29 May 1998
Chemical Physics Letters 288 Ž1998. 621–627
A direct dynamics study of the F q C 2 H 4 ™ C 2 H 3 F q H
product energy distributions
Kim Bolton a , William L. Hase a , H. Bernhard Schlegel a , Kihyung Song
b
a
b
Department of Chemistry, Wayne State UniÕersity, Detroit, MI 48202, USA
Department of Chemistry, Korea Nat’l UniÕersity of Education, Chongwon, Chungbuk 363-791, South Korea
Received 18 December 1997
Abstract
Ab initio direct dynamics was used to calculate product energy distributions for the F q C 2 H 4 ™ C 2 H 3 F q H
reaction. A broad product translational energy distribution, similar to that observed experimentally, is found when the
trajectories are initialized with a statistical vibrational energy distribution at the exit channel barrier. The trajectories show
that, on average, orbital angular momentum is conserved in going from the exit channel barrier to products, and a model
which incorporates this dynamical constraint reproduces the ensemble averaged trajectory results. q 1998 Elsevier Science
B.V. All rights reserved.
1. Introduction
An explanation of the product energy distributions
measured w1,2x for the reaction
FqC2 H4 ™C2 H3FqH
Ž R1.
remains an important issue w1–10x in unimolecular
rate theory. Crossed molecular beam experiments
w1,2x with a reactant relative translational energy Erel
of 2 to 12 kcal moly1 , indicate that the product
X
relative translational energy Erel
distribution is broad,
with an average value of f 50% of the total available energy. Infrared chemiluminescence experiments w3x show that the C 2 H 3 F product is formed
with a nonstatistical vibrational energy distribution.
These results are not in accord with a simple model
which assumes a statistical distribution of energy at
the C 2 H 3 F PPP H ‡ transition state, as predicted by
RRKM theory, followed by repulsive energy release
in which all the exit channel potential goes to product relative translation. To reconcile theory and ex-
periment, more detailed models w4–8x have been
advanced for understanding the product energy partitioning. Though these studies provided more insight
into the dynamics of the Reaction ŽR1. product
energy partitioning, they have not given quantitative
interpretations of the experimental results.
One of these previous studies w8x was a classical
trajectory simulation based on an analytic potential
energy surface ŽPES. derived in part from UHFr431G calculations. Subsequent higher level ab initio
calculations w9,10x have identified shortcomings of
this analytic PES, which may affect the product
energy partitioning. Thus, there is an interest in
performing an additional trajectory study with a more
accurate PES.
An attractive trajectory approach is ‘direct dynamics’ w11x, in which trajectories are integrated ‘on the
fly’, using forces and second-order derivatives Žwhen
required. obtained directly from an electronic structure theory at each integration step. However, such a
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 2 7 4 - 7
K. Bolton et al.r Chemical Physics Letters 288 (1998) 621–627
622
calculation is computationally expensive and there is
a limit on the level of electronic structure theory
which may be used. Fortunately, the UHFr6-31G)
level of theory, which is tractable for direct dynamics, gives stationary point, geometries and normal
mode frequencies and exit channel energetics for
Reaction ŽR1. similar to those determined at higher
levels of theory, e.g., QCISDr6-311G)) 1. In the
work presented here UHFr6-31G) direct dynamics
is used to study product energy partitioning for
Reaction ŽR1. and for the reaction
HqC2 H4 ™C2 H4 qH
Ž R2.
so that the dynamics for Reactions ŽR1. and ŽR2.
may be compared.
The trajectories are initialized at the exit channel
barrier for each of these reactions, for both practical
and fundamental reasons. First, it is computationally
prohibitive to use ab initio direct dynamics to simulate the complete reaction and calculate a sufficient
number of events. Second, even if this could be done
Že.g., by semiempirical direct dynamics w11x. it would
not be the best way to use direct dynamics to calculate the product energy distributions. By initializing
the trajectories at the exit channel barrier, quasiclassical conditions w12x with the correct zero point
energy may be assured at the barrier. In addition,
comparisons between classical and quantum dynamics have shown w13x that classical dynamics gives
accurate results for a direct process like the motion
down a potential energy barrier, if the trajectories are
initialized with the correct quasiclassical conditions.
Thus, if the correct initial ensemble of states is
chosen at the barrier and the exit channel potential is
correct, agreement between classical trajectory and
experimental results is expected.
For rovibrationally cold reactants, the conditions
in the crossed molecular beam experiments for Reaction ŽR1., the energy available to the products EX
may be expressed as
EX s Erel y D H00
Ž 1.
where Erel is the reactant relative translational energy and D H00 is the 0 K heat of reaction. The
experimental 300 K heats of formation w14x for
1
To be discussed in a subsequent paper.
C 2 H 3 F Žy33.2 kcal moly1 ., C 2 H 4 Ž12.5 kcal
moly1 ., H Ž52.1 kcal moly1 . and F Ž19.0 kcal
moly1 . and experimental anharmonic frequencies for
C 2 H 3 F and C 2 H 4 give w15x D H00 s y12.8 kcal
moly1 for Reaction ŽR1., which is similar to the
values of y11 w1x, y14 w2x and y15 " 2 w10x
assumed previously. EX may also be expressed as
EX s E ‡ q E0
Ž 2.
where E ‡ is the energy at the exit channel barrier in
excess of the zero point energy and E0 is the exit
channel barrier height with zero point energies included.
2. UHF r 6-31G) potential energy surface
The C 2 H 4 and C 2 H 3 F minimum energy geometries obtained at the UHFr6-31G) level of theory
are planar and in very good agreement with experiment w16,17x. The UHFr6-31G) geometries for the
C 2 H 4 PPP H ‡ and C 2 H 3 F PPP H ‡ transition states
are in good agreement with those determined at
higher levels of theory w18x.
The height of the exit channel barriers for Reactions ŽR1. and ŽR2. have not been converged by ab
initio calculations w10,18x. The UHFr6-31G) exit
channel barrier height, E0 , for Reactions ŽR1. and
ŽR2., with zero point energies included, are 6.39 and
3.11 kcal moly1 , respectively. Using ab initio structures and vibrational frequencies to fit the H q
C 2 H 4 ™ C 2 H 5 experimental rate constant, E0 s
2.70 kcal moly1 has been deduced for Reaction ŽR2.
w19x, in good agreement with the UHFr6-31G)
value. Since the H q C 2 H 3 F ™ C 2 H 4 F association rate has not been measured, the above approach
cannot be used to deduce a value of E0 for this
reaction. However, ab initio and experimental data
were used to estimate a value of 5–6 kcal moly1 for
this E0 w10x, in agreement with the UHFr6-31G)
value.
At the UHFr6-31G) level of theory, the two
transitional bending mode frequencies are 414 and
446 cmy1 for C 2 H 4 PPP H ‡ and 456 and 480 cmy1
for C 2 H 3 F PPP H ‡. The QCISDr6-311G)) and
MRCIrcc-pVDZ higher levels of theory give transitional mode frequencies for C 2 H 4 PPP H ‡ which are
K. Bolton et al.r Chemical Physics Letters 288 (1998) 621–627
623
propagated towards products. From Eqs. Ž1. and Ž2.,
these E ‡ values correspond to Erel of f 8, 13 and
18 kcal moly1 for Reaction ŽR1., which are in the
range of Erel values studied experimentally Ž2–12
kcal moly1 . and in the classical trajectory simulation
Ž20 kcal moly1 ..
The RRKM assumption w20,21x that vibrational
energy levels of the transition state have equal probabilities of being populated, is used to choose initial
conditions for the trajectories. The total angular momentum J and its components Jx‡ , Jy‡ and Jz‡ in the
principal rotational axes frame of the transition state
were chosen from distributions appropriate for the
molecular beam experiments. These distributions
were determined by running trajectories on the analytic PES for the complete F q C 2 H 4 ™ C 2 H 3 F q
H reaction and halting the trajectories at the exit
channel barrier Žspecified by the C 2 H 3 F PPP H ‡ dis‡
˚ .. The C 2 H 3 F PPP H ‡ moiety
tance r HC
s 1.95 A
was then rotated into its principal axis frame to find
Jx‡ , Jy‡ and Jz‡.
The total angular momentum J distribution is
quite broad, as found in the previous trajectory study,
i.e., Fig. 3c of Ref. w8x. For a F q C 2 H 4 Erel of 13.0
kcal moly1 , i.e., E ‡ s 19 kcal moly1 for the
UHFr6-31G) PES, the J distribution is peaked at
approximately 40 " with an average of 52.3 ". The
distributions of Jx‡ , Jy‡ and Jz‡ for this J distribution
are shown in Fig. 2 Ž Jx‡ is the component of J that
Fig. 1. C 2 H 4 PPP H ‡ and C 2 H 3 F PPP H ‡ transition structures
obtained at the UHFr6-31G) level of theory w9x. The out-of-plane
wag angles are: C 2 C 1 H 3 H 4 s169.78 and C 1C 2 H 5 H 6 s177.48
for the C 2 H 4 PPP H ‡ structure, and C 2 C 1 F3 H 4 s164.88 and
C 1C 2 H 5 H 6 s175.68 for the fluorinated structure. The reaction
coordinate eigenvectors are superimposed on these structures.
only ten and two percent different than the UHFr631G) values, respectively w18x. Fig. 1 shows that the
UHFr6-31G) reaction coordinate eigenvectors for
both C 2 H 4 PPP H ‡ and C 2 H 3 F PPP H ‡ include H–
C PPP H angle bending motion. The same property is
found at the higher level of theories Žsee footnote 1..
3. Direct dynamics method
Ensembles of trajectories with total energies E ‡
of 14, 19 and 24 kcal moly1 were initialized at the
exit channel barriers for Reactions ŽR1. and ŽR2. and
Fig. 2. Distributions of Jx‡ Žsolid line., Jy‡ Ždashed line. and Jz‡
Ždotted line. at the C 2 H 3 F PPP H ‡ transition state for the F q
C 2 H 4 ™ C 2 H 3 F q H trajectories run on the analytic PES with
Erel s13.0 kcal moly1 . Jz‡ is the component of J perpendicular
to the F–C–C plane and Jx‡ is the component lying approximately
along the F–C–C axis.
K. Bolton et al.r Chemical Physics Letters 288 (1998) 621–627
624
approximately lies along the F–C–C axis and Jz‡ the
component perpendicular to the F–C–C plane.. The
average values of the magnitudes of the distributions
are ² Jx‡ : s 15.6 ", ² Jy‡ : s 30.4 " and ² Jz‡ : s
31.1 ". It is interesting to note that the similar values
for ² Jy‡ : and ² Jz‡ : and the finding that the probabilities for all three components peak at zero " Žconsistent with a random projection of J w22x. do not
support the kinematic model proposed by Lee and
coworkers in which the F atom adds perpendicular to
the C 2 H 4 plane with J f Jy‡ f j X , where j X is the
C 2 H 3 F rotational angular momentum 2 .
Quasiclassical rigid rotorrnormal mode sampling
w12,23x was used to select initial conditions for the
UHFr6-31G) direct dynamics trajectories by: Ž1.
selecting J and its components from the above
distributions and calculating the transition state rotational energy from
‡
Erot
s
2
Ý Ž Ji‡ . r2 Ii‡
Ž l‡.
2
2mŽ r ‡ .
2
Ž 5.
where l ‡ is the fragments’ orbital angular momentum, m their reduced mass and r ‡ their center of
mass distance; and the sum
iso
‡
‡
Erel
s Erel
q Vcen
q E0
Ž 6.
which would be the product relative translational
energy for an isotropic exit channel potential.
4. Results and discussion
where the Ii‡ are the transition state’s principal moments of inertia; Ž2. randomly selecting a transition
state normal mode vibrational energy level in the
‡ w
range 0–Eint
23x, where
Ž 4.
and Ž3. transforming the normal mode energies to
Cartesian coordinates and momenta w24,25x.
The trajectories were propagated by integrating
Newton’s equations of motion either in Cartesian
coordinates and momenta using the Gauss-Radau
algorithm w26–28x, or by integrating the instantaneous normal modes w29–31x. The relative merits of
these techniques will be discussed elsewhere w32x.
Trajectories were integrated until the force between
the product fragments was less than 10y4 kcal moly1
˚ y1 . Trajectories were typically 50 to 100 fs, and
A
˚ A typical
the products were separated by 5.5–6.0 A.
trajectory for Reaction ŽR2. was integrated in f 176
min on an IBM RS6000r560.
X
The relative velocity scattering angle distribution u Ž Õ,Õ .
reported in Ref. w8x was not normalized by sin u and supports the
distributions of J components found here and not the model of
Lee and coworkers as deduced previously.
2
‡
Vcen
s
Ž 3.
isx , y , z
‡
‡
Eint
s E ‡ y Erot
;
Properties that were monitored are product relative translational, rotational and vibrational energies,
X
X
Erel
, Erot
and EXvib ; the orbital l and C 2 H 3 F Žor
C 2 H 4 . rotational angular momentum j at the transition state and for the products; the fragment’s rela‡
tive translational energy Erel
at the transition state;
the centrifugal potential at the transition state,
Ensembles of 140–160 trajectories were calculated for Reactions ŽR1. and ŽR2., with different
conditions for E ‡ and J. The results are listed in
Table 1. The entry in the fifth row of the table is for
a calculation in which the forces are for the C 2 H 5
PES, but with the mass of one of the ethylenic H
atoms of the H PPP CH 2 moiety changed to the mass
of a F atom to model the kinematics of the
C 2 H 3 F PPP H ‡ ™ C 2 H 3 F q H reaction. The averX
: of the available energy that goes
age fraction ² f rel
X
to Erel is slightly larger for Reaction ŽR1. than for
X
:
Reaction ŽR2.. Within statistical uncertainty, ² f rel
is the same for J s 0 and the simulations with a J
distribution to model the beam experiments, and is
also insensitive to substituting the mass of an ethyX
: s 0.50
lene H atom by the F mass. The value of ² f rel
found here for Reaction ŽR1. is larger than the 0.32
found from the previous trajectory study.
The same angular momentum constraints observed in the previous trajectory study are found in
this direct dynamics simulation. The system’s total
angular momentum J is primarily converted to product ŽC 2 H 4 or C 2 H 3 F. rotational angular momentum
X
: significantly
jX . This correlation explains why ² f rot
‡
increases for the C 2 H 4 PPP H ™ C 2 H 4 q H simulation when J changes from zero to a distribution
K. Bolton et al.r Chemical Physics Letters 288 (1998) 621–627
Table 1
Ensemble averages at the exit channel barrier and for the products
Initial conditions
J
‡
Erel
14.0
19.0
24.0
19.0
19.0
0
0
0
beam b
beam b,c
19.0
a
a
Transition state
E‡
beam
b
iso
Erel
625
Products
l‡
j‡
X
X
X
X
X
f rel
f rot
f vib
l
C 2 H 4 PPP H ‡ ™ C 2 H 4 q H
2.3
6.3
8.8
8.8
2.4
7.6
14.0
14.0
3.0
8.5
14.3
14.3
2.5
9.1
17.2
48.4
2.2
8.4
15.8
51.1
17.11
22.11
27.11
22.11
22.11
0.38
0.39
0.37
0.42
0.38
0.03
0.05
0.05
0.38
0.23
0.59
0.56
0.58
0.20
0.39
9.1
14.0
14.8
18.3
15.6
9.1
14.0
14.8
48.1
51.2
C 2 H 3 F PPP H ‡ ™ C 2 H 3 F q H
2.8
12.4
15.7
54.2
25.39
0.50
0.18
0.32
16.8
55.0
X
frot
X
f vib
y1
‡
Erel
,
‡
j
X
E
Energy is in kcal mol and angular momentum in units of ". The standard deviation in the average
and
is f 6%, and
X
X
X
iso
in Erel
, f rel , l ‡ , j ‡ , l and j is f 3%. b The J distribution simulated for F q C 2 H 4 ™ C 2 H 3 F PPP H is shown in Fig. 2. c Forces
determined from the C 2 H 5 surface, but propagation performed using masses appropriate to the fluorinated reaction Žsee text..
representing the molecular beam experiments. The
ensemble averaged orbital angular momentum is
conserved quite well when proceeding from the transition state to the products. This is illustrated by the
similarity of the ² l ‡ : and ² lX : values in Table 1.
The correlation coefficients for Ž l ‡ ,lX ., which provide
a measure of the orbital angular momentum conservation over individual trajectories, range from 0.75
to 0.99 for the conditions shown in Table 1. Substituting the fluorine mass for the hydrogen mass increases the Ž l ‡ ,lX . correlation, and integration on the
C 2 H 4 F PES instead of the C 2 H 5 PES decreases the
correlation.
Because of the conservation of ² l : in the exit
channel, the centrifugal potential at the transition
state, Eq. Ž5., is, on the average, converted to product relative translational energy. Furthermore, a
model that assumes the exit channel channel barrier
potential is transferred to product translation, Eq. Ž6.,
X
gives a good representation of the Erel
averages. The
X
iso
: larger
only differences between ² Erel : and ² Erel
y1
than 0.3 kcal mol are for the J s 0 C 2 H 4 PPP H ‡
calculations with E ‡ of 19.0 and 24.0 kcal moly1 .
X
iso
This correlation between Erel
and Erel
is only mildly
retained for individual trajectories. For Reaction ŽR2.,
X
iso
. correlation coefficient varies from
the Ž Erel
, Erel
‡
0.76 at E s 24 kcal moly1 , J s 0, to 0.84 at E ‡ s
19 kcal moly1 with J representing the beam simulations. For Reaction ŽR1. at E ‡ s 19 kcal moly1 and
X
iso
.
J representing the beam simulations, the Ž Erel
, Erel
correlation coefficient is just 0.69.
X
iso
Distributions of Erel
and Erel
are compared in
Fig. 3 for Reaction ŽR2. with E ‡ s 19 kcal moly1.
iso
distribution gives a good overall representaThe Erel
tion of the product relative translational energy disX
.. Choosing J to represent the beam
tribution P Ž Erel
simulation instead of setting it to zero has the effect
of broadening, but not significantly shifting, the
X
: in Table 1..
distribution Žsee ² f rel
X
iso .
. and P Ž Erel
Comparison of the P Ž Erel
for Reac‡
y1
tion ŽR1. with E s 19 kcal mol and with the J
distribution representing the beam simulation is given
in Fig. 4. ŽThese conditions correspond to Erel s 12.6
X
Fig. 3. Normalized direct dynamics distributions of Erel Žsolid
iso
‡
line. and Erel Ždashed line. for Reaction ŽR2. at E s19 kcal
moly1 : Ža. J is chosen from a distribution representing the
molecular beam simulations and Žb. J s 0.
626
K. Bolton et al.r Chemical Physics Letters 288 (1998) 621–627
Fig. 4. Normalized distributions for Reaction ŽR1.. Direct dynamics distributions at E ‡ s19 kcal moly1 Žcorresponding to Erel s
12.6 kcal moly1 . and J chosen from the molecular beam simulaX
X
iso Ž
tion, of Erel Žsolid line. and Erel
dashed line.. Erel distribution
deduced from experiment at Erel s12.1 kcal moly1 Ždotted line..
Error bars, which reflect standard deviations, are shown by the
vertical lines.
X
kcal moly1 .. The Erel
distribution is broader than the
iso
Erel distribution, but since the broadening is to both
higher and lower energies, there is not a significant
shift in the averages of the distributions. Also shown
X
in Fig. 4 is the Erel
distribution for Erel s 12.1 kcal
y1
mol
deduced from the crossed molecular beam
study of Reaction ŽR1. w2x. The UHFr6-31G) direct
dynamics distribution has similar broadening as the
X
: of 0.5 obexperimental distribution, and the ² frel
tained from the direct dynamics study is the same as
the experimental value. The stronger peaking in the
X
direct dynamics Erel
distribution is probably not
X
., a
significant. To deduce the experimental P Ž Erel
translation to a center of mass frame from the experimental laboratory frame was required. This transformation may be rather insensitive to peaking in
X
. as long as the distribution has the proper
P Ž Erel
breadth. Also, in the experiment there is a small
spread in the initial relative translational energy,
which has the effect of removing structure, e.g.
X
. w33x.
peaking, in the P Ž Erel
In summary, the following are the important results of the direct dynamics study of the
C 2 H 3 F PPP H ‡ ™ C 2 H 3 F q H exit channel dynamics.
1. As observed in the previous trajectory study of
the complete reaction, angular momentum con-
straints are important. Overall rotational angular
momentum is converted to C 2 H 3 F rotational angular momentum and orbital angular momentum
is, on the average, conserved in proceeding from
the exit channel transition state to the products.
As a result of this latter constraint, the centrifugal
potential at the transition state is converted to
product relative translational energy.
2. A model based on isotropic exit channel dynamics which assumes the product relative translaX
. arises from the centional distribution P Ž Erel
trifugal potential and relative translational energy
distributions at the transition state plus the exit
channel potential release, gives very good agreement with the ensemble averaged trajectory results. However, this model is not valid for individual trajectories.
3. If a statistical population of vibrational energy
levels at the exit channel transition state is assumed, and the angular momentum is chosen to
represent the molecular beam experiment, the
X
UHFr6-31G) direct dynamics Erel
distribution
is similar to the distribution deduced from crossed
molecular beam experiments, and the average
fraction of the available energy that goes to product translation is the same, i.e., 0.5.
Acknowledgements
K.B. and W.L.H. are grateful for financial support
from the National Science Foundation, CHE-9403780. H.B.S. is grateful for support from the same
source, CHE-94-00678.
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