Nascent state distributions of NO(X2∏) generated from the reaction of S(1D) with N2O: Intramolecular vibrational-energy redistribution in the reaction intermediate

Article abstract of DOI:10.1063/1.478629

The nascent internal state distribution of NO(X²∏) generated from the reaction, S(¹D) + N₂O→NO+NS, has been determined by utilizing the laser-induced fluorescence (LIF) technique. The average vibrational energy of NO relative to the statistically expected value is found to be 37%. This amount is obviously smaller than that of the fragment N¹⁶O of the isovalent reaction ¹⁸O(¹D) + N₂¹⁶O→N¹⁸O+N¹⁶O, though it is still larger than that of ¹⁸OH produced from the ¹⁶O(¹D) + H₂¹⁸O reaction. To interpret the observed difference in the product energy partitioning, we have applied the quantal intramolecular vibrational-energy redistribution (IVR) representation to the energy mixing in the collision complex. Using a local-mode vibration model with momentum couplings, we have extracted the crucial factors determining the energy partitioning in these reactions. The reaction system consisting of only heavy mass atoms generally has a large vibrational coupling and a large density of states, both of which favor the rapid energy mixing during the short-lifetime of the intermediate complex.

Full text of DOI:10.1063/1.478629

Nascent state distributions of NO (X 2 Π) generated from the reaction of S ( 1 D) with N
2 O : Intramolecular vibrational-energy redistribution in the reaction intermediate
Hiroshi Akagi, Yo Fujimura, and Okitsugu Kajimoto
Citation: The Journal of Chemical Physics 110, 7264 (1999); doi: 10.1063/1.478629
View online: http://dx.doi.org/10.1063/1.478629
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 110, NUMBER 15
15 APRIL 1999
2
Nascent state distributions of NO„X ⌸… generated from the reaction
1
of S„ D… with N2O: Intramolecular vibrational-energy redistribution
in the reaction intermediate
Hiroshi Akagi, Yo Fujimura, and Okitsugu Kajimotoa)
Department of Chemistry, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwakecho,
Sakyo-ku, Kyoto 606-8502, Japan
͑Received 21 December 1998; accepted 21 January 1999͒
The nascent internal state distribution of NO(X 2⌸) generated from the reaction, S(1D)
ϩN2O→NOϩNS, has been determined by utilizing the laser-induced fluorescence ͑LIF͒ technique.
The average vibrational energy of NO relative to the statistically expected value is found to be 37%.
This amount is obviously smaller than that of the fragment N 16O of the isovalent reaction
18O(1D)ϩN2 16O→N 18OϩN 16O, though it is still larger than that of 18OH produced from the
16O(1D)ϩH2 18O reaction. To interpret the observed difference in the product energy partitioning,
we have applied the quantal intramolecular vibrational-energy redistribution ͑IVR͒ representation to
the energy mixing in the collision complex. Using a local-mode vibration model with momentum
couplings, we have extracted the crucial factors determining the energy partitioning in these
reactions. The reaction system consisting of only heavy mass atoms generally has a large vibrational
coupling and a large density of states, both of which favor the rapid energy mixing during the
short-lifetime of the intermediate complex. © 1999 American Institute of Physics.
͓S0021-9606͑99͒01715-8͔
I. INTRODUCTION
curs, even though the lifetime of the intermediate complex is
expected to be short because of the shallow well of the PES.
This efficient energy mixing could be attributed to the con-
stituent atoms in the reaction system, that is, the absence of
light atoms. Because the 18O(1D)ϩN2 16O system only in-
cludes heavy atoms, the energy randomization in the tran-
sient 18ONN 16O should be enhanced due to the large mo-
mentum ͑G-matrix͒ couplings between the local-mode
vibrations. In addition, the low vibrational frequencies origi-
nated from the motion of heavy atoms considerably increase
the density of these states. The importance of such mass
effect is supported by the inefficient energy mixing between
the two OH products observed in the analogous reaction in-
cluding light H atoms,
Since the 1970s, the product energy distribution has
been a hot topic in reaction dynamics. In particular, the re-
lation between the shape of the potential-energy surface
͑PES͒ and the energy partitioning within the degrees of free-
dom of product molecules has been intensively studied to
obtain a fundamental understanding of the dynamics.1 Gen-
eral criteria for predicting the product energy distribution
have been proposed and the analyses using the criteria have
attained a marked success.2 The criteria mainly concern the
lifetime of the collision complex or intermediate. As the life-
time becomes longer, the product distribution becomes more
statistical. The lifetime of the complex is in turn determined
by the shape of the PES. If the potential-energy surface has a
deep well between the entrance and the exit, we expect a
long lifetime which produces a statistical distribution. On the
other hand, if the potential well is much shallower compar-
ing with the exothermicity of the reaction, a short lifetime
and a nonstatistical energy partitioning are expected.
1
16
16O D͒ϩH 18O→ OHϩ18OH,
͑
2
⌬H0 0͒ϭϪ120 kJ molϪ1
.
͑b͒
͑
Since the reaction ͑b͒ possesses a deep well, the intermediate
of this reaction should have a longer lifetime than that of the
reaction ͑a͒. Nevertheless, the energy mixing is less efficient
in the reaction ͑b͒. This evidence suggests the importance of
the mass effect over the lifetime of the intermediate complex
in contrast to the expectation in general.
In the present study, we try to confirm the critical role of
the mass effect by demonstrating another example of effi-
cient energy mixing for the reaction with a short-lived inter-
mediate, i.e.,
In our previous work,3 we determined the vibrational
state distributions of the ‘‘new’’ N 18O and the ‘‘old’’ N 16
generated from the reaction,
O
1
18O D͒ϩN 16O→͓18ON–N 16O͔→N 18OϩN 16O, ͑a͒
͑
2
which has
a
large exothermicity
⌬H0(0)ϭϪ343
͓
kJ molϪ1 and a shallow well with the N–N bond scission
͔
energy of ϳ700 cmϪ1.4 Although the available energy is
dominantly distributed to the new N 18O, the old N 16O also
possesses a considerable vibrational energy at high vibra-
tional levels. This fact suggests that the considerable energy
randomization in the reaction intermediate, 18ONN 16O, oc-
1
S D͒ϩN O→NSϩNO.
͑c͒
͑
2
Since S(1D) atom is isovalent to O(1D), this reaction is
expected to have a similar PES with no barrier and a very
shallow well. In fact, the density functional theory calcula-
tions have showed that SNNO is a cis-planer molecule and
a͒
Author to whom correspondence should be addressed.
0021-9606/99/110(15)/7264/9/$15.00
7264
© 1999 American Institute of Physics
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J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Akagi, Fujimura, and Kajimoto
7265
mixture of OCS and N2O gases was continuously supplied to
the chamber at 200 mTorr. OCS ͑Matheson 97.5%͒ and N2O
͑Showa-Denko 99.999%͒ gases were purchased and used
without further purification.
The S(1D) atoms were produced by OCS photolysis at
248 nm with a KrF Excimer laser ͑Lambda Physik
EMG53MSC͒. The KrF laser light was collected with a lens
of 1.5 m focal length, and a part of the focused light ͑ϳ0.5
cm diam͒ was used for the photolysis. At this wavelength of
OCS photolysis, the quantum yield of S(1D) generation was
already determined to be almost a unity from two photon LIF
experiments.9 The product NO generated from reaction ͑c͒
was detected by utilizing the LIF technique via the NO
A 2⌺ϩ ( ϭ0,1)←X 2⌸( ϭ0–4) transitions. For this pur-
v
v
pose, frequency doubled light from a tunable dye laser
͑Lambda Physik ScanMate 2E͒ pumped by a XeCl excimer
laser ͑Lambda Physik Compex 102͒ was used. In order to
avoid saturation, the probe laser light was expanded 2.5
times by two lenses of 8 and 20 cm focal lengths. The inten-
sities of both the photolysis and the probe laser lights were
monitored by photodiodes and used to normalize the LIF
signal. The time delay between the photolysis and the probe
lasers was set to be 500 ns and 5 ␮s for the determination of
rotational and vibrational state distributions, respectively.
These laser systems were operated with a repetition rate of
10 Hz.
The fluorescence from NO(A) was collected by two
lenses at right angles to the counterpropagating laser beams
and the fluorescence of only (0→0) and (1→1) transitions
was detected by a photomultiplier tube ͑Hamamatsu R928͒
through an interference bandpass filter ͑the center wave-
length ␭0ϭ219.5 nm, the full bandwidth at half-maximum
⌬␭ϭ12.5 nm͒. The LIF signal from the PMT was amplified
by a fast voltage amplifier ͑Comlinear CLC100͒, averaged
by a gated boxcar integrator ͑Stanford Research Systems SR
250͒, processed by an A/D converter ͑Stanford Research
Systems SR 245͒, and stored in a personal computer.
For the determination of the vibrational state distribution
of NO generated from the reaction of S(1D) with N2O, we
adopted the same procedure as Ref. 3. Briefly, the procedure
consisted of the following two measurements: ͑1͒ determina-
tion of the rotational state distribution for each vibrational
levels under the rotationally relaxed conditions with 200
mTorr total pressure and a 5 ␮s delay and ͑2͒ determination
of the relative vibrational population by comparing the peak
heights at the selected rotational lines of different vibrational
levels. To obtain reliable distribution, we utilized the rovi-
bronic transitions with the same upper levels.
FIG. 1. Schematic diagram of the experimental apparatus. PD, photodiode;
PMT, photomultiplier tube.
has a low frequency N–N stretching vibration similar to
(NO)2 .5 Moreover, this reaction system is composed of four
heavy atoms.
The reaction of S(1D) with N2O has been studied by
several groups.6,7 The reaction rate was found to be nearly
gas kinetic (1.6ϫ10Ϫ10 cm3 moleculeϪ1 sϪ1).6 The reaction
has the following two important channels other than channel
͑c͒:
1
S D͒ϩN O→SO a 1⌬ or b 1⌺ϩ͒ϩN ,
͑cЈ͒
͑cЉ͒
͑
͑
2
2
3
→S P͒ϩN O.
͑
2
These three channels proceed to a comparable extent, al-
though the S(1D) quenching process ͑cЉ͒ is slightly
preferred.7
In the present work, we have dealt with the internal state
distribution of NO(X 2⌸) produced from the reaction ͑c͒.
Because the NO molecule is the ‘‘old’’ fragment already
existed in the N2O molecule, the amount of NO internal en-
ergy is an appropriate index of energy randomization within
the reaction intermediate, SNNO. The exothermicity of this
reaction ͑c͒ ⌬H0(0)ϷϪ100 kJ molϪ1 ͑Ref. 8͒ can excite
͓
͔
the NO molecule up to the ϭ4 vibrational level. We have
v
determined the vibrational ( ϭ0–4) and rotational (
v
v
ϭ0–2) state distributions of NO generated by this reaction
utilizing the laser-induced fluorescence ͑LIF͒ technique.
The quantitative interpretation of the mass effect is also
attempted in terms of the intramolecular vibrational-energy
redistribution ͑IVR͒ in the transient intermediate. The IVR
rate is estimated on the basis of the Fermi’s Golden Rule.
The state density and the coupling among the vibrational
modes are calculated by using the local-mode approximation
with the first-order momentum coupling. The quantal time
evolution is also calculated to evaluate the rate of IVR.
It should be noted that the KrF laser light ͑248 nm͒ used
for the OCS photolysis could also photolyze N2O and pro-
duce O(1D) atoms, even though the absorption coefficient of
N2O at 248 nm is negligibly small.10,11 The O(1D) atoms, if
produced, could also produce NO by reaction ͑a͒. In fact, the
LIF signal from NO molecules was detected without the
presence of OCS. However, for NO in ϭ0–3 and ϭ4
v
v
vibrational levels, the LIF signal in the absence of OCS was
less than 3% of that with OCS and less than the noise level,
respectively. Consequently, the contribution from the N2O
photolysis at 248 nm was neglected.
II. EXPERIMENT
The experiment was performed with a conventional flow
chamber and two laser systems, as shown in Fig. 1. The 1:1
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7266
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Akagi, Fujimura, and Kajimoto
FIG. 2. Schematic diagram of the timing between the two laser triggers and
the variation of NO concentration in the reaction cell. Nascent NO ( ϭ0)
v
signal was obtained as the difference between the signals of Ϯ⌬t delays in
successive laser shots.
A mechanical booster pump and a rotary pump were
used for the flowing gas experiment. Since all NO molecules
FIG. 3. LIF spectra of NO͑X 2⌸; ϭ0 and 1͒ generated from the reaction
v
of S(1D) with N2O, together with the simulated spectrum. A←X(0←0)
and A←X(1←1) transitions were used. ͑a͒ The sum of nascent and back-
ground NO signals measured at ϩ500 ns delay. ͑b͒ Background NO signal
measured at Ϫ500 ns delay. ͑c͒ Nascent NO signal obtained by taking the
difference between signals of Ϯ500 ns delays. ͑d͒ A spectrum simulated
in the ϭ0 level could not be pumped out perfectly during
v
the shot-to-shot period under the pumping conditions, the
LIF signal from NO( ϭ0) was detected even when the
v
probe laser light irradiates the reacting gas 500 ns earlier
under the conditions of the ratio N( ϭ1)/N( ϭ0)ϭ0.25, and the rota-
than the photolysis laser light. To determine the nascent sig-
v
v
tional temperatures 2600 and 1400 K for ϭ0 and 1, respectively.
v
nal intensity of NO in ϭ0 level, the delay time between the
v
photolysis and the probe laser shots was changed alterna-
tively by a homemade switching delay circuit. The signal
intensity obtained at Ϫ⌬t delay corresponds to the back-
A. Vibrational state distribution in v؍0–4
The experimentally obtained vibrational state distribu-
tion and the distributions predicted from two statistical theo-
ries, the prior distribution1 and the distribution calculated
with phase space theory ͑PST͒,13 are listed in Table I. The
experimentally obtained population monotomically declines
as the vibrational quantum number increases. The observed
distribution is slightly colder than those predicted from the
ground NO( ϭ0) signal, and its intensity was subtracted
v
from that obtained at ϩ⌬t delay. The timing of such proce-
dure is shown schematically in Fig. 2. Utilizing the subtrac-
tion between the two data sets, Ϯ⌬t delays, we obtained
accurate LIF spectra of nascent NO molecules.
III. RESULTS
Figure 3 shows the LIF spectra of the nascent NO(
v
ϭ0,1) which is calculated by the subtraction between the
two spectra under Ϯ500 ns delay conditions, together with
the original spectra at each delay. The rotationally relaxed
LIF spectrum of NO( ϭ3) under 5 ␮s delay condition is
v
shown in Fig. 4. The simulated spectrum is also included in
each of the figures.
As the source of S(1D), we used OCS photolysis at 248
nm. Since S(1D) is removed by the every collision with the
buffer gas, OCS or N2O,6 the S(1D) atom is not translation-
ally relaxed and therefore we should take the collision en-
ergy into account. Then, adding the 13 kJ molϪ1 collision
energy12 to the heat of reaction, we have the total available
energy of 113 kJ molϪ1 which could excite the product NO
molecule to the ϭ5 level. In the present experiments, how-
v
ever, we detected the NO up to ϭ4 vibrational levels under
v
FIG. 4. Rotationally relaxed LIF spectrum of NO ( ϭ3) generated from
v
5 ␮s delay condition. For ϭ0–2 vibrational levels, the LIF
v
the reaction of S(1D) with N2O utilizing the A←X(0←3) transition ͑upper
panel͒, together with the simulated spectrum ͑lower panel͒. Spectrum taken
under 5 ␮s delay and 200 mTorr total pressure. Simulated spectrum was
calculated with a rotational temperature of 320 K.
signal intensity was sufficient to determine the nascent rota-
tional state distribution of NO molecules under the shorter
͑500 ns͒ delay condition.
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J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Akagi, Fujimura, and Kajimoto
7267
TABLE I. Vibrational state distribution of NO(X 2⌸) generated from the
reaction of S(1D) with N2O.
TABLE II. Average rotational energy
E
of NO͑X 2⌸; ϭ0–2͒ gen-
v
͗
͘
rot
v
erated from the reaction of S(1D) with N2O.
Expta
Prior ͑RRHO͒
PSTb
E
͗
v /kJ molϪ1
v
͘
rot
Expta
PSTb
0
1
2
3
4
5
0.76
0.59
0.27
0.10
0.027
0.003 2
0.000 006 3
0.50
0.29
0.14
0.055
0.012
0.000 18
v
0.19Ϯ0.04
0.033Ϯ0.008
0.011Ϯ0.003
0.0004Ϯ0.0001
¯
0
1
2
21.8Ϯ0.9
11.3Ϯ1.1
9.9Ϯ0.4
26.0
21.3
16.6
aThe experimental uncertainty corresponding to 1␴ in the least-squares fit.
bFor the PST calculation, the interaction potential for both S–N2O and
SN–NO is assumed to have the form of ϪC/r6, where C is taken to be 42
aThe experimental uncertainty corresponding to 1␴ in the least-squares fit.
bFor the PST calculation, the interaction potential for both S–N2O and
SN–NO is assumed to have the form of ϪC/r6, where C is taken to be 42
kJ Å6 molϪ1
.
kJ Å6 molϪ1
.
͑2.5 kJ molϪ1͒. The latter fact in E
implies that the old
͘
͗
rot v
NO would not behave simply as a spectator in the S(1D)
ϩN2O reaction.
statistical theories, as recognized from Fig. 5, suggesting the
incomplete vibrational energy randomization in the interme-
diate. Nevertheless, the ‘‘old’’ NO populate even in ϭ4
v
C. Energy partitioning to the ‘‘old’’ fragments
which is the upper limit for the exothermicity of this reac-
tion. This fact indicates that the considerable energy mixing
occurs during the reaction.
The mean rotational and vibrational energies of NO were
calculated as
E
ϭ
P
͒• E
͗
P
͑2͒
͑v
͑v
͒
͗
͘
rot
͘
B. Nascent rotational state distribution in v؍0–2
Ͳ
͚
͚
rot v
v
v
The rotational state distributions for each vibrational
and
level in ϭ0–2 could be described by a single Boltzmann
v
temperature. The average rotational energy for each vibra-
E
ϭ
P
͒•E
͒
P
͑3͒
͑v
͑v
͑v
͒,
͗
͘
vib
Ͳ
͚
͚
vib
tional level E
was calculated by the equation
͗
͘
rot v
v
v
respectively. In these equations, P( ) is the vibrational state
distribution and Evib( ) is the energy of each vibrational
state. To determine the E , we used the E
v
E
ϭ
͘
P
J͒•E J͒
P J͒,
͑
v
͑1͒
͑
͑
͗
Ͳ
͚
͚
rot v
v
rot
J
J
v
values for
values for
͗
͘
rot
͗
͗
͘
rot v
where Pv(J) is the rotational state distribution including the
degeneracy and Erot(J) is the energy of the rotational state.
The evaluated average rotational energies of NO are listed in
Table II, together with those predicted from PST.13 The av-
erage rotational energy decreases as the vibrational quantum
number increases similar to the statistical prediction. This
trend may attribute to the decrease of the available energy.
only ϭ0–2 vibrational levels, since the E
v
͘
rot v
Ͼ2 could not be determined. However, this omission has
little influence on the accuracy because of the minor popula-
tion in higher vibrational levels.
The average rotational and vibrational energies of the
old NO molecules can act as appropriate measures of the
energy randomization in the reaction intermediate. The aver-
v
The experimental E
of NO ( ϭ0–2) is colder than the
v
͗
͘
rot v
age energies are calculated to be
E
ϭ1600 cmϪ1 and
͘
rot
͗
average energy expected from PST, yet it is still rotationally
excited as compared with the energy at room temperature
E
ϭ550 cmϪ1, which correspond to 17% and 5.8% of the
͘
͗
vib
available energy Eavail(Ϸ113 kJ molϪ1), respectively. Since
the PST calculations predict large values ͑ E
͗
͘
rot PST
ϭ1900 cmϪ1 and Evib PSTϭ1500 cmϪ1), we can conclude
͗
͘
that the energy randomization in the intermediate of this re-
action is not completed.
In order to evaluate the energy mixing in the intermedi-
ate complex, the average rotational and vibrational energies
of the ‘‘old’’ fragment in reaction ͑c͒ were compared in
Table III with those of the reactions ͑a͒ and ͑b͒, 18O(1D)
16
ϩN216O→N18OϩN16O and 16O͑1D͒ϩH218O→ OHϩ18OH.
For the 16O(1D)ϩH218O reaction, several research groups
have determined the products rotational and vibrational state
distributions.15,16 Since their data are similar, the latest one
using the O3 photodissociation at 266 nm ͑Ref. 15͒ is listed
in Table III. The available energies Eavail and the dissociation
energies D0 of the intermediates which correspond to the
well depths of PESs are also given in Table III. Since only
the theoretical value of D0 from density functional theory
calculations is available for SNNO to our best knowledge,5
FIG. 5. Vibrational state distribution of NO(X 2⌸; ϭ0–4) produced from
v
the reaction of S(1D) with N2O. The solid line shows the distribution pre-
dicted from phase space theory and the broken line depicts the prior distri-
bution.
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7268
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Akagi, Fujimura, and Kajimoto
v2
TABLE III. Average rotational and vibrational energies of the ‘‘old’’ frag-
ments for reactions ͑a͒, ͑b͒, and ͑c͒, together with the total available energies
of the reactions and the binding energies of their intermediates ͑cmϪ1͒.
where
is the square of effective coupling and ␳ is the
density of state. That is, the extent of energy randomization
can be regarded to be proportional to the density of state and
the square of effective coupling when the time for IVR is the
same. These two universal factors are sensitive to the mo-
lecular properties such as the location and the masses of
component atoms.
The heavy components within the intermediate complex
usually generate low frequency vibrations to increase the
density of state ␳ and cause large momentum couplings
18O(1D)ϩN216O
→N18OϩN16O
16O(1D)ϩH218O
S(1D)ϩN2O
→NSϩNO
c
→
16OHϩ18OH
a
b
͓reaction ͑a͔͒
͓reaction ͑b͔͒
͓reaction ͑c͔͒
30000
700d
Ϸ4000
͑Ϸ0.63͒
͓Ϸ7000͔
11400
17400e
210
9400
͓Ϸ700͔
550
͑0.37͒
1600
͑0.84͒
Eavail
D0
f
E
͗
͘
vib expt
i
i
i
͑0.16͒
g
E
1900
͑0.68͒
͗
͘
2
rot expt
among the vibrational modes to give large
.
v
i
i
i
͓͑Ϸ1.1͔͒
E
E
6400
1300
2800
1500
1900
͗
͗
͘
vib PST
6400h
A. Mass effect on the coupling parameter v2
͘
rot PST
aReferences 3 and 14.
bReference 15.
cThis work.
2
To discuss more quantitatively the mass effects on
,
v
we adopt a model18 whose Hamiltonian is represented by the
zero-order and the coupling term as
dReference 4.
eReference 11.
HϭH0ϩV.
͑5͒
fThis value is deduced from the results using DFT calculations of Ref. 5.
gThis value is estimated from the rotational temperature of the NO( ϭ0)
v
For simplicity, we assume that ͑1͒ all zero-order vibrational
states are regarded as local-mode harmonic oscillators and
͑2͒ a harmonic momentum coupling only contributes to the
coupling between the harmonic oscillators. Then, the ele-
ments of Hamiltonian are
which obtained in the unlabeled experiments in Ref. 14.
hThis value is evaluated from the prior distribution.
iData given in parentheses are the ratios relative to the average energies
predicted from phase space theory.
the binding energy of SNNO is estimated by using this value.
The estimated energy size of SNNO is to be almost the same
as that of (NO)2 .
1
2
1
H0ϭ
Giip2ϩ kiiq2
,
͑6͒
͑7͒
ͫ
ͬ
͚
i
i
2
i
As Table III shows, the average vibrational energy
Vϭ
G p p ,
ij
͚
i j
E
of reaction ͑a͒ is much larger than the other two reac-
͘
vib
͗
iÞj
tions. Since the available energy is different among these
where pi , qi , kii , and Gij denote the momentum, the dis-
placement, the force constant of the ith oscillator, and the
element of G-matrix,19 respectively. In general, a potential
coupling should also be considered as a harmonic coupling
term of the Hamiltonian. However, for the systems with
large momentum coupling, the potential coupling makes
only a small contribution.18 Under these assumptions, the
only nonzero elements of Hamiltonian matrix corresponding
three reactions, we have to compare the energy partitioning
using the ratio of E
to the value predicted from the PST.
͘
vib
͗
The order in the energy randomization thus estimated is re-
action ͑a͒Ͼreaction ͑c͒Ͼreaction ͑b͒.
In the S(1D)ϩN2O reaction, the vibrational energy
transferred to the old product amounts to 37% of the statis-
tical prediction. Although the amount is approximately two-
third of the reaction ͑a͒, it is more than twice of that in the
reaction ͑b͒. This result shows that the energy mixing in the
S(1D)ϩN2O reaction is definitely operative in spite of the
shallow well of PES and the short lifetime of the intermedi-
ate, and confirms our conclusion in the previous paper3 that
the mass effect is more important than the well depth of the
PES. The presence of light hydrogen atoms in reaction ͑b͒
significantly reduces the state density and the momentum
coupling among the vibrational motion, resulting in the ex-
tremely inefficient energy partitioning in spite of the ex-
pected long lifetime.
to are18
v
. . . ,n , . . . ,n , . . .
͉
H
͉
. . . ,n , . . . ,n , . . .
͗
͘
a
b
a
b
1
ϭ
␣
2n ϩ1͒ϭ
n ϩ ប␻ ,
͑
ͩ
ͪ
͚
͚
ii
i
i
i
2
i
i
. . . ,n ϩ1, . . . ,n Ϫ1, . . .
͉
͉
͉
H
͉
. . . ,n , . . . ,n , . . .
͗
͗
͗
͗
͘
͘
͘
͘
a
b
a
b
1/2
ϭϪ␣ab n ϩ1͒n
͓͑
,
͔
b
a
. . . ,n Ϫ1, . . . ,n ϩ1, . . .
H
͉
͉
͉
. . . ,n , . . . ,n , . . .
a b
a
b
1/2
ϭϪ␣ab
n
n ϩ1͒
b
,
͔
͓
͑
a
IV. DISCUSSIONS
. . . ,n ϩ1, . . . ,n ϩ1, . . .
H
. . . ,n , . . . ,n , . . .
a b
a
b
Since the lifetime is not the essential factor to control the
energy partitioning in the reactions of our interest, we con-
fine our discussion to other factors in vibrational energy mix-
ing, the density of states and the coupling among the vibra-
tional states. In the statistical limit, the rate of IVR is
represented by Fermi’s Golden Rule as17
1/2
ϭ␣ab n ϩ1͒ n ϩ1͒
͓͑
,
͔
͑
a
b
. . . ,n Ϫ1, . . . ,n Ϫ1, . . .
͉
H
. . . ,n , . . . ,n , . . .
a
b
a
b
1/2
ϭ␣ab n n
,
͑8͒
͓
͔
b
a
where ni is the quantum number of the ith vibrational mode
2
2␲
␳
v
on the local-mode basis, ␻ is the vibrational frequency, and
i
kϭ
,
͑4͒
ប
␣ is the parameter of momentum coupling,
ij
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J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Akagi, Fujimura, and Kajimoto
7269
TABLE IV. The structures, local-mode vibrational frequencies, and relevant parameters of the reaction com-
plexes used for the model calculations.
18ONN16O
͓reaction ͑a͔͒
H16O18OH
͓reaction ͑b͔͒
SNNO
͓reaction ͑c͔͒
Structural parameters of A–B–C–D
A–B length/Å
B–C length/Å
C–D length/Å
ЄABC angle/deg
ЄBCD angle/deg
torsion/deg
1.152a
2.263a
1.152a
97.2a
97.2a
0.0a
0.965b
1.452b
0.965b
100.0b
100.0b
112.0b
1.552c
1.961c
1.178c
135.4c
102.3c
0.0c
Local-mode vibrational frequencies of A–B–C–D/cmϪ1 d
A–B stretch
B–C stretch
C–D stretch
ЄABC bend
ЄBCD bend
torsion
1780e
135e
1830e
325e
334e
117e
3610f
847f
1110c
151c
1770c
216c
314c
594c
3600f
1330f
1330f
313f
⌬Ecoll /cmϪ1
820g
640g
410h
2
͚
͉
␣
͉
2/͚
͉
␣
͉
ii
4.9ϫ10Ϫ2
5.2ϫ10Ϫ3
6.3ϫ10Ϫ2
ij
aReference 20.
bReference 21.
cThis value is evaluated from the DFT calculations of Ref. 5.
dThese frequencies are calculated from the normal-mode vibrational frequencies.
eReference 22.
fReference 23.
gReference 24.
hReference 12.
1/2
initial stage stores all the energy in the A–B local mode
vibration. When the energy mixing ͑IVR͒ process proceeds
from this state, we can roughly estimate the rate of IVR
Gij ប␻ ប␻
i
j
␣ijϭ
.
͑9͒
ͫ
ͬ
2
Gii Gjj
v2
For the real systems, the coupling matrices are evaluated on
the basis of the eigenvalues and eigenvectors for the most
stable structure of the individual intermediates. The struc-
tural parameters for the intermediates of the three reactions
and their local-mode frequencies are given in Table
IV.5,12,20–24 The sum of the squares of off-diagonal momen-
tum coupling parameters relative to that of diagonal term is
also given in the bottom row of the Table IV. This quantity
approximately expresses the average strength of the coupling
among the local-mode vibrations. Apparently, the hydrogen-
containing intermediate H16O18OH gives the one-order of
magnitude smaller value than the other intermediates, indi-
cating a very small coupling among the local-mode vibra-
tions. This trend originates from the small reduced mass in
the local-mode vibration containing a hydrogen atom. Since
Gii is proportional to the inverse of the reduced mass of the
local-mode, the hydrogen-containing local mode gives a
large Gii and hence a small ␣ij . In this manner, the presence
of a light atom considerably reduce the momentum coupling
and therefore reduces the rate of IVR. For the present four-
atomic reactions, 16O(1D)ϩH218O shows a ten-times less
efficient IVR compared with the 18O(1D)ϩN216O and
S(1D)ϩN2O reactions.
using Eq. ͑4͒ using the
and ␳ which can be calculated
from the model of the preceding section. Using the coupling-
scheme given in the preceding section and the vibrational
frequencies in Table IV, the densities of the zero-order vi-
brational states which can couple with the initial state, which
is defined as ␳
in Fig. 6, are calculated. The resulting
direct
values for the reactions ͑a͒, ͑b͒, and ͑c͒ are 0.0098/cmϪ1
,
0.0063/cmϪ1, and 0.0146/cmϪ1, respectively. These values
B. The state density at a relevant energy range
The reaction, AϩBCD, is initiated by the collision of the
atom A which has an intrinsic energy uncertainty of Ecoll
͑Fig. 6͒. We assume that the intermediate complex in the
FIG. 6. A model for the time-evolution of IVR in an intermediate complex.
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7270
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Akagi, Fujimura, and Kajimoto
again suggest that the reaction ͑b͒ including light atoms has
the small state density corresponding to 2/3 and 1/2 of the
reactions ͑a͒ and ͑c͒.
the energy of a finite width. In the present case of the bimo-
lecular reaction, however, the initial state is produced by the
collision between A and BCD. We assume that the coherent
width of the initial state, which corresponds to the energy
width of the ultrashort excitation pulse in the photoexcita-
tion, can be defined as the collisional energy range of the
incident atom. Since the reactant atom is generated by the
photodissociation of O3 or OCS, the collision energy of the
atom has its intrinsic energy range. Although the distribution
of this collision energy is not well-defined and the orienta-
tion of the collision is not fixed, the actual IVR can be inter-
preted as an superposition of the time evolution starting from
a variety of coherently prepared initial state with a given
coherent width. As a kind of thought experiment, we pursue
the time evolution of such a coherent initial state and try to
gain the information on the time scale of IVR in the particu-
lar reaction system.
The energy partitioning to the old NO is significantly
2
larger while ␳
and
are smaller in the reaction ͑a͒ than
v
direct
in the reaction ͑c͒. This trend could be explained in terms of
the background dark levels. The density of the background
zero-order vibrational states (␳indirect), which is not taken
into account in the present coupling scheme, is much differ-
ent between the reactions ͑a͒ and ͑c͒. Since the reaction ͑a͒
has a quite large exothermicity, the initially prepared inter-
mediate contains a large amount of energy and the state den-
sity at this high energy level is quite large. Direct counting of
the energy levels shows that ␳indirects are 5.5ϫ104 cmϪ1 for
reaction ͑a͒ and 1.1ϫ102 cmϪ1 for reaction ͑c͒.
C. The rate of IVR
Based on the above considerations, the loss of the initial
zero-order state ͉i͘ is obtained from the projection compo-
nents of ͉i͘ in the time-dependent wave function ⌿(t) as25
To estimate the rate of IVR in the three relevant reac-
tions, we use the analogy between the IVR in an intermediate
complex and the time-evolution of the zero-order state which
is prepared by an ultrashort coherent light pulse.
N
N
2
͉
2ϭ
͘
͉
cIi
͉
4ϩ2
͉
cIi͉ ͉cJi
͉
2 cos ␻ t ,
͑12͒
1. Description of a model and the method of
calculations
͉
i
͉
⌿ t͒
͑
͗
ͫ
ͬ
͚
͚
IJ
Iϭ1
IϾJ
The IVR process can be considered as the evolution of
an initial zero-order state ͉i͘, which is a superposition of
many quantum eigenstates.25 This initial zero-order state is,
in general, prepared by an ultrashort pulse of light with wide
energy range determined by the pulse duration. This ‘‘opti-
cal’’ initial state is not an eigenstate, but is represented by
the superposition of eigenstates. The projection components
of the dark zero-order states increase as the time-evolution of
the initially prepared state. This process is the IVR in an
ordinary rigid molecules excited by a fs light pulse.
In the present paper, this quantal scheme of IVR is
adopted to interpret the energy randomization of the reaction
intermediate, ABCD, generated from the bimolecular reac-
tion, AϩBCD. First, we assume that the initial zero-order
state ͉i͘ is the locally excited state, where only the A–B
local-stretching vibration possesses all the available energy,
where ␻ is equal to (EIϪEJ)/ប. The fluorescence decay
IJ
factor is omitted. The eigenvalues and the eigenvectors are
obtained under the assumption of the local-mode description
of the intermediate complex as given in the preceding sec-
tion.
As is shown in Fig. 6 the initial zero-order states for
18ONN 16O ͓reaction ͑a͔͒, H 16O 18OH ͓reaction ͑b͔͒, and
SNNO ͓reaction ͑c͔͒ are considered as the local-mode vibra-
tional states of the N–18O( ϭ17), 16O–H( ϭ8), and
v
v
N–S( ϭ8) stretching modes, respectively. In this case, all
v
eigenstates within the collision energy width ⌬Ecoll , caused
by the photodissociation of O3 or OCS, are expected to con-
tribute to the IVR. In the present calculations, the number of
zero-order states were limited by truncating the interacting
zero-order states according to the coupling strength. As the
elements of basis set, we selected 162 zero-order states in-
cluding one initial state, 26 states coupling with the initial
state in the first-order perturbation level and 135 indirect-
coupling states which become effective in the second-order
perturbation level. With these 162 states, the eigenvectors
and eigenvalues were computed by the matrix diagonaliza-
tion. Then, the time evolutions of an initial state, the sum of
direct-coupling states and the sum of indirect-coupling states
were calculated for these three reaction intermediates.
E
availϩD0 , of the reaction. Because this reaction is initiated
by the approach of the A atom to BCD, we regarded this as
a suitable assumption. The initial state ͉i͘ corresponds to a
linear combination of many eigenstates ͉I͘,
͉
i ϭ
͘
c
͉
I .
͘
͑10͒
͚
Ii
I
The time evolution of this initial state is given by
⌿ t͒ϭ
͑
c
͉
I •eϪiE t/ប
.
͑11͒
I
͘
͚
Ii
I
2. Results of the model calculations
The dephasing of these eigenstates caused by their energy
gaps leads to the increase of the projection components of
the other local-mode vibrational states, i.e., the dark zero-
order states. The energy randomization in the intermediate
complex can be considered in this way as an IVR process
similar to that in a stable molecule.
The time evolutions of the zero-order states in
18ONN 16O ͓reaction ͑a͔͒ and H 16O 18OH ͓reaction ͑b͔͒ are
shown in Fig. 7. In the 18ONN 16O system ͓Fig. 7͑a͔͒, the
population of the initial state decreases to 1/4 during the first
500 fs. At the same time, the other dark states rise with the
same rate as the decline of the initial state. This means that
the IVR in the intermediate 18ONN 16O proceeds in several
For the excitation by a laser pulse, the initial zero-order
state is prepared by a quasi-delta-function pulse of light with
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J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Akagi, Fujimura, and Kajimoto
7271
diate, the population of the initial state is expected to decline
to ϳ1/2 during the interaction time of 250 fs estimated
above. Therefore, it is likely that for the 18O(1D)ϩN2 16O
reaction the necessary time for the IVR process is compa-
rable to the minimum ‘‘lifetime’’ of the intermediate. We
can conclude that IVR in the intermediate can proceed sig-
nificantly even though the reaction is a direct process.
In the case of the 16O(1D)ϩH2 18O reaction with the
deep well, the lifetime of the intermediate could be much
longer than that of the 18O(1D)ϩN2 16O reaction. However,
the IVR within the intermediate is quite slow as we showed
in the preceding section. Using the molecular parameter
given in Table IV, we have estimated the lifetime of the
energized H 16O 18OH to be about 4 ps based on the RRK
*
theory. Even within such a long lifetime, the IVR cannot
proceed in the H 16O 18OH system because of the extremely
slow IVR rate.
FIG. 7. Time profiles of the local-mode vibrational states in the reaction
intermediates; ͑a͒ 18ONN16O and ͑b͒ H16O18OH. The solid lines show the
initial state populations, the dotted lines correspond with the sums of the
direct-coupling states, and the dotted-and-dashed lines represent the sums of
the populations of the indirect-coupling states.
V. SUMMARY AND CONCLUSIONS
We have determined the internal state distribution of
‘‘old’’ NO(X 2⌸) generated from the reaction of S(1D) with
N2O. The average vibrational energy of NO relative to the
prediction by the statistical theory ͑PST͒ is found to be 37%.
This figure suggests fairly effective energy randomization in
the intermediate of the S(1D)ϩN2O reaction in spite of its
short lifetime, and supports the previous conclusion for the
18O(1D)ϩN2 16O reaction.3 The efficient IVR in the interme-
diate can be attributed to the presence of four heavy atoms in
the system which provides low frequency vibrations to in-
crease the state density and enhances the momentum cou-
pling to facilitate the IVR.
A model calculation based on the local-mode vibrations
with harmonic momentum coupling has been performed to
estimate the mass effect on the state density and the momen-
tum coupling. The coupling strength for the heavy atom sys-
tem is found to be one-order of magnitude larger than that
for the system containing hydrogen atoms. To estimate the
rate of IVR, the quantal time evolution of the initial zero-
order state has been calculated on the analogy of the coherent
fs pulse excitation. The zero-order initial state is defined as
the local-mode vibrational state where all available energy is
accommodated in the stretching vibration of the new bond.
The decay of such initial state takes about 250 fs for the
18O(1D)ϩN2 16O reaction, which is comparable to the esti-
mated lifetime of the intermediate. We conclude from these
calculations that the efficient energy randomization can oc-
cur even within a very short lifetime of the intermediate, if
the reaction system is composed of heavy atoms. The mass
effect on the momentum coupling and the state density plays
a primary role in the energy partitioning.
hundred fs. If we take into account more background zero-
order states in higher perturbation level, the population of the
initial state should further decrease. The similar but slightly
slower decay was observed for the SNNO system. The decay
in the H 16O 18OH system ͓Fig. 7͑b͔͒ strongly contrasts with
the 18ONN 16O case. In a period of 1 ps, the initial state
decays by only 2%. Therefore, the available energy in the
intermediate H 16O 18OH would not dissipate from the
16O–H local vibrational state to other degrees of freedom.
Although the real decay of the initial local-mode vibration is
expected to occur slightly faster than these calculations be-
cause of the participation of more zero-order states, we can
roughly estimate the time required for the IVR from these
simulations.
So far we mainly focus our attention on the magnitudes
v2
of
and ␳. However, the extent of IVR and hence the
energy partition to the ‘‘old’’ product is also determined by
another factor, the lifetime of the intermediate. Even though
the IVR rate is slow, long lifetime could give a sufficient
period for the IVR to complete. Therefore, we have esti-
mated the lifetime of the intermediate and compared this
figure with the IVR time obtained above.
The lifetime of the intermediate can be considered as the
time during which the four atoms of the reaction system are
interacting with each other. For the 18O(1D)ϩN2 16O reac-
tion, we have estimated the lower limit of the interacting
time. The velocity of 18O(1D) generated from the 18O3 pho-
tolysis at 266 nm is ϳ2000 m sϪ1, corresponding to the ve-
locity of about 2 Å per 100 fs.24 Since the average van der
Waals radius of a N2O molecule is 4.9 Å, it takes 250 fs for
an 18O(1D) atom to pass the interaction region in a nonreac-
tive encounter. Thus, 250 fs could be considered as the mini-
mum ‘‘lifetime’’ of the intermediate. For the S(1D)ϩN2O
reaction, such minimum lifetime would be longer because of
the low velocity ͑ϳ1000 m sϪ1͒ due to the heavy S atom.
From the model calculation of the 18ONN 16O interme-
ACKNOWLEDGMENT
This work was supported in part by a Grant-in-Aid for
Scientific Research on Priority Areas No. 07240102 from the
Ministry of Education, Science, and Culture in Japan.
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