NO angular distributions in the photodissociation of (NO)2 at 213 nm: Deviations from axial recoil

Full text of DOI:10.1063/1.1490599

NO angular distributions in the photodissociation of ( NO ) 2 at 213 nm: Deviations from
axial recoil
A. V. Demyanenko, A. B. Potter, V. Dribinski, and H. Reisler
Citation: The Journal of Chemical Physics 117, 2568 (2002); doi: 10.1063/1.1490599
View online: http://dx.doi.org/10.1063/1.1490599
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 117, NUMBER 6
8 AUGUST 2002
NO angular distributions in the photodissociation of „NO… at 213 nm:
2
Deviations from axial recoil
A. V. Demyanenko, A. B. Potter, V. Dribinski, and H. Reislera)
Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482
͑
Received 26 March 2002; accepted 13 May 2002͒
2
ϩ
Angular distributions of selected rotational states of NO(A ⌺ ,␯ϭ0) products obtained in the 213
nm photodissociation of (NO)2 have been determined in a molecular beam by using the
photofragment ion imaging technique. Specifically, images of NO(A,␯ϭ0) products in Nϭ0, 11,
2
and 19 have been recorded, for which the maximum energies available to the NO(X ⌸) products
Ϫ1
are 2038, 1774, and 1278 cm , respectively. The recoil anisotropy parameter of the
photofragments, ␤eff , decreases significantly at low center-of-mass translational energies from its
maximum value of 1.36Ϯ0.05, and depends strongly on the rotational angular momentum of the
photoproducts. This behavior is described well by a classical model that takes into account the
transverse recoil component mandated by angular momentum conservation. For each of the
observed NO(A) N states, highly rotating NO(X) levels are produced via planar dissociation, and
the angular momenta are established at an interfragment separation of about 2.6 Å. For most of the
center-of-mass translational energy range, both corotating and counterrotating fragments are
produced, but at the lowest energies, only the latter are allowed. The correlated rotational energy
distributions exhibit deviations from the behavior predicted by phase space theory, suggesting that
exit-channel dynamics beyond the transition state influences the product state distributions. In this
study, a new method for image reconstruction is employed, which gives accurate angular
distributions throughout the image plane. © 2002 American Institute of Physics.
͓
DOI: 10.1063/1.1490599͔
I. INTRODUCTION
covalent character. The equilibrium geometry of the gas
phase dimer, determined by microwave spectroscopy, is a
cis-planar trapezoidal structure of C2v symmetry.4 The
N–N bond distance is 2.236 Å, the ONN angle is 99.6°, and
the NO bond length is 1.161 Å ͑about 0.01 Å longer than in
the monomer͒.
,5
Weakly bound molecular complexes are valuable model
systems for studying vibrational energy transfer and unimo-
1
,2
lecular decay. The reason is that their bond dissociation
energy is generally lower than the energies of the vibrational
modes of the strongly bound subunits of the complex, and
the large disparity between the intra- and intermolecular vi-
brational frequencies leads to restricted intramolecular vibra-
tional redistribution ͑IVR͒, and to state-specific couplings to
the dissociative states. Angular and energy distributions in
the fragments are the traditional fingerprints from which
mechanisms are inferred, and correlated distributions pro-
vide even deeper insights into the evolution of the system
toward the dissociation asymptote, and the forces and con-
straints that act in the exit channel.
In contrast to the extensive work carried out in the time
and frequency domains on the IR vibrational predissociation
of the dimer,6
–10
little is known about the photodissociation
dynamics of the dimer from its electronic excited states. The
intense UV absorption spectrum, first reported by Bernstein
and Herzberg,11 is broad and nearly featureless, spanning the
range 190–240 nm with a maximum at 205 nm.1
2,13
The
transition apparently involves the orbital excitation b � a
2
1
in C symmetry, with the transition dipole moment parallel
2v
to the N–N bond.
The UV photodissociation of (NO) was first studied by
The dissociation of the nitric oxide dimer, (NO) , which
is formed by the pairing of two ⌸ radicals, is particularly
2
2
Kajimoto et al.1
4–18
Experiments at 193 nm revealed two
2
open product channels:
intriguing, because it belongs to the special family of weakly
bound complexes that are covalently bound.3 The possible
arrangements of the two unpaired electrons, which are lo-
cated in ␲* orbitals in the monomers, in the nearly degen-
erate molecular orbitals of the dimer, give rise to several
–10
2
ϩ
2
͑
͑
NO͒ →NO͑A ⌺ ͒ϩNO͑X ⌸͒,
2
Ϫ1
⌬
Hϭ44 910 cm
,
͑I͒
2
2
NO͒ →NO͑B ⌸͒ϩNO͑X ⌸͒,
2
3
close-lying electronic states. Extensive spectroscopic stud-
Ϫ1
ies have provided detailed information about the ground state
structure, bonding, and dissociation dynamics. The ground
⌬Hϭ46 190 cm
.
͑II͒
The NO(A) rotational state distributions appeared statistical,
1
Ϫ1 7
1
A state is bound by D ϭ710Ϯ15 cm , reflecting its
1 0
and were fit by a constrained version of phase space theory
͑
PST͒.18 Other studies showed weak alignment and vector
a͒Electronic mail: reisler@usc.edu
correlations in the NO(A) product, and the authors proposed
0021-9606/2002/117(6)/2568/10/$19.00
2568
© 2002 American Institute of Physics
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1

2569
J. Chem. Phys., Vol. 117, No. 6, 8 August 2002
Photodissociation of (NO)2
that the dissociation occurred mostly in the plane of the
1
5–17
dimer.
The dynamics were interpreted in terms of a fast,
direct dissociation, as indicated by the structureless absorp-
tion spectrum, and the large value of the recoil anisotropy
parameter, ␤ϭ1.0–1.4.
Studies of (NO) photolysis via channel I at 210 nm by
2
femtosecond time-resolved photoelectron spectroscopy
1
9
showed that (NO) dissociation is not direct but stepwise,
2
due possibly to excited state nonadiabatic couplings. It was
proposed that the initially prepared state decays nonadiabati-
cally with a lifetime of Ϸ0.3 ps to a ‘‘dark’’ state or manifold
of levels that in turn decays to the products on the longer
time scale of Ϸ0.7 ps. Ab initio calculations reveal that in the
UV region, both charge-transfer and Rydberg states can be
accessed optically, but the exact energies of the states, and
FIG. 1. Energy diagram for the UV dissociation of (NO)2 with subsequent
2
ϩ
2
ϩ
detection of NO(A) fragments by 1ϩ1 REMPI via the E ⌺ � A ⌺
transition.
3
the interactions among them, are presently unknown. Thus,
the photodissociation processes leading to channels I and II
are currently poorly understood, and are the subject of ongo-
ing investigations.
screen coupled to a microchannel plate ͑MCP͒ ion detector.
The ion optics of the ion-acceleration stage is based on a
scheme proposed by Parker and Eppink for velocity map
2
0
In this paper, we describe our first results on the photo-
dissociation of the NO dimer via reaction I using the velocity
map imaging technique. Specifically, we report correlated
energy and angular distributions in the NO(A,␯ϭ0) frag-
ment following 213 nm dissociation; i.e., at Eav1
24
imaging, and consists of a repeller, an open extractor elec-
trode ͑50 mm hole͒, and an open ground electrode ͑25 mm
hole͒. At an optimal voltage ratio for the repeller and extrac-
tor plates, all ions with the same initial velocity are focused
onto the same spot on the MCP detector.
Nitric oxide dimer is formed in a free jet expansion of
NO:He gas mixture. A doubly skimmed pulsed molecular
beam containing typically 7% of NO seeded in 2.0 atm of He
propagates through a hole in the repeller plate. Further
Ϫ1
ϭ2038 cm . This energy is sufficient to produce either
NO(X,␯ϭ0)ϩNO(A,␯ϭ0) or NO(X,␯ϭ1)ϩNO(A,␯
ϭ0), but not NO(X,␯ϭ0)ϩNO(A,␯ϭ1). Images are ob-
tained by monitoring selected rotational levels of NO(A,␯
ϭ0), from which the correlated energy and angular distribu-
tions in NO(X,␯ϭ0,1) are determined. The speed-dependent
downstream, (NO) is photolyzed with pulsed, linearly po-
2
larized, and mildly focused ͑f.l.ϭ60 cm lens͒ UV laser ra-
diation ͑0.1–0.3 mJ͒ that intersects the molecular beam at a
right angle. Photolysis at 213 nm is achieved by using the
frequency-doubled output of an excimer-laser pumped dye
laser system ͑Coumarin 440͒.
recoil anisotropy parameter, ␤ (V), is derived from the an-
eff
gular distributions by using,21
1
P͑␪,V͒ϰͩ ͓ͪ1ϩ␤ ͑V͒P ͑cos ␪͔͒.
͑1͒
eff
2
4
␲
2
ϩ
NO(A ⌺ ) photofragments are probed state selectively
with the focused output of a Nd:YAG laser pumped dye laser
system ͑Rhodamin 640, 0.2–0.5 mJ; f.l.ϭ50 cm lens͒ by 1
Our work demonstrates strong deviations from fragment
axial recoil at low center-of-mass ͑c.m.͒ translational ener-
gies. The deviations are described with an extended model
based on the one used previously to model speed-dependent
angular distributions in unimolecular decomposition of tri-
2
ϩ
2
ϩ
ϩ1 REMPI via the E ⌺ � A ⌺ transition at ϳ600
16,25,26
nm.
The R branch of the transition is used to probe
different rotational states of the NO(A) photofragment. Both
2
2
atomic species. This model, which assumes planar disso-
ciation, relates the transverse recoil to angular momentum
constraints for fragment pairs with high rotational angular
momentum.
2
ϩ
2
ϩ
the A ⌺ and E ⌺ states are spectroscopically well char-
acterized, and the low vibrational levels of the A and E states
25–28
are not predissociative.
At the employed probe laser in-
28
tensity, the monitored transitions are saturated, but the re-
sults do not change when the laser intensity is lowered by
about a factor of 10. The photolysis and probe lasers are both
linearly polarized, and introduced coaxially at a right angle
to the molecular beam. The polarization of the photolysis
light is directed perpendicular to the molecular beam, in a
plane parallel to the detector plane. To minimize complica-
tions due to vector correlations, the polarizations of the two
lasers are fixed parallel to each other. We expect that con-
tamination of the anisotropy parameter by rotational align-
ment is negligible.16 The time delay between the pump and
probe lasers is kept at 0Ϯ5 ns. Typical signals include the
II. EXPERIMENTAL METHODS
The experiment involves the generation of (NO) in a
2
molecular beam of NO seeded in He, resonance enhanced
2
ϩ
multiphoton ionization ͑REMPI͒ for product NO(A ⌺ ) de-
tection, and photofragment velocity map imaging for record-
ing the velocity and angular distributions of state-selected
products. (NO) dimers are excited in a single photon tran-
2
sition to above the threshold of reaction I, leading to fast
dissociation.
The velocity map imaging arrangement has been de-
scribed in detail elsewhere.23 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
4
summation of (1–2)ϫ10 laser firings. The angular distri-
bution of the fragments reflects predominantly the anisotropy
in the photolysis step. Figure 1 shows the probing scheme in
relation to the dissociation mechanism.
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2570
J. Chem. Phys., Vol. 117, No. 6, 8 August 2002
Demyanenko et al.
The correspondence between the measured signal and
NO) photolysis is checked in two ways. First, the produc-
(
2
tion of the dimer in the molecular beam is confirmed by
ϩ
detecting (NO)2 signals using nonresonant two-photon ion-
2
9
ization at 281 nm, and monitoring mass 60. Second, we
verify that the dimer is the main source of the NO(A) signal
by measuring the dependence of its intensity on NO concen-
tration in the gas mixture ͑from 2% to 20%͒. The measured
dependence is close to quadratic. By measuring the REMPI
2
ϩ
2
spectrum of the NO A ⌺ � X ⌸ transition in mixtures of
7%–20% NO, we estimate that the rotational temperature in
the beam is T ϭ3–5 K.
rot
The main source of background ion signal is non-state-
selective ionization of NO(A) fragments by the pump laser
213 nm radiation. Fortunately, it gives only a small contri-
bution to the total signal, and the 600 nm REMPI probe
signal is dominant by more than an order of magnitude. To
eliminate the contribution from ionization by the pump laser,
a subtraction routine is employed, which takes into account
the background signal obtained with the pump laser only.
Good spatial and temporal overlap of the two laser beams
minimizes effects due to ionization by the pump.
III. EXPERIMENTAL RESULTS AND ANALYSIS
Following 213 nm photolysis, images were obtained for
2
ϩ
30
rotational levels of NO(A ⌺ ,␯ϭ0) Nϭ0,11,19. The
electric field vector E of the photolysis laser was maintained
parallel to the vertical direction of the image plane, and the
product recoil velocities were aligned predominantly in the
polar direction of the image. The images were symmetrized
and, after background subtraction ͑see Sec. II͒, the two-
dimensional ͑2D͒ projections were converted to three-
dimensional ͑3D͒ velocity distributions by using the basis set
expansion ͑BASEX͒ method for image reconstruction.31
The BASEX method for reconstructing 3D images with
cylindrical symmetry from their 2D projections was devel-
oped to enable us to obtain accurate velocity and angular
distributions from regions of the image near the center,
where reconstruction noise is large in the commonly used
inverse Abel transform method ͑i.e., fast Fourier transform
followed by a discrete Hankel transform͒.32 The method is
based on expanding the projection in a basis set of functions
that are analytical projections of known well-behaved func-
tions. The original 3D image can then be reconstructed as a
linear combination of these well-behaved functions, which
have a Gaussian-type shape, with the same expansion coef-
ficients as the projection. In the process of finding the expan-
sion coefficients, regularization is used to achieve a more
reliable reconstruction of noisy projections. The method is
efficient and computationally cheap, and is particularly well
suited for transforming projections obtained in photoion and
photoelectron imaging experiments. The fragment’s speed
distribution is obtained by analytical integration of the recon-
structed image, expressed as a linear combination of basis
functions, at each distance from the center over all angles.
The angular distributions are directly obtained from the ex-
pression of the image in polar coordinates.
2
ϩ
FIG. 2. 2D cuts of the reconstructed images of NO(A ⌺ ,␯ϭ0,Nϭ0), and
2
ϩ
NO(A ⌺ ,␯ϭ0,Nϭ19) obtained from (NO) photodissociation at 213
nm. The electric field vector of the photolysis laser is parallel to the vertical
direction of the image plane.
2
Figure 2 shows typical 2D cuts of the reconstructed im-
ages of NO(A ⌺ ,␯ϭ0) fragments probed in the Nϭ0 and
19 states. The total excess energy with respect to the channel
I threshold is Eavlϭ2038 cm , and the maximum energies
available to the NO(X) products are Emaxϭ2038 and 1278
cm for NO(A) Nϭ0 and 19, respectively.
The speed distribution of each NO(A,N) is converted to
a translational energy distribution P(E ) from which the cor-
related NO(X,⍀,␯,J) internal energy ͑spin–orbit, vibra-
tional, and rotational, respectively͒ distribution P(E ) can
be derived, albeit with loss of state resolution, since several
,␯,J) states can contribute to each E . While
NO(A) fragments are produced only in ␯ϭ0, NO(X) frag-
ments correlated with NO(A ⌺ ,Nϭ0) are produced in
both ␯ϭ0 and 1. The two bright inner rings in Fig. 2 (N
ϭ0) correspond to NO(X,␯ϭ1,J): the intense center ring
2
ϩ
Ϫ1
Ϫ1
t
int
2
NO(X ⌸1
/2,3/2
t
2
ϩ
2
marks the onset of NO(X ⌸3 ,␯ϭ1,J), and the larger inner
ring is associated with NO(X ⌸1 ,␯ϭ1,J). The faint out-
,␯ϭ0,J). Similar
images are obtained for the other two NO(A) states Nϭ11
/2
2
/2
2
ermost ring corresponds to NO(X ⌸1
/2,3/2
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2571
J. Chem. Phys., Vol. 117, No. 6, 8 August 2002
Photodissociation of (NO)2
2
ϩ
FIG. 4. Dependencies of the anisotropy parameter ␤eff of NO(A ⌺ ,␯
ϭ0,Nϭ0,11,19) on the c.m. translational energy E ͑squares͒. The error in
t
2
ϩ
the measured values of the anisotropy parameter is typically Ϯ0.05. The
solid lines through the data are the fits obtained by using the model de-
scribed in the text. The corresponding energy distributions are also shown
FIG. 3. Total c.m. translational energy Et distributions of NO(A ⌺ ,␯
ϭ0,Nϭ0,11,19) obtained in photolysis of (NO)2 at 213 nm ͑solid lines͒.
The dashed lines show the corresponding PST simulations obtained with the
impact parameter restricted to 2.6 Å. The arrows show the highest transla-
2
ϩ
͑
solid lines͒. The arrows indicate the highest E allowed for NO(A ⌺ ,␯
t
2
ϩ
ϭ0,Nϭ0,11,19).
tional energies allowed for NO(A ⌺ ,␯ϭ0,Nϭ0,11,19) which are 2038,
Ϫ1
1
774, and 1278 cm , respectively.
particular speed͒ are fit to Eq. ͑3͒. The results are shown in
Fig. 4. Note that each measured ␤eff is a weighted average of
2
2
and 19, except that the NO(X ⌸1
,␯ϭ1) channel is en-
the ␤ parameters associated with the NO(X ⌸
,␯,J)
/2,3/2
1/2,3/2
ergetically closed.
states whose Eint is complementary to E . In the case of
t
Figure 3 shows the P(E ) distributions for NO(A) in
NO(A,Nϭ0), ␤eff depends on both the vibrational and rota-
tional state of the NO(X,␯ϭ0,1,J) cofragment. The depen-
dence of ␤eff on NO(X) rotational level is evident in the
NO(A,Nϭ11,19) data where only NO(X,␯ϭ0) is energeti-
cally allowed. The most remarkable feature in Fig. 4 is the
t
Nϭ0, 11, and 19. The distinct peaks in the translational en-
ergy distributions correspond either to a specific NO(X) vi-
brational level, ␯ϭ0 or 1 ͑in the Nϭ0 image͒, or to high
rotational levels of NO(X) ␯ϭ0 ͑in the Nϭ11 and 19 im-
ages͒.
similarity of the functional dependence of ␤ on E for the
t
In order to derive the dependence of ␤eff on E , the
angular distributions at each distance from the center ͑i.e., a
three N values, despite the large differences in the energy
available to the NO(X) products, and hence their corre-
t
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2572
J. Chem. Phys., Vol. 117, No. 6, 8 August 2002
Demyanenko et al.
sponding rovibrational distributions. In all three cases ␤eff
gradually decreases as E decreases, but then, at a specific
t
low translational energy, it increases abruptly and later goes
down again. The causes for the variation in ␤eff are discussed
in Sec. IV.
IV. CLASSICAL MODEL FOR NONAXIAL FRAGMENT
RECOIL
A. Angular momentum conservation and transverse
recoil
In a previous paper,22 a model based on energy and an-
gular momentum conservation was developed to explain the
observed dependence of the recoil anisotropy parameter ␤eff
on the translational energy of recoiling fragments in the pho-
todissociation of triatomic species whose transition dipole
moment lies in the plane. According to this model, the rota-
tion of the diatomic fragment during dissociation must be
compensated by the orbital angular momentum of the depart-
ing fragments. Consequently, in the limit of small kinetic
energy release and substantial rotational excitation of the di-
atomic fragment, the contributions from transverse recoil be-
come large, and ␤ deviates from its limiting values ͑2 and
Ϫ1͒ even for infinitely fast dissociation. The model was ap-
plied successfully to the photoinitiated dissociation of a tri-
FIG. 5. The angular momentum vectors JNO(A) , JNO(X) , and L in planar
dissociation of nonrotating NO dimer. In ͑a͒ the two fragments counterrotate
in the plane, and their combined angular momentum can be compensated by
a small value of L. In ͑b͒, the corotating products result in Jϭ
͉
JNO(X)
ϩJNO(A) , which must be compensated by a much larger L. The critical
͉
distance Rc is the average distance between the c.m. of the fragments when
dissociation occurs. The direction of the transition dipole moment ␮ is
shown by an arrow.
atomic molecule (NO ) and a van der Waals complex
2
2
2
͑ICl-Ne͒.
We now extend this model to the more complicated case
dissociation ͑assuming J(NO)2Ϸ0͒ all values of L between
of a four-atomic species dissociating into two diatomic frag-
ments. First, such dissociation is not necessarily planar, as
out-of-plane motions are also possible. Second, unlike the
case of jet-cooled triatomic dissociation for which the atomic
fragment angular momentum is very small, here three angu-
lar momenta ͑i.e., those of the two diatomic fragments and
the orbital angular momentum͒ can acquire any value pro-
vided that total angular momentum and energy are con-
served. For NO dimer dissociation we obtain:
͉
JNO(A)ϪJNO(X)
in the case of purely planar dissociation Lϭ
. The limiting values JNO(A)ϪJNO(X) and (JNO(A)
͉
рLр(JNO(A)ϩJNO(X)) are possible, while
͉
JNO(A)
ϮJNO(X)
͉
͉
͉
ϩJNO(X)) correspond, respectively, to fragments counterro-
tating and corotating in the parent molecular plane ͑see Fig.
5
JNO(X) states must correlate with low c.m. recoil energies
(
͒. Thus, for a fixed photolysis energy and NNO(A) , high
NO(X)
E ϭEt
NO(A)
ϩEt
) and a high average L. For corotating
t
fragments L may be very high.
J͑NO͒2ϭJNO͑A͒ϩJNO͑X͒ϩL,
͑2͒
There can be several reasons for the creation of highly
rotating NO fragments in the dissociation. In direct dissocia-
tion, the anisotropy in the angular potential is usually the
main source of torque, while in vibrational predissociation,
the main source of diatom angular momentum is bending or
torsional vibrations in the transition state. In the latter case,
for example, the mapping of symmetric and antisymmetric
bending levels can give rise to both corotating and counter-
rotating fragments, unless additional constraints are imposed
͑e.g., angular momentum constraint; see the following͒.
Irrespective of the source of angular momentum, the
fundamental conservation laws must be obeyed, and they
impose restrictions on product trajectories. In the c.m. coor-
dinate system, the classical orbital angular momentum is
given by
NO͑A͒
NO͑X͒
NO͑A͒
NO͑X͒
Eavlϭh␯ϪD ϭE
ϩEint ϩEt
ϩEt
, ͑3͒
0
int
NO(A,X)
where D is the dissociation threshold of channel I; E
are the c.m. recoil energies of the fragments; Eint
corresponding internal energies ͑i.e., vibrational, rotational,
and spin–orbit energies͒; J(NO)2, JNO(A,X) are the angular
0
t
NO(A,X)
are the
momenta of the parent (NO) molecule, and the NO(A,X)
2
fragments; and L is the orbital angular momentum.
The orbital angular momentum L is due to the rotation
of the c.m. of the recoiling fragments around the c.m. of the
parent molecule and is predominantly directed perpendicular
to the parent molecular plane. The latter means that only
components of the fragments’ angular momenta that are per-
pendicular to the parent molecular plane can be compensated
by L. For purely out-of-plane ͑helicopter-like͒ rotation L
Ϸ0, and JNO(X)ХJNO(A) must dominate in jet-cooled
samples.
Lϭm
r
ϫVNO͑A͒ϩm
r ϫVNO͑X͒
NO NO͑X͒
,
͑4͒
NO NO͑A͒
where rNO(X,A) is a vector directed from the c.m. of (NO) to
2
the c.m. of the NO(X,A) fragments, and VNO(X,A) is the c.m.
velocity of NO(X,A). Angular momentum conservation dic-
tates that
In our case, J(NO)2 is determined by the rotational tem-
perature of the parent, while NNO(A)ϭ0,11,19 are selected in
the experiment. Since both J(NO)2 and JNO(A) are known,
JNO(X) must be related to L, as per Eq. ͑2͒. In the case of 3D
͉
L
͉
ϭ
͉
mNORϫVNO
͉
ϭmNO•R•V • sin ␸,
͑5͒
NO
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2573
J. Chem. Phys., Vol. 117, No. 6, 8 August 2002
Photodissociation of (NO)2
where R is the distance between the center-of-masses of the
departing NO(A) and NO(X) fragments (RϭrNO(X)
ϪrNO(A)), and ␸ is the angle between R and VNO . If we
define axial recoil as the direction of recoil of the diatomic
fragments with no rotational angular momentum, then this
2
2
direction must coincide with R.
Using,
͉L͉ϭប
NO(A)
NO(X)
•
ͱ
L•(Lϩ1), and E ϭEavlϪEint ϪEint , the relation
t
between the recoil angle ␸ and the translational energy of
diatomic fragments can be expressed as
2
ប L•͑Lϩ1͒
2
sin ␸ϭ
.
͑6͒
2
C
NO͑A͒
int
ϪE Ni n Ot ͑X͒͒
R •m •͑E ϪE
NO
avl
Here RC is a critical interfragment distance where the rota-
tional and orbital angular momenta become established. Its
value can be loosely related to the value of R at the transition
state; i.e., R уR . Note that we treat RC as the distance
FIG. 6. Comparison of simulations of the change in the anisotropy param-
eter ␤eff with translational energy (Et) for the case of 3D ͑dashed line͒ and
planar ͑solid line͒ dissociation. The corresponding experimental dependence
␤eff(E ) for Nϭ19 is depicted with squares. The abrupt increase in ␤
C
TS
t
eff
between the center-of-masses of the two NO fragments,
rather than as the N–N bond length, since at this distance we
assume that the N–N bond is already broken.
observed at low E is reproduced only by assuming planar dissociation. See
t
the text for details.
The angle ␸ is related to the recoil anisotropy parameter
by
NO(X)
␤
best fit Erot
͑after convolution with the instrument resolu-
tion function͒ were derived next ͑see Sec. V͒, and the ␤eff
values were finally calculated as the weighted averages of
the individual ␤ values of those NO J states, whose energies
␤
ϭ2•P ͑cos ␸͒.
͑7͒
2
From Eq. ͑6͒ it is evident that at each E , a specific com-
avl
3
3
bination of fragments’ quantum states (JNO(A) ,JNO(X)) and
orbital momentum L will be associated with a distinct trans-
verse recoil angle ␸, and the deviations from axial recoil will
increase with increasing Erot of NO(X). For each
correspond to E values that fall within a small range ⌬E .
t
t
The L values allowed by Eqs. ͑2͒, ͑3͒, and ͑8͒ were deter-
mined for planar dissociation by using R as a fit parameter.
C
Since the maximum measured recoil anisotropy (␤ ϭ1.36
eff
(
JNO(A) ,JNO(X)) pair, however, several L values are possible,
e.g., рLр(JNO(A)ϩJNO(X)) for 3D disso-
ciation, and Lϭ
Ϯ0.05) was less than the limiting value (␤ϭ2), the final
anisotropy was scaled to provide a best fit to the experimen-
tal data. Possible causes for the reduction in the maximum
anisotropy value are discussed in Sec. V. The best fits of the
anisotropy dependencies were achieved by assuming planar
͉
JNO(A)ϪJNO(X)
͉
͉
JNO(A)ϮJNO(X)
͉ for planar dissociation. We
note that ͑i͒ although the measured anisotropy parameter is
an average over all possible ␤(JNO(A) ,JNO(X) ,L), for
JNO(X)ӷJNO(A) the allowed L values cluster around the value
of JNO(X) , and therefore large deviations from axial recoil
are expected for high Erot of NO(X); ͑ii͒ the variation of ␤
with E depends on the particular dissociation geometry, i.e.,
whether or not it is confined to a plane ͑see the following͒;
and ͑iii͒ in accordance with angular momentum conserva-
dissociation and R ϭ2.6Ϯ0.4 Å, and are shown in Fig. 4.
C
The increase in ␤eff near the threshold for the opening of
␯ϭ1 observed in the NO(A,Nϭ0) image illustrates that for
NO(X)
t
the same Eint , fragments generated in ␯ϭ1 have much
less rotational excitation than those in ␯ϭ0, and therefore
give rise to higher values of ␤. Such behavior has been ob-
served before in the photodissociation of NO2 .2
2
tion, for a given Eavl , JNO(A) , and R , the condition
C
2
The abrupt increase of ␤ at small translational energies
observed in the Nϭ11, 19 images has a different origin. It is
caused by the fact that at low fragments’ translational ener-
gies, the high L values that correspond to corotating frag-
ments become energetically forbidden ͓cf. Eq. ͑8͔͒. In planar
ប •L•͑Lϩ1͒
E у
t
͑8͒
2
C
2
•m •R
NO
must be satisfied, which constrains the maximum allowed
value of L for a particular pair (JNO(X) ,JNO(A)).
dissociation, only two values, L ϭ
͉
JNO(A)ϪJNO(X)
͉ and
Ϫ
L ϭ
͉
JNO(A)ϩJNO(X)
͉
, are possible for each (JNO(A) ,JNO(X))
ϩ
B. Application of the model
pair ͑assuming parent JХ0͒, and the corresponding anisot-
ropy parameter ␤eff is an average of the values corresponding
The above-described model is general, but its implemen-
tation is not always straightforward, because the geometrical
configuration at which it is applied is ill defined. In particu-
lar, the use of RC is approximate and its value is unknown,
and must be either inferred from fits to the data or obtained
from calculations. Also, we do not know a priori whether or
not the dissociation is confined to the plane.
to LϪ and L , with the higher L resulting in a smaller value
ϩ
of ␤eff . Evidently, the exclusion of fragments with the
smaller ␤ values results in a substantial increase in the ob-
served value of ␤eff . In contrast, when all values of L be-
tween Lϩ and LϪ are allowed ͑i.e., when the dissociation is
not confined to a plane͒, this becomes only a minor effect, as
shown in Fig. 6. We conclude that the increase of ␤eff at very
Simulations of the variation of ␤ (E ) obtained from the
eff
t
NO(A,N) images were performed as follows. From the Et
low E is merely a manifestation of angular momentum con-
straints in planar dissociation that allows the formation of
only counterrotating products.
t
NO(X)
distribution for each image, Erot
was generated by energy
conservation. The NO(X) rotational state distributions that
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2574
J. Chem. Phys., Vol. 117, No. 6, 8 August 2002
Demyanenko et al.
V. DISCUSSION
decay time of the excited electronic state of the (NO) elec-
2
tronic transition under study is ␶ϳ0.3 ps,19 the rotational
structure in the upper state is unresolved, and a wave packet
of rotational states is excited. The evolution of this wave
packet in time is the origin of the effect of parent rotation on
the anisotropy, as described by Eq. ͑9͒.
A. Angular distributions of NO fragments: Variation
of ␤eff
Unlike the well-characterized ground state of (NO)2 ,
very little is known about its electronically excited states.
Nevertheless, according to molecular orbital analysis and ab
The effect of out-of-plane motion in the dissociation on
1
initio calculations the transition dipole moment of the B
2
␤
is harder to evaluate. Such motions may be a result of a
1
�
A UV system lies in the molecular plane, parallel to the
1
symmetry requirement for surface crossing, Franck–Condon
overlap, or vibrational energy flow during dissociation. They
may contribute here as well.
1
5–17
N–N bond.
This geometry corresponds to a limiting
value of ␤ϭ2, whereas the maximum value of the anisotropy
parameter, ␤maxϭ1.36Ϯ0.05, measured at high E for non-
rotating products is much smaller.
t
The value of ␤ can also be reduced if rotation of the
recoil axis takes place during the dissociation due to the evo-
lution of the molecular geometry. This may happen, for ex-
ample, if the geometry of the electronically excited state dif-
fers substantially from that of the ground state, as happens in
Several factors can lead to a reduction in the limiting
3
4
value of ␤, among them: ͑i͒ simultaneous excitation of par-
allel and perpendicular transitions; ͑ii͒ parent rotation during
dissociation ͑lifetime effect͒; ͑iii͒ out-of-plane motions of the
dissociating molecule; ͑iv͒ deviations from axial recoil due
to in-plane motions; and ͑v͒ contributions from background
signals with low ␤ values.
2
2
the case NO photodissociation. Recall that ␤ϭ2.0 can be
2
obtained for nonrotating fragments only when both the tran-
sition dipole moment ␮ and R lie parallel to the a axis of a
⑀
prolate top. When R deviates from the a axis, ␤Ͻ2.0 will be
obtained even for fast dissociation generating nonrotating
fragments.22
Additionally, we cannot exclude the possibility that a
small background signal whose ␤ is small leads to some
reduction in ␤eff . As discussed previously, background sub-
traction is essential to the data processing, and although done
carefully, a small reduction in ␤ due to this factor cannot be
totally excluded.
In conclusion, it is clear that some of the reduction in the
maximum value of ␤ for nonrotating products is due to the
finite dissociation lifetime. Out-of-plane motions and rota-
tion of the recoil axis due to the evolution of the molecular
geometry during the dissociation are likely causes of further
reduction. Contamination from signals with lower ␤ values
cannot be definitively excluded, but is expected to make only
a minor contribution. In contrast, for highly rotating frag-
ments, the large reduction in ␤ is caused by the transverse
recoil component in planar dissociation, as described in Sec.
IV.
If an out-of-plane perpendicular transition is also ex-
cited, it is expected to add a constant negative contribution to
␤, which is independent of E , and will therefore reduce ␤eff
t
equally for all fragment speeds. However, neither experiment
nor theory support the existence of a perpendicular transition
correlated with NO(A). Moreover, the contribution of such
transition is expected to change as the excitation wavelength
is varied, but measurements of ␤ in the range 193–220 nm
do not reveal a large variation.1 An in-plane perpendicular
transition would not result in the observed dependence of
␤eff on E , as it would contribute values of ␤ ranging from
7,35
t
eff
2,36
2
Ϫ1 at large E to positive ones at small E .
t
t
Parent rotation during dissociation is known to cause a
reduction in ␤. When the dissociation is not infinitely fast on
the time scale of parent rotation, the parent may rotate
through an angle ␺ϭ␻
the angular velocity of rotation of the parent molecule and
͗ ͘
␶ prior to dissociation, where ␻ is
͗
␶͘ is its average lifetime. In the case of a dissociating di-
3
7
atomic this leads to the well-known expression,
1
ϩ͑␻
͗
␶
͘
͒2
When comparing the simulated and measured distribu-
␤
͑E ͒ϭ
.
͑9͒
t
͒2
tions of ␤eff as a function of E , it must be borne in mind
1
ϩ4•͑␻
͗
␶
͘
t
that for low NO(X) rotational levels ͑high E ͒, no change in
t
Taking into consideration the rotational temperature of
the parent ͑3–5 K͒, this formula gives ϭ1.8–2.0 ps for
ϭ1.33–1.41. The measured buildup time of NO(A) at 210
␤ is expected. It is only by examining the region of high
͗
␶
͘
NO(X) rotational states ͑low E ͒ that the mechanistic origin
t
␤
of the ␤ variation can be inferred. Our study shows that the
high J states of NO(X) must be produced via planar disso-
ciation. For low NO(X) rotational states, our work cannot
distinguish between planar and nonplanar dissociation
events, or between corotating and counterrotating fragments.
The ␤eff distributions obtained from images of low and
1
8
nm is ␶Ϸ1 ps, which corresponds to ␤Ϸ1.70, clearly
higher than the measured value. Using the classical method
of Yang and Bersohn gives similar results.38 Since the prod-
uct buildup time has been measured at 210 nm, we have
checked to see whether the maximum value of ␤ depends on
excitation energy. At 216 and 220 nm, the measured values
are 1.30Ϯ0.05 and 1.16Ϯ0.05, respectively, which would
correspond at most to a modest increase in lifetime. We con-
clude that lifetime reduction contributes to the reduction in
high N states of NO(A) can all be fit by using R ϭ2.6 Å
C
Ϯ0.4, which places the transition state ͑TS͒ at an internu-
clear distance close to the equilibrium N–N distance in the
ground state ͑2.26 Å͒. Therefore, the TS can be considered
‘‘tight,’’ i.e., it has a vibrational structure. In such a case, the
projection ͑mapping͒ of symmetric and antisymmetric in-
plane bending wave functions into NO angular momentum
states will give rise to both corotating and counterrotating
fragments. Only for very highly rotating NO fragments,
␤, but it is not the only factor.
The above-mentioned treatment uses a classical ͑or
semiclassical͒ approach to parent rotation, which has its limi-
tations, especially for the case of small parent rotational
3
9
quantum numbers. However, considering that the measured
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2575
J. Chem. Phys., Vol. 117, No. 6, 8 August 2002
Photodissociation of (NO)2
tive to the higher ones. The global NO(A) rotational state
distribution measured by us at 213 nm and shown in Fig. 7 is
also colder than predicted by PST.
The picture that emerges from the NO(A) images re-
corded at 213 nm is that the NO(X) rotational distributions
correlated with specific NO(A) N states also deviate from
statistical ͑PST͒ predictions. Most noticeably, the population
of high NO(X) rotational levels is always overestimated by
PST compared to the experiment. Since unconstrained PST
did not give a good fit, we used an impact parameter con-
straint in which the maximum impact parameter was 2.6 Å,
the interfragment separation that best fits the anisotropy data.
This somewhat reduced the population of high J states, but
the fit still was not satisfactory ͑Fig. 3͒.
Assuming that the dissociation is constrained entirely to
the molecular plane for all rotational states makes the agree-
ment poorer. The failure of PST to describe the rotational
distribution is not surprising, since the dissociation involves
a rather tight TS, a situation for which PST does not apply.
In order to obtain the correlated NO(X) rotational distri-
butions, the speed distributions were fit by a set of Gaussian
FIG. 7. Rotational energy distribution of NO(A,␯ϭ0) fragments after 213
nm photolysis (Eavlϭ2038 cm ).
Ϫ1
would the production of corotating products become forbid-
den by angular momentum constraints ͓Eq. ͑8͔͒. On the other
hand, the repulsive forces between the fragments already
start to act near the Franck–Condon region. In such a situa-
tion, the angular anisotropy of the potential energy surface,
and in particular the coupling between the angular and the
dissociative N–N coordinates, are expected to influence the
final state distributions as well, causing deviations from sta-
tistical behavior ͑see the following͒.
functions, each centered at the position of
a
2
NO(X ⌸1
,J) state. The widths of the Gaussian func-
/2,3/2
tions were optimized to give the best fit for each speed dis-
tribution. The relative populations of the J states were then
determined by a least-squares fit, using Tikhonov regulariza-
tion to stabilize the coefficients in the overlapped region of
low J states. This procedure, which was found to be stable
and superior to iterative methods, is similar to the one used
in the image reconstruction.31 Because of the small separa-
tion between low J states (Jр5) in the energy distribution,
the accuracy of extraction of their populations is rather low
͑Ϯ20%͒, while for higher J’s it is about Ϯ5%. The extracted
rotational distributions show that the two spin–orbit states of
NO(X) are approximately equally populated in all the corre-
lated distributions. Therefore, in the rotational state distribu-
tions shown in Fig. 8, which are compared to PST predic-
tions, the spin–orbit levels were suppressed for clarity of
presentation.
Inspection of Fig. 8 reveals that the rotational state dis-
tributions are not described well by PST, and appear to ex-
hibit signatures of exit channel dynamics. The distributions
are likely to include contributions from one or more of the
following sources: ͑i͒ mapping of TS wave functions ͑bend-
ing and torsional͒ into fragment rotations; ͑ii͒ exit-channel
repulsion and the angular anisotropy in the potential energy
surface; and ͑iii͒ rotational energy flow beyond the TS.40 The
nature of the dynamical forces can be clarified by extending
the measurements to photolysis wavelengths closer to the
threshold of reaction I, and such measurements are currently
in progress.
It is also worth noting that the model presented here is in
agreement with the results of Naitoh et al. obtained with 193
1
7
nm photolysis. These authors inferred the value of ␤eff
from Doppler profiles of selected N states of NO(A) in the
range Nϭ7–23, and found that the average ␤eff value in-
creased for higher N states. Recall that with jet-cooled
samples, LХJNO(A)ϩJNO(X) , and the fragments have a
broad rotational state distribution extending to the energetic
limit (JNO(X)р60).17 The peak in the Doppler profile corre-
sponds to low fragment velocities ͓i.e., to high J states of
NO(X)͔, resulting in, on average, a large deviation from
axial recoil, and a small average ␤eff . As the N level of the
monitored NO(A) state increases ͑but is still not too high͒,
the maximum allowed NO(X,J) level decreases, and there-
fore the average ␤eff should increase, as indeed observed.17
B. Correlated energy distributions
This paper is concerned with the stereodynamical as-
pects of (NO) photodissociation, while the corresponding
2
scalar properties ͑correlated energy distributions͒ will be the
subject of a separate report. Here we comment only on those
dynamical aspects that follow from the anisotropy measure-
ments.
The global rotational state distributions of the NO(A,␯)
fragments measured before at 193 nm were fit by a con-
strained version of PST, in which restrictions due to pre-
ferred in-plane dissociation for higher N’s were included.18
The relative contribution of in- and out-of-plane motions was
adjusted for each NO(A) rotational state until a fit to the data
was obtained. Such a fit was carried out for vibrational levels
vϭ0–2. The unconstrained PST distributions underesti-
mated the contributions of low NO(A) rotational states, rela-
VI. SUMMARY
͑1͒ A classical model that describes quantitatively the
deviations from axial recoil in photodissociation of triatomic
species is applied to the dissociation of the tetra-atomic,
weakly bound molecule (NO) . It is shown that the devia-
2
tions are governed by the conservation of angular momen-
tum.
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2576
J. Chem. Phys., Vol. 117, No. 6, 8 August 2002
Demyanenko et al.
ACKNOWLEDGMENTS
Support by the National Science Foundation and the Do-
nors of the Petroleum Research Fund, administered by the
American Chemical Society is gratefully acknowledged. The
authors benefited greatly from discussions with Albert
Stolow, Carl Hayden, and David W. Chandler.
1
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FIG. 8. Correlated distributions P(J) of NO(X,J) states extracted from the
observed energy distributions P(E ) ͑see Fig. 3͒ for NO(A,Nϭ0,11,19) are
25
t
given as a bar graph. The corresponding PST distributions are shown for
comparison ͑solid line͒.
2
6
͑
2͒ The application of the model to the 213 nm photo-
27
2
8
dissociation of (NO) describes well the dependence of ␤
2
eff
on NO(X,J) for the monitored NO(A,␯ϭ0,N) levels. The
2
9
best fit in all cases is obtained at an interfragment separation
Dethlefs, and V. Bondybey, J. Chem. Phys. 96, 7171 ͑1992͒.
of R ϭ2.6Ϯ0.4 Å, close to the N–N distance in the ground
30
2
ϩ
2
ϩ
C
Probed A ⌺ ͑as well as the intermediate E ⌺ ͒ state belongs to Hund’s
case b, i.e., the rotational and spin–orbit motions are uncoupled. The
rotational energy is best described by the ‘‘good’’ quantum number N,
ErotϭB•N•(Nϩ1) ͑ignoring the spin–rotational splitting͒. As a conse-
state ͑2.26 Å͒. For high rotational levels of NO(X), the dis-
sociation must take place in the plane, and it produces both
corotating and counterrotating NO fragments, possibly as a
result of mapping of symmetric and antisymmetric transition
state wave functions into product rotational states. For the
highest allowed J states, due to angular momentum con-
straints, only counterrotating products are allowed.
2
ϩ
2
ϩ
2
ϩ
quence, in the E ⌺ � A ⌺ transition, an A ⌺ state with a specific N
͑
and two different JϭNϮ1/2͒ is probed.
31
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