ar
X
iv
:0
91
0.
09
52
v2
[
ph
ys
ics
.ch
em
-p
h]
3
N
ov
20
09
An analytic model of the stereodynamics of rotationally inelastic molecular collisions
Mikhail Lemeshko∗ and Bretislav Friedrich†
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany
(Dated: November 3, 2009)
We develop an analytic model of vector correlations in rotationally inelastic atom-diatom collisions
and test it against the much examined Ar–NO (X2Π) system. Based on the Fraunhofer scattering of
matter waves, the model furnishes complex scattering amplitudes needed to evaluate the polariza-
tion moments characterizing the quantum stereodynamics. The analytic polarization moments are
found to be in an excellent agreement with experimental results and with close-coupling calculations
available at thermal energies. The model reveals that the stereodynamics is governed by diffraction
from the repulsive core of the Ar–NO potential, which can be characterized by a single Legendre
moment.
Observing correlations among the vectors that charac-
terize a collision can disclose all there is to know about
how the collision proceeds [1]. Dudley Herschbach [2]
likened vector correlations to “forbidden fruit” whose
“tasting” reveals what would otherwise remain hidden.
An example he frequently cites is the undoing of the az-
imuthal averaging about the initial relative velocity vec-
tor via a three-vector correlation, which reveals stereo-
dynamical features lost by averaging over the initial dis-
tribution of impact parameters. The pioneering work
of Herschbach and coworkers [3] on vector correlations
in the domain of molecular collisions spurred an effort
to extract the information hidden in molecular dynam-
ics computations, both quasiclassical and quantum, as
these contain vector correlations as a default bonus [4].
However, even when characterized to the full by vec-
tor correlations, the why of collision stereodynamics can
only be answered as well as the theoretical method ap-
plied to treat the collisions allows. In the present work,
we extract vector correlations from an analytic model of
direct rotationally inelastic atom–diatom collisions, and
thereby gain a particularly simple, yet perspicacious in-
sight into their stereodynamics.
The collision model employed is based on the Fraun-
hofer scattering of matter waves [5], recently extended
to treat collisions in fields [6]. In contrast to classi-
cal or semiclassical theories, the Fraunhofer model fur-
nishes complex scattering amplitudes needed to extract
the characteristics of vector correlations that reflect the
quantum stereodynamics. Owing to its quantum nature,
the model accounts for diffraction, interference, and other
nonclassical effects. As an example, we deal with inelastic
collisions of closed-shell atoms with rotationally polarized
symmetric-top-equivalent linear molecules, represented
by the much examined Ar–NO (X2Π) system[7, 8]. The
vector correlations obtained from the Fraunhofer model
are found to be in an excellent agreement with the re-
sults of experiments and close coupling calculations of
Wade et al. [9]. This allows interpreting the collision
stereodynamics of the Ar–NO (X2Π) system in terms of
the Fraunhofer model.
The Fraunhofer model was described in detail in
refs. [5, 6]. It is based on the sudden approximation,
which treats the rotational motion as frozen during the
collision and thereby allows expressing the inelastic scat-
tering amplitude in terms of the elastic one. The elastic
scattering amplitude is, in turn, expressed in terms of
the amplitude for Fraunhofer diffraction of matter waves
from a sharp-edged, impenetrable obstacle acting in place
of the molecular scatterer. At collision energies of hun-
dreds of cm−1, consistent with the sudden approxima-
tion, the shape of the scatterer is approximated by the
repulsive core of the atom–molecule potential, with the
attractive part disregarded. The Fraunhofer model ren-
ders fully state- and energy-resolved scattering ampli-
tudes and all the quantities that unfold from them in
analytic form.
The stereodynamics of an atom–diatom collision is
usually described by a set of four vectors: the initial and
final relative velocities, k and k′, and the initial and final
rotational angular momenta of the diatomic molecule, j
and j′. We use the initial and final relative velocities k
and k′ to define the XZ plane of the space-fixed coordi-
nate system, with the initial relative velocity k pointing
along the Z axis. In keeping with the convention of Orr-
Ewing and Zare [10], we characterize the spatial distribu-
tion of the angular momenta relative to the XZ plane by
the polarization moments A(k)q± , which arise as coefficients
in the expansion of the density operator over the state
multipoles [11]. Since within the Fraunhofer model the
scatterer is two-dimensional, the model can only account
for alignment, but not for orientation [6]. As a result, all
polarization moments with odd k or q vanish within the
model.
For this case study, we chose the k− k′ − j′ three-
vector correlation in the Ar–NO (j = Ω = 1/2 → j′,Ω′ =
1/2) collisions, as this can be compared with the re-
sults of experiments and close-coupling calculations of
Wade et al. [9]. In addition, we illustrate the scope of
the model by treating the k− j− k′ and k− j− k′ − j′
correlations in the Ar–NO (j = Ω = 3/2 → j′,Ω′ = 3/2)
scattering which, to date, have not been measured or
evaluated. We restrict our considerations to the two low-
est rotational channels, j′ = 9/2 and 17/2, reported

2-1
-0.5
0
0.5
1
1.5
j'=9/2
-1
-0.5
0
0.5
1
0o 5o 10o 15o 20o 25o 30o 35o
!
j'=17/2
A(
2)
(
j')
p
ol
ar
iz
at
io
n
m
om
en
t
0
FIG. 1: Polarization moment A(2)0 (j′) pertaining to the
k− k′ − j′ three-vector correlation in Ar–NO (j = Ω =
1/2,→ j′,Ω′ = 1/2) collisions at 520±70 cm−1. The analytic
results furnished by the Fraunhofer model (red solid line) are
compared with the experiment (black dots) and close-coupling
calculations (blue dashed line) of ref. [9].
in ref. [9], and average over the e/f parity states as
these have not been resolved in the experiment. We take
into account the energy spread of the molecular beams,
Ecoll = 520±70 cm−1, by averaging our results over three
collision energies corresponding to the most probable en-
ergy and to energies at half-maximum of an essentially
Gaussian collision energy distribution. In determining
the Ar–NO potential, we rely on the most recent po-
tential energy surface (PES) obtained by Sumiyoshi et
al. [12] and make use of only the average potential, Vsum,
since the differential and depolarization cross sections are
found to be only weakly affected by the difference PES,
Vdif [13]. The PES of ref. [12] comes close to that of
Alexander [8] and both PES’s yield essentially the same
polarization moments.
TABLE I: Meaning of the A(2)0 (j) and A
(2)
2+(j) alignment polar-
ization moments. The Z axis points along the initial relative
velocity k. The final relative velocity k′ lies in the X > 0
half of the XZ plane. The indicated ranges of the moments
correspond to the high-j limit. The A(2)2−(j) moment vanishes
identically.
Moment A(2)0 (j) A
(2)
2+(j)
Meaning j along Z j along X or Y
Range j ⊥ Z → -1 j ‖ X → -1
j ‖ Z → 2 j ‖ Y → 1
-1
-0.5
0
0.5
1
j'=9/2
-1
-0.5
0
0.5
0o 5o 10o 15o 20o 25o 30o 35o
!
j'=17/2
A(
2)
(
j')
p
ol
ar
iz
at
io
n
m
om
en
t
2+
FIG. 2: Polarization moment A(2)2+(j′) pertatining the
k− k′ − j′ three-vector correlation in the Ar–NO (j = Ω =
1/2,→ j′,Ω′ = 1/2) collision at 520±70 cm−1. The analytic
results furnished by the Fraunhofer model (red solid line) are
compared with the experiment (black dots) and close-coupling
calculations (blue dashed line) of ref. [9].
In order to characterize the k− k′ − j′ three-vector
correlation, we make use of the alignment moments
A(2)0 (j′) and A
(2)
2+(j′) of the diatomic’s final rotational an-
gular momentum j′ with respect to the XY plane. The
A(2)0 (j′) moment accounts for alignment of j′ with respect
to the initial relative velocity k and, in the high-j′ limit,
ranges between −1 and 2. Positive (negative) values of
A(2)0 (j′) correspond to j′ ‖ k (j′ ⊥ k, in which case j′ lies
in the XY plane). The A(2)2+(j′) moment varies from −1
to 1 (in the high-j′ limit). Its positive (negative) values
correspond to alignment of j′ along the Y -axis (X-axis).
This is summarized in Table I.
Figures 1 and 2 display the A(2)0 (j′) and A
(2)
2+(j′) mo-
ments obtained in analytic form from the Fraunhofer
model along with the results of experiment and close-
coupling calculations of Wade et al. [9]. The agreement
between the Fraunhofer model and the close-coupling cal-
culation is compelling.
Unfortunately, only two experimental points are avail-
able for the scattering angles concerned. Therefore, it is
not clear whether the oscillatory behavior at small-angles
would indeed show up in an experiment. We hope that
the present work will inspire an experiment whose reso-
lution will suffice to clarify this issue.
One can see that for zero scattering angle, ϑ = 0,
A(2)0 = −1 and A
(2)
2+ = 0. The reason is geometric: in
pure forward scattering, the j′ vector must be perpendic-
ular to k. Also, since k is roughly parallel to k′ for very
small ϑ, the j′ vector is approximately perpendicular to

3-0.8
-0.4
0
0.4
0.8
j'=9/2
-0.8
-0.4
0
0.4
0o 5o 10o 15o 20o 25o 30o 35o
!
j'=17/2
A(
2)
(
j)
p
ol
ar
iz
at
io
n
m
om
en
t
0
FIG. 3: Polarization moment A(2)0 (j) pertaining to the
k− j− k′ three-vector correlation in Ar–NO (j = Ω = 3/2,→
j′,Ω′ = 3/2) collisions at 520±70 cm−1 obtained from the
Fraunhofer model.
-1
-0.5
0
0.5
1
1.5 j'=9/2
-1
-0.5
0
0.5
1
1.5
0o 5o 10o 15o
j'=17/2
!
A(
2)
(
j')
p
ol
ar
iz
at
io
n
m
om
en
t
0
FIG. 4: The k− j− k′ − j′ four-vector correlation in the Ar–
NO (j = Ω = 3/2,→ j′,Ω′ = 3/2) collisions at 520±70 cm−1
in terms of the dependence of the final polarization moment
A(2)0 (j′) on the initial alignment A
(2)
0 (j) = 0 (black line),
A(2)0 (j) = −0.8 (blue dashed line), and A
(2)
0 (j) = 0.8 (red dot-
ted line). Obtained analytically from the Fraunhofer model.
k′. For small but nonzero scattering angles, ϑ ∼ 5◦, the
A(2)0 moment becomes positive, both for j′ = 9/2 and
17/2, indicating that j′ tends to align along k.
The A(2)2+ moment, on the other hand, exhibits narrow
positive oscillations at very small angles (ϑ ≈ 1◦), but is
in general negative, which corresponds to alignment of j′
along the X-axis.
Interestingly, the polarization moments presented in
this paper are in a quantitative agreement with the ac-
curate, close-coupling calculations, while the differen-
tial cross sections for the Ar–NO scattering, evaluated
in ref. [6], agree only qualitatively. From this we draw
the conclusion that the polarization moments are mainly
due to the hard-core part of the potential. This con-
clusion is also supported by purely classical arguments
based on the conservation of the projection of angular
momentum on the collision’s kinematic apse, see, e.g.,
ref. [14]. The phase shift of the moment’s oscillations as
derived from the model with respect to those obtained
from the close-coupling calculation is likely due to ne-
glecting in the model the attractive part of the Ar–NO
potential. This explanation is supported by a general-
ized Fraunhofer model that accounts for both attraction
and repulsion [15] and which shifts the oscillations back
toward smaller ϑ.
The alignment moments of Figs. 1 and 2 were normal-
ized by the differential cross sections obtained from the
Fraunhofer model. Since the Fraunhofer differential cross
sections decrease faster with the scattering angle than
their close-coupling counterparts [6], the oscillations of
the analytic polarization moments are left relatively un-
damped at large scattering angles.
Since the j vector with j = 1/2 can only be oriented
but not aligned, the only-to-alignment-sensitive Fraun-
hofer model cannot handle vector correlations involving
the j-vector with j = 1/2. However, since the j vec-
tor with j = 3/2 can be aligned, we worked out the
k− j− k′ and k− j− k′ − j′ vector correlations for the
Ar – NO (j = 3/2,Ω,→ j′,Ω) scattering within the
Ω = 3/2 manifold. Figure 3 displays the A(2)0 (j) polariza-
tion moment for the k− j− k′ three-vector correlation.
One can see that the small-angle scattering is favored by
positive values of the A(2)0 (j) moment, which corresponds
to j ‖ k, i.e., to a “broadside” approach of NO with re-
spect to k, which enhances the scattering cross section.
However, for larger scattering angles, the A(2)0 (j) moment
becomes slightly negative, attesting to a preference for an
“edge-on” approach with j ⊥ k.
Figure 4 exemplifies the k− j− k′ − j′ four-vector cor-
relation in terms of the alignment moment A(2)0 (j′) of the
final j′ for different polarizations A(2)0 (j) of the initial
j. For an unpolarized initial state (black line), j′ tends
to align perpendicular to k (“broadside” recoil) for very
small ϑ, but reverses to a slight alignment in the parallel
direction (“edge-on” recoil) for larger scattering angles.
Initial polarization of NO such that j ⊥ k is seen to result
in only small changes of the final alignment (blue dashed
line). However, in the case of a “broadside” approach,
j ‖ k, the stereodynamics changes significantly (red dot-
ted line). The A(2)0 (j′) moment remains slightly negative
throughout the range of scattering angles, indicating a

4propensity for an “edge-on” recoil, with j′ ⊥ k.
The Fraunhofer model readily explains the above re-
sults: the analytic scattering amplitudes are proportional
to the Bessel functions, which is a signature feature of
diffraction. It is thus a diffractive oscillatory pattern that
determines the angular dependence of the polarization
moments. While the shape and frequency of the angular
oscillations are entirely determined by the hard core of
the PES, their position is somewhat influenced by the
PES’s attractive branch [15]. The Clebsch-Gordan co-
efficients that appear in the scattering amplitude bring
about selection rules that constrain the final parity of
the states and the projections of the angular momentum
j′ on k. Within the model, the shape of the scatterer
enters through the Legendre moments of a series expan-
sion of the hard-core PES in terms of Legendre polyno-
mials, Pκ(cos θ). The angular momentum algebra that
the model entails gives rise to additional selection rules
which allow for nonzero contributions to the polarization
moments to arise only from Legendre moments of order
κ ≥ j′ − j. Therefore, the vector correlations for the
j = 1/2, 3/2 → j′ = 9/2 channels are governed by the
Legendre moment with κ = 4, whereas the κ = 8 Leg-
endre moment governs the polarization moments of the
j = 1/2, 3/2→ j′ = 17/2 channels.
Moreover, since the Fraunhofer model can account for
collisions in an electrostatic field [6], we investigated the
effect of the field on the polarization moments. A field
of 16 kV/cm, sufficient to significantly orient the NO
molecule in the space-fixed frame, was found to cause
only a tiny difference in the parity-resolved polarization
moments as compared with the field-free ones. Upon
averaging over the e/f states, the effect of the field was
found to be altogether negligible.
In summary, we made use of the Fraunhofer model
of direct rotationally inelastic atom–diatom collisions to
study vector correlations in such collisions analytically.
The vector correlations obtained from the model closely
reproduce those extracted from close-coupling calcula-
tions which, in turn, agree well with experiment. The
Fraunhofer model of vector correlations demonstrates
that the stereodynamics of the Ar–NO rotationally in-
elastic collisions is contained solely in the diffractive part
of the scattering amplitude which is governed by a single
Legendre moment characterizing the anisotropy of the
hard-core part of the system’s PES. Given the “geomet-
ric” origin of this behavior – ordained by the angular
momentum algebra – we expect to find a similar behav-
ior in other systems.
We thank Marcelo de Miranda and Pablo Jambrina for
helpful discussions, Elisabeth Wade and David Chandler
for making available to us the results of their experiments
and computations, and Millard Alexander for the Ar–NO
PES. We are grateful to Gerard Meijer for discussions,
encouragement, and support.
∗ Electronic address: mikhail.lemeshko@gmail.com
† Electronic address: brich@fhi-berlin.mpg.de
[1] L. C. Biedenharn, 1960, Nuclear Spectroscopy, Part
B, ed. F. Azjenberg-Selove (Academic Press), p. 732;
A. J. Alexander, M. Brouard, K. S. Kalogerakis, J. P. Si-
mons, Chem. Soc. Rev, 1998, 27, 405; M. L. Costen,
S. Marinakis, K. G. McKendrick, Chem. Soc. Rev, 2008,
37, 732.
[2] D. Herschbach, Eur. Phys. J. D, 2006, 38, 3.
[3] D. A. Case, D. R. Herschbach, Mol. Phys., 1975, 30,
1537; D. A. Case, G. M. McClelland, D. R. Herschbach,
Mol. Phys., 1978, 35, 541; J. D. Barnwell, J. G. Loeser,
D. R. Herschbach, J. Phys. Chem., 1983, 87, 2781.
[4] S. K. Kim, D. R. Herschbach, Faraday Disc. Chem.
Soc., 1988, 84, 159; M. P. de Miranda, F. J. Aoiz,
Phys. Rev. Lett., 2004, 93, 083201; M. P. de Mi-
randa, F. J. Aoiz, V. Sa´ez Ra´banos, M. Brouard,
J. Chem. Phys., 2004, 121, 9830; J. Aldegunde,
M. P. de Miranda, J. M. Haigh, B. K. Kendrick, V. Sa´ez-
Ra´banoz, F. J. Aoiz, J. Phys. Chem. A, 2005, 109, 6200.
[5] S. I. Drozdov, Soviet. Phys. JETP, 1955, 1, 591,
588; J. S. Blair, 1966, Nuclear Structure Physics, eds.
P. D. Kunz, D. A. Lind, W. E. Brittin (The University
of Colorado, Boulder), Vol. VII C, p. 343-444; M. Faubel,
J. Chem. Phys., 1984, 81, 5559.
[6] M. Lemeshko, B. Friedrich, J. Chem. Phys., 2008, 129,
02430; M. Lemeshko, B. Friedrich, Int. J. Mass. Spec.,
2009, 280, 19; M. Lemeshko, B. Friedrich, Phys. Rev. A,
2009, 79, 012718; M. Lemeshko, B. Friedrich,
J. Phys. Chem. A, 2009, in press; arXiv: 0906.0443.
[7] M. H. Alexander, Chem. Phys., 1985, 92, 337; A. Gi-
jsbertsen, H. Linnartz, C. A. Taatjes, and S. Stolte,
J. Am. Chem. Soc., 2006, 128, 8777; F. J. Aoiz,
V. J. Herrero, V. Sa´ez Ra´banos, J. E. Verdasco,
Phys. Chem. Chem. Phys., 2004, 6, 4407.
[8] M. H. Alexander, J. Chem. Phys., 1999, 111, 7426.
[9] E. A. Wade, K. T. Lorenz, D. W. Chandler, J. W. Barr,
G. L. Barnes, J. I. Cline, Chem. Phys., 2004, 301, 261.
[10] A. J. Orr-Ewing, R. N. Zare, Annu. Rev. Phys. Chem. 45,
315 (1994); A. J. Orr-Ewing, R. N. Zare, in The Chem-
ical Dynamics and Kinetics of Small Radicals, edited
by K. Liu and A. Wagner, World Scientific, Singapore
(1995).
[11] K. Blum, Density matrix theory and applications,
Plenum, New York (1996).
[12] Y. Sumiyoshi, Y. Endo, J. Chem. Phys., 2007, 127,
184309.
[13] F. J. Aoiz, J. E. Verdasco, V. J. Herrero,
V. Sa´ez Ra´banos, M. H. Alexander, J. Chem. Phys.,
2003, 119, 5860; P. J. Dagdigian, M. H. Alexander,
J. Chem. Phys., 2009, 130, 204304.
[14] V. Khare, D. J. Kouri, D. K. Hoffman, J. Chem. Phys.,
1981, 74, 2275.
[15] M. Lemeshko, B. Friedrich, to be published.