Product quantum-state-dependent anisotropies in photoinitiated unimolecular decomposition

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JOURNAL OF CHEMICAL PHYSICS

VOLUME 111, NUMBER 16

22 OCTOBER 1999

Product quantum-state-dependent anisotropies in photoinitiated

unimolecular decomposition

A. V. Demyanenko, V. Dribinski, and H. Reisler^a⁾

Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482

H. Meyer

Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602-2451

C. X. W. Qian

Department of Chemistry, University of Victoria, P.O. Box 3065, Victoria,

British Columbia, Canada V8W 3V6

͑

Received 8 June 1999; accepted 22 July 1999͒

Angular distributions of state-selected NO and O products in the photoinitiated unimolecular

decomposition of jet-cooled NO have been measured by using both the photofragment ion imaging

2

technique with velocity map imaging and ion time-of-ﬂight translational spectroscopy. The recoil

anisotropy parameter of the photofragments, ␤, depends strongly on the rotational angular

3

momentum of the photoproduct. O( P_j_ϭ₂_,₀) angular distributions are recorded at photolysis

wavelengths 371.7, 354.7, and 338.9 nm. At these wavelengths, respectively, vibrational levels v

ϭ0, vϭ0,1 and vϭ0–2 of NO are generated. In addition, ␤ values for NO(vϭ2) in speciﬁc high

rotational levels are determined at ϳ338 nm. The experimental observations are rationalized with a

classical model that takes into account the transverse recoil component mandated by angular

momentum conservation. The model is general and applicable in cases where fragment angular

momentum is large, i.e., a classical treatment is justiﬁed. It is applied here both to the experimental

NO results, and results of quantum calculations of the vibrational predissociation of the Ne–ICl van

2

der Waals complex. It is concluded that deviations from the limiting ␤ values should be prominent

in fast, barrierless unimolecular decomposition, and in certain dissociation processes where a large

fraction of the available energy is deposited in rotational excitation of the diatom. The application

of the model to NO dissociation suggests that the nuclear dynamics leading to dissociation involves

2

a decrease in bending angle at short internuclear separations followed by a stretching motion. This

Physics. ͓S0021-9606͑99͒00239-1͔

I. INTRODUCTION

the laser polarization vector E, and P (x) is the second Leg-

2

endre polynomial. In the limit of fast dissociation and axial

recoil, ␤ϭ2 and Ϫ1 are obtained when the transition dipole

moment is parallel or perpendicular, respectively, to the frag-

ment velocity vector. Both classical and quantum mechanical

models have been developed to describe ␤.¹

In most of the systems for which ␤ has been measured,

photofragmentation has been accompanied by signiﬁcant

translational energy release. The work described here is dif-

ferent in that the fragments possess high internal excitation

accompanied by small kinetic energy release, a situation

characteristic of unimolecular decomposition. We report

studies of the photoinitiated unimolecular decomposition of

Measurements of product angular distributions in photo-

dissociation dynamics can provide detailed information on

the symmetry of the dissociative electronic transition, the

subsequent dynamics of product evolution, and the time

–6

1

scale of the dissociation. Since the pioneering work of Zare

2

͑a͒

and Herschbach, these measurements have become a ver-

satile, albeit complex, tool in the arsenal of experimental

techniques of chemical physics. The angular distributions re-

ﬂect several well-known vector correlations, among which

the most common is the correlation between the electronic

transition dipole moment ␮ and the fragment recoil velocity

␧

v (

͗

␮ •V

͘

). The

͗

␮ •V

͘

correlation is characterized in

␧

NO which demonstrate that as a result of a signiﬁcant trans-

2

terms of the recoil anisotropy parameter, ␤, and is deter-

mined in the space ﬁxed frame ͑SFF͒ of coordinates. The

product angular distribution is described by the standard re-

coil anisotropy function,¹

verse recoil component, ␤ depends sensitively on the rota-

tional state of the NO fragment, and to a lesser degree on NO

vibration. The larger the rotational excitation, the larger the

transverse recoil component that accompanies the fragmen-

tation.

,2

1

P͑␪͒ϭ

͓1ϩ␤•P ͑cos ␪͔͒,

͑1͒

2

It is well established both experimentally and theoreti-

cally that molecular rotation prior to dissociation can affect

4

␲

where ␪ is the angle between the recoil velocity vector and

7,8

the observed value of ␤. It has also been pointed out that

parent vibrational motion prior to, and nuclear motions dur-

ing the dissociation can inﬂuence ␤ under conditions which

^a^͒Electronic mail: reisler@chem1.usc.edu

0021-9606/99/111(16)/7383/14/$15.00

7383

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1

7384

J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Demyanenko et al.

are related to the speciﬁc dissociation dynamics.⁹

ergy, E_t.²⁸In addition, an effect due to the spin-orbit state

Product state-dependent ␤ in photodissociation has been

reported before. For example, Mordaunt et al. developed a

pseudotriatomic model for the 193-nm photodissociation of

was observed: the angular distribution of the highest spin-

3

orbit state, P , was characterized by a lower ␤ at all trans-

0

lational energies, compared to the other spin-orbit states. The

3

NH that explained the observed changes in ␤ as a function

energies of the spin-orbit states are: P : P : P

3

2 1 0

1

0

Ϫ1

of the N quantum number of NH₂. Their classical model,

based on angular momentum conservation and impulsive dis-

sociation, assumed that dissociation commenced at the point

of the conical intersection, and involved out-of-plane

torques. Kable et al. used a model involving Renner–Teller

interaction to interpret the observed ␤ changes in the photo-

dissociation of HCO.¹¹Beswick and co-workers, in a quan-

tum mechanical calculation of the photoinitiated predissocia-

tion of Ne–ICl and Ne–Cl₂complexes, demonstrated a

substantial change of ␤ as a function of the rotational quan-

ϭ0:158:226 cm . Huber and co-workers, who carried out

TOF measurements of oxygen ͑without state selection͒ at

355 and 351 nm, also obtained Et-dependent ␤ parameters.²⁹

To date, no model has been offered to explain these obser-

vations.

3

In this paper, we report studies of O( Pjϭ2,0) angular

anisotropy as a function of fragment recoil velocity carried

out at photolysis wavelengths 371.7, 354.7, and 338.9 nm.

These wavelengths correspond to energies in excess of D0 of

Ϫ1

1776, 3065, and 4380 cm where, respectively, vϭ0, v

4

tum number of the diatomic fragment. In a classical trajec-

ϭ0,1, and vϭ0–2 vibrational levels of NO are open. In

addition, ␤ values for NO(vϭ2) associated with high rota-

tory calculation, Loock et al. predicted that a triatomic par-

ent bending motion will inﬂuence the fragment angular

distribution in a way that depends on the rotational state of

tional levels J ͑low E ) were determined at ϳ338 nm.

t

The experimental observations are rationalized with a

classical model that takes into account ͑i͒ transverse recoil;

9

the diatomic fragment. A dependence of ␤ on fragment

͑ii͒ the evolving nuclear dynamics. The model is general and

speed was observed in ozone photodissociation, and on N

2

1

2

applicable in cases where fragment angular momentum is

large ͑i.e., a classical treatment is justiﬁed͒, but can also be

related to the quantum mechanical treatment. It is applied

rotational state—in N O photodissociation. The depen-

2

dence of ␤ on the NH rotational level in HN photolysis has

3

been interpreted as resulting from out-of-plane motions dur-

ing the dissociation.¹³

here both to the experimental NO results, and to results of

2

quantum calculations on the vibrational predissociation of

the Ne–ICl van der Waals complex. We note that in contrast

to direct photodissociation, where product state distributions

provide clear signatures of exit-channel dynamics, the statis-

The photoinitiated unimolecular decomposition of NO₂

has been investigated extensively, and serves as a benchmark

for statistical models of unimolecular reactions.¹⁴This paper

is concerned with the angular distributions of the photofrag-

ments and their dependence on the fragment quantum state,

and a model is proposed to explain the observed dependen-

cies. Dissociation takes place predominantly on the ground

potential energy surface ͑PES͒, following excitation to the

tical NO state distributions in NO decomposition are not

2

sensitive to the shape of the PES. A dependence of ␤ on the

internal energy of the diatomic product can then serve as a

useful probe of the dynamics.

2

The paper is organized as follows. The experimental ob-

mixed 1 B /1 A state via a parallel transition. Strong vi-

2

1

servations of the fragment angular distributions in NO dis-

bronic coupling between the states results from a conical

2

14–16

sociation are described ﬁrst. Two techniques are used: pho-

tofragment ion imaging and ion TOF translational

spectroscopy. A simple model based on conservation laws is

then presented, which gives a physical interpretation of the

experimental observations. This model is also applied to the

vibrational predissociation of the Ne–ICl complex, and ␤ vs

J_I_C_lis compared to the corresponding results obtained by

intersection located near the bottom of the B₂state.

The dissociation rate of jet-cooled NO is fast, and the prod-

2

_u_c_t_s_e_x_h_i_b_i_t_s_p_a_t_i_a_l_a_n_i_s_o_t_r_o_p_y_.1

7,18

The NO state distribu-

tions, despite severe ﬂuctuations, can be described on aver-

age using statistical theories, implying dissociation on a

1

4,19,20

21,22

barrierless PES.

cal calculations

Rate measurements

and theoreti-

1

5,16,23,24

have led to the same conclusion.

4

quantum mechanical calculations. We end this paper with a

Fragment angular distributions in NO photodissociation

2

discussion of the implications of the ␤ variation to NO dis-

2

were determined ﬁrst by Busch and Wilson, who used 370-K

_s_a_m_p_l_e_s_a_n_d₃₄₇_-_n_m_p_h_o_t_o_l_y_s_i_s_.₁8 Their results of the aver-

sociation mechanism, and an evaluation of other dissociative

systems where a similar ␤ variation is expected.

_a_g_e_v_a_l_u_e_o_f_␤_w_e_r_e_c_o_n_ﬁ_r_m_e_d_b_y_o_t_h_e_r_i_n_v_e_s_t_i_g_a_t_o_r_s_.25,26

Subsequent studies with jet-cooled samples ͑Ͻ5 K͒ demon-

strated the effect of parent rotation on decreasing the magni-

tude of ␤, and concluded that the dissociation rate of jet-

II. EXPERIMENTAL METHODS

A. Experimental arrangement

cooled NO is faster than the characteristic time of parent

2

1

7

rotation.

A puzzling dependence of ␤ on the fragment rotational

The experiments at the University of Georgia and the

University of Southern California ͑USC͒ both use molecular

state was ﬁrst reported by Butenhoff and Rohlﬁng,²⁷who

Ϫ1

found that near dissociation threshold (D ϭ25 130 cm ),

beams of NO seeded in either Ne or He/Ne, and resonance

0

2

the highest NO rotational levels exhibited markedly dimin-

ished anisotropy. More recently, Liu and co-workers re-

ported time-of-ﬂight ͑TOF͒ studies of the O( P_j_ϭ₂_,₁_,₀) pho-

tofragments at 355 nm that showed a gradual decrease in ␤

with decreasing total center of mass ͑c.m.͒ translational en-

enhanced multiphoton ionization ͑REMPI͒ for product detec-

tion. NO is detected in a one-color experiment, while O at-

oms are detected in experiments where the pump and probe

frequencies are different.

3

Details of the Georgia apparatus have been described

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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Photoinitiated unimolecular decomposition

7385

previously.³⁰Brieﬂy, the NO beam is generated by expand-

2

ing a mixture of 4% NO in Ne through a piezoelectric mo-

2

lecular beam source ͑60-␮s duration, 10 Hz͒ at a backing

pressure of 1.8 atm. The gas pulses pass through a skimmer

into the scattering chamber where they are intersected at

right angle by the photolysis/probe laser. The interaction

with the laser pulse takes place in an electric drift-ﬁeld ar-

rangement. Product ions are accelerated ͑10 V/cm͒ toward a

microchannel plate ͑MCP͒ detector mounted off-axis. The

velocity component of the neutral fragments is determined

from the arrival time of the ions at the detector. The laser

system is a Nd:YAG pumped dye laser ͑DCM dye͒, whose

frequency near 680 nm is doubled in a KDP crystal. The

resulting ultraviolet ͑UV͒ output ͑ϳ0.6 mJ͒ is focused with a

lens (fϭ500 mm) onto the molecular beam.

FIG. 1. (2ϩ1) REMPI spectrum of NO resulting from the photodissocia-

The ion-imaging arrangement has been described in de-

2

tion of NO₂in the region of the H ⌺,HЈ ⌸(vЈϭ0)� X ⌸(vЉϭ2) hot

band transition ͑top trace͒. The bottom trace represents a simulation assum-

ing a rotational temperature of 2000 K. See text for details.

3

1

tail elsewhere. In brief, it consists of an ion-acceleration

stage, a 60-cm-long drift tube, and a charge coupled device

͑CCD͒ camera that monitors a phosphor screen coupled to a

MCP ion detector. The ion optics of the ion-acceleration

molecular beam accounts for ions generated from residual

gases. ͑ii͒ Signal obtained at the temporal maximum of the

molecular beam pulse, but with the pump laser off accounts

for background generated by the probe alone. Both types of

stage is based on a scheme proposed by Parker et al. for

_v_e_l_o_c_i_t_y_m_a_p_i_m_a_g_i_n_g_,₃2 and consists of a repeller, an open

extractor electrode ͑50-mm hole͒, and an open ground elec-

trode ͑25-mm hole͒. At an optimal voltage ratio for the re-

peller and extractor plates, all ions with the same initial ve-

locity are focused onto the same spot on the MCP detector.

A doubly skimmed pulsed molecular beam containing

background give spatially isotropic signals. The second type

3

of contamination is substantial only when O( P ), whose

0

fractional population is low, is probed, since higher probe

energies are required for its detection ͓0.2–0.4 mJ, as com-

3

% NO and 5% O seeded in 1.8 atm of a He͑30%͒/

2 2

3

pared with 0.03–0.1 mJ for O( P )]. Good spatial and tem-

2

Ne͑70%͒ mixture propagates through a hole in the repeller

poral overlap of the two laser beams minimizes effects due to

the probe.

plate. The NO mixture is prepared by passing the carrier gas

2

mixture over NO kept at Ϫ29 °C ͑o-xylene slush͒. Further

2

downstream, NO (T р2 K) is photolyzed with pulsed, lin-

2

rot

2

B. Two-photon spectroscopy of the NOH ⌺,HЈ ⌸„vЈ

early polarized, and focused (fϭ250 mm) laser radiation

0.6–1.6 mJ͒ intersecting the molecular beam at right angle.

2

؍

0…—X ⌸„vЉ

؍

2… transition

͑

Photolysis at 338–372 nm is achieved by using the

frequency-doubled output of a Nd:YAG laser pumped dye

laser system. The polarization of the photolysis light is di-

rected perpendicular to the molecular beam, in a plane par-

allel to the detector plane.

In the photolysis wavelength range around 340 nm, vi-

brationally excited NO X ⌸(vϭ2) products are detected by

(2ϩ1) REMPI via the H ⌺,HЈ ⌸(vЈϭ0) state. The two-

photon spectroscopy of this Rydberg state has been dis-

cussed in detail previously.³³The one-color NO(vϭ2)

2

In the two-color ion-imaging experiments, the output of

REMPI spectrum resulting from NO photodissociation is

2

an excimer-laser pumped dye laser system ͑Coumarin 450͒ is

shown in the top part of Fig. 1. The spectrum in the bottom

part is calculated assuming a rotational temperature of 2000

K, truncated to exclude rotational states whose energies ex-

ceed the available energy. The positions of the various rota-

tional lines in the Q₁₁_dand Q₂₁_dbranches are displayed at

3

frequency doubled to probe the O( P , jϭ0,2) fragments by

j

3

2

ϩ1 REMPI via the 3p P� 2p P transitions at 225.68

and 226.28 nm, respectively. Due to the Doppler width of the

O( P ) fragments, the probe frequency is scanned over the

3

j

3

4

entire bandwidth to ensure that the images include all prod-

uct velocities. The dissociation and probe lasers are both

linearly polarized, and introduced coaxially at right angle to

the molecular beam. To minimize complications due to vec-

tor correlations, the polarizations of the two lasers are ﬁxed

parallel to each other. The time delay between the pump and

the top, following the notation of Huber and Miescher. The

X–H two-photon transition is dominated by a zeroth rank

tensor which gives rise to rotational Q-branch structures. In

previous work, a much smaller contribution due to a second-

rank tensor component was also identiﬁed.³³While in an

isotropic ensemble this contribution can usually be ne-

glected, a different situation arises for an ensemble of prod-

uct molecules, which is characterized by an anisotropic dis-

tribution of the rotational angular momentum. In this case,

the two-photon signal is sensitive to the quadrupole and

probe lasers is kept at ␶ ϭ0Ϯ5 ns. Typical signals include

d

4

summation of 1–2ϫ10 laser ﬁrings.

Experimental complications in the two-color experi-

ments arise from dissociation by the probe radiation, and

photolysis of residual gases. To eliminate this contamination,

a subtraction routine is employed, which takes into account

two types of background signals: ͑i͒ signal obtained with the

pump and probe lasers on, but delayed with respect to the

hexadecapole moments of the m distribution.

j

When the angular distribution of the photofragments is

resolved, Beswick et al. have shown that the state multipole

moments of the rotational angular momentum distribution

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7386

J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Demyanenko et al.

must be considered functions of the recoil direction.⁴^,³⁵With

linearly polarized photolysis light, the angular distributions

of the fragments are uniquely characterized by the recoil

angle ⌰ with respect to the pump laser polarization direction.

In this case, it can be shown that the (2ϩ1) REMPI signal is

a function of both the direction of the probe laser polariza-

tion and the recoil angle. The two-photon intensity for a

particular rotational NO product state J will depend on the

(

0)

(2)

(4)

alignment moments T₀,T₀, and T₀, which in turn will

be functions of ⌰,

2

FIG. 2. Image of NO( ⌸₁,vϭ2, Jϭ17.5) and its Abel transform obtained

/2

via (2ϩ1) REMPI at 337.59 nm in a one-color experiment. The Abel trans-

form represents a 2D cut ͑including the axis of cylindrical symmetry͒

through the 3D velocity distribution.

_B͑

2͒

n͑J͒

͑

0

0͒_͑_⌰_͒_ϩ

͑2͒

Iϭ

T

͑⌰͒P ͑cos ␦͒

2

_B͑

0͒

ͭ

_B͑0͒

0

_B͑

4͒

0͒

͑

0

4͒

ϩ

T

͑⌰͒P ͑cos ␦͒ .

͑2͒

III. EXPERIMENTAL RESULTS AND ANALYSIS

4

ͮ

The results obtained at USC and the University of Geor-

gia are complementary in that the imaging experiments have

their highest accuracy for products with the largest transla-

tional energy release, while in the Georgia experiments, the

best accuracy is achieved when products’ translational en-

ergy is low. In addition, in the USC experiments the angular

(

Q)

Here we have used the state multipole moments T

as de-

0

3

6

ﬁned by Blum, which are proportional to the more familiar

(

0

Q)

state multipole moments A

introduced by Greene and

3

7

Zare. The angle ␦ describes the direction of the probe laser

polarization with respect to the photolysis laser polarization.

For a one-color photodissociation experiment, we necessarily

3

distributions of the O( Pjϭ2,1,0⁾^p^r^o^d^u^c^t^s^a^r^e^o^b^t^aⁱⁿ^e^d^a^s

(

Q)

well.

have ␦ϭ0. The coefﬁcients B

depend on the angular mo-

menta involved in the two-photon detection step. For the Q

branches of the X–H transition, we ﬁnd approximately

A. Photofragment ion imaging

(

2)

(0)

(4)

(0)

Images were obtained for selected rotational levels of

B

/B ϭϪ0.3 and B /B ϭϪ0.04, independent of J.

3

(

Q)

NO(vϭ2, J) with ϳ338-nm photolysis, and for O( P ) and

As shown in the Appendix, the multipole moments T₀are

2

3

O( P ) fragments with 371.7-, 354.7-, and 338.9-nm pho-

expanded in a series of Legendre polynomials P (cos ⌰)

0

␭

(

Q)

tolysis. The electric ﬁeld vector E of the photolysis laser is

maintained parallel to the vertical direction of the image

plane, and the product recoil velocities are aligned predomi-

nantly in the polar direction of the image. The images are

symmetrized and, after background subtraction as described

in Sec. II, are Abel-transformed using established

procedures.³⁹

with coefﬁcients C . In this notation, the anisotropy pa-

rameter ␤ is described in terms of the coefﬁcients C₂

␭

(

Q)

.

In a one-color experiment, it is impossible to disentangle

the contributions due to the different state multipole mo-

ments. Nevertheless, the sensitivity of our experimental data

to the vector correlations can be estimated from the contri-

butions of higher order Legendre polynomials to the mea-

sured angular distributions. Within experimental uncertainty,

no such contributions are obvious. Although it is well known

that the NO products are aligned rotationally, we must con-

Shown in Fig. 2 is the raw image, as well as a two-

dimensional ͑2D͒ cut of the three-dimensional ͑3D͒ Abel-

2

transformed image of NO( ⌸₁,vϭ2, Jϭ17.5) obtained

/2

with 337.59-nm photolysis in a one-color experiment. The

three spin-orbit states of the correlated O fragment are

clearly resolved. The ring correlated with O( P ) collapses

to a spot at the center, since almost no translational energy is

associated with this channel. Despite the brightness of this

spot, the integrated intensity of the O( P ) channel is very

small. Similar images were obtained for NO(vϭ2, J

ϭ14.5–20.5).

clude that the data presented here are not seriously affected

_b_y_r_o_t_a_t_i_o_n_a_l_a_l_i_g_n_m_e_n_t_.₃8 Neglecting the small contribution

3

(

4)

0

due to the hexadecapole moment T , we ﬁnd the effective

0

anisotropy parameter ␤_e_f_fto be

3

͑

2͒

0

␤_e_f_fϭ␤Ϫ0.3C₂

,

͑3͒

independent of fragment rotational level. The importance of

While probing NO provides the angular distribution for

each fragment in a well-deﬁned quantum state, the oxygen-

photofragment images allow an overview of the total NO

rovibrational distribution correlated with each oxygen spin-

orbit state, albeit with loss of state resolution, since at each

(

2)

the coefﬁcient C , which is associated with the quadrupole

2

(

2)

moment T ͑see Appendix͒, can only be determined in an

0

experiment where the photolysis and probe laser polarization

directions are controlled independently. Applying the

method of laser-induced grating spectroscopy ͑LIGS͒ to the

E several NO(⍀,v,J) states can overlap. In Fig. 3, images

t

3

NO(vϭ0) product detection near D , Butenhoff and Rohlf-

of O( P ) and O( P ) obtained with 338.9-nm photolysis

0

2 0

ing conclude that ␤_e_f_fdoes not change signiﬁcantly when the

are shown. NO in vϭ0,1,2 is produced: the bright central

spot corresponds to NO(vϭ2, J), the intense inner ring

marks the onset of NO(vϭ1, J), while the faint outermost

2

7

probe laser polarization is rotated by 90°; thus, the coefﬁ-

(

2)

cient C must be negligible. We expect that the contamina-

2

tion of the anisotropy parameter due to the rotational align-

ment will be small in our case as well.

ring is associated with NO(vϭ0, J). Similar images are ob-

3

tained at the other two wavelengths. Images of O( P ) are

1

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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Photoinitiated unimolecular decomposition

7387

3

TABLE I. ␤ values corresponding to O( P ) and O( P ) fragments ob-

2

0

tained from the NO(vϭ2, J) angular distributions. The uncertainty in ␤ is

Ϯ0.1

Q₂₁_d(J)

J͑NO͒

Corresponding

3

Ϫ1

O( P ) state

E , cm

␤͑Imaging͒

␤͑TOF͒

j

tr

3

Jϭ21.5

Jϭ20.5

Jϭ19.5

Jϭ18.5

P₂

P₁

P₂

P₁

P₂

P₁

P₂

P₁

P2

P1

P₂

P₁

21.27

78.51

134.42

187.47

29.47

¯

Ϫ0.1

Ϫ0.2

0.2

1.1

Ϫ0.1

1.0

Ϫ0.2

1.2

Ͼ0.3

1.3

Ͼ0.3

1.3

Ͼ0.4

1.3

3

0.23

0.68

0.96

0.04

1.05

0.47

1.47

0.84

1.34

1.09

1.48

1.03

¯

3

Jϭ17.5

Jϭ16.5

Jϭ15.5

Jϭ14.5

Jϭ13.5

237.5

79.5

3

284.94

126.94

329.35

171.35

370.84

212.84

411.22

253.22

3

FIG. 3. Images of the O( P ) photofragments and their Abel transforms

j

3

obtained from NO photodissociation at 338.9 nm. O( P ), jϭ0 and 2 were

2

j

3

detected by (2ϩ1) REMPI at 225.68 and 226.28 nm, respectively.

3

¯

Ͼ0.4

3

qualitatively similar to those of O( P₂_,₀) and therefore were

not explored in detail. The fragment speed and angular dis-

tributions are derived from the Abel-transformed images.

The speed distribution is obtained by integrating the signal at

each distance ͑or velocity͒ over all angles. To obtain the

angular distributions, the velocities within a prescribed small

velocity range ⌬V are summed.

Similar conclusions are drawn from analyses of the

3

O( P ) images. The speed distribution of each O( P ), ob-

j

tained by integration over all angles, has been converted to a

translational energy distribution P(E ) which is related to the

t

NO internal energy ͑vibrational, rotational, and spin-orbit͒

distribution P(E ) by energy conservation. The P(E ) dis-

int

t

For NO images, the angular distributions associated with

3

tributions of O( P ) and O( P ) obtained at 338.9-, 354.7-,

3

2

0

O( P ) and O( P ) were determined by integrating over

1

2

and 371.7-nm are shown in Fig. 5.

The distinct peaks in the translational energy distribu-

tions are associated either with speciﬁc NO vibrational levels

each ring’s velocity width. Note that in these one-color ex-

periments, probing each NO(v,J) state is associated with a

slightly different excess energy and this was taken into ac-

͑

whose thresholds are indicated by arrows͒, or with oscilla-

count in the analysis. The dependence of ␤ on E in the range

t

tory structures in the rotational energy distribution of each

NO(v). The latter, which depend sensitively on excitation

energy, are well documented and interpreted as signatures of

bending vibrational structure of the transition state ͑TS͒.¹⁴

Ϫ1

3

0

–400 cm for NO(J) correlated with O( P ) and O( P )

2 1

is summarized in Fig. 4 and Table I. Two trends are mani-

fest: ͑i͒ the recoil anisotropy parameter ␤ decreases with in-

creasing rotational and decreasing translational energy; and

The distributions obtained at 354.7 nm are in good agree-

3

͑

ii͒ at the same E , NO at higher J ͓correlated with O( P )]

t

2

7,28,29

ment with previous results.

They reﬂect the structures

gives rise to lower ␤ than NO at lower J ͓correlated with

of the NO(vϭ0,1) rotational distributions, which have been

3

O( P )]. Thus, ␤ depends both on rotational and transla-

1

17,19,38

measured directly by monitoring NO.

In particular, the

tional energy of the fragments.

bimodal energy distribution of NO(vϭ1) at this wavelength,

which has two distinct peaks at low and high J’s, is well

reproduced in the image. The differences between the

3

O( P ) and O( P )E distributions obtained at each pho-

2

0

t

tolysis wavelength are another manifestation of the ﬂuctua-

1

4

tions that are pervasive in NO dissociation.

2

In order to derive the dependence of ␤ on E , the angu-

t

lar distributions of narrow rings of width ⌬V are ﬁt to Eq.

͑1͒. The results are shown in Fig. 6. Note that each measured

␤

is a weighted average of the ␤ parameters associated with

the NO(⍀,v,J) states whose E_i_n_tis complementary to E_t.

Also, the measured data result from a convolution with the

apparatus resolution function.

The most remarkable feature in Fig. 6 is the similarity of

2

FIG. 4. Dependence of the anisotropy parameter ␤ of NO( ⌸₁,vϭ2, J)

the functional dependence of ␤ on E at the three photolysis

t

/2

on the c.m. translational energy E for NO dissociation around 338 nm. The

wavelengths, despite the large differences in excess energy

and the corresponding NO rovibrational distributions. Like-

t

2

3

squares and circles represent NO fragments correlated with O( P ) and

2

3

O( P ), respectively. The solid line is the ﬁt using the model described in

the text; the highest NO J state associated with O( P ) is not included in the

1

3

wise, at each of the three photolysis energies, the curves for

1

3

the two measured oxygen spin-orbit states, O( P ) and

ﬁt, because it can be generated only from NO , whose rotational state is

2

3

greater than 0.5.

O( P ), are similar. This is signiﬁcant, since while O( P )

0

2

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1

7388

J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Demyanenko et al.

3

FIG. 6. The dependence of ␤ obtained from the O( P ) and O( P ) Abel-

2

0

transformed images on c.m. translational energy at the same dissociation

wavelengths as in Fig. 5.

FIG. 5. Photofragment c.m. translational energy distributions obtained from

3

O( P ) and O( P ) Abel-transformed images at the speciﬁed dissociation

2

0

wavelengths. The arrows indicate the opening of each correlated NO vibra-

tional level.

simulated by Gaussian distributions of appropriate width. A

strong correlation between the velocity direction of the par-

ent molecule and the actual position of photolysis is as-

sumed. The overall effect of these deviations from an ideal

experiment can be well described by a convolution of the

original TOF distribution with a Gaussian of full width at

half-maximum ͑FWHM͒ 8%. In order to completely deter-

mine the kinematics of the fragmentation, we calculate the

translational energy from energy conservation neglecting the

3

can be aligned, the projection of the O( P ) angular momen-

0

tum on the quantization axis is zero and no alignment is

3

possible. In a previous study at ϳ355 nm, O( P ) was found

0

3

28

to exhibit less anisotropy than O( P ), but this behavior is

2

not reproduced in our study ͑see Fig. 6͒. Possibly, due to the

very small population of O( P ), some isotropic contami-

nation contributed to the observed reduction in ␤ in the pre-

vious study. The clear dependence of ␤ on NO rotational

level is evident in the 371.66-nm photolysis data where only

NO(vϭ0) is energetically allowed.

3

14

0

͑small͒ internal energy of the parent. With the current reso-

lution, it is not possible to detect any indication of the modu-

lation due to Legendre polynomials of degree larger than 2.

A series of TOF proﬁles and their simulations are shown in

Fig. 7.

B. TOF spectra

The TOF data obtained at Georgia conﬁrm the results

obtained at USC. The one-color photolysis and detection re-

quires focusing of the laser beam, which results in a small

interaction volume. Therefore, a forward convolution proce-

dure is applied to the analysis of the TOF spectra. Assuming

an angular c.m. distribution described by even Legendre

polynomials, the arrival time of the ions at the detector for

all scattering angles ⌰ and ␾ is calculated. Because the ion

optics does not preserve completely the cylindrical symme-

try, it is necessary to include the ␾ dependence. The ﬁnite

angular and velocity spreads of the molecular beam are

The ␤ parameter is determined by a least-squares ﬁt to

the data. Near the energetic threshold, the TOF spectra are

sensitive to the different ﬁne-structure states of the O atom.

In these cases, we simulate the spectra as a superposition of

two corresponding c.m. distributions with independent ␤ val-

ues and amplitudes. With increasing E , the different spin-

t

orbit states are less well resolved, resulting in strong corre-

lation between the parameters. It is for this reason that we

can only determine a lower limit for the anisotropy param-

eters for NO vϭ2, JϽ17.5. Furthermore, the resolution of

the TOF experiment was not sufﬁcient to resolve structures

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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Photoinitiated unimolecular decomposition

7389

2

FIG. 8. TOF spectra of NO(X ⌸ ,vЉϭ2, Jϭ18.5) recorded at the indi-

2

1/2

FIG. 7. TOF spectra of NO(X ⌸ ,vЉϭ2, J) resulting from the photodis-

1

/2

cated laser pulse energies. The dashed lines represent simulations of the

TOF spectra as described in the text. For the simulated spectra in the bottom

part, we assumed the limiting anisotropy parameters ␤ϭϪ1.0 and ␤ϭ1.3,

respectively. All spectra are calculated using the amplitude ratio

sociation of NO . The dashed curves represent ﬁts to the experimental spec-

tra assuming the indicated anisotropy parameters ␤( P ) and ␤( P ) asso-

ciated with the two ﬁne-structure components of the O( P) fragment. The

spectra are taken with an acceleration voltage of 10 V/cm, laser energy 0.6

mJ, and 500-mm focal length lens. The available translational energy E ͑in

cm ͒ for the NOϩO( P ) channel is indicated for each spectrum.

2

3

2

1

3

␴

( P )/␴( P )ϭ0.4.

1 2

t

Ϫ1

3

2

fects, we measured product TOF spectra for different laser

pulse energies. Typical results for Jϭ18.5 are shown in Fig.

due to the opening of the exit channel associated with the

3

production of O( P ) fragments. Therefore this channel,

0

8

0

4

. The derived anisotropy parameters changed by less than

.1 when the laser pulse energy was increased by a factor of

. Furthermore, a substantial part of the reduction was due to

which is open for NO(vϭ2, JϽ16.5), is not included in the

data analysis. The uncertainty in the ␤ parameters is deter-

mined by the range of values that ﬁt the experimental spec-

tra, and is estimated at Ϯ0.1.

A summary of the ␤ parameters is given in Table I. As

in the imaging data, with increasing NO(J), a substantial

reduction in the anisotropy parameter is evident. The devia-

space-charge effects. This somewhat puzzling result can be

readily understood when the classical optical pump rate is

compared with the dissociation rate. The radiative lifetime

40

for the parallel electronic transition is Ͻ1 ␮s, and we esti-

mate for our experiment an optical pump rate corresponding

to a pump cycle of at least 15 ps. This value must be com-

tions from the low-J limit can be detected only at E

t

Ϫ1

3

Ͻ300 cm . very close to the thresholds for each O( P ),

j

pared with the subpicosecond lifetime of NO with excess

2

we ﬁnd very small and even negative ␤ values, indicating

substantial deviations from axial recoil.

Ϫ1 21

energies Ͼ1500 cm

. At the higher excess energies inves-

tigated here, the lifetime is even shorter. Thus, dissociation is

clearly the fastest process, and the deviations from the cubic

power dependence must therefore be attributed to a depletion

C. Saturation behavior

of the NO density in the laser photolysis/probe volume,

which should not affect the value of ␤.

In estimating the accuracy of the measured ␤ param-

eters, especially in the one-color experiments that require

high power densities, the effect of saturation has to be evalu-

ated. Under the conditions of the present experiment, NO

2

IV. CLASSICAL MODEL FOR NONAXIAL FRAGMENT

RECOIL

(

2ϩ1) REMPI detection is characterized by quadratic power

dependence indicating the two-photon absorption as the rate

limiting step. On the other hand, a power dependence of less

than 2.5 is found for the production of NO fragments in the

A. Transverse recoil derived from angular momentum

conservation

one-color photodissociation of NO . The deviation from the

ideal cubic power dependence could be attributed to satura-

2

The classical model presented below for NO dissocia-

tion is based on energy and angular momentum conservation,

2

tion of the one-photon step in NO photolysis. Depending on

2

NO

O

t

O

el

the competition between the optical pumping and dissocia-

tion rates, we expect a loss of anisotropy for the state re-

solved fragment velocity distributions. To check these ef-

hvϪD ϭE ϩE_tϩE ϩE ,

͑4͒

͑5͒

0

int

J_N_O₂ϭJ_N_OϩJ ϩL,

O

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7390

J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Demyanenko et al.

tion laws must be obeyed, and they impose restrictions on

product trajectories. In the c.m. coordinate system, the clas-

sical orbital angular momentum is given by ͓Fig. 9͑b͔͒,

Lϭm

r

ϫV_N_Oϩm r ϫV ,

͑6͒

NO NO

O O

O

where r_N_Ois the vector directed from the c.m. of NO to the

2

c.m. of the NO fragment, V_N_Ois the c.m. velocity of NO,

and r_Oand V_Oare the corresponding quantities for the O

atom. Momentum conservation dictates that

Lϭm RϫV ,

͑7͒

O

which is equivalent to the classical expression for the orbital

angular momentum in terms of the long-range impact param-

eter b, Lϭ␮Vb, where ␮ϭm_N_Om /(m ϩm ) is the re-

O

NO

O

duced mass and V is the relative velocity of the recoiling

fragments. If we deﬁne axial recoil as the recoil of a diatomic

fragment in a direction that correlates with no angular mo-

mentum in the fragment, then the direction of the recoil must

coincide with R. Consequently, we relate axial and trans-

verse recoil to the direction of R.

Using

͉

J

͉

ϭប

ͱ

J(Jϩ1)

and

͉L͉ϭm R V sin ␸

O C O

ϭ␮VR sin ␸, where ␸ is the angle between R and V , and

C

O

R is a critical distance where the angular momenta become

C

established, we obtain

FIG. 9. ͑a͒ Geometry of the NO molecule prior to photoexcitation indicat-

2

ing the orientation of the transition dipole moment vector ␮ , the laser

␧

m R V sin ␸ϭប

ͱ

J͑Jϩ1͒.

͑8͒

O

C

O

polarization vector E, and the initial angle ␣ between R and E. ͑b͒ Geom-

i

O

el

NO

vib

etry and orientation of NO₂after evolving from the initial to the critical

conﬁguration. The angle ␸ dictated by angular momentum conservation de-

termines the deviation from axial recoil. Notice that the change in ␥ deter-

mines the change in the angle ␣.

Deﬁning E_a_v_lϭhvϪD ϪE ϪE ϭE ϩE as the total

0

t

rot

fragment rotational and translational energy associated with

each NO(v) product, this expression can be written as

2

␮_N_O_E•r

2

sin ␸ϭ

.

͑9͒

avl

2

␮

•R •

ͩ

Ϫ1

ͪ

NO

O

C

where D is the dissociation threshold of NO , E and E_t

E_r_o_t

0

2

t

O

are the c.m. recoil energies of the fragments; E is the elec-

^t^r^oⁿⁱ^c^eⁿ^e^r^g^y^o^f^t^h^e^s^pⁱⁿ^-^o^r^bⁱ^t^s^t^a^t^e^o^f^t^h^e^O^a^t^o^m^;^Eint in-

el

Equation ͑9͒ highlights the conditions necessary for

large deviations from axial recoil ͑large ␸͒: ͑i͒ E /E_r_o_tcan-

not be too large compared to 1. In other words, E_r_o_tmust

constitute a large fraction of Eavl ; and ͑ii͒ INOϭ␮NOr must

NO

avl

cludes the vibrational, rotational, and spin-orbit energies of

the NO fragment; J_N_O₂, J_N_Oand J are the angular momenta

2

O

2

C

of the parent NO molecule, and NO and O fragments, re-

2

be comparable to I_N_O₂ϭ␮R at the critical distance R_C

spectively; and L is the orbital angular momentum. The

model applies to triatomic dissociation induced by optical

transitions that lie in the molecular plane. In common with

all classical models, it is valid when the product angular

momentum is high. In our case, J_N_O₂ϭ1/2 and 3/2 and thus

where the angular momenta have reached their ﬁnal values.

In applying the model to NO dissociation, we adopt the

2

following geometrical conﬁguration of the ground state. The

equilibrium ONO bond angle is 134°, and the NO bond

42

lengths are both equal to rϭ1.189 Å ; this corresponds to

^JNO₂and J can be neglected, and J is balanced mainly by

O

NO

Rϭ1.692 Å and ␥ϭ152°. The choice of the critical distance

R_Cis less certain. We deﬁne it as the distance at which the

NO product has developed its ﬁnal free rotational state, and

consequently the orbital angular momentum must be estab-

lished as well. This value can be loosely related to the value

of R at the TS; i.e., R уR . Relying on recent calculations

L. Note that for a ﬁxed photolysis energy, high J

states

NO

O

correlate with low recoil energy (E ϭE ϩE ) and high L.

t

Jacobi coordinates are chosen to describe the dissocia-

tion in the molecule ﬁxed frame ͑MFF͒; i.e., R is the distance

from the departing O atom to the c.m. of NO, r is the inter-

nuclear distance in NO, and ␥ is the angle between R and r

C

TS

of the NO₂PES and statistical ͓variational Rice–

Ramsperger–Kassel–Marcus ͑RRKM͔͒ calculations that

͑Fig. 9͒. During dissociation, NO rotation, which is in the

parent plane, is accompanied by a counter-rotation of R

have determined the location of the TS and its geometry as a

function of excess energy,²

3,43,44

we use R у2.7 Å, r

around the c.m. of NO , in order to conserve angular mo-

2

C

mentum.

ϭ1.189 Å, and ␥ϳ120° ͑corresponding to ONO TS angle

of ϳ109°͒.⁴³

Notice that in contrast to direct triatomic dissociation,

where the anisotropy in the angular potential is the main

source of torque, in vibrational predissociation from a barri-

erless PES, the main source of diatom angular momentum is

bending vibrations in the TS.¹⁴^,⁴¹Irrespective of the source

of angular momentum, however, the fundamental conserva-

From Eqs. ͑8͒ and ͑9͒ it is evident that at each E_a_v_l, a

speciﬁc fragment quantum state will be associated with a

distinct transverse recoil angle ␸, and the deviations from

axial recoil will increase with increasing E_r_o_t͑and concomi-

tantly decreasing E ). Also, in accordance with angular mo-

t

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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Photoinitiated unimolecular decomposition

7391

FIG. 11. Dependence of ␤ in the vibrational predissociation of Ne–ICl on

the rotational state of the ICl fragment, obtained from the quantum calcula-

tions of Ref. 4 ͑squares͒, and calculated from the classical model ͑circles͒.

See the text for details.

unknown, and must be either inferred from ﬁts to the data, or

obtained from calculations. Below, we discuss two cases,

one in which we compare our model to results of quantum

calculations for Ne–ICl predissociation, and the other—an

FIG. 10. Final fragment recoil angle ␸ ͑top͒ and anisotropy parameter ␤

͑

bottom͒ plotted as a function of the c.m. translational energy E , calculated

t

for different values of the available energy E_a_v_l͑deﬁned as the total avail-

able translational and rotational energy of the fragments͒. R ϭ3.3 Å and

C

application to the NO decomposition results reported here.

2

␣_Cϭ27° were used in these simulations ͑see the text͒.

1. Ne–ICl vibrational predissociation

^m^eⁿ^t^u^m^c^oⁿ^s^e^r^v^a^tⁱ^oⁿ^,^a^t^a^gⁱ^v^eⁿ^Eavl and R , there is a

maximum value of product J that is allowed.

C

As a ﬁrst application, we calculate the expected devia-

tions from axial recoil in the predissociation of the Ne–ICl

The dependence of ␸ on E is illustrated in Fig. 10 for a

t

complex following photoexcitation to its B electronic state

that correlates with the B ⌸₀state of the ICl product,⁴

3

^s^e^rⁱ^e^s^o^f^Eavl ^,^u^sⁱⁿ^g^t^h^e^N^O2 parameters given above.

Clearly, this dependence becomes much steeper at low E_t.

At the bottom, we illustrate the corresponding changes in ␤,

assuming ␤ϭ1.6 for pure axial recoil ͑see below͒. As ex-

ϩ

Ne–ICl͑X,vϭ0, J_N_e_–_I_C_lϭ0͒→Ne–ICl͑B,vϭ2, J_N_e_–_I_C_lϭ1͒

→

NeϩICl͑B,vϭ1, J_I_C_l͒. ͑11͒

pected, ␤ decreases strongly at small E , and may even as-

t

For this complex, the ICl rotational distributions were

determined experimentally.⁴⁵Quantum calculations of the

ICl rotational distribution, as well as the dependence of the ␤

sume negative values.

The limiting value of ␤ associated with axial recoil

needs comment. In an electronic transition with B symme-

2

parameter on the ICl ﬁnal rotational state, J , were carried

try, the transition dipole moment ␮ lies parallel to the ﬁgure

ICl

␧

out using an empirical PES.⁴

axis, or the a axis of a prolate top. When R is parallel to the

a axis, ␤ϭ2.0, but when R deviates from the a axis, ␤

Ͻ2.0 will be obtained even for fast dissociation, generating a

diatom with no angular momentum. Deﬁning ␣ as the angle

Optical excitation prepares an isolated resonance cen-

Ϫ1

tered at 42.31 cm^a^b^o^v^e^t^h^e^I^C^l⁽^B^,^v^ϭ²^,^JIClϭ0) asymp-

tote via a transition whose dipole moment is nearly parallel

to R of the bent Ne–ICl complex. The resonance in the ex-

cited state is coupled to the continuum via vibrational pre-

dissociation, and projects onto J_I_C_lof the ICl(B) state. The

_b_e_t_w_e_e_n_R_a_n_d_t_h_e_a_a_x_i_s_,_w_e_o_b_t_a_i_n_t_h_e_u_s_u_a_l_e_x_p_r_e_s_s_i_o_n_,1

,18

␤

ϭ2P ͓cos͑␣ϩ␸͔͒.

͑10͒

2

calculated ICl(vϭ1)

rotational distribution

⁽^Eavl

Thus, in a parallel transition with ␣Þ0, ␤Ͻ2.0 is observed

even for pure axial recoil (␸ϭ0). The results obtained by

using Eq. ͑10͒ with ␣ϭ24° and positive ␸ ͑corresponding to

clockwise rotation͒ are shown in Fig. 10͑b͒.

Ϫ1

ϭ140.4 cm ) is similar to the one obtained experimentally

and spans Jϭ0–35.⁴The ␤ parameter calculated quantum

mechanically is nearly 2.0 for low J’s, but decreases substan-

tially for higher J’s ͑see Fig. 11͒. Since Ne–ICl predissocia-

tion falls nicely within the criteria identiﬁed for nonaxial

recoil ͓Eq. ͑9͔͒, the classical model should show how much

of the ␤ variation with J obtained in the quantum calcula-

tions is due to transverse recoil.

,45

B. Applications of the model

Although the model described above is general, its

implementation for predissociating systems is not always

straightforward. On the one hand, the large degree of rota-

tional excitation justiﬁes the classical treatment. On the other

hand, the geometrical conﬁgurations at which the model is

applied are often ill deﬁned. In particular, R_Cand ␣_Care

In our calculations we used R ϭ4.95 Å, which is the

location of the top of the calculated potential barrier. We

assume ␣ϭ0 throughout the dissociation process, since ␤

ϭ2 is found for low J_I_C_l. The comparison with the results of

C

4

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7392

J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Demyanenko et al.

the quantum calculations is shown in Fig. 11. The good

agreement between the quantum calculations at J_I_C_lϾ10 in-

dicates that the predominant reason for the strong decrease in

␤

as a function J_I_C_lis the transverse recoil that must accom-

pany the dissociation into high fragment J ͑and low E_t)

states. Angular momentum conservation is included explic-

itly in the quantum calculations.⁴

2

. Implementation of the model to NO unimolecular

2

reaction

The main differences between the Ne–ICl example and

the NO case are that for NO , ␣ and R are treated as ﬁtting

2

C

parameters, and the transverse recoil angle ␸ can be either

added to or subtracted from ␣. In the former case, a decrease

in ␤ for all values of ␸ should be observed. In the latter case,

␤

will increase until ␣-␸ reaches zero ͑or V coincides with

0

R͒, and then decrease as V and R diverge. In the absence of

0

dynamical information, we chose to use an equal weighting

of both, namely,

␤_a_vϭ͓P ͑cos͑␣ϩ␸͒͒ϩP ͑cos͑␣Ϫ␸͔͒͒.

͑12͒

2

This choice is sensible, since the main source of fragment

rotational angular momentum in NO dissociation at energies

2

1

4

not too close to D is bending vibrations in the TS. Invok-

0

ing a separation of the motions perpendicular to the reaction

coordinate and the harmonic approximation, we give equal

weighting to inward and outward bending motions ͑corre-

sponding to positive and negative ␸͒, which evolve into

clockwise and counterclockwise rotation of R, respectively.

We note that the use of Eq. ͑12͒ yields better ﬁts to the data

than using solely ϩ␸ or Ϫ␸, although the ﬁts are not sensi-

tive to the exact weighting factors in Eq. ͑12͒. The data

shown in Fig. 4 are best ﬁt ͑using J_N_O₂ϭ1/2) with ␣_C

ϭ24°, and R ϭ3.3 Å.

C

3

The model simulations of the ␤ variation in the O( P )

j

3

images were performed as follows. The corresponding NO

internal energy distributions were taken either from phase

FIG. 12. Simulations of the dependencies of ␤ obtained from O( P ) im-

2

ages on c.m. translational energy for the three experimental dissociation

wavelengths as in Figs. 5 and 6. The values of R_Cand ␣_Cthat best ﬁt the

data for each wavelength are indicated. See the text for details. The error

bars represent uncertainties in ␤ in the region of high and low E_t.

space theory ͑PST͒ calculations using the separate statistical

ensembles method ͑SSE͒,⁴

6,47

or from experiments in which

the NO rovibrational distributions were measured

1

4,17,19,38

directly.

The NO internal energy distributions were

convoluted with the instrument resolution function, and the

ﬁnal ␤ values were calculated as the weighted averages of

the individual ␤ values for the NO v,J states, whose ener-

has much less rotational excitation than vϭ0) gives rise to a

slightly higher value of ␤. The 338.9-nm data are best ﬁt

with ␣ ϭ27°. Here, vϭ2,1, and 0 are generated, and the

C

gies correspond to E values that fall within a small range

distributions are calculated by PST/SSE, since no experi-

mental data are available. In summary, the simulations ﬁt

t

4

8

49

⌬

E_t.

The simulations of the O( P ) results are shown in Fig.

3

the data with rather similar choices of R_Cand ␣ , and ex-

2

C

1

3

2. R ϭ3.3Ϯ0.3 Å gives the best ﬁt for all the data sets. At

hibit large contributions of transverse recoil for high J states.

C

71.7 nm we obtain ␣ ϭ30°, and since only NO(vϭ0, J)

C

can be produced, the NO rotational distributions in the simu-

lations are generated by PST. At 354.7 nm, experimental

data for the NO rovibrational distributions are available, and

the distributions measured by Reisler and co-workers are

used as input in the simulation.¹However, using PST/SSE

to calculate the NO distributions yields an equally good ﬁt,

V. DISCUSSION

A. Implications to NO photodissociation dynamics

2

The two parameters of the model, R_Cand ␣ , are rel-

evant to the mechanism of NO dissociation. Recall that in

C

4,19

2

order to calculate ␤, ␣ and R must be related to the space

ﬁxed frame ͑SFF͒, which is determined by the laser polar-

ization vector E.¹Our model is based only on conservation

laws and does not take into account ͑at least explicitly͒ any

particular nuclear motions during dissociation.

both with ␣ ϭ24°. The small increase in ␤ near the thresh-

C

,2

old for the opening of vϭ1, which is apparent both in the

experimental data and the simulations, and has been ob-

2

8

served before, illustrates that at the same E_a_v_l,vϭ1 ͑which

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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Photoinitiated unimolecular decomposition

7393

In barrierless dissociation, the location of the TS is not

sharply deﬁned and is known ͑e.g., by variational RRKM

calculations͒ to move inward with increasing excess

ration along the MEP expressed relative to the SFF. For a

given change in ␥, the corresponding change in ␣ depends on

the ratio between I_N_O_–_Oand I_N_O. In other words, substantial

rotation of R can be achieved only when I_N_O_–_Ois compa-

rable to I_N_O. Thus, we must assume that the change in ␥

from its equilibrium value ͑152°͒ to its ﬁnal value ͑ϳ125°͒

occurs at a relatively small R, and is then followed by addi-

tional stretching of R ͑or the dissociating NO bond͒ until the

critical distance R_Cis reached.

2

3,24

energy.

around 2.7 Å,

For NO at high excess energies it is located

2

4,43

and thus the value of R_Cthat best ﬁts the

data, R ϭ3.3 Å, is reasonable. It is located between the cal-

C

culated value of the TS ͑2.7 Å͒ and the location of the cen-

trifugal barriers ͑4–5 Å͒, beyond which no further energy

exchange among fragment states is expected. This R_Cvalue

also allows the population of the highest observed NO rota-

We can now compare the motions suggested by our

tional levels. Note that at high E , ␤ values close to the

t

model to those predicted or calculated for NO . The changes

limiting axial-recoil value are obtained, and these are not

very sensitive to the choice of R_C.

2

in nuclear conﬁguration during dissociation can be related to

2

The angle ␣_Cbetween the recoil velocity vector and E

the geometry of the conical intersection that couples the A

1

2

obtained from the ﬁts is ␣ ϭ27Ϯ3°. In order to understand

and B states, and/or to the curvature of the ground-state

C

2

its origin, it is useful to distinguish between the excitation

and dissociation steps. The ground-state equilibrium geom-

etry of NO determines the initial angle ␣ between R and

MEP in the (R,␥) plane. Calculations show that the conical

15,16

intersection occurs at an angle ␥Х120°,

while recent ab

2

i

initio calculations of the ground-state PES ﬁnd that in the TS

the a axis at the moment of excitation ͓Fig. 9͑a͔͒. The initial

43

region ␥ϭ123°, a value close to that required for the coni-

bending angle ␥ ϭ152° ͑see above͒ corresponds to ␣_i

i

cal intersection. Moreover, in the most recent ab initio cal-

culations, it is shown that the MEP in the (R,␥) plane is

signiﬁcantly curved only for RϽ2.1 Å reaching angles of

ϭ7.4°, and therefore R must rotate during the dissociation

with respect to the SFF to achieve the value of ␣_Cthat best

ﬁts the data. This added rotation should correlate to nuclear

motions during the dissociation, which, in turn, must con-

serve angular momentum.

1

20–130°, while at larger R, the MEP can be described by

4

pure stretching motion up to the TS. Thus, most of the

change in ␥ occurs at small R, and the bending angle for the

conical intersection happens to be similar to that in the TS

region. Trajectory calculations using the method described

The most likely source of the change in ␣ is a change in

␥

during dissociation. Since the parent angular momentum is

nearly zero, the relative rotations of r and R in the molecular

plane that determine ␥ must be correlated. In other words, in

order to conserve angular momentum, any rotation of R

9

by Loock et al., and carried out on an analytical PES whose

shape has similar characteristics to the NO PES calculated

2

͑

with angular velocity ␻ ϭd␸ /dt) must be compensated

R R

ab initio, also conﬁrm that the expected changes in ␤ on such

a PES are similar to those observed experimentally.

Does the closing of ␥ reﬂect dynamics on the ground

by a counter-rotation of r ͑with angular velocity ␻_r

ϭd␸ /dt). For a nonrotating parent molecule we obtain

r

d␸_R␮R²

I_N_O_–_Od␸_R

2

A state, excited B state, or both? Near threshold, it is

1 2

2

d␸_r

ϭϪ

• _␮

ϭϪ

• _d_t

,

͑13͒

r²

I_N_O

established that dissociation evolves mostly on the ground

electronic state, and the dissociation time is longer than a

picosecond. Thus, at the near threshold energies ͑0–128

dt

NO

where I_N_O_–_Oand I_N_Oare the relevant moments of inertia.

The time rate of change of ␥ is the combined rate of change

of R and r, or,

Ϫ1

27

cm ͒ explored by Butenhoff and Rohlﬁng, changes in ge-

ometry during dissociation must relate to the dynamics on

␮R²

␮_N_Or²

2

d␥

^d^␸R

^d^␸r

^d^␸R

A . The situation is more complicated at the high excess

1

ϭ

Ϫ

ϭ

ͩ

1ϩ

ͪ

^•dt

.

͑14͒

Ϫ1

dt

energies ͑2000–4000 cm ͒ employed here. Since the disso-

ciation lifetime is much shorter than a picosecond, many

overlapping resonances are excited coherently, even with a

nanosecond laser. Therefore, the excited dissociation wave

Since rotation of R gives rise to a change in ␣ ͓Fig. 9͑b͔͒,

Eq. ͑14͒ yields

^␮NOr²

^␮NO_r₂_ϩ_␮_R2

2

function may initially contain a signiﬁcant B₂character,

d␣ϭd␸ ϭ

d␥.

͑15͒

R

which will dephase into the ground state. Recent time-

dependent calculations estimate the dephasing time at 50–90

fs ͑the duration of few bending periods͒,⁵⁰which is probably

not much shorter than the dissociation time. Consequently, it

is not guaranteed that the initial coherent superposition of

resonances launched at the crossing seam will behave in an

ergodic fashion en route to dissociation. In other words, the

dissociative trajectories may be inﬂuenced by the PES at the

crossing seam. Our experiments cannot distinguish between

these different scenarios; they do suggest, however, that the

reduction in the bending angle ␥ must occur near the

Franck–Condon region.

If the minimum energy path ͑MEP͒ in the (R,␥) plane is

known ͓i.e., RϭR(␥) and rϭr(␥) are known͔, it is then

possible to integrate Eq. ͑15͒, provided ␥(t) is known, and

obtain the cumulative change in ␣,

␮_N_Or²

␥

C

⌬

␣ϭ

͵

d␥,

͑16͒

␮_N_Or²ϩ␮R²

␥

i

where ␥_Cis the angle ␥ at the moment of bond cleavage.

Equation ͑16͒ relates the deformation of the molecule in

the MFF to the rotation of R in the SFF. The ﬁnal value of

␣_Cϭ␣ ϩ⌬␣ depends on the change in the nuclear conﬁgu-

i

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7394

J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Demyanenko et al.

4

B. Deviations from axial recoil in unimolecular

decomposition

involving excitation by circularly polarized light. They also

provided a simpliﬁed formula of ␤ corresponding to the ini-

tial parent molecular state with a zero angular momentum

J_iϭ0. In this case, ␤ can be expressed as

In this section we discuss the requirements for observing

changes in ␤ in photodissociation dynamics due to transverse

recoil. Clearly, an in-plane optical transition ͑parallel or per-

pendicular͒ is required. Several conditions are apparent by

inspection of Eq. ͑9͒. First, large deviations from axial recoil

should be observed when the c.m. axial-recoil velocity com-

ponent is small compared to the tangential velocity compo-

nent induced by the rotation of the diatomic fragment. This

means that in fast barrierless unimolecular dissociation,

where typically the kinetic energy release is small and frag-

ment rotational excitation often extends to E_a_v_l, such devia-

tions should be routinely observed. Second, the nonaxial re-

coil, which has a sin ␸ dependence, increases sharply as E_r_o_t

approaches E_a_v_l; in other words, the highest J states will

exhibit particularly large deviations. Third, the moment of

inertia of the diatom should be comparable to the moment of

2

͑

2␮ Ϫ␮ Ϫ␮ ͒

0 ϩ Ϫ

␤

͑J͒ϭ

,

͑17͒

2

͑␮ ϩ␮ ϩ␮ ͒

0

ϩ

Ϫ

2

where ␮ ϭ

͉

͗

0^ʈ

^␮e

ʈ

1,J,0

͘

͉

, and ␮ ϭ

͉

͗

^␮␧

͘

0 1,J,Ϯ1 ͉ are

ʈ ʈ

the reduced matrix elements of the transition dipole moment

as deﬁned in Ref. 4, ͗0͉ is the initial parent wave function,

͘

and ͉Jf ,J,⍀ is the dissociation wave function with ﬁnal

state total angular momentum Jfϭ1, angular momentum of

the diatomic fragment J, and a projection ⍀ϭ0 or Ϯ1 of

Jfϭ1 on the z axis ͑along R of the dissociation fragments͒.

Based on this expression, it can be concluded that when

␮ ϭ0, ␤ϭϪ1 is obtained ͑a perpendicular transition͒,

0

Ϯ

2

0

2

Ϯ

2

0

while ␮ ϭ0 but ␮ Þ0 corresponds to ␤ϭ2 ͑a parallel

inertia associated with the orbital motion; i.e., R cannot be

transition͒. These two limiting cases correspond to transition

dipole moments that are perpendicular and parallel to R, re-

spectively. It is important to point out that Eq. ͑17͒ has no

explicit dependence on product J state. A J-dependent ␤

therefore implies that the reduced transition dipole matrix

elements are functions of J.

In quantum mechanical treatments, observables are usu-

ally related to the quantization axis rather than to the parent

molecular plane. Therefore, the physical interpretation of the

reduced transition dipole matrix elements obtained in the

close coupling calculations is not readily apparent, and it is

difﬁcult to glean the origin of the dependence of ␤ on J. The

classical treatment presented here, although valid only in the

high J limit, nevertheless gives a simple and clear physical

picture of the ␤ changes. Assuming no rotation of R during

C

too large when the reduced masses related to r and R are

similar. This condition is also necessary for the generation of

a signiﬁcant change in the angle ␣ during dissociation.

Pursuant to the above, it is straightforward to predict

systems for which a correlation of ␤ with diatomic rotation

will be observed. The change of ␤ is expected to be large for

vibrational predissociation, and for direct dissociation ac-

companied by substantial fragment rotational excitations.

Some examples are Ne–ICl, NO , O , N O, SO , and Cl O.

2

3

2

The 193-nm photolysis of NH via a conical intersection also

3

falls nicely into this category, and the model suggested by

Mordaunt et al. demonstrates how ␤ can change from a lim-

iting value of Ϫ1 to positive values.¹⁰Their model is also

based on angular momentum conservation, and has common

features with the model presented here. In certain cases, the

ratio of the moments of inertia given in Eq. ͑9͒ is not favor-

able; for example, in ICN, BrCN, CINO, and BrNO. Indeed,

in the 266-nm photodissociation of ICN and the 355-nm pho-

tolysis of CINO, no dependence of ␤ on fragment rotation

has been observed.⁵

dissociation, the quantum mechanical treatment for J ϭ1

f

implies that when the transition dipole moment lies along R

2

͑the z axis͒, ␮ Þ0 and ␮ ϭ0. The dissociation process may

0

Ϯ

2

lead to a decrease in ␮ and an increase in the ␮ compo-

0

Ϯ

nent with increasing fragment rotational quantum number.

1,52

This mixing of the ⍀ϭ0 and Ϯ1 components has been

4

We emphasize that the effect described here is by no

means the only one to cause deviations from the limiting

value of ␤. It has already been demonstrated that speciﬁc

exit-channel dynamics either involving surface crossings or

in- and out-of-plane nuclear motions during the dissociation,

called Coriolis coupling in the quantum treatment, whereas

in the classical picture we show that it is a result of conser-

vation of angular momentum in the molecular frame. When

the initial transition dipole moment lies in the molecular

plane, but is perpendicular to the direction of ﬁnal recoil,

2

as well as alignment effects and polarization of the m dis-

␮ Þ0 and ␮ ϭ0 are expected in the absence of ransverse

J

Ϯ 0

tributions, can cause large deviations from the limiting val-

ues. For example, in a recent work on the photodissociation

recoil. Likewise, if dissociation proceeds with extensive

transverse recoil, it is expected that there will be an increase

2

of NO at 212.8 nm, orbital alignment in the oxygen atom

in ␮ and a decrease in ␮ as the fragment rotational quan-

2

0 Ϯ

5

3

has been observed. However, even when other effects are

important, the deviations from axial recoil resulting from

conservation of angular momentum must be subtracted from

the data before the importance of other parameters can be

assessed.

tum number is increased. This corresponds to a change of ␤

from Ϫ1 toward 2, as J increases. Both cases were demon-

strated in the quantum mechanical calculations of Beswick

and co-workers ͑as displayed in Fig. 3 of Ref. 4͒. As shown

above for the Ne–ICl dissociation, the dependence of ␤ on

ICl rotation obtained with the quantum mechanical calcula-

tion agrees well with the results of our classical model ͑Fig.

11͒. It is important to point out that each fragment J state is

associated with a different recoil direction. A detailed quan-

tum mechanical treatment is beyond the scope of this paper,

but new calculations that treat this effect explicitly would

Finally, a comparison of the classical treatment of ␤ with

the quantum mechanical formulation is enlightening. G. G.

Balint-Kurti and M. Shapiro provided one of the ﬁrst quan-

tum mechanical expressions of ␤ for a triatomic dissociation

initiated with linearly polarized light.³Beswick and co-

workers later gave a more general formula including cases

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J. Chem. Phys., Vol. 111, No. 16, 22 October 1999

Photoinitiated unimolecular decomposition

7395

(

Q)

provide additional insights into the quantum origin of the

effect.

C

that depends on the angular momenta of the parent

␭

molecule, the fragments, as well as the different possible

electric dipole transition moments.³⁵

VI. SUMMARY

͑

0

0͒

͑0͒

T

͑⌰͒ϭC₀ϩC₂P ͑cos ⌰͒,

2

A classical model is presented that describes nonaxial

fragment recoil in unimolecular decomposition of triatomic

molecules. The model is based on conservation of angular

momentum, and predicts large deviations from axial recoil

when ͑i͒ the c.m. translational energy of the recoiling prod-

ucts is small compared with the rotational energy of the di-

atomic fragment ͑i.e., E ӶE ); and ͑ii͒ the moment of in-

tertia of the diatomic fragment is comparable to that

associated with the orbital motion. The two parameters

needed to determine the nonaxial recoil are the interfragment

͑

²^͒^͑^⌰^͒^ϭ^C ^͑0²^͒^ϩ^C ^͑2²^͒^P^͑^c^o^s^⌰^͒^ϩ^C4 P ͑cos ⌰͒,

͑2͒

0

2

4

͑4͒

0

͑4͒

͑⌰͒ϭC₂P ͑cos ⌰͒ϩC₄P ͑cos ⌰͒

2

4

͑

6

4͒

ϩC P ͑cos ⌰͒.

6

It is interesting to note that without resolving the angular

distribution, the rotational alignment moment T is deter-

^mⁱⁿ^e^d^b^y^t^h^e^c^o^e^f^ﬁ^cⁱ^eⁿ^t^C0 , whereas the angular anisot-

ropy can only be inﬂuenced by a nonvanishing rotational

^a^lⁱ^gⁿ^m^eⁿ^t^t^h^r^o^u^g^h^t^h^e^c^o^e^f^ﬁ^cⁱ^eⁿ^t^s^C2 ^aⁿ^d^C4 . Therefore,

our ﬁndings are consistent with the possibility of detecting

^s^t^r^oⁿ^g^r^o^t^a^tⁱ^oⁿ^a^l^a^lⁱ^gⁿ^m^eⁿ^t^d^u^e^t^o^t^h^e^c^o^e^f^ﬁ^cⁱ^eⁿ^t^C0 while

the corresponding state and angularly resolved product dis-

tributions do not show a signiﬁcant contribution from a

fourth Legendre polynomial P (cos ⌰). On the other hand,

the effective anisotropy parameter will contain a contribution

from the rotational quadrupole alignment moment due to the

t

rot

(

2)

0

(

2)

(

2)

(2)

separation R at the moment when the angular momenta are

C

established, and the corresponding average angle ␣_Cbe-

tween R and E ͑or ␮ at the moment of excitation͒. Devia-

(

2)

␧

tions from the limiting ␤ values should be prominent in fast

unimolecular decomposition, and in direct dissociation when

a large fraction of E_a_v_lis deposited in E_r_o_t. In these cases the

contribution of axial recoil should be evaluated before other

vector correlations are examined.

4

(

2)

^c^o^e^f^ﬁ^cⁱ^eⁿ^t^C2

.

The application of the model to the unimolecular decom-

position of NO describes well the dependence of ␤ on

2

1

NO(J) and O(E ), as observed in our work as well as by

See, for example, R. J. Gordon and G. E. Hall, Adv. Chem. Phys. XCVI,

͑1996͒.

a͒ R. N. Zare and D. R. Herschbach, Proc. IEEE 51, 173 ͑1963͒; ͑b͒ R. N.

t

1

͑

other investigators. The best ﬁt is obtained with R ϭ3.3

C

2

Ϯ0.3 Å and ␣ ϭ27Ϯ3°, implying that the dynamics during

C

Zare, Mol. Photochem. 4, 1 ͑1972͒.

3

4

dissociation involve a reduction of the ONO bending angle

in the Franck–Condon region followed by bond stretching.

Such reduction is compatible with the geometry of the coni-

G. G. Balint-Kurti and M. Shapiro, Chem. Phys. 61, 137 ͑1981͒.

O. Roncero, P. Villarreal, G. Delgado-Barrio, N. Halberstadt, and J. A.

Beswick, J. Phys. Chem. 98, 3307 ͑1994͒.

S. Mukamel and J. Jortner, J. Chem. Phys. 61, 5348 ͑1974͒.

T. J. Butenhoff, K. L. Carleton, R. D. van Zee, and C. B. Moore, J. Chem.

Phys. 94, 1947 ͑1991͒.

C. Jonah, J. Chem. Phys. 55, 1915 ͑1971͒.

S. Yang and R. Bersohn, J. Chem. Phys. 61, 4400 ͑1974͒.

H.-P. Loock, J. Cao, and C. X. W. Qian, Chem. Phys. Lett. 206, 422

5

6

2

cal intersection between the B and A states, and with the

2

1

calculated anisotropy of the ground-state PES.

3͒ Comparison of the classical model with quantum me-

7

8

9

͑

chanical calculations of the predissociation of the Ne–ICl

complex indicates that the main source of the transverse re-

coil of the ICl fragment is the orbital angular momentum

associated with high product J states. This case illustrates the

correspondence between the classical and quantum treat-

ments.

͑

1993͒.

1

0

1

D. H. Mordaunt, M. N. R. Ashfold, and R. N. Dixon, J. Chem. Phys. 104,

460 ͑1996͒.

6

S. H. Kable, J. Loison, D. Neyer, P. L. Houston, I. Burak, and R. N.

Dixon, J. Phys. Chem. 95, 8013 ͑1991͒.

R. J. Wilson, J. A. Mueller, and P. L. Houston, J. Phys. Chem. 101, 7593

12

͑

1997͒; J. A. Syage, ibid. 105, 1007 ͑1996͒; D. W. Neyer, A. J. R. Heck,

and D. W. Chandler, ibid. 110, 3411 ͑1999͒.

K.-H. Gericke, M. Lock, R. Fasold, and F. J. Comes, J. Phys. Chem. 96,

ACKNOWLEDGMENTS

1

3

4

422 ͑1992͒.

The authors wish to thank A. Sanov, I. Bezel, W. Kim,

and C. Wittig for helpful discussions. This work was sup-

ported by NSF Grant Nos. CHE-9632519 ͑H.R.͒ and CHE-

S. A. Reid and H. Reisler, J. Phys. Chem. 100, 474 ͑1996͒; S. A. Reid, A.

Sanov, and H. Reisler, R. Soc. Chem. Faraday Discuss. 102, 129 ͑1995͒.

L. Burnelle, A. M. May, and R. A. Gangi, J. Chem. Phys. 49, 561 ͑1968͒;

R. A. Gangi and L. Burnelle, ibid. 55, 851 ͑1971͒.

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J. Buenker, and C. Petrongolo, Mol. Phys. 70, 835 ͑1990͒.

V. P. Hradil, T. Suzuki, S. A. Hewitt, P. L. Houston, and B. J. Whitaker,

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15

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