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Classical dynamics of rotationally inelastic scattering of atoms with molecules
21
Chemical Physics 111 (1986) 21-31 North-Holland. Amsterdam
CLASSICAL DYNAMICS OF ROTATIONALLY OF ATOMS WITH MOLECULES
iNELASTIC
SCATTERING
D. POPPE Institut ftirPhysikaiische Chemie, Freie Universitiit Berlin, Takustrasse 3, 1000 Berlin 33, FRG Received 4 July 1986
Rotational excitation in collisions of atoms with diatomic molecules is investigated using classical mechanics. The structure of the fully resolved cross sections with respect to the final molecular angular momentum, its projection onto the quantization axis, and the scattering angle are studied numerically using simple model potentials. In particular the influence of isotropic and anisotropic attractive forces is investigated. In the first case the structure of the cross section is still similar to that for repulsive scattering. Anisotropic attraction introduces new phenomena whose relations to the properties of the potential are explored.
1. Intmduction
Rotational excitation in collisions of atoms with diatomic molecules has been investigated in the last few years with increasing effort. For recent reviews see the articles by Thomas [l] .and Schinke and Bowman [2]. Particularly the rotational rainbow as a prominent feature in the differential cross section has been studied both theoretically and experimentally. In case of purely repulsive interactions between the molecule and the atom the detailed analysis of the scattering process has provided us with important information on the mechanism of rotational excitation. The infiniteorder-sudden(IOS) and the centrifugal-sudden(CS) approximation [2] have proven to be an excellent tool in this field. Rotational excitation in purely repulsive potentials is now a well understood process. Little, however, is known about the dynamics of rotational transitions in cases where the assumptions of the 10s or the CS approximations are not fulfilled. This is is the case if at least one of the following conditions are met: (1) An attractive possibly spatially anisotropic well is present whose depth is not small compared with the collision energy.
(2) The collision is not sudden, i.e. the interaction time between the colliding species is not short with respect to the rotational period of the molecule. (3) The amount of energy stored in the rotational motion during the collision exceeds the collision energy so that the atom is captured in a possibly long-lived complex. Another topic which can hardly be treated in the approximations mentioned above is the reorientation of the molecular angular momentum. It is intimately connected with the conservation of the total angular momentum which is violated in 10s as well as in CS. For impulsive scattering it can be shown that the final molecular angular momentum j’ for initially non-rotating molecules is given by j’=
-RxAp.
(1) The transferred momentum is denoted by Ap and R is the point of contact where the excitation takes place. It has been argued that j’ is uniformly distributed in the plane perpendicular to Ap independent of j’ and the scattering angle 8. Thus, for impulsive collisions, the distribution of the magnetic quantum number m’ can already be deduced from m’ unresolved quantities using sim-
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
22
D. Poppe / Rotutionully
inehstic utom - molecule sccrttering
ple geometric arguments [3-81. No additional dynamical information beyond the impulsive collision mechanism can be derived from the m’ resolved cross sections. For realistic potentials this result is not a priori valid due to the finite range of the intermolecular interaction. In the present article we try to contribute to a theoretical understanding of rotational excitation in such non-IOS cases which are discussed above. There is of course a considerable amount of work in the literature dealing with particular systems, or which are addressed only to integral cross section, etc. It is our goal to deliver a systematic investigation of fully resolved differential cross sections discussing different cases with the same theoretical approach. For this first step it is useful to treat the excitation process within the framework of classical mechanics. Besides its simplicity it usually allows for an uncomplicated backtracing of properties of the cross sections to those of the interactions. Quantum effects cannot be accounted for, however, the analysis of the classical data will provide us with qualitative arguments where quantum interference effects have to be expected. The plan of the paper is as follows. In section 2 the model potentials are introduced. Results are presented in section 3.
2. Potential models Here two potential models (hereafter referred to as model I and model II, respectively) are introduced. Model I consists of a purely isotropic attractive potential of Lennard-Jones (8,4)-type superimposed by an anisotropic repulsive core
+A[(R,+
B
cos2a)/R]*.
(2)
This model serves as an example for the interaction between non-reactive neutral species where the anisotropy is likely to be short ranged and repulsive. Due to the cos2a-anisotropy the torque W/h changes sign whenever (Y passes (Y= 0 or (Y= 7r/2. For vanishing z we obtain a purely repulsive potential. In this case and under 10s
Table 1 Parameters
for potential
models
I and II
Model I
Model II
2.38 0.6
2.38
0.5
0.5
conditions the structures of sections are well understood tional rainbows. We shall analysis at least qualitatively case of model I. The second model, model tional excitation in repulsion well V(R,
a,=,{[(
R,+ -2[(R,+B
the differential cross in terms of the rotaextend this type of to the more general II, allows for rotaand in attraction as
B cos2a)/R]* CosS+!]4),
(3)
where the equilibrium distance of the Lennard-Jones (8,4) potential depends on the relative orientation. The torque ~V/&Y changes sign not only when OLgoes through a! = 0 and a = 7r/2 but also when the equilibrium distance R,,= R, + B cos2ais passed. The potential is designed to model long-range anisotropies as they are typical for ionic systems. Both potentials are not directly related to a particular system. However, to make an easy comparison to the well studied system Na,-He [9-131, the masses have been chosen from the latter (the mass of the atomic projectile M = 4 and the mass of atoms in the molecule m = 23). The values of the parameters for both potentials are summarized in table 1.
3. Nunierlcal
results
Calculations have been performed within the framework of classical scattering theory using a program package that has been discussed elsewhere [14]. Usually batches of 2000 trajectories were run. For reasons of simplicity we confine ourselves to the case of initially non-rotating molecules. The collision energy E is fixed to E = 0.1 eV.
D. Poppe / Rotationally inelastic atom-molecule
scattering
23
3.1. Model I
‘b*
’
1
.’
J
f
IL8
Fig. 1. Density distributions for 2000 trajectories. (a) Initial impact parameter b* (abscissa) and scattering angle 0 (ordinate). (b) m’/j’ and j’. m’ is the projection of final molecular angular momentum j’ onto the space fixed z axis. (c) Scattering angle 0 and m’/j’. (d) Orientation of j’ parameterized by/3 and cos 6 and (see fig. 12). (e) B and j’. (f) b* and j’. (g) 3d plot of the final state distribution. The sizes of the symbols increase with increasing (ml/j’). The distributions areshownforO
We begin with model I. In figs. l-3 results are shown as a function of the parameter E which controls the well depth of the potential. The 3d plots for the final molecular angular momentum j’, its orientation with respect to the space-fixed axis (ml/j’), and the scattering angle 8 illustrate the scattering outcome. Though these representations do,not allow for a quantitative analysis they are useful in detecting generic structures of the scattering process. More quantitative information is provided by the projected 2d distributions. Details on the dependence of the final states of the trajectories on the initial conditions are also given. For vanishing z (fig. 1) the potential is purely repulsive and short ranged (a Rp8). The deflection distribution exhibits only positive scattering angles * reflecting the repulsive character of the interaction. The excitation distribution shows how j’ depends on the impact parameter b. It is a generic property of repulsive interactions that small b leads to large j’ and vice versa. The &-averaged differential cross section is also visualized in a 2d-density plot for the trajectories. It exhibits the well-known structure whose most prominent feature is the rotational rainbow. We have also studied the reorientation of the molecular angular momentum. The distribution in the (r&/j’-@) plane summed over j’ has a well defined shape. The limiting curve defining the largest ( m’/j’ 1 as a function of 0 follows the hard shell result (see appendix). For a detailed comparison with the hard shell result the spatial distribution of j’ with respect to Ap. is shown in the cos S-p plane (both angles are defined in fig. 12 below). The overwhelming majority of the trajectories lie in the vicinity of cos 6 = 0 as predicted for the hard shell model. The azimuthal
* Prior to the collision space-fixedyz plane
the atomic projectile moves in the (v > 0) with the relative momentum parallel to the z axis. The scattering angle is counted positive (negative)if the y component of the final relative momentum is positive (negative). This definition folds the true scattering angles in the interval (- T, IT). For example hue scattering angles with -2~ i B < - TT appear to be positive,etc (see figs.la-3a).
D. Pope
24
/ Rotationally inelasiic atom-molecule
distribution is nearly uniform. This result has been conjectured earlier and verified computationally for impulsive collisions [6,15]. Closer inspection reveals systematic deviations from the impulsive
scattering
results. The density distribution follows an overall sinusoidal shape with a slight preference of /3 = 7/2 and /3 = 3n/2. We shall come back to this point later on.
cos6
c
b*
m/i .I
J
Fig. 2. Same as fig. 1 but for c = 0.1 eV. j,& A.
- 22.2, b,,,, = 5.8
Fig. 3. Same as fig. 1 but for c = 0.3 eV. j/,,, A.
= 32.2, b,,,, = 7.0
D, Poppe / Rotationally inelastic atom-molecule
25
scattering
.I
PIj’l
Jmax
O.’7
0.01
0.1
1.0
VJeVI
Fig. 5. Largest classically allowed molecular angular momentum jA, as a function of the averaged well depth of potential I. E = 0.1 eV. M = 4, m = 23. The-line corresponds to j&, = ~0.16 0.
Fig. 4. Integral cross sections a(0 + j’) for potential I in arbitrary units. From top to bottom: z = 0.9 eV, j&, = 47.4. z = 0.6 eV, j& = 42. c = 0.3 eV, J&, = 32. c = 0, jAax = 18.
The deviations from cos S = 0 and from a uniform /I in the present case are somewhat larger than for a model potential for He-Na, discussed earlier [15]. The latter is also a repulsive R-’ potential. Its anisotropy is a single P,(cos o) term while our present model contains higher-order Legendre polynomials P,(cos a). The reorientation of the molecular angular momentum appears to be rather sensitive to the shape of the anisotropy. As we increase the well depth four effects are clearly evident: (1) The deflection distribution is extended to negative scattering angles 0 as a consequence of the dominating attraction for large impact parameter 6. The averaged dependence of the distribution on b is essentially determined by the isotropic part of the interaction. The ordinary rainbow angle defined by the “minimum” of the deflection distribution can be estimated from elastic scattering theory. Diminished scattering intensity is observed in the forward direction. This effect has been termed rotational shadow [16] and is expected for rotational excitation proceeding via a direct collision (no complex formation). (2) The excitation distribution j’(f~*) is struct-
urally unchanged with respect to the purely repulsive case. The largest dynamically accessible j’ increases with rising well depth. Results are shown in fig. 5. Notice that there is a fairly well defined smooth curve j;(b*) giving the largest possible j’ as a function of b*. With increasing e the function ji(b*) becomes less steep for small b and increasingly steeper for large b. The numerical results also indicate that the trajectories are uniformly distributed under the curve j,l(b*) for all b*. Under these circumstances the integral cross section for rotational excitation a( j’) is basically determined by j,$b*). This result has an interesting aspect for several empirically found laws on the j’ dependence of u( j’). The highest energetically allowed j; = 72 is in all cases much larger than the dynamically accessible j’ levels so that rotationally bound complexes do not exist. (3) Plots of the (j’, @)-distributions illustrate the m’-summed differential cross sections. The well-known rotational rainbow line (type CYof ref. [ll]) exhibiting large scattering intensity is extended to negative 8 as the well depth increases. The overall smooth rainbow line is interrupted by the rotational shadow which causes diminished forward scattering as well as enhanced excitation of large j’ at small negative 8. The triangle shaped area is also a rainbow structure (type p of ref. [ll]) which stems from large impact parameter scattering. (4) The 3d-plots have been projected onto the (m’/j’-0) plane to visualise the orientation of the
26
D. Poppe / Rotationally melmtic atom _ molecule scattering
final molecular angular momentum. Negative and positive 8 show a quite different behaviour. The latter give rise to an m' distribution which is similar to the corresponding distribution for the repulsive case (see fig. 1). Negative B values (undistinguishable from positive 8 in a scattering experiment) lead to a more random distribution covering the whole (m’/j’-0) plane. For c = 0.3 eV, figs. 6a and 6b display the distributions in the cos 8-p plane for positive and negative 8, respectively. Deviations from the hard shell result for positive 8 are minor and occur mostly with small j’. Negative 0 values fill the whole cos S-/3 plane, however, still with a preference of cos 6 = 0. Strong deviations from the impulsive case are also witnessed. Moreover the /3 population is not uniform preferring B = 7/2 and /3 = 31~/2 the more the larger c is. This carries over to P( m’ 1j’s) (defined in the appendix) for 8 < 0 which is expected to be still doubly peaked similar to the hard shell case. The maxima, however, are shifted to larger 1m' I. We note as an aside that p = n/2 and /3 = 3n/2 correspond to a j' which lies in the YZ plane (see fig. 12 below) being the plane of the initial approach of the atom. The corresponding trajectories are extremely non-planar with I’ > 1. The apparently symmetric pattern in the cos 8-p plane expresses the fact that the m' distribution is symmetric with respect to the sign inversion of m'. The differences with respect to the hard shell case are mainly due to the influence of the isotropic attractive forces. They lead to a rotation of the relative momentum vector prior and after the angular momentum transfer occurs. Due to conservation of the total angular momentum the corresponding rotation axes are the initial and final orbital angular momentum 1 and I’, respectively. From our calculations we conclude that these rotations are of minor importance for positive scattering angles. Integral cross sections a( j’) are shown in fig. 4. For vanishing well depth we obtain a monotonic exponential decrease of u with increasing j’. With rising c the cross sections decrease slower and exhibit a sharp drop off at the largest classically accessible j;, . This result can be well understood within a two-step model of the collision. In the first step the isotropic well enhances the momen-
. : . 1 :. . I
a’
I
I
P I
cos
: y
-.<.;.; ! .
.‘-.’ 1.: ‘.-L’ ; ..:1,+ . ;: .. .. . .
‘..L’ $>> .*_: . - :‘:
: . i..?:
..+:.:;.: : 3.
. ..::
’
4;
.a. .
.. 1, :: , .
I.
’ :
..*..:
_.:-.,.
:-
.. :.
..s : - * :.. . .
Y...
;, . . . ....’
:: &..!.’ .:
: .;:.
’
b . ..;. I..?.. . .
.::
.
.
..-;‘;, .
P Fig. 6. Orientation of j’ in the cos S-/3 plane for potential I for positive (a) and negative (b) scattering angles. E = 0.1 eV. M = 4, m = 23. The distributions are shown for 0 < /3 < 2n and -16~0~661.
turn of the atom and reduces therefore the effective impact parameter responsible for the rotational excitation. For example a well depth V, reduces b by a factor of [E/( E + Vo)]‘/2. The second step is the rotational excitatibn in the repulsive part of the potential. It takes place with an enhanced collision energy (E + V,), however, only for the reduced impact parameter. In comparison with collisions without a well, large impact parameters are now missing which favour small angular momenta. In this way a focusing effect for the effective impact parameter offers an explanation for the j’ dependence of u.
D. Poppe / Rotationally inelastic atom -molecule
For 4 = 0.6 eV (averaged well depth = 0.23 eV) complex formation is observed (see fig. 7). About 2% of the trajectories exhibit a complex living longer than 2.4 ps. The long lifetime is caused by
.I
J
scattering
27
strong intermediate rotational excitation, which hinders the particles to separate since they cannot overcome the centrifugal barrier. The decay is only allowed after redistribution of the molecular energy and/or lowering of the centrifugal barrier. We have also studied the mass dependence of the excitation. All calculations discussed so far were performed for m = 23 and M = 4 which we refer to as the HHL case. The replacement (m = 2 and M= 46) gives the LLH case. It leaves the reduced mass of the projectile with respect to the target unchanged. The target, however, is now lighter by a factor of 11.5. The motion prior to the rotational excitation is still the same for the HHL and LLH case. Changes are introduced by the enhanced rotational velocity of the lighter molecule and the larger rotational energy associated with a given j’. The largest energetically allowed molecular angular momentum is j; = 23. Fig. 8 presents results for c = 0.6 eV. The anisotropy is strong enough to excite j; = 23. In comparison with the HHL case we observe extensive complex formation (= 25%) now also due to the fact that the anisotropy allows for rotational excitation beyond the energetic threshold. In general all structures of the HHL case are also present for LLH, however, less prominent. The integral cross section a(j’) depends only weakly on j’ with a steep decrease just below j;. 3.2. Model II
.I
1
L$
tL
1
8
Fig. 7. Same as fig. 1 but for c = 0.6 eV. j,&_ - 42.0, b,,,= = 7.35 A.
In contrast to the previous case, model II allows for rotational excitation by the long-range attractive part of the potential. The spatial form of the torque av/aar has completely altered compared to model I (see section 2). Again we have studied in particular the influence of the well depth now given directly by c. Results for three values of E are shown in figs. 9-11. Characteristic differences between model I and model II: (1) The excitation distribution j’(b*) exhibits now two regions of efficient excitation. The scattering with small b leads to positive deflection angles and is mainly governed by repulsive forces. The “minimum” of j’(b2) is caused by trajectories that experience dominantly the equilibrium geometry of the potential where the anisotropy is
D. Poppe / Rotationally inelastic atom-molecule
28
small. Here also trajectories contribute along which the torque oscillates in time leading to an overall small excitation. The excitation function witnesses strong angular momentum transfer for medium
impact parameters b. Comparison with the deflection distribution shows that they are associated with negative B values near the common angular rainbow. Large values of j’ are produced by
m ..
:: ‘I,.;‘;
.,.:. .
;
.
,-,.;.;:,
i
.:‘. i’ ., ,,._.,.r ‘_:.:‘*r, .. ..
\ .
1,;
.p,.
.._,
,
..
:
‘..‘..:; .,.p.:>: .s-.,
:: .f,&.‘-
.
;
:
“,f..,
:,
p,’
rz~;-,‘~C$::.~. .a.‘?
.‘,.i.. :,.\ ,
.y:.:,.;., ::
\:
,:,.:. ,..*, ‘. . T. ‘,. : ‘.
.,,,’
,..$
,.-
z_ ,
f. ..;.
j
.,
‘-5:
:
1 b2
j’
..:_.
,;., :;,;.
_._
L.
. . .;
.
:
..:>f:..
scattering
,y
5:‘: .:.,...14’ : .: ..‘. .
_,.-a.. ,.
: _‘,.,_
..
.,L..
; i:,. ‘. ‘. :_:, ..:,*. .,..$
.
:
j:...;.:
.,‘, :‘:~.:,y. :; _.:
:
c;y:;‘:
;’
,’
.%.!
e
b2
Fig. 8. Same as fig. 1 but for c = 0.6 e-V. m = 2, M = 46. j&, - 42.0, b,,,, = 7.35 A.
Fig. 9. Same as fig. 1 but for c = 0.1 ey. Potential II. j,& - 12.1, bmax - 6.91 A.
D. Poppe / Rotationally inelastic atom-molecule
trajectories that pass the equilibrium R, at (Y= 0 or a = v/2 thereby maintaining the sign of the torque. The tail of j’(b*) at large b is generated
scattering
29
by the attractive forces of the potential. The shape of the distribution is determined by the relative magnitude of attractive and repulsive
b2
.I
J
$ J
IL8
Fig. 10. Same as fig. 1 but for c = 0.2 eV. Potential II. jA - 28.6, b,,,, = 6.9 A.
.I
. . ... : ..
..
-
.
.. ...
I
1
!I$ 1
t/
0
Fig. 11. Same as fig. 1 but for c = 0.6 eV. Potential II. jA - 44.5, b_ = 11 A.
30
D. Poppe / RotationaNy inelastic atom - molecule scattering
anisotropies and depends therefore also on the radial form of the potential (compare with fig. 2 of ref. [14]). (2) The projection onto the j’-8 plane displays now two rotational rainbow lines. There is a rotational rainbow with large scattering intensity at negative 8 values starting at B = 0. It stems from trajectories with relatively large impact parameters. These trajectories experience only the attractive forces. The structure is analogous to the rotational rainbow for excitation in a repulsive potential under 10s conditions. To explore the generic properties it may still be appropriate to apply the 10s assumptions to these large impact parameter collisions. This can be approximately justified by the fact that a large fraction of the final molecular angular momentum is transferred in a particular configuration R and (Y (to be identified with the 10s orientation tires) near the classical turning point. The other weaker and less singular rotational rainbow generates a step in the m’-summed differential cross section along the function j&(e) which is the largest j’ for a given B. There is no simple dynamical interpretation for j&(e). We note, however, that with increasing c the attractive forces become more and more important for rotational excitation (see the j’-b2 distribution). Simultaneously the j;(e) curve flattens. In the region of orbiting collisions it becomes diffuse and is not well defined. (3) The spatial orientation of j’ for small c =G 0.1 eV is similar to that for model I. Positive scattering angles show j’ I Ap while negative 0 values give rise to a random m’ distribution (see &/j’--e plot in fig. 9). For c > 0.1 eV we found that the m’ distribution for positive B values is more or less uniform, while negative e prefer m’ = kj’ the more the larger j’ and ) 8 1 is. For m’ = kj’ the molecular angular momentum is directed along the z axis which is the direction of the initial approach of the projectile. The molecular axis r lies then in the xy plane and is still near its initial configuration, since the rotational period is large compared to the collision time even for the largest j’. Particularly the non-planar geometries with r around the bisector of the x and y axis lead to m’ = kj’ with a large j’. If the molecule is light enough to allow for
appreciable rotation during the collision the orientation of j’ changes as can be seen from the LHH case with E = 0.2 eV (not shown here). Then m’ = 0 is preferred the more the larger j’ is. This finding is in accordance with the results of Barg and Toennies [18] for Li+-Hz. This effect is also present for model I however it is very weak (see fig. 8).
Acknowledgement The author gratefully acknowledges financial support from Sonderforschungsbereich 161 “Hyperfeinwechselwirkungen” and lfrom “Fonds der Chemischen Industrie”.
Appendix The classical (conditional) probability P( m’ 1 ej’) for m’ with given 8 and j’ is related to the distribution in the cos 8-p plane by P( m’ 1f3jf) 1 cc 2”d(cos 6) dfi s(m’-m(/3, JJ -1 a x NW
I W7
(A-1)
where Q expresses the (conditional) p and S for given B and j’. From that m(P,
y, 6))
Y, 8) =j’(sin
probability of fig. 12 we see
P sinysin6+cos6cosy). (A.?)
For the rotational excitation of a hard ellipsoid with a point particle one has exactly cos 6 = 0 and an uniform 8. Then eq. (A.l) can be evaluated analytically to give P( 172’I ej’) a (1 - m’/m,)-1’2,
(A-3)
with Im’I cm,. The largest classically and B m. = j’ sin y,
(A.4 accessible
m depends
on j’
(A-5)
31
D. Poppe / Rotationally inelastic atom - molecule scattering
eq. (A.2), we find that 1~2’1Gj’ max( Icos(S+y)
I, Icos(S-_y) I). (A-9)
For a given scattering angle 0, eq. (A.9) relates the largest m’ to the possible deviations 6 # 7r/2 from the hard shell result. Y
Fig. 12. Geometry of the scattering. The space fixed coordinate system is denoted by xyz. The z’ axis coincides with the transferred momentum Pp. The angle /3 is chosen such that j’ is in the x’z’ plane. The angle S is between j’ and Ap.
since cos y= -(p-p’
cos e)
x [ p2 + (pt)’ - 2pp’
cos
81 -l’*,
(~.6)
where p and p’ are the initial and final relative momenta, respectively. Neglecting the rotational energy transfer ( p = p’) we simplify the equation above to y=n/2+8/2.
(A-7)
Eq. (AS) reduces to m =j’ cos( d/2).
(A.8)
The projected 2d distributions for 8 and m’/j’ in fig. 1 exhibit a limiting curve which is well described by eq. (A.8). Returning to the general case,
References [l] L.D. Thomas, in: Potential surfaces and dynamics calculations, ed. D.G. Truhlar (Plenum Press, New York, 1981) p. 731. [2] R. Schinke and J.M. Bowman, in: Molecular collision dynamics, ed. J.M. Bowman (Springer, Berlin, 1983) and references therein. [3] D. Richards, J. Phys. B 14 (1981) 4799; B 15 (1982) 3025. [4] V. Khare, D.E. Fitz and D.J. Kouri, J. Chem. Phys. 73 (1980) 2802. [5] R. Schinke and H.J. Korsch, Chem. Phys. Letters 74 (1980) 449; P. Eckelt and H.J. Korsch, J. Phys. B 10 (1977) 741. [6] D. Beck, U. Ross and W. Schepper, Z. Physik. A 293 (1979) 107. [7] A. Mattheus, A. Fischer, G. Ziegler, E. Gottwald and K. Bergmann, preprint. [8] M.A. Treffers and J. Korving, Chem. Phys. Letters 97 (1983) 342. [9] K. Bergmann, R. Engelhardt, U. Hefter and J. Witt, J. Chem. Phys. 71 (1979) 2726. [lo] P.L. Jones, U. Hefter, A. Mattheus, K. Bergmann, W. Mliller, W. Meyer and R. Schinke, Phys. Rev. A 26 (1982) 1283. [ll] R. Schinke, H.J. Korsch and D. Poppe, J. Chem. Phys. 77 (1982) 6005. [12] H.J. Korsch and R. Schinke, J. Chem. Phys. 73 (1980) 1222; 75 (1981) 3580. [13] K. Bergmann, U. Hefter and J. Witt, J. Chem. Phys. 72 (1980) 4777. [14] D. Poppe and R. Battner, Chem. Phys. 30 (1978) 375. [15] D. Poppe, Chem. Phys. 84 (1984) 243. [16] D. Poppe and H.J. Korsch, Chem. Phys. 85 (1984) 267. [17] R. Schinke, J. Chem. Phys. 75 (1981) 5205(L). [18] G.D. Barg and J.P. Toennies, Chem. Phys. Letters 51 (1977) 23.
Chemical Physics 111 (1986) 21-31 North-Holland. Amsterdam
CLASSICAL DYNAMICS OF ROTATIONALLY OF ATOMS WITH MOLECULES
iNELASTIC
SCATTERING
D. POPPE Institut ftirPhysikaiische Chemie, Freie Universitiit Berlin, Takustrasse 3, 1000 Berlin 33, FRG Received 4 July 1986
Rotational excitation in collisions of atoms with diatomic molecules is investigated using classical mechanics. The structure of the fully resolved cross sections with respect to the final molecular angular momentum, its projection onto the quantization axis, and the scattering angle are studied numerically using simple model potentials. In particular the influence of isotropic and anisotropic attractive forces is investigated. In the first case the structure of the cross section is still similar to that for repulsive scattering. Anisotropic attraction introduces new phenomena whose relations to the properties of the potential are explored.
1. Intmduction
Rotational excitation in collisions of atoms with diatomic molecules has been investigated in the last few years with increasing effort. For recent reviews see the articles by Thomas [l] .and Schinke and Bowman [2]. Particularly the rotational rainbow as a prominent feature in the differential cross section has been studied both theoretically and experimentally. In case of purely repulsive interactions between the molecule and the atom the detailed analysis of the scattering process has provided us with important information on the mechanism of rotational excitation. The infiniteorder-sudden(IOS) and the centrifugal-sudden(CS) approximation [2] have proven to be an excellent tool in this field. Rotational excitation in purely repulsive potentials is now a well understood process. Little, however, is known about the dynamics of rotational transitions in cases where the assumptions of the 10s or the CS approximations are not fulfilled. This is is the case if at least one of the following conditions are met: (1) An attractive possibly spatially anisotropic well is present whose depth is not small compared with the collision energy.
(2) The collision is not sudden, i.e. the interaction time between the colliding species is not short with respect to the rotational period of the molecule. (3) The amount of energy stored in the rotational motion during the collision exceeds the collision energy so that the atom is captured in a possibly long-lived complex. Another topic which can hardly be treated in the approximations mentioned above is the reorientation of the molecular angular momentum. It is intimately connected with the conservation of the total angular momentum which is violated in 10s as well as in CS. For impulsive scattering it can be shown that the final molecular angular momentum j’ for initially non-rotating molecules is given by j’=
-RxAp.
(1) The transferred momentum is denoted by Ap and R is the point of contact where the excitation takes place. It has been argued that j’ is uniformly distributed in the plane perpendicular to Ap independent of j’ and the scattering angle 8. Thus, for impulsive collisions, the distribution of the magnetic quantum number m’ can already be deduced from m’ unresolved quantities using sim-
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
22
D. Poppe / Rotutionully
inehstic utom - molecule sccrttering
ple geometric arguments [3-81. No additional dynamical information beyond the impulsive collision mechanism can be derived from the m’ resolved cross sections. For realistic potentials this result is not a priori valid due to the finite range of the intermolecular interaction. In the present article we try to contribute to a theoretical understanding of rotational excitation in such non-IOS cases which are discussed above. There is of course a considerable amount of work in the literature dealing with particular systems, or which are addressed only to integral cross section, etc. It is our goal to deliver a systematic investigation of fully resolved differential cross sections discussing different cases with the same theoretical approach. For this first step it is useful to treat the excitation process within the framework of classical mechanics. Besides its simplicity it usually allows for an uncomplicated backtracing of properties of the cross sections to those of the interactions. Quantum effects cannot be accounted for, however, the analysis of the classical data will provide us with qualitative arguments where quantum interference effects have to be expected. The plan of the paper is as follows. In section 2 the model potentials are introduced. Results are presented in section 3.
2. Potential models Here two potential models (hereafter referred to as model I and model II, respectively) are introduced. Model I consists of a purely isotropic attractive potential of Lennard-Jones (8,4)-type superimposed by an anisotropic repulsive core
+A[(R,+
B
cos2a)/R]*.
(2)
This model serves as an example for the interaction between non-reactive neutral species where the anisotropy is likely to be short ranged and repulsive. Due to the cos2a-anisotropy the torque W/h changes sign whenever (Y passes (Y= 0 or (Y= 7r/2. For vanishing z we obtain a purely repulsive potential. In this case and under 10s
Table 1 Parameters
for potential
models
I and II
Model I
Model II
2.38 0.6
2.38
0.5
0.5
conditions the structures of sections are well understood tional rainbows. We shall analysis at least qualitatively case of model I. The second model, model tional excitation in repulsion well V(R,
a,=,{[(
R,+ -2[(R,+B
the differential cross in terms of the rotaextend this type of to the more general II, allows for rotaand in attraction as
B cos2a)/R]* CosS+!]4),
(3)
where the equilibrium distance of the Lennard-Jones (8,4) potential depends on the relative orientation. The torque ~V/&Y changes sign not only when OLgoes through a! = 0 and a = 7r/2 but also when the equilibrium distance R,,= R, + B cos2ais passed. The potential is designed to model long-range anisotropies as they are typical for ionic systems. Both potentials are not directly related to a particular system. However, to make an easy comparison to the well studied system Na,-He [9-131, the masses have been chosen from the latter (the mass of the atomic projectile M = 4 and the mass of atoms in the molecule m = 23). The values of the parameters for both potentials are summarized in table 1.
3. Nunierlcal
results
Calculations have been performed within the framework of classical scattering theory using a program package that has been discussed elsewhere [14]. Usually batches of 2000 trajectories were run. For reasons of simplicity we confine ourselves to the case of initially non-rotating molecules. The collision energy E is fixed to E = 0.1 eV.
D. Poppe / Rotationally inelastic atom-molecule
scattering
23
3.1. Model I
‘b*
’
1
.’
J
f
IL8
Fig. 1. Density distributions for 2000 trajectories. (a) Initial impact parameter b* (abscissa) and scattering angle 0 (ordinate). (b) m’/j’ and j’. m’ is the projection of final molecular angular momentum j’ onto the space fixed z axis. (c) Scattering angle 0 and m’/j’. (d) Orientation of j’ parameterized by/3 and cos 6 and (see fig. 12). (e) B and j’. (f) b* and j’. (g) 3d plot of the final state distribution. The sizes of the symbols increase with increasing (ml/j’). The distributions areshownforO
We begin with model I. In figs. l-3 results are shown as a function of the parameter E which controls the well depth of the potential. The 3d plots for the final molecular angular momentum j’, its orientation with respect to the space-fixed axis (ml/j’), and the scattering angle 8 illustrate the scattering outcome. Though these representations do,not allow for a quantitative analysis they are useful in detecting generic structures of the scattering process. More quantitative information is provided by the projected 2d distributions. Details on the dependence of the final states of the trajectories on the initial conditions are also given. For vanishing z (fig. 1) the potential is purely repulsive and short ranged (a Rp8). The deflection distribution exhibits only positive scattering angles * reflecting the repulsive character of the interaction. The excitation distribution shows how j’ depends on the impact parameter b. It is a generic property of repulsive interactions that small b leads to large j’ and vice versa. The &-averaged differential cross section is also visualized in a 2d-density plot for the trajectories. It exhibits the well-known structure whose most prominent feature is the rotational rainbow. We have also studied the reorientation of the molecular angular momentum. The distribution in the (r&/j’-@) plane summed over j’ has a well defined shape. The limiting curve defining the largest ( m’/j’ 1 as a function of 0 follows the hard shell result (see appendix). For a detailed comparison with the hard shell result the spatial distribution of j’ with respect to Ap. is shown in the cos S-p plane (both angles are defined in fig. 12 below). The overwhelming majority of the trajectories lie in the vicinity of cos 6 = 0 as predicted for the hard shell model. The azimuthal
* Prior to the collision space-fixedyz plane
the atomic projectile moves in the (v > 0) with the relative momentum parallel to the z axis. The scattering angle is counted positive (negative)if the y component of the final relative momentum is positive (negative). This definition folds the true scattering angles in the interval (- T, IT). For example hue scattering angles with -2~ i B < - TT appear to be positive,etc (see figs.la-3a).
D. Pope
24
/ Rotationally inelasiic atom-molecule
distribution is nearly uniform. This result has been conjectured earlier and verified computationally for impulsive collisions [6,15]. Closer inspection reveals systematic deviations from the impulsive
scattering
results. The density distribution follows an overall sinusoidal shape with a slight preference of /3 = 7/2 and /3 = 3n/2. We shall come back to this point later on.
cos6
c
b*
m/i .I
J
Fig. 2. Same as fig. 1 but for c = 0.1 eV. j,& A.
- 22.2, b,,,, = 5.8
Fig. 3. Same as fig. 1 but for c = 0.3 eV. j/,,, A.
= 32.2, b,,,, = 7.0
D, Poppe / Rotationally inelastic atom-molecule
25
scattering
.I
PIj’l
Jmax
O.’7
0.01
0.1
1.0
VJeVI
Fig. 5. Largest classically allowed molecular angular momentum jA, as a function of the averaged well depth of potential I. E = 0.1 eV. M = 4, m = 23. The-line corresponds to j&, = ~0.16 0.
Fig. 4. Integral cross sections a(0 + j’) for potential I in arbitrary units. From top to bottom: z = 0.9 eV, j&, = 47.4. z = 0.6 eV, j& = 42. c = 0.3 eV, J&, = 32. c = 0, jAax = 18.
The deviations from cos S = 0 and from a uniform /I in the present case are somewhat larger than for a model potential for He-Na, discussed earlier [15]. The latter is also a repulsive R-’ potential. Its anisotropy is a single P,(cos o) term while our present model contains higher-order Legendre polynomials P,(cos a). The reorientation of the molecular angular momentum appears to be rather sensitive to the shape of the anisotropy. As we increase the well depth four effects are clearly evident: (1) The deflection distribution is extended to negative scattering angles 0 as a consequence of the dominating attraction for large impact parameter 6. The averaged dependence of the distribution on b is essentially determined by the isotropic part of the interaction. The ordinary rainbow angle defined by the “minimum” of the deflection distribution can be estimated from elastic scattering theory. Diminished scattering intensity is observed in the forward direction. This effect has been termed rotational shadow [16] and is expected for rotational excitation proceeding via a direct collision (no complex formation). (2) The excitation distribution j’(f~*) is struct-
urally unchanged with respect to the purely repulsive case. The largest dynamically accessible j’ increases with rising well depth. Results are shown in fig. 5. Notice that there is a fairly well defined smooth curve j;(b*) giving the largest possible j’ as a function of b*. With increasing e the function ji(b*) becomes less steep for small b and increasingly steeper for large b. The numerical results also indicate that the trajectories are uniformly distributed under the curve j,l(b*) for all b*. Under these circumstances the integral cross section for rotational excitation a( j’) is basically determined by j,$b*). This result has an interesting aspect for several empirically found laws on the j’ dependence of u( j’). The highest energetically allowed j; = 72 is in all cases much larger than the dynamically accessible j’ levels so that rotationally bound complexes do not exist. (3) Plots of the (j’, @)-distributions illustrate the m’-summed differential cross sections. The well-known rotational rainbow line (type CYof ref. [ll]) exhibiting large scattering intensity is extended to negative 8 as the well depth increases. The overall smooth rainbow line is interrupted by the rotational shadow which causes diminished forward scattering as well as enhanced excitation of large j’ at small negative 8. The triangle shaped area is also a rainbow structure (type p of ref. [ll]) which stems from large impact parameter scattering. (4) The 3d-plots have been projected onto the (m’/j’-0) plane to visualise the orientation of the
26
D. Poppe / Rotationally melmtic atom _ molecule scattering
final molecular angular momentum. Negative and positive 8 show a quite different behaviour. The latter give rise to an m' distribution which is similar to the corresponding distribution for the repulsive case (see fig. 1). Negative B values (undistinguishable from positive 8 in a scattering experiment) lead to a more random distribution covering the whole (m’/j’-0) plane. For c = 0.3 eV, figs. 6a and 6b display the distributions in the cos 8-p plane for positive and negative 8, respectively. Deviations from the hard shell result for positive 8 are minor and occur mostly with small j’. Negative 0 values fill the whole cos S-/3 plane, however, still with a preference of cos 6 = 0. Strong deviations from the impulsive case are also witnessed. Moreover the /3 population is not uniform preferring B = 7/2 and /3 = 31~/2 the more the larger c is. This carries over to P( m’ 1j’s) (defined in the appendix) for 8 < 0 which is expected to be still doubly peaked similar to the hard shell case. The maxima, however, are shifted to larger 1m' I. We note as an aside that p = n/2 and /3 = 3n/2 correspond to a j' which lies in the YZ plane (see fig. 12 below) being the plane of the initial approach of the atom. The corresponding trajectories are extremely non-planar with I’ > 1. The apparently symmetric pattern in the cos 8-p plane expresses the fact that the m' distribution is symmetric with respect to the sign inversion of m'. The differences with respect to the hard shell case are mainly due to the influence of the isotropic attractive forces. They lead to a rotation of the relative momentum vector prior and after the angular momentum transfer occurs. Due to conservation of the total angular momentum the corresponding rotation axes are the initial and final orbital angular momentum 1 and I’, respectively. From our calculations we conclude that these rotations are of minor importance for positive scattering angles. Integral cross sections a( j’) are shown in fig. 4. For vanishing well depth we obtain a monotonic exponential decrease of u with increasing j’. With rising c the cross sections decrease slower and exhibit a sharp drop off at the largest classically accessible j;, . This result can be well understood within a two-step model of the collision. In the first step the isotropic well enhances the momen-
. : . 1 :. . I
a’
I
I
P I
cos
: y
-.<.;.; ! .
.‘-.’ 1.: ‘.-L’ ; ..:1,+ . ;: .. .. . .
‘..L’ $>> .*_: . - :‘:
: . i..?:
..+:.:;.: : 3.
. ..::
’
4;
.a. .
.. 1, :: , .
I.
’ :
..*..:
_.:-.,.
:-
.. :.
..s : - * :.. . .
Y...
;, . . . ....’
:: &..!.’ .:
: .;:.
’
b . ..;. I..?.. . .
.::
.
.
..-;‘;, .
P Fig. 6. Orientation of j’ in the cos S-/3 plane for potential I for positive (a) and negative (b) scattering angles. E = 0.1 eV. M = 4, m = 23. The distributions are shown for 0 < /3 < 2n and -16~0~661.
turn of the atom and reduces therefore the effective impact parameter responsible for the rotational excitation. For example a well depth V, reduces b by a factor of [E/( E + Vo)]‘/2. The second step is the rotational excitatibn in the repulsive part of the potential. It takes place with an enhanced collision energy (E + V,), however, only for the reduced impact parameter. In comparison with collisions without a well, large impact parameters are now missing which favour small angular momenta. In this way a focusing effect for the effective impact parameter offers an explanation for the j’ dependence of u.
D. Poppe / Rotationally inelastic atom -molecule
For 4 = 0.6 eV (averaged well depth = 0.23 eV) complex formation is observed (see fig. 7). About 2% of the trajectories exhibit a complex living longer than 2.4 ps. The long lifetime is caused by
.I
J
scattering
27
strong intermediate rotational excitation, which hinders the particles to separate since they cannot overcome the centrifugal barrier. The decay is only allowed after redistribution of the molecular energy and/or lowering of the centrifugal barrier. We have also studied the mass dependence of the excitation. All calculations discussed so far were performed for m = 23 and M = 4 which we refer to as the HHL case. The replacement (m = 2 and M= 46) gives the LLH case. It leaves the reduced mass of the projectile with respect to the target unchanged. The target, however, is now lighter by a factor of 11.5. The motion prior to the rotational excitation is still the same for the HHL and LLH case. Changes are introduced by the enhanced rotational velocity of the lighter molecule and the larger rotational energy associated with a given j’. The largest energetically allowed molecular angular momentum is j; = 23. Fig. 8 presents results for c = 0.6 eV. The anisotropy is strong enough to excite j; = 23. In comparison with the HHL case we observe extensive complex formation (= 25%) now also due to the fact that the anisotropy allows for rotational excitation beyond the energetic threshold. In general all structures of the HHL case are also present for LLH, however, less prominent. The integral cross section a(j’) depends only weakly on j’ with a steep decrease just below j;. 3.2. Model II
.I
1
L$
tL
1
8
Fig. 7. Same as fig. 1 but for c = 0.6 eV. j,&_ - 42.0, b,,,= = 7.35 A.
In contrast to the previous case, model II allows for rotational excitation by the long-range attractive part of the potential. The spatial form of the torque av/aar has completely altered compared to model I (see section 2). Again we have studied in particular the influence of the well depth now given directly by c. Results for three values of E are shown in figs. 9-11. Characteristic differences between model I and model II: (1) The excitation distribution j’(b*) exhibits now two regions of efficient excitation. The scattering with small b leads to positive deflection angles and is mainly governed by repulsive forces. The “minimum” of j’(b2) is caused by trajectories that experience dominantly the equilibrium geometry of the potential where the anisotropy is
D. Poppe / Rotationally inelastic atom-molecule
28
small. Here also trajectories contribute along which the torque oscillates in time leading to an overall small excitation. The excitation function witnesses strong angular momentum transfer for medium
impact parameters b. Comparison with the deflection distribution shows that they are associated with negative B values near the common angular rainbow. Large values of j’ are produced by
m ..
:: ‘I,.;‘;
.,.:. .
;
.
,-,.;.;:,
i
.:‘. i’ ., ,,._.,.r ‘_:.:‘*r, .. ..
\ .
1,;
.p,.
.._,
,
..
:
‘..‘..:; .,.p.:>: .s-.,
:: .f,&.‘-
.
;
:
“,f..,
:,
p,’
rz~;-,‘~C$::.~. .a.‘?
.‘,.i.. :,.\ ,
.y:.:,.;., ::
\:
,:,.:. ,..*, ‘. . T. ‘,. : ‘.
.,,,’
,..$
,.-
z_ ,
f. ..;.
j
.,
‘-5:
:
1 b2
j’
..:_.
,;., :;,;.
_._
L.
. . .;
.
:
..:>f:..
scattering
,y
5:‘: .:.,...14’ : .: ..‘. .
_,.-a.. ,.
: _‘,.,_
..
.,L..
; i:,. ‘. ‘. :_:, ..:,*. .,..$
.
:
j:...;.:
.,‘, :‘:~.:,y. :; _.:
:
c;y:;‘:
;’
,’
.%.!
e
b2
Fig. 8. Same as fig. 1 but for c = 0.6 e-V. m = 2, M = 46. j&, - 42.0, b,,,, = 7.35 A.
Fig. 9. Same as fig. 1 but for c = 0.1 ey. Potential II. j,& - 12.1, bmax - 6.91 A.
D. Poppe / Rotationally inelastic atom-molecule
trajectories that pass the equilibrium R, at (Y= 0 or a = v/2 thereby maintaining the sign of the torque. The tail of j’(b*) at large b is generated
scattering
29
by the attractive forces of the potential. The shape of the distribution is determined by the relative magnitude of attractive and repulsive
b2
.I
J
$ J
IL8
Fig. 10. Same as fig. 1 but for c = 0.2 eV. Potential II. jA - 28.6, b,,,, = 6.9 A.
.I
. . ... : ..
..
-
.
.. ...
I
1
!I$ 1
t/
0
Fig. 11. Same as fig. 1 but for c = 0.6 eV. Potential II. jA - 44.5, b_ = 11 A.
30
D. Poppe / RotationaNy inelastic atom - molecule scattering
anisotropies and depends therefore also on the radial form of the potential (compare with fig. 2 of ref. [14]). (2) The projection onto the j’-8 plane displays now two rotational rainbow lines. There is a rotational rainbow with large scattering intensity at negative 8 values starting at B = 0. It stems from trajectories with relatively large impact parameters. These trajectories experience only the attractive forces. The structure is analogous to the rotational rainbow for excitation in a repulsive potential under 10s conditions. To explore the generic properties it may still be appropriate to apply the 10s assumptions to these large impact parameter collisions. This can be approximately justified by the fact that a large fraction of the final molecular angular momentum is transferred in a particular configuration R and (Y (to be identified with the 10s orientation tires) near the classical turning point. The other weaker and less singular rotational rainbow generates a step in the m’-summed differential cross section along the function j&(e) which is the largest j’ for a given B. There is no simple dynamical interpretation for j&(e). We note, however, that with increasing c the attractive forces become more and more important for rotational excitation (see the j’-b2 distribution). Simultaneously the j;(e) curve flattens. In the region of orbiting collisions it becomes diffuse and is not well defined. (3) The spatial orientation of j’ for small c =G 0.1 eV is similar to that for model I. Positive scattering angles show j’ I Ap while negative 0 values give rise to a random m’ distribution (see &/j’--e plot in fig. 9). For c > 0.1 eV we found that the m’ distribution for positive B values is more or less uniform, while negative e prefer m’ = kj’ the more the larger j’ and ) 8 1 is. For m’ = kj’ the molecular angular momentum is directed along the z axis which is the direction of the initial approach of the projectile. The molecular axis r lies then in the xy plane and is still near its initial configuration, since the rotational period is large compared to the collision time even for the largest j’. Particularly the non-planar geometries with r around the bisector of the x and y axis lead to m’ = kj’ with a large j’. If the molecule is light enough to allow for
appreciable rotation during the collision the orientation of j’ changes as can be seen from the LHH case with E = 0.2 eV (not shown here). Then m’ = 0 is preferred the more the larger j’ is. This finding is in accordance with the results of Barg and Toennies [18] for Li+-Hz. This effect is also present for model I however it is very weak (see fig. 8).
Acknowledgement The author gratefully acknowledges financial support from Sonderforschungsbereich 161 “Hyperfeinwechselwirkungen” and lfrom “Fonds der Chemischen Industrie”.
Appendix The classical (conditional) probability P( m’ 1 ej’) for m’ with given 8 and j’ is related to the distribution in the cos 8-p plane by P( m’ 1f3jf) 1 cc 2”d(cos 6) dfi s(m’-m(/3, JJ -1 a x NW
I W7
(A-1)
where Q expresses the (conditional) p and S for given B and j’. From that m(P,
y, 6))
Y, 8) =j’(sin
probability of fig. 12 we see
P sinysin6+cos6cosy). (A.?)
For the rotational excitation of a hard ellipsoid with a point particle one has exactly cos 6 = 0 and an uniform 8. Then eq. (A.l) can be evaluated analytically to give P( 172’I ej’) a (1 - m’/m,)-1’2,
(A-3)
with Im’I cm,. The largest classically and B m. = j’ sin y,
(A.4 accessible
m depends
on j’
(A-5)
31
D. Poppe / Rotationally inelastic atom - molecule scattering
eq. (A.2), we find that 1~2’1Gj’ max( Icos(S+y)
I, Icos(S-_y) I). (A-9)
For a given scattering angle 0, eq. (A.9) relates the largest m’ to the possible deviations 6 # 7r/2 from the hard shell result. Y
Fig. 12. Geometry of the scattering. The space fixed coordinate system is denoted by xyz. The z’ axis coincides with the transferred momentum Pp. The angle /3 is chosen such that j’ is in the x’z’ plane. The angle S is between j’ and Ap.
since cos y= -(p-p’
cos e)
x [ p2 + (pt)’ - 2pp’
cos
81 -l’*,
(~.6)
where p and p’ are the initial and final relative momenta, respectively. Neglecting the rotational energy transfer ( p = p’) we simplify the equation above to y=n/2+8/2.
(A-7)
Eq. (AS) reduces to m =j’ cos( d/2).
(A.8)
The projected 2d distributions for 8 and m’/j’ in fig. 1 exhibit a limiting curve which is well described by eq. (A.8). Returning to the general case,
References [l] L.D. Thomas, in: Potential surfaces and dynamics calculations, ed. D.G. Truhlar (Plenum Press, New York, 1981) p. 731. [2] R. Schinke and J.M. Bowman, in: Molecular collision dynamics, ed. J.M. Bowman (Springer, Berlin, 1983) and references therein. [3] D. Richards, J. Phys. B 14 (1981) 4799; B 15 (1982) 3025. [4] V. Khare, D.E. Fitz and D.J. Kouri, J. Chem. Phys. 73 (1980) 2802. [5] R. Schinke and H.J. Korsch, Chem. Phys. Letters 74 (1980) 449; P. Eckelt and H.J. Korsch, J. Phys. B 10 (1977) 741. [6] D. Beck, U. Ross and W. Schepper, Z. Physik. A 293 (1979) 107. [7] A. Mattheus, A. Fischer, G. Ziegler, E. Gottwald and K. Bergmann, preprint. [8] M.A. Treffers and J. Korving, Chem. Phys. Letters 97 (1983) 342. [9] K. Bergmann, R. Engelhardt, U. Hefter and J. Witt, J. Chem. Phys. 71 (1979) 2726. [lo] P.L. Jones, U. Hefter, A. Mattheus, K. Bergmann, W. Mliller, W. Meyer and R. Schinke, Phys. Rev. A 26 (1982) 1283. [ll] R. Schinke, H.J. Korsch and D. Poppe, J. Chem. Phys. 77 (1982) 6005. [12] H.J. Korsch and R. Schinke, J. Chem. Phys. 73 (1980) 1222; 75 (1981) 3580. [13] K. Bergmann, U. Hefter and J. Witt, J. Chem. Phys. 72 (1980) 4777. [14] D. Poppe and R. Battner, Chem. Phys. 30 (1978) 375. [15] D. Poppe, Chem. Phys. 84 (1984) 243. [16] D. Poppe and H.J. Korsch, Chem. Phys. 85 (1984) 267. [17] R. Schinke, J. Chem. Phys. 75 (1981) 5205(L). [18] G.D. Barg and J.P. Toennies, Chem. Phys. Letters 51 (1977) 23.