Unimolecular reaction dynamics from Kinetic Energy Release Distributions. III. A comparative study of the halogenobenzene cations

Article abstract of DOI:10.1063/1.477934

The translational kinetic energy release distribution (KERD) in the halogen loss reaction of the chloro-, bromo-, and iodobenzene cations has been experimentally determined in the microsecond time scale and theoretically analyzed by the maximum entropy method. The KERD is constrained by the square root of the translational energy, i.e., by the momentum gap law. This can be understood in terms of quantum-mechanical resonances controlled by a matrix element involving a localized bound state and a rapidly oscillating continuum wave function, as in the case of a vibrational predissociation process. The energy partitioning between the reaction coordinate and the set of the remaining coordinates is nearly statistical, but not quite: less translational energy is channeled into the reaction coordinate than the statistical estimate. The measured entropy deficiency leads to values of the order of 80% for the fraction of phase space sampled by the pair of fragments with respect to the statistical value. In the case of the dissociation of the chlorobenzene ion, it is necessary to take into account a second process which corresponds to the formation of the chlorine atom in the excited electronic state ²P_1/2 in addition to the ground state ²P_3/2. The observations are compatible with the presence of a small barrier (of the order of 0.12 eV) along the reaction path connecting the D²A₁ state of C₆H₅Cl⁺ to the Cl(²P_1/2) + C₆H₅⁺(X¹A₁) asymptote.

Full text of DOI:10.1063/1.477934

Unimolecular reaction dynamics from kinetic energy release distributions. III. A
comparative study of the halogenobenzene cations
P. Urbain, B. Leyh, F. Remacle, A. J. Lorquet, R. Flammang, and J. C. Lorquet
Citation: The Journal of Chemical Physics 110, 2911 (1999); doi: 10.1063/1.477934
View online: http://dx.doi.org/10.1063/1.477934
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 110, NUMBER 6
8 FEBRUARY 1999
Unimolecular reaction dynamics from kinetic energy release distributions.
III. A comparative study of the halogenobenzene cations
P. Urbain, B. Leyh, F. Remacle, and A. J. Lorquet
D e´ partement de Chimie, Universit e´ de Li e` ge, Sart-Tilman, B-4000 Li e` ge 1, Belgium
R. Flammang
D e´ partement de Chimie Organique, Universit e´ de Mons-Hainaut, 19, avenue Maistriau, 7000 Mons,
Belgium
J. C. Lorquet
D e´ partement de Chimie, Universit e´ de Li e` ge, Sart-Tilman, B-4000 Li e` ge 1, Belgium
͑
Received 26 May 1998; accepted 4 November 1998͒
The translational kinetic energy release distribution ͑KERD͒ in the halogen loss reaction of the
chloro-, bromo-, and iodobenzene cations has been experimentally determined in the microsecond
time scale and theoretically analyzed by the maximum entropy method. The KERD is constrained
by the square root of the translational energy, i.e., by the momentum gap law. This can be
understood in terms of quantum-mechanical resonances controlled by a matrix element involving a
localized bound state and a rapidly oscillating continuum wave function, as in the case of a
vibrational predissociation process. The energy partitioning between the reaction coordinate and the
set of the remaining coordinates is nearly statistical, but not quite: less translational energy is
channeled into the reaction coordinate than the statistical estimate. The measured entropy deficiency
leads to values of the order of 80% for the fraction of phase space sampled by the pair of fragments
with respect to the statistical value. In the case of the dissociation of the chlorobenzene ion, it is
necessary to take into account a second process which corresponds to the formation of the chlorine
2
2
atom in the excited electronic state P1 in addition to the ground state P . The observations are
/2
3/2
compatible with the presence of a small barrier ͑of the order of 0.12 eV͒ along the reaction path
˜
2
ϩ
2
ϩ
5
˜
1
connecting the D A state of C H Cl to the Cl( P )ϩC H (X A ) asymptote. © 1999
1
6
5
1/2
6
1
American Institute of Physics. ͓S0021-9606͑99͒00906-X͔
I. INTRODUCTION
In this paper, we adopt an approach based on the analy-
sis, by the maximum entropy formalism,1
5–20
of experimen-
Does a unimolecular chemical reaction proceed statisti-
cally or is it possible to favor a chosen reaction channel by
selectively exciting some internal modes? This central ques-
tion in chemical physics has been for about 60 years an in-
centive to the development of a variety of experimental tech-
niques and theoretical methods.
tal fragment energy distributions for ionic unimolecular dis-
sociations. More precisely, we measure by sector tandem
mass spectrometry, under collision-free conditions, the trans-
lational kinetic energy released to the dissociation fragments
in the metastable time window (10 Ϫ10 s). As a conse-
quence of this time selection, the dissociating parent ions are
characterized by a relatively wide ͑about 0.5 eV͒ bell-shaped
internal energy distribution, centered at a given energy, here-
Ϫ6
Ϫ5
Mass spectrometric experimentation and modeling have
contributed to this debate. The vast majority of unimolecular
dissociations involving ions have been interpreted as statis-
tical processes. Ultrafast internal conversions and rapid in-
tramolecular vibrational energy redistribution ͑IVR͒ are as-
sumed to be completed before the dissociation step. Stated in
other terms, phase space is expected to be sampled in a uni-
after denoted E . The measured kinetic energy release dis-
S
tribution is then, in a second step, compared with the corre-
sponding purely statistical estimate, called the prior
distribution, for which all exit channels at a given total en-
ergy are considered to be equiprobable. From the observed
deviations from this microcanonical equilibrium, the maxi-
mum entropy method makes it possible to infer ͑i͒ the nature
of the dynamical constraints which are responsible for the
deviations and ͑ii͒ the fraction of phase space which is ef-
fectively sampled.
form way. Statistical models1 have been developed
–4
5–12
to
study unimolecular rate constants: variational transition state
theory, transition state switching model, phase space theory,
orbiting transition state theory, and statistical adiabatic chan-
nel model.
However, product energy distributions are much more
The set of reactions chosen for the present work is the
loss of the halogen atom from ionized halogenobenzene cat-
sensitive to any deviation from
a
pure statistical
3
,13,14
ϩ
situation.
Compared with the rate constants, they pro-
ions ͑C H X with XϭCl, Br, I͒, which have previously
6
5
21–38
vide us with complementary information, because they re-
veal the sampling of the phase space associated with the
dissociation fragments.
received extensive attention.
A first account dealing
only with the bromobenzene case has already been
published.39 As far as bromo- and iodobenzene are con-
0021-9606/99/110(6)/2911/11/$15.00
2911
© 1999 American Institute of Physics
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1

2912
J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
Urbain et al.
˜
cerned, these studies have concluded that the dissociation
proceeds statistically, via an extremely loose transition state.
the center-of-mass frame͒, P(␧), can be obtained by differ-
entiation of the measured ion kinetic energy spectrum fol-
lowed by a transformation of variables from the laboratory
coordinates to the center-of-mass coordinates.
ϩ
However, in the case of the loss of Cl from C H Cl , the
6
5
data of Pratt and Chupka24 and of Yim and Kim were best
accounted for by a model involving a nontotally loose tran-
sition state. As will appear in the present paper, the latter
dissociation turns out to be a much more challenging prob-
lem.
37
43–45
Carrying
out this differentiation either numerically or using the
Holmes–Osborne procedure43 led to no significant differ-
ence. In the decompositions investigated in this paper, the
amount of kinetic energy released to the fragments in the
laboratory frame is negligible ͑Ͻ50 eV͒ compared with the
kinetic energy of the center of mass ͑a few keV, depending
on the accelerating voltage͒, so that discrimination effects
can be neglected. This eliminates the need to calculate ex-
plicitly the basis functions describing the response of the
kinetic energy analyzer to a hypothetical kinetic energy re-
lease distribution expressed by a Dirac delta function. The
electrostatic analyzer exit slit ͑␤-slit͒ has been closed to 0.25
mm to reach an energy resolution ⌬E/E of 10 , which
makes any deconvolution procedure unnecessary.
Accelerating voltage scan spectra have been recorded for
fragment ion translational energies ranging between 1 and 5
keV ͑in the laboratory frame͒. B /E as well as magnet scan
spectra have been measured with an accelerating voltage
equal to 8 kV. The following ion source conditions were
applied. Trap current: 30␮A. Electron energy: 70 eV, except
for the magnet scan spectra for which the electron energy has
been reduced to 14.6 eV in order to eliminate overlap with
more intense signals coming from fragment ions produced in
the ion source. Ion source pressure ͑measured at an ion
gauge located approximately 15 cm from the ionization
chamber͒: 10 Pa. The pressure in the field-free regions
was kept in the 10 Pa range, in order to avoid collision-
induced dissociations. Chloro-, bromo-, and iodobenzene
provided by Fluka ͑99.5% stated purity͒ were carefully de-
gassed but otherwise used without further purification.
Furthermore, although the parent ion internal energy dis-
tributions are quite wide, as previously mentioned, some
control on the time selection and thus on the average internal
energy is nevertheless possible by working at different ion
source accelerating voltages or by using different field-free
regions. In the present work, we could shift the internal en-
ergy distribution by about 0.15–0.2 eV. This shift has been
instrumental for the unambiguous identification of the dy-
namical constraint involved in the dissociation process. In
principle, it makes it also possible to investigate how phase
space sampling varies with internal energy.
The paper is organized as follows. The experimental
method is described briefly in Sec. II and the results are
presented in Sec. III. Next, the salient points of the maxi-
mum entropy formalism relevant to the present experimental
framework are reviewed in Sec. IV. The dissociations of the
bromo- and iodobenzene ions show a very similar behavior
and are discussed together in Sec. V. The influence of the
internal energy on the efficiency of phase space sampling is
studied in Sec. VI. Section VII is devoted to the more com-
plex case of chlorobenzene. Concluding remarks are pre-
sented in Sec. VIII.
Ϫ3
2
Ϫ4
Ϫ6
II. EXPERIMENT
A. Kinetic energy release distributions
Our spectra have been recorded with a forward geometry
two-sector ͑AEI-MS9͒ mass spectrometer, i.e., an instrument
where an electrostatic analyzer is followed by a magnetic
sector. As we are interested in metastable dissociations, it is
very desirable to carry out experiments with various time
windows and thus with various internal energy contents of
the dissociating ions. For this purpose, dissociations taking
place in both the first field-free region ͑between the ion
source exit slit and the electrostatic analyzer͒ and the second
field-free region ͑between the electrostatic analyzer and the
magnet͒ and in different accelerating voltage ranges have
been monitored. Either the accelerating voltage scan
B. Internal energy distribution of the metastable
parent ions
The dissociations sampled in our experiments corre-
spond to a time window defined by the entry and exit times,
␶1
and ␶ , in the field-free region. These times can be easily
2
calculated using elementary physics based on the spectrom-
eter geometry parameters and operating conditions. Accord-
ingly, the measured kinetic energy release distributions do
not correspond to a well defined energy but to a distribution
of internal energies, T(E). Indeed, the corresponding collec-
tion efficiency T depends on the rate constant which, in turn,
depends on the internal energy E:
4
0
method ͑V-scan͒ or the linked scanning of the electrostatic
2
41
analyzer and the magnet at constant B /E ratio were used
to record fragment ion kinetic energy spectra for dissocia-
tions taking place in the first field-free region. Magnet
T͓k͑E͔͒ϭB͓exp͑Ϫk͑E͒␶ ͒Ϫexp͑Ϫk͑E͒␶ ͔͒.
͑2.1͒
1
2
4
2
scanning was used to investigate dissociations in the sec-
ond field-free region.
Throughout this paper, the internal energy, E, is ex-
pressed with respect to the dissociation asymptote. B is a
normalization constant. The functions k(E) have been ex-
perimentally determined by various authors.2
The translational kinetic energy released to the frag-
ments during a dissociation brings about a signal broadening
which is detected in the corresponding ion kinetic energy
spectrum measured in the laboratory reference frame. From a
physical point of view, one is interested in the kinetic energy
released in the center-of-mass frame. When instrumental
broadening as well as discrimination in the direction perpen-
dicular to the ion flight path can be neglected, the KERD ͑in
1–31,33,34,36,37
Their energy dependence in the domain covered by T(E) has
been fitted to the following empirical form using the differ-
ent experimental values available from the literature:
␯
E
k͑E͒ϭkoptͩ ͪ
,
͑2.2͒
E
s
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J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
Urbain et al.
2913
FIG. 1. Collection efficiency T(E) ͓Eq. ͑2.1͔͒ for the metastable dissocia-
FIG. 2. Collection efficiency T(E) ͓Eq. ͑2.1͔͒ for the metastable dissocia-
ϩ
ϩ
tion C H I →C H ϩI taking place in the two field-free regions of the
6
5
6
5
ϩ
ϩ
tion C H Cl →C H ϩCl taking place in the two field-free regions of the
6
5
6
5
AEI-MS9 mass spectrometer under different experimental conditions de-
fined in the figure.
AEI-MS9 mass spectrometer under different experimental conditions de-
fined in the figure.
where kopt is the value of the rate constant at the energy E at
which T(E) reaches its maximum.
S
cation in the first field-free region with a fragment ion kinetic
energy of 1 and 3 keV ͑in the laboratory frame͒ are presented
in Figs. 3 and 4. ͑Similar data for the bromobenzene ion can
The excitation process is by electronic impact and thus
inherently nonselective. The experimental data are therefore
the result of an average over the range of internal energies
defined by T(E). The measured distribution of translational
39
be found in Part II of this series. ͒ These results are aver-
ages of 12 distributions recorded over a period of one year.
1/2
2
͗ ͘ ͗ ͘
The values of the first few moments ͑͗␧͘, ␧ , and ␧ ͒
˜
energies P(␧) is given by:
are presented in Table I. The average kinetic energies ͗␧͘ are
ϱ
in good agreement with the previous values obtained by Bur-
˜
P͑␧͒ϭ
͵
T͑E͒P͑␧
͉
E͒dE,
͑2.3͒
28
gers and Holmes. In particular, we confirm their slow
␧
variation with the internal energy.
where P(␧
͉E) is the translational kinetic energy release dis-
tribution at a given internal energy E.
IV. THE MAXIMUM ENTROPY METHOD
Figures 1 and 2 show the T(E) functions obtained under
three different sets of experimental conditions, correspond-
ing to three different time windows, for the loss of I or Cl
from ionized iodobenzene or chlorobenzene, respectively.
These data illustrate the extent of the upward shift of T(E)
with decreasing lifetime. The values of kopt , E , ␯, ␶ and
An analysis of a product state distribution by the maxi-
mum entropy method consists of comparing the actual, ex-
perimentally observed distribution P(␧͉E) with the prior dis-
0
15–20
tribution P (␧
͉
E) which serves as a reference.
The
decisive advantage of this formalism is that its functional
form relating the experimentally observed KERD to the prior
distribution is not simply an empirical parametrization. It
involves constraints which are related to observables and
which, therefore, possess a physical significance. A different
set of constraints implies a difference in the reaction mecha-
nism.
S
1
␶2
are summarized in Table I for the three halogenoben-
zenes.
III. RESULTS
The kinetic energy release distributions have been re-
corded under various experimental conditions corresponding
to different internal energy distributions. In the first field-free
region, we investigated the following processes using the
V-scan technique. In some cases, the results have been
The prior distribution is the least biased distribution, that
is the most statistical distribution in phase space, where no
15–20
dynamical effects have been included.
Its entropy is
therefore maximal. This distribution is defined as that for
which all quantum states at a given total energy are equi-
probable, i.e., every group of product states that is energeti-
cally allowed is populated with a probability proportional to
the number of quantum states of that group. Therefore, the
only constraint that is included in the counting of the quan-
tum states is the conservation of the total energy. The con-
servation of the total angular momentum is an additional
general constraint ͑good constant of motion of the total
Hamiltonian͒ that limits the number of final states that are
accessible. However, the inclusion of the conservation of the
total angular momentum requires a theoretical decision about
2
checked by the B /E linked scanning method.
ϩ
ϩ
5
C H Cl →C H ͑5 and 1 keV͒ϩCl,
6
5
6
ϩ
ϩ
5
C H Br →C H ͑4 and 1 keV͒ϩBr,
6
5
6
ϩ
ϩ
5
C H I →C H ͑3 and 1 keV͒ϩI.
6
5
6
These dissociations have also been all monitored in the
second field-free region,
ϩ
ϩ
5
C H X ͑8 keV͒→C H ϩX͑XϭI,Br,Cl).
6
5
6
As typical examples, the kinetic energy release distribu-
˜
46
tions P(␧) obtained for the dissociation of the iodobenzene
the ‘‘cutoff’’ ͑one has to decide what the value of Jmax is͒
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2914
J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
Urbain et al.
ϪDS
TABLE I. Time window and most probable internal energy E , first moments of the KERDs, and values of ␭ , DS, and e
for EϭES obtained for the
S
0
different metastable decompositions investigated in this work. For first field-free region ͑1st FFR͒ experiments, the fragmentation translation energy in the
laboratory frame is mentioned. Experiments in the second field-free region ͑2nd FFR͒ have been conducted with an accelerating voltage of 8 kV. The relative
ϪDS
uncertainty on the average KER ͗␧͘ is estimated to be of the order of Ϯ3%. The uncertainty on e
is equal to Ϯ0.03.
ϩ
ϩ
5
ϩ
ϩ
5
ϩ
ϩ
5
C H Cl →C H ϩCl
C H Br →C H ϩBr
C H I →C H ϩI
6
5
6
6
5
6
6
5
6
1
st FFR
1 kV
1st FFR
5 kV
1st FFR
1 kV
1st FFR
4 kV
1st FFR
1 kV
1st FFR
3 kV
2nd FFR
2nd FFR
2nd FFR
␶1
␶2
͑␮s͒
͑␮s͒
9.89
13.3
1.13
0.863
8.95
4.16
8.22
1.23
1.68
1.53
3.35
1.30
4.3
11.75
15.80
0.67
0.73
6.30
0.219
0.059
0.0069
5.4
Ϫ1.36
0.16
4.16
8.22
0.75
1.68
6.05
0.221
0.060
0.0073
6.2
Ϫ1.56
0.22
1.74
3.77
0.85
3.8
6.00
0.221
0.060
0.0069
6.7
13.35
17.95
0.47
0.64
5.40
0.196
0.046
0.0038
5.8
Ϫ1.33
0.15
4.16
8.22
0.55
1.68
4.80
0.194
0.045
0.0037
7.2
Ϫ1.65
0.24
2.00
4.34
0.62
3.3
4.50
0.206
0.051
0.0049
6.4
ES ͑eV͒
kopt (10 sϪ1)
␯
5
8.15
7.47
␧1/2
␧͘ ͑eV͒
͑eV1/2
͒
0.269
0.090
0.0156
0.275
0.096
0.0183
0.269
0.091
0.0161
͗
͗
͗
͘
2
2
␧
͘
͑eV ͒
͑eV
Ϫ1/2
a
a
a
␭1
␭0
͒
5.1
4.9
6.0
a
a
a
Ϫ1.47
0.19
0.83
Ϫ1.45
0.19
0.83
Ϫ1.74
0.27
0.76
Ϫ1.72
0.26
0.77
Ϫ1.55
0.21
0.81
a
a
a
a
a
DS
ϪDS
a
e
0.85
0.80
0.86
0.79
a
ϩ
Data corresponding to the main reaction channel of C H Cl .
6
5
and therefore an assumption about the dynamics. In the
present work, following the point of view expressed in Ref.
with
A ϭ␮3/2/͑21/2␲ h ͒,
2
3
͑4.2͒
T
17, we choose to retain for the prior distribution the least
biased distribution without including the conservation of the
total angular momentum. Moreover, since in the studied re-
actions the reduced mass along the reaction coordinate is
sufficiently large to lead to high values of the orbital angular
momentum, the discrepancy between the experimental and
the prior distributions is not expected to be due to the con-
straint imposed by the conservation of the total angular
where ␮ is the reduced mass along the reaction coordinate.
At a given total energy E, the rovibrational density of
states N(EϪ␧) can be approximated by a continuous func-
tion. The prior distribution is therefore given by
0
E͒ϭA͑E͒␧1/2N͑EϪ␧͒,
P ͑␧
͉
͑4.3͒
where A(E) is a normalization factor.
4
7
This functional form for the prior distribution is equiva-
lent to that derived for the actual distribution in phase space
theory49 in the case of a loose complex bound by long-range
momentum.
The actual distribution P(␧
͉
E) measures the transla-
tional energy of the pair of fragments. Therefore, a definition
of the prior distribution for translational states is needed. The
50
polarizability forces. The difference is one of methodology,
i.e., we first get the functional form ͑4.3͒ by taking for the
prior distribution the least biased one, without making any
assumption on the form of the potential along the reaction
coordinate or on the ‘‘cutoff’’ value of the total angular mo-
mentum. Then, in a second stage discussed below ͓Eqs. ͑4.4͒
three-dimensional density of translational states per unit vol-
ume and per unit energy4
distribution,
,17,19,48
is, for an isotropic angular
␳T͑␧͒d␧ϭA ␧1/2d␧,
͑4.1͒
T
FIG. 3. Experimental ͑solid line͒ and fitted ͑dots͒ kinetic energy release
FIG. 4. Experimental ͑solid line͒ and fitted ͑dots͒ kinetic energy release
ϩ
ϩ
ϩ
ϩ
distributions obtained for the metastable dissociation C H I →C H ͑1
distributions obtained for the metastable dissociation C H I →C H ͑3
6
5
6
5
6
5
6
5
keV͒ϩI ͑first field-free region͒. The prior distribution ͓Eq. ͑4.3͔͒ at EϭE is
keV͒ϩI ͑first field-free region͒. The prior distribution ͓Eq. ͑4.3͔͒ at EϭE is
S
S
drawn in dashed lines.
drawn in dashed lines.
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1

J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
Urbain et al.
2915
and ͑5.1͔͒, we parametrize the discrepancy between the ac-
tual ͑experimental͒ and the prior distributions using the
maximum entropy formalism.
Lagrange parameters, together with the averaging over the
distribution of internal energies ͑2.3͒, was found to fit ad-
equately all our data in the case of both the iodo- and bro-
mobenzene ions. Figures 3 and 4 show the fits obtained for
The rovibrational density of states, N(EϪ␧), is gener-
ated by a Beyer–Swinehardt algorithm2
,4,51
where the rota-
ϩ
the C H I ion in the first field-free region with a fragment
6
5
tional parameters and the vibrational frequencies have been
ion kinetic energy of 1 and 3 keV in the laboratory frame. In
the case of chlorobenzene, the functional form ͑5.1͒ involv-
ing a single constraint did not lead to satisfactory fits, as
discussed in detail in Sec. VII.
calculated ab initio at the RHF/6-31G(d) level.3
8,39
In the
case of the chlorobenzene cation, a rovibronic density of
states is calculated, because, as discussed in Sec. VII below,
2
the Cl atom is assumed to be generated in both of its P
As we now elaborate, the rationale for Eq. ͑5.1͒ is that
the reaction can essentially be thought of as a vibrational
predissociation. The definition of the latter term requires
3/2
2
and P electronic states.
1
/2
In the second step of our analysis, we seek a quantitative
parametrization of the discrepancy between the prior and the
4
qualifications. In the particular case of van der Waals com-
actual
formalism,
distributions.
In
the
maximum
entropy
plexes, the rate of decay associated with this process has
been extensively studied. These complexes fragment in a
very nonstatistical way and their rate constants are orders of
magnitude slower than the statistical limit because the in-
tramolecular energy redistribution is hindered by the fre-
1
6,17,19,20
the actual distribution is shown to be re-
lated to the prior distribution in the following way:
n
0
P͑␧
͉
E͒ϭP ͑␧
͉
E͒exp Ϫ␭ Ϫ
␭rAr
,
͑4.4͒
ͩ
0
͚
ͪ
rϭ1
quency mismatch between the weak van der Waals bond and
the stronger bonds of the molecular moieties.4
,54–58
The frag-
where the quantities A are physical observables and the ␭
are the conjugated Lagrange parameters. The average values
r
r
mentation mechanism of the metastable halogenobenzene
cations studied here presents obvious differences with re-
spect to the case of van der Waals complexes, the main one
being that IVR is far more complete, with the result that their
͗
Ar
P(␧
͘
͉
of the observables Ar over the actual distribution
E) are the constraints. P(␧ E) in Eq. ͑4.4͒ is the unique
͉
͗ ͘
normalized distribution that satisfies the n constraints Ar
and that is otherwise of maximum entropy.
The entropy deficiency DS is the difference between the
entropy of the prior distribution ͑denoted S ͒ and that of the
dissociation is much closer to the statistical
limit.2
1–26,30,31,35–37,39,59
Quantum mechanically, however, in
0
spite of these differences, both types of vibrational predisso-
ciations can be described in terms of metastable states or
resonances, excited above the lowest threshold for dissocia-
tion. The rate of decay of these quasibound states is con-
actual one ͑denoted S͒,
E
P͑␧
͉
͉
E͒
0
DS͑E͒ϭS ͑E͒ϪS͑E͒ϭ
͵
P͑␧
͉
E͒ln
ͫ
ͬ
d␧. ͑4.5͒
0
P ͑␧
E͒
0
trolled by matrix elements, denoted
͗ ͘
n͉V͉␧ , between the
localized wave function, , of a bound state excited above
͉n
͘
The entropy deficiency DS is always positive or equal to
ϪDS
the lowest fragmentation asymptote and an oscillating wave
function along the reaction coordinate for the pair of frag-
ments, ͉␧͘. Each particular predissociation process involves a
zero and the quantity e
measures the fraction of phase
space effectively sampled by the pair of fragments.5
2,53
specific coupling operator V and different types of
͉
n
͘
and
V. THE DISSOCIATION OF THE BROMO AND
IODOBENZENE CATIONS
͉
␧͘ wave functions. However, one common feature is that
54–56,60
these matrix elements
are found to decay exponen-
ϩ
tially as a function of the square root of the kinetic energy, ␧,
and become already negligible for values of ␧ smaller than
the highest value allowed by conservation of the total en-
ergy, E. Although this was first recognized in the study of
van der Waals complexes, the fact that, well above the dis-
sociation threshold, the decay widths obey the so-called ‘‘ex-
ponential momentum gap law’’5 indicates that the propen-
sity rule applies to other predissociation cases as well.
The momentum gap law can be related to the functional
For the unimolecular fragmentation of the C H Br ion
6
5
previously investigated,39 very good least-squares fits led us
to argue that the appropriate functional form involves a
single constraint, viz., ␧1 , so that Eq. ͑4.4͒ becomes
/2
0
E͒exp͓Ϫ␭ Ϫ␭ ␧1/2͔.
P͑␧
͉
E͒ϭP ͑␧
͉
͑5.1͒
0
1
4,56
We emphasize that this conclusion concerning the nature
of the informative constraint results not only from the quality
of the individual fits but also from self-consistency checks
based on the analysis of KERDs corresponding to different
energy windows. The shape of the experimental distribution
does not show any fine structure, so that a single constraint
can be expected to generate a good fit, at least in the case of
bromo- and iodobenzene. In principle, ␭1 depends on the
internal energy. However, this energy dependence was ne-
glected in the energy domain covered by a given transmis-
sion function ͑see Figs. 1 and 2͒. The average value of both
form of the constraint, ␧ ͓Eq. ͑5.1͔͒ that we have identified
ͱ
in the maximum entropy analysis of the experimental distri-
butions. It implies that the final states with a high value of ␧
are not produced during the predissociation process and are
excluded from the sampling of phase space. Final states with
a low value of ␧ and comparatively more energy in the rota-
tional and vibrational degrees of freedom of the fragments
are favored with respect to the prior distribution, which leads
1
/2
␧
and ␧ were indeed observed ͑Table I͒ to be nearly the
to a positive value of the Lagrange multiplier ␭ in Eq. ͑5.1͒.
1
same in the different energy windows, a fact which is con-
firmed by independent measurements carried out by other
authors. The functional form ͑5.1͒ with energy independent
The average kinetic energy of the experimental distribution,
͗␧͘, is smaller than the purely statistical one, computed with
the prior distribution ͓Eq. ͑4.3͔͒.
2
8
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2916
J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
Urbain et al.
Information on the statistical nature of the reaction can
be obtained via an analysis of the efficiency of phase space
pressions for the moments of the KERD. The latter can be
expressed61 as generalized hypergeometric series of the di-
ϪDS
1/2
sampling. The values obtained for ␭ , DS, and e
at
mensionless reduced variable ␭ E . Therefore, when either
1
1
EϭE in the case of bromo- and iodobenzene dissociations,
E or ␭ is equal to zero, these polynomials reduce to their
S
1
ϪDS
ϪDS
are presented in Table I. The quantity e
measures the
constant term and e
reaches its maximum value of one,
fraction of phase space sampled by the pair of fragments
with respect to the statistical value. Values of the order of
in agreement with the discussion above. It decreases in an
2
approximately Gaussian way as ␭ E increases.
1
80% are obtained. In these simple bond cleavage reactions,
These expectations can be compared with the experi-
the potential energy curve along the reaction coordinate is
mental findings reported in Table I where, in the quite nar-
ϪDS
thought to increase smoothly without giving rise to an energy
row energy range investigated, e
is observed to undergo
barrier and hence to an obvious transition state.3
0,31
This has
a slight decrease as E increases. Although slight, these trends
are believed to exceed the error limits.
38
been confirmed by Klippenstein’s ab initio calculations.
The discussion given in part II of this series39 should be
Equation ͑6.1͒ has been applied to the case at hand. The
value of s that best fits the rovibrational density of states of
the pair of fragments calculated by the Beyer–Swinehardt
ϩ
repeated here. Long-range forces between the planar C H5
6
fragment and the bromine or iodine atoms presumably con-
tain negligible anisotropic terms and are hardly expected to
generate torques. Since the fragments separate with a very
small relative velocity, no energy exchange takes place be-
tween the receding fragments. Hence, the above-mentioned
values are thought to be related to the fraction of phase space
accessible to the transition state. This confirms the general
consensus about the statistical character of the reaction.
Ϫ1/2
algorithm is equal to 10. From Table I, a value of 6 eV
was adopted for the parameter ␭ . This leads to values of
1
ϪDS
e
equal to 0.85, 0.83, and 0.81 at Eϭ0.67, 0.75, and 0.85
eV, respectively. The decrease is not as steep as that ob-
served in the experiment, but the order of magnitude is cor-
ϩ
rect and the trend observed in C H Br is reproduced. To
6
5
account for that in iodobenzene, a slight decrease of ␭ as E
1
increases has to be assumed, which is reasonable.
This general behavior seems to be borne out by indepen-
dent experiments on the vinyl bromide ion, extending over a
broader range of internal energies, to be reported in a future
VI. INFLUENCE OF THE INTERNAL ENERGY ON
PHASE SPACE SAMPLING
The quantity eϪDS is the ratio between the volume of
phase space effectively sampled and the total accessible vol-
ume. Does the efficiency of phase space exploration increase
or decrease as the internal energy E varies?
62
paper.
VII. THE DISSOCIATION OF THE CHLOROBENZENE
CATION
Right at threshold, the discrimination introduced by the
The KERD of the chlorobenzene cation cannot be fitted
by the equations which satisfactorily describe the behavior of
the bromo- and iodobenzene ions ͑Fig. 5͒. Equation ͑5.1͒,
together with ͑2.3͒, leads to a poor fit of the experimental
KERD, as shown in the top panel of Fig. 5. In addition, the
momentum gap law is inoperant because ␧ϭ0. Hence, DS
ϪDS
ϭ0 and e
ϭ1. Phase space is totally explored when it is
reduced to a single cell. As the internal energy increases, the
highest possible value of ␧ increases too, and the ratio be-
tween the volume in phase space allowed by the momentum
gap law and the total volume in phase space computed from
the prior distribution decreases. However, at sufficiently high
energies, the momentum gap law can be expected to be less
restrictive on purely statistical grounds. The number of states
characterized by a dispersion of the entire excess energy E
among the subset of internal degrees of freedom increases
much more rapidly than those where a significant part of E
has flown into the reaction coordinate. In addition to this
purely statistical argument, the influence of anharmonicity,
and thus the efficiency of IVR, is expected to increase with
˜
averaged surprisal, I,
˜
P͑␧͒
0
˜
IϭϪln
,
ͫ
ͬ
˜
P ͑␧͒
where
ϱ
˜
0
0
P ͑␧͒ϭ
͵
P ͑␧
͉
E͒T͑E͒dE,
␧
˜
with P(␧
͉
E) given by Eq. ͑2.3͒, is plotted in the lower panel
˜
of Fig. 5 as a function of
local measure of the relevance of the constraint. Its graph
should remain linear in its entire range if ␧ is the only
ͱ
␧. The averaged surprisal I is a
E, leading to a decrease of ␭ and thus to an easing of the
1
59
constraint.
ͱ
Insight into this problem is provided by modelization. If
the density of states N(E) is assumed to increase as the s-th
constraint present in the range of values of ␧ sampled by the
experimental distribution. ͑Average surprisal graphs were
also calculated for the bromo- and iodobenzene ions. They
are indeed observed to fulfill this condition.͒ Thus, both pan-
els of Fig. 5 suggest that a second, rather weak component
contributes to the KERD for translational energies larger
than about 130 meV.
power of the energy, as in the classical approximation,1
,2,4
then the analytical approximation to the KERD is given by
P͑␧ E͒ϭ͓⌫͑sϩ2.5͒/⌫͑1.5͒⌫͑sϩ1͔͒
͉
1/2
ϫeϪ␭0EϪsϪ1.5␧1/2͑EϪ␧͒se
Ϫ␭ ␧
,
͑6.1͒
1
where s is related to the effective number of degrees of free-
dom.
This second component cannot be due to more energetic
dissociations induced by collision with the residual gas in the
field-free region. Appearance energy measurements con-
ducted by scanning the energy of the ionizing electrons did
The classical parametrization of the density of states has
well-known shortcomings,1 but it provides analytical ex-
,2,4
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J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
Urbain et al.
2917
be generated both in their ground and electronic excited
2
2
2
states, P3 and P , while only the ground state, P , is
/2
1/2
3/2
ϩ
ϩ
accessible for the C H ϩBr and C H ϩI fragments. The
6
5
6
5
maximum entropy analysis including this second channel
leads to tight fits and indicates that for this second channel
the release of the internal energy into the translation degrees
of freedom is favored. We then discuss how this could be
related to the presence of a barrier along the reaction coor-
dinate.
؉
5
A. Structure of the C H fragment
6
In the case of chlorobenzene, the internal energy content
of the dissociating parent ion is larger by about 0.5 eV than
that of both ionized bromo- and iodobenzene ͑see Table I͒.
ϩ
This could open an isomerization pathway for the C H5
6
fragment ion, resulting in a different variation of the density
of states with internal energy and thus to a different prior
distribution. The problem would then have to be treated as a
two-exit-channel dissociation.
Tandem mass spectrometric techniques are particularly
suitable to probe ionic structures. The major problem which
ϩ
5
arises to carry out mixture analysis on C H is the absence
6
ϩ
of well-established structurally pure C H standards. From
6
5
the available literature,6
3,64
it can be concluded that either
acetylene condensation reactions at high pressure or ion-
assisted dehalogenation reactions of halogenobenzene are
ϩ
thought to lead to a majority of C H5 ions with the cyclic
6
phenylium structure. Ab initio calculations at the self-
65
consistent-field ͑SCF͒ 4-31G level, as well as MINDO/3
calculations,66 show the occurrence of stable open-chain
structures which are 0.65–0.7 eV more energetic than the
phenylium cation.
FIG. 5. ͑Top͒ solid line: experimental kinetic energy release distribution
ϩ
ϩ
obtained for the metastable dissociation C H Cl →C H ͑5 keV͒ϩCl ͑first
6
5
6
5
field-free region͒. The dotted line represents the ͑unsuccessful͒ fit to the
one-parameter equation ͑5.1͒. The prior distribution ͓Eq. ͑4.3͔͒ at EϭES is
drawn in dashed lines. ͑Bottom͒ nonlinear surprisal plot showing the occur-
rence of an additional contribution to the KERD.
According to Cooks et al.63 and to Eyler and Campana,64
the phenylium structure favors hydrogen loss ͑leading to a
fragment ion at m/zϭ76͒ and charge stripping ͑doubly
charged ion at m/zϭ38.5͒. The open-chain structure favors
the loss of C2H2 , leading to a signal at m/zϭ51. As a con-
sequence, the ͓m/zϭ76͔/͓m/zϭ51͔, ͓m/zϭ50͔/͓m/z
ϭ51͔, and ͓m/zϭ38.5͔/͓m/zϭ37͔ intensity ratios are diag-
nostic for the cyclic structure.
not show any component below the appearance energy of the
ϩ
5
C H ϩCl fragments, as would be expected if collision-
6
induced dissociations were present.
The additional component could be related to either a
second dissociation channel or to an additional dynamic con-
straint. We have systematically investigated these possibili-
ties, using the maximum entropy method.
The introduction of an additional dynamic constraint
turned out to be unsuccessful. Neither a modification of the
power to which ␧ is raised, nor trying the vibrational or the
rotational energies as constraints, nor simultaneously adopt-
ing two constraints, nor allowing the Lagrange parameters to
vary with the internal energy led to an improvement of the
In order to check for the ionic structure reached in the
metastable dissociations considered in this work, a series of
collisionally activated dissociation ͑CAD͒ experiments were
ϩ
carried out on C H5 ions produced either in the ion source,
6
or resulting from a metastable decomposition. These experi-
ments have been carried out with the six-sector
67
(E B E E B E ) AutoSpec 6F mass spectrometer at the
1
1
2
3
2
4
University of Mons-Hainaut. This spectrometer has been re-
cently modified by the insertion of an r.f.-only quadrupole
6
8
fit. However, at the higher internal energies sampled for
collision cell between E and E . This instrument has the
2 3
ϩ
3
chlorobenzene, the C H5 ion could conceivably isomerize,
MS capabilities necessary to perform CAD on ions result-
ing from a previous metastable dissociation. The MS ex-
periments were carried out as follows. C H X ions were
6
3
thereby leading to a different density of states for the prod-
ucts. We therefore carefully investigated the structure of the
ϩ
6
5
ϩ
5
C H ion, conducting additional experiments which are de-
transmitted through the E B E sectors and underwent uni-
6
1 1 2
scribed in Sec. VII A below. From these experiments, we
conclude that only the phenylium ion is formed when the
halogenobenzene cations fragment. In Sec. VII B, we show
molecular dissociation in the quadrupole cell floated at a
voltage similar to the 8 kV accelerating voltage. The so-
produced m/z 77 ions were then reaccelerated at ϳ8 keV,
mass-selected by E B , and subjected to collisional activa-
ϩ
that for the fragmentation of the C H Cl ions, due to the
6
5
3
2
small value of the spin-orbit coupling, the chlorine atoms can
tion with nitrogen in a collision cell between B2 and
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2918
J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
Urbain et al.
ϩ
TABLE II. Peak height ratios measured in MIKES-CAD spectra of C H5
6
ions.
͓
m/zϭ76͔
͓m/zϭ50͔
͓m/zϭ51͔
͓
m/zϭ51͔
a
b
b
b
a
b
b
b
C H Cl
0.20 /0.74
0.52 /1.13
6
5
a
a
C H Br
0.21 /0.84
0.56 /1.12
6
5
a
a
C H I
0.23 /0.94
0.50 /1.11
6
5
a
ϩ
5
ϩ
The C H ions result from dissociation of C H X in the ion source. The
6
6
5
CAD spectra ͑target gasϭO ͒ are recorded by scanning E .
2
3
b
ϩ
ϩ
The C H ions result from metastable dissociation of C H X in the quad-
6
5
6
5
rupole located between E2 and E . CAD ͑target gasϭN ͒ takes place be-
3
2
tween B2 and E4 .
ϩ
E . MIKES spectra were then recorded by scanning E .
FIG. 6. Schematic set of potential energy curves for the C H Cl
4
4
6
5
ϩ
Such a procedure preserves, therefore, the resolution and the
sensitivity whatever the mass difference between the
→C6H5 ϩCl reaction. Experimental data about the position of electronic
states of the ion and about the dissociation asymptotes are represented by
heavy dots and full lines, respectively. The dashed lines represent mere
correlations. Inset: results of ab initio calculations ͑see text͒ on the action of
a point charge on a chlorine atom.
ϩ
ϩ
C H X and C H ions. The results obtained with the three
6
5
6
5
ϩ
5
halogenobenzene precursors for C H ions produced either
6
in the ion source or in a metastable dissociation are summa-
rized in Table II.
The criteria discussed above lead us to the conclusion
ϩ
ϩ
The production of a significant amount of C H5 in its
6
that the C H ions resulting from a metastable dissociation
6
5
excited triplet state can be discarded by a statistical argument
contain a larger proportion of phenylium ions than those pro-
based on the substantial energy gap ͑Ϸ0.8–1.0 eV͒6
9–71
be-
duced in the ion source under higher internal energy condi-
ϩ
tween the ground and excited states. At values of the internal
energy close to E , the density of the vibrational-rotational
states of C H5 is much smaller ͑by at least three or four
orders of magnitude͒ in the triplet than in the ground singlet
state.
tions. Furthermore, the CAD spectra of C H5 ions resulting
6
S
from metastable dissociations with the three different precur-
sors are identical within experimental error, so that these
ions have the same structure ͑or, at least, correspond to the
same structural mixture͒ whatever the selected precursor. A
rapid interconversion between cyclic and acyclic structures is
not very probable, as CAD allows us to distinguish between
ϩ
6
However, very good results were obtained when the gen-
eration of the Cl atom in both of its states, P3 and P
2
2
,
/2
1/2
ϩ
5
was taken into account. In contradistinction to the Br and I
atoms, the small value of the spin-orbit splitting in the Cl
atom ͑0.11 eV͒ with respect to the average internal energy
makes it statistically possible to generate both states in com-
petition. This higher threshold energy channel induces a sec-
ond component in the kinetic energy distribution. Since the
available information is the kinetic energy distribution of the
different ways of producing C H . As the ͓m/z
6
ϭ50͔/͓m/zϭ51͔ ratio is even larger than the ratio measured
by Eyler and Campana in the case of ion-assisted dehaloge-
nation of fluorobenzene, we may conclude that pure pheny-
lium ions result from the dissociation of the three investi-
gated halogenobenzene cations. At this point, it must also be
mentioned that appearance energy measurements of metasta-
ϩ
5
pair of fragments irrespective of the electronic state of the
bly produced C H from the three precursors are compatible
6
72,73
chlorine atom, a ‘‘synthetic’’ analysis
of the experimen-
with a unique structure for the three cases. All these obser-
vations confirm the high activation barrier for cycle
tal results was carried out. It was assumed that each rovibra-
tional state of the phenyl cation can be accompanied by ei-
ther a ground or an excited chlorine atom. This leads to the
following expression for the kinetic energy distribution,
where ESO is the energy difference between the two elec-
tronic states of the chlorine atom, resulting from the spin-
orbit coupling (ESOϭ0.11 eV),
6
6
opening. With ionized chlorobenzene undergoing meta-
stable decomposition, the internal energy content is of the
order of 1.3 eV for an accelerating voltage of 8 kV: this
provides us with a lower limit for the isomerization barrier.
These experiments lead us to the conclusion that only the
phenylium ion structure has to be considered in our maxi-
mum entropy analysis of the kinetic energy release distribu-
tions of chloro-, bromo-, and iodobenzene cations.
P͑␧
͉
E͒ϭC͑E͓͒pP ͑␧
͉
E͒ϩ͑1Ϫp͒P ͑␧
͉
E͔͒
1
2
for EуESO and ␧рEϪESO
E͒ϭC͑E͒P ͑␧ E͒ otherwise,
,
B. The composite nature of the reaction
͑7.1͒
P͑␧
͉
͉
An attempt was then made to interpret the KERD as
resulting from a two-channel process, involving electronic
excitation. A correlation diagram between the electronic
1
p and (1Ϫp) define the branching ratio between the two
channels. C(E) is the normalization factor. Using ␧ as a
ϩ
1/2
states of C H Cl determined by photoelectron spectroscopy
6
5
and those of its fragments is presented in Fig. 6 to obtain a
schematic set of adiabatic potential energy curves.
single constraint in the two channels leads to the following
equations for P (␧͉E) and P (␧͉E):
1
2
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J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
Urbain et al.
2919
FIG. 8. Solid line: experimental kinetic energy release distribution obtained
FIG. 7. Solid line: experimental kinetic energy release distribution obtained
ϩ
ϩ
for the metastable dissociation C H Cl →C H ͑5 keV͒ϩCl ͑first field-free
6
5
6
5
ϩ
ϩ
for the metastable dissociation C H Cl →C H ͑5 keV͒ϩCl ͑first field-free
6
5
6
5
region͒. Dotted line: fit assuming the existence of a potential energy barrier
Eqs. ͑7.3͒ and ͑7.4͔͒. Dashed lines: prior distribution ͓Eq. ͑4.3͔͒ at
region͒. Dotted line: fit including two Lagrange parameters ͓Eqs. ͑7.1͒ and
͓
͑7.2͔͒. Dashed lines: prior distribution ͓Eq. ͑4.3͔͒ at EϭES .
EϭES .
0
E͒exp͑Ϫ␭ Ϫ␭ ␧1/2͒,
P ͑␧
͉
͉
E͒ϭP ͑␧
͉
͉
ϱ
1
0
1
˜
0
E͒exp͑Ϫ␭ Ϫ␭ ␧1/2͒dE
P͑␧͒ϭp
͵
T͑E͒C͑E͒P ͑␧
͉
͑7.2͒
0
1
␧
0
EϪESO͒exp͑Ϫ␭ЈϪ␭Ј␧1/2͒,
0 1
P ͑␧
2
E͒ϭP ͑␧
ϱ
0
where the primed Lagrange parameters ␭Ј and ␭Ј refer to the
ϩ͑1Ϫp͒
͵
T͑E͒C͑E͒P ͑␧
͉
EϪESO͒dE,
͑7.3͒
0
1
␧
ϩ⌬
second channel.
The experimental results have been fitted to Eqs. ͑7.1͒
and ͑7.2͒ ͓taking into account the averaging over the collec-
tion efficiency T(E) as in Eq. ͑2.3͔͒. The nice fit shown in
Fig. 7 leads to the following interpretation. Two independent
valid when EуESOϩ⌬ and when ⌬р␧рEϪESO , and
ϱ
˜
0
E͒exp͑Ϫ␭ Ϫ␭ ␧1/2͒dE,
P͑␧͒ϭ
͵
T͑E͒P ͑␧
͉
͑7.4͒
0
1
ϩ
␧
reaction channels are open in the fragmentation of C H Cl .
6
5
The main contribution (pϭ0.85Ϯ0.03) is very similar to
ϩ
ϩ
when EрESOϩ⌬ or when ⌬у␧уEϪESO
.
The fit to the experimental data is shown in Fig. 8, while
the values of ␭ , DS, and e
nel are given in Table I. ⌬ can be estimated to be about 0.12
eV. A value of 0.9Ϯ0.03 was obtained for the branching
ratio p. This is in good agreement with the branching ratio
obtained from the fit to Eqs. ͑7.1͒ and ͑7.2͒. The discontinu-
ity observed when ␧Ϸ⌬ has little physical significance. It
results from our neglect of tunneling.
that observed in C H Br and in C H I . It leads to a halo-
6
5
6
5
2
gen atom in its P3 state with a positive ␭ Lagrange pa-
/2
1
ϪDS
obtained for the first chan-
rameter ͑implying a release of kinetic energy less than the
1
statistical estimate͒. The observed value for eϪDS ͑Ϸ80%͒ is
ϩ
quite similar to that found in the case of C H Br and of
6
5
ϩ
C H I .
6
5
In addition, a weaker contribution is also observed. It is
characterized by a negative Lagrange parameter ␭Ј , whose
1
absolute value has the same order of magnitude as ␭ ͑and
1
The existence of a small barrier along the reaction path
thus is well outside the uncertainties͒, but which is now
negative. The negative sign indicates a release of transla-
tional energy larger than the statistical expectation. There-
fore, unlike the component leading to the Cl atom in its
ground electronic state, the second component is not con-
trolled by the momentum gap law ͑which disfavors the high
values of the kinetic energy͒ but by a process which favors
the release of the internal energy to the relative translational
motion of the fragments.
Such a process could be due to the presence of a small
barrier along the reaction coordinate, if the barrier energy is
preferentially converted into relative translational energy of
the fragments. To estimate the height of this barrier, we
made the further assumption that the energy of the barrier
˜
of state D is predicted by ab initio calculations carried out by
Klippenstein.38 It is compatible with the theory of long-range
forces. The interaction between a fluorine atom and a posi-
tive point charge has been predicted by Gentry and Giese74
to generate two attractive potential energy curves correlating
2
with the P state and one curve with a hump correlating
3/2
2
with the P state of F. The same model was applied to the
1/2
chlorine atom. A series of ab initio calculations of increasing
accuracy were carried out using Dunning’s correlation con-
sistent basis sets75 to determine its quadrupole moment and
its anisotropic polarizabilities. The best calculations76
UMP2/AUG-cc-pVQZ ͑80 basis functions͒ and B3LYP/
AUG-cc-pV5Z ͑131 basis functions͔͒ led to nearly identical
sets of potential energy curves, reported in the inset of Fig. 6.
Two attractive curves are found to connect the P state of
Cl to the X B1 and B B2 states of C6H5Cl without any
barrier. A third one connects the P1 state of Cl to D A .
͓
͑denoted ⌬͒ is completely converted into translational energy
2
of the fragments. Equations ͑7.1͒ and ͑7.2͒ can then be re-
3/2
˜
2
˜
2
ϩ
placed by the following ones, in which ␭Ј has been set equal
1
2
˜
2
to zero and in which the internal energy averaging is explic-
itly introduced,
/2 1
Its remarkably horizontal shape ͑together with Klippen-
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2920
J. Chem. Phys., Vol. 110, No. 6, 8 February 1999
Urbain et al.
stein’s results͒ implies that the postulated barrier is deter-
mined by higher terms in the multipole expansion and by
incipient exchange terms.
leading to a value of the fraction of the sampled phase
space volume smaller than the one expected on grounds
of the prior distribution.
Coming back to the reaction mechanism, it should be
remembered that one is studying a process with a lifetime of
the order of a few microseconds. Hence, it would be unrea-
͑2͒ Not surprisingly, the family of the halogenobenzene cat-
ions is homogeneous and the values obtained for the
entropy deficiency and phase space sampling are very
similar. The analysis carried out here and the obtained
values for the entropy deficiency concern the energy par-
titioning between the reaction coordinate and the set of
the remaining coordinates ͑which may be denoted as a
bath, although no thermal equilibrium is implied͒. It can
be concluded that this partitioning is nearly statistical,
but not quite: less translational energy is channeled into
the reaction coordinate than the statistical estimate. Un-
fortunately, as we are not able to extract more than one
constraint from these relatively smooth KERD, no con-
clusion can be derived concerning the energy partition-
ing within the bath.
˜
sonable to assume that the population of state D has escaped
efficient internal conversions for which a lifetime of the or-
Ϫ13
77
der of 10
s is usually proposed. Instead, it can be as-
sumed that the entire population has relaxed to the ground
˜
2
X B electronic state. Then, during the final stretch of the
1
reaction coordinate that leads to products, a small fraction
˜
͓
(1Ϫp)Ϸ0.1͔ of the population of the ground state X un-
˜
dergoes a transition back to state D. This is made plausible
by noting the existence of an avoided crossing along the
˜
˜
reaction path between states X and C. The ground electronic
˜
2
state X B has five ␲ electrons in its electronic shell and
diabatically correlates to the triplet state of the C H frag-
1
ϩ
5
6
͑3͒ The method is able to supply further information on the
5
1
ment ͑which has a ␲ n configuration͒, not to its ground
reaction mechanism by demonstrating the composite na-
6
singlet state ͑which has a ␲ configuration͒. Conversely, the
ϩ
ture of the fragmentation of C H Cl . The major disso-
6
˜
6
5
diabatic correlation for the ␲ configuration connects state C
with the lowest dissociation asymptote. The resulting nona-
diabatic interaction could be responsible for a transfer of
ciation channel ͑90%͒ leads to the chlorine atom in its
2
ground electronic state, P3 , and displays a behavior
/2
very similar to the other halogenobenzene cations. For
the additional minor channel ͑10%͒, the kinetic energy
released on the fragments is larger than the prior esti-
mate. Our data suggest the probable existence of a small
barrier ͑Ϸ0.12 eV͒ along this reaction path, which leads
˜
˜
population from state X to C, followed by a predissociation
˜
˜
of state C by D via a conical intersection. It should further-
more be noted that the electronic configurations of the three
˜
˜
˜
states X, C, and D are singly excited with respect to each
other. This circumstance is compatible with a substantial in-
teraction.
ϩ
5
2
to C H ϩCl( P ). The appearance of this second
6
1/2
24,37
component may be related to earlier findings
that the
ϩ
ϩ
In the case of C H Br and C H I , the back transition
6
5
6
5
chlorobenzene ion decomposition appears to require a
nontotally loose complex.
˜
to state D is highly improbable because of the large magni-
tude of the spin-orbit splitting in the bromine and iodine
atoms.
ACKNOWLEDGMENTS
The authors thank Professor R. D. Levine for helpful
discussions on the maximum entropy formalism and Dr. M.
Godefroid for his advice on the calculation of atomic prop-
erties. P.U. is grateful to the F.R.I.A. for a research fellow-
ship. This work has been supported by the ‘‘Fonds National
de la Recherche Scientifique,’’ by the ‘‘Fonds de la Recher-
che Fondamentale Collective’’ and by research grants from
VIII. CONCLUDING REMARKS
Three conclusions can be drawn from the present analy-
sis of the unimolecular dissociation of the halogenobenzene
ions.
the University of Liege. B.L. and F.R. thank the FNRS ͑Bel-
`
gium͒ for a research associateship.
͑
1͒ The theoretical analysis of the KERD is carried out at an
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noncommittal concerning the properties and even the ex-
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tion coordinate. However, information on the reaction
mechanism can be obtained by identifying the nature of
the informative variable responsible for its nonstatistical
character. The functional form of the relevant constraint,
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