modes.

The rate constant curve, Fig. 7, requires an energy level

to be added at 240 cm , 21 cm above the 219 cm

estimated from isotope ratios. The minimum in Fig. 16 at

exactly this frequency does not suggest that there is much

around 28 500 and 28 600 cm in curve ͑a͒ are reproduced

well by the calculation, curve ͑b͒, when the vibrational fre-

Ϫ1

quency of the C–C–O bend is 250 cm . The ﬁrst sharp

Ϫ1

peak at 28 500 cm is assigned as the ﬁrst excited C–C–O

Ϫ1

bending level and the peak at 28 600 cm as the combina-

tion with one quantum of hindered rotation. This agrees very

Ϫ1

well with the scaled value of 252 cm for the C–C–O bend-

CCO bend character associated with the 240 cm energy

ing mode.

level. The data do not provide a convincing assignment for

the 185 and 240 cm levels.

Ϫ1

The rate constant curve for CH CO requires a level to be

2

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129.170.65.78 On: Mon, 28 Oct 2013 05:59:40

3216

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

FIG. 18. The RRKM ﬁt to rate constants of the CD CO dissociation. The

open circles are measured rate constants. The dotted line is the RRKM ﬁt

2

FIG. 17. The RRKM ﬁt to CH CO dissociation rate constants. The open

2

circles are measured rate constants. The dotted line is the RRKM ﬁt using a

step function for the transmission probability as shown in Fig. 6. The solid

line is the RRKM ﬁt including the one-dimensional tunneling with an imagi-

using a step function for the transmission probability as shown in Fig. 7. The

solid line is the RRKM ﬁt including the one-dimensional tunneling with an

Ϫ1

imaginary frequency of 40i cm . The parameters listed in Table V are used

Ϫ1

nary frequency of 40i cm . The parameters listed in Table V are used for

for the RRKM ﬁts.

the RRKM ﬁts.

ment nor the simple one-dimensional tunneling treatment of

the reaction coordinate is satisfactory at this level of detail

for these experiments.

4

. Density of reactant states, ␳(E)

The density of vibrational states of the reactant is ad-

Recently, sharp structures with pronounced peaks in

k(E) have been observed for intramolecular carbon atom

exchange in ketene.³³These are attributed to quantum me-

chanical resonances for motion along a reaction coordinate

with a dip at the top of the barrier, a reaction coordinate

which also involves changing contributions from at least two

vibrational coordinates. A quantitative understanding of the

detailed step shapes must await rotationally resolved experi-

ments, a detailed ab initio reinvestigation of the potential

energy surface in the region of the transition state and a

justed to ﬁt the step sizes in the k(E) data of Figs. 6 and 7 to

the RRKM rate with f/g ϭ1.11 for CH CO and 1.19 for

t

2

CD CO, respectively, where ␳(E)ϭfϫ␳_W_R(E), as listed in

2

Table V. The Whitten–Rabinovitch approximation predicts a

1

0% increase of the density of states as the energy increases

Ϫ1

from 28 250 to 28 700 cm . This is consistent with the ex-

periment as shown in the RRKM ﬁts. The WR approxima-

33

tion gives a density of states 3.5 times higher for CD CO

2

than for CH CO compared to 3.7Ϯ0.5 to ﬁt the rate data. For

2

Ϫ1

g ϭ3, the densities of states derived for 28 250 cm are

t

.6ϫ10 ͑/cm _͒_a_n_d₁_.₇_ϫ₁₀5 ͑/cm ͒ for CH CO and

4

Ϫ1

thorough,₂

treatment

tion state.

multidimensional

of the dynamics of passage through the transi-

quantum

mechanical

4

2

–7

CD CO, respectively. This result presumes that all three trip-

2

let spin states are fully active in the reaction. If only one or

two are coupled to the S levels, then the above densities

0

must be multiplied by 1/3 or 2/3, respectively.

6. Temperature dependence of rate constants

The temperature of the jet affects the rate constants

through the thermal distribution of rotational quantum num-

bers. At 4 K the amount of rotational energy in the ground

state ͑2 or 3 cm ͒ is negligible compared to the ϳ10 cm

resolution of features in k(E). JЉ values are limited to about

5

. Tunneling

A much lower imaginary frequency ͑100Ϯ40i cm^Ϫ¹͒

Ϫ1

than predicted by ab initio calculation ͑523i cm ͒ is nec-

essary to ﬁt the ﬁrst step in k(E) for CH CO dissociation. In

3 and KЉ to 0 for para and 1 for ortho CH CO. In the upper

2

order to reproduce the sharp steps in k(E) which are associ-

ated with vibrationally excited states of ketene at the transi-

tion state, the classical step function had to be employed.

Figures 17 and 18 show that even a frequency as low as 40i

smooths out the steps to a greater extent than observed. The

state the combination of selection rules permits higher J val-

ues and K values. Since the A rotational constant is an order

of magnitude larger than B or C, molecules passing through

‡

the transition state with K ϭJ must carry tens of wave num-

bers of energy as rotation, energy which is inaccessible to the

reaction coordinate. At 4 K the calculated difference between

the RRKM rate with and without K mixing is too small to be

experimentally measurable. However, at 30 K where JЉ val-

rate and PHOFEX data actually show signiﬁcant peaks in

Ϫ1

rate versus energy ͑e.g., 28 360 and 28 410 cm in CH CO

2

Ϫ1

and 28 365 cm in CD CO͒. Thus neither the classical treat-

2

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Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3217

vibrationally excited D CO.¹⁵A stronger test of K mixing

2

will be possible with complete resolution of initial angular

momentum states.

V. CONCLUSION

The stepwise increase of rate constants with increasing

energy observed in ketene dissociation gives direct experi-

mental evidence for the utility of the fundamental tenets of

the RRKM theory that ͑1͒ the rate of reaction is controlled

by ﬂux through quantized transition-state thresholds, ͑2͒ the

vibrational energy in the excited molecule is distributed sta-

tistically among all the vibrational degrees of freedom, and

͑3͒ the intramolecular vibrational redistribution ͑IVR͒ rate is

much faster than the reaction rate. Further, and in contrast to

the usual assumption in applying RRKM theory, K is seen to

be a good quantum number for Jр6. The experimental ob-

servation of the quantized transition-state thresholds as dis-

tinct steps in rate constants conﬁrms the original concept of

transition-state theory that passage through the transition

state is ‘‘vibrationally adiabatic’’ in the sense that energy is

tied up in vibrational modes orthogonal to the reaction coor-

dinate during the brief passage through the transition state

region. The comparison of RRKM calculations to the experi-

ment also supports the basic tenet of transition state theory

that systems cross the transition state directly from reactants

and proceed to products without recrossing. The relative step

heights are quite well reproduced by RRKM calculations

when unity is used for the transmission coefﬁcient for each

quantized transition state. In principle a value signiﬁcantly

less than unity is also possible in the unlikely circumstance

that it is identical for every transition-state vibrational level

threshold.

However, straightforward application of RRKM theory,

even with one-dimensional tunneling along the reaction co-

ordinate, does not reproduce the detailed shape of each step.

Several steps exhibit small peaks that are clearly inconsistent

with the monotonically increasing function of Eq. ͑20͒. The

threshold step for the zero-point vibrational level exhibits a

less rapid rise than those for excited vibrational levels. The

latter rise more rapidly than could be consistent with one-

dimensional tunneling for any reasonable value for the

imaginary frequency for a barrier of the height observed. At

this level of detail a quantum mechanical treatment of the

reaction coordinate coupled to at least one other coordinate

appears to be needed. Ultimately, it will be valuable to com-

pare such a theory with rates for fully resolved initial rota-

tional states.

FIG. 19. RRKM calculations for CH CO dissociation rate constants for K

conserved, T_r_o_tϭ4 K ͑solid line͒ and T_r_o_tϭ30 K ͑dashed line͒.

2

ues up to 9 are signiﬁcant, the rate calculated for K mixing is

much slower than that without. Figure 19 shows that with K

conserved there is only a small difference between RRKM

rates calculated at 4 and 30 K; while in Fig. 20 for K mixed

the rate is much slower. The CO͑vϭ0, Jϭ12͒ PHOFEX

curves taken at 4 and 30 K are almost identical, Fig. 10. This

is consistent with the RRKM calculations in Fig. 19, which

implies that K quantum number is conserved in the photo-

dissociation of ketene. A similar conclusion has been drawn

from the observed density of states as a function of J for

The ab initio calculations for the geometry and vibra-

tional frequencies of ketene at the transition state³⁸have

proven to be extremely useful in the interpretation of experi-

mental results. The impulsive model calculation based on the

ab initio transition-state geometry predicted the CO product

rotational distribution quantitatively³⁰thus conﬁrming the ab

initio CCO angle of 116°. The scaled ab initio value for the

Ϫ1

C–C–O bend of the CH CO transition state, 252 cm , is in

2

Ϫ1

FIG. 20. RRKM calculations for CH CO dissociation rate constants for K

2

excellent agreement with the value of 250Ϯ15 cm found

mixed, T_r_o_tϭ4 K ͑solid line͒ and T_r_o_tϭ30 K ͑dashed line͒. The steps in the

here. The theoretical frequency derives further support from

the observed³⁰width of the CO rotational distribution. The

CH wagging frequency at the CH CO transition state ap-

4

K rate rise somewhat less sharply than for K conserved in Fig. 19 but are

otherwise nearly identical. The 30 K rate is markedly smaller for K mixed

than for K conserved.

2

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3218

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3

pears to be 20% lower than calculated. The barrier for the

hindered internal rotation of ketene at the transition state is

found to be 240 cm when a sinusoidal potential function as

D. C. Chatﬁeld, R. S. Friedman, D. G. Truhlar, B. C. Garrett, and D. W.

Schwenke, J. Am. Chem. Soc. 113, 486 ͑1991͒.

W. H. Miller, Proceedings of The Robert A. Welch Foundation 38th Con-

ference on Chemical Research ͑to be published, 1995͒.

W. H. Miller, Acc. Chem. Res. 26, 174 ͑1993͒.

T. Seideman and W. H. Miller, J. Chem. Phys. 97, 2499 ͑1992͒.

U. Manthe and W. H. Miller, J. Chem. Phys. 99, 3411 ͑1993͒.

C. Leforestier and W. H. Miller, J. Chem. Phys. 100, 733 ͑1994͒.

J. M. Bowman, Chem. Phys. Lett. 141, 545 ͑1987͒.

D. Wang and J. M. Bowman, J. Phys. Chem. 98, 7994 ͑1994͒.

S. L. Mielke, G. C. Lynch, D. G. Truhlar, and D. W. Schwenke, J. Phys.

Chem. 98, 8000 ͑1994͒.

R. A. Marcus and O. K. Rice, J. Phys. Colloid Chem. 55, 894 ͑1951͒; R.

A. Marcus, J. Chem. Phys. 20, 359 ͑1952͒.

E. Abramson, R. W. Field, D. Imre, K. K. Innes, and J. L. Kinsey, J.

Chem. Phys. 83, 453 ͑1985͒; R. L. Sundberg, E. Abramson, J. L. Kinsey,

and R. W. Field, ibid. 83, 466 ͑1985͒.

4

Ϫ1

in Eq. ͑11͒ is assumed. This is 35% lower than the best ab

5

6

Ϫ1 38

initio value of 384 cm

.

Similarly, the barrier height for

7

the reaction itself, 3.06 kcal/mol, is a kcal or so less than the

8

_b_e_s_t_t_h_e_o_r_e_t_i_c_a_l_e_s_t_i_m_a_t_e_.₃8 Thus the actual transition state is

9

found to be somewhat looser than predicted by the ab initio

calculations.

10

11

The density of vibrational states of the excited reactant

1

2

3

and the number of strongly coupled triplet levels, g , control

t

the step height in the RRKM ﬁts to measured rate constants.

The Whitten–Rabinovitch approximation gives a good esti-

mation for the ratio of the densities of CH CO and CD CO

2

1

4

5

reactant states. The absolute values for the density of states

Y. S. Choi and C. B. Moore, J. Chem. Phys. 97, 1010 ͑1992͒.

W. F. Polik, D. R. Guyer, and C. B. Moore, J. Chem. Phys. 92, 3453

for CH CO and CD CO are found to be 1.11g and 1.19g

2

t

͑

1990͒.

times higher than predicted by the Whitten–Rabinovitch ap-

16

A. Geers, J. Kappert, F. Temps, and W. Wiebrecht, Ber. Bunsenges. Phys.

Chem. 94, 1219 ͑1990͒.

J. Miyawaki, K. Yamanouchi, and S. Tsuchiya, J. Chem. Phys. 99, 254

Ϫ1

proximation for energies between 28 200 and 28 700 cm

.

1

7

8

Recent ab initio work on S ketene gives densities of states

0

57

͑

1993͒.

in the range of 1–1.5 times the harmonic density. This

S. I. Ionov, H. F. Davis, K. Mikhaylichenko, L. Valachovic, R. A. Beaudet

and C. Wittig, J. Chem. Phys. 101, 4809 ͑1994͒.

C. E. Dykstra, Annu. Rev. Phys. Chem. 32, 25 ͑1981͒; C. W. Bauschlicher,

S. R. Langhoff, and P. R. Taylor, Adv. Chem. Phys. 77, 103 ͑1990͒.

F. F. Crim, Annu. Rev. Phys. Chem. 35, 657 ͑1984͒.

H. Reisler and C. Wittig, Annu. Rev. Phys. Chem. 37, 307 ͑1986͒.

W. H. Green, C. B. Moore, and W. F. Polik, Annu. Rev. Phys. Chem. 43,

591 ͑1992͒.

E. R. Lovejoy, S. K. Kim, and C. B. Moore, Science 256, 1542 ͑1992͒.

L. R. Khundkar, J. L. Knee, and A. H. Zewail, J. Chem. Phys. 87, 77

would suggest that g is closer to 1 than to 3. A g value of 3

t

would imply a density of states some 3.3 times larger than

harmonic. Those few molecules for which experimental data

19

20

1

3–18

exist,

exhibit even greater excesses in density of states,

2

1

2

ﬁve to ten times harmonic. Thus there is no ﬁrm basis on

which to select values for g and ␳(E).

t

2

3

4

It is clear from comparison of the experimental data with

RRKM theory that there is signiﬁcantly more to be learned

about the dynamics of passage through the transition state. A

quantum mechanical treatment of the dynamics including

coupling of the reaction coordinate to the coupled rovibra-

tional degrees of freedom orthogonal to it will be necessary.

For this treatment to be meaningfully compared with experi-

ment, the ab initio potential energy surface must be mapped

out through the full spread of energetically accessible bond

angles near the transition state. In order to interpret experi-

ments in which the initial rotational and nuclear spin states

are fully selected, ab initio results are needed for the direc-

tion of the transition dipole moment vector for the UV exci-

tation and for the magnitudes of the intersystem crossing

matrix elements. Ultimately, the shapes of steps, their depen-

dence on J and K, and the symmetries of transition-state

vibrational levels along with accurate dynamical calculations

will provide a clear picture of molecular dynamics in the

region of the transition state.

͑

1987͒; S. J. Klippenstein, L. R. Khundkar, A. H. Zewail, and R. A. Mar-

cus, ibid. 89, 4761 ͑1988͒.

25

G. A. Brucker, S. I. Ionov, Y. Chen, and C. Wittig, Chem. Phys. Lett. 194,

301 ͑1992͒; S. I. Ionov, G. A. Brucker, C. Jaques, Y. Chen, and C. Wittig,

J. Chem. Phys. 99, 3420 ͑1993͒.

26

J. Miyawaki, K. Yamanouchi, and S. Tsuchiya, Chem. Phys. Lett. 180, 287

͑

1991͒.

27

W. H. Green, I-C. Chen, and C. B. Moore, Ber. Bunsenges. Phys. Chem.

2, 389 ͑1988͒; I-C. Chen, W. H. Green, and C. B. Moore, J. Chem. Phys.

9

89, 314 ͑1988͒.

28

W. H. Green, A. J. Mahoney, Q.-K. Zheng, and C. B. Moore, J. Chem.

Phys. 94, 1961 ͑1991͒.

S. K. Kim, Y. S. Choi, C. D. Pibel, Q.-K. Zheng, and C. B. Moore, J.

Chem. Phys. 94, 1954 ͑1991͒.

I-C. Chen and C. B. Moore, J. Phys. Chem. 94, 269 ͑1990͒.

I-C. Chen and C. B. Moore, J. Phys. Chem. 94, 263 ͑1990͒.

E. R. Lovejoy, S. K. Kim, R. A. Alvarez, and C. B. Moore, J. Chem. Phys.

29

3

0

1

2

95, 4081 ͑1991͒.

33

E. R. Lovejoy and C. B. Moore, J. Chem. Phys. 98, 7846 ͑1993͒.

I. Garcia-Moreno, E. R. Lovejoy, and C. B. Moore, J. Chem. Phys. 100,

34

8890 and 8902 ͑1994͒.

3

5

6

7

R. Hilbig and R. Wallenstein, IEEE J. Quant. Electron. 17, 1566 ͑1981͒.

C. D. Pibel, Ph.D. thesis, University of California at Berkeley, 1992.

R. N. Dixon and G. H. Kirby, Trans. Faraday Soc. 62, 1406 ͑1966͒; R. G.

W. Norrish, H. G. Crone, and O. Saltmarsh, J. Chem. Soc., 1533 ͑1933͒.

W. D. Allen and H. F. Schaefer, III, J. Chem. Phys. 89, 329 ͑1988͒; 84,

2212 ͑1986͒.

J. W. Rabalais, J. M. McDonald, V. Scherr, and S. P. McGlynn, Chem.

Rev. 71, 73 ͑1971͒.

W. F. Polik, D. R. Guyer, C. B. Moore, and W. H. Miller, J. Chem. Phys.

ACKNOWLEDGMENTS

It is a pleasure to thank Professors Wesley D. Allen,

Stephen J. Klippenstein, and William H. Miller for many

stimulating discussions and for early access to their results

cited herein. This work was supported by the Director, Ofﬁce

of Energy Research, Ofﬁce of Basic Energy Sciences,

Chemical Sciences Division of the U.S. Department of En-

ergy, under Contract No. DE-AC03-76SF0098.

38

39

40

92, 3471 ͑1990͒; R. Hernandez, W. H. Miller, C. B. Moore, and W. F.

Polik, ibid. 99, 950 ͑1993͒.

4

1

2

C. G. Stevens and J. C. D. Brand, J. Chem. Phys. 58, 3324 ͑1973͒; J. C. D.

Brand and C. G. Stevens, ibid. 58, 3331 ͑1973͒.

C. C. Lin, Rev. Mod. Phys. 31, 841 ͑1959͒.

E. B. Wilson, J. Chem. Rev. 27, 11 ͑1940͒; B. L. Crawford, J. Chem. Phys.

8, 273 ͑1940͒.

1

43

H. Eyring, J. Chem. Phys. 3, 107 ͑1935͒; E. Z. Wigner, Phys. Chem. Abstr.

B 15, 203 ͑1932͒; M. G. Evans and M. Polanyi, Trans. Faraday Soc. 31,

44

8

75 ͑1935͒.

G. Brocks, A. van der Avoird, B. T. Sutcliffe, and J. Tennyson, Mol. Phys.

50, 1025 ͑1983͒.

2

J. Z. H. Zhang and W. H. Miller, J. Chem. Phys. 91, 1528 ͑1989͒.

This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

J. Chem. Phys., Vol. 102, No. 8, 22 February 1995

129.170.65.78 On: Mon, 28 Oct 2013 05:59:40

Kim, Lovejoy, and Moore: Dissociation of triplet ketene

3219

4

5

52

P. A. Stockman, R. E. Bumgarner, S. Suzuki, and G. A. Blake, J. Chem.

Phys. 96, 2496 ͑1992͒.

W. H. Miller, J. Am. Chem. Soc. 101, 6810 ͑1979͒.

P. D. Mallinson and L. Nemes, J. Mol. Spectrosc. 59, 470 ͑1976͒.

H. S. Johnston, Gas Phase Reaction Rate Theory ͑Ronald, New York,

W. H. Miller, J. Am. Chem. Soc. 105, 216 ͑1983͒.

W. Nadler and R. A. Marcus, Chem. Phys. Lett. 144, 24 ͑1988͒.

S. J. Klippenstein and R. A. Marcus, J. Chem. Phys. 91, 2280 ͑1989͒; 93,

2418 ͑1990͒.

P. R. Bunker, P. Jensen, W. P. Kraemer, and B. Beardsworth, J. Chem.

Phys. 85, 3724 ͑1986͒; P. Jensen and P. R. Bunker, ibid. 89, 1327 ͑1988͒.

P. R. Bunker and P. Jensen, J. Chem. Phys. 79, 1224 ͑1983͒.

S. J. Klippenstein, A. L. L. East, and W. D. Allen, J. Chem. Phys. 101,

9198 ͑1994͒ and A. L. L. East, W. D. Allen, and S. J. Klippenstein, ibid.

53

4

6

7

8

54

55

1

966͒.

4

5

9

0

1

56

57

C. Eckart, Phys. Rev. 35, 1303 ͑1930͒.

G. Z. Whitten and B. S. Rabinovitch, J. Chem. Phys. 38, 2466 ͑1963͒.

C. B. Moore and G. C. Pimentel, J. Chem. Phys. 38, 2816 ͑1963͒.

This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

J. Chem. Phys., Vol. 102, No. 8, 22 February 1995

129.170.65.78 On: Mon, 28 Oct 2013 05:59:40

Article Doi

Transition state vibrational level thresholds foe the dissociation of triplet ketene

DOI: 10.1063/1.468631

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Article abstract of DOI:10.1063/1.468631

Full text of DOI:10.1063/1.468631

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