A new mechanism for the enhancement of activated bimolecular reactions by rotational excitation

Article abstract of DOI:10.1063/1.481496

The kinematic mass model (KMM), which has been developed to examine the dynamics of activated bimolecular reactions, has here been adapted to examine how orientational effects associated with reagent rotation influence the rotational state dependence of reaction cross-section. It is shown that, for reactions where the critical dividing surface (CDS) and the equipotential contours near to the CDS are prolate, i.e., elongated in the direction of the longitudinal molecular axis, rotation favors impact on the CDS near collinear geometries where the barrier to reaction is least, with the result that reaction cross-sections are enhanced with increasing reagent rotation. In the case of the rotational velocity being comparable with, or greater than, the relative translational velocity, this enhancement can greatly exceed that due to part of the rotational energy being available for barrier crossing. The KMM model, allowing for this orientational effect, has been applied to the reactions O+HCl (DCl) and O+H₂ on well-known model potential energy surfaces (PESs) where both the CDSs and the equipotential contours near the CDS are prolate. The results agree well with those from trajectory calculations. The role of the above effects of reagent rotation in the case of surfaces of nonprolate shapes is discussed qualitatively.

Full text of DOI:10.1063/1.481496

A new mechanism for the enhancement of activated bimolecular reactions by rotational
excitation
Adolf Miklavc, Marko Perdih, and Ian W. M. Smith
Citation: The Journal of Chemical Physics 112, 8813 (2000); doi: 10.1063/1.481496
View online: http://dx.doi.org/10.1063/1.481496
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 20
22 MAY 2000
A new mechanism for the enhancement of activated bimolecular reactions
by rotational excitation
Adolf Miklavc and Marko Perdih
National Institute of Chemistry, Hajdrihova 19, 61115 Ljubljana, Slovenia
Ian W. M. Smith
School of Chemistry, The University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom
͑
Received 13 October 1999; accepted 3 March 2000; publisher error corrected 1 August 2002͒
The kinematic mass model ͑KMM͒, which has been developed to examine the dynamics of
activated bimolecular reactions, has here been adapted to examine how orientational effects
associated with reagent rotation influence the rotational state dependence of reaction cross-section.
It is shown that, for reactions where the critical dividing surface ͑CDS͒ and the equipotential
contours near to the CDS are ‘‘prolate,’’ i.e., elongated in the direction of the longitudinal molecular
axis, rotation favors impact on the CDS near collinear geometries where the barrier to reaction is
least, with the result that reaction cross-sections are enhanced with increasing reagent rotation. In
the case of the rotational velocity being comparable with, or greater than, the relative translational
velocity, this enhancement can greatly exceed that due to part of the rotational energy being
available for barrier crossing. The KMM model, allowing for this orientational effect, has been
applied to the reactions OϩHCl ͑DCl͒ and OϩH on well-known model potential energy surfaces
2
͑
PESs͒ where both the CDSs and the equipotential contours near the CDS are prolate. The results
agree well with those from trajectory calculations. The role of the above effects of reagent rotation
in the case of surfaces of nonprolate shapes is discussed qualitatively. © 2000 American Institute
of Physics. ͓S0021-9606͑00͒02120-6͔
I. INTRODUCTION
ing that selectivity in the face of facile collisional relaxation.
The most successful experiments, which are cited in the re-
view by Grote et al., employ pulsed laser pumping to excite
HF or HCl to specific rotational levels in their first vibra-
tional level and observe the reaction of these excited mol-
ecules with metal atoms to yield the metal fluoride and H
atoms.
The measurement and understanding of how reaction
cross-sections depend on the energy present in different mo-
5
tions of the reagents is a crucial element of the field of mo-
lecular reaction dynamics.1,2 For direct activated bimolecular
reactions, the relative importance of translational and vibra-
tional energy has been quite well-understood, at least for
three-atom systems of the AϩBC→ABϩC type, for many
The great majority of information about the effects of
reagent rotation on chemical reactivity comes from quasi-
classical trajectory ͑QCT͒ calculations. Nevertheless, there
has been no systematic effort to unravel the separate effects,
for example, of the form of the PES and of the kinematics or
mass combination for the reaction. Most QCT studies have
aimed to quantify the rotational effects for a particular reac-
tion. The dependence of the reaction cross-section on the
rotational level (j) of the molecular reagent has often been
found to be quite complex. Moreover, it has not proven easy
from QCT calculations to identify the crucial elements of the
collision dynamics and it remains a challenge to theory to
explain how reaction cross-sections depend on j.
Broadly speaking, the explanations that have been put
forward to explain reagent rotational effects can be classified
as either ‘‘energetic’’ or ‘‘orientational.’’ Sathyamurthy3
pointed out that, as the reagent rotational energy is increased,
more product states become accessible so, at least in the
phase space limit, the reaction cross-section might be ex-
pected to increase. However, direct reactions rarely, if ever,
obey phase space theory, so more dynamical explanations
should be sought. One such explanation would be that rota-
tional motion in the (BC) reagent assists the system to sur-
3
years via the ‘‘Polanyi rules.’’ Reagent vibrational excita-
tion selectively enhances the reaction cross-section in the
case of a ‘‘late’’ barrier on the potential energy surface
͑
PES͒, a situation which is expected for an endothermic
3
reaction, whereas translational energy selectively promotes
reaction where the barrier is ‘‘early,’’ as is usually found for
exothermic reactions. In each case, the effect is easily un-
3
derstood, since the effective motion is the one which most
effectively couples to the motion along the reaction coordi-
nate which carries the system over the potential energy bar-
rier.
The situation with respect to the effects of reagent rota-
tional excitation is less satisfactory. However, a corollary
may be that such effects provide a more sensitive probe of
the PES for any particular reaction, or at least of that part of
the PES leading to and including the critical dividing
1
surface ͑CDS͒ between reagents and products. Extensive re-
views of the effects of reagent rotation on reaction cross-
4
5
sections have been given by Sathyamurthy and Grote et al.
As they point out, the experimental data base on rotational
energy effects is quite sparse, reflecting the difficulty of pre-
paring molecules in defined rotational states and of maintain-
0021-9606/2000/112(20)/8813/6/$17.00
8813
© 2000 American Institute of Physics
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1

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J. Chem. Phys., Vol. 112, No. 20, 22 May 2000
Miklavc, Perdih, and Smith
mount the potential energy barrier to reaction. Although, as
pointed out previously, on a simple basis it is difficult to see
be most suitable for comparison with the result of KMM
6
calculations when discussing rotational effects.
In subsequent formulations1
2,13
of the KMM, it was ex-
reactivity can be enhanced in this manner for a reaction
dominated by a collinear minimum energy path, since rota-
tional motion would be orthogonal to the motion along the
reaction coordinate, rotational energy might assist in those
many nonlinear collisions that lead to reaction, even in those
cases where the lowest barrier to reaction is for a collinear
geometry.
Orientational effects of reagent rotation have largely em-
phasized the possibility that increased reagent rotational mo-
tion will disrupt the progress of a trajectory towards the most
tended to include the effects of vibration in the BC molecule.
In these calculations, the barriers at the CDS were calculated
to include not only the electronic energy but also the adia-
batic vibrational energy corresponding to the state of the BC
reagent; i.e., of vϭ0 ͑Ref. 12͒ allowing for changes in the
zero-point energy in passing from isolated AϩBC to the
CDS, and of vϭ1 ͑Ref. 13͒ to explore the effects of vibra-
tional excitation in the BC molecule. These modifications
made it possible to compare the KMM results with the nu-
merous results of quasiclassical trajectories. In the case of
the OϩHCl ͑DCl͒ reactions, reasonably good agreement was
again obtained in regard to the dependence of the reaction
cross-sections on both translational and rotational energy of
the reagents, for both vϭ0 and vϭ1, although the agree-
ment was somewhat less good than in the case of the CT
calculations. This poorer agreement might be due to addi-
tional simplifications introduced in estimating the vibra-
tionally adiabatic barriers and to the neglect of recrossing
effects, which become larger the greater the internal excita-
tion of the BC reagent, particularly for H atom transfer reac-
tions. Of relevance to the present work was the observation
that the reaction cross-sections increased for higher initial j
levels but the enhancement predicted by the KMM calcula-
tions was too weak in all the H atom transfer reactions that
were considered.
favorable, i.e., lowest energy, geometry for reaction. Classi-
cal trajectory ͑CT͒ studies7
–10
have shown that this is par-
ticularly the case for ‘‘oblate’’ PESs on which the long-range
interactions tend to guide the trajectory towards the lowest
energy reaction path, thus increasing the reaction cross-
section above the value it would have in the absence of these
forces and hence in the case of straight line trajectories. Ro-
tational excitation reduces this funneling effect and causes a
decline in reactivity with rotational quantum number in the
case of such PESs.
For several reasons, including the ease with which dif-
ferent dynamical effects can be separated out, calculations
using simplified models have an important part to play in
problems of reaction dynamics. Over the past few years we
1
1–13
have developed
a simple kinematic mass model ͑KMM͒
to treat activated bimolecular reactions which is an extension
In the present work we show, using modified KMM cal-
culations, that the orientational effect associated with reagent
rotation can give rise to an enhancement of the cross-sections
with increasing reagent rotation which can greatly exceed
that due to part of rotational kinetic energy being available
for barrier crossing. The analysis of the reactions
of the angle-dependent line-of-centers ͑ADLOC͒ model
originally proposed by Smith.1
,14
The KMM, like the AD-
LOC treatment, assumes straight line trajectories up to the
CDS on the PES, and then examines whether the energy
associated with some critical component of the velocity of
approach exceeds the potential energy at that point on the
CDS. However, in contrast to the ADLOC treatment, the
effects of the anisotropy of the PES and of BC rotational
motion are included in the KMM.
OϩHCl͑DCl͒ and OϩH which we present clearly demon-
2
strates this behavior and leads to a convincing explanation of
the strong j-dependence of the reaction cross-section.
The KMM was first used to calculate opacities, reaction
cross-sections, and the effects of reagent rotation for the re-
actions OϩHCl͑DCl͒→OH͑OD͒ϩCl and OϩHBr→OH
ϩBr. Calculations for OϩHCl were performed on two
II. THE KINEMATIC MASS MODEL
The KMM has been described in detail elsewhere.11–13
Therefore only an outline of the model will be given in the
present article emphasizing those aspects which are espe-
cially relevant to the present calculations.
1
5,16
PESs,
one oblate, and one prolate, and the model results
9
were compared with the results of CT calculations run on
the same surfaces. When applied to the OϩHCl͑DCl͒ reac-
tions on the oblate surface, the cross-sections were about an
order-of-magnitude smaller than those obtained from the CT
calculation, presumably because of the extensive effects of
funneling. However, the same reactions were modeled quite
satisfactorily by the KMM applied on the prolate PES. In all
cases where a prolate surface was used, both the translational
energy and the j-dependence of the reaction cross-sections
In the KMM, attention is focused on relative motion of
the reagents within a small element of space at the CDS. It is
assumed that the reagents follow straight paths up to this
point. This situation is represented in Fig. 1 where some of
the parameters which are used in the implementation of the
model are illustrated: v is the relative collision velocity; the
coordinates R and ␥ are the values of the Jacobi coordinates
at the point of impact in the collision on the CDS, and n is a
unit vector normal to the equipotential contours at R, ␥. Re-
action is assumed to occur if the kinetic energy associated
with the relative velocity (v*) perpendicular to the equipo-
tential contours at the point of impact on the CDS exceeds
the potential energy barrier or the adiabatic potential energy
barrier (Ebar) at that point, i.e.,
9
were in fair agreement with the CT results, although at high
j the model results were lower than those from the trajecto-
ries. In view of the success in modeling dynamics on prolate
surfaces, only such surfaces have been used in subsequent
work including that reported in the present article. It should
be noted that, in the CT calculations reported in Refs. 7–9,
coplanar collisions were studied, with the vibrational effects
on the barriers and these trajectory results therefore appear to
2
1/2␮*v* уE
.
͑1͒
bar
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J. Chem. Phys., Vol. 112, No. 20, 22 May 2000
Enhancement of activated bimolecular reactions
8815
mentum having the values initially selected. Consequently,
any effects of the rotational motion on the distribution of the
points of impact were neglected. As pointed out in Ref. 12,
such an approximation may not be adequate for reactions
where the CDS is nonspherical and the rotational velocity is
comparable to, or larger than, the relative translational veloc-
ity. This is the case for OϩHCl͑DCl͒ at high initial values of
j, and even more so for OϩH . This conclusion was
2
reached12 on the basis of simple model calculations using
just one value ͑1 Å͒ of the impact parameter and a rotating
ellipse as the CDS.
In the present calculations, we have confirmed this cru-
cial hypothesis by analyzing the distribution of points of im-
pact with proper averaging over the impact parameter so that
the distributions obtained correspond to those that should
occur in calculations of the reaction cross-sections. The main
modification to the standard KMM model is that now the
approach to the CDS is not assumed to be instantaneous.
Rather the reagents approach along straight line trajectories
with a relative velocity corresponding to the selected colli-
sion energy and assumed to be the same all the way to the
CDS. During this time the diatomic species ͑and therefore
the ellipse representing the CDS͒ rotates with an angular
momentum corresponding to the selected rotational state j. In
general, the KMM model has been formulated for three-
dimensional collisions between an atom ͑A͒ and a diatomic
FIG. 1. Schematic representation of an AϩBC collision. The heavy line
represents part of the critical dividing surface for reaction, the lighter lines
represent equipotential contours. R and ␥ are the coordinates of the point
where the straight line trajectory impacts on the critical dividing surface; R
being the distance between the centers-of-mass of BC and A at this point, ␥
the angle between R and rBC , where the internuclear axis rBC is assumed to
lie along the line defined by ␥ϭ0.
1
2,13
molecule (BC).
For present purposes, the coplanar form
was judged to be adequate.11 Moreover, the forms of the
CDS and of the equipotential contours in the vicinity of this
surface were assumed to be elliptical, the major and minor
axes of these two ellipses being estimated from information
about the PESs. The centers of these ellipses which should
approximate the shapes of the CDSs and of the equipotential
contours in the region relevant for the reactions in question
coincide in each case with the center-of-mass of the diatomic
reagent, consistent with the requirements of the basic laws of
mechanics upon which the model has been developed.
However, a crucial feature of the KMM is that rotational
energy effects can be included, because v* depends not only
on the relative translational velocity but also on the rotational
motion of the molecular reagent. It has been shown1
1–13
that
␮
* and v* are given by
␮
␮
*ϭ
,
͑2a͒
͑Rϫn͒2
␮
1
ϩ
I
and
III. RESULTS AND DISCUSSION
j͑Rϫn͒
v*ϭv Ϫ
.
͑2b͒
For the OϩH reaction, the parameters of an ellipse, a
n
2
I
ϭ1.59 Å and bϭ0.75a, approximating the shape of the true
CDS, were derived from the analysis17 ͑see Fig. 5 in Ref. 17͒
of the PES used in QCT calculations.18 To represent the
equipotential surface, an ellipse was used with aЈ, the length
of the major axis, set equal to a but with bЈϭ0.92aЈ. These
parameters were chosen to match the shape of the CDS near
the saddle point on the PES for collinear geometries ͑see Fig.
2 in Ref. 17͒. Figure 2 displays the distribution of the points
of impact on the elliptical CDS. These results clearly dem-
onstrate that, at higher j, this distribution is shifted towards
the longitudinal axis where the barrier height is lower thus
enhancing the reaction cross-section.
In these equations, ␮ is the usual reduced mass of the colli-
sion partners, I is the moment of inertia of the molecule, j is
its rotational angular momentum perpendicular to the plane
of the collision, and v is the component of the relative ve-
n
locity normal to the equipotential contours. It should be
noted that ␮*/␮р1 and the deviation of this ratio from unity
reflects the importance of rotational recoil in the system. v*,
on the other hand, can greatly exceed v , depending on the
n
rotational velocity and reflecting the role of rotational energy
in facilitating barrier crossing.
In implementing the KMM in studies of several reac-
tions occurring on prolate PESs with the molecular reagent
In Fig. 3, the results of standard KMM calculations are
compared with those which take into account the change in
the distribution of points of impact for each value of j. Both
sets of model calculations were performed using the adia-
1
2
13
in vϭ0 and vϭ1, Monte Carlo methods were used to
generate the impact parameter and the angles defining the
initial orientation of the reagents for individual collisions.
The reaction cross-sections were then calculated as if the
points of impact on the CDS were reached instantaneously
with the translational energy and the rotational angular mo-
batic barrier heights associated with the reaction of H (v
2
ϭ0). That is, the height of the barrier on the CDS included
zero-point energy. In Fig. 4, it can be seen that the inclusion
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8816
J. Chem. Phys., Vol. 112, No. 20, 22 May 2000
Miklavc, Perdih, and Smith
FIG. 2. The distribution of collisions on the critical
dividing surface for the OϩH2 reaction, for jϭ0 ͑᭹͒
and for jϭ10 ͑᭺͒. The collision energy was Etrans
ϭ12 kcal/mol in each case. The shape of the critical
dividing surface was approximated by an ellipse as de-
scribed in the text.
of the factor allowing for the modified distribution of impact
points over the CDS considerably improves the agreement of
the model calculations with those from QCT calculations.
Since the coplanar version of the KMM was used, the reac-
tion probabilities were normalized to the QCT cross-section
at jϭ0 and with Etransϭ20kcal/mol, as in Ref. 8.
IV. SUMMARY AND CONCLUSIONS
The calculations reported in this article indicate that re-
agent rotational excitation can significantly modify the dis-
tribution of points on the CDS of a PES which are reached
by potentially reactive trajectories. For the surfaces for
which calculations have been carried out in the present work,
for which both the CDS and the equipotential contours are
prolate in shape, the effect of increasing rotational excitation
Similar calculations to those on OϩH were carried out
2
on OϩHCl and OϩDCl and the results confirmed the effects
of reagent rotation on reactivity. In these cases, the KMM
results could be compared directly with those from coplanar
1
0
CT calculations, that is, with the BC vibration ‘‘frozen.’’
1
2
The energies at the CDS were those from the PES, with no
zero-point energies added, and the KMM results required no
normalization since both sets of calculations represented the
results of coplanar collisions. The CDS and the equipotential
surfaces were again modeled by ellipses with aϭ2.57 Å, b
ϭ0.69a, and aЈϭ2.57 Å, bЈϭ0.84aЈ, respectively. The
centers of these ellipses coincide with the center-of-mass of
the molecule as described at the end of the previous Sec. II.
In the CT calculations,10 the internuclear distance rHCl(rDCl
)
was held constant at its value at the saddle point on the
collinear PES. The chosen values of a, b, aЈ, and bЈshould
therefore be regarded as describing the anisotropy of the PES
at these values of rHCl(rDCl).
It can be seen from Fig. 5 that the standard KMM treat-
ment reproduces the j-dependence of the cross-section for the
OϩHCl reaction quite well for values of j up to about 8 but
does not reproduce the steep increase which occurs at higher
j. However, if the KMM calculations are modified to allow
for the effects of rotation on the distribution of collisions on
the CDS, the model results agree well with those from CTs
over the whole range of j. As shown in Fig. 6, the situation is
similar for OϩDCl, although the j-dependence is now much
less pronounced, due to the rotational velocity of DCl being
less than that of HCl for the same j. It should be noted that
the ␥-dependence of the barrier heights is much steeper for
FIG. 3. KMM cross-sections for the OϩH2 reaction at three different colli-
sion energies: Etransϭ12 kcal/mol ͑᭝,᭡͒, Etransϭ15 kcal/mol ͑ᮀ,᭿͒, and
Etransϭ20 kcal/mol ͑᭺,᭹͒. Open symbols show the results of standard
KMM calculations and therefore include only the energetic effects of re-
agent rotation; closed symbols show the results which include the efects of
the distribution of collisions on the critical dividing surface.
OϩHCl than for OϩH so that in the latter case reactive
2
collisions occur over a much wider range of ␥. It is for this
reason that the rotational effects only become strong at
higher values of j in the OϩHCl reaction.
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J. Chem. Phys., Vol. 112, No. 20, 22 May 2000
Enhancement of activated bimolecular reactions
8817
FIG. 6. The same as Fig. 5 but for the OϩDCl reaction.
many different molecular orientations as it approaches the
diatomic reagent. In the limit where the rotational velocity is
very much greater than the collision velocity, the first impact
on the CDS will be wherever the separation R on this surface
is greatest. In the case of a prolate CDS, R is greatest where
the barrier to reaction is least, so reactivity is enhanced. On
the other hand, if R is greatest where the barrier to reaction is
greatest, which is the case when the CDS is oblate, reactivity
is likely to be diminished by increasing reagent rotational
excitation. This suggests that information about the actual
j-dependence of cross sections for a particular reaction will
provide valuable information about the shape of the PES for
that system near the configurations critical for reaction.
We reiterate that when the equipotential contours are
oblate, no simple modeling is possible due to the strong fun-
neling effects which lead to trajectories being reoriented on
their approach to the CDS. The following qualitative obser-
vations can however be made: when j increases, the favor-
able reorientation effects may be disrupted, leading to a de-
crease of the reaction cross-sections. At higher j, the effects
discussed here may prevail so that the cross-section would
start to increase with j. The variation of the cross-sections
with j might then go through a minimum, behavior which has
been observed in several cases.4
FIG. 4. Comparison of the QCT cross-sections for the OϩH reaction ͑open
2
symbols͒ with those calculated by the modified KMM treatment ͑closed
symbols͒. The collision energies are: Etransϭ12 kcal/mol ͑᭝,᭡͒, Etransϭ15
kcal/mol ͑ᮀ,᭿͒, and Etransϭ20 kcal/mol ͑᭺,᭹͒.
is to concentrate the points of impact near the collinear ar-
rangement of the three atoms, where the barrier to reaction is
least, and therefore to enhance the reaction cross-sections.
Qualitatively, this effect can be understood in the following
way. When the rotational velocity is much greater than the
relative translational velocity, the attacking atom can sample
Although the modified KMM reproduces the effects of
rotational excitation on reaction cross-sections quite well, as
judged by comparison with the results of QCT calculations,
the agreement is less than perfect. Any remaining differences
might be caused by approximations in the energies and shape
of the CDSs used in the KMM calculations. A more interest-
ing possibility is that our model calculations underestimate
the effect of rotation because they make no allowance for the
fact that the approach of the reagents will become slower as
they get close to the CDS and the potential energy rises. It
seems likely that the rotational velocity will be affected less
by such intermolecular forces. Consequently, the time avail-
able for a favorable orientation to be found may be longer
than that given by a calculation which assumes that the re-
FIG. 5. Rotational dependence of the reaction cross-sections for the OϩHCl
reaction at Etransϭ10 kcal/mol calculated from QCTs ͑᭹͒, from the present
KMM method ͑᭿͒, and from the standard KMM method ͑᭺͒ in which only
the energetic effects of reagent rotation are taken into account.
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1

8818
J. Chem. Phys., Vol. 112, No. 20, 22 May 2000
Miklavc, Perdih, and Smith
agents follow straight line trajectories up to the CDS with
their initial relative velocity undiminished by intermolecular
forces. If this is correct, the impacts on the CDS will cluster
even more strongly about the point where R is greatest and
Ebar least.
Loesch, in Selectivity in Chemical Reactions, edited by J. C. Whitehead
͑
Kluwer, The Hague, 1988͒, p. 47.
6
7
8
9
H. R. Mayne, J. Am. Chem. Soc. 112, 8165 ͑1990͒.
H. Loesch, Chem. Phys. 104, 213 ͑1986͒.
H. Loesch, Chem. Phys. 112, 85 ͑1987͒.
H. Kornweitz, A. Persky, I. Schechter, and R. D. Levine, Chem. Phys.
Lett. 169, 489 ͑1990͒.
10
͑
a͒ R. D. Levine J. Phys. Chem. 94, 8872 ͑1990͒; ͑b͒ H. Kornweitz, A.
Persky, and R. D. Levine ibid. 95, 1621 ͑1991͒. The terms ‘‘oblate’’ and
‘prolate’’ used to describe the PES for reaction relate to the form of the
ACKNOWLEDGMENTS
‘
Financial support of this work from the Ministry of Sci-
ence and Technology of the Republic of Slovenia and from
the British Council is gratefully acknowledged. Adolf
Miklavc is grateful to Professor Sture Nordholm for stimu-
lating discussions.
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