Observation of a transition state resonance in the integral cross section of the F+HD reaction

Observation of a transition state resonance in the integral cross section of the F+HD

reaction

Rex T. Skodje, Dimitris Skouteris, David E. Manolopoulos, Shih-Huang Lee, Feng Dong, and Kopin Liu

Citation: The Journal of Chemical Physics 112, 4536 (2000); doi: 10.1063/1.481041

View online: http://dx.doi.org/10.1063/1.481041

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JOURNAL OF CHEMICAL PHYSICS

VOLUME 112, NUMBER 10

8 MARCH 2000

Observation of a transition state resonance in the integral cross section

of the F¿HD reaction

Rex T. Skodje

Institute for Atomic and Molecular Sciences, Academia Sinica, Taipei, Taiwan 106

and Department of Chemistry, University of Colorado, Boulder, Colorado 80309

Dimitris Skouteris and David E. Manolopoulos

Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, United Kingdom

Shih-Huang Lee, Feng Dong, and Kopin Liu

Institute for Atomic and Molecular Sciences, Academia Sinica, Taipei, Taiwan 106

͑Received 14 September 1999; accepted 8 December 1999͒

We have studied the reaction FϩHD at low collision energies using a combination of experimental

and theoretical methods. Clear evidence for a reactive resonance is found in the integral cross

section for the reactive channel FϩHD→HFϩD. Using a crossed molecular beam apparatus, the

total reactive cross sections for the HFϩD and DFϩH channels were obtained in the collision

energy range of 0.2–5 kcal/mol. In addition, Doppler proﬁles were obtained over this range of

energies, which provide information about the angularly resolved distribution of ﬁnal vibrational

states. The cross section shows a distinctive steplike feature near 0.5 kcal/mol. Furthermore, the

Doppler proﬁles reveal a dramatic change in the angular distribution of products over a narrow

energy range centered at 0.5 kcal/mol. This feature is shown to arise from a reactive resonance

localized near the transition state. Theoretical scattering calculations have been carried out using the

Stark–Werner potential energy surface, which accurately reproduce the shape of the resonance

feature. A detailed analysis of quantum dynamics using the spectral quantization method reveals that

a quantum resonance exists near 0.52 kcal/mol, which is localized about the collinear FHD

geometry. At collision energies below 1 kcal/mol, the reaction was found to proceed almost

exclusively through resonant tunneling with very little contribution from direct, over the barrier,

reaction. The properties of the quantum resonance, such as the position, lifetime, and partial widths

were found to correlate well with the experimental results and the quantum scattering calculations.

I. INTRODUCTION

be sought in the differential cross section for collision ex-

periments since impact parameter averaging is expected to

smear out resonance features in the total reactive cross

section.^7,8This stands in contrast to resonances seen in, e.g.,

nuclear physics where strong resonant signatures are often

apparent in the integral cross sections.⁹The issue seemed to

be put to rest in the recent discussions of reactive resonances

in the HϩH₂system where experiment¹⁰and theory¹¹coa-

lesced to show the resonances to be completely obscured in

the integral cross section, ␴(E), but perhaps survived in the

differential cross section, d␴(E)/d⍀.

The FϩH₂reaction has served as an archetypal system

for the study of chemical reaction dynamics¹and as a focal

point in the search for reactive resonances.²A series of suc-

cessive improvements in experimental and theoretical meth-

ods has led to a detailed understanding of many aspect of this

reaction. Although this system has been studied for decades,

much of the more recent interest in this problem was inspired

by the crossed molecular beam experiments of Lee and

co-workers.³An anomalous forward scattering peak in the

state-resolved product distribution for HF( ϭ3) was ob-

v

Ј

An alternative approach is to search for resonances using

transition state spectroscopy, which is dynamically a half-

collision process and circumvents the impact parameter-

averaging problem. Neumark and co-workers¹²have recently

observed the photodetachment spectrum of FH^Ϫ₂. The spec-

trum revealed structure, but, at the available level of energy

resolution, the peaks could not be deﬁnitely assigned to tran-

sition state resonances.^6,13Theoretical analysis assigned

some of the spectral peaks to van der Waals type states or

threshold type states¹⁴associated with the well in the long

range entrance and exit channels but not to states trapped

near the barrier. At this point in time, the only conclusive

evidence for reactive resonances in bimolecular AϩBC re-

actions is for the IϩHI system, which was studied by photo-

detachment of IH I^Ϫ.¹⁵

served at collision energies in the range, E_C

ϭ1.84–3.42 kcal/mol, which seemed to provide evidence

that reactive resonances may exist in this system. Additional

molecular beam studies of FϩH₂and its isotopomers have

subsequently provided higher levels of resolution and further

details about the reaction and possible resonance features.⁴

The evidence for reactive resonances in the beam experi-

ments was not conclusive, however, since there are other

dynamical explanations for the forward scattering peak when

considered on the realistic potential surface.^5,6Furthermore,

the contemporaneous theoretical simulations could not repro-

duce the experimental results in a way that would conﬁrm

the existence of a resonance. Nevertheless, since this work, it

has become the conventional wisdom that resonances should

0021-9606/2000/112(10)/4536/17/$17.00

4536

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4537

J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Transition state resonance in FϩHD

The theoretical investigation of FϩH₂reactive reso-

nances has a long history, dating back to the collinear studies

of Wu et al.¹⁶and Schatz et al.,¹⁷and since then a wide

variety of dynamical methods and potential energy surfaces

͑PES͒ have been applied. In the simplest physical terms, the

existence of reactive resonances was found to trace back to

dynamical trapping in wells on the vibrationally adiabatic

potential curves.¹⁸While a more sophisticated analysis is

certainly possible, the vibrationally adiabatic model seems to

provide the correct zero-order dynamical picture. The reso-

nance energies are, of course, sensitive to the PES, and ac-

curate predictions had to wait for a sufﬁciently accurate sur-

face to be constructed.

Recently, a highly accurate PES has been developed by

Stark and Werner ͑SW͒ ͑Ref. 19͒ that makes possible quan-

titative comparison with experiments. The SW-PES exhibits

a highly bent saddlepoint and long range van der Waals

wells in the entrance and exit channels. The dynamics on this

PES has been studied using quasiclassical trajectory ͑QCT͒

simulations, by Aoiz et al.,²⁰and also with full quantum

due to a kinematic tunneling effect.⁶An analogous peak was

obtained in the FϩHD reaction, which was tentatively attrib-

uted to the same sort of effect.²²In this work, we shall re-

open the interpretation of peak A for the FϩHD case and

come to the opposite conclusion.

In particular, we present an experimental and theoretical

investigation of the reaction

→HF , j ͒ϩD

͑v

Ј Ј

FϩHD ϭ0,jϭ0,1͒

͑v

→DF , j ͒ϩH

͑v

Ј Ј

at low collision energies, E_Cϭ0.2–5 kcal/mol. There has

been considerably less attention given to this isotope and to

the low energy regime in particular. At such low energies,

small features in the entrance channel of the potential energy

surface may have a large effect on the reaction dynamics.

Even if the van der Waals well does not trap resonance

states, it still may signiﬁcantly deﬂect incoming trajectories.

Indeed, we shall observe a ‘‘steering’’ effect toward collision

in the T-shaped geometry due to the long range potential.

Low collision energies are also interesting since the number

of partial waves contributing to the cross section is greatly

reduced. Resonance features occurring at low energies may

not be washed out to the extent seen at higher energies.

In the present work we return to the crossed molecular

beam study of the reaction dynamics using an apparatus de-

scribed in detail previously.²⁷Due to technological improve-

ments incorporated into the beam machine, it is possible to

measure reliable cross sections to signiﬁcantly lower colli-

sion energies than achieved in previous measurements on

this system. In particular it is possible to go below the

scattering

calculations,

by

Manolopoulos

and

co-workers,^6,21,22and others.²³The photodetachment spec-

trum of Neumark and co-workers has also been quantum

mechanically simulated.^13,24The comparison of the quantum

simulation to the results of Lee and co-workers, while not

perfect, was found to be reasonably good. The simulations of

the FH^Ϫ₂photodetachment spectrum are also in fairly good

agreement with experiment. These successes build conﬁ-

dence in the belief that the SW-PES is, in fact, sufﬁciently

accurate to describe a variety of experimental observables.

While the SW-PES combined with accurate quantum

mechanical simulations seemed to be getting close to the

experimental observations, the role of transition state reso-

nances was not totally clear. Using three-dimensional quan-

tum simulations, Manolopoulos and co-workers^13,24and

Takayanagi and Kurosaki²⁶have identiﬁed numerous reso-

nance states trapped in the van der Walls wells for FϩH₂,

FϩHD, and FϩD₂, but not states localized near the saddle-

point. The van der Waals resonances are far too narrow to

signiﬁcantly affect the reaction probability and are probably

not a good explanation for the results of Lee and co-workers

although they probably inﬂuence the photodetachment spec-

trum. Of course earlier studies on other surfaces has yielded

reactive resonances trapped near the saddlepoint.²⁵While

such resonances have not been reported on the SW-surface

there is no reason to believe that they will not exist. Transi-

tion state resonances appear on all reasonable surfaces and

appear to be quite robust with respect to details of the PES

and dynamical approximation.

HF( ϭ3) energetic threshold which occurs at E_C

v

Ј

ϭ1.1 kcal/mol. Since theoretical calculations on various PES

have predicted reactive resonances below the ϭ3 thresh-

v

Ј

old, it will be interesting to investigate this regime. Some-

what complicating the analysis of the results is the possible

contribution of reaction involving the excited state atom,

F(²P_1/2). Since the excited state lies only 1.15 kcal/mol

above the ground state F(²P_3/2), there is no way to eliminate

*

this species from the beam source. While reaction of F with

H₂has been observed at low collision energies,²⁸there is

good reason to believe that the associated cross section is

fairly small.^29,32Consequently, we feel conﬁdent that the

dominant contribution to the reaction at low energy occurs

through the ground atomic state.

Our contention in this paper is that a feature seen in the

integral cross section for FϩHD is due to the existence of a

reactive resonance. Since this assertion runs contrary to con-

ventional wisdom, it is useful from the onset to state how the

resonance might survive the averaging effects. In a sense, the

result is consequence of some fortuitous features of the

FϩHD system. First, the resonance in question lies just

above the entrance channel threshold. This minimizes the

number of partial waves contributing at the lowest energies.

Second, and most important, at the resonance energy there is

little or no direct ͑or background͒ reactive scattering. All the

reactive ﬂux apparently passes through the resonance state.

Thus, the integral cross section shows a sharp step, ͑roughly

the low energy ‘‘half’’ of a Lorentzian͒ with a half-width

equal to the resonance half-width. The half-Lorzentian can

Manolopoulos and co-workers⁶and Takayanagi and

Kurosaki²⁶also have observed a broader peak, ‘‘peak A,’’ in

the cumulative reaction probability for individual partial

waves. For Jϭ0, this peak lies at collision energy 0.4 kcal/

mol in the FϩH₂reaction and moves steadily upward with

the total angular momentum, J. The high J versions of this

feature are apparently responsible for a large part of the for-

ward scattering peak in the HF( ϭ3) product. Castillo and

v

Ј

Manolopoulos have argued, based on a time-delay analysis,

that peak A is not a resonance in the FϩH₂reaction, but is

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Skodje et al.

be ﬁt to the Jϭ0 resonant reaction probability. At higher

energy, but still below the direct reaction threshold, J-shifted

resonances contribute to the cross section. However, these

shifted Lorenztians combine to give a characteristic proﬁle

that is easily identiﬁable in the experimental cross section.

Thus, although the resonance is J-shifted to higher energy,

the absence of background reaction permits a direct analysis

of the integral cross section.

Ultimately, the goal of both the experimental and theo-

retical analyses is a detailed understanding of the reaction

process. By combining the experimental and theoretical in-

vestigations of this system, the hope is to provide a picture of

the dynamical effects in this system. The work in this paper

is divided into three components. First, the experimental

measurements are carried out to identify potentially interest-

ing phenomena. Second, quantum scattering calculations are

performed to establish that the SW-PES combined with ac-

curate quantum dynamics can reproduce the observed ef-

fects. And third, classical trajectory, wave packet, and spec-

tral quantization calculation are employed to physically

interpret the phenomena. All three components of this work

are required to establish the existence of a reactive reso-

nance.

resulting uncertainties in the relative cross section measure-

ments were all within Ϯ5% and the overall errors in the

experiments ͑including uncertainties in the collision energy,

as discussed below͒ were estimated to be less than Ϯ10%.

To span the collision energy range of this work (E_C

ϭ0.2–5 kcal/mol), for each beam two different sources

were employed. To produce the F-atom beam, a discharge

was passed through 5% F₂in either He or Ne at 8 atm. It is

well known that a F-atom beam produced through a dis-

charge contains signiﬁcant amounts of the spin–orbit excited

F ( P_1/2) atoms.^28,31However, there is experimental evi-

2

*

dence that the reactivity of F is, in fact, quite small com-

pared to that of the ground state F(²P_3/2) atom.³²Thus, the

results presented below should not be signiﬁcantly affected

*

by the negligibly small contribution from F . To produce the

target HD beam, either a reverse-seeded HD with ϳ15% Ar

or the neat HD expansion was used. The beam speeds were

determined by either the Doppler-shift or the fast ionization

gauge measurement. Although the different beam sources

cover different collision energy ranges, there are substantial

intervals of overlap among them. The consistency of the

steplike structure in the resulting excitation functions ͑see

below͒ from different sets of source combinations conﬁrms

the accuracy of the speed determination and that the results

are devoid of experimental artifacts after the density to ﬂux

transformation was implemented. The previous characteriza-

tion of the neat expansion of the HD beam indicated an av-

erage rotational energy of 0.05 kcal/mol, which corresponds

to a rotational temperature of 50 K with 82% in jϭ0 and

18% in jϭ1.²⁷A similar, or somewhat colder, rotational

distribution is anticipated for the reverse-seeded HD beam.

The translational energy resolution of the experiment was

governed by the velocity spreads and angular divergences of

the two doubly-skimmed molecular beams. Near E_C

ϳ0.6 kcal/mol, which is the region of interest of the report,

the resolution was determined to be about 15% (⌬E_C/E_C)

while the uncertainty in absolute energy was less than 0.1

kcal/mol.

In Sec. II, the experimental methods and results are pre-

sented. In Sec. III, the theoretical treatment of the reaction

dynamics is presented. In the discussion in Sec. IV, the the-

oretical and experimental results are compared and the evi-

dence in favor of the existence of a reactive resonance is

summarized.

II. EXPERIMENT

A. Apparatus and method

The experiment was carried out in a crossed-beam scat-

tering apparatus using the same procedures as described

previously.²⁷In brief, a skimmed F-atom beam was gener-

ated from a discharge source³⁰of F₂passing through a pulsed

valve. The subsequent supersonic expansion translationally

cooled the F-beam, which then collided with a target HD-

beam from a second pulsed valve. By alternating the HD gas

pulses and taking the difference signal, the H- or D-atom

reaction product was interrogated using a (1ϩ1) resonance-

enhanced multiphotom ionization ͑REMPI͒ detection scheme

combined with time-of-ﬂight mass spectrometry. One conve-

nient feature of the apparatus is the ability to control the

collision energy simply by changing the intersection angle of

the two molecular beams without the need to change any

other source parameter. Therefore, at a given intersection

angle, the probe laser wavelength can be scanned and the

resulting difference signals integrated over the entire Dop-

pler proﬁle to yield a quantity proportional to the integral

reactive cross section. The reaction excitation function, i.e.,

the translational energy dependence of the reactive cross sec-

tion, was then obtained by repeating the measurements at

different intersection angles. To normalize the relative cross

sections for the two isotopic channels, several back-to-back

wavelength scans at ﬁxed intersection angles were performed

with appropriate measures taken to ensure that the H- and

D-atom products were detected with equal efﬁciency. The

B. Results

Figure 1 shows the normalized excitation functions for

the HFϩD and DFϩH product channels. The results of a

previous QCT simulation on the SW-PES, presented by Aoiz

and co-workers,²⁰are also plotted for reference. It is clear

that while there is reasonably good agreement for the DFϩH

channel, a very signiﬁcant discrepancy between QCT and

experiment is evident for the HFϩD channel. The most strik-

ing observation of the experimental results is a pronounced

steplike feature peaking at around 0.5 kcal/mol, which are

completely absent in the QCT simulation. In view of the

success of the SW-PES in predicting many other dynamical

attributes for this and isotopically analogous reaction,³³the

discrepancy with the classical prediction suggests the quantal

nature of this feature. Since the adiabatic barrier to reaction

is about 1.0 kcal/mol on the SW-PES, the 0.5 kcal/mol fea-

ture in the HFϩD channel must be mediated through tunnel-

ing, and must only produce HF( р2) product states.

v

Ј

The isotope branching ratio, ␴(DFϩH)/␴(HFϩD), ob-

served in the experiment is qualitatively reproduced by the

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Transition state resonance in FϩHD

ʈ

FIG. 2. A series of unnormalized -Doppler spectra of the H/D-atom prod-

ucts taken for the two isotopic channels at tunneling energies. The shapes of

ʈ

these -Doppler proﬁles provide an immediate view of the global angular

distribution, as indicated by ‘‘b’’ ͑backward hemisphere͒ and ‘‘f’’ ͑forward

hemisphere͒. Note the sharp contrast in product angular distributions for the

two isotopic channels. Energetic considerations lead to the product vibra-

tional state assignments of the observed features. These assignments are

conﬁrmed by the on-going c.o.m. differential cross section measurements.

FIG. 1. The normalized excitation functions ͑upper panel͒ and the branching

ratio ͑low panel͒ of the two isotopic channels for the FϩHD reaction. The

QCT simulations from Ref. 20 are plotted for comparison. The experiment

is normalized to theory by a single scaling factor for both channels.

QCT calculations at the higher energies. Thus, the propensity

in favor of the DF product appears consistent with classical

theory at these energies. The physical origin of the branching

ratio may be similar to that seen in the ClϩHD reaction

discussed recently.³⁴At lower collision energies, the QCT

branching ratio remains roughly constant while the experi-

mental ratio decreases linearly in E_Cexcept a ‘‘bump’’ near

1.2 kcal/mol. There are two identiﬁable features occurring at

nearby energies that could be the cause of the bump, viz.,

is particularly favorable to demonstrate the rich dynamical

information using this simple technique with the rotating

sources machine.

ʈ

Plotted in Fig. 2 are a series of -Doppler proﬁles of the

D-atom product from the HFϩD channel at energies in the

vicinity of the 0.5 kcal/mol feature. The proﬁles begin pre-

dominantly backward peaked at 0.48 kcal/mol, then protrude

into the forward hemisphere and become roughly isotropic at

0.73 kcal/mol ͑Ref. 38͒ and ﬁnally shift to a somewhat for-

ward weighted distribution at higher collision energies. The

dramatic evolution of the Doppler proﬁles over such a nar-

row energy range is truly startling and unprecedented. From

the extent of the Doppler shift, all these Doppler features

either a threshold anomaly for the opening of the HF(

v

Ј

ϭ3) channel or an effect of the adiabatic barrier to reaction.

However, at this point the nature of this bump is unclear.

In addition to the facile control of collision energies in

integral cross section measurements using this rotating

source machine, further advantage can be taken to obtain

differential cross section information as a continuous func-

tion of energy. A technique, referred to as Doppler-selected

time-of-ﬂight, has recently been developed and applied to a

number of reactions^30,35,36to directly map out the three-

dimensional ͑3D͒ c.o.m. differential cross section. The ﬁrst

step of this new approach is the conventional ͑1D͒ Doppler-

appear to arise mainly from the HF( ϭ2) product state.

v

Ј

On-going experiments using the Doppler-selected TOF tech-

nique clearly demonstrate that this is indeed the case. The

quantitative information on the rotationally state-resolved

differential cross sections will be reported in the future. For

ʈ

comparison, a few -Doppler proﬁles of the H-atom product

from the DFϩH channel in the threshold region are also

shown in Fig. 2. By contrast, all the proﬁles now are nearly

identical and display two backward peaks. Again, the

Doppler-selected TOF measurements prove that these two

peaks correspond to the DF( ϭ4) and DF( ϭ3) product

ʈ

shift technique operated in the -conﬁguration, where the ini-

tial relative velocity of the colliding species lies along the

propagation axis of the probe laser. It can readily be shown,

and has been demonstrated previously,³⁷that with the com-

bination of energy conservation and the state-to-state probe,

this 1D-projection measurement yields directly the product

state-speciﬁc angular distribution. The kinematics of FϩHD

v

Ј

states, respectively. A backward angular distribution con-

ﬁrms the previous study of this isotope channel at higher

energies,³and is the expected behavior for a direct abstrac-

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4540

J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Skodje et al.

TABLE I. Properties of SW-PES for FϩHD.

tion reaction with a rebound mechanism when viewed within

the classical framework. The striking contrast between the

vibrationally-speciﬁc angular distributions for the two iso-

tope channels, seen in Fig. 2, strongly suggests that the 0.5

kcal feature might be a byproduct of the elusive dynamical

resonance ﬁrst predicted from the collinear model studies of

this reaction over 25 years ago.^16,17The theoretical investi-

gation in the next section not only conﬁrms this conjecture

unequivocally, but also provides much deeper insights into

the nature of the quasibound state.

a

b

c

Saddlepoint

R^a

r ^a

␥

V₀

V_ad

FHD-bent

FDH-bent

FHD-collinear

FDH-collinear

3.467

3.166

3.911

3.431

1.458

1.441

131.6

55.1

180

0

1.53

1.92

1.06

1.24

1.79

1.87

Normal modes^d

Saddlepoint

␻

v

rxn

bend

FHD-bent

FDH-bent

FHD-collinear

FDH-collinear

3216

3294

3161

3347

i 714

i 526

i 502

i 474

227

276

262

231

III. THEORY

To help interpret the experimental ﬁndings, theoretical

dynamics calculations have been performed in the low en-

ergy regime. The calculations were done on the SW-PES

which does not explicitly include the spin–orbit coupling.

The properties of the SW-PES have been discussed in detail

elsewhere.¹⁹For convenience, here we review the character-

istics of this PES specialized to the isotope FϩHD and to the

coordinate system we employ. For most depictions of the

dynamics, we employed the ␣-channel Jacobi coordinates,

Other properties

Exothermicity 31.32 kcal/mol

r_e͑HD͒

1.400

3774

45.6

r_e͑HF͒

1.735

␻ ͑HD͒

␻ ͑HF͒ 4123

␻ ͑DF͒ 2988

0

B_e͑HD͒

E_vϭ0,jϭ0͑HD͒ 5.36 kcal/mol

B_e͑HF͒

20.9

B_e͑DF͒

11.0

E_vϭ3,jϭ0͑HF͒ϭ38.32 kcal/mol

^aJacobi coordinates expressed in units of bohr and deg.

^bEnergy expressed in kcal/mol measured relative to the minimum of the

entrance channel well.

^cAdiabatic barrier in kcal/mol computed using V_adϭប⌬␻_v/2ϩប⌬␻_bend/2

ϩV₀for the bent barrier and V_adϭប⌬␻_v/2ϩប⌬␻_bendϩV₀for the collinear

barrier.

(R,r,␥), where R is the F–͑HD͒

distance, r is the HD

com

bond length, and ␥ is the angle between the R and r vectors.

We deﬁne ␥ϭ0° to the the FDH collinear arrangement and

␥ϭ180° to be the FHD arrangement. The saddlepoints are

located at bent geometries, (R,r,␥)ϭ(3.498,1.457,133.1°),

in units of bohr and deg, for the FHD case and ͑3.186, 1.457,

53.3°͒ for the FDH case. The barrier height is 1.53 kcal/mol

relative to the potential minimum in the entrance channel.

The collinear barriers are located at (R,r,␥)

ϭ(3.911,1.442,180°) for FHD and ͑3.431, 1.442, 0°͒ for

FDH with a height of 1.92 kcal/mol. A normal mode analysis

at the bent saddlepoints reveals that the reaction coordinate

has large angular and translational components and relatively

little vibrational, r, component. If the harmonic zero point

energies of the vibration and bending modes are included,

the adiabatic barrier height is 1.06 and 1.24 kcal/mol for

FHD and FDH, respectively. The collinear conﬁgurations are

unstable with respect to bending motion, since the collinear

geometry is a local potential maximum vs ␥, which suggests

the collinear approach is dynamically unfavorable. In the en-

trance channel, a van der Waals well of depth 0.36 kcal/mol

is located at the T-shaped geometry at Rϭ4.77 a₀. Thus, the

long range part of the interaction should tend to steer the

dynamics into a sideways approach. The important proper-

ties of the surface are summarized in Table I.

^dThe frequencies and rotation constants are in cm^Ϫ1

.

the collision energies investigated by Lee and co-workers.³It

was a straightforward extension of this earlier work to per-

form the present calculations at more collision energies and

for both quantum mechanical parity blocks so as to obtain

the collision energy dependence of the cross sections for

FϩHD( ϭ0,jϭ0) and FϩHD( ϭ0,jϭ1).³⁹

v

The resulting cross sections for these two initial states of

the HD-molecule are shown in Fig. 3, and the averaged cross

sections simulating the experimental beam populations, 80%

jϭ0 and 20% jϭ1 are compared with the present experi-

mental results in Fig. 4. It is clear from the comparison that

the simulation is in excellent agreement with experiment for

the FϩHD→DFϩH reaction, and also in fairly good agree-

ment with experiment for FϩHD→HFϩD. In particular, the

distinct peak in the experimental FϩHD→HFϩD cross sec-

tion at 0.5 kcal/mol is also very apparent in the simulation.

However, this peak is somewhat larger in the simulation than

in the experiment.

The discrepancy between theory and experiment pre-

sumably relates to inaccuracies in the SW-PES since the dy-

namical calculations are converged. The peak in the

FϩHD→HFϩD cross section at 0.5 kcal/mol occurs below

the classical threshold for the reaction, which has been de-

termined to be around 1 kcal/mol in the QCT calculations on

the SW-PES, and therefore involves quantum tunneling.

Thus, we suspect that the SW-PES may permit too much

tunneling. One source of error could be in the barrier height.

Indeed, the adiabatic inclusion of spin–orbit coupling in the

Hartke–Stark–Werner ͑HWS͒ surface²¹does predict an in-

crease in the barrier energy by about 0.35 kcal/mol. A more

detailed critique of the SW-PES will be deferred to Sec. IV.

The tacit assumption from this point on will be that the SW-

PES surface is sufﬁciently realistic, in spite of this discrep-

We have investigated the reaction dynamics for low en-

ergy FϩHD collisions using a variety of theoretical tech-

niques including converged time-independent scattering cal-

culations, collisional wave packet dynamics, spectral

quantization, and quasiclassical trajectories. Together, these

approaches can provide insight in the reaction dynamics.

A. Reactive scattering calculations

In order to simulate the experimental results described in

Sec. II, we have calculated the quantum mechanical cross

sections for the FϩHD reaction on the SW-PES. These cal-

culations were performed using the same quantum reactive

scattering methods used previously²²to study this reaction at

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Transition state resonance in FϩHD

FIG. 4. Comparison of the computed cross sections for 80% HD(jϭ0) and

20% HD(jϭ1) with the experimental results multiplied by 8/9.

FIG. 3. Computed integral cross sections for the FϩHD( ϭ0,jϭ0) and

v

FϩHD( ϭ0,jϭ1) reactions on the SW-PES vs collision energy.

v

Fig. 6. It is clear from this plot that the peak energy almost

perfectly satisﬁes the equation one would expect for a linear

ancy, to give valuable insight into the experimental results

present in Sec. II. In particular, we shall focus on the 0.5

triatomic. Furthermore, the rotational constant

B

kcal/mol peak in the FϩHD( ϭ0,jϭ0)→HFϩD cross sec-

v

ϭ0.0051 kcal/mol that is obtained from Fig. 6 is too large to

tion in Fig. 4 and assume that whatever can be learned about

this peak from the calculations applies equal well to the peak

that is seen in the experiment.

One striking feature of the 0.5 kcal/mol peak that can be

seen from Fig. 3 is that it is highly sensitive to the initial

rotational state of the HD-molecule. The computed cross sec-

tion for the FϩHD( ϭ0,jϭ0)→HFϩD reaction shows the

v

peak very clearly, but it is much smaller and shifted to lower

energy in the cross section for the FϩHD( ϭ0,jϭ1)

v

→HFϩD. In order to investigate the peak further, we shall

therefore concentrate exclusively on the FϩHD( ϭ0,jϭ0)

v

→HFϩD cross section, which is resolved into its partial

wave ͑total angular momentum͒ components in Fig. 5. This

resolution of the partial wave sum clearly shows the effect of

partial wave averaging, which tends to ‘‘smear out’’ the peak

in the integral cross section. The interesting thing is that the

feature, although modiﬁed from pure Lorentzian, survives

this averaging in both the experimental cross section mea-

surement and the simulations.

Further insight into the origin of the peak can be ob-

tained by using the data in Fig. 5 to plot the energy, E_J, of

the peak maximum as a function of J(Jϩ1), as is done in

FIG. 5. Computed partial cross sections for the FϩHD( ϭ0,jϭ0)

v

→HFϩD reaction as a function of the total angular momentum quantum

number, J, up to collision energies of 3 kcal/mol.

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

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FIG. 6. The evolution of the energy, E_J, of peak A as a function of J(J

ϩ1).

be consistent with anything other than a fairly compact ge-

ometry close to the transition state geometry of the reaction.

Castillo and Manolopoulos²²reached the same conclusion in

an earlier study but were unable to determine whether or not

the peak ͑referred to as peak ‘‘A’’ in Ref. 22͒ was due to a

resonance. This issue will be settled conclusively in follow-

ing sections, where we shall show that the peak is due to a

quantum reactive scattering resonance with a collinear FHD

geometry localized in the transition state region of the reac-

tion.

FIG. 7. Two ensembles of quasiclassical trajectories with Jϭ0, ϭ0, and

v

jϭ0 plotted in the (R,␥) coordinates. The top panel is at a translational

energy of 0.6 kcal/mol, which is below the classical reaction threshold,

while the lower panel is at E_Cϭ1.0 kcal/mol, which is just above the reac-

tion threshold.

B. Classical dynamics

A classical analysis of the reaction dynamics provides a

useful complement to the quantum studies in two ways. First,

the quasiclassical trajectories give us physical insight into

the nature of the reaction coordinate. Second, a semiclassical

treatment of the resonance using a periodic orbit analysis

provides a useful approximation for the characteristics of the

scattering resonance.

were termed ‘‘migratory’’ by Aoiz et al.²⁰As noted previ-

ously, addition of even one quantum of HD-rotational energy

greatly suppresses the migratory reaction mechanism. During

the ‘‘internal rotation’’ phase of these collisions, most of the

kinetic energy resides in the rotational mode. Trajectories

impacting near the T-shaped geometry, ␥ϭ90°, show little

deﬂection and are never reactive. It is very interesting that

almost all the reactive trajectories going into the HFϩD

channel ﬁrst hit the D-atom, while for the DFϩH channel

they ﬁrst hit the H-atom. Clearly, the reaction coordinate for

passage over the transition state ͑especially into the HFϩD

channel͒ shows a very signiﬁcant angular component. In this

regard, the low energy FϩHD collisions resemble those for

the heavy–light–heavy system IϩHI that show angular

bottlenecks to reaction.^40,41One can easily imagine that at

collision energies below the classical barrier the reaction

may occur by tunneling through an angular barrier initiated

by a rotational scattering event.

Aoiz and co-workers²⁰have carried out a thorough QCT

study of FϩHD on the SW-PES. The results shown in Fig. 1

reveal that the experimental ␴(E) is in qualitative agreement

with QCT at higher energies, especially for the DFϩH chan-

nel. The 0.5 kcal/mol feature is not reproduced and is appar-

ently a quantum mechanical effect mediated through tunnel-

ing. Nevertheless, the behavior of low energy trajectories

gives some insight for dynamics in the entrance channel. In

Fig. 7 we plot two quasiclassical ensembles with quantum

numbers ϭ0, jϭ0, and Jϭ0 in the coordinates ͑R,␥͒ ͑Ref.

v

40͒ at energies just below and just above the reaction thresh-

old. Since vibrational adiabaticity in the entrance valley is

essentially exact, the vibrational coordinate r is a spectator

until the transition state is crossed. It is seen from the ﬁgure

that the dynamics exhibits a very strong rotational-

translational coupling. The collinear geometries, FHD and

FDH, are seen to be very unstable with respect to rotational

͑bending͒ motion. Trajectories hitting the D-atom tend to be

scattered toward the H-atom side while those impacting the

H-atom are scattered toward the D-atom. Such trajectories

A semiclassical treatment of reactive resonances can be

obtained through the use of periodic orbits.⁴²It has been

observed that often a single orbit at the center of the classical

complex region can provide reasonable predictions of reso-

nance energies. For collinear resonance states, the so-called

‘‘resonating periodic orbits’’ ͑RPOs͒, ﬁrst discussed by Pol-

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Transition state resonance in FϩHD

lak and Child⁴³have proved particularly useful. The quanti-

zation condition for the RPO is

C. Collisional wave packet study

To investigate the dynamics at tunneling energies, it is

necessary to adopt a quantum mechanical treatment of the

reaction. Since the rotational ͑or bending͒ motion played a

central role in the classical reaction mechanism, the use of a

collinear or other two-dimensional representation of the re-

action would be unwise. Therefore, we have carried out a

three-dimensional time-dependent wave packet simulation

for the case Jϭ0. It is hoped that the dynamics of the lowest

partial wave will be sufﬁcient to gain a physical understand-

ing of the dynamical effects at low energy.

The quantum wave packet calculations were carried out

using a method described previously.⁴⁵The following is a

brief review. The system is described by the standard Jϭ0

Hamiltonian for an AϩBC reaction expressed in Jacobi

coordinates.⁴⁶The wave function was represented on a spa-

tial grid in the radial coordinates ͑R, r͒ and in a Legendre

basis for the angular coordinate ␥. Typically, the size of the

representation (N_R,N_r,N_␥) is between ͑128,128,12͒ to

͑256,256,20͒. The wave packets were propagated using a

Feit–Fleck split operator representation for the time-

development operator, U(⌬t). The R and r contributions to

kinetic energy are evaluated using fast-Fourier-transform

͑FFT͒ algorithm. The correct behavior at the origin is en-

forced using a sine-FFT representation. Using this propaga-

tor, the size of the representation is limited by the storage of

U(⌬t), which consists of N_RϫN_rmatrices of size N²_␥. The

value chosen for ⌬t was 2 a.u.

p•dqϭh

ϩ

_pϩ1͒,

v

͑1͒

͑v_R

Ͷ

where (

, _R) are quantum numbers for the exit and en-

v

v_P

trance channel ‘‘local modes’’ of the resonance. For asym-

metrical reactions, the quantum numbers for which an actual

quantum resonance exists generally corresponds to a situa-

tion of near degeneracy between the reactant and product

local modes.⁴⁴For the FϩHD reaction, it is not obvious that

the RPO’s would prove useful in identifying resonances

since the reaction path follows a bent geometry.

Using a Newton–Raphson search procedure, the RPO

was found over a range of energies from slightly above the

classical barrier to nearly the three-body dissociation limit.

A quantized orbit in roughly the correct energy range

and with an approximate degeneracy between the local

modes was found to correspond to the quantum numbers

(

, _R)ϭ(3,0). This orbit was located using a numerical

v

v_P

root search procedure with Eq. ͑1͒, was found to have an

energy E_CϭϪ0.45 ͑i.e., a total energy 4.91 kcal/mol͒, a pe-

riod 22.5 fs, and Lyapunov numbers 4.73 and 0.24. Because

the exit channel reaction path is strongly collinear, the RPO

remains collinear. The orbit is shown later, in Fig. 14, along

with a quantum resonance state of similar energy. Restricted

to the collinear subspace, the RPO is stable at total energies

above 9.3 kcal/mol. However, in three-dimensions the RPO

is never stable which again points to the bending instability

of the classical dynamics. Unlike for systems with a collinear

reaction path, the local bending stability parameters of the

RPO are not useful to obtain a bending frequency since the

bend potential is a double well along much of the orbit.

Instead, a bending frequency was obtained by time-

averaging the bend potential, V(␾;R_FH,R_HD), i.e., the full

potential with the R_FHand R_HDbond lengths held constant,

Unphysical reﬂection of the wave packet at the edge of

the grid was suppressed using an absorbing optical potential

situated in the exit channels. A linear ramping form of the

optical potential was used. The parameters were optimized to

minimize reﬂection for each packet using analog collinear

calculations as a guide. This was necessary since the low

translation energy modes in the entrance channel are difﬁcult

to absorb.

The quantum dynamics of the low energy FϩHD reac-

tion can be visualized with the aid of quantum wave packets

that propagate inward toward the barrier from the entrance

channel. We have employed initial wave packets localized at

R₀in the entrance channel with average initial momentum

ϪP₀of a Gaussian form,

and the G-matrix element, G , along the periodic orbit.

Thus, we can use the quantities

␾␾

␶

1

V ␾͒ϭ

͑

V ␾;R t͒,R

t͒͒dt,

͑

HD

͑

͵

FH

␶

0

͑2͒

␶

1

2

⌿ R,r,␥,tϭ0͒ϭexp Ϫ RϪR ͒²/2␴

͑

͓

͑

0

Gϭ

G

␾;R t͒,R

t͒͒dt

͑

HD

͵

␾␾

FH

␶

0

ϪiP R/ប ␾ r͒L ␥͒.

͑4͒

͔

͑

0

j

to deﬁne a semiclassical action and quantum number through

Here, ␾ is the ground vibrational eigenstate of HD and L_jis

0

the jth-Legendre polynomial. The value of ␴ determines the

energy dispersion of the packet. We wish to make ␴ large to

minimize the spread around the mean collision energy, how-

ever it cannot be made too large without losing the physical

insight into the reaction dynamics. In Fig. 8 we show snap-

ͱ_{2 EϪV ␾͒͒/G ␾͒d␾ϭh}

_bendϩ1/2͒. ͑3͒

͑v

I E͒ϭ

͑

Ͷ

The bend frequency was deﬁned using the energy difference

between the ϭ0 and ϭ1 solutions to Eq. ͑3͒, i.e., E

v

1

ϪE₀ϭប ␻_bend. The result is ␻_bendϭ320 cm^Ϫ1. A prediction

for the resonance energy in 3D can be made by adding the

2D RPO energy to the bending zero-point energy. The value

obtained, E_Cϭ0.46 kcal/mol, is close to the 0.5 kcal/mol

feature seen in the experiment and the quantum scattering

calculation. We shall later see that a quantum resonance ex-

ists that is closely related to this classical orbit.

shots of a wave packet with jϭ0, E ϭ0.6 kcal/mol and

͗ ͘

⌬E ϭ0.25 kcal/mol at various stages of the collision pro-

͗

͘

cess. For this wave packet, most of the reaction is due to

2

tunneling. Two representations of the ͉⌿͉ are given; the

ﬁrst in the ͑R, ␥͒ coordinates with r integrated out, and the

second is a slice through the FHD collinear subspace, i.e.,

␥ϭ180°. At early times, the packet is uniformly distributed

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

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FIG. 8. Snapshots of a quantum wave

packet started as a broad Gaussian in

the FϩHD( ϭ0,Jϭ0) channel with

v

E

ϭ0.6 kcal/mol

and

⌬E

͗

͘

C

ϭ0.25 kcal/mol. The ﬁrst set of panels

show the packet in the ͑R, ␥͒ coordi-

nates with the r-coordinate integrated

out. The second set of panels show the

packet at the same times except in the

FHD collinear subspace. The ͑R, ␥͒

plots show the packet focusing into the

T-shaped geometry and then tunneling

through the barrier along an angular

coordinate. The ͑R, r͒ plots show the

creation of

a persistent probability

density near the transition state.

in angle, ␥, and localized far from the barrier. As the packet

approaches the barrier, the effects of the van der Waals well

and of the collinear angular barriers tend to focus the wave

packet into the T-shaped geometry, ␥ϭ90°. This behavior is

clearly reminiscent of the low energy classical dynamics.

During the strong part of the collision, the packet spreads

into the product wells mostly through the ␥-coordinate mo-

tion. This angular tunneling path was also presaged from the

classical mechanics. The density mainly enters into the

HFϩD reactive channel, with the much smaller DFϩH con-

tribution coming mostly from the higher energy tail of the

wave packet. The collinear plots show the creation of a per-

sistent density just past the saddlepoint with three plainly

visible nodes. This density build up is localized at the col-

linear geometry showed no nodes in the ␥-direction. This

density pattern persists out to the longest times in the simu-

lations. The results of this simulation are consistent with the

collisional formation of a reactive resonance with a linear

geometry.

We have also considered collisions with jϭ1, which

make about a 20% contribution in the experiment. The es-

sential difference with the jϭ0 dynamics seems to trace

back to the ␥ϭ90° node in the initial wave function for j

ϭ1. This node, which persists during the evolution, inhibits

the focusing of the wave packet at the T-shaped geometry

and, hence, the subsequent angular spreading into the reac-

tive channel. However, one still observes the formation of a

three-node amplitude in the FHD linear arrangement, but

there is much more reﬂection of the packet back into the

FϩHD entrance channel. The ratio of reaction probabilities

at E ϭ0.6 kcal/mol, P(jϭ0):P(jϭ1), is about 2.5:1,

͗ ͘

which is consistent with the results of the scattering calcula-

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Transition state resonance in FϩHD

FIG. 8. ͑Continued.͒

tion presented above when averaged over the energy spread

of the packet.

candidate for the reactive resonance. The wave function as-

sociated with this peak was extracted from the optimal wave

packet using the Fourier transform

D. Spectral quantization

ϱ

⌿ R,r,␥,t͒•e^{iE /ប}dt.

͑5͒

res

⌽

R,r,␥͒ϰ

͑

res

͑

͵

To investigate the possibility of a reactive resonance

near 0.5 kcal/mol, we have carried out an analysis of the

Ϫϱ

reaction

using

the

spectral

quantization

͑SQ͒

The probability density for the resonance wave function is

shown in Fig. 10. It is seen to have the same nodal pattern as

the state created in the collisional wave packet study, namely

three-nodes in the asymmetric stretch direction and zero-

nodes in the bending and symmetric stretch directions. To

further verify that this state is a resonance, we have propa-

gated numerous packets that show a peak ﬁxed at this energy

with similar linewidths. Final, convincing evidence of the

resonance character of this state was obtained using a time

decay calculation. The resonance wave function pictured in

procedure.^14,45,47,48The transition state was probed using a

set of initial wave packets propagated from near the saddle-

point. Background was reduced by optimization of the initial

packet. Using a control algorithm described previously, the

spectral background was minimized using an initial state ob-

tained variationally from the sum of ﬁve Gaussian basis

functions localized at points near the collinear FHD RPO. A

typical spectrum ͑using an unoptimized initial packet͒ is

shown in Fig. 9. The intense peak at E_Cϭ0.52 kcal/mol is a

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Skodje et al.

FIG. 11. The natural logarithm of the survival probability of the resonance

state shown in Fig. 10 as a function of time. The ﬁgure clearly shows an

exponential decay after an induction period.

FIG. 9. A typical energy spectrum employed by the spectral quantization

method that is used to identify resonance states.

the Fourier extracted wave function. When the incoming

wave is depleted, pure exponential decay is established. The

lifetime obtained from the decay calculation is 109 fs. This

value is consistent with the lifetime obtained by a Lorenztian

ﬁt to the spectral peak shown in Fig. 9. It is also consistent

with the linewidth of peak ‘‘A’’ from the scattering calcula-

tions.

Fig. 10 was used as an initial state for a new wave packet

calculation. The decay of the survival probability, plotted in

Fig. 11, shows exponential decay after an induction period.

As discussed in our previous work, the induction period of

slower decay is due to the existence of incoming waves in

The branching ratios of the decay products, FϩHD(

v

Ј

ϭ0,j ) and HF( , j )ϩD are calculated by computing the

v

Ј

Ј Ј

decay ﬂux projected onto the various exit channels. The ﬂux

into the DFϩH reactive channel was very small and was not

accurately measured. To obtain the branching ratios in the

HF( , j )ϩD channels it was necessary to switch to the

v

Ј Ј

DϩHF channel Jacobi coordinates. Unlike time-independent

calculations, for the time-dependent decay of an unstable

state the ﬂux into the decay channels has an intrinsic depen-

dence on the translational coordinate R at which the ﬂux is

evaluated.⁴⁹Effectively, the ﬂux at a value R depends on the

decay rate of the resonance at an earlier time, tϭR/ , where

v

is the channel velocity. Thus, due to this retardation effect,

v

the ﬂux increases with R since the resonance has a greater

population at earlier times. This R-dependence can be elimi-

nated by multiplying the ﬂux by exp(ϪR/ ␶), which is con-

v

sistent with the continuity equation for an exponentially de-

caying state with lifetime ␶. As an illustration, in Fig. 12, the

ﬂux into each HF ( ) product vibration state is plotted vs

v

Ј

time and the rotational channels have been summed over.

The ﬁnal results, shown in Table II, indicate that the reso-

nance decays into both the nonreactive and reactive channels

but shows a strong preference for the FHϩD reactive chan-

nel. The vibrational product distribution is dominantly into

the FH ( ϭ2) channel, with a probability of 97%. The HF

v

Ј

(

ϭ1) channel has a 3% probability and HF ( ϭ0) is

v

Ј

v

Ј

ϳ0.1%. The rotational distributions are somewhat more dif-

ﬁcult to numerically converge due to need to represent the

dynamics further into the exit channels. In the FϩHD chan-

nel, the branching ratio, P (jϭ0): P (jϭ1) is 2.4:1. The j

ϭ2 state is energetically closed at the resonance energy. The

FIG. 10. The probability density of the reactive resonance at E_C

ϭ0.52 kcal/mol. In the top panel the FHD collinear subspace is shown using

the coordinates ͑R, r͒. In the bottom panel, the probability density is sliced

at rϭ2 bohr and is shown in the ͑R, ␥͒ coordinates. The plot clearly shows

a state with three-nodes along the asymmetric stretch and zero nodes in the

symmetric stretch and bend.

rotational distribution for decay into the HF ( ϭ2,j)ϩD

v

Ј

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Transition state resonance in FϩHD

FIG. 12. The natural logarithm of the quantum ﬂux into the vibrational

channels HF( ϭ2), HF( ϭ1), and HF( ϭ0) as a function of time.

vЈ vЈ vЈ

The ﬂuxes show exponential decay corresponding to the lifetime of the

state. The ﬂux is divided by the survival probability and corrected for the

channel radius to obtain the branching ratios report in Table II.

FIG. 13. The resonance decay probability into the rotational channels

HF( ϭ2, j ) and the Franck–Condon factors of the resonance state. The

vЈ

curve marked ‘‘wave packet’’ is the asymptotic normalized decay probabil-

ity projected onto the HF( ϭ2,j ) rotational channels. The curve marked

Ј

vЈ

Ј

‘‘scattering’’ was obtained from the S-matrix S( ϭ0,jϭ0; ϭ2, j ) at

E_Cϭ0.6 kcal/mol. The ‘‘FC-factor’’ curve was obtained by projecting the

resonance wave function unto the asymptotic rotational states of HF.

v

vЈ

Ј

channels, shown in Fig. 13, is seen to be extremely broad.

The average rotational energy release is about 3 kcal/mol,

which is nearly a third of the available energy in the ϭ2

v

Ј

channel. The Jϭ0 scattering calculations at 0.6 kcal/mol

also exhibit a similar distribution of rotational populations as

shown in the ﬁgure.

The very hot rotational distribution of the HF-product

might, at ﬁrst sight, be simply attributed to the rotational

However, we have computed the FC-factors by projecting

the asymptotic free-rotor states onto the resonance wave

function, and the result is not nearly so broadly distributed.

As shown in Fig. 13, the FC-factors also fail to account for

the peaking at j ϭ9. We believe the broad and inverted

Ј

Franck–Condon factors of

a

linear-triatomic to

rotational distribution is further evidence of the strong role of

rotational ͑or bending͒ motion in the reaction dynamics. The

decay of the resonance state has major contribution from

bending dynamics, as did the collisional formation of the

atomϩdiatomic free rotor decay process for the resonance.

TABLE II. Theoretical characteristics of transition state resonance.

reactive complex. Angular tunneling into the high-j states

may be an important effect.

Ј

3D(Jϭ0)

The rotational properties of the triatomic complex can be

investigated by computing the rotation constant B₀for the

resonance state. Based on the assumption of a linear com-

plex, B₀was computed by taking the expectation value of

the inverse moment of inertia with the resonance state, and

then

E_res

5.88 kcal/mol

0.52 kcal/mol

109 fs

E_CϭE_resϪE_vϭ0,jϭ0͑HD͒

Lifetime

B₀

1.8 cm^Ϫ1

Vib. branching ratios ͑HFϩD͒

HF( ϭ2)ϭ0.97,

vЈ

HF( ϭ1)ϭ0.03,

vЈ

HF( ϭ0)ϭ0.001

vЈ

Rot. branching ratios ͑FϩHD͒

HD(jϭ0)ϭ0.7,

HD(jϭ1)ϭ0.3

͑HFϩD͒ϭ0.6,

͑FϩHD͒ϭ0.4,

͑DFϩH͒р0.01

ប²

1

B₀ϭ

.

͑6͒

ͳ ʹ

Channel branching ratios

2

I

Since we do not want to include the contribution from the

decay products, the integration in the matrix element is cut-

off beyond Rϭ4 a₀. The result is B₀ϭ1.8 cm^Ϫ1. This is

close to the value of 1.62 cm^Ϫ1obtained classically by time-

averaging the moment of inertia along the quantizing peri-

odic orbit. It is also identical to the value B₀

Quantum numbers ͑local mode picture͒

Quantum numbers ͑normal mode picture͒

Bending frequencyϭE_res͑3D͒ϪE_res͑2D͒ 349 cm^Ϫ1

ϭ3,

v_FH

ϭ0, _bendϭ0

v_HD

v

ϭ3, ϭ0, _bendϭ0

v_asv_ss

v

Collinear

E_res

4.88 kcal/mol

Ϫ0.48 kcal/mol

͑below threshold͒

1300 fs

ϭ0.0051 kcal/molϭ1.8 cm^Ϫ1

obtained

from

the

E_CϭE_resϪE_vϭ0͑HD͒

J-dependence of peak ‘‘A’’ in the scattering calculations.

Although the resonance at 0.52 kcal/mol is our primary

interest since it serves as an explanation for the peak in the

cross section, for completeness we have also located several

other transition state resonances that lie outside of the energy

range of interest. In the FHD collinear conﬁguration, we

have found resonances at E_Cϭ9.25 and 11.50 kcal/mol.

Lifetime

Product branching ratios

HF͑2͒ϭ0.91, HF͑1͒ϭ0.09,

HF͑0͒ϭ0.003

4.91 kcal/mol

1.62 cm^Ϫ1

E_res͑semiclassical͒

B₀͑semiclassical͒

Bend frequency ͑semiclassical͒

320 cm^Ϫ1

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4548

J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Skodje et al.

pared resonance state, was found to be quite long, 1.3 ps,

more than an order of magnitude larger than the 3D result.

This makes clear once again that the bending instability of

the dynamics plays a major role in the decay of the reso-

nance. The branching ratio of HF( ) products was similar

v

Ј

to the 3D case although the ϭ0,1 probabilities are three

v

times higher for the collinear case.

IV. DISCUSSION

The exhaustive study of the FϩH₂reaction and the re-

cent successes of quantum dynamical simulations had led to

the speculation⁵¹that the last chapter had been written on the

nature of the FH₂transition state. In this work, perhaps an

appendix has been added to this volume of research. We

believe that the sort of reactive resonance inferred by Lee

and co-workers, and analyzed through many theoretical stud-

ies, has ﬁnally been observed in a beam experiment. The

remarkable aspect of this discovery is that the resonance sur-

vives in the integral cross section. It is a fortunate accident

that the resonance for the FϩHD isotope lies so near the

threshold that direct reaction does not appear to interfere

with the resonant scattering. The reaction mechanism at low

energies is essentially pure resonant tunneling which gives

rise to a distinctive excitation function even when averaged

over an angular momentum. In this section, we shall explore

the nature of a resonance-mediated reaction in more detail,

and attempt to correlate the results obtained theoretically and

experimentally. We shall argue that all the experimental

ﬁndings are consistent with the view that the reaction near

E_Cϭ0.5 kcal/mol is mediated through a reactive resonance

and that for E_CϾ1.1 kcal/mol the reaction becomes increas-

ingly dominated by over the barrier dynamics and the open-

FIG. 14. The probability density for the ͑3, 0͒ resonance of the collinear

FϩHD reaction. The quantizing periodic orbit, RPO, is overplotted on the

wave function.

These resonances are also spatially localized in the region of

the transition state with 1 node along the HD coordinate and

4 ͑and 5͒ nodes in the FH coordinate. In the FDH arrange-

ment, a resonance was found at E_CϭϪ0.46 kcal/mol with

four-nodes in the FD coordinate, zero-nodes in the HD co-

ordinate, and zero-bending nodes. We also identiﬁed reso-

nance states at E_Cϭ0.3, 1.1, and 1.6 kcal/mol which are the

previously discovered^22,26‘‘uninteresting’’ resonances lo-

cated in the channel wells.

To label the resonance states, it is necessary to adopt a

zero-order reference system that mimics the dynamics near

the transition state. Several schemes are possible. One might

use the vibrational states of a linear triatomic molecule using

local or normal modes. Alternatively, hyperspherical modes

have been proposed to model heavy–light–heavy type sys-

tems. Here, we ﬁnd it convenient to use a local mode de-

scription, which was used previously for DϩH₂.⁴⁴For the

FHD collinear arrangement, we can use the quantum num-

ing of the HF( ϭ3) channel.

v

Ј

A. Resonance mediated reaction

bers (

,

)

and for the FDH arrangement,

, _bend). In this scheme, the resonance at 0.52 kcal/

v_FDv_HDv

v_FHv_HDv_bend

(

,

For a resonance-mediated reaction, the reaction proceeds

by the collisional formation of a resonance and subsequent

decay into product channels. This sort of mechanism is visu-

alized through the wave packet evolution shown in Fig. 8.

Unlike other systems, which may have reactive resonance

but usually at energies above the barrier energy, for FϩHD

there is no direct contribution to the reactive cross section at

low energies. The resonance thus serves as a ‘‘gateway-

state’’ and, in fact, the only pathway through which all the

reactive ﬂux must pass. Such a reaction mechanism was im-

plicit in the studies of collinear AϩBA model systems by

Bisseling et al.⁵²In a resonance mediated reaction many of

the characteristics of the reactive cross section can then be

tied to the properties of the resonance. Speciﬁcally, the reso-

nance energy, width, and partial widths can be used to char-

acterize the full reaction dynamics at low energies. Further-

more, at sufﬁciently low temperatures the thermal rate

constant will be dominated by the resonance contribution to

the reaction.

mol is ͑3, 0, 0͒. The resonances at 9.25 and 11.50 are ͑4, 1,

0͒ and ͑5, 1, 0͒. The resonance at Ϫ0.46 kcal/mol is the ͑4, 0,

0͒ state referenced to the FDH geometry.

Since the resonance states we have obtained are local-

ized in the FHD linear geometry, it is useful to consider the

collinear version of the reaction on the SW-PES. We have

previously investigated¹⁴the collinear dynamics of FϪH₂on

the SEC6-PES of Truhlar and co-workers⁵⁰and so here we

merely redo similar calculations on a different surface. The

analogous resonance state for the collinear reaction is found

below the FϩHD threshold at E_CϭϪ0.48 kcal/mol (E_total

ϭ4.88 kcal/mol), which is quite close to the value predicted

by the RPO. The resonance wave function, shown in Fig. 14,

is similar in appearance to the collinear slice of the 3D state

pictured in Fig. 10. The RPO, also pictured here, is seen to

lie in the region of high probability density. If the energy

difference between the 2D and 3D states is interpreted as due

to bending zero point energy, i.e.,

E_res3D͒ϭE 2D͒ϩប␻

,

bend

͑7͒

͑

res

B. Reactive cross sections

then the bending frequency is 349 cm^Ϫ1, close to the result

obtained semiclassically. The lifetime of the collinear reso-

nance, obtained from the exponential decay rate of a pre-

We ﬁrst consider the issue of the inﬂuence of an isolated

near-threshold resonance on the excitation function. A reso-

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Transition state resonance in FϩHD

nance has not been seen in an integral cross section for a

chemical reaction before, and it is have often been argued

that resonances will not be observed in the integral cross

sections because of the effect of partial wave averaging. It is

therefore interesting to investigate the resonance contribution

to the computed FϩHD→HFϩD cross section in a little

more detail. The most effective way we have found to do this

is to combine the expected resonance contribution to the re-

action probability at Jϭ0 with the usual J-shifting

approximation.⁵³The reaction probability going from the

␣-channel to the ␤-channel is

ͱ ͱ

⌫_␣⌫_␤E_␣E_␤

P_␣→␤E,Jϭ0͒ϭconst•

,

͑8͒

͑

EϪE Jϭ0͒͒²ϩ⌫²/4

͑

res

where E_␣and E_␤are the channel translational energies and

and ⌫_␤are the partial widths of the resonance, which

⌫

␣

satisfy ⌺⌫_␥ϭ⌫. Equation ͑8͒ is the Breit–Wigner expres-

sion for the contribution from a single pole in the S-matrix at

E

_resϪi⌫/2, although usually the kinematic energy factors are

assumed to be constant over the width of the resonance.

However, it is known that for near-threshold resonances one

term should be retained for low translational energy.^9,40

When this expression is combined with the standard

J-shifting approximation to the cross section,

ប²␲

␴

E͒ϭ

2Jϩ1͒P

EϪB J Jϩ1͒,

͑ ͑

0

͑

͚

␣→␤

2uE_␣

Jϭ0

Jϭ0),

͑9͒

we obtain the resonance contribution to the cross section.

The resonance parameters required in this formula may be

obtained either from the spectral quantization results or from

a ﬁt to peak A in the Jϭ0 scattering calculation; and both

methods yield nearly equal results. For deﬁniteness, we opt

to ﬁt peak A. The top panel of Fig. 15 shows the ﬁt obtained

to the Jϭ0 reaction probability using Eq. ͑8͒ with E_␤as-

sumed constant, and bottom panel shows the corresponding

resonance contribution to the integral cross section. The

resonance parameters obtained were ⌫ϭ0.148 kcal/mol,

FIG. 15. Analysis of the contributions of peak A to the computed Jϭ0

reaction probability ͓top panel, see Eq. ͑8͔͒, and to the integral cross section

͓bottom panel, see Eq. ͑9͔͒ for the FϩHD( ϭ0,jϭ0)→HFϩD reaction. In

v

ﬁtting the resonance peak, the exit channel velocity term was held ﬁxed.

ing of HD. The energy level spacing between jϭ0 and j

ϭ1 is 0.26 kcal/mol which is, in fact, equal to the peak shift

between the two cross sections seen in Fig. 3. The ratio of

peak heights is roughly 3:1 which is close to the ratio of

E

_resϭ0.498 kcal/mol, and B₀ϭ0.0054 kcal/molϭ1.9 cm^Ϫ1

.

In Fig. 15 we plot the resonance contribution to the cross

section along with the calculated cross section. Below about

1 kcal/mol, it is seen that this expression reproduces nearly

exactly the calculated cross section and also the shape ͑but

not the magnitude͒ of the experimental excitation function.

The clear message from the bottom panel of the ﬁgure is that

the resonance survives partial wave averaging to appear as a

distinctive feature in the integral cross section because it oc-

curs so far below the classical threshold for the reaction.

From this analysis we can conclude that the 0.5 kcal/mol in

the experimental cross section is in fact the resonance con-

tribution.

partial widths, ⌫(FϩHD( ϭ0,j ϭ0))/⌫(FϩHD( ϭ0,j

Ј

v

Ј

ϭ1))ϭ2.5, when the translational energy factor in Eq. ͑8͒

and the helicity are taken into account.

We note the small oscillations evidenced in the calcu-

lated cross sections in Fig. 3 at energies above 1 kcal/mol.

Comparing Fig. 3 to the bottom panel of Fig. 15 suggests

that the small oscillations from the high-J resonance terms in

Eq. ͑9͒. Indeed, a comparison between the spacing in the

oscillation with the spacing between the J-shifted versions of

peak A is consistent with that view. The experimental exci-

tation function is not resolved to the point where such small

oscillations might be detected.

The calculated cross section for FϩHD( ϭ0,jϭ1)

v

shown in Fig. 3 also shows a steplike feature at low energy

but is not so distinct as for the jϭ0 case. We can use Eq. ͑8͒

to compute the resonance contribution to this process as

well. The only difference with the previous jϭ0 analysis is

an overall factor of the partial width appropriate to each

channel and an energy shift ⌬E equal to the rotational spac-

The remaining, nonresonant, contribution to the theoret-

ical FϩHD→HFϩD cross section is seen, in Fig. 15, to

show a typical threshold type behavior between 1 and 2 kcal/

mol.

A similar threshold is observed in the case

FϩHD→DFϩH, which of course has no resonance contri-

bution at low energy. This pattern is consistent with the onset

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Skodje et al.

of direct reaction at collision energies higher than the adia-

batic barrier height. The experimental FϩHD→DFϩH cross

section threshold agrees almost perfectly with the results of

the scattering calculation. For the FϩHD→HFϩD reaction

over the same energy range, the experimental cross section

deviates slightly from the theoretical prediction in shape, but

still seems consistent with a direct reaction threshold.

Finally, we note that if the resonance cross section, Eq.

͑9͒, is evaluated using the classical approximation that the

J-sum can replaced with an integral, an analytical expression

may be obtained,

The Doppler proﬁles we have measured suggest that the

vibrational product distribution, HF( )ϩD, is highly domi-

v

Ј

nated by the HF( ϭ2) state at energies below the ϭ3

v

Ј

threshold. This is unlike higher energy collisions studied

here and in previous work, or for the DF( )ϩH isotopic

v

Ј

channel, which showed a somewhat more ‘‘democratic’’ ﬁ-

nal state distribution. This highly inverted product distribu-

tion is consistent with the partial widths of the resonance

which predict only 3% decay into the ϭ1 state and 0.1%

v

Ј

decay into the ϭ0 state. A practical consequence of this

v

Ј

distribution is that perhaps the FϩHD reaction can be used

to create a more efﬁcient chemical laser. In terms of a physi-

cal picture, the resonance can be roughly viewed as a state

trapped in a well on an adiabatic potential curve that corre-

⌫_␣⌫_␤

␪ E ͒ϩ1

͑

␣

␴

E͒ϭconst •

Im ␪ E ͒ln

,

͑

Ј

ͫ

ͩ

ͪ

ͬ

␣→␤

␣

␪ E ͒Ϫ1

͑

ͱ

B₀⌫ E_␣

␣

͑10͒

lates with HF( ϭ3). To decay into the ϭ2 state, a vi-

v

Ј

brational transition must occur to the next lowest curve; to

where

decay into the ϭ1 state a double transition to the next

v

Ј

E

Jϭ0͒Ϫi⌫/2͒

͑

lower curve must occur, etc. If the coupling between the

vibrationally adiabatic curves is weak, one expects the tran-

sition probability to fall off rapidly with larger quantum

number changes.

The rotational product distribution from the experiment

is not yet available. However, the calculated rotational dis-

tributions are quite interesting. The probability distribution

res

␪ E ͒ϭ

ͱ

.

͑11͒

͑

␣

E_␣

It is interesting to note that in this approximation the value of

B₀only enters the cross section as an overall-scaling factor.

As seen in Fig. 15, the classical approximation seems quite

accurate. A similar expression was obtained in a previous

work.⁵⁴

of the states HF( ϭ2,j ) is seen from Fig. 14 to be quite

v

Ј

broad and actually peaks at j ϭ9, close to the energetic

Ј

threshold. Such a distribution argues for a fairly strong bend

component in the decay dynamics.

C. Product distributions

The product distribution in a resonance-mediated reac-

tion should follow the branching ratios for resonance decay,

i.e., the ratio of partial widths. We note ﬁrst the dominance

of the HFϩD channel over the DFϩH channel at low energy

in both experiment and theory. Indeed, it is not until E_C

ϭ1 kcal/mol that the DFϩH channel becomes signiﬁcant. In

terms of the resonance picture this behavior is completely

understandable. Since the resonance state is highly localized

in the FHD collinear arrangement, forming DFϩH products

would require an internal rotation of HD, which is improb-

able. In the resonance decay calculations, the ﬂux into the

DFϩH channel was, in fact, very small. There was no evi-

dence for other mechanisms signiﬁcantly contributing to the

DFϩH channel. Speciﬁcally, there are no other resonances

with the arrangement FDH above between 0 and 1 kcal/mol

to feed this channel. The direct, i.e., nonresonant, reaction

into the DFϩH channel is small for several reasons. An ob-

vious factor is that the higher mass of the D-atom inhibits

tunneling. Furthermore, the adiabatic barrier height for this

channel is 0.2 kcal/mol higher than for HFϩD, which also

makes the tunneling smaller. Finally, there is the effect of the

steric factors in the reaction. As seen from the classical

analysis, the trajectories that react to form HFϩD predomi-

nately ﬁrst attack the D-atom, and vice versa. The cone of

acceptance interpretation of Johnson et al.⁵⁵predicts the

DFϩH channel should have the higher steric factor since

there is a greater range of angles where the D-side of HD is

attacked. Because of the migratory character of the trajecto-

ries, this argument must be turned around to favor the HFϩD

channel. Indeed, the low energy classical simulations con-

ﬁrm this.

D. Angular distributions

Perhaps the most dramatic observation in the experiment

was the rapid variation of the Doppler proﬁle at energies

around the position of the resonance. Over an interval of

only about 0.2–0.3 kcal/mol, the proﬁles shifted continu-

ously from backward to sideways and forward distributed.

We expect this rapid change to also be manifested in the

differential cross section for the HF( ϭ2)ϩD channel. In

v

Ј

the resonance picture, the phase shift of an individual partial

wave changes by ␲ in an interval of roughly ⌫ centered at

E_res. In the present problem there is negligible direct reac-

tion to interfere with the resonant amplitude. Thus, it is the

coherent superposition of resonant amplitudes in many par-

tial waves that combine to determine the differential cross

section. From the scattering calculation, we have been able

to compute the differential cross section at energies around

E

_res. In Fig. 16, the theoretical differential cross section is

seen to go from backward peaked at energies below E_res, to

more forward and sideways peaked at energies above E_res

.

This is certainly at least in qualitative agreement with the

experimental results. A quantitative comparison must wait

until the experimental differential cross section is available.

In physical terms, the ͑3, 0, 0͒ state should be regarded

as a broad resonance since its width ͑0.15 kcal/mol͒ is much

larger than the rotation constant ͑0.0051 kcal/mol͒. This im-

plies that the complex will only rotate a fraction of a period

during its lifetime. For example, if we select Jϭ6, which

shows the maximum of the semiclassical deﬂection

function⁵⁶at E_Cϭ0.7 kcal/mol, the rotation period is 1420

fs. Thus, over the course of one resonance lifetime of 109 fs,

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4551

J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

Transition state resonance in FϩHD

of spin–orbit coupling alluded to earlier. However, the the-

oretical prediction would be too large if the SW-barrier were

too narrow and thus gave excess tunneling. Since the reac-

tion coordinate is largely angular, the barrier may be too

narrow with respect to ␥. Of course, since tunneling is expo-

nentially sensitive to the barrier width, the error may be

rather small. From the results available so far, it appears that

the spin–orbit correction in the HSW-PES may put the bar-

rier too high to be consistent with experiment. When the

experimental product distributions are available, a more sen-

sitive test may be available.

ACKNOWLEDGMENTS

R.T.S. is grateful to Y. T. Lee, S.-H. Lin, and K. Liu for

their support and hospitality during his stay at the Institute

for Atomic and Molecular Sciences. The work of R.T.S. was

partially supported by a grant from the National Science

Foundation. The work of K.L., H.S.L., and F.D. was sup-

ported by the National Science Council of Taiwan and the

Chinese Petroleum Corporation. The work of D.E.M. and

D.S. was supported by EPSRC of the United Kingdom.

FIG. 16. The differential cross section in square Angstroms obtained from

the quantum scattering calculation at six collision energies in the vicinity of

the resonance energy. By convention, backward scattering is at 180° and

forward scattering is at 0°. The curves are labeled by the collision energy in

kcal/mol.

the complex rotates about 28°. This value seems somewhat

small in light of the distributions shown in Fig. 16 and the

experimental results. However, this sort of simple classical

analysis ignores the subtler phase relations between the con-

tributing amplitudes that must underlie the behavior of the

quantum angular distribution. A more thorough physical

analysis of differential cross section will be deferred to a

later work.

¹J. B. Anderson, Adv. Chem. Phys. 41, 229 ͑1980͒.

²Resonances in Electron-Molecule Scattering, van der Waals Complexes,

and Reactive Chemical Dynamics, edited by D. G. Truhlar, ACS Sympo-

sium Series Vol. 263 ͑1984͒; J. N. L. Connor, W. Jakubetz, and J. Manz,

Mol. Phys. 35, 1301 ͑1978͒; J. C. Polanyi and D. C. Tardy, J. Chem. Phys.

51, 5717 ͑1969͒. J. H. Parker and G. C. Pimintel, ibid. 51, 91 ͑1969͒.

³D. M. Neumark, A. M. Wodtke, G. N. Robinson, C. C. Hayden, and Y. T.

Lee, J. Chem. Phys. 82, 3045 ͑1985͒; D. M. Neumark, A. M. Wodtke, G.

N. Robinson, C. C. Hayden, K. Shobatake, R. K. Sparks, T. P. Schafer,

and Y. T. Lee, ibid. 82, 3067 ͑1985͒.

E. Critique of the potential function

⁴M. Faubel, S. Schlemmer, F. Sondermann, U. Tappe, and J. P. Toennies,

J. Chem. Phys. 94, 4676 ͑1991͒; M. Faubel, L. Rusin, S. Schlemmer, F.

Sondermann, U. Tappe, and J. P. Toennies, ibid. 101, 2106 ͑1994͒; G.

Dharmasena, K. Copeland, J. H. Young, R. A. Lasell, T. R. Phillips, G. A.

Parker, and M. Keil, J. Phys. Chem. 101, 6429 ͑1997͒; J. Chem. Phys.

106, 9950 ͑1997͒.

The combination of highly resolved experimental cross

sections and converged quantum mechanical dynamics cal-

culations that we have presented, allows us to comment on

the accuracy of the SW-PES. The ability of quantum simu-

lation to reproduce the resonance feature at 0.5 kcal/mol to

within 0.1 kcal/mol, is strong evidence for the accuracy of

the SW-PES in the collinear portion of the potential surface

where the resonance state is built up. Furthermore, the half-

width of the step in the excitation function in the range E_C

ϭ0.3–0.5 kcal/mol also seems in reasonable agreement with

the scattering calculation and with the SQ resonance lifetime.

This suggests that the decay coupling, which is largely me-

diated through the bend, is also reproduced adequately by the

SW-PES. Excellent agreement is also observed for the inte-

gral cross section for FDϩH production. Indeed, as seen in

Fig. 4, the computed cross section reproduces the experiment

almost exactly from threshold up to 3 kcal/mol of transla-

tional energy. The QCT calculations also do well for this

reaction. Since we interpret reaction into this channel as a

dominantly direct, over the barrier reaction, this agreement

suggests that characteristics of the transition state are being

correctly modeled. One point of disagreement, as mentioned

earlier, is the absolute height of resonance feature. Theory

predicts a peak that is about a factor of 2 too large. Since the

threshold behavior for the FϩHD→FDϩH reaction is pre-

dicted with great accuracy, we suspect the barrier height may

be correct after all, despite the reservations about the neglect

⁵F. J. Aoiz, L. Banares, V. J. Herrero, V. Saez Ravanos, K. Stark, and H.-J.

Werner, J. Phys. Chem. 98, 10665 ͑1994͒; J. Chem. Phys. 102, 9248

͑1994͒.

⁶J. F. Castillo, D. E. Manolopoulos, K. Stark, and H.-J. Werner, J. Chem.

Phys. 104, 6531 ͑1996͒.

⁷G. S. Schatz, Annu. Rev. Phys. Chem. 39, 317 ͑1988͒.

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ʈ

the drastic change in the appearance of the -Doppler proﬁles as a function

of energy.

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Observation of a transition state resonance in the integral cross section of the F+HD reaction

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