Rotational rainbows in atom-rigid rotor scattering: A comparison of the classical ios approximation with classical trajectory calculations

Chemical Physics 69 (1982) 99-108

North-Holland Publishing Company

ROTATIONAL. BAINBOWS IN ATOM-RIGID ROTOR SCATTBRING: A COMPARISQN OF THE CLASSICAL IOS APPROXIMATION WITEI CLASSICAL TRAJBCTQRY CAXULATIONS H_J. KORSCH Fachbereich Physik. Universitlir Kaiserslautenr, D-6 750 Kaiserslaurern,

West Germany

and D. POPPE Iivun 1V.Stmnski- Institutfir Physikalischeund TkeoretischeChemieder TecimischenUniversit&Berlti, D-I 000 Berlin. WestGermany

Dedimted to Professor Ernst Lippert on the occasibn ofhis6OtJ1birthday Received 10 November

1981;in

final form 22 hlarch 1982

The classicalequivalent of the quanta1 IOS approximation for rotationally inelastic atom-molecule collisiocs is derived. A detailed comparison of Integral and different% cross sections with exact classical trajectory caIcuIations for the model He-Naa potential Y(R, 7) = ARma [ I+ ef’z (cos -y)] shows verygood agranent. Such a comparison can be used as a test of the validity of the 10s assumptions for systems, which do not yet allow a direct cemparisnn between exact and IOS quantal results. The generic structure of the classical differential cross sections is discussed and compared with the 10s results. Furthermore, the validity of the IOS factorization formula for initially rotationally excited molecules is tested. Good agreement is found.

1. Introduction One of the most important recent findings in IOtationaliy inelastic atom-molecule collisions is ‘he experimental observation [l-4] of rotational rainbow structures in differential state-to-state cross sections, which had been predicted previously from theoretical reasoning [5-91 (for a recent review on iOtatiO%tl rainbows see the articles by Thomas [P] and Schinke and Bowman [lo]). Most of the theoretical treatments are based on rather crude approximations of the interaction potential or the collision dynamics: The rotational excitation of a hard-shell molecule can be treated almost in closed form in classical mech~s [ 11,121 (for a two-dimensional quantum treatment see ref. [ 131). The most widely used dynamical approximation is the infinite order sudden (IQS) approximation (for review see ref. [14]). The IOS approximation is mainly used as a quantum mechanical decoupling scheme, 0301-0104/82/0000-OOOOl$OZ.75

0 1982 North-Holland

which reduces the inelastic atom-molecule system essentially.to elastic scattering from a spherically symmetric potential (the anisotropy enters only via a parameter). The JOS approximation is intuitively based on the assumption that the collision is sudden (steep potentials, high energy or - in other words - short collision times compared to the rotational period of the molecule) and almost elastic (only a small fraction of the energy is transferred to the molecule). The socalled centrifugal sudden approximation (i.e. the orbital angular momentum of the colliding atom is replaced by a conserved effective one) and energy sudden approximation (i.e. the rotational states of the target molecule are energetically degenerate) achieve a complete decoupling of the coupled equations. For noble gas-Na2 collisions at about 1 .O e%’the 10s cross sections show a remarkable agreement with experimental measurements [2,15]. In contrast to the large number of applications of

159 Korsck, D. PoppefRotan~~~mirl~w~:cfcssiculIOSappro.rhation

ICI0

the IOS approximation its cksiicai.equivaIent has only

J = 0 +J’ cross sections by means of the quantal fac-

attracied very little attention. Relatively few paperr discuss dynamical decoupling schemes in classical mechanics (see, e.g. refs. [16--231 ar,d references given therein), but the precise classical version of the 10s approximation has only been obtained very recently in context with a more elaborate investigation of the full semiclassical limit of the quantal IOS scat-

totiation formula [28] OI iti classical z&g [29]. For the present derivation it is most convenient to use body fzed coordinates (for details see ?&lloney and Schatz [20] and references given therein) where L’ and J2 are given by

tering amplitude [24] {an application of this semiclassical trea;ment to the case of rigid-shell molecules [ 11,12] can be found in ref. [25]). In the present article a derivation of the classical 10s approtiation is presented completely in the

framework of classical mechanics. The validity of the approximation is tested by comparison with exact cIassical trajectory calculations for a model He-Na., potential. The nature of the dynamical simplificati&s in the 10s scheme has nothing to do with the level of description (classical or quantal). Therefore a comparison of the classical results can be valuable as a test of the accuracy of the 10s approximation for the large number of systems, which caMOt be treated by means of numerically exact close-coupling programs at present time.

L’ = J&t iJ’-2&t s

= P? + Pi j&y

_

muthal angle of the rotor with conjugate momentum P.:, _Py is the momentum conjugate to 7. Following ref. 1201 we now introduce the centif& gal sudden appro.ximation, i.e. we replace the_dynamical variable L by an effective constant value L, which reduces the ten equations of Imotion to the hi coupled equations [20] fi=P/tTl, P=Lqm~3

(4) -

acqkryaR

We consider the collision of az atom A with a diatomic mdecule BC (or? more generally, a linear molecule) which w-ill be assumed to be a rigid rotor (an extension of the present work to nonrigid molecules including vibrational excitation will be presented in 1 forthcoming article [27J). We will closeljl follow the recent article by Mulloney and Ssfistz 1201. The hamiltonian is f J2/21 + VCR, y> ,

)

^j=P$,

6) (6)

av(x,7)iay _

(7)

The deflection angle can be obtained from

2. Classic& IOS spprotiation

+ L’j2mR’

(3

Jtot is the total angular momentum and $J is the azi-

k7 = P; cot r/f sin+ -

H = P’j3n

(2)

‘J,

(1)

where m is the reduced mass of A-K, lis the moment of inertia of the molecule and y is the angle between the vector R from the center of mass of the molecule to th2 atom A and the molecular axis 2. P is ;he momentum conjugate to R = [RI, L is the orbital angular momentum and J is the rotational momentum of the moIecu!e. We will furthermore assume that the molecule is nonrotating initially, mallg use of the fact that within the IOS approtiation the cross section for J -+ J’ transitions can be obtained from the

e’=&R2.

(8)

We furthermore have P, = 0, so that P+ is a constant of motion, which is zero in our case, because the molecule is initially at rest. The next step - the energy sdden approximation assumes that the molecule can only absorb rotational momentum and no rotational energy, i.e. the moment of inertia I is wry large and the terms containing 1 in the denominator can be neglected. Eq. (6) irmnediateIy gives y = 7 = constant in this approximation, i-e_ the rotor is fixed during the collision. This approximation leads to a complete decoupling of eqs. (4) and (5) determining the trajectory of the atom from eq. (7) for the rotor. Closed form expressions for the deflection angle 6 and the foal rotational momentum J of the molecule [eq. (3) gives J= PT because of Pfi = 0] c;lil be obtied from eqs. (8) and (7) by removing the time dependence with help of eq. (4), and we end up with the classical deflection function (for convenience we use

HJ; Korseh. D. ~oppe~Rotationri?ainbovJs: the

notationL,

y instead

of L, 7

in the following)

f?(L,y)=a-z

X 1

dRR-*{Zm[E-

V(R,y) - L7/2??2@]}-1~~ (9)

RO and the classical excitation function

x y

al? a$ (2m[E - V(R,y)

- Ly2??2@] j--1/2, (10)

&l

in complete agreement with the expressions derived previously in a semiclassical analysis of the quantal IOS scattering amplitude [24]. In the following we restrict ourselves to homonuclear molecules and repulsive potentials V(R, 7). An extension to the heteronuclear case is obvious (see for instance ref. 1241). The effects produced by anisotropic potentials possessing a minimum are more intricate and will be discussed in a forthcoming article [25]. Because of the symmetry V(R, 7) = V(R, B - 7) in the homonuclear case we can restrict ourselves to orientation angles 0 G y < irj2 and - in qnantum mechanics to even Aj transitions j= 0 +j’ = 0,2,4, . .. . The classical double differential 10s c:oss section is given by d*o --_= dSUJ

l o&sin8

_I_LdL. 0

cLxticdlGS

7,)intheregionO~y,~~/2~dO~L,<~_Inthe present case we have two such roots in the classically

101

allowed region (compare the detailed discussion in refs. [10,24]). Explicit expressions for the derivatives ofB(L,~)andJ(L,~)appearingin(ll)as~ell3sa discussion of the numerical methods can be found in ref. 1241. Integrating eq. (11) over J intervals of length 2fi centered at J = (j’ f $) fi(j’ = 0,2,4, ...) we obtain the quasiclassical cross sections

forj = 0 -+ j’ transitiorIs in complete agreement with

the result obtained in ref. [24] from a semic!assical analysis of the quantum 10s amplitude. The differential cross section (12) can be calculated by Monte Carla evaluation of the integral in eq. (11) or alternatively by direct computation of eq. (12), which requires two-dimensional root searching. Numerical calculations (compare section 4) show, that the last procedure is far more efficient, which is due to the bad statistical convergence of the Monte Carlo integration. For the integral cross sections, however, a Monte Carlo integration is more convenient. The classical 10s approximation leads therefore to closed form expressions for the differential cross sections for rotationally inelastic collisions, which - contrary to the remarks in ref. [20] - lead to a drastical reduction in computation time in comparison with a full classiczl integration of the equation of motion combined with a Monte Carlo sampling technique. It shou!d furthermore be pointed out, that quantum effects can be included in a straightforward manner by classical S-matrix techniques (see ref. [24]). The classical differential IOS cross section (12) diverges at the classical rotational rainbows determined by the vanishing of the Jacobi detenniuant W,WXL,

where the sum NIX over all simultaneous solutions (L,, 7,) of the equations 0 = B(L,, 7,) and J=J(L,,

approximation

‘y)lr&~w = 0

03)

(compare the discussion in ref. [24]) at the boundary of the classically allowed region. Eq. (13) determines the rotational rainbow curve JR = .JR (6) (JR is the maximum final molecular angular momentum for scattering angle e) and the corresponding rainbow orientation 7R and orbital angular momentum L,. It can be easily shown from (12) that the classical 10s cross sections show a typical square root singularity at the rainbows, which is an artifact of the IOS approximation and which is structurally changed into

a st2p and a logarithmic singutity in a more exact treatment (see ref. [23] and the discussion in section 4 of the present iirticle).

corresponding

to a confidence

level of 95% [32].

4. Applications 3. Classical trajectory

calculations

The validity of the classical IOS approximation is most conveniently tested by comparison with exact classical trajectory calculations. We used a program package described elsewhere [30-321 in detail. The main features cf the present numerical calculations are: (i) The equations of motion are solved numerically in carte&n coordinates by means of a Hamming predictor-corrector routine. Conserv&ion of energy and total angular momentum is always better than

l%o .

(ii) The vibrational degree of freedom was also taken into account_ The Na+teraction was approximated by a Morse-poteniiaI fitted to the spectroscopic data from Henberg [33]. Calcuiatiors for different initial vibrations @v-epractically the same results. (iii) Trajectories are analyzed with respect to deflection angle 8 and molecular angular momentum J. The rotational quantization is done by means of a boxing procedure, where the final molecular angular momentum is converted into a noninteger quantum number h via J2 = R’X(A + 1). We assume, that the final statej’ is excited, if m&x(0,X--i}
(14)

a prescription, which obviously underestimates the cross section for j’ = 0, which is, however, of Little interest in the present study_ The anguiar distribution is devided into equidistant boxes (n - 1) A0 < B < nA6 (IZ= 1,2, ___). In all calculations reported here we used &I = iOQ. (iv) Cross sections for transitions to the foal state f were calculated by Monte Carlo averaging from Of = T&&NffN

>

v(R, r) = v&I&)

>

(17)

with g(y) = I + E co&

(18)

and

V,(R)=AR-“.

(1%

All our calculations

were done for the He-Naz system at energies of about 0.1 eV. W2 used the potential parameterse=3,n=8andA=1250eV&g,whichisa rotigh approximation to the ab initio potential surfaca [EJ]. The same system with a much smaller anisotropy e was studied previously by means of classical prturbation theory [23]. in this study a characteristic difference in the structure of the classicaUy exact cross sections and 10s cross sections has been predicted. It is thus of interest to compare the computational fmdings of a classical exact trajectory computation with these structural predictions, as well as, of course, wiii’th the classical 10s results.

4. I. Ana& tied expressions Within the 10s approximation tional momentum transfer (recall molecule is nonrotating and that Iy repulsive) occurs for backward For the potential (17) we fmd J(L = 0,~) = -2me

the maximum rotathat initially the the potential is purc(I, = 0) scattering.

@g/d?)

WI

where IV$V is the fraction of trajectories ending up in the desired fd box f, and b,, is the maximum impact parameter. Error e&n&s (&en as error bars in the next section) are calculated by means of uf’ = of + 2+-l/

We have performed numerical computations for a model potential with a P2-anisotropy and.an inverse power R-dependence

[Xf(1X - iv,)lniL] 112 )

(W

which can be treated ti closed form for an exponential radial potential YO(r) i= e--& gi-ving a m&urn possible rotational momentum of [9]

103

H.1. Korzch, D. Poppe/lZotatiorzrl rainbows: ckzsxicalIO.9appmximation

JR@) = 2&a - b) sin 8/2

2i

TR = c@s-‘(2 f +1/2

,

(22)

where p = (~JKQ~/~ is the classical momentum; i.e. the rainbow orientation rR does only depend on the potential anisotropy e and J,, is proportional to the square root of the energy, just as in the case of hard shell scattering [26]. For the inverse power potential VO(r) = ARen we obtain J(L = 0,7) = 2Pf,(@

’

C&7)$)‘”

(23)

with (24)

For g(7) = 1 + E ~0~~7, eq. (21) has a maximum at 7R = cos-’ {f n(l/E + 1/2 - l/n) - [$ &l/e

+ l/2 - l/n)2 - n/2e] q=

)

(25)

which reduces for very steep potentials n +- to (22) and the maximum rotational momentum is given by J max=J(L=0,7=7R).

(26)

J maT is proportional to E1/2-11n, which gives J,, =EW for steep potentiak - in agreement with the

hard shell result [26 1. In the small

scattering

ande

(Le.

high

ai+ar

mo-

mentum) region an analytical expression for the def’lection and excitation functions can be derived for inverse power potentials V,(R) = AR-” [lo], and the $-dependence of the rotational rainbow is obtained as (ref. [lO];seealso theappendixofref. [31]) JR(O) = ~t’~-~/” ,

(30)

where a and b are the semi-axes of the ellipsoid. 4.2. Differemtii

0 + j’ cress sections and rainbobv

StrUCtUWS

In the following sections we report some numerical results for the model He-Na2 system described in the beginning of section 4. The 10s rotational excitation function J(L, 7) in fig. 1 shows the welI known behaviour [10,24,26] for purely repulsive potentials dominated by a CO& anisotropy: As function of L, J(L, 7) has a maximum at L = 0 and decreases smoothly to zero with increasing values of L. J(L, -y) is zero for 7 + 0 and 7 = 90” and shows a maximum at an intermediate angle. The position of this maximum is slightly and the height is strongly L-dependent. The r-dependence of J(L, 7) clearly indicates that there will be two roots of the simultaneous equations 6 = 6 (L, 7) and J = J(L, 7), even for a deflection function decreasing monotonically in L and 7.8 (L, 7) is pfctted in fig. 2 for 7 = 0’ and 90’. Also shown in fig. 2 are the L-values solving 0 = 0 (L,7) andJ=J(L,y)foraf=edvalueofJ=(j’+1/2)8 with j’ = 4. This classical 10s deflection curve e (L IJ) for fixed J shows the typical maximum, which produces - in the case of purely elastic scattering from potentials with a well - the well-known ordinary rainbow. At the extrema of 0 (L [J, and J(L, 7) the jacobian of the mapping (L, 7) + (e, J) vanishes, because of

(27)

with

where rR is given by eq. (25) and f2(?z) = TGr((?z

i- 1)12)/r(n/2)

.

(29

Eq. (2S).is valid at small scattering angles. In the limit of a very steep potential (n -+ “)JR is proportional to 0, again in agreement with the-approximate hard shell result 112,261 which gives for the case of hard ellipsoid scattering [ 121

Fig. 1. Cl&s&l 10s rotstional excitation function J(J~,7) WISUS orbital an@ar momentum L for fwed molecular &entation

angle 7. The inset shows

the 7 dependence

for fiuedl.

104

H.J. K.mch, D. PappefRotatimaZ rainbows: ckssiml IQS approximatin

F&. 2. Cla_tiicalES deflection function s(L, 7) for fLved maleculu orientation urge 7 (dashed lines). The solid curve show the 10s deflection function 63(L I/) for fixed final rotational momenkm J= (? f l/3) h(j’ = 4). The minimum CZISX the rotational rainbow. The (e, .L)-u~zz which Ieads to 2 j = 0 + j’ = 4 transition in the exact classical tmjectory computations is shaded.

i; Fig. 3. Esaet &.ssical Monte Carlo computations: Density plats in the scatteting antie-impact parameter plane for freed j = 0 + j’ = 0,2,4,6 +&tiors. The numbers are proportional to the number

of trajectories

which start with given impact pa-

;uneter b and end-with ,$e scattering outcome 0, j’ (alI quan(Fcr a more detailed discussion of the zeros of the

jacobian (31) as well as of the relationship between ordinary and rotationaI rainbows see refs. [9,10,25, 34,351.) Also investigated in fig. 2 is ‘&e outcome of the exact classical trajectory calculations: The shaded area shows schematically the scattering angles 8 found for those trajectories, which lead to excitation of the final rotor state j’ = 4 as a fkmction of the initial orbital angular momentum L. This fully class&I 0 (L IJ) dependence is, of course, not a simple function but a density distribution P(tl, L IJ) where an average is taken over the additional initial variables of the exact classical calculation. The maxima of P@, L IJ) show a double-ridge structure which coincides nearly with the IOS curve 0 (L [J). In this way the 10s approximation samples quite well the most important part of the classical density. This effect is most clearly seen for smalli’ f 4. For larger values ofj it is less pronounced. Fig. 3 shows more detailed classical density plots in the (8, Q-plane for various values of the foal rotational momentum of the mole&e. 8 and 6 are scaled quantitiesaccordingtog=(@-_1)/@.,-8l)andg=(b - bl)/(& - bl), respectively. 6t and 8, (bl and bz) are the n&imum and maximum v&es of .3(b) observed in the Monte Car10 computation. [In fig. 3 b,, h2, 01, 01 are given by 039,5.0,5.5”, 164°C = 0);

tides are boxed). 6 and b xe the scaled scattering impact pvameter (see text), respectively.

z&e

axd

0.15,5.0,7.0”, 171° o’= 2);0.iX,4.5,16.8°, 178’ (i= 4); 0.16,4.i, 29O, 170” (j = 6), where the values of b are given in A.] For small values of j’ the double ridge structure is clearly demonstrated. At higheri’ values the valley in the middle is more and more filled up and there is evidence for an additionaI new ridge in the center (see fig. 3 fori’ = 6), which is not described by the IOS approximation and demands further investigations. The differential state-to-state cross sections for 0 + 4 and 0 + 14 transitions are shown in fig. 4. Classical IOS results are compared with exact classical trajectory computations. -4 total number of lo4 chssical trajectories were computed in this case. On the first sight the agreement seems to be excellent, which shows the quality of the IOS approximation under the present collision conditions

and explains in part

remarkable success of the quanta1 IOS approximation in comparison with the experiment [2]. A more detailed inspection of fig. 4 shows, however, a characteristic structural difference between the classica!.Iy exact Monte Carlo computations and the 10s cross sections: The typical 10s square root singularity at the IOS chssical rainbow is changed into a somewhat more smooth behaviour. The exact classithe

H.J. Kmch,

D. POppe/ROmtiOMf

minbo ws: chicalIOS

105

approxinlalion

-I

0"

F& 4. CIasicaI 10s (solid Iines) and exact classical (Monte Carlo) differential cross sections for rotatiomzl ~ansiti0ns j = 0 +j’ = 4 and j = 0 +j’ = 14. The 10s cross sections show a square root sing&&y at the rotationP rainbow.

cal trajectory cross sections show a step 3: the boundary of the &s&ally allowed region and a more or less pronouncedmaximum. A more detailed investigation using smaller A&boxes and more trajectories showed that this behaviour is not an artifact of the limited number of trajectories and the Af3 = IO” angle boxing in fig. 4. The same kind of behaviour was recently discussed by Korsch and &chards [23], who actually predicted a breakup of the 10s square root singulality into a step and a logarithmic singularity. The logarithmic singularity is, however, very weak and does not show up in the present computations because of the quasiclassicalj’-boxing procedure. Fig. 5 shows a compa&on of the IOS rotational rainbow curve J,(Q), i.e. the boundary of the classically allowed IOS transitions, and the smallest scattering angle found in the c&i&l trajectory computation. for a given transition 0 +i’ = 0,2,4, . . . [recall that we have J= 0” + 1/2)!)n]. The agreement is excellent. Furthermore fig. 5 gives the maximum possible J-K& ue (18.6) - for backward scattering - given by the analytical expressions (23)-(26) with n = 8. Ako shown is the smaJl angle approximation (JR = a7/*) calculated by eqs. (27)-(29) with good agreement. The energy dependence of&&e maximum classical rotational momentum transfer JmaY is shown in fig. 6, where the IOS result Jmz = CEl/*- l/n [eqs. (23)(26)] is compared with the maximum J found for all N trajectoriks computed for a given energy (obviously this maximum is only a lower bound for the classics J maxand tends to increase with IV)_ The classical results

300

600

900

1200

1500

1500

rainbow JR VBISUSscattering result [eqs. (26)] for backward scattering and the broken line is the smallscattering angle approximation [eq. (27)J. Also shown are the rotational rainbows found in exact classical trajectory computafions. The minimum scattering angle found for a Fig. 5. Classical (IOS) rotational

angle 0 (solid curve). The triangle (a) is the analytical

given j = 0 -j’

transition is marked (0).

show the same E1/z-I/n dependence they are, however, somewhat lower.

as the IOS results,

4.3. In tegraij + j ’ crosssections and dynamical thresholds The integrai state-to-state cross sections for rotational ground state are shown in fig. 7. Tne IOS cross

32 28 2L 20 i

I

I

0.05

I

0.1

,

0.2

I

0.3

E (eV] Fig. 6. Double Iogarithmic plot of the energy dependence of the cIassiczxl10s mzi&num rotational momentum cansfer [eq. (26)] (solid line) compared with the mMmum found in the exact classical trajectory (Monte Carlo) computation withN= 103-lo4 tiajectories (o), which tends toincrease with N.

‘;; 5.

I

..

y-+2;_

:i

1

1

I-

Fig. 7. Inte& 10s ~!rosss&ions ~(0 4 j’) forE = 0.1 eV (--0-1 snd E = 0.3 eV I-n-). ForE.= 0.1 eV the resulis from an exact classical (Monte Carlo) comptltation shown (o).

are also

___

._._._T-_-I:

__.L?q_

-I

.Y

(

I

J

OL ’ 0

0.2

0.1

0.L

03

EieV)

F&. 8. Energy dependence of selected ~(0 -7)

sections are computed by a Monte Carlo sampling procedure (compare section 2) and zre given for two energies (E = 0.1 eV and E = 0.3 ev). The ciassical integral crosssections diverge for elastic scattering due to the non-fmite range of the potential. There is a steep decrease of the cross sections for small values of j’ and a more moderate decIine at intermediate j’-values up to a maximum &&., where the cross sections drop down to zero. This typical behaviour (similar 00”) distributions have been reported and discussed. e.g., in ref. [9,15,26]) is easily esplzined by inspection of fig. 5: With increasingj’ the classical accessible angular region BR(i’) < 8 < pi is more and more restricted and for j’ exceeding J,,,lfi - I/:! the transitions are classically forbidden. The steep raise and the singularity at smallj’ are due to the forward peak of the classical differential cross sections, which is produced by the long range part of the potential, This smaU angle region contributes only

to the smzll i’ transitions. F&o plot:ed in fig. 7 are the exact c&sical

trajectov computations (IO4 trajectories). The agreement is almost perfect. The energy dependence of the a(0 -I’) cross sections is investigated in fig. S fori’ = 6, 1Z snd 18. The IGS cross sections show t_hetypical dynzmicsl thresholds, as discussed, recently by S&i&e f36J (we aIso ref. [ 151): The transition 0 -+j’ is dyncm&aily forbidden up to JmauQ, which increases with ihc energy E, reaches the value 0“ + l/2) h (the

integral cross sections. The IOS results (solid and broken curves) are compared with exact classical [Monte Carlo) trajectory computations for j’ = 6 (e), j = 12 (0) andj’ = 18 (A). The dynamical thesholds are marked by an =ow_

dynamical threshold). At this value of E the cross sections show a steep rise, pass through a maximum and decrease again, because of the overall decrease of the cross sections with the energy. A comparison with *the

classical trajectory results given again good apeement. In the remainder of this section we wiU discuss integral cross sections u(i +i’) for initially rotating 0 f 0) molecuIes. Within the IOS approximation these cross sections can be obtained from the a(0 -j’) cross section by means of the 10s factorization formula. Quantum mechanically we have a Clebsch-Gordul weighted sum [ZS] aQ’ -+j)=F

C2Q j”j’lOO0)

aQ” + 0)

(32)

and classicai.ly we fmd [29] j’+j

u(i'+-j)=J v”-ii

dj" Wj, j”, j’) a(?’ + 0) ,

(33)

where the classical weight function

WQ,j”, j’) = @‘/l;) X {[(j+j’)’

_

j"]b"

-Q'-j)2]}-t/2

,

(34)

H.J. Borsch, D. Poppe/RotrtionaI rainbows: cl~&alIOS

is the classical limit of the Clebseli-Gordan weight factor in eq. (32). Eq. (34) can be derived completely within the framework ofclassicalmechanics [29]. The 10s factorization formulas simply express l &e fact, that the rotational momentum transfer hlis - within the sudden approximation - independent of the initial J-value, but we have to average over ali directions of the rotational momentum J compatible w&h a given rotor direction r”(these directions are obviously given by&i=O). In fig. 9, a typical aG’ +j) distribution is plotted for E = 0.1 eV andj = 8. The 10s results are most conveniently calctited by means of the quantum formula (32), which avoid eventual difficulties due to the numerical

integration

of the square-root

sing&&ties

in

the classical equation (33). Also shown in fig. 9 are the exact classical trajectory computations. All 10s results are within the statistical error of the classical calculations. The 10s approximfition seems, however, to overestimate upward 0” >j) transitions and to underestimate

deexcitation

have been recently

(i’
reported

Similar results

by Agrawal and Raff

[37], who also gave a receipe to diminish tbis difference (multiplication by the ratio of the channel velocities ui,/ui= o). Fig. 10 finally investigates the energy dependence of the 00“ f- ~1 cross sections for j = 8 and selected j’

20

I.1.

.

.

.

.

.

.

.

.

j-0

0.2.

Fig. 10. Energy dependence of selected o(i-+i’) integral cmss sections. The IOS results (solid and broken curves) are compared with exact classical [Monte Carlo) trajectory computations. The dynamical thresholds marked by an arrow are the

same as in fig. 7 (see text). with Aj = -6,6,1&l& The cross sections show again the typical dynamical thresholds discussed above for the case j = 0. A comparison-with fig. 8 shows, that the positions of the thresholds (marked by arrows) are the same ti both cases. This is immediately clear, because within the IOS approximation AJ is independent of the initial J. If a transition 0 +j’ is dynamically allowed, then the transition j + j + j’ is also possible. This is also obvious from the IOS factorization formulas (32) and (33).

.

c-i-

OQ = IO

‘Y

concluding remark

*

j =8

._

-I

E (ev)

5. T

107

approximation

:\

As an obvious but neverthelessnecessary final reinark it should be pointed out that the validity of the classical 10s approximation as well as the exact trajectory results are of course limited by the validity of classical mechanics itself. Therefore important quantum effects - like interference 01 tunneling into classically forbidden regions - cannot be described. These effects can be approximately treated, however, by -semiclassical techniques [24,25].

Acknowledgement Fig. 9. Integrd IOS cross sections a(i-+ j')forE = 0.1 eV and j = 8 (-e-) calcuhted by mems of the IOS factorization

formula (32). The open circles (0) are the results from an

exact classical (Monte Carlo) tiajectory computation.

Acknowledgement is made to the Deutsche Forschungsgemeinschaft for fmancial support of this research. The authors are furthermore very grateful to

Drs. 31. her,

D. Richards 2nd R. Schinke for stimuJ.at-

ing Gsscussions,

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