ALIGNMENT, VIBRATIONAL, AND STERIC EFFECTS IN
UNIMOLECULAR AND BIMOLECULAR SYSTEMS:TOWARDS AN UNDERSTANDING OF CHEMICAL REACTIONS IN
QUANTUM-STATE DETAIL

byJoanna R. Fair
B.A., Rice University, 1995

A thesis submitted to the Faculty of the
Graduate Schoolof the University of Colorado
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
Chemical Physics
2000

2

Fair, Joanna R. (Ph.D., Chemical Physics)
Alignment, Vibrational, and Steric Effects in Unimolecular and Bimolecular Systems:
Towards an Understanding of Chemical Reactions in Quantum-State Detail
Thesis directed by Adjoint Professor David J. Nesbitt

This thesis describes a series of experimental and theoretical projects designed
to explore the dynamics of molecules and molecular collision systems in quantumstate resolved detail. First, the unimolecular dynamics of HOD are explored via
photoacoustic spectroscopy of the 3νOH and 4νOH overtone bands. Analysis of theovertone series provides the vibrational dependence of rotational constants as well as
evidence that the transition dipole moment vector tilts away from the OH bond withincreasing excitation, inconsistent with simple local-mode, bond-dipole model
predictions but in excellent agreement with full 3D quantum variational calculations.
Second, the generation of radical clusters from the photolysis of Ar-H2S and Ar2-H2S
is investigated via quasiclassical trajectory calculations, providing a comparison withearlier experiments in supersonic jet expansions. The calculations confirm the overall
efficiency of radical cluster formation as well as significant intracluster excitation;clusters with excess rotational energy above the dissociation limit are found to be
trapped by an angular momentum barrier that prevents dissociation.

Third,

quasiclassical trajectory calculations on experimentally determined intermolecularpotentials for He-O2, He-CO, and He-CO2 simulate the collisional formation ofrotationally aligned molecular distributions in a supersonic expansion.

These

calculations verify a preference for j perpendicular to the expansion axis, with3

asymptotic alignment increasing monotonically with j.

This alignment reflects

comparable contributions from elastic and inelastic collisions; in addition, moleculeswith j aligned parallel to the expansion axis have faster average velocities thanmolecules with j perpendicular.

Fourth, the role of intramolecular vibrational

redistribution on the timescale of a reactive encounter is explored with a reduceddimensionality model of Cl + H2O → HCl + OH, with isotopic variations. The
vibrational eigenstates of H2O/HOD/D2O are calculated versus Cl-water separation,
generating adiabatic potential energy curves and nonadiabatic coupling terms fortime-dependent dynamics calculations.
resonance

of

the

H2 O

symmetric

For the vibrational eigenstates, the nearandasymmetric stretches

rotatesthesymmetric/asymmetric stretch towards/away from the Cl atom as it approaches in thevibrationally adiabatic limit. For Cl+HOD, vibrational excitation in the OH modeenhances HCl + OD reaction probability by more than an order of magnitude morethan excitation in OD.

4

To my parents, Frank and Janet, and my brother, Ken,andin memory of my grandparents

5

ACKNOWLEDGEMENTS

While heading down the winding road leading from Huntsville, TX, to adoctorate in chemical physics, I have had the fortune of crossing paths withmany great teachers over the years. Before coming to graduate school, I was
inspired by Pat Nash, Pat Williams, Robert Jones, and especially John
Hutchinson, my mentor and friend. Since my arrival in Boulder, my advisor
David Nesbitt has challenged me and encouraged my development as ascientist—not always a painless process, but one that I now truly value.
During my graduate career, I was privileged to work and commiseratewith the members of the Nesbitt group, including my labmates Ondrej Votava,
Stuart Mackenzie, and Tanya Myers, as well as Bill Chapman, Brad Blackmon,
Scott Davis, David Anderson, Dairene Uy, Michal Farnik, Miles Weida, Ken
Kuno, David Fromm, Sergey Nizkorodov, and Hendrick Hamann. I am also
thankful to all the folks in the JILA Instrument Shop, Electronics Shop,
Computing, SRO, Reception, and Supply Office for skillful and friendly support,
especially Hans Green for help with the Blackbody Radiation Pump and Chela
Kunasz for rescuing me from numerous computer glitches.
Finally, I would like to thank my officemates, who kept me sane andaccompanied me to Dairy Queen during thesis-writing stress: Hilary Eaton,
Sebastian Heinz, Elizabeth Mirowski, Amy Van Engen, and especially myhonorary officemate Anthony Lupinetti, without whom this thesis would nothave been possible.

6

CONTENTS
CHAPTER
1. INTRODUCTION………………………………………..………………1
References for Chapter 1……………….…………….…………………. 6
2. OH STRETCH OVERTONE SPECTROSCOPY AND
TRANSITION DIPOLE ALIGNMENT OF HOD………………………7
2.1 Introduction…………………………………………………….. 7
2.2 Experimental…………………………………………………… 7
2.3 Results………………………………………………………….. 13
2.3.1 HOD: 3νOH…………………………….……………. 13
2.3.2 HOD: 4νOH …………………………………………. 20
2.4 Discussion…………………………………………………... 26
2.4.1 OH stretch dependence of HOD rovibrational
constants……………….……………………………26
2.4.2 Intramolecular alignment of the transition
dipole moment vector………………………………. 29
2.5 Summary………………………………………………………. 40
References for Chapter 2…………….………………………………… 42

3. OPEN-SHELL CLUSTER FORMATION VIA LASERPHOTOLYSIS: QUASICLASSICAL TRAJECTORY STUDIESOF Arn-H2S AND Arn-SH CLUSTER DYNAMICS…………………… 45
3.1 Introduction…………………………………………………….. 45
3.2 Computational Details………………………………………….. 51~
3.2.1 Ground-state potential surfaces: Arn-H2S ( X ) ……...51~
3.2.2 Excited-state potential surfaces: Arn-H2S ( A) ………52
3.2.3 Initial-condition sampling……………………………54
3.2.4 Photoexcitation and trajectory calculations…………60

7

3.3 Results………………………………………………………….. 61
3.3.1 Time-dependent radical concentrations…………….. 62
3.3.2 Energetics of radical cluster formation…………….. 65
3.3.3 Dynamics of radical cluster formation……………… 69
3.3.4 Efficiency and mechanisms of cluster formation:comparison with experiment………………..……… 77
3.5 Conclusions…………………………………………………….. 85
References for Chapter 3……………….…………….…………………. 87

4. DYNAMICS OF COLLISIONAL ALIGNMENT IN SUPERSONICEXPANSIONS: TRAJECTORY STUDIES OF HE + CO, O2
AND CO2 ……………………………………………………………… 90
4.1 Introduction…………….………………………………………..90
4.2 Computational Approach………………………………………..98
4.3 Results………………………………………………………….. 105
4.3.1 Multicollision alignment……………………………. 105
4.3.2 Dependence on Rotor Angular Momentum…………. 110
4.3.3 Velocity dependence………………………………... 119
4.4 Discussion……………………………………………………... 123
4.5 Summary..……………………………………………………... 130
References for Chapter 4………………………………………………..132

5. EXPLORING THE ROLE OF VIBRATIONAL NONADIABATICEFFECTS IN VIBRATIONALLY MEDIATED CHEMICALREACTIONS………………………………………………………….. 137
5.1 Introduction…………………………………………………... 137
5.2 Computational Approach…………………………………….. 140
5.2.1 Calculating vibrational wavefunctions at eachCl-water distance……………………………………143
5.2.2 Adiabatic coupling terms…………………………… 149
5.2.3 Wavepacket propagation…………………………… 1528

5.3 Results………………………………………………………….. 157
5.3.1 Vibrational eigenfunctions, adiabats, andcoupling elements……………………………………158
5.3.2 Cl + H2O/D2O/HOD wavepacket propagation…….. 168
5.4 Discussion ……………………………………………………… 180
5.5 Conclusions…………………………………………………… 182
References for Chapter 5………………………………………………..184
BIBLIOGRAPHY……………………………………………………………...186

9

TABLESCHAPTER 2
2.1 Observed 3νOH transitions out of low-JKaKc states in HOD via highresolution photoacoustic spectroscopy…………………………………… 15
2.2 HOD vOH=3 spectroscopic constants from centrifugally distorted
asymmetric top least-squares fit (in cm-1)…………………………………. 16
2.3 HOD spectroscopic constants (in cm-1 unless otherwise noted) from rigid
rotor fits to Av, Bv, Cv……………………………………………………... 19
2.4 Observed 4νOH transitions in HOD………………………………………... 23
2.5 HOD 4νOH spectroscopic constants (in cm-1)…………………………….

25

2.6 HOD Dunham coefficients (in cm-1) for OH stretch excitation…………...

28

2.7 Experimental and theoretical angle between the transition dipole moment
and the OH bond.……….………………………………………………... 32
CHAPTER 3
3.1

Parameters of the potential………………………………………………. 53

3.2

Calculated product branching ratios for Ar-H2S and Ar2-H2S photolysis
at 193 nm and 248 nm…………………………………………………… 78

CHAPTER 4
4.1

Intramolecular harmonic potential parameters and effective “hardsphere” He + rotor cross sections, used to normalize the number ofcollisions………………………………………………………………… 101

10

CHAPTER 5
5.1

LEPS parameters for the Cl + H2O potential surface……………………. 144

5.2

Table of parameters from quadratic fit of optimal damping parameters,c, for Cl + H2O: c = Ax2+Bx+C………………………………………… 157

5.3

vnm, curve-crossing velocities from Eqn. 5.16, in cm/s……………….… 164

11

FIGURES
CHAPTER 2
2.1

Sample high-resolution photoacoustic scan over the 3νOH overtone regionin (a) H2O and (b) H2O, HOD and D2O (≈1:2:1) photoacoustic
cells………………………………………………………………………... 17

2.2

Sample of the photoacoustic spectra in the 4νOH overtone region of the
OH stretch for (a) H2O and (b) H2O/HOD/D2O (≈1:2:1) cells…..…….. … 21

2.3

Rotational constants and vibrational origins obtained from least-squares
fits of a rigid-rotor Watson asymmetric top Hamiltonian to ground-statedata and νOH, 3νOH, and 4νOH transitions in HOD…….…………………... 27

2.4&
Diagram of the alignment of the transition dipole moment, M v'-v" , in theplane of the HOD molecule………………………………………………... 31

2.5&
Plot of the experimentally and theoretically determined angle of M v'-v"
with the OH bond for a series of OH stretch excitations………………….. 34

2.6

Plot of the 2D OH/OD stretch wavefunctions for vOH = 0, 1, 3, and 4
calculated on the potential energy surface of H2O………………………... 35

2.7

Components of the dipole moment surface of HOD parallel (µ||) andperpendicular (µ⊥) to the OH bond as a function of rOH, at the equilibrium
values of the OD bond length and HOD bend angle……….……………… 39

CHAPTER 3
3.1

Schematic representation of the production of Arn-SH (n = 1, 2) clusters
from photolysis of Arn-H2S precursors………………………………….. 48

3.2

Initial condition sampling for the Ar position relative to the H2S subunit
for 500 trajectories of Ar-H2S……………………………….…………... 58
12

3.3

Initial condition sampling for the Ar momentum relative to the H2Ssubunit for 500 trajectories of Ar-H2S, with coordinates defined as in
Figure 3.2………………………………………………………………... 59

3.4

Time-dependent radical concentrations from photolysis of Arn-H2S
clusters at two wavelengths. a) Ar-H2S photolysis at 193 nm.
b) Ar2-H2S photolysis at 193 nm. c) Ar-H2S photolysis at 248 nm.
d) Ar2-H2S photolysis at 248 nm………………………………………… 63

3.5

Histogram of the energy remaining in 5000 Ar-SH clusters after
photolysis, separated into bound and unbound at the end of 1 ns……….. 67

3.6

Histograms of the energy partitioning between a) EvdW, van-der-Waalsstretch vibrational modes and b) Erot, end-over-end rotational of the ArSH cluster………………………………………………….…………….. 68

3.7

Distribution of total radical-cluster angular momenta for bound
Ar-SH clusters from 248 nm photolysis, partitioned into energeticallybound (E < De) and quasibound (E > De)…………………………….…. 70

3.8

Effective potential, VJeff, for several representative values of J as afunction of R, with r fixed to its equilibrium value……………………... 73

3.9

Tunneling lifetimes of quasibound clusters versus ∆E/Ebarrier fromphotolysis of Ar-H2S at 248 nm, where ∆E is the energy in the cluster
above the dissociation limit and Ebarrier is the barrier height………..…… 76

3.10 COM cluster energy versus collision partner mass for photolysis of H2O
and H2S clustered with a single collision partner……………………..… 84
CHAPTER 4
4.1

A schematic depiction of collisional alignment in supersonic
expansions………………………………………………………………… 94

4.2

Contour plots of the potential energy surfaces for a) He-O2,13b) He-CO, and c) He-CO2………………………………………………... 100
4.3

A plot of the alignment of He + O2 versus effective collision number(Zeff) at three collision energies: 266 cm-1, 89 cm-1, and 18 cm-1…….….. 106

4.4

Results of fitting the alignment versus collision number data for
He + O2, CO2, and CO to a2 = a2∞ [1-e-Zeff/β]…………………………….. 108

4.5

The distribution of rotor angular momenta (j) after a single collision at
266 cm-1 (300 K)…………………………………………………………. 111

4.6

Alignment (a2) as a function of effective collision number (Zeff) for low(j/<j> = 0-0.4), intermediate (j/<j> = 0.8-1.2), and high (j/<j> = 1.6-2.0)
angular momentum values for the He + O2 collision system.……………. 113

4.7

Asymptotic alignment (a2∞) for He + O2, CO, and CO2 for low(j/<j> = 0-0.4), intermediate (j/<j> = 0.8-1.2), and high (j/<j> = 1.6-2.0)
angular momentum values……………………………………………….. 114

4.8

Plot of initial mj/j versus final mj/j for an ensemble of 500,000
He + O2 collisions at 266 cm-1 for a) low (j/<j> = 0-0.4), b) intermediate(j/<j> = 0.8-1.2), and c) high (j/<j> = 1.6-2.0) angular momentum
values…………………………………………………….………………. 116

4.9

Distributions of a) ∆mj/<j> and b) ∆j/<j> for an ensemble of 500,000
He + O2 collisions at 266 cm-1…………………………………………… 118

4.10 The percent difference of the final rotor velocity from the alignment–
averaged final rotor velocity, ∆v/<v> versus (mj/j)f………………….…... 120
4.11 Comparison of the experimental jet-expansion data of Harich and Wodtke
to the overall average ∆v/<v> versus (mj/j)f obtained by averaging theclassical-trajectory data over all j and both potentials…. ………………… 122

14

CHAPTER 5
5.1

Coordinate system for Cl + HOH reactive scattering calculations………… 142

5.2

Adiabatic potential energy curves for Cl + HOH reactive scattering,
generated by calculating the H2O and OH/HCl vibrational eigenfunctions
as a function of parametric RCl-HOH………………………………………... 159

5.3

Fourth and fifth excited-state vibrational eigenfunctions for
Cl + HOH outside the interaction region (R = 4.23 Å, Fig. 5.3a and b,
respectively) and near the interaction region (R = 2.8 Å, Fig. 5.3c and d,respectively)……………………………………………………….….…… 161

5.4

Adiabatic potential energy curves for a) Cl + HOH, b) Cl + DOD, and c)
Cl + HOD near the interaction region……………………………………... 163

5.5

Fourth and fifth excited-state vibrational eigenfunctions for Cl + HOD
outside the interaction region (R = 4.23 Å, Fig. 5.5a and b, respectively)
and near a very strongly avoided crossing (R = 2.8 Å, Fig. 5.5c and d,respectively)………………………………………………………….……. 167

5.6

Sample wavepacket propagation for Cl + H2O(a=1), monitoring thescattering into a) Cl + HOH(a=1), b) Cl + HOH(s=1), andc) HCl(v=0) + OH(v=1)…………………………………………………... 169

5.7

Accumulated fluxes versus time for the scattering froma) Cl + H2O(s=1) and b) Cl + H2O(a=1) for p = 10 au……………….……. 171

5.8

Accumulated flux for p ~ 10 au wavepackets into non-elastic channels for
Cl + HOH, Cl + DOD, and Cl + HOD, summed separately over inelastic
and reactive channels………………………………………………………. 173

5.9

Wavepacket probability in each channel as a function of time fora) Cl + HOH(s=1) and b) Cl + HOH(a=1) at p ~ 10 au…………………… 175

15

5.10 Wavepacket probability in each channel as a function of time fora) Cl + DOD(a=1) and b) Cl + HOD(vOD=1), damped to zero after leaving
the interaction region………………………………………………………. 177
5.11 Momentum dependence of output fluxes in each channel froma) Cl + H2O(s=1) and b) Cl + H2O(a=1)…………………………………... 178

16

CHAPTER 1
INTRODUCTION

A physical chemist’s ultimate goal might be to understand and manipulate
each aspect of a reactive encounter, i.e., the initial rovibrational states of the reactants,
relative orientations, direction of approach, and collision velocities, guiding the
breaking and formation of bonds at a state-to-state level of detail.

While suchdetailed understanding and control of chemical reaction is not currently feasible
experimentally, a number of strides have nevertheless been made in this direction.
Two well-developed areas of this study included vibrationally mediated chemistry1
and stereodynamical control of chemical reactions,2 with progress in these areas more
fully detailed in subsequent chapters.
The crossroads between these two areas can be beautifully illustrated by aseries of experiments by the Zare group on the reaction of CH4 and CD3H with Cl
atoms.3-5 In these studies, CH4, coexpanded with Cl2, was vibrationally excited tov3=1 (asymmetric stretch); the Cl2 was then photolyzed at 355 nm to produce
collisions between Cl and CH4 at a collision energy of 1280 cm-1. The state-to-state
differential cross sections for the reaction of Cl with CH4(v3=1) were measured and17

compared with results from ground-state CH4 and CD3H(v1=1, CH stretch). This
series of experiments determined sufficient detail about the nature of the reactive
encounter to propose a mechanism for the H-atom abstraction, namely, that small,medium, and large impact parameters produce backwards, forwards, and sideways
scattering of the HCl, respectively. CH4 vibrational excitation was shown to increase
the cone angle of acceptance for reaction. This mechanism was then confirmed bystudies of steric effects in the reaction of Cl with spatially aligned CD3H.
This example illustrates three major steps in understanding chemical reactionin quantum-state detail: preparation and interrogation of a specific quantum state of
the reactant molecule; control and understanding of the interaction between thereactants; and collection and analysis of the products of the reaction. In a series offour projects, we follow this path toward quantum-state understanding of chemical
reactions, beginning with the preparation of a molecule in a single quantum state andconcluding with a study of how such a prepared quantum state influences chemical
reaction.

Over the course of this thesis, the interchange between theory andexperiment is emphasized, and in particular, the opportunities for theory to provide
insight initially unavailable from experiment alone are stressed.
The first project, described in Chapter 2, involves the experimental
investigation of the unimolecular dynamics of HOD via OH vibrational overtone
excitation.6 From an experimental point of view, vibrational excitation of a particularbond, which can be used to prepare this molecule in a single rovibrational quantum
state, provides a path towards selective reaction. Since the probability of absorbing aphoton varies as the cosine squared of the angle between the transition dipole moment18

of the molecule and the polarization vector of the light, such photoexcitation can
prepare an aligned distribution of reactive molecules for stereochemical studies.
Importantly, this study provides evidence that for OH overtone transitions, thetransition dipole moment vector tilts away from the OH bond with increasing vOH
excitation, qualitatively inconsistent with simple local-mode, bond-dipole model
predictions of the vibrational transition moments. This observation thus provides theopportunity for exchange between theory and experiment: the relative importance ofelectrical (i.e., in the dipole surface) versus mechanical (i.e., in the potential surface)anharmonicity effects are investigated by 2D and 3D quantum variational
calculations.
The second project, described in Chapter 3, extends the study of unimolecular
dynamics to unimolecular collisions via photodissociation of a reactive precursor
within a cluster. Specifically, the production of radical clusters from the photolysis of
Ar-H2S and Ar2-H2S is investigated via quasiclassical trajectory calculations,providing a comparison with earlier experiments on these clusters in supersonic jet
expansions.7,8 If the cluster is considered as a single unit, this process is a means ofpreparing a wide range of cluster quantum states for subsequent dynamical studies.
Alternatively, photodissociation within clusters of potentially reactive species
provides the possibility of controlling the geometry of the reactive encounter. Thus,
this study provides the next step on the path towards understanding reactive
bimolecular collisions.

Once again, this work allows for interchange between

experiment and theory through investigating the details of the experimentallyobserved photodissociation process. Not only do the photolysis calculations confirm
19the overall efficiency of radical cluster formation, but they also provide evidence forbound clusters with excess rotational energy above the dissociation limit, trapped byan angular momentum barrier that prevents dissociation. These calculations thus
contribute additional understanding of the cluster dynamics unavailable from theexperiment.
The third project, described in Chapter 4, extends the study of collision
dynamics to bimolecular, nonreactive scattering with a computational study ofcollisional alignment in supersonic expansions. Experimentally, collisions between alight “diluent” gas and heavier “seed” rotor molecules in a seeded supersonic
expansion have been shown to produce a distribution of rotor molecules with apreference for j perpendicular to the expansion axis. This phenomenon has been
demonstrated for a wide variety of rotor molecules and collision partners, aligning amore general class of molecules than can be accomplished with AC and DC field
methods. Despite the promising possibility that collisional alignment could be used
to prepare reagents for studies of steric effects in chemical reactions, it has yet to see
practical application in this way. In part, this results from a current lack of agreement
among experimental and theoretical studies about the details of this alignment
process. The goal of this chapter thus is to elucidate the mechanism of alignment via
classical trajectory calculations on experimentally determined intermolecularpotentials. We focus on particular areas of controversy such as the velocity- and jdependence of alignment, as well as on determining the relative contributions fromboth elastic (mj-changing) and inelastic (j-changing) collisions.

20

The final project, described in Chapter 5, concludes the study of collision
dynamics with an investigation of vibrational effects in chemical reaction, focusing
on the reaction of Cl with water and its isotopes. While vibrationally mediatedchemistry has been investigated in great detail both experimentally and theoretically,
few studies have directly probed the effect of the reactive atom on the vibrational
state of the molecule. Consequently, we explore the role of the reactive atom inintramolecular vibrational redistribution on the timescale of a reactive encounter via
time-dependent quantum reactive scattering.

In these studies, the vibrational

eigenstates of a reduced-dimensionality model of Cl + H2O → HCl + OH (andisotopic variations) are calculated as a function of Cl-H2O center of mass separation,
generating adiabatic potential energy curves and the nonadiabatic coupling matrix
elements between these curves for use in time-dependent dynamics simulations.
These calculations allow us to look in detail at the nature of the vibrational
eigenfunctions for H2O/D2O/HOD with chlorine-atom approach. In this study, we
have thus come full circle from our initial work on the overtone spectroscopy of HOD
to probe how an initially prepared quantum state is influenced and changed by thereactive encounter.

21

References for Chapter 1

1. F. F. Crim, J. Phys. Chem. 100, 12725 (1996).
2. A. J. Orr-Ewing, J. Chem. Soc. Faraday Trans. 92, 881 (1996).
3. W. R. Simpson, A. J. Orr-Ewing, and R. N. Zare, Chem. Phys. Lett. 212, 163(1993).
4. W. R. Simpson, T. P. Rakitzis, S. A. Kandel, A. J. Orr-Ewing, and R. N. Zare, J.
Chem. Phys. 103, 7313 (1995).
5. W. R. Simpson, A. J. Orr-Ewing, S. A. Kandel, T. P. Rakitzis, and R. N. Zare, J.
Chem. Phys. 103, 7299 (1995).
6. J. R. Fair, O. Votava, and D. J. Nesbitt, J. Chem. Phys. 108, 72 (1998).
7. S. R. Mackenzie, O. Votava, J. R. Fair, and D. J. Nesbitt, J. Chem. Phys. 105,
11360 (1996).
8. S. R. Mackenzie, O. Votava, J. R. Fair, and D. J. Nesbitt, J. Chem. Phys. 110,
5149 (1999).

22

CHAPTER 2OH STRETCH OVERTONE SPECTROSCOPY AND
TRANSITION DIPOLE ALIGNMENT OF HOD

2.1 Introduction
In recent years, a wealth of experimental studies have been aimed at
elucidating photofragmentation and chemical reaction dynamics of vibrationallyquantum state selected molecules. As a result of its small size, large vibrational
spacings, and relatively convenient access to high overtone absorption bands (i.e., vOH= 3, 4, and 5) with near-IR and visible dye lasers, H2O and its isotopomers have beena focus of significant attention. Of particular relevance has been the highly localized
versus delocalized nature of the OH stretch vibrational levels in HOD, which has
made it an especially attractive candidate for experimental and theoretical studies ofvibrationally mode specific reaction dynamics. Indeed, as beautifully demonstrated
in the Crim,1-4 Rosenwaks,5-7 and Zare8 groups, vibrational excitation of HOD has
been used to control which bond is broken by subsequent UV photolysis1,5-8 as wellas to manipulate bond specific reaction probabilities.2-4

23

Despite the elegant dynamical studies performed on vibrationally excited
HOD, until recently surprisingly little was known about its high-resolution overtone
spectroscopy. There have been far-IR and microwave studies of HOD in the groundvibrational state9 as well as near-IR studies of HOD in the fundamental region thathave analyzed single-quantum (i.e., νOH, νOD) excitation from the ground to firstexcited OH or OD stretching states.10 However, there have been no rotationallyresolved studies on HOD excited beyond vOD = 2, and, until fairly recently, norotationally analyzed overtone studies whatsoever in the pure OH stretch manifold.
The difficulty of such a rotational analysis is easily appreciated, due to the large
rotational constants, anomalous centrifugal effects, and widely spaced, irregular level
patterns of a highly asymmetric top. These issues are further exacerbated in light
hydrides such as HOD by the anomalously large changes in rotational constants with
OH vibrational excitation.
This absence of high overtone data on HOD has recently been changed byefforts from several directions. The first is an overtone study by Bykov et al. on the
3νOH and 5νOH bands of HOD obtained via intracavity laser spectrometer andoptoacoustic dye laser spectrometer, respectively.11

This study determinedvibrational origins and rotational constants for both bands, with a frequency precisionof 0.03 cm-1 for 3νOH.

However, as a prerequisite to vibrationally mediatedphotolysis studies of HOD12 and HOD-containing clusters13 under supersonic jet
conditions, we have more recently performed a high-resolution overtone study in ourlaboratory on the 3νOH band of HOD.

This study, obtained by photoacoustic

detection in a room-temperature cell with an injection-seeded optical parametric24

oscillator, was performed to determine rovibrational frequencies for 3νOH transitions
out of the ground (i.e., JKaKc = 000) rotational state with greater precision.
In addition to this work, high-resolution OH stretch overtone data on HOD via
H + HOD “action” spectra and photoacoustic spectra for 4νOH has been obtained byCrim and co-workers, for which a rotational analysis has not previously been
attempted. The successful completion of our work in the 3νOH region stimulated us torotationally analyze this unassigned 4νOH data.

In the meantime, vibrationallymediated photolysis studies on HOD in the 4νOH region have been reported by
Brouard and Langford14 in which the primary aim was toward characterizing thedetailed OD photofragment distributions from photolysis of quantum-state-selected
HOD. Nevertheless, the work clearly necessitated sufficient understanding of the
4νOH rotational structure to label the HOD quantum states, which allowed estimatesof the band origin and rotational constants to be obtained. However, as some of theseresults differ substantially from the present analysis, a more detailed spectroscopic
investigation of the HOD 4νOH overtone band is still warranted. Thus, the detailed
analysis of both our 3νOH spectra and the 4νOH data of the Crim group forms theinitial thrust of this chapter.
The second area addressed in this chapter is the alignment of the vibrational
overtone transition dipole moment vector with respect to the HOD molecular
framework. This proves to be a surprisingly subtle issue, one whose resolution isquite sensitive to the topology of the potential energy and dipole moment surfaces.
For example, if we think in terms of a simple “bond-dipole” model for a purely
localized vibration, the transition dipole moment vector would be aligned perfectly25along the axis of the vibrating bond. Conversely, if electrical anharmonicity effects
lead to a redistributed charge distribution with nuclear displacement, the transition
dipole moment vector can be rotated significantly away from the bond axis by anamount which depends on the degree of overtone excitation. Clever experimental
techniques for study of such vibrational-transition-moment anisotropies have beendeveloped for large organic molecules aligned in stretched films.15 However, thisinformation has proven challenging to obtain for much smaller species such as water,which rotate nearly freely and therefore cannot be effectively aligned by the film. A
more traditional alternative is via rotationally resolved spectroscopy of these species
in the gas phase. Specifically, this information can be extracted by measuring thefractional a-, b-, and c-type character of each vibrational band, which from a rigid
asymmetric top model determines where the transition moment lies relative to theprincipal inertial axes of the molecule. Consequently, with rotationally resolved data
for the OH stretch (νOH, 3νOH, and 4νOH) series in HOD, one can not only determine
the direction of this transition moment vector but also probe how that vector changes
as a function of OH vibrational excitation. This is of particular interest for suchfundamental species as water, where high quality ab initio dipole moment andpotential energy surfaces exist which can be used with exact 3D quantum variational
calculations for rigorous comparison with experiment.
The remainder of this chapter is organized as follows. Section 2.2 briefly
reviews the experimental apparatus for 3νOH and 4νOH photoacoustic spectroscopy of
HOD. In Sect. 2.3, the results from the 3νOH analysis are used to facilitate rotational
assignment of the 4νOH overtone spectrum of HOD of Crim and co-workers, and26

relative a- and b-type band intensities are determined for the νOH, 3νOH and 4νOH
overtone series.

These intensity ratios probe the vibrational dependence of thetransition dipole moment vector and are used in Sect. 2.4 to test theoretical models of
the H2O dipole moment surface. The conclusions of this chapter are summarized in
Sect. 2.5.

2.2 Experimental
The source of light for the 3νOH photoacoustic signals is a single-mode opticalparametric oscillator (OPO) generating narrow-bandwidth infrared radiation, which is
described elsewhere.16 Briefly, the tripled output at 355 nm of an injection-seeded,pulsed Nd:YAG laser is used to pump two β-barium borate (BBO) crystals in a fourmirror ring resonator to generate “signal” and “idler” beams via parametricamplification. The ring resonator configuration is exploited by injection seeding the
OPO cavity with a tunable, single-mode cw dye laser prior to the UV pump, which
therefore generates near Fourier transform limited, tunable output (3.5 ns, ∆ν = 160
MHz) for both signal and idler pulses. Day-to-day reproducibility and frequency
stability at the MHz level17 for the single-mode 355 nm pump source is maintainedby servo loop locking the injection seed for the Nd:YAG laser fundamental to astabilized optical transfer cavity,18 which is locked to a polarization-stabilized HeNelaser.19 Based on the stability of the optical transfer cavity, frequency differences canbe routinely measured to a precision of better than a few ten-thousandths of a cm-1
over several hundred cm-1, which is far narrower than the Doppler broadened line
width for HOD at room temperature (∆νFWHM ≈ 0.030 cm-1). The spectral precision27

of these measurements is quantitatively verified by comparison with well-known
HITRAN |03-> H2O overtone transitions20,21 simultaneously observed in thephotoacoustic cells.
For the 4νOH band work, a similar photoacoustic cell arrangement is used in
the Crim group but with a conventional pulsed dye laser operating in the 13,750 cm -1
region. The bandwidth of the dye laser is 0.05 cm-1, which is now in excess of room
temperature Doppler widths for HOD. This multimode dye laser has the considerable
advantage over the single mode OPO light source of being able to scan relatively
quickly over the entire photoacoustic spectrum.

However, the corresponding

disadvantage is a much lower overall accuracy of frequency measurement.
Consequently, the dye laser wavelength scan has been calibrated to 56 |04- > H2O
overtone transitions embedded in the photoacoustic spectrum, based on the H2Otransition frequencies reported in the HITRAN database.20,21 It is worth noting that
this calibration plot is quite linear but shifted by nearly 10 cm-1 from values obtained
simply from wavelength readings on the tuning mechanism. Left uncorrected, thiswould lead to errors of comparable magnitude in the least-squares fit band origins,with smaller but also significant effects on the reported rotational constants. Thissuggests the dye laser calibration to be the predominant reason for the significant
discrepancies between the recently reported spectroscopic values of Brouard andLangford14 and our current results from the calibrated 4νOH photoacoustic data. Fromthe scatter in this calibration, the frequency precision of the dye laser is estimated tobe 2σ = 0.07 cm-1, dominated by residual nonlinearity in the dye laser wavelength
scan.
28

These light sources (either the pulsed OPO or dye laser output) are thensequentially passed through two photoacoustic cells, the first containing 10 Torr ofpure H2O, the second containing 10 Torr of a mixture of H2O/HOD/D2O in a nearstoichiometric 1:2:1 ratio. Photoacoustic signals from each cell are simultaneously
digitized via sample and hold circuits and recorded as a function of frequency.
Transitions for both pure HOD 3νOH and 4νOH are readily identified by comparison
between spectra recorded for the two cells. Absolute frequencies are obtained bycalibration against H2O second and third overtone transitions in this spectral region
tabulated in the HITRAN database.20,21

2.3 Results
2.3.1 HOD: 3νOH
The transition frequencies, rotational assignment and asymmetric top spectral
fits of the 3νOH band of HOD are obtained in the following way. In order to minimize
the effects of centrifugal distortion, it is useful to restrict the photoacoustic scans toregions corresponding to low JKaKc.

The initial search is guided by theoretical

calculations by Tennyson and co-workers22 on the H2O potential surface of
Polyansky et al.,23 which predict a 3νOH band origin at 10632.5 cm-1. Due to the light
H/D masses, one must contend with an anomalously large vibrational dependence of
the A, B, and C rotational constants, which can be estimated for vOH = 3 by linear
extrapolation from the ground and vOH = 1 rotational constants.

A simulated

asymmetric top spectrum for the 3νOH band generated from these predictions enables
the search to be constrained to maxima in the P and R branch regions at low JKaKc,
29

which proves sufficiently reliable to make initial rotational assignments of theovertone spectra. These assignments are then rigorously confirmed by ground-state
combination differences with the microwave data of De Lucia et al.,9 which also
agree to within 100 MHz, i.e., <10% of the Doppler broadened line widths (∆νDoppler≈ 1 GHz) observed in a room temperature photoacoustic cell. A summary of the low
JKaKc transitions (J<5) observed and assigned in the P-branch (10631-10656 cm-1) andR-branch (10680-10690 cm-1) regions are reported in Table 2.1.
The 3νOH spectral data for HOD have been analyzed by least squares fitting toa Watson24 asymmetric top Hamiltonian (A-type reduction, Ir representation),including up to quartic centrifugal distortion terms; the results are summarized inTable 2.2. The resulting simulation of the HOD spectrum in this region, shown in
Fig. 2.1c, agrees essentially quantitatively in position and intensity with the observed
transitions in Fig. 2.1.a. As unambiguous confirmation of this rotational assignment,
we have performed vibrationally mediated photodissociation studies of HOD in
supersonic jets, where by virtue of supersonic cooling the spectrum collapses to asingle 3νOH 101←000 transition out of the 000 ground rotational state.12 Of special
note, Tennyson’s prediction22 of the HOD band origin is within 0.8 cm-1 of theexperimental result, which provides strong quantitative confirmation of the Polansky
potential surface23 for H2O. Furthermore, this 3νOH data set allows quite reliable
extrapolation to the 4νOH overtone manifold, permitting detailed rotational
assignment of the third overtone OH stretch HOD data of Crim and coworkers. To
facilitate comparison with fits of other OH stretch transitions of HOD, these fits have30

Table 2.1. Observed 3νOH transitions out of low-JKaKc states in HOD via highresolution photoacoustic spectroscopy.

J'

Ka '

Kc'

J"

Ka "

Kc"

ν (cm-1)

1

1

0

1

1

1

10631 314(0)a

2

1

1

2

1

2

10636.353(3)

3

1

2

3

1

3

10643.752(-1)

1

1

0

1

0

1

10645.613(-1)

1

0

1

0

0

0

10646.917(0)

2

0

2

1

1

1

10647.092(1)

2

1

1

2

0

2

10648.302(-1)

3

1

2

3

0

3

10652.813(-1)

4

1

3

4

1

4

10653.236(0)

2

1

2

1

1

1

10655.996(1)

3

3

0

3

2

1

10680.378 (0)

3

1

3

2

0

2

10680.689(-11)

5

3

3

4

3

2

10680.689(6)

4

1

4

3

1

3

10680.763(-1)

5

3

3

5

2

4

10682.533(-1)

4

2

2

3

2

1

10683.851(1)

4

0

4

3

0

3

10686.072(0)

4

1

4

3

0

3

10689.822(-1)

5

0

5

4

1

4

10689.908(0)aNumbers in parentheses are residual differences from predicted transition
frequencies from the fit.
b Blended peak; weight = 0.1 in fit.
31

Table 2.2. HOD vOH=3 spectroscopic constants from centrifugally distorted
asymmetric top least-squares fit (in cm-1).

A

20.3873(24)aB

9.0739(7)C

6.1667(7)∆J

4.81(13)×10-4

∆JK

0b

∆K

1024(24)×10-2

δJ

1.69(15)×10-4

δK

2.95(22)×10-3

ν010631.679(2)σfit

0.0017

a Uncertainties represent two standard deviations from least-squares fit.
b Zero within uncertainty and thus constrained to be zero.

32

Figure 2.1. Sample high-resolution photoacoustic scan over the 3νOH overtone
region in (a) H2O and (b) H2O, HOD and D2O (≈1:2:1) photoacoustic cells.
Transitions present in only the mixed isotope cell are due exclusively to HOD (andenhanced five-fold for visual clarity). (c) The results of a least squares fit to an
asymmetric top Watson Hamiltonian are shown for HOD 3νOH in the lowest panel.
33

been repeated with parameters limited to the lowest order A, B, and C constants and avibrational origin. The results of such a rigid asymmetric top fit of HOD for 3νOH are
listed in Table 2.3.
For a rigid asymmetric top, the ratio of a-type, b-type, and c-type band
intensities is determined by the square modulus of the transition dipole momentcomponent along the corresponding inertial axes of the molecule. For 3νOH-excited
HOD, only a-type and b-type transitions are observed, consistent with a transition
moment in the plane of the molecule. To specify the intramolecular alignment of thein-plane transition moment, integrated a-type and b-type line strengths are measured
and compared with predicted line intensities from the Watson Hamiltonian.
Averaged over all line strengths measured, the 3νOH second overtone band of HOD
corresponds to an a-type/b-type ratio of 2.7(10):1, where the uncertainty represents
two standard deviations.

34

Table 2.3. HOD spectroscopic constantsa (in cm-1 unless otherwise noted) from rigid
rotor fits to Av, Bv, Cv.

vOH
0b

1c

3 (this work) 4 (this work) 4 (previous)dAv

23.256(95) 22.292(31)20.235(13)19.357(6)19.465Bv

9.084(36)9.072(12)9.041(9)9.029(6)

9.042

Cv

6.398(29)6.318(7)6.163(5)6.089(5)6.092

3707.32(9)10631.75(6)13853.50(4) 13844.8(10632.5)e(13853.4)e

ν0 (v←0)∆v (amu.Å2)

0.054(15)0.054(4)0.037(4)0.030(3)

σfit

0.10

0.17

0.067

0.085

a
b
c
d
e

(13853.4)e0.2

Uncertainties represent two standard deviations obtained from the least-squares fit.
Fit to J ≤ 5 data of De Lucia et al.9
Fit to J ≤ 5 data of Benedict et al.10
From Brouard and Langford.14
Predictions by Tennyson and coworkers based on the Polyansky et al. potential
surface.23

35

2.3.2 HOD: 4νOH
Scans over the 4νOH band have been obtained by tandem photoacoustic
spectroscopy in cells containing i) H2O and ii) isotopically mixed H2O/HOD/D2O.
Since pure D2O absorptions in this region correspond to even higher (∆vOD = 5)
overtone excitation10 and can be neglected, comparison of the two spectra readily
permits contributions from H2O and HOD to be identified separately. Sample data
for H2O and HOD overtone absorptions in the 4νOH region from the H2O and the
H2O/HOD/D2O photoacoustic cells are shown in Fig. 2.2a, demonstrating relatively
clean rotational resolution of the H2O and HOD overtone signals. The region scanned
is between 13760 cm-1 and 13858 cm-1, which from our rotational assignment covers
most of the P branch and the Q branch up to and including the H2O vibrational origin.
The analysis of the 4νOH spectra for HOD follows an identical strategy to the
3νOH band. However, with vOH = 0, 1 and 3 manifolds rotationally characterized,
considerably less uncertainty exists in the extrapolation procedure.

Specifically,

linear extrapolations of the principal rotational constants for vOH = 0, 1, and 3 produce
excellent predictions for A4, B4, and C4. Similarly, a Birge-Sponer extrapolation ofthe νOH and 3νOH band origins yields the necessary information to estimate the 4νOHband origin. Unlike the previous analysis of 3νOH, the much smaller extrapolation up
to 4νOH is now sufficiently accurate to demonstrate an immediate and unambiguous
correspondence between predicted and experimentally determined spectra. Based on
this simulation, preliminary rotational assignments for the low-JKaKc transitions arereadily made and rigorously confirmed by ground state combination differences.

36

Figure 2.2. Sample of the photoacoustic spectra in the 4νOH overtone region of the
OH stretch for (a) H2O and (b) H2O/HOD/D2O (≈1:2:1) cells. Transitions present in
only the H2O/HOD/D2O cell are exclusively from HOD. (c) Simulation of the HOD
spectrum using A, B, and C rotational constants and vibrational origin obtained froma least-squares fit of a Watson asymmetric top Hamiltonian to the transitions in Table
2.4. The constants from the fit are given in Table 2.5.

37

This procedure quickly leads to a complete rotational assignment of HOD transitionsin the spectral region scanned.

The measured HOD 4νOH overtone absorption

frequencies and rotational assignments are provided in Table 2.4.
By analogy to analysis of the 3νOH overtone band, these transitions are leastsquares fit to a Watson asymmetric top Hamiltonian (Ir representation) including up
to quartic terms. While the frequency precision is lower than for 3νOH, the transitions
are fit by these parameters to within 2σ = 0.09 cm-1 (see Table 2.5). The quality of fit
is demonstrated in Fig. 2.2b and c, comparing a 9 cm-1 section of the photoacoustic
spectrum with the least-squares prediction.

As with the 3νOH band, the J ≤ 5transitions are also fit to a rigid rotor Hamiltonian, with the results given in Table 2.3.
Also reported in Table 2.5 are the 4νOH band origin predictions from
Tennyson and co-workers,22 as well as the previous estimates of the 4νOH band origin
and rotational constants from Brouard and Langford.14 Note that the agreement
between theoretical and experimental band origins is extremely good, to within
0.1 cm-1. Conversely, there is 8.7-cm-1 discrepancy between our results and thepreviously reported values of Brouard and Langford. As anticipated, the agreement
between current and previous values is far better for the rotational constants, though
there still are appreciable discrepancies outside our experimental uncertainties for A
and B. Again, given the significant differences in our own work between uncorrected
dye laser readings and the HITRAN database for the 4νOH H2O lines, it is apparent
that independent calibration of the dye laser scan is absolutely essential in obtaining
accurate transition frequencies.

38

Table 2.4. Observed 4νOH transitions in HOD.

J' Ka' Kc' J" Ka" Kc"ν (cm-1)

J' Ka' Kc' J" Ka" Kc"ν (cm-1)

5

1

5

6

0

6 13760.92(-3)

6

3

4

6

3

3 13811.37(3)

4

1

3

5

1

4 13764.65(4)

5

3

3

5

3

2 13814.58(-1)

4

0

4

5

1

5 13771.89(1)

4

0

4

4

1

3 13814.74(-3)

3

2

1

4

2

2 13772.59(4)

2

1

2

3

0

3 13815.135(-2)

3

2

2

4

2

3 13774.78(5)

1

1

0

2

1

1 13815.92(-2)

4

1

4

5

1

5 13774.99(2)

4

3

2

4

3

1 13816.52(-2)

4

0

4

5

0

5 13775.85(5)

5

3

2

5

3

3 13816.99(-4)

4

1

4

5

0

5 13778.94(5)

4

3

1

4

3

2 13817.17(0)

3

1

2

4

1

3 13781.81(0)

3

3

1

3

3

0 13817.79(0)

3

0

3

4

1

4 13785.46(-1)

3

3

0

3

3

1 13817.86(-1)

7

4

4

7

4

3 13786.00(1)

6

3

3

6

3

4 13818.20(2)

6

4

3

6

4

2 13787.83(-1)

1

1

1

2

1

2 13820.95(-8)

6

4

2

6

4

3 13788.21(1)

5

2

4

5

2

3 13822.14(0)

5

4

2

5

4

1 13789.33(0)

1

0

1

2

0

2 13822.53(-8)

5

4

1

5

4

2 13789.37(-4)

0

0

0

1

1

1 13823.81(-2)

2

2

0

3

2

1 13789.87(0)

3

0

3

3

1

2 13825.31(-8)

4

4

0

4

4

1 13790.62(2)

4

2

3

4

2

2 13829.96(-3)

4

4

1

4

4

0 13790.62(2)

3

1

3

3

1

2 13830.67(-2)

39Table 2.4. Observed 4νOH transitions in HOD. (continued)

J' Ka' Kc' J" Ka" Kc"ν (cm-1)

J' Ka' Kc' J" Ka" Kc"ν (cm-1)

3

1

3

4

1

4 13790.77(1)

3

2

2

3

2

1 13834.72(0)

2

2

1

3

2

2 13791.02(2)

1

0

1

1

1

0 13836.31(2)

3

0

3

4

0

4 13791.71(6)

2

2

1

2

2

0 13837.11(-1)

3

1

3

3

2

2 13791.89(14)

2

2

0

2

2

1 13838.01(0)

3

1

3

4

0

4 13796.89(-10) 0

0

0

1

0

1 13838.13(0)

2

0

2

3

1

3 13798.20(8)

3

2

1

3

2

2 13839.00(0)

2

1

1

3

1

2 13798.93(3)

2

1

2

2

1

1 13840.36(5)

5

1

4

5

2

3 13805.59(-6)

4

2

2

4

2

3 13841.92(-7)

2

1

2

3

1

3 13806.07(-2)

1

1

1

1

1

0 13846.64(-3)

2

1

1

2

2

0 13806.19(10)

5

2

3

5

2

4 13847.34(2)

2

0

2

3

0

3 13807.14(-5)

1

1

0

1

1

1 13852.33(2)

1

0

1

2

1

2 13810.63(-2)

2

1

1

2

1

2 13857.27(4)aNumbers in parentheses are residual differences from predicted transition
frequencies from the fit.

40

Table 2.5. HOD 4νOH spectroscopic constantsa (in cm-1).

A

19.443(9)B

9.054(8)C

6.094(4)∆J

3.5(18)×10-2∆JK

5(8)×10-2∆K

8.7(9)×10-1

dJ

1.9(14)×10-2

dK-2(17) ×10-2

ν013853.636(28)a Uncertainties represent two standard deviations obtained from the least-squares fit.

41

Of special relevance to the direction of the overtone transition dipole moment
in HOD, the ratio of a-type to b-type intensities is determined from the photoacoustic
scans to be 6.7(5):1. It is worth noting that this ratio is more than twice as large as the2.7(10):1 value observed for 3νOH, thus indicating significant changes in thealignment of the transition dipole moment vector with OH stretch excitation. We will
return to this point later in the discussion section.

2.4 Discussion
2.4.1 OH stretch dependence of HOD rovibrational constants
The vibrational dependence of the rotational constants can be most simply
characterized by a low-order Dunham expansion in the OH stretch quantum, i.e.,

A v = A e + α eA (v OH + ½)

(2.1a)

Bv = Be + α Be (v OH + ½)

(2.1b)

C v = C e + α Ce (vOH + ½)

(2.1c)As shown in Fig. 2.3a, the data quantitatively corroborate a linear dependence on vOH.
These Dunham expressions can be extrapolated to the OH vibrational minimum atvOH ≈ -½; the corresponding “equilibrium” HOD rotational constants (corrected forzero-point motion in the OH stretch coordinate only) and values for αeA, αeB, and αeC
are listed in Table 2.6.

Note the anomalously large sensitivity of Av to vOH,

decreasing by nearly 20% from vOH = 0 to vOH = 4. The results of these rotational42

Figure 2.3. Rotational constants and vibrational origins obtained from least-squares
fits of a rigid-rotor Watson asymmetric top Hamiltonian to ground-state data and νOH,
3νOH, and 4νOH transitions in HOD (see Table 2.3). (a) Least-squares fits of a loworder Dunham expansion (Eqns. 2.1a-c) to Av, Bv, and Cv reveal the strong linear
dependence of these rotational constants on OH vibrational quantum. Note inparticular the ≈20% change in Av from vOH = 0 to 4. (b) The νOH, 3νOH, and 4νOH
vibrational origins are fit to a Birge-Sponer expression (ν0 / vOH) = -(ωexe)vOH + (ωe ωexe). The linearity of this plot indicates near Morse oscillator behavior in the OH
stretch of HOD. The “equilibrium” rotational constants (corrected for zero-point
motion of OH only), αe's, harmonic frequency ωe, and vibrational anharmonicity ωexe
obtained from these fits are given in Table 2.6.
43

Table 2.6. HOD Dunham coefficientsa (in cm-1) for OH stretch excitation.

Ae

23.749(12)Be

9.092(3)Ce

6.435(3)α Ae-0.9751(41)α Be-0.0141(10)α Ce-0.0772(11)ωe3869.9(19)ωexe

81.4(6)a Uncertainties represent two standard deviations obtained from the least-squares fit.

44

analyses can also be stated in terms of inertial defect (∆ = IC - IB - IA), which for arigid planar molecule should essentially vanish. As listed in Table 2.3 for vOH=0 to 4,
the inertial defects for HOD are consistently ≈0.05 amu⋅Å2 and yet all slightly
positive due to large amplitude in-plane zero-point vibrational motion.25,26
The series of HOD band origins for νOH can be fit to the standard BirgeSponer expression

E v = ω e (v OH + ½) - ω e x e (v OH + ½)2(2.2)

to obtain harmonic frequencies and vibrational anharmonicities. A corresponding
Birge-Sponer plot of the νOH, 3νOH, and 4νOH band origins given in Table 2.3 andshown in Fig. 2.3b yields ωe = 3869.9(19) cm-1 and ωexe = 81.4(6) cm-1 (see Table
2.4). For a simple 1D Morse oscillator model of the OH stretch vibration, theequilibrium bond strength, De, is related to the vibrational anharmonicity by ωexe =ωe2/(4De). This translates into an equivalent Morse oscillator dissociation energy ofD0 ≈ De - ½ωe ≈ 126(3) kcal/mol, in reasonable agreement with the experimental D0
value for H2O of 119(1) kcal/mol.27

2.4.2 Intramolecular alignment of the transition dipole moment vector
The relative a-type to b-type intensity ratios for 3νOH and 4νOH are 2.7(10):1
and 6.7(5):1, respectively. These values can be further augmented by data from
Benedict et al. for νOH,10 which yield an a-type to b-type ratio of 1.4(1):1. These dataclearly indicate a strong dependence of intensity ratios, and thus the intramolecular45

alignment of the HOD transition dipole moment vector, on OH vibrational quanta.
Our analysis proceeds as follows.

For sufficiently small amplitude vibrations,vibrational and rotational motion can be approximately decoupled; in this limit, thetransition dipole moment vector is defined by&& &M v'-v" = v' µ ( R) v"(2. 3)&& &In Eqn. 2.3, µ (R) is the dipole moment operator, R represents all internal&
displacement coordinates, and thus M v'-v" is solely a function of initial (v") and final(v') vibrational state. For vibrational excitation in a triatomic asymmetric top such as
HOD, this transition dipole moment vector lies in the molecular plane and therefore
supports “hybrid” a- and b-type bands. The fractional intensity of each type isdetermined by the angle between the transition moment and the a and b inertial axes,
respectively. Figure 2.4 depicts the location of these inertial axes as referenced to the
HOD equilibrium geometry.
&
In terms of M v'-v" , the fractional a-type or b-type intensities of a vibrationalband can be written as&& & 2
M v'-v" ⋅ (a / b)
Ia / b = &&2&2 &M v'-v" ⋅ a + M v'-v" ⋅ b

(2.4)

&
&
where a and b are unit vectors along the a and b axes. This expression can be more46

&Figure 2.4. Diagram of the alignment of the transition dipole moment, M v'-v" , in theplane of the HOD molecule. The geometry of HOD with respect to the a and b axesof the molecule is determined from the equilibrium bond lengths and bond angles&
given by Benedict et al.10 The angle θv'-v" between M v'-v" and the OH bond is listed
in Table 2.7 for the OH vibrational series νOH, 3νOH, and 4νOH.

conveniently rewritten in terms of experimentally measured a-type to b-type intensity
ratios as&&2
M v'-v" ⋅ aIa
cos 2 θ a= &== cot 2 θ a&22Ibcos θ b
M
⋅b

(2.5)

v'-v"

&
&
&
where θa and θb are the angles between M v'-v" and the a and b axes, with θa + θb =π/2. Based on the known HOD equilibrium geometry10 and θa determined from Eqn.
2.5, the angle between the transition dipole moment vector and the vibrationally47

excited OH bond is given by θv'-v" ≈ 59° - θa (see Fig. 2.4). Table 2.7 summarizes the&results for the series of OH stretch overtone transitions. Interestingly, M v'-v" does notlie along the OH bond, i.e., θv'-v" ≠ 0. Furthermore, there is a clear counterclockwise
rotation of the transition dipole moment away from the OH bond with increasing
vibrational excitation.

Table 2.7. Experimental and theoretical angle between the transition dipole moment
and the OH bond.

Band

2D, bond-dipole 2D, dipole surface 3D, dipole surface

Experiment

νOH

3°

25°

24(1)°

19(1)°

3νOH

1°

31°

29(1)°

28(2)°

4νOH

1°

40°

36(1)°

38(1)°a Uncertainties represent two standard deviations obtained from the least-squares fit.

In order to gain physical insight into these trends, we next predict the&
intramolecular alignment of M v'-v" at various levels of theory. This requires solving&&
for the wave functions Ψv'( R ) and Ψv"( R ) for each vibrational level and choosing a& &trial form of the dipole moment surface, µ (R) . The transition dipole moment&
M v'-v" can then be calculated directly from Eqn. 2.3.
The first approach we explore in an attempt to reproduce the behavior of theexperimental transition moment vector is based on a simple linear “bond-dipole”
48

model of the vibrational excitation.28-30

Explicitly, this calculation involves

approximating the dipole moment surface of the molecule by placing point charges at
each of the atoms (+δ at H and D, -2δ at O) that are not allowed to redistribute withvibrational excitation. Stated simply, this model predicts a purely linear dependenceof dipole moment on OH/OD stretch excitation and therefore explicitly neglects
contributions from “electrical anharmonicity” in the dipole moment surface. The
potential energy surface for nuclear displacements is taken from Polyansky et al.23
The vibrational wave functions on this potential surface are then obtained by solving
for the two-dimensional (2D) OH/OD stretching wavefunctions with the bend
coordinate fixed at 104.5°.

Such a model includes only the “mechanicalanharmonicity” inherent in the OH stretching coordinate and furthermore neglects
any mixing between pure stretching excitations and bend-stretch combinations.
As can be seen from Fig. 2.5 and Table 2.7, this calculation qualitatively fails&
to reproduce the experimental results. Specifically, θv'-v" ≈ 0; that is, M v'-v" ispredicted to lie nearly along the OH bond for each overtone transition. This behavior
is easily understood by examining in more detail the OH/OD stretch nature of thisexcitation. The dependence of the 2D wave functions on number of OH stretch
quanta is shown in Fig. 2.6 from the ground state up to vOH = 4. These plots show
extremely little mixing with the OD stretch coordinate, rOD, consistent with a pure“local-mode” stretch excitation along the OH bond. In this limit, the dipole moment& &vector µ (R) can be expanded in a Taylor series in rOH to produce the following

49

&
Figure 2.5. Plot of the experimentally and theoretically determined angle of M v'-v"
with the OH bond for a series of OH stretch excitations (see Table 2.7). Despite thelocal-mode nature of the OH stretch excitation, this angle grows to almost 40° for thehighest measured vibrational band (4νOH). This figure clearly demonstrates thefailure of the bond-dipole approximation to reproduce the experimentally determined
angles, as well as the near-quantitative agreement between experiment andcalculations, which include the dipole surface of water from Gabriel et al.30

50

Figure 2.6. Plot of the 2D OH/OD stretch wavefunctions for vOH = 0, 1, 3, and 4
calculated on the potential energy surface of H2O from Polyansky et al.23 The
uniformity in the OD coordinate among OH vibrational levels demonstrates thenearly pure “local-mode” character of the OH stretch in HOD.

51

&
expression for the transition moment M v'-v" :&&M v'-v" = µ r

OH = rev' v"&+ dµ drOH r

OH = rev' (rOH - re ) v"

+ . . . (6)&For anything but a pure rotational transition with ∆v=0, the first term in M v'-v" is zerodue to orthogonality of the different vibrational wave functions. In the local-mode,&bond-dipole approximation, the only nonvanishing contribution to M v'-v" arises from&&dµ drOH . Since only the component of µ parallel to the OH bond changes with rOH,&
M v'-v" would thus lie parallel to the OH bond, with θv'-v" = 0. These predictions are inclear disagreement with experiment, indicating that a more sophisticated model of the
HOD dipole moment surface is necessary. The small but nonvanishing values of θv'-v"predicted from this model simply reflect weak mixing of OH and OD stretch
vibrations due to a perturbative breakdown of the local-mode picture.
A much improved dipole moment surface can be obtained from the ab initio& &work of Gabriel et al.31 which conveniently provides µ (R) as an explicit function of
intramolecular coordinates. A second set of predictions for the overtone transition
moment vector is performed with this alternate dipole surface but still for 2D OH/OD
stretch degrees of freedom. Thus, this model incorporates “electrical anharmonicity”
in the dipole surface while still neglecting possible coupling between bend and stretch
excitations. The results of these calculations are reported in Table 2.7 and plotted inFig. 2.5. Both the magnitude of θv'-v" and the vibrationally dependent “tilt” awayfrom the OH bond are now in quite reasonable agreement with experiment. These
results provide strong confirmation of the dipole moment surface of Gabriel et al. and52

indicate that electrical anharmonicity in the H2O dipole moment surface plays the&dominant role in intramolecular alignment of M v'-v" in the OH overtone stretch
manifold of HOD.
In principle, there could be additional contributions to this intramolecular&alignment of M v'-v" from mixing between HOD bend and OH/OD stretch motion on
the potential energy surface. Indeed, the role of 2:1 Fermi resonant bend-stretch
coupling has proven quite important in theoretical studies of intramolecularvibrational relaxation (IVR) in overtone CH excited hydrocarbons.31-37 Furthermore,
it is conceivable that overall rotation of the HOD molecule can be responsible for anenhanced Coriolis-induced coupling of the different vibrational modes. As a final
test, therefore, we relax all constraints on vibrational mode mixing by repeating these
HOD calculations in all 3 intramolecular and 3 overall rotational degrees of freedom.
To do this, we use the TRIATOM suite of programs developed by Tennyson et al.38
to calculate rovibrational wave functions on the full 3D potential energy surface of
Polyansky et al.23 These wave functions are then used with the same dipole moment
surface of Gabriel et al. to predict the series of OH stretch overtone spectra in HOD,
explicitly calculating intensities of JKaKc-resolved HOD rovibrational transitions.
These rovibrational transitions can in turn be readily assigned and their intensities
used to determine the theoretical alignment, θv'-v", of the transition moment vector as afunction of OH overtone excitation via Eqn. 2.5. The results in Table 2.7 and Fig. 2.5
demonstrate that this potential and dipole surface also reproduce θv'-v" quite
adequately as well as the experimentally observed tilt away from the OH bond withincreasing OH vibrational excitation. However, the agreement is not appreciably
53

improved from the simpler 2D calculations for the OH stretch manifold, indicatingthat stretch-bend interactions are not as important as electrical anharmonicity incontrolling the alignment of the transition dipole moment vector in HOD.
Finally, with the ab initio dipole moment surface for water, one can now
explicitly investigate why the transition moment vector “tilts” away from the OH
bond with overtone stretch excitation. This is difficult to represent for a 2D vector
quantity as a function of three intramolecular coordinates.

However, as one

compromise, Fig. 2.7 displays the two nonvanishing orthogonal projections of the full
dipole surface i) µ|| parallel to the OH bond and ii) µ⊥ perpendicular (but in plane) to
the OH bond, where a positive sense of µ|| and µ⊥ corresponds to the O-H and ≈O-D
bond directions, respectively. These two projections are plotted as a function of rOH,
with the OD bond length and HOD bend angle fixed near the equilibrium values. To
establish the relevant length scale, the equilibrium OH separation and classicalturning points for vOH = 4 are indicated. In qualitative agreement with the linear&
bond-dipole model, the component of µparallel to the OH bond varies

approximately linearly with rOH, though with some additional curvature at high rOH.
However, in clear contrast to bond-dipole predictions, the corresponding&
perpendicular component of µ is far from independent of the OH bond length.
Indeed, the magnitude of µ⊥ decreases by nearly 30% of the increase in µ|| over therange of vOH = 4 classical turning points. The latter implies dµ⊥/drOH is nonzero, and&specifically that the component of M v'-v" perpendicular to the OH bond is nonzero.
&
Consequently, M v'-v" must rotate away from the OH bond, in direct contrast to thebond-dipole picture.
54

Figure 2.7. Components of the dipole moment surface of HOD (= dipole surface ofH2O from Gabriel et al.30) parallel (µ||) and perpendicular (µ⊥) to the OH bond as afunction of rOH, at the equilibrium values of the OD bond length and HOD bend
angle. The equilibrium OH bond length and the classical turning points for v OH = 4
are noted. As expected, µ|| is roughly linear in OH bond displacement for sufficiently
small rOH, i.e., consistent with a simple bond-dipole model. However, over the samerange of displacements, µ ⊥ strongly decreases with rOH, which is in direct contrast
with bond-dipole predictions and responsible
for counterclockwise rotation of the&vibrational transition moment vector ( M v'-v" ) away from the OH bond axis.

55

As a final note, the combination of Fig. 2.7 and a nearly perpendicular HOD
bond angle imply that the component of the dipole moment along the OD bond must
be decreasing with OH stretch excitation over the range of rOH < 1.6 Å. However,&this trend must eventually reverse, since in the limit of infinite rOH separation, µasymptotically approaches the dipole moment of isolated OD radical. This would
yield µ⊥=sin(104.5º)· 1.65 D ≈ 1.60 D,39 which is substantially larger than the ≈1.35 D
value theoretically predicted at rOH = 1.6 Å. This behavior further underscores thedramatic importance of electrical anharmonicity in the dipole moment surface andtherefore the relevance of detailed comparisons between ab initio theory andexperiment for such fundamental molecular systems as H2O.

2.5 Summary
High-resolution infrared and visible photoacoustic spectroscopy is used toinvestigate the second and third OH stretch overtone bands of HOD. Rotational
analysis of these data permits Av, Bv, and Cv rotational constants and vibrational
origins to be determined for vOH = 3 and 4. In conjunction with earlier studies of theground state and first OH stretch excited manifold, the vibrational dependence of
these spectroscopic data can be well fit by simple Dunham and Birge-Sponer analyses
in vOH. More quantitatively, the 3νOH and 4νOH vibrational origins for HOD (in the
Born-Oppenheimer approximation) can be used to test the accuracy of current ab
initio/empirical potential energy surfaces by Polyansky et al. Agreement between
experiment and full 3D quantum variational calculations on this surface is excellent:
0.8 cm-1 and 0.1 cm-1 for 3νOH and 4νOH, respectively.
56

Experimental intensities of a-type and b-type transitions within each of theνOH, 3νOH, and 4νOH manifolds are analyzed to determine the intramolecularalignment of the transition dipole moment vector.

This vector is found to lie

significantly away from the OH bond and furthermore “tilts” systematically withincreasing OH stretch excitation. These results are in clear contrast with simple
bond-dipole model predictions, which for a local-mode excitation would be perfectlyaligned along the OH bond, suggesting the importance of electrical anharmonicity inthe dipole moment surface. This is explicitly tested by further comparison withpredictions from an ab initio dipole moment surface of Gabriel et al., which are inexcellent qualitative agreement with experimental results. The absence of significant
contributions to this transition dipole alignment from bend-stretch rotationallyinduced vibrational mixing is explicitly tested by comparing 2D OH/OD stretch
calculations (i.e., at fixed HOD bend angle) with full QM variational rovibrational
calculations (with both 3D vibrational and external rotational motion included).
These overtone results underscore the importance of nonlinear effects in the dipole
moment surface (i.e., electrical anharmonicity) as the predominant influence on
intramolecular alignment of the transition dipole moment vector.

57

References for Chapter 2

1. R. L. Vander Wal, J. L. Scott, F. F. Crim, K. Weide, and R. Schinke, J. Chem.
Phys. 94, 3548 (1991).
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4. J. D. Thoemke, J. M. Pfeiffer, R. B. Metz, and F. F. Crim, J. Phys. Chem. 99,
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Chem. Phys. 95, 3341 (1991).
7. Y. Cohen, I. Bar, and S. Rosenwaks, J. Chem. Phys. 102, 3612 (1995).
8. M. J. Bronikowski, W. R. Simpson, B. Girard, and R. N. Zare, J. Chem. Phys. 95,
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and L. N. Sinitsa, J. Mol. Spec. 153, 197 (1992).
12. D. F. Plusquellic, O. Votava, and D. J. Nesbitt, J. Chem. Phys. 109, 6631 (1998).
13. Votava, O., Plusquellic, D. F., Mackenzie, S. R., Fair, J. R., Nesbitt, D. J. (inpreparation).
14. M. Brouard and S. R. Langford, J. Chem. Phys. 106, 6354 (1997).
15. J. Michl and E. W. Thulstrup, Spectroscopy with polarized light: solute alignmentby photoselection in liquid crystals, polymers, and membranes (VCH, 1995).
16. O. Votava, J. R. Fair, D. F. Plusquellic, E. Riedle, and D. J. Nesbitt, J. Chem.
Phys. 107, 8854 (1997).
58

17. D. F. Plusquellic, O. Votava, and D. J. Nesbitt, Appl. Opt. 35, 1464 (1996).
18. E. Riedle, S. H. Ashworth, J. T. Farrell, Jr., and D. J. Nesbitt, Rev. Sci. Instrum.
65, 42 (1994).
19. T. M. Niebauer, J. E. Faller, H. M. Godwin, J. L. Hall, and R. L. Barger, Appl.
Opt. 27, 1285 (1988).
20. Rothman, L. S., HITRAN database (ONTAR, 1996)
21. L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D.
C. Benner, V. M. Devi, J.-M. Flaud, C. Camy-Peret, A. Perrin, A. Goldman, S. T.
Massie, L. R. Brown, and R. A. Toth, J. Quant. Spectrosc. Radiat. Transfer 48, 469(1992).
22. Tennyson, J., private communication
23. O. L. Polyansky, P. Jensen, and J. Tennyson, J. Chem. Phys. 105, 6490 (1996).
24. J. K. G. Watson, J. Chem. Phys. 46, 1935 (1967).
25. T. Oka and Y. Morino, J. Mol. Spec. 6, 472 (1961).
26. J. K. G. Watson, J. Chem. Phys. 98, 5302 (1993).
27. J. A. Kerr, Chem. Rev. 66, 465 (1966).
28. I. Schek, J. Jortner, and M. L. Sage, Chem. Phys. Lett. 64, 209 (1979).
29. M. L. Sage and J. Jortner, Adv. Chem. Phys. 47, 293 (1981).
30. M. S. Child and L. Halonen, Adv. Chem. Phys. 57, 1 (1984).
31. W. Gabriel, E.-A. Reinsch, P. Rosmus, S. Carter, and N. C. Handy, J. Chem.
Phys. 99, 897 (1993).
32. D. J. Nesbitt and R. W. Field, J. Phys. Chem. 100, 12735 (1996).
33. E. L. Sibert, III, W. P. Reinhardt, and J. T. Hynes, Chem. Phys. Lett. 92, 455(1982).
34. E. L. Sibert, III, W. P. Reinhardt, and J. T. Hynes, J. Chem. Phys. 81, 1115
(1984).
35. E. L. Sibert, III, W. P. Reinhardt, and J. T. Hynes, 1984 81, 1135 (1984).
5936. M. Quack, Ann. Rev. Phys. Chem. 41, 839 (1990).
37. M. Quack and W. Kutzelnigg, Ber. Bunsenges. Phys. Chem. 99, 231 (1995).
38. J. Tennyson, J. R. Henderson, and N. G. Fulton, Comput. Phys. Commun. 86, 175(1995).
39. D. D. Nelson, Jr., A. Schiffman, D. J. Nesbitt, J. J. Orlando, and J. B. Burkholder,
J. Chem. Phys. 93, 7003 (1990).

60

CHAPTER 3
OPEN-SHELL CLUSTER FORMATION VIA LASER PHOTOLYSIS:QUASICLASSICAL TRAJECTORY STUDIES OF
Arn-H2S AND Arn-SH CLUSTER DYNAMICS

3.1 Introduction
Over the past decade, the study of radical van der Waals complexes has been
the focus of considerable interest. An especially active area in this field has been the
study of rare gas-radical hydride cluster species, with particular emphasis oncomplexes of rare gases such as Ne, Ar, and Kr with OH(OD),1-22 and SH(SD)23-28
radicals, which have been mapped out both experimentally and theoretically. Suchprototype systems are ideal for developing an understanding of radical solvation andclose-range inelastic energy transfer effects, as well as providing the fundamental
basis for further studies of radical complexes with reactive species.1,29-31
Typical preparation methods of such rare-gas/radical clusters involve
generation of the radical from an appropriate hydride precursor (e.g., SH/OH from
H2S, H2O) via UV photolysis or discharge, followed by coexpansion of the radicalwith the rare-gas partner in a supersonic jet. The spectroscopy and dynamics of these61

clusters can then be probed via laser-induced fluorescence (LIF) or resonanceenhanced multiphoton ionization (REMPI) by pumping to rovibrational levels of an
excited electronic state, usually many centimeters downstream from the source.
High-resolution studies of these vibrationally and rotationally cold clusters haveprovided a wealth of spectroscopic information about intermolecular potential energy
surfaces in the electronically excited-state; a recent example is the development ofhigh quality empirical potential energy surfaces by Miller and coworkers for R-SH~

complexes (R = Ar, Kr, Ne)26,27 out of the first excited ( A) state. Such studies donot, however, significantly explore the ground-state cluster potential surface, since
the collisional cooling process produces population overwhelmingly in the lowest
rovibrational levels. While this cooling clearly aids in the spectroscopic analysis of
the excited state, it also limits the range of ground-state quantum states available fordynamical studies. Alternative spectroscopic methods for characterizing the groundstate potential surface have been stimulated emission pumping studies by Lester andcoworkers on Ar/Ne-OH13,32,33 as well as the dispersed fluorescence studies of Miller
and coworkers on Ne-OH19 and Ne/Ar/Kr-SH.28
At the same time, a number of studies have been carried out investigating
photodissociation dynamics in small clusters. Systems sampled by these techniques
include complexes of one or few rare-gas atoms with small molecules, in an effort tolink gas-phase photolysis of monomer precursors with photodissociation in solvatedenvironments. These studies raise the intriguing possibility that the photolyzed atomcan, in some circumstances, leave behind a bound, highly excited open-shell cluster.
Several groups have previously provided indirect experimental evidence for such62

products; examples include I-HI from (HI)n photodissociation34 and Ar-I from Ar-HI
photodissociation.35 Theoretical calculations by Juanes-Marcos and Garcia-Vela of
Ar-Cl from Ar-HCl photolysis36,37 and Bowman and coworkers of Ar-OH from ArH2O have also predicted these open-shell cluster products.38
More recently, direct spectroscopic evidence from our group has beenpresented for the production of bound, rovibrationally excited radical clusters fromphotolysis of closed-shell precursor clusters.38-40 Via this technique, the rare gas andUV chromophore are coexpanded in a slit supersonic jet to form cold, closed-shellclusters. The monomer hydride is then photolyzed within the cluster, the fast H atom
escapes with high efficiency, and the resulting radical cluster is probed via LIF. The
time delay (typically 100 ns) between photolysis and probe lasers is shorter than thetimescale for subsequent collisions in the jet, thus probing strictly unimolecular
dynamics for formation of open-shell clusters.
Specifically, this technique has been demonstrated to be an efficient source of
hot Ar-SH and Ar2-SH radical clusters from photolysis of Ar-H2S and Ar2-H2S
followed by H atom ejection. Since these open-shell clusters are formed with highinternal excitation, this in principle opens up a wide range of the ground-state
potential surface for detailed investigation. An important preliminary anddynamically intriguing question to address, however, is how such a relatively“violent” photolysis event can produce weakly bound radical clusters so efficiently.
Figure 3.1 outlines the key features of this process. Arn-SH (n = 1, 2) clusters areformed from Arn-H2S photolysis via the following sets of pathways:

63

Figure 3.1. Schematic representation of the production of Arn-SH (n = 1, 2) clusters
from photolysis of Arn-H2S precursors. Pathways 1 and 3 depict “nonfragmentary”
photolysis, in which excess energy is accommodated by the cluster, and pathway 2
depicts a “boil-off” mechanism, in which excess energy is removed by the escape ofan Ar atom.

Arn-H2S + hν → Arn-SH + H(3.1)

Arn-H2S + hν → Arn-1-SH + Ar + H(3.2)

Arn-SH radical clusters are in effect produced by two mechanisms: i) “boil off,” inwhich one Ar atom escapes to remove excess energy, and ii) “nonfragmentary,” inwhich this excess energy is successfully accommodated by the remaining radical
cluster.

Indeed, it is initially surprising that the clusters survive this photolysis

process at all: the excess photolysis energy is as much as two orders of magnitudelarger than the radical-cluster binding energy in this interaction.
64

The simple

dynamical explanation is that by conservation of momentum, the light H atom cancarry off as much as 95% of the photolysis energy. The remaining excess energy
from H2S photolysis is further reduced by transformation into the Arn-SH center-ofmass (COM) frame, which further increases the probability that internal energy will
not exceed the binding energy of the open-shell cluster.
While simple conservation principles begin to explain how the energy isdisposed, several dynamical aspects warrant further investigation.

First of all,previous experimental work has provided estimates of the efficiency of eachformation process by comparison with LIF signals from SH monomer populations in
the jet. However, uncertainty in radiative lifetimes of the monomer and cluster
species, as well as the extent of Arn-H2S clustering in the jet, distribute theseestimates over approximately an order of magnitude range. Second, we would like toprobe the energetics of cluster formation, with particular attention to the“nonfragmentary” photolysis pathways (i.e., where no additional Ar atom is boiled
off to stabilize the nascent complex). Experimentally, the dominant features in thecluster LIF spectra have been attributed to intermolecular progressions (e.g. van der
Waals stretch and librational modes) out of the lowest vibrational states of Ar-SH and
Ar2-SH. However, this cannot explain the strong intensity peaking at the highestenergy end of the spectrum, which one speculates arise from highly internally excitedclusters near the dissociation limit in the ground electronic state. Thus, it is necessary
to show that the intracluster photolysis process can produce highly rovibrationallyexcited (albeit long-lived) molecules, as well as to study how that excessintermolecular energy is distributed. Furthermore, once these hot free-radical clusters
65are generated by unimolecular photolysis, they experimentally exhibit a much slower
cooling on a 10-µs timescale. It is therefore interesting to determine whether this is apurely due to bimolecular cooling by collisions (which would imply remarkably highenergy-transfer efficiencies), or if any noncollisional processes (such as unimoleculardissociation of clusters with E > De) might contribute to the decrease in averageinternal excitation.
In order to address these issues, we have used quasiclassical trajectory
calculations to simulate the unimolecular photolysis dynamics in Ar-H2S and Ar2-H2S
photolysis. In these simulations, the ground-state Arn-H2S wavefunction is used togenerate and sample from an appropriate distribution of initial coordinates andmomenta. This initial wavefunction is projected (in a Franck-Condon sense) onto an
excited-state, repulsive surface on which the photolysis dynamics are calculated. The
resulting trajectories allow us to explore the dynamics of H atom photoejection andsubsequent stabilization of the open-shell van der Waals cluster. The remainder ofthis chapter is organized as follows.

Section 3.2 describes the details of thecomputational methodology for generating initial conditions and calculating thephotolysis trajectories.

Section 3.3 describes the results of these quasiclassicaltrajectories, including the lifetimes of radical clusters, the energetics and dynamics ofcluster formation, and branching ratios among photolysis products. Section 3.4
summarizes the key conclusions of this study.

66

3.2 Computational Details~

3.2.1 Ground-state potential surfaces: Arn-H2S ( X )In order to generate the initial conditions for Arn-H2S and subsequently
calculate the photodissociation dynamics, two potential surfaces are required: one for~the ground electronic state, Arn-H2S ( X ) , and one for the first excited electronic state
accessed by the photolysis pulse. Neglecting three-body terms, one could calculate
what one needs with only the relevant two n=1 (i.e., Ar-H2S) surfaces; unfortunately,
neither dimer potential surface is currently available. However, for our current
purpose we can obtain a qualitatively correct description of the initial state sampling
simplybysubstitutingtheV(R,θ,φ)Ar-H2O

potential

of

Cohen

and

Saykally(AW2),41 where R represents the distance from the Ar atom to the center ofmass of the H2S (i.e., H2O) subunit; θ, rotation of the Ar in the H2S plane; and φ, Ar
rotation out of this plane. In these coordinates, (θ, φ) = (0°, 0°) is defined with Ar
along the C2v axis of the H2S, proximal to the H atoms. The minimum energy
configuration is located at (R, θ, φ) = (3.636 Å, 74.3°, 0°), i.e., rotated 74.3° awayfrom the C2v axis in the plane of the H2O, with an energy of 143 cm-1. The groundstate wavefunction for this system is highly angularly delocalized with near freerotation in θ.

The sampling for the larger Ar2-H2S clusters requires an Ar-Ar

potential as well. This interaction is described by the semiempirical Ar-Ar interaction
potential of Aziz and Chen (HFD-C42), with a well depth of 99.5 cm-1 and an
equilibrium Ar-Ar distance of 3.759 Å.

67~

3.2.2 Excited-state potential surfaces: Arn-H2S ( A)In the excited state, the interaction between the Ar and H2S subunits can nolonger be approximated by the corresponding Ar-H2O surface, since the dissociative
H-SH interaction lies at the heart of the excited-state potential. Furthermore, anintermolecular vibrational SH potential must be included for any dynamics
calculations. Since the H2S is thus fragmented into two subunits, the dissociative H
atom and the remaining SH radical, the interaction between the Ar and H2S is
likewise separated into Ar-SH and Ar-H interactions.
The dissociative potential of the H2S monomer is modeled after the ab initio
calculations of Theodorakopoulos et al. for the lowest dissociative state, the 11A"
state.43 In these calculations, the non-breaking SH bond distance (r) and HSH bondangle are fixed at their equilibrium values (2.52 Å and 92.2°), and the excited-state
potentials are calculated as a function of the breaking SH bond distance. The lowest
adiabatic dissociative state represents two strongly coupled diabatic states (1A2 and1B1) crossing near 2.2 a0 and as such is not well modeled by a single exponential fit.44

To approximate this state, we fit the ab initio points outside the crossing region, i.e.,
those with r > 2.2 a0, to a single exponential. The parameters obtained from thisexponential fit are given in Table 3.1. As noted above, this potential is essentiallyonly valid for a fixed non-breaking SH bond distance and HSH bond angle; however,for a rapid photodissociation process, the H atom should depart before any significant
SH rotation or vibration occurs. Thus, we calculate this potential based solely on the
dissociative SH bond distance. The remaining, nondissociating bond in the H2S

68

subunit is modeled by a Morse oscillator potential with parameters obtained from
Ramsay (see Table 3.1).45

Table 3.1. Parameters of the potential.
HS-H Dissociation: V = A exp(-BrHS-H)
A

482417 cm-1B

2.36/Å
SH Stretch: V = De(1-exp(-α(r-re))2

re1.34 Å

De

30419 cm-1

α1.86/Å

No full potential is available for the Ar-SH interaction; we have thus chosen
to adapt the spin-orbit averaged Ar-OH potential of Lester, Clary, and coworkers13 by
replacing the mass of oxygen with the mass of sulfur. This potential is defined in
Jacobi coordinates (R, θ), with R as the distance from the Ar to the OH center ofmass, and θ as the angle between R and the OH internuclear axis (θ = 0° for H
pointed toward the Ar). The minimum energy is 127 cm-1 in the linear OH-Ar
configuration, and the equilibrium distance between the Ar and the OH center of massis 3.7 Å. This surface overestimates the binding energy of Ar-SH: D0 for Ar-OH is
107 cm-1,3 while D0 for Ar-SH is estimated to be 82 cm-1.24 In addition, a recent

69model potential developed by Yang et al. demonstrates that the Ar-SH ground-state
potential is weaker, but more isotropic, than the Ar-OH potential.28 We thus expectthat our calculations may overestimate the number of bound Arn-SH clusters formed
from Arn-H2S photolysis; however, the effects on the relative efficiencies for Ar-SH
and Ar2-SH production should be less significant. The remaining interaction for ArH2S, i.e., Ar-H, is modeled by the potential of Tang and Toennies.46
For Ar2-H2S, the Ar-Ar interaction is described by the potential of Aziz andChen used for the ground electronic surface (Sect. 3.2.1).

3.2.3 Initial-condition sampling
The initial conditions for the quasiclassical trajectory calculations are
generated in two steps; the first involves sampling the ground-state potential surface
in a manner approximately consistent with the ground-state wavefunction.

Thisprocess is initially detailed for the Ar-H2S calculations, after which the necessary
modifications for the Ar2-H2S photolysis simulations are described.
To sample the ground-state potential energy surface also requires a roughestimate of the ground-state zero-point energy. The zero-point energy of Ar-H2O is44.65 cm-1 above the potential minimum.47 In the pseudodiatomic approximation,
this energy is proportional to the inverse square root of the reduced mass; thus, we
scale the Ar-H2O zero-point energy by µ Ar − H 2O µ Ar − H 2S for Ar-H2S, resulting in anenergy eigenvalue of 36.68 cm-1. The zero-point energy for the trimer (Ar2-H2S) is
estimated as twice that of the dimer (Ar-H2S), since there are now two such
pseudodiatomic modes.
70Next, we generate a microcanonical ensemble of relative Ar-H2S positions,
i.e., with each initial condition having a total energy equal to this ground-state zeropoint energy. Most quasiclassical sampling techniques based on the ground-state
eigenfunction typically require first calculating the eigenfunction, then weighting thesampling of coordinates and momenta consistent with the probability distribution.
However, for the present purposes, the ground-state potential is quite approximate,thus a more qualitative sampling method will suffice. To that end, we sample therelative positions of the Ar and H2S subunits with a probability that decreases
exponentially with potential energy.

This distribution mimics the relationshipbetween the ground-state potential energy function and probability distribution of aharmonic oscillator.
For a one-dimensional, quantum-mechanical harmonic oscillator, the groundstate probability distribution is described by

P(R) = ΨH.O.

2= Ne − µωR !
2

(3.5)

where N is a normalization constant, µ is the reduced mass of the oscillator, and ω isits vibrational frequency. Since the potential energy of the harmonic oscillator isgiven by V(R) = ½µω2R2, and its ground-state energy, Eg.s., is equal to !ω/2, thisprobability distribution can be rewritten as

P(R) = Ne-V(R)/E g.s.

(3.6)

71

with V(R) = 0 at the potential minimum.

Thus, the probability distribution incoordinate space for a harmonic oscillator is an exponentially decreasing function of
the potential energy, with P maximized at the potential minimum and down by afactor of e at the classical turning points (i.e., where V = Eg.s.). This probability
distribution thus reproduces the classically allowed region of a 1D harmonic
oscillator and goes to zero in the classically forbidden regions.
In a similar vein, we describe the probability in coordinate space for an Ar at aparticular point on the Ar-H2S surface by:P(R,θ , φ ) = Ne Ar - H 2S
-V

(R,θ ,φ ) /E g.s.

(3.7)

where Eg.s. = 44.65 cm-1 and VAr -H 2S (R,θ , φ ) is the Ar-H2S potential energy.
Operationally, we sample the Ar-H2S surface by randomly placing the Ar within a (10Å)3 box centered on the H2S. If the potential energy at this point is greater than the
ground-state eigenenergy, this initial condition is rejected; otherwise, its relative
probability is calculated from Eqn. 3.7 (with N fixed to 1), and compared with arandom number on the interval [0,1]. If the calculated probability is greater than thisrandom number, this initial position is accepted. A similar procedure is carried out
for the Ar2-H2S clusters. Since for two separable harmonic oscillators, the total
probability is given by the product of the probabilities for the individual oscillators,
the total Ar2-H2S probability is estimated as-V

P = Ne  Ar - H 2S
1

+V



Ar 2 - H 2S /E g.s.

(3.8)
72where Eg.s. is the ground-state energy of a single Ar-H2S interaction. A plot of 500
initial Ar-H2S positions is given in Fig. 3.2. In this plot, x and y are coordinates inthe H2S plane defined with respect to its COM: y lies along the C2v axis, with positive
y above the S atom, and z is perpendicular to the plane. The maximum density of
initial positions peaks at the Ar-H2S minimum, but the spread in points adequately
represents the “floppy” nature of the intermolecular potential.
Once the relative Ar-H2S positions, and thus the Ar-H2S potential energy, hasbeen determined, the remaining zero-point energy is placed in kinetic energy relative
to the center of mass of the cluster. In the frame of a stationary H2S, the magnitude ofthis momentum for the single-Ar case is approximately

p Ar = m Ar

2 (E

g.s.

- VAr -H 2S)µ Ar,H 2S(3.9)

where µ Ar,H 2S is the reduced mass of the Ar and H2S subunits. The orientation of the
Ar velocity with respect to the H2S subunit is then randomly selected. A plot of this
distribution of momenta relative to the H2S is given in Fig. 3.3. For the two-Ar case,the relative orientation of the two momenta also affects total energy in the COM
frame. Consequently, the magnitudes and relative orientations of the momenta are
randomly sampled up to the maximum value given by Eqn. 3.9. The final momenta
are then each scaled such that the resulting kinetic energy in the COM frame isconsistent with the chosen zero point and potential energies.

73

INITIAL Ar POSITIONS
5
4
3

y (Å)

2
1
0
-1

S
H

H

-2
-3
-4
-5
5
4
3

z (Å)

2
1
0

H S H

-1
-2
-3
-4
-5
-5 -4 -3 -2 -1

0

1

2

3

4

5

x (Å)
Figure 3.2. Initial condition sampling for the Ar position relative to the H2S subunit
for 500 trajectories of Ar-H2S. With respect to the COM of H2S, x and y are in the
H2S plane, with positive y above the S atom and along the C2v axis; z is perpendicular
to this plane. The sampling probability decreases exponentially with potential
energy, so that the maximum density is at the Ar-H2S minimum and the spread inpoints mimics the “floppy” nature of the intermolecular potential.
74

INITIAL Ar MOMENTA
8py (atomic units)

6
4
2
0
-2
-4
-6
-8
8

pz (atomic units)

6
4
2
0
-2
-4
-6
-8
-8

-6

-4

-2

0

2

4

6

8

px (atomic units)
Figure 3.3. Initial condition sampling for the Ar momentum relative to the H2Ssubunit for 500 trajectories of Ar-H2S, with coordinates defined as in Figure 3.2. All
excess ground state energy above the Ar-H2S potential energy is placed in Ar
momentum, with random orientation with respect to the H2S subunit.
75

3.2.4 Photoexcitation and trajectory calculations
Once the initial positions and momenta for the Ar and H2S subunits are
determined, each initial condition is promoted to the excited-state surface, and theinternal energy is adjusted to equal the excess energy from the photolysis. If thisprocess were strictly vertical (in a Franck-Condon sense), the relative positions andmomenta of the atoms would be unchanged by the photoexcitation.

Simplysubstituting the ground-state potential with the excited-state potential, however,would not ensure the appropriate excited-state internal energy, i.e., the sum of theinitial ground-state and photolysis energies minus the SH bond dissociation energy.
~

Since the part of the A -state potential surface most relevant to the photolysis is thedissociative H-SH bond potential, we optimize the dissociative H-SH bond distanceso that the total energy in the cluster is equal to the appropriate excited-state internalenergy. In effect, this step initially places all of the photoexcitation into potential
energy. Two cases are examined: photolysis at 248 nm, which provides 8855 cm-1
above the bond dissociation energy, and at 193 nm, which provides 20,300 cm-1 ofexcess energy.
Once the initial conditions are projected onto the excited-state surface in thismanner, the photodissociation trajectories are calculated by integrating Hamilton's
equations of motion in Cartesian coordinates with a variable-order, variable-stepsize,
Adams-method numerical integrator.48 Trajectories are typically integrated until all
Ar-SH distances exceed 20 a0, or until a fixed integration time has elapsed (1 ns for
Ar-H2S photolysis, 250 ps for Ar2-H2S). Conservation of energy is verified to bebetter than one part in 106, more typically in 107, over the duration of each trajectory.
76

Approximately 500 trajectories are initially calculated for each initial cluster size andphotolysis wavelength; additional trajectories are subsequently calculated as detailed
in the next section. For each trajectory, the momenta and positions of all atoms arerecorded as a function of time, typically in 60-fs steps, for subsequent analysis ofproduct branching ratios, lifetimes of cluster species, energetics of bound clusters,
and dynamical pathways.

3.3 Results
Our first step in examining the results of these photolysis trajectory
calculations is to determine the timescale on which the radical clusters are formed andfall apart, as well as to take a preliminary look at which species are generated. Next,
we analyze the energetics of the radical clusters, first by investigating the overall
energy partitioning among the various rotational and vibrational modes of the cluster
for the whole product distribution, then by focusing on individual trajectories. This
examination leads to a discussion of the dynamics of cluster formation, and more
specifically, the role of cluster angular momentum. Finally, we bring these pieces
together in order to determine overall product branching ratios for comparison withexperimental results.

3.3.1 Time-dependent radical concentrations
We begin the trajectory calculations by examining the 193-nm photolysis of
Ar-H2S and Ar2-H2S, for which the simplest picture emerges. The concentrations ofradical species as a function of time after photolysis for Ar-H2S and Ar2-H2S
77

photolysis at this wavelength are depicted in Figs. 3.4a and b. Both calculations
converge on this 1-ns timescale: the concentration of each radical species levels off
such that little change occurs past the 10-ps mark, i.e., few additional radical clusters
would be predicted to fall apart if the calculations were continued for additional time.
As seen in Fig. 3.4a, the concentration of bound Ar-SH formed from Ar-H2S drops
dramatically such that by the end of the calculation, little or no bound radical cluster
product remains.
In contrast, for Ar2-H2S photolysis, a significant amount of Ar-SH remains
bound: the calculation levels off with roughly 20% of the trajectories following thispathway. Yet few or no trajectories produce Ar2-SH product. Taken together, thesetwo calculations demonstrate that for 193-nm photolysis of Ar-H2S and Ar2-H2S, theprimary product is Ar-SH, formed from “boiling off” one Ar atom from Ar2-H2S
photolysis. This result is not surprising: 193-nm photolysis releases 20,300 cm-1 ofexcess energy; even a dynamical reduction by 20 (≈ MHS/MH) is an order ofmagnitude larger than can be accommodated in the cluster without boiling off at least
one subunit.
The photolysis of Ar-H2S and Ar2-H2S at 248 nm reveals more complex
behavior (Figs. 3.4c and d). In both cases, significant numbers of radical clusters
survive for the duration of the calculation (1 ns for Ar-H2S and 250 ps for Ar2-H2S
precursors). These cluster concentrations level off for Ar-H2S photolysis (note thelog (time) axis), suggesting satisfactory convergence in the calculations. For the ArH2S precursor, Ar-SH appears to be formed at 248 nm with roughly 20% efficiency,

78

79
Figure 3.4. Time-dependent radical concentrations from photolysis of Arn-H2S clusters at two wavelengths.
a) Ar-H2S photolysis at 193 nm. b) Ar2-H2S photolysis at 193 nm. c) Ar-H2S photolysis at 248 nm. d) Ar2H2S photolysis at 248 nm.

providing additional confirmation that the nonfragmentary photolysis channel for ArH2S can indeed result in significant open-shell cluster formation. By way of contrast,
the radical cluster species concentrations for 248 nm Ar2-H2S photolysis do notappear to stabilize asymptotically on the 250-ps timescale of this calculation. Rather,
the Ar2-SH concentrations decrease approximately linearly with logarithmic time,such that ~25% of the initially formed Ar2-SH clusters fragment with each 10-foldincrease in time.
These calculations clearly support a significant channel for open-shell radical
cluster (Ar-SH or Ar2-SH) formation at both 193 nm and 248 nm. The dominantchannel at 193 nm is Ar-SH formed by fragmentary photolysis, i.e., ejection of a hot
H atom from Ar2-H2S followed by boil-off a single Ar atom. The efficiency for thisopen-shell cluster formation process is much higher at 248-nm photolysis, since there
can now be contributions from both nonfragmentary Ar-H2S and Ar2-H2S photolysis
as well as from Ar2-H2S photolysis followed by subsequent boil off. Since the Ar2H2S photolysis at 248 nm has not converged, it cannot yet be determined how many
additional Ar2-SH clusters will fall apart, nor what products they will form. At thisstage, then, it is difficult to probe the relative importance of the photolysis pathways.
In principle, this issue could be resolved by carrying out the calculations for alonger time. However, based on the near-linear dropoff of Ar2-SH concentration withlogarithmic time, a calculation nearly two orders of magnitude longer would be
required for convergence. Rather than simply extending the computational time, wefirst examine the energetics and dynamics of cluster formation to look for a way to
predict propensities towards various pathways and to estimate final branching ratios.
80

In the ensuing analysis, we primarily focus on the photolysis of Ar-H2S at 248 nm;
results for clusters of all types will then be summarized in Sect. 3.3.4.

3.3.2 Energetics of radical cluster formation
In the corresponding experimental work, our group observed that photolysis
of weakly bound closed-shell precursors produced rotationally and vibrationally hotradical clusters. Accordingly, we will explore the energetics of bound radical clusters
formed from Ar-H2S photolysis at 248 nm, looking at the total energy left in thecluster as well as how it is partitioned among the various rotational and vibrationalmodes.
As can be observed in Fig. 3.4c, the trajectories divide into two groups: thosein which the radical cluster falls apart to form Ar and SH, and those which remainbound on the timescale of the calculation (1 ns). Since this calculation is nearly
converged (i.e., the radical concentrations level off), we can then separately examine
the energetics of bound and unbound clusters, assuming that those clusters remaining
bound on this timescale will be bound for all time. In doing this, we take advantageof the rapid departure of the H atom; once it has left the precursor closed-shell cluster,no additional forces act upon the resulting radical cluster, and thus all energy in the
COM frame of the radical cluster remains there for all time. Consequently, we cancalculate the COM energy of the radical cluster immediately after the H atom has
departed. Thus, the calculation can be halted when the H atom interaction with thecluster is negligible; this point is chosen to be 20 a0 away from the S atom, where the81H-SH interaction is less than 10-14 cm-1. To obtain better statistics for the subsequent
analysis, we have calculated an additional 5000 trajectories of Ar-H2S at 248 nm.
Figure 3.5 contains a histogram of the energy remaining in the Ar-SH COM
frame after photolysis, separated into bound and unbound at the end of the 1-ns
calculation. Superimposed is a dividing line at the dissociation energy of the Ar-SH(i.e., Ar-OH) potential surface, De = 127 cm-1. As expected, all unbound clusters
have internal energy greater than De; however, only approximately half of the boundclusters (henceforth “energetically bound”) have E less than De, and half (henceforth“quasibound”) have E greater than De. This result is initially puzzling: it would seemthat all clusters with E > De should ultimately fall apart.
To investigate further, we first determine the partitioning of energy in all
bound Ar-SH clusters (i.e., energetically bound and quasibound) by separating theinternal energy into EvdW (cluster van der Waals stretch energy), Erot (cluster endover-end rotation), and ESH (vibration and libration of the SH subunit). Since ESH issignificantly smaller than EvdW and Erot (i.e., ≤ 10%), we can focus exclusively on the
EvdW and Erot, with histograms of the energy partitioning given in Fig. 5.6. The
clusters are quite vibrationally hot; however, the EvdW distribution ends abruptly at the
dissociation limit.

The excess cluster energy lies rather in rotation, with the end-

over-end rotational energy distribution extending significantly beyond thedissociation limit (Fig. 5.6b).
The reason behind this unequal partitioning is clear: only EvdW (not Erot) > Dewill cause the cluster to dissociate. However, for quasibound clusters (i.e., with total
E > De), sufficient transfer from Erot to EvdW might seem inevitable. In contrast, the82

ENERGY IN Ar-SH COM
AFTER PHOTOLYSIS

Fraction of trajectories

0.40

De(Ar-SH) = 127cm-1

0.35

Bound
Unbound

0.30
0.25
0.20
0.15
0.10
0.05
0.00
40

80

120

160

200

240

280

320

360

400

Energy (cm-1)

Figure 3.5. Histogram of the energy remaining in 5000 Ar-SH clusters after
photolysis, separated into bound and unbound at the end of 1 ns. The line at 127 cm -1
denotes the dissociation energy (De) of the Ar-SH. Notably, half of the boundclusters at the end of this calculation have E > De.

83

Fraction of trajectories

Fraction of trajectories

BOUND Ar-SH INTERNAL ENERGY
0.25a) EvdW

0.20
0.15
0.10
0.05
0.00
0.25

b) Erot
0.20
0.15
0.10
0.05
0.00
0

50

100

150

200

-1

Energy (cm )Figure 3.6. Histograms of the energy partitioning between a) EvdW, van-der-Waalsstretch vibrational modes and b) Erot, end-over-end rotational of the Ar-SH cluster.
While the vibrational energy distribution tails off at the dissociation limit (De =127 cm-1), the rotational energy distribution clearly exceeds this limit.

84

convergence of the cluster calculations (Fig. 3.4c) indicates that relatively few, if any,of the clusters remaining bound at 1 ns will ultimate fall apart. The quasiboundclusters must therefore be subject to a dynamical constraint preventing intraclusterrotational-to-vibrational energy transfer; this possibility is explored in the next
section.

3.3.3 Dynamics of radical cluster formation
As noted above, the “quasibound” clusters, i.e., those with E > De that remainbound at the end of the 1-ns calculation, can contain energy above the dissociation
limit as long as that energy is trapped in rotation. Since this must correlate to highangular momentum states, we next look at the angular momentum distributions forboth energetically bound and quasibound trajectories.
Figure 3.7 contains a histogram representing the total radical-cluster angularmomenta for bound Ar-SH clusters from 248-nm photolysis. A clear distinction
between the energetically bound and quasibound distributions exists: the center of theangular momenta of quasibound clusters, near 40!, is shifted approximately 10!
higher than that of energetically bound clusters. Importantly, the center of each
distribution exceeds the average pre-photolysis angular momentum (from the initialcondition sampling) by > 20!; thus, most of the angular momentum observed in thesebound clusters is generated by photolysis. In addition, it represents an unusually hot
distribution of clusters: for a peak value of J = 40!, the rotational temperature would
be greater than 300 K, much higher than temperatures typically available bycoexpanding rare-gas atoms with pre-generated radicals. Correspondingly, we might85

Ar-SH ANGULAR MOMENTA

Fraction of trajectories

0.25

Energetically Bound
Quasibound

0.20

0.15

0.10

0.05

0.00
0

10

20

30

40

50

60

Angular momentum / hbar

Figure 3.7. Distribution of total radical-cluster angular momenta for bound Ar-SH
clusters from 248 nm photolysis, partitioned into energetically bound (E < De) andquasibound (E > De). The center of the angular momenta of quasibound clusters isapproximately 10! higher than that of energetically bound clusters; both haverotational temperature ≥ 300 K.

86expect these high angular momentum states to constrain cluster dynamics in a way
not typically observed in jet expansions.
Such a constraint can be shown to exist for diatomic molecules, in which
incorporating the kinetic energy from orbital angular momentum (L) into an effective
potential, VLeff(R) = V(R) + L2/(2µR2), can produce a barrier at a given value of
R = R*. Any trajectory with total energy less than VLeff(R*) and R < R*, will remainclassically bound, regardless of whether its total energy exceeds the dissociation limitof the diatomic. Indeed, Garcia-Vela has demonstrated that such a barrier can prevent
Ar-Cl dissociation subsequent to Ar-HCl photolysis.36 For the triatomic Ar-SH, since
the SH bond distance is smaller than the Ar-SH distance and the H atom is muchlighter than the S, we might anticipate behavior similar to that of a pseudodiatomic.
Thus, we next explore the possibility that the quasibound trajectories are held boundby an angular momentum barrier.
First, we must determine how to calculate such a barrier for the triatomic ArSH complex.

In the diatomic case, VLeff(R) is obtained by invoking the orbital

angular momentum as a constant of the motion; in the triatomic case, the total angularmomentum (J) is instead conserved.

Pollack has shown that for triatomics, aneffective potential can be written which may contain a rigorous barrier todissociation;49 for Ar-SH, this effective potential can be written as

VJeff(R, r, θ) = V(R, r, θ) + J2/2I,
I = mArmSH/mAr-SH R2 + mSmH/mSH r2

87(3.10)

where R is the distance between the Ar and the center of mass of SH, r is the SH bonddistance, θ is the angle between these two vectors, and J is the total angularmomentum.
Finding the total angular momentum barrier, defined as Ebarrier, requires two
steps. First, VJeff is minimized as a function of (r, θ) at parametric values of R,
producing VJeff(R).

Second, Ebarrier is determined by maximizing VJeff(R) for R

outside the well region, i.e., Ebarrier ≡ VJeff(R*), where R = R* is the location of thismaximum.

Operationally, since the minima in VJeff(R) should occur near theequilibrium value of r (see Table 3.1) and θ = 0°, we set VJeff(R) = VJeff(R, r = req
θ = 0ο), then find Ebarrier from this potential. As in the diatomic case, if photolysis
generates an Ar-SH cluster with R < R* and total energy less than Ebarrier, it willremain classically bound. Importantly, the VJeff obtained in this fashion produces aminimum barrier, which corresponds to all angular momentum lying in end-over-end
tumbling of Ar-SH; when SH angular momentum is nonzero, the width of the barrieris larger.
To estimate the barrier height for trajectories in our calculations, we havecalculated VJeff(R) for several representative values of J; these effective potentials areplotted in Fig. 3.8. Notably, while Ebarrier is only on the order of a few wavenumbers
for the lowest angular momentum (20!), it rises nearly to 100 cm-1 for an angularmomentum of 60!. Clearly, the angular momenta sampled by the Ar-SH clusters, asseen in Fig. 3.7, provide barriers sufficient to accommodate energies far above thedissociation limit. Accordingly, we expect to find a significant number of photolysis

88

ANGULAR MOMENTUM BARRIER
150*

Energy (cm-1)

100
50

*
*

0

J=0
J = 20J = 40J = 60

-50
-100
-150

2

4

6

8

10

12

R (Å)

Figure 3.8. Effective potential, VJeff, for several representative values of J as afunction of R, with r fixed to its equilibrium value (see Eqn. 10). The asterisks
designate R= R*, i.e., the location of the effective angular momentum barrier. Thisbarrier rises nearly to 100 cm-1 for an angular momentum of 60!.

89

trajectories with E > De but less than Ebarrier. Any such dynamically bound trajectories
will be designated as “quasibound.”To determine which trajectories are quasibound, we return to those shown in
Fig. 3.4. These trajectories are separated into three categories (energetically bound,
quasibound, and unbound) by calculating the total internal energy and comparing to
Ebarrier, which is in turn obtained from maximizing Eqn. 3.10 with respect to R foreach trajectory. We find that for photolysis of Ar-H2S at 248 nm, nearly all the
trajectories with E > De that survive to 1-ns (i.e., quasibound trajectories) have E <Ebarrier and are thus dynamically bound. Thus, we can conclude that the angularmomentum barrier plays a significant role in the classical dynamics of cluster
formation.
The question remains, however, whether such high rotational states will live
for sufficient time to be experimentally detected.

From a quantum-mechanicalperspective, the finite height of the angular momentum barrier provides a finite
probability of tunneling, although tunneling might seem unlikely for a system
involving heavy Ar and S masses. We follow the general procedure of Child50 toobtain a JWKB estimate of the tunneling lifetime, τ:τ=π  [κ 2 (1 + κ 2 )]1 / 4 ω  (1 + κ 2 )1 / 2 − κ 

3.11

where
c
κ = exp − ∫ k(R)dR  b

k(R) = {2 µ Ar -SH [E − V(Ar-SH)]}

1/ 2

90

/!,

ω is the classical attempt frequency, and b,c are inner and outer points, respectively,
in R at which VJeff = Etot. We set ω at 40 cm-1, which is approximately the vibrationalfrequency of Ar-SH.24

This tunneling lifetime can then be compared to theexperimental delay between the photolysis and probe pulses (typically 100 ns) toestimate how many quasibound clusters would fall apart before being detected.
Figure 3.9 contains the tunneling lifetimes calculated for quasibound clusters
resulting from photolysis of Ar-H2S at 248 nm. These lifetimes are plotted as afunction of ∆E/Ebarrier, where ∆E is the energy above the dissociation limit such thatas ∆E/Ebarrier → 0, the total energy decreases toward the dissociation limit. X’srepresent Ar-SH clusters whose tunneling lifetimes are shorter than 100 ns(i.e., tunneling on a timescale shorter than the photolysis-probe delay), and o’s
represent clusters whose tunneling lifetimes are greater than that delay.

Not

surprisingly, the shortest tunneling lifetimes are clustered near ∆E/Ebarrier → 1, i.e.,
when the total internal energy of the cluster approaches the barrier height.
Intriguingly, tunneling lifetimes shorter than the 100-ns experimental timescale
represent approximately half of the quasibound clusters; from this initial estimate, wewould expect that nearly half thus fall apart prior to detection.
As discussed above, angular momentum of the SH subunit increases the widthof the barrier “seen” by the molecule thus drastically increasing the tunneling lifetimeof a given cluster. Accordingly, we would expect fewer clusters to tunnel than
indicated by this 1D estimate. Nevertheless, this effect opens the possibility thattunneling may contribute to the dramatic cluster cooling observed experimentally as
the photolysis-probe delay increases up to 10 µs. However, since our calculations
91

QUASIBOUND Ar-SH TUNNELING LIFETIMES

Tunneling Lifetime (s)

1e+80

1e+60

X

Lifetime < 100 ns
Lifetime > 100 ns

1e+40

1e+20

1e+0
X
X
XX XXX
X
XX
X
X
XX
X
XX
X
X
X XX XX
XX
XX
X
X
XX
X
X
X XXX
X
X
X
X
X
X
X
XXX
X
X
X
X
XX
XX
X
X
X
XX
X
X
XX
X
X
X
X
X
XX
X
XX
X
X
X
X
X
XX
X
X
X
X
X
X
X
XXX
X
X
XX
X
X
X
X
X
X
X
XX
X

1e-20

0.0

0.2

0.4

0.6

0.8

1.0

∆E/Ebarrier

Figure 3.9. Tunneling lifetimes of quasibound clusters versus ∆E/Ebarrier fromphotolysis of Ar-H2S at 248 nm, where ∆E is the energy in the cluster above thedissociation limit and Ebarrier is the barrier height. The x’s represent lifetimes shorter
than 100 ns (the photolysis-probe delay), and o’s represent lifetimes greater than thatdelay. While trajectories with tunneling lifetimes less than 100 ns are clustered near
∆E/Ebarrier → 1, these represent approximately half of these quasibound trajectories.

92

confirm a significant population of clusters with E < De (i.e., which cannot undergo
quantum mechanical tunneling), loss of the hottest clusters by tunneling is not the
only source of the observed cooling. In addition, the large dynamic range observed in
the tunneling lifetimes (from one vibrational period to billions of years) indicates thatfew clusters will fall apart in the range of experimental delays (<10% of the
trajectories in Fig. 3.9 have lifetimes in the 100-ns to 10-µs). Thus, while tunnelingmay contribute to an overall loss of signal in comparison with classical predictions, it
is not likely to be the source of the dramatic cooling on the experimental timescale.

3.3.4 Efficiency and mechanisms of cluster formation: comparison with experimentWith an understanding of the energetics and dynamics of radical cluster
formation, we have a basis for predicting the relative contributions of various
photolysis mechanisms to the experimental spectra. For this purpose, we summarize
the branching ratios between photolysis products in Table 3.2. These branching ratios
include all classically bound clusters (i.e., energetically bound and quasibound), since
the number of clusters predicted to tunnel cannot be precisely determined (asdiscussed in the previous section).

The frequency of bound versus unboundtrajectories can be estimated from relative populations in calculations given in Fig.
3.4.

However, some calculations had either not converged by the end of thecalculation or had not produced measurable products.
To aid in determining branching ratios among the photolysis products, we thus
perform additional calculations on systems nonconverged results. We first focus on
Ar-H2S, for which we can calculate the internal energy and angular momentum of the93Table 3.2. Calculated product branching ratios for Ar-H2S and Ar2-H2S photolysis at
193 nm and 248 nm.
Product distribution, %Ar-SH
λphotolysis (nm) Precursor
193

248

Ar2-SHTotal

E < DeDe < E < Ebarrier

Ar-H2S

0.06(2)a00.06(2)

Ar2-H2S

17(2)

12(2)

4(1)

Ar-H2S

24(2)

13(2)

11(1)

Ar2-H2S

51(3)

50(3)

0.8(4)

Total

00.8(4)

a) Numbers in parentheses represent 2σ.

cluster as soon as the H atom flies off and thus obtain branching ratios among bound,
quasibound, and unbound clusters. Calculations at both wavelengths converged in
the original 1-ns timescale, but no Ar-SH product was observed at 193 nm. Thus, wehave calculated an additional 20,000 trajectories at 193 nm, halting the trajectories
after H-atom flyoff and determining if any radical clusters will survive. Out of these,
none are energetically bound, but 11/20,000 are quasibound.
For Ar2-H2S, we can also use the understanding of cluster energetics anddynamics to explore the photolysis in more detail, although the picture is not quite as
simple, i.e., the classical fate of the cluster is not necessarily determined after H atom
flyoff. If the total internal energy of the cluster is less than the dissociation energy for
either Ar-Ar or Ar-SH, then we expect Ar2-SH to remain classically bound. This
94

allows interrogation of the Ar2-H2S trajectories at 248 and 193 nm to determine if any
form bound Ar2-SH. For 248-nm photolysis, 4/500 have energy below the barrier toboiling off any subunit and thus form Ar2-SH, while none out of 5500 at 193 nm do.
The remaining clusters contain sufficient energy to boil off one or more of thesubunits (Ar or SH), with SH boil-off resulting in an unbound cluster and Ar boil-off
in a bound, quasibound, or unbound Ar-SH cluster. Since prior to the boil off of a
subunit, the fate of the cluster cannot be determined, the calculation must continue
until one subunit escapes.
While cluster concentrations from the 193-nm Ar2-H2S trajectories level off
during the timescale of the original calculation, the 248-nm trajectories do not.
Correspondingly, we propagate forward those 248-nm trajectories still containing
Ar2-SH at the end of the previous 250-ps calculation for an additional decade of time,i.e., 2.5 ns.

Based on the observation of a ~25% drop in concentration in each

decade of time (see Fig. 3.4), we might expect an additional ~25% loss during thisextended calculation.

Instead, the dissociation rate decreases such that only an

additional ~11% fall apart (i.e., approximately 25% remain after 2.5 ns). Thisdecreased rate may imply that some ultimately unbound clusters will survive longenough to be probed; however, the binding energy of the Ar-OH potential used islarger than the binding energy of Ar-SH, and our estimates represent an upper limit to
the dissociation lifetimes that would be observed experimentally.
More important for our branching ratio estimates, all of these clusters boil off
an Ar to form Ar-SH. We thus anticipate all of the remaining trajectories in thisgroup (i.e., those which still contain Ar2-SH after 2.5 ns) will ultimately fall apart to95

Ar + Ar-SH, rather than Ar2 + SH or Ar + Ar + SH. Accordingly, we have estimatedthe net product branching ratios in Table 3.2 based on this assumption.
This estimate thus completes the branching ratios in Table 3.2, providing thebasis for elucidating the dominant cluster formation mechanisms as well as forcomparing with experimental results. By measuring relative Ar-SH/Ar2-SH signals
as a function of reciprocal argon concentration, the relative importance of the various
photolysis channels were experimentally determined at both photolysis wavelengths.
The key results are highlighted below and discussed in the context of the trajectory
calculations.
First of all, even at the highest photolysis energy, some fraction of the Ar-SH
species observed experimentally resulted from Ar-H2S photolysis, i.e., from a
nonfragmentary pathway. This result is supported by our trajectory calculations, witha nearly 20% survival rate of Ar-SH from Ar-H2S at 248 nm and a small but nonzero
survival rate at 193 nm. Incidentally, all of the 193-nm clusters are quasibound (i.e.,with De < E <Ebarrier), compared to half of the 248-nm trajectories. This must result
from accommodation of a much larger excess photolysis energy, producing hotter
clusters.

Also notable is the efficiency of the “boil-off” mechanism for Ar-SH

production from Ar2-H2S; 17% and 51% of Ar2-H2S precursors follow that path at
193 nm and 248 nm, respectively. In each of these cases, fewer Ar-SH clusters are
quasibound than bound, presumably because the extra Ar atom typically carries away
sufficient energy so that the remaining cluster contains E < De.
In contrast to the importance of the nonfragmentary pathways for Ar-H2S,
nonfragmentary photolysis of Ar2-H2S (i.e., producing Ar2-SH) is not a major channel
96

at either wavelength. At first, this might seem surprising: given a particular amountof energy to accommodate, Ar2-SH has more modes of motion for energy disposal
than Ar-SH. However, while additional modes may delay dissociation, the excessenergy can eventually build up in a particular mode and lead to boil off of one
subunit. In addition, if H2S dissociation were strictly uncoupled from interactions
with the Ar atom(s), less kinetic energy would be “lost” by transforming the velocityof the dissociating SH into the COM frame of the radical cluster with two Ar atoms
present than with one Ar present. In this limit, an Ar2-SH cluster must accommodate
more energy than Ar-SH; thus, the boil-off mechanism dominates.
The relative importance of photolysis mechanisms in the trajectory
calculations leads us to two addition points of comparison with experiment. First,
experimental evidence showed that at 193 nm, fragmentary photolysis of Ar2-H2S ismore dominant over nonfragmentary photolysis than at 248 nm. In other words, an
Ar2-SH cluster is more likely to boil off an atom at the higher photolysis energy,
consistent with the greater excess energy generated at 193 nm. This observationagrees with our calculations, which yield no Ar2-SH product from 193-nm photolysis
of Ar2-H2S; however, we cannot predict whether additional trajectories would
produce any such product and thus cannot exactly determine the relative importanceof these pathways.

Second, the experiments showed that the probability of

nonfragmentary photolysis of both Ar2-H2S and Ar-H2S decreases with photolysis
energy, but that nonfragmentary photolysis of Ar2-H2S decreases more slowly than
Ar-H2S. Clearly, while the calculations agree that nonfragmentary photolysis
decreases as a function of collision energy for both precursor clusters, the second
97point cannot be addressed without any Ar2-SH product from 193-nm photolysis of
Ar2-H2S.
Having explored the dynamics of the Arn-H2S photolysis system, we now
expand our focus to investigate more generally the feasibility of radical cluster
formation from photolysis of closed-shell precursors. To that end, we work throughthe simple ballistic model proposed previously39 to estimate the energetics of radical
cluster formation, focusing on precursors containing clusters of H2S/H2O (designated
H2X) with a single additional subunit, typically a rare gas (designated Rg).
In this model, the first assumption is that the H2X is uncoupled frominteractions with the accompanying subunit during the photolysis step. Consequently,
by conservation of momentum, the departing H atom must carry away most of theexcess energy from the photolysis, leaving the following in XH translational energy:

m XH ⋅ v 2XH hν − D 0 (HX−H)
TXH =
=
2
(1 + m XH m H )(3.12)For H2S, this reduces the photolysis energy imparted to the SH by ~34-fold from theexcess photolysis energy; for H2O, by ~17-fold. The recoil energy of the XH isfurther reduced by transformation into the COM frame of the remaining radical
cluster, with a larger reduction for lighter collision partners:

E Rg − XH =µ Rg − XH ⋅ v 2XH
2=µ Rg − XH hν − D 0 (HX − H)⋅
m XH
(1 + m XH m H )98(3.13)

Eqn. 3.13 thus represents an upper limit on the COM energy imparted to the radical
cluster by photolysis of the monomer.
Using Eqn. 3.13, we calculate this COM cluster energy for photolysis of H2O
and H2S with a variety of cluster partners, plotted in Fig. 3.10 as a function ofcollision partner mass. The photolysis wavelengths represented are 193 nm and 248
nm for H2S, and 193 nm for H2O; for comparison, the experimental dissociationenergies of the radical clusters are also shown. Several interesting trends can be seenin this figure. First of all, decreasing the mass of the collision partner does indeed
decrease the COM energy. However, a similar trend is observed in the binding
energies of the radical clusters, so simply reducing the mass of the collision partner
would not ensure an increased probability of radical cluster formation. Second, thephotolysis products of H2S and H2O at 193 nm nearly overlap on this chart. This
results from two competing effects: the lighter mass of O versus S causes the H atom
to carry away less energy, but the H-OH bond is stronger than the H-SH bond,leaving less excess energy from the photolysis within the cluster.
Third, and most importantly, it is clearly feasible to produce bound open-shellclusters even when the COM energy estimated by this ballistic model is larger thanD0. Indeed, even though ECOM for Ar-H2S photolysis at 193 nm is nearly fourfold
larger than the binding of Ar-SH, free-radical clusters are observed bothexperimentally and theoretically. This raises the possibility that any of the clusters
found in Fig. 3.10 might be formed via photolysis of the closed-shell precursor.

99

COM COLLISION ENERGY (Rg-XH)
AFTER MONOMER PHOTOLYSIS
500

400

ECOM (cm-1)

Kr

H2S, 193 nm
H2S, 248 nm
H2O, 193 nm

300

Ar
Ne

200
D0, Ar-OH

He
100

D0, Kr-SH
D0, Ne-OH D0, Ar-SH

H2

0
1

D0, Kr-OH

10

D0, Ne-SH

100

Mass of collision partner (amu)
Figure 3.10. COM cluster energy versus collision partner mass for photolysis of H2O
and H2S clustered with a single collision partner (calculated from Eqn. 3.13).
Superimposed are the dissociation energies for appropriate rare-gas/radical clusters(denoted with a + and labeled). These calculations assume no potential-energy
interaction between the collision partner and radical during the photolysis process andthus represent an upper limit to the COM energy.

1003.5 Conclusions

We have carried out quasiclassical trajectory calculations for the photolysis of
Ar-H2S and Ar2-H2S at 193 nm and 248 nm. These calculations serve as a basis forcomparison to our earlier experiments on photolysis of these clusters in a supersonicjet expansion, in which we observed a remarkable efficiency for formation of theradical cluster species Ar-SH and Ar2-SH from photolysis of the closed-shellprecursors.
First, the quasiclassical photolysis calculations confirm the overall efficiencyof radical cluster formation. Indeed, for photolysis of Ar-H2S at 248 nm, roughly
20% of the precursors form bound Ar-SH clusters. Even at the shorter photolysis
wavelength of 193 nm, the fragmentary photolysis of Ar2-H2S forms Ar-SH clusters
with roughly 20% efficiency. The timescales on which unbound radical clusters fallapart are typically much shorter than the experimental timescale, ranging from ~5 psfor 193 nm photolysis of Ar-H2S to ~10 ns for 248 nm photolysis of Ar2-H2S. The
remarkable efficiency of radical cluster formation results in part from the fact that therelatively light H atom, by conservation of momentum, carries away the majority ofthe excess photolysis energy in kinetic energy.
However, the calculations confirm that the clusters retain sufficient energy intheir COM frame for significant intracluster rovibrational excitation. The energy
found in these clusters is primarily partitioned between relative rotation and vibrationof the Ar and SH subunits. Ar-SH vibrational states up to the dissociation limit are
populated, indicating that the photolysis process yields bound clusters that sample

significant regions of the potential energy surface. In addition, rotational energies up
to and beyond the dissociation limit are found in bound clusters, pointing to theexistence of an angular momentum barrier to dissociation. Indeed, nearly half of thebound Ar-SH clusters from photolysis of Ar-H2S at 248 nm, and all of the bound ArSH from photolysis at 193 nm, contain total energy greater than the dissociation limit.
However, that energy, trapped in rotation, is unavailable for transfer into vibration;
thus, the clusters are classically bound. Quantum mechanical tunneling calculations
indicate that as an upper limit, roughly half of these quasibound clusters may fallapart prior to probing on the experimental timescale (i.e., 100-ns delay between
photolysis and probe pulses).
These calculations provide a basis for comparing the relative efficiencies of
two radical-cluster formation mechanisms: nonfragmentary (in which no Ar atoms arelost) versus “boil off” (in which one Ar atom carries away the excess energy).
Overall, the nonfragmentary production of Ar-SH from photolysis of Ar-H2S isdemonstrated to be remarkably efficient at the longest photolysis wavelength (248nm), while the “boil off” mechanism produces a large fraction of bound Ar-SH
clusters (≈ 20%) from Ar2-H2S, even at the shortest wavelength (193 nm). Finally,
the overall efficiency of this method for bound closed-shell cluster production from
Arn-H2S photolysis provides evidence that this technique could be extended to boundclusters of other hydrides (e.g., H2O) and clustering species (e.g., H2, Kr, Ne).

102

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105

CHAPTER 4
DYNAMICS OF COLLISIONAL ALIGNMENT IN SUPERSONICEXPANSIONS: TRAJECTORY STUDIES OF HE + CO, O2 AND CO2

4.1 Introduction
Probing and ultimately controlling the dynamics of chemical reactions at thequantum state-to-state level has long been a goal of chemical physics research. At thehighest level of detail, studies of bimolecular reaction dynamics would permit thestereochemistry to be manipulated, influencing what region of the potential surface issampled as the reagent molecules approach and collide. One key step in this overall
scheme is to orient or align the colliding molecules with respect to some lab-/bodyfixed axis frame, either by static (DC) fields, optical (AC) fields, or simply directedcollisions in an anisotropic flow.1 Depending on the efficiency of these processes,
such techniques ultimately offer the opportunity to probe the detailed topology of areactive potential energy surface, or, in other words, the “steric effect” in a chemical
reaction.
This general area of molecular control of stereodynamics has witnessedenormous growth over the past 30 years, primarily exploiting alignment and/or106

orientation of rotational angular momentum states in external fields. Mathematically,
orientation and alignment refer to ordered moments of the mj distribution (the z-axis
projection of j). Oriented samples correspond to <mj> ≠ 0, with the lowest order
quantum-mechanical orientation given by2= < jm j j z jm j >O (1)0j(j + 1)

(4.1)

where the outer brackets represent an average over the mj distribution. Alternatively,aligned samples can have <mj> = 0 but require a non-uniform distribution in |mj|, witha lowest order alignment given by= < jm j 3 j z2 − j 2 jm j >A (2)0j(j + 1)

(4.2)

In the classical limit, this orientation and alignment can be related to the first andsecond Legendre moments, respectively, of the corresponding angular distribution of
j. For collision systems symmetric with respect to the relative velocity vector (i.e., in
the absence of any external fields), the final mj and -mj populations are exactly equal,
which precludes any orientation effects. However, the distributions in |mj| can be
non-uniform and therefore result in finite rotational alignment.

In general, a

distribution with an excess of j states parallel or perpendicular to the alignment axis
corresponds to A0(2) > 0 or A0(2) < 0, respectively.
The use of angular momentum quantum states to influence or control
bimolecular collision geometries was originally developed by the Bernstein3 and107

Brooks4 groups.

In these studies, a |jkm> state-selected molecular beam ofsymmetric-top CH3I molecules was focused with an inhomogeneous hexapole electricfield and subsequently oriented along a specific laboratory direction with ahomogeneous electric field. By crossing this oriented beam of CH3I with a beam of
Rb or K atoms, it was demonstrated that reaction preferentially occurs when theiodine end of CH3I is pointed toward the reactive alkali species, thus providing firstexperimental insights for steric effects in simple atom-abstraction processes.
Although the hexapole focusing methods have enjoyed widespread use,5-8 thistechnique is typically constrained to molecules exhibiting a large, first-order Stark
effect. Alternatively, electric-field techniques have been developed that exploit thesecond-order Stark effect to align the rotational states of linear molecules.9-11 Morerecently, Loesch and coworkers12-14 and Friedrich et al.15,16 have utilized strong
homogeneous electric fields to create highly oriented “pendular” states of rotationallycooled linear molecules by the so-called “brute force” technique. Such methods havebeen particularly successful in linear hydrogen-bonded clusters,17-21 where the large
dipole moments and small rotational constants make them especially convenient tomanipulate into the pendular regime.
The use of DC electric field methods for alignment and orientation requires
the molecule to have a large permanent dipole moment, which significantly limits therange of applicability. Alternatively, magnetic-field alignment techniques developed
by Herschbach and coworkers can be applied more generally to both polar andnonpolar molecules, but with the restriction that these species must be
paramagnetic.22,23 In principle, optical (AC) field methods offer more generality; for108example, polarized light can be used to orient or align molecules by single-photon
excitation,24-30 selective photodissociation,31-33 Raman scattering,34,35 laser-induced
AC Stark effect,36,37 or intense non-resonant laser fields.38-40 Indeed, much of thework in the area of rotational alignment has been carried out simply using singlephoton excitation followed by polarized fluorescence detection, with the necessary
theoretical framework developed by Zare and coworkers and Meyer for parallel bandsof linear and symmetric-top molecules,41-45 and by Weida and Parmenter for
perpendicular transitions of symmetric-top molecules as well as a-, b-, and c-typetransitions of asymmetric tops.46

Unlike static DC field methods, rotationalalignment via single-photon excitation requires no permanent dipole moment;
however, it does require significant transition strengths and a relatively long-lived
upper state.
A more general alternative is the use of collisions in an anisotropic velocity
distribution to create the alignment, as pictured schematically in Fig. 4.1 for linear
molecules without internal degrees of freedom. When a velocity gradient exists
between such molecules, collisions along the direction of flow will preferentiallyalign their angular momenta (j) perpendicular to the flow. Stated physically, themolecules with j parallel to the flow present a larger cross sectional area forcollisional reorientation, while molecules with j perpendicular to the flow are lesslikely to be hit. As modeled by Zare and coworkers,47 even if all such collisions
simply randomize the direction of j, this will lead to a net alignment of angularmomenta perpendicular to the gas flow (i.e., negative alignment). This concept of109

Figure 4.1. A schematic depiction of collisional alignment in supersonic expansions.
The lighter, faster-moving “diluent” atoms (white circles) collide with heavier,slower-moving “seed” rotor molecules (gray ovals) along the expansion axis. Rotor
molecules with j parallel to the expansion axis present a larger collisional crosssection than those with j perpendicular and thus are more subject to reorienting
collisions.

110alignment by reorienting collisions, first presented by Gorter,48 can be easily
extended to alignment in supersonic expansions. For an expansion mixture containing
a lighter “diluent” gas and a heavier “seed” gas, velocity gradients cause the slowermoving seed gas molecules to undergo multiple, anisotropic collisions with the fastermoving diluent gas molecules.

These directed collisions align seed molecule

rotations by creating an anisotropic distribution of mj along the expansion axis.
Indeed, such anisotropically aligned distributions of seed molecules in asupersonic expansion were observed as early as the 1970s by the Zare group, who
studied directed collisions in a neat expansion of Na atoms (“diluent”) with small
concentrations of Na2 (“seed”) to align the rotations of the dimers.47 Since then,many experimental studies have demonstrated collisional alignment of rotor
molecules for a wide range of collision partners and expansion conditions, producingaligned distributions of Na2,49,50 I2,51-55 CO2,56 CO,57 O2,58-60 and N2.61 These studies
demonstrate the powerful generality of collisional alignment: in principle, any nonspherical molecule (i.e., with an anisotropic state-to-state collision cross section) can
be rotationally aligned by collisions in a directed flow.
While several studies have attempted to characterize the alignment process,contrasting descriptions of the collisional mechanism have arisen. For example,
experimental IR-based studies by Weida and Nesbitt demonstrate that alignment of
CO2 seeded in a He expansion becomes increasingly negative as a function of CO2
rotational angular momentum;56 this is quite similar to the observations of Hefter etal.49 in Na expansions, which indicate increasingly negative alignment of Na2 with j
up to j = 10. Conversely, under certain expansion conditions Herschbach and111

coworkers find that alignment of I2 in Ne becomes less negative with I2 rotational
angular momentum,53,54 though only a limited number of rotational levels wereaccessed with their laser excitation source. Quite recently, studies of the alignment
dependence on the final velocity of the “seed” molecule have produced morestrikingly contradictory results.

For example, experiments performed by the

Aquilanti group on O2 and N2 with a range of diluent gases including He indicate thatrotor molecules moving faster than the average are increasingly aligned with jperpendicular to the expansion axis.58,61 On the other hand, studies by Harich andWodtke in He + CO expansions lead to exactly the opposite conclusion, i.e., the faster
CO molecules are preferentially aligned with j parallel to the expansion axis.57
In parallel with this experimental work, there have been several theoreticalstudies of collisional alignment in supersonic expansions. Early efforts by Sanders
and Anderson62 and more recent ones by Pullman et al.63 have explored alignment of
I2 by Ar collisions. Sanders and Anderson specifically address the evolution ofrotational alignment over multiple collisions but do not identify the mechanism for
how this alignment is created on a collision-by-collision basis. Conversely, Pullman
et al. focus exclusively on the single-collision regime, exploring the alignment
dependence on impact parameter, collision energy, and I2 initial angular momentum
as a function of an Ar-I2 model interaction potential. For all potentials studied,classical trajectories fall primarily into two regimes: trajectories at small impact
parameters produce more molecules with j perpendicular to the collision axis, while
those at large impact parameters produce more with j parallel. In addition, potentials
with an attractive well exhibit “sticky,” long-lived collision complexes.
112

This

ultimately leads to aligned molecules with j perpendicular to the relative velocity
vector, although with overall alignments typically smaller in magnitude than direct
collisions.
The recent experimental results highlight important new theoretical issues notadequately addressed by previous work. For example, the final j-state and velocity
dependence of the rotational alignment remain relatively unexplored. Furthermore,
the contradictory velocity dependences reported for He + O2 and He + CO alignment
studies raise the possibility of extreme sensitivity to the molecular interaction
potential. Given these qualitative differences and the importance of the interaction
potential suggested by the Ar-I2 collisional alignment simulations, further study of therole of the interaction potential in these particular systems would appear to be inorder.
The focus of this chapter is a detailed investigation of collisional alignment
mechanisms in supersonic expansions. To facilitate comparison with experiment,
three collision systems (He + CO2, He + CO, and He + O2) have been studied that are
based on realistic interaction potentials. Using classical trajectory calculations, thesupersonic jet expansion conditions are simulated by producing multiple collisions
between He “diluent” gas atoms and single “seed” molecules along the expansion
axis. To address the possible dependence of alignment effects on initial collisionvelocity, these studies are performed for three different collision energies
corresponding roughly to three different temperature regions (≈ 300 K, 100 K and 20
K) in the expansion. These simulations permit direct comparison with experimental

113

results, as well as an opportunity to investigate the mechanism for alignment
formation on a collision-by-collision basis.
The remainder of this chapter is organized as follows. In Sect. 4.2, we
describe the details of the computations used to model the supersonic expansions. In
Sect. 4.3, we discuss the results of these computations, addressing both
multicollisional alignment effects and single-collision mechanistic details. In Sect.
4.4, we compare these results to experiments and discuss the implications. The
conclusions of our studies are summarized in Sect. 4.5.

4.2 Computational Approach
To investigate the mechanism of collisional alignment in supersonic
expansions, classical trajectory calculations are used to simulate multiple-collision
dynamics between light “diluent” gas atoms and heavier “rotor” molecules. To
address the dependence of alignment on relative velocity, three different regions of
the expansion are represented, corresponding to 300 K (Ecom = 266 cm-1), 100 K (Ecom= 89 cm-1) and 20 K (Ecom = 18 cm-1) collision conditions. The choice of O2, CO, andCO2 “seed” rotor molecules and He “diluent” atoms enables direct comparison ofthese trajectory calculations to the experiments of Aquilanti et al.,58 Harich andWodtke,57 and Weida and Nesbitt.56 A multicollision simulation consists of a single
rotor molecule undergoing many successive but uncorrelated collisions (up to Ncoll ≈200) with He atoms at the same energy (details provided below). From several
thousand of these multicollision simulations, we group the results together in an
ensemble of rotors (i.e., Nensemble ≈ Nsim · Ncoll ≈ 500,000), from which the relevant
114statistics for collisional development of the rotational alignment can be explored foreach choice of rotor molecule and region of the expansion.
For each multicollision simulation, we first need an appropriate He-rotor
interaction potential. While details of the diluent-seed potentials have been neglectedin early hard-sphere theories of collisional alignment,47 in light of the qualitative
discrepancies observed for He + O2 versus He + CO collision systems,57,58,61 wechoose to model the He-rotor interactions as accurately as possible, with potentials
experimentally determined from crossed molecular beam scattering or cluster IRspectra studies. Figure 4.2 shows contour maps of the potentials used for He-O2,64
He-CO,65 and He-CO2.66 At first glance, the He-O2 and He-CO potentials appear
qualitatively similar in terms of well depth and degree of anisotropy. However,unlike the symmetric He-O2 potential, the He-CO potential differs slightly from one
end of the rotor molecule (θ = 0o) to the other (θ = 180o). Furthermore, the He-CO
well is slightly shallower than for He-O2 (De ≈ 23 versus 26 cm-1). The He-CO2
potential differs from the other two by its significantly greater anisotropy and much
deeper well (De ≈ 41 cm-1).
While the He-rotor intermolecular potentials may or may not prove criticallyimportant to the collisional dynamics, it is reasonable to assume that the dynamics areless sensitive to rotor intramolecular potentials. For this reason, we approximate therotor intramolecular degree of freedom as harmonic oscillator(s), with parameters
given in Table 4.1. Furthermore, the O2 and CO vibrational frequencies are reducedby roughly threefold to expedite longer time steps in the numerical integration.

115θ (degrees)

He-ROTOR POTENTIALS
150

a)

He-O2

b)

He-CO

c)

He-CO210050

θ (degrees)

0
150
100
50

θ (degrees)

0
150
100
50
0
3.0

4.0
o
R (A
)

5.0

Figure 4.2. Contour plots of the potential energy surfaces for a) He-O2, b) HeCO, and c) He-CO2. R represents the distance between the He atom and the centerof mass of the rotor, while θ represents the angle between the rotor axis and theposition of the He atom. θ equal to 0o or 180o corresponds to a collinear
configuration, while θ equal to 90o corresponds to a T-shaped configuration (forHe-CO, the C points toward the He at 0o). The solid contours represent 20-cm-1
increments in the positive region of the potential, while the dashed contours
represent 2-cm-1 increments in the negative region. The well depths areapproximately a) 26 cm-1 for He-O2, b) 23 cm-1 for He-CO, and c) 41 cm-1 for HeCO2.

116

Table 4.1. Intramolecular harmonic potential parameters and effective “hard sphere”
He + rotor cross sections, used to normalize the number of collisions (see text for
details).

σ(Å2)

Potential Parametersµ(amu)re (Å)

ω1 (cm-1) ω2 (cm-1) ω3 (cm-1) 266 cm-1 89 cm-1 18 cm-1

O2

32

1.207

600

21

25

28

CO

28

1.128

600

24

29

33

CO2

44

1.16

1351.2

26

29

31

672.2

2396.4Since these vibrational spacings are still more than twofold larger than the highest
collision energies sampled, translational-vibrational (T-V) energy transfer effects arestill anticipated (and numerically verified) to be insignificant. However, since thelow-frequency bending mode of CO2 (ν2 ≈ 667 cm-1) is already close to this limit, theknown experimental harmonic frequencies are used for each CO2 vibrationalfrequency.67 In all cases, vibrational excitation from a single collision is verified tobe ≤ 2% and more typically ≈ 0.1% of the zero-point energy for the lowest-frequency
vibration.
Initial conditions for the He and rotor are selected following the procedure ofRaff and Thompson.68 For each expansion region, we select a single collision energy
corresponding to ½µ<v 2rel > , where<v 2rel > is the rms relative speed of the rotor andHe gas at the characteristic temperature, placed entirely in He atom velocity towards a117

stationary rotor atom. This temperature is also used for Boltzmann sampling of initialrotational states, so that the collisions do not significantly heat up or cool theensemble. The initial coordinates of the rotor are obtained by random selection of its
orientation in the rotational plane. The result is monoenergetic collisions with anisotropic, Boltzmann distribution of rotors at the given temperature with no initialalignment. Trajectories are initiated at sufficiently large He-rotor distances so thatintermolecular interactions are negligible (≤0.1% of the well depth), with the Heplaced at a random distance (10-20 Å for O2 and CO, 15-30 Å for CO2) “upstream” of
the rotor along the collision axis. Initial impact parameters for the collision are also
randomly sampled from a uniform distribution (i.e., weighted by 2πbdb), with amaximum value (bmax) of 5 Å for He + CO and He + O2 and 7 Å for He + CO2.
Convergence with respect to maximum impact parameter is explicitly verified foreach collision system. In the He + O2 system, for example, trajectory ensembles havebeen studied with bmax ranging from 2.0 Å to 10.0 Å; the resulting collisionalalignment grows rapidly with bmax up to ≈2.8 Å, then grows by ≤10% for a 3.6-foldfurther increase in bmax from 2.8 Å to 10 Å. For multiple collisions with a single rotor
molecule, the rotor’s center-of-mass translation is reset to zero at the end of eachcollision, while its rotational speed and angle with respect to the collision axis are
retained.
The trajectories are calculated by integrating Hamilton's equations of motion
in Cartesian coordinates with a variable-order, variable-stepsize, Adams-method
numerical integrator.69 To ensure that a collision has had the opportunity to occur,
each trajectory is run for a minimum time period τ = 2Ro/vo, where Ro is the initial118He-rotor distance and vo is the initial relative velocity. The trajectory terminates
when the He-rotor interaction potential is negligible, i.e., less than 10-4 of the total
kinetic energy. Step size constraints are chosen to maintain conservation of energy
better than 10-4 and typically 10-5 for a single-collision trajectory. This permits
sufficient computational speed without significantly affecting the predictedalignments: for example, a test ensemble of 100 multicollision simulations of He-O2
at 266 cm-1 calculated with 100-fold tighter constraints yields asymptotic alignments
differing by less than the statistical uncertainty (<3%). For every collisional event ina multicollision simulation, we record the impact parameter (b), the magnitude of theclassical initial and final rotor angular momenta (ji, jf), the initial and final projectionsof j along the collision axis {(mj)i, (mj)f}, and the final rotor velocity along that axis.
These values provide the necessary information for analysis of alignment as afunction of number of collisions, collision energy, angular momentum, and final rotor
velocity for comparison with experimental results. In particular, this enables thealignment formation to be examined on a collision-by-collision basis.
Once a trajectory in an ensemble has been generated, the final alignment iscalculated as follows. For a given mj state, the classical angle between the rotor
angular momentum vector, j, and the collision axis is obtained from θ = cos-1(mj/j).
For a distribution of final mj/j states and hence values of θ, the alignment can be
described by an expansion in even-order Legendre polynomials,

ρ(cosθ) = ½[1 + a2P2(cosθ) + a4P4(cosθ) + . . .],

119

(4.3)

where ρ is the probability density of molecules with j pointing at an angle θ withrespect to the collision axis.

For this normalized probability distribution, i.e.,∫ρ(cosθ) d(cosθ) = 1, the coefficients are obtained simply from an =(2n+1)<Pn(cosθ)>. Although higher order (n = 4, 6, …) alignment information inthese distributions is clearly available, only the lowest (n = 2) term can be determined
experimentally from a one-photon absorption study.

Thus, we focus on the a2alignment values, calculated from the expectation value of P2(cosθ), i.e.,a 2 = 5< P2 (cos θ)> = [5 N sim ] ∑i =sim
P2 (cos θ i )
1
N

(4.4)

where Nsim ≈ 2000-5000 represents the number of trajectories averaged. As expected,
all odd moments of the Legendre expansion explicitly vanish within statisticaluncertainty for all multicollision simulations. Since each rotor in a simulation has
undergone multiple collisions with He, the net a2 alignment of the ensemble isobtained by calculating <P2> over all of the multicollision simulations at thatcollision number.

The net result is the a2 alignment as a function of number ofcollisions for each ensemble of trajectories.

For comparison with experimental

results, the quantum mechanical quantity A0(2) approaches
semiclassical limit.2

120

2

5

a2 in the high-j,4.3 Results
4.3.1 Multicollision alignment
In this subsection, the results of the multicollision simulations are presented,
with a focus on i) the rate with which the alignment develops with collision number,and ii) how the magnitude of this alignment depends on collision energy andpotential. To make things simpler, we focus initially on the He + O2 simulations,which provide the least computationally intensive system for sampling and testing
various parameters. As described in Sect. 4.2, the a2 alignment parameter can becalculated as a function of the number of collisions (Ncoll) by averaging P2(cosθ) for
each collision over all simulations within a given ensemble. It is more meaningful torenormalize Ncoll to the effective number of hard-sphere collisions by

Zeff = Ncoll σ/(πbmax2)(4.5)

where σ represents the effective hard-sphere collision cross section at each collisionenergy as determined from the potential (see Table 4.1) and bmax is the maximum
impact parameter sampled.
A plot of a2 for the three collision energies (266 cm-1, 89 cm-1, and 18 cm-1) ispresented in Fig. 4.3 as a function of Zeff. First of all, within statistical error, thealignment parameter decreases more or less smoothly from isotropic (i.e., a2 ≈ 0) toan asymptotic value near a2 ≈ -0.4. The negative sign of this asymptotic alignment isin good agreement with experiment and indeed confirms that more molecules are

121

COLLISION ENERGY DEPENDENCE OF ALIGNMENT:
He + O2
0.0

266 cm-1 (300K)
89 cm-1 (100K)
18 cm-1 (20K)

-0.1

a2

-0.2
-0.3
-0.4
-0.5
0

10

20

Zeff
Figure 4.3. A plot of the alignment of He + O2 versus effective collision number(Zeff) at three collision energies: 266 cm-1, 89 cm-1, and 18 cm-1. The three traces
demonstrate a remarkable insensitivity to collision energy: all three reach asymptoticalignments within 30% of each other, despite a 15-fold variation in collision energy.

aligned with j perpendicular than j parallel to the collision axis. Secondly, thisasymptotic behavior is typically achieved in ≤10 effective hard sphere collisions.
Finally, even though the final rotational distributions vary dramatically for the threecollision energies, the asymptotic alignments are within ≈ 30%. In other words, theclassical magnitude of the final rotor alignment is rather insensitive to collisionenergy.

122

For a quantitative comparison of these He + O2 alignment results with theother potentials, the alignment versus collision number has been fit to a rising
exponential of the form a2 = a2∞ [1-e-Zeff/β], where a2∞ corresponds to the asymptotic
value of a2 and β corresponds to the number of collisions to achieve (1-1/e) ≈ 63% of
this asymptotic alignment. The results of these fits for simulations on all threepotentials are summarized in Figs. 4.4a and b. As expected, a2∞ < 0 for all potentials,
indicating a net excess of rotors with j perpendicular versus parallel to the expansion
axis. Moreover, the asymptotic alignments obtained for all three potentials (but mostespecially He + O2 and He + CO) reflect a surprising insensitivity to a nearly 15-foldchange in collision energy.

However, significant variations are observed whencomparing results between different potentials, specifically in i) the magnitude ofasymptotic alignment (a2∞) and ii) β, the 1/e number of collisions required to reachthis alignment. For example, |a2∞| increases substantially from to O2 to CO to CO2.
Furthermore, roughly an order of magnitude as many collisions are required to reachthese asymptotic alignment values for CO versus O2 and CO2.
Closer inspection of the three collision potentials indicates angular anisotropy
to be the source of these effects, i.e., the variation in R on a given isoenergy contour
for a full revolution of the rotor (see Fig. 4.2). In particular, this angular anisotropy
increases from He-O2 to He-CO to He-CO2: the zero-energy contour for He-O2 varies
by only ∆R ≈ 0.4 Å, which should be compared with the much larger anisotropy
values of ≈ 0.8 Å and 1.0 Å exhibited for He-CO and He-CO2 potentials, respectively.
Since this angular anisotropy is the only way the dynamics distinguish side versus

123a2∞ (ASYMPTOTIC ALIGNMENT)
β (COLLISIONS TO 1/e OF a2∞)

0.0

-0.5

a)

He + O2
He + CO-1.0

He + CO2266 cm-1 (300K)
89 cm-1 (100K)
18 cm-1 (20K)b)

30

He + COHe-CO220

10

He + O2

He + CO20

Figure 4.4. Results of fitting the alignment versus collision number data for He +O2, CO2, and CO to a2 = a2∞ [1-e-Zeff/β]. Errors bars correspond to ±2σ in the fit. a)
The asymptotic alignment, a2∞, is insensitive to collision energy for all threepotentials and increases from O2 to CO to CO2, reflecting the increased anisotropy ofthe potentials (see Fig. 4.2). b) β, the number of collisions to achieve (1-1/e) ≈ 63%of this asymptotic alignment, increases dramatically from He + O2/CO2 to He + CO,reflecting presence of odd terms in addition to even terms of a Legendre expansion of
the He + CO potential.

end-on collisions, one therefore expects and observes the net alignment to increase
from He + O2 to He + CO/CO2.
Also notable is the substantial decrease in alignment for the lowest energy
collisions of He + CO2 versus the relatively energy independent alignments observedfor He + CO and He + O2 (see Fig. 4.4a). This effect can be attributed to the greater
well depth for He-CO2.64-66 Since 18-cm-1 collision energies correspond to less thanhalf the depth of the He-CO2 well, one expects far more long-lived collisions thanobserved for He-O2 and He-CO. The presence of such “sticky” collision trajectories
ultimately results in memory loss of the initial direction of the relative velocity
vector, thereby diminishing the net alignment.63

Indeed, additional He + O2

trajectories have been performed at Ecom = 9 cm-1 (i.e., less than half the He-O2 welldepth) that indicate resulting O2 alignments to be similarly reduced at these lower
collision energies.

In general, however, these results indicate a remarkably low

sensitivity to collision energy, which motivates our focus on trajectory ensembles at asingle collision energy in the following sections.
More striking is the large difference in rates at which these three systems
come to the asymptotic alignment value, as seen in Fig. 4.4b. While the asymptoticalignment magnitudes can be rationalized by anisotropy differences in the potentials,
the reason for the larger number of collisions needed to align CO is less apparent.
However, one feature of the CO potential that distinguishes it from the others is theasymmetry of the two rotor ends, which contributes additional odd versus even terms
in the Legendre expansion of the potential. As a result, the potential is significantly
asymmetric with respect to He-CO and He-OC configurations, with a well closer to125the C end. In addition, the shift between the bond midpoint and center of mass makes
the CO rotational axis asymmetric with respect to hard-sphere potential edges of the
O and C atoms. A slowly approaching He atom therefore preferentially collides with
the C end of the rotor, sampling the potential well region non-uniformly. Thisasymmetric sampling consequently requires a much larger number of collisions for
CO to reach asymptotic alignment than for either CO2 or O2.

4.3.2 Dependence on Rotor Angular Momentum
As mentioned in Sect. 4.1, several experimental studies indicate a dependenceof alignment on final rotational quantum state. Both Hefter et al.49 and Weida andNesbitt56 have found alignment to increase with j;interestingly, this alignment

appears to vanish when extrapolated to j = 0, which has led to the suggestion that thisj-dependence might result from the 2j + 1 quantum mechanical degeneracy of therotational level. Indeed, a simple geometric model based on these ideas qualitatively
predicts a monotonic increase in A0(2) to an asymptotic value with j.56 This makes it
especially intriguing to see if these j-dependent trends are reproduced from a purely
classical perspective.
To illustrate how the alignment depends on the rotor angular momentum, j, weagain focus on the He + O2 ensemble data at a single collision energy (266 cm-1),
though similar trends at all other collision energies are observed as well.

As

described previously, a2 is calculated after each collision by averaging P2 over all of
the multicollision simulations. However, prior to the calculation of a2, each rotor isfirst sorted into a bin according to its postcollision angular momentum, j. Three bin
126

ranges are chosen to examine the alignment behavior at “low,” “intermediate,” and“high” j values relative to a 300 K distribution. These bins are j/<j> = 0.0-0.4, 0.81.2, and 1.6-2.0 (where <j> ≈ 10.2 for O2 at 300 K) and correspond to the darkened
regions shown in Fig. 4.5. At each collision number in a multicollision ensemble, the1.6-2.0}

0.0-0.4

0.8-1.2

POSTCOLLISION j-DISTRIBUTION}}NUMBER OF ROTORS (ARB. UNITS)

j-dependent alignment is extracted from <P2> for all the rotors in a given j bin.

0

1

2

3

j/<j>

Figure 4.5. The distribution of rotor angular momenta (j) after a single collision at
266 cm-1 (300 K). This is divided into “low” (j/<j> = 0.0-0.4), “intermediate” (j/<j>= 0.8-1.2), and “high” (j/<j> = 1.6-2.0) angular momentum bins for calculating thealignment dependence on j.

127

A plot of this j-dependent alignment of O2 as a function of collision number isgiven in Fig. 4.6. Notably, the asymptotic alignment depends quite strongly on j,
starting at essentially zero (within uncertainty) for the lowest range of j values andbecoming increasingly negative with increasing j. This result is consistent with thework of Weida and Nesbitt in He + CO2 coexpansions,56 as well as the work of Hefter
et al. in Na + Na2 expansions.49 This is a particularly interesting result to obtain frompurely classical calculations, since vanishing alignment at low j has previously been
rationalized in terms of j-dependent degeneracies.56 However, there is a similar jdependence in the semiclassical density of rotational states, which also vanishes in
the limit of j = 0. Thus, while it is not surprising that such alignment trends can be
reproduced by purely classical trajectory calculations, this will necessitate further
discussion in Sect. 4.4 of alternative, non-quantum mechanical explanations for this jdependence.
With the j-dependent alignments corroborated by He + O2 simulations, we
extend our study to the other two potentials. Specifically, the distributions of a2
versus collision number are sorted into the same j/<j> bins for the multicollisionensembles of He + CO and He + CO2 at 266 cm-1 (with <j> = 8.8 and 20.0 at 300 K
for CO and CO2, respectively) and fit to a rising exponential. The results of this fit,
given in Fig. 4.7, show that the trends are the same for all three potentials, with thealignment monotonically increasing with rotor angular momentum. The variation inmagnitude among potentials within each j/<j> range simply reflects the variation in
overall alignment noted previously in Fig. 4.4a.

128

J DEPENDENCE OF ALIGNMENT:
He + O2
0.4

j/<j>=0.0-0.4
j/<j>=0.8-1.2
j/<j>=1.6-2.0

0.2
0.0

a2

-0.2
-0.4
-0.6
-0.8
-1.0
0

10

20

Zeff

Figure 4.6. Alignment (a2) as a function of effective collision number (Zeff) for low(j/<j> = 0-0.4), intermediate (j/<j> = 0.8-1.2), and high (j/<j> = 1.6-2.0) angularmomentum values for the He + O2 collision system. The magnitude of the asymptoticalignment increases with j in this system.

129

J DEPENDENCE OF ALIGNMENT:ALL POTENTIALS
0.0

a2

-0.5

-1.0

-1.5

He + O2
He + COHe + CO2
0.0-0.4

0.8-1.2

1.6-2.0j/<j>Figure 4.7. Asymptotic alignment (a2∞) for He + O2, CO, and CO2 for low (j/<j> =0-0.4), intermediate (j/<j> = 0.8-1.2), and high (j/<j> = 1.6-2.0) angular momentum
values. Error bars represent ±2σ in the exponential fit. While the magnitudes of thealignments vary among the potentials, the trend for all three potentials is thatalignment increases with j.

130

The next step is to investigate how this alignment is created, specifically byidentifying correlations between initial and final states in the production of an aligned
distribution. One important observation in this regard is evident in Fig. 4.4b, i.e., thealignment grows in gradually over several collisions. Thus, one plausible mechanismis that the incoming alignment of a rotor molecule (i.e., classically mj/j) influences theprobability of a given final alignment. From this perspective, the rotor alignment isformed by a series of mj/j-changing collisions that ultimately funnel the angularmomentum states into an aligned distribution.

To explore this possibility, theindividual collision trajectories are analyzed to obtain a probability map for all(mj/j)i→(mj/j)f transitions. In order to improve the statistics, approximately 500,000
single-collision simulations are included for each of the potentials. Furthermore, thesingle-collision (mj/j)i→(mj/j)f distributions are additionally split into the same j/<j>bins used above, but where j now refers to the angular momentum state prior to thecollision.
Sample He + O2 probability maps of (mj/j)f versus (mj/j)i for the three rangesof j/<j> are presented in Fig. 4.8 and indicate several features worth noting. First ofall, for the lowest values of j/<j> (Fig. 4.8a), single collisions from each (mj/j)i
produce a large spread in (mj/j)f, i.e., little correlation exists between the initialalignment of an O2 molecule and its final alignment after a collision. However, as jincreases from Fig. 4.8a to c, this spread in (mj/j)f for a given (mj/j)i narrows
dramatically, collapsing into a tight band along (mj/j)f ≈ (mj/j)i. Thus, the rotor’s
initial alignment with respect to the collision axis is largely conserved for high j.

131

GYROSCOPIC STABILITY
1.0a)(mj/j)f

0.5"low" j

0.0

-0.5

-1.0
1.0

b)

(mj/j)f

0.5

0.0

-0.5

-1.0
1.0

c)

(mj/j)f

0.5"high" j

0.0

-0.5

-1.0
-1.0

-0.5

0.0

0.5

1.0

(mj/j)i

Figure 4.8. Plot of initial mj/j versus final mj/j for an ensemble of 500,000 He + O2
collisions at 266 cm-1 for a) low (j/<j> = 0-0.4), b) intermediate (j/<j> = 0.8-1.2), andc) high (j/<j> = 1.6-2.0) angular momentum values. As j/<j> increases, the spread in(mj/j)f for each (mj/j)i narrows considerably, reflecting the increased gyroscopic
stability of higher j states.

132

This result is not limited to the He + O2 system: He + CO and He + CO2 studies also
demonstrate that the spread in (mj/j)f decreases by roughly an order of magnitude
from the lowest to the highest j range. Simply stated, single collisions are far lesseffective at scrambling molecular alignment at high j than low j; this physicallyreflects the increased gyroscopic stability of more rapidly spinning rotor molecules.
This also begins to explain the strong j dependence of the asymptotic alignments
observed, since any alignment created by a series of collisions is far more easily
erased by additional collisions for low j than for high j rotors.
The next stage in investigating how alignment is created is to determine thetypes of collisions that change mj/j, i.e., to explore whether mj- or j-changing
collisions predominate. To this end, the single-collision He + O2 trajectories are
sorted by size of the collisional change in mj (i.e., ∆mj) and in j (i.e., ∆j); the resulting
distributions can be analyzed by i) what fraction of collisions produce significant
changes in mj or j and ii) the magnitude of these changes.

To facilitate thiscomparison, “non-collisional” events are excluded by limiting the maximum impact
parameter to be the average hard-sphere collision radius of O2 (2.9 Å), and both ∆mjand ∆j are scaled by <j>. Figure 4.9 reports histograms of the resulting distributions.
Notably, >70% of the collisions produce non-negligible changes for both mj and j(i.e., greater than 5% of <j>). More importantly, the ∆mj and ∆j distributions arequite similar, implying that neither type of collision predominates in developing therotor alignment. The relevance of this finding to previous studies will be explored in
Sect. 4.4.

133

FRACTION OF COLLISIONS FRACTION OF COLLISIONS

-1.0
0.3

-0.5

0.0

0.5

1.0

0.0

0.5

1.0

a) ∆mj/<j>
0.2

0.1

0.0

b) ∆j/<j>
0.1

0.2

0.3
-1.0

-0.5

Figure 4.9. Distributions of a) ∆mj/<j> and b) ∆j/<j> for an ensemble of 500,000 He+ O2 collisions at 266 cm-1. The similarity in these distributions indicates that both∆mj ≠ 0 and ∆j ≠ 0 collisions are significant in creating alignment.

134

4.3.3 Velocity dependence
In addition to providing insight into j-dependent alignment, the singlecollision regime can be used to examine the dependence of the alignment on the final
rotor velocity. Indeed, discrepancies between the studies of Aquilanti et al.58 andHarich and Wodtke57 on this velocity dependence provided the initial incentive forthese calculations, specifically to address whether subtle differences in the twointermolecular surfaces could be responsible for such clear differences inexperimental results.
Since the multicollision simulations demonstrate the alignment to beinsensitive to collision energy, we simply consider the 500,000-trajectory, singlecollision ensembles at 266 cm-1 (300 K) for each potential.

To carry out thiscomparison, the change in rotor velocity component along the expansion axis
(i.e., from zero initially to its post-collision value) is recorded for each trajectory.
The average final velocity (normalized to the velocity predicted for a head-on elastic
collision) is then calculated for each bin of (mj/j)f. In order to prevent statisticalskewing by non-collisions at large impact parameter, we exclude events with a final
velocity ≤10% of the elastic upper limit. To examine the results for a possible jdependence, the single-collision data are separated into j bins defined previously.
Fig. 4.10 contains a plot of the post-collision rotor velocity versus final mj/jstate, with the velocity expressed as a percent difference from the average final
velocity integrated over all alignments, i.e., ∆v/<v>. The upward curvature at theends of this distribution clearly demonstrates that on average, molecules with |mj/j| ≈1 are faster than |mj/j| ≈ 0. Stated explicitly, molecules with j parallel to the135

ALIGNMENT DEPENDENCE ON FINAL ROTOR VELOCITY∆v/<v> (%)

10

5

0

-5

-10
-1.0

-0.5

0.0

0.5

1.0

(mj/j)f

Figure 4.10. The percent difference of the final rotor velocity from the alignment–
averaged final rotor velocity, ∆v/<v> versus (mj/j)f. The data are binned by j/<j> andHe + rotor potential as follows: ● = 0.0-0.4, He + O2; ▲ = 0.8-1.2, He + O2; ■ = 1.62.0, He + O2; O = 0.0-0.4, He + CO; ∆ = 0.8-1.2, He + CO;  = 1.6-2.0, He + CO.
Error bars represent ±2σ of the mean. The upward curvature of this data indicates
that, on average, molecules with |mj|/jf ≈ 1 (i.e., j parallel to the collision axis) are
faster than those with |mj|/jf ≈ 0 (i.e., j perpendicular). This trend is the same for all
potentials and ranges of j shown.

expansion axis are on average moving faster than those with j perpendicular, with aroughly 15% spread in velocity around the average velocity after a single collision. In
addition, this velocity-dependent alignment is virtually identical for He + O2 and He +CO, thus ruling out any major dependence on subtle differences in the twointermolecular potentials. Furthermore, this trend is robustly independent of angularmomentum, i.e., the data for all j ranges in Fig. 4.10 overlap within statisticaluncertainties.

In summary, these calculations predict the faster (slower) rotor

molecules to exhibit a more positive (negative) alignment, i.e., j preferentiallyparallel (perpendicular) to the expansion axis, with little sensitivity to the interaction
potential or rotational quantum number.
These velocity-dependent trends are most consistent with the results of Harich
and Wodtke. To quantify the comparison between theory and experiment, Harich andWodtke’s velocity dependent alignment data for He + CO(j=6) have been reanalyzed
as a classical distribution of mj/j states as a function of final velocity. For each mj/j,
an average final rotor velocity is calculated and plotted in Fig. 4.11 as ∆v/<v>.
Superimposed on this line is the final velocity distribution of rotor molecules obtained
from our calculations, averaging over all j values and both potentials. The trends inthe experimental and calculated data are clearly quite similar: molecules aligned withj perpendicular to the collision axis (|mj/j| ≈ 0) travel more slowly than those with jparallel (|mj/j| ≈ 1). The implications of these results will be further developed below.

137

VELOCITY-DEPENDENT ALIGNMENT:COMPARISON WITH EXPERIMENT10TRAJECTORIESEXPERIMENT∆v/<v> (%)

5

0

-5

-1.0

-0.5

0.0

0.5

1.0

(mj/j)f

Figure 4.11. Comparison of the experimental jet-expansion data of Harich andWodtke to the overall average ∆v/<v> versus (mj/j)f obtained by averaging theclassical-trajectory data in Fig. 4.11 over all j and both potentials. Both themulticollision experimental results and the single-collision classical trajectories
demonstrate the same trend: on average, molecules with |mj|/jf ≈1 are faster than thosewith |mj|/jf ≈ 0.

138

4.4 Discussion
Several of the results obtained in this study warrant further discussion. Firstof all, the overall trends in the theoretical alignments are in excellent agreement withexperiment.

The multicollision simulations predict a2 < 0 for all potentials,

supporting the simple picture that rotor molecules with j parallel to the expansion axis
present a larger cross section to the colliding He diluent and thus are more likely toexperience realigning collisions.

Somewhat more surprisingly, this alignment

appears to be quite insensitive to collision energy.

Specifically, the asymptoticalignments indicate less than 40% changes over an order of magnitude range in Ecomfor all three collision systems tested. Indeed, only at energies significantly lower than
the well depth for each collision system do the alignments decrease noticeably, a
trend that can be ascribed to increased probability of long-lived versus direct collision
trajectories. This indicates that high velocity slip conditions (e.g., at the throat of the
expansion) or much slower collisions (e.g., further downstream) can both contributeto the rotor alignment, provided that the relative velocities are sufficiently wellaligned with respect to the expansion axis. This is especially relevant to comparisons
between experimental results, where different expansion conditions yield different
final jet temperatures, velocity slips, and thus relative scalar speeds of the diluent gas
and rotor species.
In the multicollision ensembles, the maximum alignments ranged from alower limit of a2 ≈ -0.46(1) (for He + O2), to a2 ≈ -0.73(1) (for He + CO), to thehighest value of a2 ≈ -1.03(1) (for He + CO2). By way of contrast, experimental
studies for He + CO2 indicate maximum CO2 alignment of A0(2) ≈ -0.115,56 which in139

the classical limit corresponds to a2 = -0.29, i.e., ≈ threefold smaller than obtained
from the current work. Similarly, the experimental CO alignment (integrated over all
velocities) for He + CO studies is A0(2) = -0.08;57 this corresponds classically to a2 ≈-0.2, which is also roughly threefold smaller than suggested by the multicollision
simulations. In essence, despite excellent qualitative agreement for alignment trends
as a function of both velocity and rotor angular momentum, the maximum alignment
values theoretically predicted are quantitatively much higher than those measured
experimentally.
These discrepancies reflect the different nature of collisions in the simulations
versus the jet expansions. In the theoretical simulations, all collisions occur at thesame center of mass energy and are directed initially along the expansion axis. In
contrast, the collisions in a supersonic expansion take place over a range of relative
energies as the molecules cool and, more importantly, over a wider range of collision
angles. The distribution of relative energies is not responsible for these differences,since the alignment results are quite insensitive to collision energy. Likely to bemuch more relevant is the subsequent influence of this cooling on the distribution ofcollision velocities in the expansion. At large values of velocity slip, most diluent
gas-seed gas collisions are strongly aligned along the expansion axis; however, as thisvelocity slip approaches zero, collisions with the seed gas molecule will be far lessaligned. For example, the studies of Weida and Nesbitt indicate that velocity slip forCO2 in the late stages of a He + CO2 expansion is as low as a few percent; thus, mostcollisions are essentially isotropic with respect to the laboratory-frame expansion
axis. This excess of non-directional collisions can lead to efficient scrambling of the140rotational alignment produced up to that stage; indeed, Halpern et al.70 and Schade etal.71 have demonstrated such collisional loss of alignment for acetylene and NO,
respectively. This effect has been explicitly noted in the He + CO2 studies: as afunction of distance downstream, the net CO2 alignment reaches a maximum and thensteadily decreases due to nominally unaligned collisions late in the expansion. The
neglect of velocity slip/angular effects in the current theoretical studies clearly leads
to an overestimation of the asymptotic alignment values, i.e., in the direction
consistent with the observed differences.
Despite these quantitative differences between the magnitudes of theoreticaland experimental alignments, the agreement is qualitatively quite good and provides
support for a closer examination of the underlying alignment mechanism.

For

example, when alignment is further partitioned by rotor angular momentum, theclassical trajectory simulations demonstrate that the asymptotic values increase
monotonically with j for each of the three intermolecular potentials. Contrary toprevious interpretations, this j-dependent alignment can not be ascribed entirely toquantum effects contributing at low j, since the same trends are clearly evident also inclassical trajectory calculations. This j dependence is most likely due to increased
gyroscopic stability, i.e., the faster a classical rotor is spinning, the more stability it
exhibits toward collisional torques changing the alignment.

This point is mostdramatically made in Figs. 4.6a-c, which indicate a significant decrease in the
distribution of mj/j with increase in j after a single collision. As a result, once high-jmolecules become aligned, they are less likely to be tilted away from this alignment
by subsequent collisions. This concept of the stability of higher j states against mj/j141changing collisions is supported by studies of pressure-broadening coefficients, which
generally indicate a rapid decrease with increasing j state.72-76 Furthermore, Hulsman
and Korving report that cross sections for reorienting collisions between Ar and Na2
decrease strongly with angular momentum, with a dependence nearly proportional to
1/j2.50This discussion of j-dependent alignments would seem to suggest thatmultiple, mj-changing collisions are the primary mechanism for alignment insupersonic expansions. Indeed, previous work by Weida and Nesbitt56 demonstrates
that alignment of CO2 in a supersonic expansion becomes increasingly negative withj, a phenomenon they attribute to mj-changing collisions. Likewise, a quantum closecoupled calculation by Werner and coworkers77 on collisional alignment of N2+ ionswith He demonstrate cross sections for mj-changing collisions (at fixed j) that are
significantly larger than j-changing collisions (at fixed mj).

Consequently, thealignment observed by Dressler et al.78 in the corresponding He drift tube experimenthas been attributed to mj-changing collisions at more or less fixed j. More recent
work by Anthony et al. on these He-N2+ drift tube experiments demonstrates that themost negative alignments occur at high j.79
Herschbach and coworkers have developed an alternate explanation of thecollisional alignment involving j-changing collisions. Experimentally, they monitor
alignment of I2 as a function of final jet temperature, observing that as the final
temperature decreases, the negative alignment of a given j state reaches a maximum
and begins to decrease again, eventually becoming positive under certain expansion
conditions.54

They ascribe this phenomenon to a contribution from anisotropic142rotational cooling: molecules with j perpendicular to the collision axis experience agreater torque than those with j parallel and are thus cooled more rapidly.
Consequently, this would predict an excess of rotors with j predominantly
perpendicular to the expansion axis at low j (i.e., cooler temperatures) and relatively
fewer molecules with j perpendicular in the high j states, and thus a net a2 alignmentthat becomes more positive with increasing j. Since the collision energy used in eachof our calculations is the mean temperature used to sample the rotational distribution(i.e., the two are roughly equivalent), these studies cannot directly assess the validityof the anisotropic rotational cooling model relative to the other models. However, it
is clear from Fig. 4.9 that both mj- and j-changing collisions are occurring at similarfrequencies and contribute to the mj/j alignment.
The velocity-dependent alignments presented in Fig. 4.10 demonstrate several
interesting points. First of all, the sign of the resulting velocity-averaged alignment iscorrectly predicted to be negative, irrespective of the magnitude of the velocity slip.
However, when one investigates alignment as a function of deviations away from theaverage speed, the results are more subtle. Specifically, the rotor molecules moving
faster than the average (∆v > 0) have a greater propensity for j perpendicular to the
expansion axis than the more slowly moving velocity groups (∆v < 0).

Mostimportantly, this behavior is exhibited for all values of j for both He + CO and He +O2 collision systems and so does not appear to be sensitive to the detailed shape of theintermolecular potential. Over multiple collisions in a supersonic expansion, thiseffect predicts a net negative alignment (a2 < 0) for the slower rotor molecules andpositive alignment (a2 > 0) for the faster molecules, i.e., exactly the same trend143

reported by Harich and Wodtke in supersonic expansions of He + CO. Such behavior
supports the simple physical picture discussed by Harich and Wodtke that rotor
molecules with j parallel to the expansion axis present a larger collisional crosssection to the diluent gas, preferentially experiencing more directed collisions andhence more forward acceleration with respect to the average rotor speed.
In further support of this interpretation, both the computational results and thework of Harich and Wodtke can be compared with the work of Anthony et al. on
velocity-resolved collision-induced alignment of N2+ in a He drift tube.79,80
Interestingly, these studies reveal that faster N2+ drift velocities are associated withmore negative alignment (the alignment even becomes positive at the lowest
velocities for the lowest j values). At first glance, this greater propensity for jperpendicular versus j parallel to the drift axis at the highest ion velocities would
appear to be in disagreement with both the supersonic jet He + CO studies and thecurrent theoretical results. However, as pointed out by Harich and Wodtke, the rotor
molecules in supersonic expansions are accelerated by collisions with the fastermoving diluent gas, while the ions in a drift tube, pulled through the buffer gas by an
external electric field, are decelerated by collisions with the buffer. As a result, thefast end of the velocity distribution reflects those ions which have suffered fewer
collisions, which is therefore consistent with the more negative alignments (i.e., jperpendicular preferred over j parallel) observed experimentally at higher velocities.
By way of contrast, experiments on He + O2 and He + N2 by Aquilanti andcoworkers indicate that alignment in a supersonic expansion becomes more negative
for the higher velocity molecules, i.e., the concentration of rotors with j perpendicular144

to the expansion axis increases with ∆v > 0. This is qualitatively different from the
He + CO results of Harich and Wodtke, the He + N2+ ion alignment studies of
Anthony et al., and the theoretical predictions from the current work. There are atleast two factors that should be considered as possible sources for this discrepancy.
First of all, not all the experiments of Aquilanti and coworkers have been performedon collision systems (e.g., He + N2) sampled in other studies.

However, thetheoretical results obtained in this work are robustly independent of theintermolecular potential for all three systems (He + CO, He + O2, He + CO2); it
would be quite surprising if the behavior for He + N2 were somehow significantly
different. Furthermore, the theoretical trends observed in the current study for thetwo systems investigated by Harich and Wodtke and Aquilanti and coworkers, i.e.,
He + CO and He + O2, are essentially identical.
Secondly, the experiments on He + O2 by Aquilanti and coworkers have been
performed under much stronger expansion conditions, thus sampling lower j values
than in the He + CO experiments. In this regard, it is worth noting that calculations inthis work have been carried out over nearly a 15-fold range of jet temperatures,
though none as low as Aquilanti and coworkers. Indeed, quantum cross section
calculations have been performed by Aquilanti and coworkers for the He-O2 collisionsystem81 in which authors attribute the quite different velocity effects between theirstudies and those of Harich and Wodtke to differences in expansion conditions.
Specifically, Aquilanti and coworkers probe in only a very small solid angle along theforward-/backward-scattered direction while Harich and Wodtke necessarily averageover a broader range of angles. Therefore, one possible explanation might be that the145different scattering angles sampled in the two experiments produce different velocitydependent alignments. However, this would also suggest a strong dependence of
these velocity effects on impact parameter, whereas our calculations indicate
essentially identical alignment trends (i.e., j perpendicular faster than j parallel) for
all impact parameters, with bmax as low as 0.1 Å. In essence, these classical trajectory
calculations do not provide a satisfactory explanation for the experimental
discrepancies in velocity dependent alignments but do indicate that differences inscattering for the two intermolecular potentials (He + O2 vs. He + CO) are notsufficient to resolve this issue.

4.5 Summary
Classical trajectory calculations have been carried out on experimentally
determined potentials for He-O2, He-CO, and He-CO2 to explore the collisional
mechanism for rotational alignment and to facilitate comparison with previous
experimental studies. These calculations demonstrate that for multiple collisions
between a light “diluent” gas, He, and a heavy “seed” rotor, the rotor molecules
become anisotropically aligned with a clear propensity towards j perpendicular to the
expansion axis.

This alignment grows in with an approximately exponential

dependence on the number of collisions and a 1/e collision number that varies from ≈2 to 30. The sign and magnitude of this asymptotic alignment are found to beremarkably insensitive to collision energy over more than an order of magnitude
dynamic range, as well as to the detailed shape of the interaction potential.

146

In addition to this examination of the overall alignment from multiplecollision simulations, j-dependent effects have also been explored. These simulations
reveal the asymptotic alignment to depend strongly on rotor angular momentum,
increasing in magnitude with j for all three potentials studied and conversely
vanishing as j tends toward zero. The evidence of this behavior in a purely classicalsimulation indicates that experimental trends of vanishing alignment for low j statescannot be simply ascribed to quantum effects near j = 0. Investigated on a collisionby-collision basis, the higher j states appear to be far more stable with respect tocollisional destruction of the alignment. Specifically, the statistical spread in the finalalignment (mj/j)f for a given initial alignment (mj/j)i narrows dramatically with j,consistent with the simple physical picture of greater classical gyroscopic stability formore rapidly spinning rotors.
With regard to velocity dependent alignments, the calculations indicate thatmolecules moving faster than the average velocity have their angular momenta
preferentially aligned parallel to the expansion axis, in agreement with theexperimental observations of He + CO collisions by Harich and Wodtke.

Thisprovides theoretical support for this physical picture of the alignment process: rotor
molecules with j parallel to the relative velocity axis are struck from behind more
often by seed gas molecules than rotors with j perpendicular to this axis and thus are
preferentially accelerated ahead of the average rotor velocity distribution. These
simple models are also consistent with the velocity dependent He + N2+ ion drift
experiments of Anthony et al. but as yet do not appear to be consistent with the
behavior reported by Aquilanti for He + O2 and He + N2 collision dynamics.
147

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152

CHAPTER 5
EXPLORING THE ROLE OF VIBRATIONAL NONADIABATIC EFFECTS
IN VIBRATIONALLY MEDIATED CHEMICAL REACTIONS

5.1 Introduction
Over the past decade, the possibility of influencing chemical reactions byvibrationally mediated chemistry has enjoyed a great deal of interest and success.
Originally pioneered by Crim and coworkers and Zare and coworkers for the reactionof H with water and its isotopes, it has since been extend to variety of reactive atoms(H, Cl, O) and molecular reaction partners (H2O/HOD,1-7 HCN,8-10 CH411-13). Suchsystems provide the tantalizing possibility of pre-excitation of a particular bond in amolecule leading to selective reaction.
One means of exploring the degree of selectivity imparted by vibrational
excitation has been to examine the branching ratios among possible products. For
example, in the reaction of H with vibrationally excited HOD, Bronikowski showedthat the relative branching ratio for OD/OH production was at least 25 from vOH=1,
while the branching ratio for OH/OD was at least 8 from vOD=1.1 Similar effects
have been demonstrated for the reaction of HOD with other atoms and at other levels
153

of vibrational excitation, both experimentally2-7 and theoretically.14-20 In addition,
the Crim group’s studies of nearly isoenergetic vibrational states of H2O containing
differing amounts of local-mode stretch and bend excitation demonstrated increased
reactivity with H atoms when all quanta were placed in the stretch excitation.2,3
Another approach to determining vibrational selectivity has been to monitor
how excitation in the nonreactive bond translates into product-state excitation. For
example, the Crim group monitored the OH vibrational states resulting from reactionof H and Cl atoms with H2O in two states, |04-> and |13->, where |nm(+/-)> denotes
symmetric and antisymmetric combinations, respectively, of local mode stretcheswith n quanta in one bond and m quanta in the other.3,4 These studies revealed amarked propensity toward production of OH in v=0 in the first case and OH in v=1 inthe second.
From these demonstrations of mode-specific reaction a “spectator bond”
paradigm has emerged, i.e., the concept that the nonreactive bond does not participate
in the reaction dynamics, and any initial excitation in that bond should be retained in
the final product distribution.

Underlying this paradigm is an assumption ofvibrational adiabaticity in the reactive species, i.e., that the initial vibrational state
prepared in the reactive molecule is essentially unaffected by the approach of thereactive atom. This view of these reaction systems has increased in popularity suchthat reduced-dimensionality 4-atom reactive scattering calculations often fix thelength of the nonreactive bond.21
However, a significant amount of experimental and theoretical evidence
demonstrates that the nonreactive bond may not always play the role of a spectator in154

mode-specific reactions. Such an effect is quite dramatically demonstrated in recent
experiments by the Crim group on Cl + HCN, which produced similar excitation of
the product CN stretch from both Cl + HCN(004) and Cl + HCN(302),3 where thefirst quantum number corresponds to a normal mode of predominately CN stretch
character, and the third to a normal mode of predominantly CN character. Theirinitial explanation for this phenomenon was that the Cl + HCN reaction proceeds via
a complex, resulting in intramolecular vibrational redistribution (IVR) prior toreaction.

However, Schatz and coworkers have carried out quantum scattering

calculations on the related H + HCN reaction which allow the CN bond to vibrate
while constraining HHC to be collinear, a geometry that removes the possibility ofcomplex formation.22 They determine that excitation to HCN(100), i.e., one quantumof mostly CN stretch character, significantly enhances reaction cross section to H2 +CN and lowers the threshold energy relative to H + HCN(000); in fact, this thresholddecreases by nearly 100% of input vibrational energy.
Similar effects, albeit on a smaller scale, can be observed in the earlier studiesof the reactivity of water and its isotopes mentioned previously. For example, Crimand coworkers noted a decrease in relative OH v=1/v=0 production from Cl +H2O|13-> versus H + H2O|13->.4 Similarly, Schatz and coworkers noted in an early
quasiclassical trajectory study that excitation of the OD bond in HOD or of highly
excited bend states of H2O (5-7 quanta) provides some enhancement of the H
reactivity above the ground state.14 A later study by this group, examining lower
levels of vibrational excitation in HOD, again demonstrated that excitation of the OD
stretch, in addition to enhancing the reaction probability of the D atom, increases H
155

reactivity to a lesser extent.15 More recent 6D quantum calculations of H + HOD by
Zhang and Light also indicate that excitation of the OD bond does cause a slightenhancement of reaction probability for the OH bond.20
At the heart of these studies is the apparent breakdown of the local modenature of the initial excitation of the reactant. The approach of the reactive atom mustcause a redistribution of vibrational energy in the reactant molecule, occurring on thetimescale of a chemical reaction.

Correspondingly, the question emerges: what

happens to the excited vibration as the atom approaches?

In each of the previous

studies, the primary focus has been on examining the product distribution rather thanthe details of the interaction of the atom with the molecular vibrations.
To address these issues, we explore vibrationally nonadiabatic effects inchemical reactions, focusing on a model system for the collinear reaction of Cl with
reduced-dimensionality, linear water isotopes. Our first goal will be to determine in atime-independent fashion how the vibrational eigenfunctions of this model water
system are affected as the Cl atom distance is parametrically decreased, focusing
primarily on the v=1 stretch polyad in water. Our second goal will then be to follow
the dynamics of the reaction in a time-dependent fashion via wavepacket propagation.
In doing so, we hope to demonstrate the insight that can be gained from such a timedependent approach.

5.2 Computational Approach
To study the reaction dynamics of Cl with water isotopes in various
vibrational levels, the computational process can be divided up into three major steps.
156

For simplicity, these computations are primarily described for Cl + H2O but can beapplied to the other systems to be studied, i.e., Cl + D2O and Cl + HOD (with theapproach of the Cl atom along the OH bond). First, we calculate the eigenstates andeigenenergies of the water molecule as a parametric function of the Cl-H2O
internuclear distance. In doing so, we assume that the vibrational motion is separablefrom the relative Cl + H2O motion, similar to the Born-Oppenheimer separation of
nuclear and electronic motion,23 in order to calculate an adiabatic basis of vibrational
eigenfunctions.

Thus, the total wavefunction is composed of vibrational andinternuclear parts,

Ψ total = ∑ ψ n χ n

(5.1)

n

where Ψ represents the total wavefunction, ψ represents the internuclear part, and χ
represents the vibrational part. The vibrational eigenstates, χn, are determined bysolving the time-independent Schrödinger equation for the OH bonds with parametricdependence on the Cl-H2O distance:H(r1, r2; R) χnvib(r1, r2; R) = En(R)* χnvib(r1, r2; R)(5.2)

where r1 is the distance between the O atom and the H atom proximal to the Cl, r2 isthe distance between the O atom and the H atom distal to the Cl, and R is the distancefrom the Cl to the center of mass of the water. A plot of this coordinate system isgiven in Fig. 5.1. The changes in these eigenfunctions with R provide insight into
157

Figure 5.1. Coordinate system for Cl + HOH reactive scattering calculations. R
represents the distance from the Cl atom to the center of mass of water, while r1 is thelength of the proximal OH bond (i.e., closest to the Cl), and r2 is the length of the
distal OH bond. With appropriate isotopic substitutions, this coordinate system isalso used for Cl + DOD and Cl + HOD (replacing H with D on the distal OH bond).

how the water vibrations are affected by the approach of Cl. In addition, the En’s
provide one-dimensional (1D) adiabatic potential surfaces for relative Cl-H2O
motion. Accordingly, along an uncoupled adiabatic potential curve, the Schrödinger
equation describing the intermolecular Cl-H2O motion can be exactly written as

 − !2
∂2+ E n ( R )ψ n ( R ) = Eψ n ( R )2 2µ Cl ,HOH ∂R


(5.3)

In the second step of the computational process, we must be able to transfer
probability among the adiabatic curves in order to allow for inelastic or reactive
scattering. These curve-crossing probabilities are governed by the nonadiabatic terms
in the intramolecular Hamiltonian, essentially, a breakdown in the separation between

158

vibrational and intermolecular motions. In this adiabatic basis, the couplings between
the curves depend on the rate of change of the vibrational eigenstates as a function ofR, thus depending on the velocity of approach of the Cl atom in a classicalinterpretation. The transition probabilities can therefore be used to estimate acharacteristic curve-crossing velocity for the transition between low-velocity,
adiabatic behavior and high-velocity, diabatic behavior.
In the final step of the computational process, we carry out time-dependent
wavepacket propagation for the relative Cl-H2O motion, starting with an initialentrance channel (e.g. the symmetric or antisymmetric vibration of H2O), following
the dynamics, and probing the products. These three steps are described in detailbelow.

5.2.1. Calculating vibrational wavefunctions at each Cl-water distanceThis calculation first requires a potential surface describing the interactions of
the four atoms, as well as means of representing and calculating the eigenfunctions.
The potential surface used to describe the Cl + H2O interaction is based on thetriatomic O + HCl London-Eyring-Polanyi-Sato (LEPS) potential surface given by
Persky and Broida,24 with parameters given in Table 5.1. This surface is used todescribe the interaction of one OH stretch with the approaching Cl; this stretch isdesignated the “proximal” OH stretch, r1. The other OH stretch is solely described
the Morse potential part of the LEPS surface and is designated the “distal” OH
stretch, r2. Thus, in the subsequent calculations, the Cl atom is only allowed toapproach the proximal end of the molecule. Physically, this potential surface does159not accurately describe the Cl + H2O interaction because products and reactants arenearly isoenergetic, while Cl + H2O should be endothermic by 5640 cm-1.25 The
surface was rather designed to lower artificially lower the reactive channels so thatthe reactive channels were more accessible at the v=1 level, simplifying our
computations by limiting the number of vibrational eigenfunctions calculated.
Clearly, this feature of the potential surface, as well as the linear configuration chosen
for water, indicates that no quantitative comparison with experiment will be made.
Rather, this system will provide a simple model for the dynamics of reaction with aheavy-light-heavy molecule; in addition, we will discuss which conclusions are valid
more universally.

Table 5.1. LEPS parameters for the Cl + H2O potential surface.

Parameter

O-H

Bond
H-Cl

β(Å )

2.295

1.868

2.290

De(cm-1)

37,23037,25023,600

re(Å)

0.970

1.275

1.570

k

0.170

0.158

-0.250

r‡

1.154

1.453

-1

O-Cl

After choosing the potential surface for the four-atom interaction, a
representation for the vibrational eigenfunctions, χn, must be designated. For thispurpose, we use a Fourier expansion of the wavefunction on an evenly spaced grid,160

such that the expansion coefficients are equivalent to the amplitude of the
wavefunction in momentum space.26

The momentum wavefunction can thus beobtained via a Fourier transform of the coordinate wavefunction; with FFTtechniques, the length of this computation scales as NglogNg (Ng is the number of grid
points). Such a representation takes advantage of the local nature of the kinetic
energy operator in momentum space, where calculation of the kinetic energy
operation requires only a simple multiplication step and scales as Ng. The complete
Hamiltonian operation thus scales approximately as NglogNg, rather than the Ng2
typically necessary for calculating derivatives of the wavefunction in position space.
Altogether, the Hamiltonian in the COM frame of H2O for the vibrational
eigenfunctions, with parametric dependence on the distance between the Cl and H2O
center of mass, is exactly given by

H=T+V
V = V(r1, r2, R) − ∂2
T = !2 2 2µ OH ∂r1+− ∂2
2µ OH ∂r22+∂2
m O ∂r1 ∂r2 (5.4)

where V is the potential energy from the LEPS potential, T is the kinetic energy of
the OH stretches, and r1, r2, and R are as defined previously. The size of the gridchosen to represent these eigenfunctions is 16×128 points, with spacings (∆r1, ∆r2) =(0.079, 0.057) Å. The larger number of points in r1 is designed to encompass both161

product and reactant wells; the spatial extent represented is thus (r1, r2) =([0.63, 7.94], [0.63, 1.90]) Å.
With this representation of vibrational eigenfunctions, we use a relaxation
method to determine the ground-state vibrational eigenfunction,27,28 propagating an
initial trial wavefunction via the time-propagation operator in imaginary time, τ = it:χ trial ( τ) = e − Hτ / ! χ trial (0)(5.5)

Since any trial wavefunction represents a superposition of all the eigenstates of the
Hamiltonian, each eigenstate will be damped out exponentially at a rate proportional
to its energy; thus, the ground-state eigenfunction will die out most slowly.
Accordingly, as τ Æ ∞, χtrial will collapse to the ground-state eigenfunction. Thus,
obtaining the ground state eigenfunction requires repeated application of the
propagation operator e −Hτ / ! to the trial wavefunction. Since the relaxation process
does not preserve the norm of the wavefunction, the length of the timestep is limitedby the need to renormalize χtrial repeatedly to avoid reaching machine precision.
Next, the excited-state eigenfunctions are calculated via the same relaxation
method. However, for each excited state, all previously determined lower eigenstates
are first projected out from the trial wavefunction, so that an expansion of the trial
wavefunction in the eigenfunctions of the system contains only excited-state
wavefunctions.

Accordingly, propagation in imaginary time will collapse thewavefunction to the lowest excited state included, i.e., the next-higher state in energy.

162

In practice, all lower eigenstates are projected out at each imaginary timestep to avoid
collapsing to a lower state via numerical instabilities.
To employ this relaxation technique for determining the eigenstates, we
expand the propagation operator in imaginary time in a series of Chebychev
polynomials, i.e.,Ne −Hτ / ! ≅ ∑ a n ( τ) Tn (-Hτ/! )

(5.6)

n =0

where the Tn are real Chebychev polynomials and the an are the expansion
coefficients described below.27

Because Chebychev polynomials are exponentiallyconvergent on [-1, 1], the Hamiltonian must be renormalized such that its eigenvalues
span that range, with

∆E = Emax - Emin = (Tmax + Vmax) - Vmin

(5.7)

Eavg = (Emax + Emin)/2,(5.8)

where Tmax and Tmin are the upper and lower limits on kinetic energy, and Vmin is theminimum potential energy. Eqn. 5.6 is then rewritten ase −Hτ / ! ≅ e avg
E

τ !

∆Eτ

N

∑ a ( 2! ) T (
n

n

n =0

163

- 2[H - E avg ]
∆E

)

(5.9)

the multiplicative factor preceding the expansion is a phase factor resulting from therenormalization of the Hamiltonian. The expansion coefficients an are from modified
Bessel functions, In,

an = (2-δ0n)In(∆Eτ/2!) ,(5.10)and the Chebychev expansion is calculated from the recursion relation

Tn ( X ) = 2 X Tn -1 ( X ) − Tn - 2 ( X ) ,(5.11)

where X = -2(H-Eavg)/∆E, T0(X) = 1, and T1(X) = X. This series exponentiallyconverges for N > (∆Eτ/2!)½.

Since ∆E increases as the range of eigenvalues

increases, more polynomials must be included for exponential convergence.
Consequently, we truncate the potential and kinetic energies so that the upper limit is
two times the estimated maximum eigenenergy (set to be ~20,000 cm-1 above thepotential minimum).
Once an eigenstate has been calculated, its quality can be estimated bycomputing the energy dispersion29

D ( τ ) = < χ( τ ) | H 2 | χ ( τ ) > − < χ( τ ) | H | χ ( τ ) > 2

which equals zero for a pure eigenstate.

164(5.12)

We first employ the Chebychev propagation to calculate the vibrational
eigenstates at a Cl-H2O separation of 7.94 Å, i.e., in a region where coupling causedby the Cl atom among vibrational states of water should be negligible. Eigenstatesfrom both the Cl + H2O and HCl + OH wells are included. Subsequently, we
incrementally vary the Cl-H2O distance from 7.94 to 2.12 Å on a uniform spatial gridof 512 points in R. For efficient convergence, we use the eigenstate calculated at theprevious R as the initial guess for the calculation at the next R. The convergence of
these eigenstates is verified for each R to approach machine precision (limited from
16 digits to 8 digits by the square root in Eqn. 5.12), with D < 0.008 cm-1. This
process generates adiabatic potential curves as a function of R, with one curve foreach vibrational wavefunction. These vibrational potential energy surfaces, En(R),thus govern the Cl-H2O intramolecular motion. In additional calculations, similar
surfaces are obtained for Cl + D2O and Cl + HOD on the same Cl + H2O potential
surface, i.e., varying only the mass of the isotope.

5.2.2. Adiabatic coupling terms
To calculate the coupling elements between adiabatic curves, an assumptionof separate timescales for vibrational and nuclear motion is no longer valid. Thus,
terms involving the derivative of the vibrational motion with respect to the nuclear
coordinate must be added back into the nuclear Schrödinger equation given in Eqn.
5.3, which can now be written as30

165

 − !2∂2+ E n (R ) − E ψ n (R ) =2 2 µ Cl − HOH ∂R

∂


∑
τ nm (R ) ∂R + ρ nm (R )ψ m (R ) ,m(5.13)

where En(R) is an adiabatic vibrational potential energy surface calculated as above,
E is the total energy of the system, and τnm and ρnm are the coupling terms

τ nm (R ) =ρ nm (R ) =!2µ Cl − HOHχn!2
2 µ Cl− HOH

∂
χm
∂R

∂2
χn
χm
∂R 2

(5.14).

(5.15)

Operationally, we follow the method of Tuvi and Band31 to symmetrize these
coupling terms to preserve the hermiticity of the Hamiltonian. This process requires
rewriting the Schrödinger equation as − !2
∂2+ E n (R ) − E ψ n (R ) =2 2 µ Cl − HOH ∂R

~
∂
∂


τ
τ
ρ
(
R
)
(
R
)
+
+
∑
nm
nm (R ) ψ m ( R )
 nm
∂R ∂R

m

(5.16)

where~ρ nm (R ) = ρ nm (R ) −− !2=
2 µ Cl− HOH∂[τ nm (R )]
∂R
∂
∂
χn
χm
∂R
∂R

166

(5.17)

Equation 5.16 is analytically equivalent to 5.13; however, all derivatives of the~vibrational eigenfunctions with respect to R in τnm and ρ nm are now first derivatives,
facilitating numerical computation. We calculate these derivatives to second order,~

additionally taking advantage of the hermiticity of ρ nm and anti-hermiticity of τnm toaverage each element with its transpose or the complex conjugate of its transpose,respectively.
In addition to using the coupling terms for calculating the wavepacket
dynamics, we can estimate relative curve-crossing velocities in the following way.
The most significant coupling term is typically τnm; as described by Child,30 ∂ ∂R isrelated to the momentum operator, such that τ nm ∂ ∂R is related semiclassically toequal µ Cl − HOH v rel (R) /! , where vrel(R) is the relative Cl-H2O velocity at a given R.
τ nm ∂ ∂R becomes most significant as it approaches the magnitude of the energy
spacings between nearest adiabats, i.e., |En(R)-Em(R)|. We can thus estimate thecharacteristic transition velocity by setting these two terms equal to each other andsolving for vrel(R) to obtain a characteristic transition velocity, vnm(R):v nm ( R ) =Em (R) − En (R)
∂
χn
χm
∂R

(5.18)

vnm(R) is thus the characteristic vrel for the transition between the nth and mth adiabats,similar to a Landau-Zener velocity for crossing between diabats; its minima (i.e., themost likely crossing points) occur at values of R corresponding to avoided crossings
between adiabats. At relative Cl-H2O velocities smaller than the minimum in vnm(R)167at a crossing region, we would thus expect the system to proceed adiabatically.
Correspondingly, these velocities should aid in our predictions for the curve-crossing
dynamics.

5.2.3. Wavepacket propagation
The final step in this process is determining how the choice of initial H2O
vibrational state and relative Cl-H2O velocity influence the dynamics of the Cl + H2O
collision by performing time-dependent wavepacket scattering calculations. In brief,
our approach is to begin a Gaussian wavepacket on one of the adiabats at a large ClH2O separation and propagate forward in real time, using a Chebychev expansion of
the propagation operator.

When the wavepacket reaches the Cl-H2O interaction

region, the coupling terms between the adiabats allow for transfer of probability toother states. The flux in each output channel is calculated as the wavepacket exits theinteraction region to determine the relative amounts of each product state. After theregion of flux calculation, an appropriately scaled damping operator is placed on eachof the output channels so that the wavefunction does not reflect or transmit at the endof the grid. The details of this phase of the computation are given below.
The time-dependent technique resembles time-independent methods bypermitting determination of the branching ratios among elastic, inelastic and reactive
scattering channels after the wavepacket has entered and exited the interaction region.
Unlike time-independent methods, this technique has the important advantage offollowing the flow of probability among channels as a function of time to determine
how the wavepacket makes transitions at crossing regions. The disadvantage of this168

technique is that rather than providing monoenergetic scattering probabilities, thewavepacket must (by the Heisenberg uncertainty principle) contain some spread in
momentum. While the energy-dependent scattering can nevertheless technically berecovered, it requires calculating the outgoing flux for extended periods of time (i.e.,
without damping). For our purposes, this difficulty can more simply be overcome bymaking the wavepacket appropriately broad in coordinate space and thus narrow inmomentum space. Thus, the initial conditions for the wavepacket propagation are
selected in the following way.
For each of the three reaction systems, the initial Gaussian wavepackets havecenter momenta at a near-thermal velocity, with additional momenta explored for Cl+ H2O. For Cl + H2O, Cl + D2O, and Cl + HOD, initial momenta are 10.0, 10.4, and10.2 au, respectively, scaled to have equivalent kinetic energy in all three collisionsystems. The additional momenta selected for Cl + H2O are 15.0 and 20.0 au; thus,
the three Cl + H2O momenta correspond to relative energies of 510, 1100, and 2000
cm-1 and speeds of 101,000, 152,000, and 202,000 cm/s (for comparison, the mean
relative speed of Cl and H2O at 300 K is 73,000 cm/s).

The spatial extent of all

wavepackets is chosen to be 0.26 Å; for Cl + H2O, the Gaussian width in momentum
space (∆p = 2.0 au) is less than the spacing between the three momenta (5.0 au).
These wavepackets are then centered at R = 6.9 Å, away from the interaction region.
Convergence of the calculation with respect to this choice of distance is explicitly
verified for each momentum and mass combination by comparison with the output
flux from wavepackets begun at 6.35 Å; the flux in each channel varies by ≤ 1% of its
own value.
169

As with the eigenfunction relaxation method described above, a Chebychev
expansion is used for the propagation operator. However, since the wavepacket
propagation is in real time, the expansion is modified in this way:e −iHτ / ! ≅ e avg
iE

t !

∆Et

N

∑ a ( 2! ) φ (
n

n

n =0

- 2[H - E avg ]∆E)(5.19)

where the φn are imaginary Chebychev polynomials defined on [-i,i]. The expansion
coefficients, an, are calculated from Bessel functions, Jn, via26

an = (2-δ0n)inJn(∆Et/2!)(5.20)The recursion relation for the Chebychev polynomials is similar to the one given in
Eqn. 5.11, with modifications for damping out the wavefunction in the exit channel
described below; however, this expansion converges exponentially for N > (∆Eτ/2!)(note the lack of square root). The Chebychev propagator preserves the norm of the
wavepacket, and in principle, the timestep can be of unlimited size, at the cost ofadditional terms in the expansion. However, in order to be able to monitor the flux
and observe the wavepacket propagation, the timestep is chosen to be 12.5 fs;
correspondingly, the typical size of the Chebychev expansion is on the order of 300400 polynomials.

With this expansion, preservation of the total norm of the

wavepacket is verified to be better than one part in 108 per timestep, withapproximately 100 timesteps required for each propagation.

170

After the wavefunction exits the interaction region, the flux out of each exit
channel is averaged over a series of points prior to the absorbing boundary. The
accumulated flux at each point in a given channel is thus obtained from32

F = ∑ J(t)∆t =
∆t

∆t



! ∂



∑ Re ψ (R, t ) i ∂R ψ (R , t ) ,*µ Cl − HOH Λt(5.21)

where ∆t is the size of the timestep. Since the initial norm of the wavepacket equals
1, the accumulated flux summed over all channels should also equal 1; we thus
explicit verify that this accumulated flux agrees to ≤ 1% with the wavepacket
probability lost by the end of the calculation.
After the wavepacket passes through the region in which the flux is
calculated, a damping operator is applied at each time propagation step to absorb the
wavepacket, preventing reflection from or transmission through the end of the grid.
Following Mandelshtam and Taylor,33,34 this damping operator has the forme − γ (R ) ,(5.22)

whereγ(R)= c(R-R0)/(Rmax-R0), R ≥ R0
= 0, R < R0

(5.23)

171

with Rmax equal to the largest Cl-H2O radius (7.94 Å), and c and R0 as adjustableparameters such that the damping operator decreases fromexp{-γ(R0)} = 1 toexp{-γ(Rmax)] = exp(-c). The operator is incorporated into the Chebychev expansionof the propagation operator byφn ( X ) = e − γ ( R ) [2Xφn -1 ( X ) − e − γ ( R ) φn - 2 ( X )] ,(5.24)

where X = -2(H-Eavg)/∆E, φ0(X) = 1, and φ1(X) = exp{-γ(R)}X. The operator thus
acts on the wavepacket via multiplication by exp{-γ(R)} in the region [R0,Rmax].
We determine the optimal parameters for this operator by fixing R0 such that
(Rmax - R0) = 1.32 Å, i.e., significantly larger than the deBroglie wavelength of thelowest input momentum (for p = 10 au, λdeBroglie = 0.32 Å), and varying c. Trial
wavepackets with one center momentum are propagated towards this absorbing
boundary on a single potential curve (with all couplings turned off); c is optimized byminimizing the survival probability for this wavepacket (i.e., reflection plustransmission). This procedure is repeated for a range of center momenta; in all cases,
the survival probability is robustly insensitive to ~20-30% changes in c. Accordingly,
the optimal choice of c for any center momentum can be estimated from a quadratic
least-squares fit of these data points. The parameters from this least-squares fit are
given in Table 5.2.

172

Table 5.2. Table of parameters from quadratic fit of optimal damping parameters, c,
for Cl + H2O: c = Ax2+Bx+C.
Parameter
A
B
C

1.47E-6
9.23E-4
3.46E-3

To dampen the elastically scattered output wavepacket, the c parameter usedcorresponds to the initial center momentum.

However, the portions of the

wavepacket that make transitions to other states (via the adiabatic coupling terms),
will not have the same center momentum. Rather, the kinetic energy of the fractionof the wavepacket on a different output channel should approximately equal the input
channel kinetic energy plus the difference in the asymptotic potential energies of thetwo states. Based on this difference, the output momentum on any other state can be
estimated and the damping parameters selected accordingly. In addition, since the
wavepackets are typically started near the damping region, the damping operator is
initially fixed to 1 (i.e., no damping) across its entire range for sufficient time
(typically 250-500 fs) for the wavepacket to propagate out of that region.

5.3 Results
The results of these calculations are divided into two parts. In Section 5.3.1,
we examine the static picture, i.e., the vibrational eigenstates calculated as a functionof R, the resulting adiabatic potential curves, and the couplings between these curves,
focusing on developing predictions for the subsequent dynamics calculations. In

173

Section 5.3.2, we discuss the wavepacket propagation on these potential curves,
determining how the dynamics proceed and how the initial vibrational state
influences the subsequent dynamics. In these, we primarily focus on the results for Cl+ H2O, comparing with Cl + D2O and Cl + HOD when appropriate.

5.3.1. Vibrational eigenfunctions, adiabats, and coupling elements
As described in Section 5.2.2, we have calculated the H2O and OH/HCl
vibrational eigenfunctions as a function of parametric Cl + H2O distance. We first
calculate the asymptotic eigenfunctions (i.e., at large Cl-H2O separation), including
eigenstates up to the sixth excited state, to serve as an initial basis for our subsequentparametric variation of R.

The eigenenergies calculated at each step in R thus

generate the adiabatic potential curves for Cl + H2O shown in Fig. 5.2.
We can determine the states represented by adiabatic curves by examining theasymptotic eigenstates, which can then used to label each of the curves. Theseeigenstates encompass the ground vibrational state, the lowest (v=1) stretching
manifold in Cl + H2O and OH + HCl, and the next excited state (corresponding toHCl(v=2) + OH(v=0) asymptotically) to ensure the availability of a higher energy
output channel. These adiabatic curves by definition always follow the lowest seveneigenstates; thus, these labels may not accurately represent the nature of the
wavefunction in the Cl + H2O interaction region.

However, asymptotically, thefourth excited state corresponds to the symmetric stretch of water, and the fifth
excited state, the antisymmetric stretch; the adiabats corresponding to these states willthus serve as the input channels for the dynamics calculations.
174

ADIABATS: Cl + HOH-60000

Energy (cm-1)-62000-64000-66000-68000-70000-72000
2

3

4

5

6

7

8

R (Å)
Figure 5.2. Adiabatic potential energy curves for Cl + HOH reactive scattering,
generated by calculating the H2O and OH/HCl vibrational eigenfunctions as afunction of parametric R. Asymptotic eigenfunctions (at large Cl-HOH separation)
serve as an initial basis for the parametric variation.

175

As seen in Fig. 5.2, the adiabats begin to vary in energy as R approaches
3.5 Å, indicating that they have reached the region where the couplings among
various states should become quite significant. Accordingly, we re-examine thecharacter of the vibrational eigenstates in this region. In Fig. 5.3a-d, we focus inparticular on the symmetric and antisymmetric stretch eigenstates at two Cl-H2O
distances: outside the interaction region (R = 4.23 Å, Fig. 5.3a-b), where theeigenstates resemble the asymptotic eigenstates, and near the interaction region (R =2.8 Å, Fig. 5.3c-d). As mentioned previously, the “proximal” OH bond, r1, points
toward the Cl, the “distal” bond, r2, is on the opposite side of the molecule; thus,
eigenstates at short r1 correspond to Cl + H2O states, and at long r1, to HCl + OH
states.
Intriguingly, the eigenfunction that corresponds asymptotically to thesymmetric stretch (Cl + H2O(s=1), the fourth excited state) becomes directedprimarily along the proximal OH bond (r1) with the Cl approach, i.e., it has rotatedtoward the Cl. Thus, the vibrational excitation is now pointed toward the reactive
atom. In contrast, the antisymmetric stretch (Cl + H2O(a=1), the fifth excited state)
rotates to localize primarily along the distal OH bond (r2), i.e., away from the reactive
Cl atom. Thus, the approach of the Cl atom breaks the symmetry of the potential toconvert normal-mode states into local-mode stretches.

The localization of thesymmetric stretch excitation toward the Cl might lead us to predict this state to be
more reactive than the antisymmetric stretch. To further explore how the changing
character of these eigenstates will affect reactivity, we next examine the adiabatic
curves in more detail, looking at all three systems studied (Cl + H2O/D2O/HOD).
176

Figure 5.3. Fourth and fifth excited-state vibrational eigenfunctions for Cl + HOH
outside the interaction region (R = 4.23 Å, Fig. 5.3a and b, respectively) and near theinteraction region (R = 2.8 Å, Fig. 5.3c and d, respectively). The fourth excited-state
eigenfunction, which corresponds asymptotically to the symmetric stretch (5.3a),becomes directed primarily along the proximal OH bond (r1) with Cl approach (5.3c),so that the vibrational excitation points toward the reactive atom. In contrast, the fifth
excited-state wavefunction, which corresponds asymptotically to the antisymmetricstretch (5.3b), rotates to localize primarily along the distal OH bond (r2) (5.3d).
177

Figure 5.4a-c contains plots of the adiabats in the interaction region for thesethree systems, with the corresponding asymptotic states labeled (although not shown
here, all curves are calculated to R = 7.94 Å for use in the dynamics calculations).
Avoided crossings are numbered, with the characteristic minimum velocity for eachcrossing (calculated from Eqn. 5.18) given in Table 5.3.

For convenience, theclassical turning points for the low (designated plo, momentum = 10 au, vrel ~ 1×105cm/s) and high (phi = 20 au, vrel ~ 2×105 cm/s) momenta used in subsequent dynamics
calculations are also designated. First, we focus on the general shape and trends inthe curves, as well as discussing the locations of the turning points for the lowest
momentum relative to the interaction region.
For Cl + H2O (Fig. 5.4a), there are numerous avoided crossings between
states; thus, it is not surprising that the character of the adiabatic wavefunctions
changes dramatically. Notably, since the symmetric stretch on this surface is ~200
cm-1 lower in energy than the antisymmetric stretch, it is correspondingly closer inenergy to the HCl + OH product states in the v=1 manifold.

In addition, thesymmetric stretch also becomes even closer in energy to the ground state withdecreasing R, resulting in two narrowly avoided crossings between s=1 and OH(v=1)+ HCl(v=0) in the region of R = 3.0 and 2.7 Å; this behavior is not exhibited by theantisymmetric stretch adiabat.
By examining the avoided crossings in more detail, we can predict theaccessibility of each transition based on its characteristic velocity and distance fromthe classical turning point of the central wavepacket velocity. Nearly all of the
avoided crossings for Cl + H2O have lower characteristic crossing velocities than178

Energy (cm-1)-62000a)-64000phi-66000

6 phi

5
4

plo

3

2

1

-68000HCl(v=2)+OH
Cl+HOH(a=1)
Cl+HOH(s=1)
HCl+OH(v=1)
HCl(v=1)+OH

plo-70000Cl+HOH
HCl+OH-72000

Energy (cm-1)-62000b)-64000-66000

5p

6

hi

2

-68000

4 3

phi

plo

1

plo

-70000

Cl+DOD
DCl+OD-72000-62000

Energy (cm-1)

6-64000

c)

5

7

phi

-66000-68000

4 phi

2

plo

1

3

plo

-70000

mixed OH/HCl v=2
Cl+HOD (vOH=1)
Cl+HOD (vOD=1)}mixed OH/HCl v=1
Cl+HOD
HCl+OD-720002.0

DCl(v=2)+OD
Cl+DOD(a=1)
Cl+DOD(s=1)
DCl+OD(v=1)
DCl(v=1)+OD

2.5

3.0

3.5

R (Å)

Figure 5.4. Adiabatic potential energy curves for a) Cl + HOH, b) Cl + DOD, and c)
Cl + HOD near the interaction region. Each number designates an avoided crossing
between two curves, with the characteristic velocities (calculated from Eqn. 5.16)
given in Table 5.3. The asymptotic eigenstates to which each curve corresponds areshown on the right side of the figure. For convenience, the arrows designate classicalturning points for the low (designated plo, momentum = 10 au, vrel ~ 1×105 cm/s) andhigh (phi = 20 au, vrel ~ 2×105 cm/s) momenta used in subsequent dynamics
calculations on the fourth and fifth excited states.
179

Table 5.3. vnm, curve-crossing velocities from Eqn. 5.16, in cm/s (see Fig. 5.4a-c fordesignation of crossing points).
Crossing

Cl + HOH

Cl + DOD

Cl + HOD

1

<50

100

42,000

2

800

2300

16,000

3

>1E6

9000

180,000

4>1E6

18,000

2000

5

3200

5600

6000

6

1000

2500

22,0007

15,000the asymptotic value of the central velocity (see Table 5.3). However, the classicalturning points for the p = 10 au wavepacket occur before the primary interaction
region, thus only the leading fringes of the wavepacket in momentum space will reachthose crossings. The strongest coupling links s=1 to OH(v=1) + HCl(v=0), with thefirst crossing between these states occurring just outside the reach of the central
wavepacket velocity; from this, we would expect some possibility of reaction froms=1. In contrast, the only apparent reactive path for the a=1 state requires twotransitions, first crossing near R = 2.6 Å into the symmetric stretch, then crossing
through one (not both) of the avoided crossings between this state and OH(v=1). In
light of these differences, the symmetric stretch should be the more reactive mode at
low collision velocities because it ultimately places the vibrational excitation along
the reaction coordinate and thus couples into product states.

180

For the Cl + D2O adiabats (Fig. 5.4b), a picture emerges with subtle yet
important differences. While the adiabatic potential curves have the same generalshapes as those of Cl + H2O, the curve-crossing velocities are generally 2-3 times
higher (see Table 5.3).

These velocities reflects the larger energy separations

between neighboring states in the same stretch manifold (e.g., the asymptotic energy
difference for s=1 and a=1 is 287 cm-1 versus 204 cm-1 for H2O), and slightly smaller
coupling terms (i.e., the wavefunctions vary more slowly with R).

The major

exception to this observation is the coupling to the DCl(v=2) state. The heavier massof D versus H lowers the DCl(v=2) stretch relative to the v=1 manifold so that it
approaches the antisymmetric stretch; consequently, the coupling velocities between
the antisymmetric stretch and DCl(v=2) are approximately two orders of magnitude
smaller than the equivalent velocities for H2O. For this reason, we might expect tosee an increase in the reactive scattering relative to inelastic scattering from Cl + D2O
over Cl + H2O. In general, however, the velocity turning points on a=1 and s=1 in Cl+ D2O are further away from the interaction region, because 1) they occur at larger R,
and 2) the crossing points among curves occur at smaller R. Consequently, theoverall reactive and inelastic scattering for D2O should be less than H2O at the sameenergy.
For Cl + HOD (with the Cl approaching from the H side of the molecule), theadiabatic curves no longer resemble those of H2O/D2O and as such should produce
substantially different dynamics.

One important distinction between HOD andH2O/D2O is that the asymptotic product states in HOD do not correspond to localized
vibration in HCl or OD and are simply designated by their total vibrational quanta.
181

Although they cannot be treated as uncoupled product states, this designation allows
for distinguishing among elastic, inelastic, and reactive scattering. Of the two input
stretches (vOD and vOH), the vOD is lower in energy and thus closer to the productstates below. However, the approaching Cl atom lowers the energy of vOH (relative tothe ground state) because the path of the Cl lies along the OH bond. As a result, theclassical turning point for vOH (plo) is much further into the interaction region than theturning points for either stretch in H2O, and crossings with other states are moreaccessible. Correspondingly, we expect the total product in Cl + HOD to be larger
than in Cl + H2O for the low-momentum wavepacket.
For further comparison of Cl + HOD with Cl + H2O, we examine thewavefunctions at the approach to the interaction region (see Fig. 5.5a-b). First, at thesame R = 4.23 Å as examined for Cl + H2O (i.e., corresponding to the asymptotic
wavefunctions), the symmetric and antisymmetric stretches seen for Cl + H2O are
instead local-mode OH and OD stretches. While the symmetric stretch of H2O rotates
towards the Cl as it approaches, the OH stretch is localized towards the Cl axis for theentire approach of the reactive atom. However, the OH and OD stretches are notcompletely uncoupled. The adiabats demonstrate a very strong avoided crossing
between these two states at R = 2.8 Å, indicating that the nature of the adiabatic
wavefunctions changes rapidly with R. Accordingly, at that Cl-HOD distance, wehave examined the fourth and fifth excited state wavefunctions (Fig. 5.5c-d), whichcorrespond asymptotically to vOD = 1 and vOH = 1, respectively. Intriguingly, thecharacter of the wavefunctions nearly reverses at this point: while both haveprobability in the product well, the fourth state resembles vOH = 1 in the reactant well,
182

Figure 5.5. Fourth and fifth excited-state vibrational eigenfunctions for Cl + HOD
outside the interaction region (R = 4.23 Å, Fig. 5.5a and b, respectively) and near avery strongly avoided crossing (R = 2.8 Å, Fig. 5.5c and d, respectively). The
asymptotic eigenfunctions correspond to local-mode OH (Fig. 5.3a) and OD (Fig.
5.5b) stretches. However, at the strongly avoided crossing, the character of the
wavefunctions nearly reverses: in the reactant well, the fourth state (Fig. 5.5c)
resembles vOH = 1, while the fifth state (Fig. 5.5d) resembles vOD = 1.

183

while the fifth state resembles vOD = 1. Correspondingly, we might expect sometransfer between those states, with greater transfer from vOH = 1 to vOD = 1 becausevOH is higher in energy.

5.3.2. Cl + H2O/D2O/HOD wavepacket propagation
Having calculated the vibrationally adiabatic potential energy curves and thecoupling elements between those curves, we probe the dynamics by propagating
wavepackets on each incoming state of interest.

As detailed in Section 5.2.3,

Gaussian wavepackets at relative velocities of approximately 10,000, 15,000, and20,000 cm/s are employed to explore the dynamics and branching ratios amongelastic, inelastic, and reactive scattering. This analysis focuses first on Cl + H2O
dynamics, primarily at the lowest collision energy, then on additional energies andcollision systems.
We first examine a sample wavepacket propagation into the Cl + H2O
interaction region, monitoring transfer into other channels. Figure 5.6a follows thecourse of a wavepacket initiated on the Cl + H2O(a=1) state, with t = 0 at the bottomof the graph and each step higher indicating an additional 100-fs step.

Most

noticeably, the majority of the wavepacket probability remains on this curve for theduration of the interaction region, disappearing only when it is absorbed at the end of
the grid by the damping operator. Thus, the dominant output channel is elastic
collision between Cl and the antisymmetric stretch of H2O with no effect on thevibrational eigenstate.

184

a)

b)

c)

Figure 5.6. Sample wavepacket propagation for Cl + H2O(a=1), monitoring thescattering into a) Cl + HOH(a=1), b) Cl + HOH(s=1), and c) HCl(v=0) + OH(v=1).
The initial wavepacket is given at the bottom of the graph; each incremental stephigher indicates an additional 100-fs step.
For visual clarity, the wavepackets in
5.6b and c are increased in amplitude by factors of 50 and 250, respectively, relative
to 5.6a. Most of the wavepacket probability remains in the elastic Cl + HOH(a=1)
state; however, this sufficiently couples with other states to transfer probability into
inelastic (5.6b) and reactive (5.6b) scattering.

185

While this vibrationally adiabatic picture describes the majority of the reaction
probability for Cl + H2O(a=1) at this low collision energy, the wavepacket
experiences some coupling with other states, with a small amount of transfer to othercurves. For example, Fig. 5.6b demonstrates the growth of probability on the Cl +H2O(s=1) state, with the magnitude increased by 50 times relative to the elasticallyscattered wavepacket for visual clarity.

This figure illustrates that as the initial

wavepacket reaches the potential well, the state that asymptotically correlates to theantisymmetric stretch sufficiently couples to the asymptotic symmetric stretch state toproduce a redistribution of vibrational energy. Similarly, Fig. 5.6c shows the transfer
of probability to the reactive HCl(v=0) + OH(v=1) state, magnified by a factor of250. Interestingly, at this energy, vibrational redistribution (Fig. 5.6b) is more likely
to occur than chemical reaction (Fig 5.6c). While these pictures do not provide
quantitative information about the product distributions, they do provide a firstglimpse into the dynamics of this system.
In order to provide quantitative determinations of probability in each exit
channel, the fluxes as a function of time are monitored as described in Sect. 5.2,
terminating when the remaining wavepacket probability is less than 0.01. Fig. 5.7a-b
gives the accumulated fluxes versus time for the p = 10 au calculations of Cl +H2O(s=1) and Cl + H2O(a=1), respectively. Both figures demonstrate that these are
predominantly direct reactions, i.e., without long-lived resonance states; the nearexponential rise in flux likely results from the spread in the Gaussian wavepacket.
Notably, this direct behavior is similar among all input states and momenta calculated
for each of the water isotopes. In addition, for both states of Cl + H2O at this input
186

OUTPUT FLUX

Accumulated Flux

0.05

a)

Cl + HOH(s=1)[input channel]

0.04
HCl + OH(v=1)

0.03

HCl + OH
HCl(v=1) + OH

0.02

Cl + HOH(a=1)

0.01

Cl + HOH
HCl(v=2) + OH

0.00

Accumulated Flux

0.05

b)

Cl + HOH(a=1)[input channel]

Cl + HOH(s=1)

0.04
0.03
0.02
0.01
0.00
500

HCl + OH(v=1)
HCl(v=1) + OH

1000

1500

2000

Time (fs)Figure 5.7. Accumulated fluxes versus time for the scattering from a) Cl + H2O(s=1)
and b) Cl + H2O(a=1) for p = 10 au (510 cm-1 collision energy, and 101,000 cm/s
relative Cl-H2O speeds). These plots indicate that at these energies, the interactions
are predominantly elastic (i.e., >90% probability of exiting on the entrance channel).
Of the remaining channels, reaction dominates over inelastic scattering from
H2O(s=1) to H2O(a=1) (Fig. 5.7a); in contrast, inelastic scattering from H2O(a=1) toH2O(s=1) dominates over reaction (Fig. 5.7b).
187

momentum, the elastic channel dominates, with a total probability near 0.95 of
exiting on the input channel. Among those channels which alter Cl + H2O state,
reactive channels dominate over inelastic scattering from H2O(s=1) to H2O(a=1) (Fig.
5.7a); in contrast, inelastic scattering from H2O(a=1) to H2O(s=1) dominates overreactive scattering (Fig. 5.7b). Thus, as predicted from the adiabatic potentials, thesymmetric stretch is more reactive than the antisymmetric stretch of H2O.
Monitoring the accumulated flux out of each channel in this fashion allows acomparison of the products of the Cl + H2O reaction with the other collision systems
studied, namely, Cl + D2O and Cl + HOD (each calculated for p ~ 10 au). The flux
out of each channel is summed over reactive and inelastic product states and plottedin Fig. 5.8. Overall, these data support the predictions made from the adiabatic
curves, as discussed in Section 5.3.2. First, as predicted from the rotation of thesymmetric stretch wavefunction toward the Cl atom, the symmetric stretches of both
H2O and D2O are more reactive than the antisymmetric stretches. In addition, theoverall amount of inelastic and reactive scattering from H2O is larger than from D2O,
consistent with the larger coupling velocities between states of D2O and the greater
distance of the classical turning points from the interaction region. Finally, the OH
vibrational excitation of HOD is the most reactive of all states of H2O/D2O/HOD and,indeed, is more reactive than vOD by nearly an order of magnitude, consistent with alocal-mode picture of this reaction.
However, two more subtle comparisons of reactive and inelastic flux warrant
additional investigation. First, despite the apparent similarity in adiabatic potential
curves between H2O and D2O, the antisymmetric stretch states of these two molecules
188

TOTAL REACTIVE AND INELASTIC FLUX
p~10au0.14

Accumulated Flux

0.12

Inelastic
Reactive

0.10
0.08
0.06
0.04

1)

D
(v

O

H=

1)

O

H
O

D
(v

H
O
l+

l+

C

C

l+
C

D=

)
D
(s
=1

D
O

D
(a
=1

)
C

l+

D
O

H
(s
=1
H
O

l+
C

C

l+

H
O

H
(a
=1

)

0.00

)

0.02

Input Channels

Figure 5.8. Accumulated flux for p ~ 10 au wavepackets into non-elastic channels
for Cl + HOH, Cl + DOD, and Cl + HOD, summed separately over inelastic andreactive channels. The symmetric stretches of both H2O and D2O are more reactive
than the antisymmetric stretches, with overall amount non-elastic scattering from H2O
greater than from D2O, while the OH vibrational excitation of HOD is the mostreactive state.

189

produce opposite relative amounts of inelastic and reactive products. Second, despite
the excitation of the “wrong” bond in HOD, reactive scattering from vOD = 1 issignificantly larger than inelastic scattering. This is somewhat surprising, in part
because the curve-crossing velocity with Cl + HOD(vOH=1) given in Table 5.3 issmaller than the velocity for coupling with the highest mixed v=1 wavefunction, thus
one might expect that inelastic channel to dominate.
Fortunately, time-dependent wavepacket propagation allows for monitoringof probability in each channel as a function of time, i.e., for directly observing
transfers between states. Accordingly, we examine the probability in each channel asa function of time for several initial states, plotted in Fig. 5.9a-b and 5.10a-b.
Importantly, the diminution of the total probability at the end of the reactive
encounter results from application of the damping operator near the end of the grid.
For Cl + H2O(s=1) and (a=1), comparisons of the dynamics can be madebetween Fig. 5.9a-b. For the input channel Cl + H2O(s=1) (Fig. 5.9a), three major
pathways (i.e., HCl(v=0) + OH(v=1), HCl(v=1) + OH(v=0), and Cl + H2O(a=1))
remove probability nearly simultaneously from that state, indicating that probability
transfers directly to each of the dominant product states. In comparison, for the initialstate Cl + H2O(a=1) (Fig. 5.9a), probability first transfers from a=1 to s=1, as
evidenced by the growth of probability in s=1. This transfer must occur as the tail of
the wavepacket reaches the avoided crossing between these two states (designated as
avoided crossing 5 on the adiabatic potential curves in Fig. 5.3a). As the probability
in s=1 subsequently drops near 500 fs, the probability in HCl(v=0) grows, indicating

190

OUTPUT PROBABILITY
0.08

HCl + OH(v=1)a)

Probability

0.07
Cl + HOH(s=1)[input channel]

0.06
0.05
0.04
HCl(v=1) + OH

0.03
Cl + HOH(a=1)

0.02
HCl + OH
Cl + HOH
HCl(v=2) + OH

0.01
0.00
0.08

Cl + HOH(a=1)[input channel]

Probability

0.07

b)

0.06
Cl + HOH(s=1)

0.05
0.04
0.03
0.02
0.01

HCl + OH(v=1)
HCl(v=1) + OH

0.00
0

500

1000

1500

Time (fs)Figure 5.9. Wavepacket probability in each channel as a function of time for a) Cl +HOH(s=1) and b) Cl + HOH(a=1) at p ~ 10 au. At the end of the timescale, theprobability in each channel approaches zero due to damping of the wavefunction
outside the interaction region. For Cl + HOH(s=1) (Fig. 5.9a), probability transfers
directly into the asymmetric stretch and two HCl/OH product states, while for Cl +H2O(a=1) (Fig. 5.9b) probability first transfers into the symmetric stretch, then toHCl(v=0).

191

that the path from the input antisymmetric stretch state to reaction must pass throughthe symmetric stretch.
As predicted in Sect. 5.3.2, the dynamics of Cl + D2O(a=1) are altered fromthose of Cl + H2O(a=1) by the presence of a low-lying state from the v=2 manifold.
Fig. 5.10a demonstrates that as with H2O(a=1), some probability transfers from a=1
to s=1 and subsequently into the DCl + OD v=1 product states. However, moreprobability flows directly from D2O(a=1) into DCl(v=2) via avoided crossings 3 and4 shown on Fig. 5.3b. This behavior indicates that the lowering of the v=2 manifold
relative to v=1 by isotopic substitution effectively opens an additional productchannel unavailable to H2O.
For Cl + HOD, the probabilities as a function of time from the vOD input
channel are given in Fig. 5.10b. This probability plot shows amplitude oscillating outof vOH=1 as the wavepacket passes through the couplings at crossings 1 and 2 (Fig.
5.3c). Since the probability for the v=1 reactive product grows in between those two
peaks, it must result from the coupling at crossing 3. This result is intriguing, sincethat coupling velocity is quite larger than most of the other coupling velocities yet
clearly contributes to the formation of the product channels.
The last part of this analysis is to examine how a greater collision momentum
affects the probability distribution in each product channel, focusing on Cl + H2O.
Figure 5.11a-b plots the output fluxes from Cl + H2O(s=1) and Cl + H2O(a=1),
respectively, as a function of increasing momentum. Three major points can begleaned from a comparison of these two figures. First of all, not surprisingly, thetotal amount of inelastic/reactive scattering increases with increasing momentum in
192

OUTPUT PROBABILITY
0.008

a)

Probability

DCl(v=2) + OD
Cl + DOD(a=1)[input channel]

0.006

0.004
DCl(v=1) + OD

Cl + DOD(s=1)

0.002

DCl + OD(v=1)
DCl + OD
Cl + DOD

Probability

0.000
0.008

b)
Cl + HOD(vOD=1)[input channel]

0.006

mixed v=1 product (upper)

0.004
Cl+HOD(vOH=1)

0.002

mixed v=1 product (lower)

0.000
0

500

1000

1500

Time (fs)Figure 5.10. Wavepacket probability in each channel as a function of time for a) Cl+ DOD(a=1) and b) Cl + HOD(vOD=1), damped to zero after leaving the interaction
region. For D2O (5.10a), as with H2O(a=1), some probability transfers from a=1 to
s=1 to products; however, more probability flows directly from D2O(a=1) intoDCl(v=2) due the lowering of the v=2 manifold relative to v=1 by isotopic
substitution. For Cl + HOD (5.10b), amplitude leaves vOH=1 and transfers to the v=1
reactive product as the wavepacket passes through the avoided crossings 1 and 2.

193

OUTPUT FLUX
1.0

Accumulated Fluxa) Cl + HOH(s=1)
0.8
0.6
0.4

p=10aup=15aup=20au0.2

Accumulated Flux

0.0
1.0

b) Cl + HOH(a=1)
0.8
0.6
0.4
0.2

H
C
l+
O
H
C
l+
H
O
H
C
H
l(v
=1
)+
O
H
C
H
l+
O
H
(v
C
=1
l+
)
H
O
H
(s
=1
C
l+
)
H
O
H
(a
H
=1
C
)
l(v
=2
)+
O
H

0.0

Output Channels
Figure 5.11. Momentum dependence of output fluxes in each channel from a) Cl +H2O(s=1) and b) Cl + H2O(a=1). In both cases, total inelastic and reactive scattering
increase with increasing momentum; for H2O(s=1), inelastic scattering grows larger
than reactive, while for Cl + H2O(a=1), reactive grows larger than inelastic. As aresult, at high collision momenta, excitation of the antisymmetric stretch forms more
HCl + OH product than does the symmetric stretch.

194both cases. This increase reflects the greater number of avoided crossings accessedby wavepackets at the higher momentum, as evidenced by the location of the classicalturning points further into the interaction region (denoted phi in Fig. 5.4a).
Second, the relative amounts of inelastic and reactive scattering reverse
between low and high momentum: for scattering from s=1, inelastic grows larger thanreactive, while for scattering from a=1, reactive grows larger than inelastic. The
adiabatic curves for this system (Fig. 5.4a) provide an explanation for thisphenomenon.

As mentioned previously, as R decreases, the s=1 adiabatic state

begins to resemble a local-mode OH stretch directed toward the Cl atom, while thea=1 adiabat becomes a local-mode stretch away from the Cl. Correspondingly, thetransition between these two adiabats represents a transition between local-mode
stretches. For high-momentum scattering from s=1 and a=1, the classical turning
point lies directly on the avoided crossing between these two states (avoided crossing5); accordingly, a larger amount of wavepacket probability transfers than at low
momentum. For input on Cl + H2O(s=1) (Fig. 5.11a), this diabatic transfer to the
local-mode stretch away from the Cl leaves probability “stuck” in the a=1 exit
channel, which has no subsequent crossings into reactive states as the wavepacket
exits the interaction region. For input on Cl + H2O(a=1) (Fig. 5.11b), the opposite
occurs: more probability transfers into the local-mode stretch toward the Cl and cansubsequently transfer into the HCl(v=0) + OH(v=1) product state.

The higher

momentum also produces an increased probability of this transfer to the reactive
channel such that probability in the product channel grows.

195Third, as a result of these high-momentum inversions in inelastic versus
reactive channels for both input states, the relative amounts of reaction product fromthe two input channels also inverts: excitation of the antisymmetric stretch now forms
more HCl + OH product than does excitation of the symmetric stretch. In otherwords, the antisymmetric stretch becomes more reactive than the symmetric stretchwith increased collision velocity. The implications of this result are discussed in thenext section.

5.4 Discussion
The preceding section outlines the key results of the Cl + H2O/D2O/HOD
reactive scattering calculations, exploring in detail how the adiabatic curves generated
for this prototypical set of reaction systems can be used to predict the subsequent
dynamics. Our final step is to reexamine the specifics of this model system todetermine what insights can be formed that are independent from the details of thepotential surface and the collinear water geometry.
Perhaps the most intriguing outcome of these calculations is the rotation of thesymmetric-stretch eigenstate of H2O towards the approaching Cl atom and thecorresponding rotation of the antisymmetric stretch away from the Cl. This result
indicates that, adiabatically, the antisymmetric stretch correlates with vibrational
excitation in the non-reacting bond.

This concept is not new; indeed, Schatz 35

postulated that this correlation should generally exist in systems with symmetric andantisymmetric stretches for the following reason. The local-mode, reactive bondstretch is lowered in energy relative to the unreactive bond by the approaching196reactant; it thus adiabatically couples to the lower-energy symmetric stretch, thereby
increasing the reactivity of the symmetric stretch relative to the antisymmetric stretch.
This principle was demonstrated by Schatz with classical trajectory
calculations for O + CS2: vibrational adiabaticity in this reaction yields more OS +CS product from the CS2 symmetric stretch than the antisymmetric stretch.35 This
result is consistent with our results for Cl + linear H2O: at the lowest translationalenergy calculated, the symmetric stretch of H2O is more reactive. It is intriguing that
this effect appears in both O + CS2, in which O-S-C is a light-heavy-light atom
combination, and in Cl + H2O, with a Cl-H-O as a light-heavy-light combination, in
calculations at comparable collision energies. Since the vibrational energies of thesymmetric and antisymmetric stretch in CS2 are much more widely separated than in
H2O, the adiabaticity in that system should persist to higher collision velocities. In
contrast, the adiabaticity of our model H2O breaks down at a collision velocity only
three times higher than the average thermal velocity, indicative of a transition todiabatic behavior. Notably, the spacing between the experimental symmetric andantisymmetric stretch energies in water is approximately half that spacing in ourmodel.36 Correspondingly, we anticipate that the vibrational adiabaticity of water
should break down at even lower, i.e., near-thermal, collision velocities.
A related result, observed in all three collision systems, is that the local-mode
stretch in the reactive bond is lowered in energy with respect to the ground state as
the reactive atom approaches.

In H2O and D2O, this lowers the energy of theproximal OH stretch (which corresponds asymptotically to the symmetric stretch)toward the reactive channels. In Cl + HOD, however, this lowers the higher-energy
197

OH stretch toward the OD stretch, producing a strong mixing between vOH and vOD
character in the HOD well region. While the magnitude of this mixing will vary for
different systems, the trend should be the same for any high-frequency vibration
approached by a reactive atom.
Finally, the lowering of the next excited vibrational manifold (v=2) in D2O
relative to D2O(v=1) opens up a higher-energy product channel for the antisymmetricstretch of D2O, one which is unavailable to H2O. This emphasizes that isotopic
substitution affects not only the vibrations within a single manifold, but also thepositioning of those vibrational manifolds with respect to one another.

5.5 Conclusions
The role of intramolecular vibrational redistribution (IVR) on the timescale ofa reactive encounter is explored via time-dependent quantum reactive scattering on alinear model system with isotopic variations.

In these studies, the vibrational

eigenstates of Cl + H2O → HCl + OH are calculated as a function of Cl-H2O centerof-mass separation using a Chebychev expansion of the time-dependent propagation
operator in imaginary time. This calculation generates adiabatic potential energy
curves, as well as the coupling matrix elements for transitions between these curves,for use in time-dependent dynamics calculations. For the vibrational eigenfunctions,the near-resonance of the H2O symmetric and antisymmetric stretch causes thesestates to couple strongly as the Cl atom approaches. These couplings rotate thesymmetric stretch of H2O into a local-mode excitation pointing towards the Cl, andthe antisymmetric stretch away from the Cl.
198

Scattering calculations from thesymmetric and antisymmetric stretch thus indicate greater probability for productformation from the symmetric stretch than the antisymmetric stretch at low collisionvelocities. This vibrational adiabaticity breaks down at higher velocities, so that theantisymmetric stretch becomes more reactive than the symmetric stretch. For thesimilar Cl + D2O simulation, the reduced stretching vibrational frequencies provide
less energy towards reaction, hence the reaction probability decreases for both modes,
while the lowering of the DCl(v=2) product state relative to the antisymmetric stretch
states increases reactive scattering over inelastic scattering from the symmetricstretch. For Cl + HOD, vibrational excitation in OH enhances HCl + OD reaction
probability by an order of magnitude more than excitation in OD.

199

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