Wave packet study of gas phase atom–rigid rotor scatteringYan Sun and Donald J. Kouri Citation: The Journal of Chemical Physics 89, 2958 (1988); doi: 10.1063/1.455001 View online: http://dx.doi.org/10.1063/1.455001 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/89/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Time dependent integral equation approaches to quantum scattering: Comparative application to atom–rigidrotor multichannel scattering J. Chem. Phys. 96, 5039 (1992); 10.1063/1.462747 Body frame close coupling wave packet approach to gas phase atom–rigid rotor inelastic collisions J. Chem. Phys. 90, 241 (1989); 10.1063/1.456526 Virial theorem for inelastic molecular collisions. Atom–rigid rotor scattering J. Chem. Phys. 73, 3823 (1980); 10.1063/1.440613 Neumann Series Solution for the AtomRigid Rotor Collision J. Math. Phys. 13, 1485 (1972); 10.1063/1.1665867 Molecular Collisions. XV. Classical Limit of the Generalized Phase Shift Treatment of Rotational Excitation:Atom—Rigid Rotor J. Chem. Phys. 55, 3682 (1971); 10.1063/1.1676649

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Wave packet study of gas phase atom-rigid rotor scattering Van Sun and Donald J. Kouri Department of Chemistry and Department of Physics, University of Houston, Houston, Texas 77004

(Received 4 September 1987; accepted 18 May 1988)

The close coupling wave packet (CCWP) method has recently been extended to treat gas phase atom-diatom collisions. The total angular momentum representation reduces the complexity of the coupled wave packet equations. In this paper, the theory is extended by modifying the form of the initial packet so that, even though the (JjIM) representation is used, a single wave packet propagation provides complete information for scattering out of a particular j, mj initial rotor state with total angular momentum J. We present results of further testing the method using the Lester-Bernstein model atom-rigid rotor system for various numbers of coupled channels N, including N = 25, 64, 144, 256, 969. The results for 969 channels show clearly the transition from the "I-dominant" regime at lower energies, where the scattering is dominated by the long-range attraction, to a more sudden regime at higher energies, where the scattering is dominated by the short-range repulsive interaction. The dependence on the final orbital angular momentum at higher energy is interpreted in terms of orbital angular momentum rainbow scattering. The results are very encouraging indicating that the wave packet method can treat gas phase collisions involving very large numbers of quantum states.

I. INTRODUCTION

Recently, we have developed a time-dependent (wave packet) method to treat three-dimensional gas phase atom­diatom scattering. I In order to take advantage of the iso­tropy of space, we introduce a partial wave expansion in the total angular momentum (JjIM) representation. This en­ables the wave packet propagation for the relative motion to be reduced from three dimensions to one.

In our earlier paper, we carried out an example compu­tation using the well-known Lester-Bernstein model of an atom-rigid rotor collision.2

•3 In order to demonstrate the correctness and accuracy of the method, we only treated a 16 channel example for which accurate time-independent close coupling (CC) results at a single energy were available for comparison.2•

3 In this paper, we carry out several calcula­tions designed to test more stringently the power of the spherical wave close coupling wave packet (CCWP-J) method. First, calculations were done to examine how the CCWP-J scales with the number of quantum states N includ­ed in the basis. In order to do this, the parameter determin­ing the effective rotor energy constant was taken to be very small. For the largest calculation done, a large value (30) of the total angular momentum J was taken so that the number of coupled (j,/) states would be large. Finally, values for the maximum rotor j, ranging up to 76, were taken. The calcula­tions have been done for 25,64, 144,256, and 969 channels. Results as a function of energy are given explicitly for the calculation done with 969 channels.

In addition to our computational studies we have shown that, even though the collision is treated in the Arthurs­Dalgarn04 coupled angUlar momentum (J,j,I,M) represen­tation, one does not have to carry out separate wave packet propagations for initial-l satisfying 'J - jo' .;;;.10 .;;;1 + jo in or­der to obtain transition amplitudes out of a specific initial state J,jo,mo, where mo is the initial Z component of rotor

angular momentum. This possibility was not realized in our earlier study. J

In analyzing the results for the 969 channel calculation, we find that the variation of the transition probabilities with final I suggests that a rainbow mechanism may be involved. In order to consider this we present an lOS approximation analysis of such orbital angular momentum rainbows. The approach is very similar to that used by Schinke in a study of rotational rainbows in diatom-surface collisions. 5

This paper is organized as follows. In Sec. II we give a brief description of the CCWP-J theory developed earlier and present the new method for preparation of a packet to simplify the calculation of cross sections for scattering out of a well-defined initial rotational state jo,mo' In Sec. III, we give the results of the computational tests of the method, emphasizing the variation of the CCWP-J method with the number of coupled rotational channels. Explicit transition probabilities as a function of energy are given and discussed for the largest calculation, which included 969 coupled channels. In this section we also present an lOS approximate analysis of the S matrix for transitions in the orbital angUlar momentum to show that rainbow scattering involving this quantum number may be involved. Finally, in Sec. IV, we summarize our findings and indicate future directions for this work.

II. THEORY

The atom-diatom system's quantum state ''''> is propa­gated in time according to (in atomic units) the usual equa­tion6

,,,,(t» = exp( - iHt) ,,,,(0». (1)

We assume the spectra of H are between Emin and Emax ' and define the operator Cd by 7

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Y. Sun and D. J. Kouri: Atom-rigid rotor scattering 2969

OJ = (H _ Emax ; Emin )/(Emax ~ Emin ),

such that the eigenvalues of the operator OJ satisfy

OJ"E[ - 1,1]

for all A. We therefore have

exp( - iHt) = exp[ - it(Emax + Emin )12]

. exp [ - iOJt(Emax - Emin )/2] .

(2)

(3)

(4)

The first part on the right-hand side above is just a constant and, following Tal-Ezer and Kosloff,7 the second (operator) part can be expanded in Chebychev polynomials:

00

exp[ -iOJt(Emax -Emin)/2] = L ak'Tk(OJ), (5) k=O

with

ak = (2-c5ok)ikJdt(Emax -Emin )12], (6)

where Jk is a cylinder Bessel function. The Chebychev polynomials satisfy the recurrence rela­

tion

Tk (OJ) = 2OJTk_ l - Tk_2

and also

To(OJ) = 1, Tl(OJ) = OJ.

If we define Itf') = Tk(OJ)I¢'(O», then

I¢,(t» = const· i ak I¢'k), k=O

where Itf') is obtained by the recurrence relation

I¢'k) =2OJItf'-I) -1tf'-2) and

(7)

(8)

(9)

(10)

(11)

The summation over k is truncated when higher order con­tributions are negligible.

The Hamiltonian for an atom-rigid rotor system is giv­en by

H= _p~ +_/2_+1 + V 2f.l 2f.lR 2 21

=Ho+ V, (12)

where 12 is the orbital angular momentum squared and 1 is the rigid rotor angular momentum squared.

We define the total angular momentum J as

J=l+j. (13)

It is clear that J 2, Jz ,I, and 12 commute with one another, so we can construct their simultaneous eigenstates IJM jl ). Since the free Hamiltonian Ho also commutes with these operators, there exist states IJM jlE ) as simultaneous eigen­states of Ho; i.e.,

HoIJMjIE) =EIJMjIE). (14)

When evaluating the Chebychev expansions, OJ Itf') is the main part of the calculation. OJ and Itf') are expressed in the representation «J2, Jz , I, 12, R », where R is the distance between the atom and the center of mass of the rotor. In such a representation, the only differential operator in OJ is that

arising from the radial kinetic energy d 2/ dR 2 whose action is obtained by applying a Fourier sine transformation via FFT, as discussed in our earlier work 1 (see also the seminal work of Kosloff and Kosloff 8). The free Hamiltonian eigenstates can be simply expressed as

(JIM'j'I'R 'IJMjIE)

= c5JJ ·c5MM ·c5j/c5/l' j[(kR '), (15)

where j[ is a spherical Bessel function of order I, and k is given by

k 2 = 2f.l( E _ j(j ~ 1) ). (16)

Because J2 commutes with the full Hamiltonian H, J and M are good quantum numbers. When H is expressed in a repre­sentation containing J2 and Jz , it is diagonal in these two quantum numbers, thus reducing the size of the matrices appearing in the problem. In addition, only states with the same parity of j + 1 will be coupled by the Hamiltonian. In order to obtain the state to state S matrix, 1 the initial wave packet is chosen to be IJM jo/o) Ig), where g is a spherical Gaussian wave packet along with a well-defined total angu­lar momentum state IJM jo/o). It can be expressed as

(MjIR I¢'(O»

= (JMj/IJMjo/o) (R Ig)

=c5jj"c5 11" ioo

k2dkA(k)j~,(kR)

= c5jj"c511" • (21Tc52 ) l/4exp( _ (R ~~o)2 - ikoR ) /R.

(17)

After the wave packet is propagated out of the interaction region, we may project out the free Hamiltonian eigenstates IJM jlE) and calculate the S matrix for the jo/o -+ jl transi­tion, given byl

SJ(jI,jo/o) =.!!.... fI 1~(kjlljo/o) 'exp(iEot)/A(ko), 2 '/ ko

(18)

where

1~(kjlljo/o) = (JMjIE I¢,(t»

and

= ioo

R 2 dRA(kR)'(JMjIR I¢,(t»

(19)

(20)

We have developed a stable and accurate Hankel transfor­mation method earlier to carry out the integration in Eq. ( 19).

The physical scattering amplitUde f(jmljomolk) for scattering in the final direction k (the incident direction be­ing along the positive z axis) is given by4

f(jmUomolk)

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2960 Y. Sun and D. J. Kouri: Atom-rigid rotor scattering

where

TJ(j/ljo/o) =Ojj"OIl" -SJ(j/ljo/o). (22)

The differential cross section is given as usual by

!!!:r = ~ 1 /(jmjjomolk) 12. (23) dk ko

The triple summation in Eq. (21) can be reduced to two sums if we simply prepare the initial wave packet properly. We can rewrite Eq. (21) as

/(jmljomolk)

=~ ~ J~+j i-I+1y (k) ~ [m-mo

kko J=O 1= J-jl

X (Imo - mjmIJmo)TJ(j1 Uomo) , (24)

where the "mixed representation" Tmatrix is defined bylo

TJ(j/ljomo)

J+jo

= '). CI"TJ(j/ljo/o)' ~,= !J-)"I

(25)

Then, according to Eq. (22), TJ(j/ljomo) is determined according to

TJ(j/ljomo)

= CIOjj" - L C~,SJ(j/ljo/o) I"

= Co .. -!!... ~ exp(iEot) ~ C /J(k '/1 . l ). I))" 2 k A(k) 4..~, 1100

o 0 ~, (26)

We know

/J(kjl Uo/o) = (JMjIE I",(t»

= (JMjIE lexp( - iHt) ",(0) ),

where

1",(0» = IJMjo/o)lg)·

If we choose a new initial wave packet as

1'11(0» = L C~,"'(O» I"

we can also propagate it and project out the free Hamilto­nian eigenstates as above. The integral in Eq. (19) with the new wave packet will directly lead to l:~, C/o / J (k jll jio) in Eq. (26). By using this approach, we can evaluate the differ­ential cross section more efficiently, since only a single wave packet must be propagated rather than 2 min (J, jo) + 1 sep­arate packets (corresponding to the number of initial 10 val­ues possible for given J,jo)'

III. COMPUTATIONAL TESTS

The above described wave packet method in the total angular representation (termed previously the eewP-J) has previously been tested for a model first studied by Lester and Bemstein2 and later by Johnson et al.3 They used differ­ent approaches, both, however, based on the time-indepen­dent close coupling method. For this system, the potential is given as

V(R,R'r) = B[ (1/R 12) - 20/R 6) J[ 1 + aP2(R'r)],

where a is a measurement ofthe asymmetry,

B= 2PER~,

P is the reduced mass for the relation motion, Rm is the radial distance at the minimum of the potential, and E is the depth of the potential well. The energy is measured in units of 1/2pR ~, and the parameter A is defined by

E=EA.

We report results of our calculations using parameter values:

B= 1000,

a =0.25.

Thus, changing A varies the energy of the system. The first aspect of interest in the eeWp-J approach is the verification of the scaling behavior with number of coupled channels. In Table I, we present results of computation times on the eRA Y2 supercomputer at NASA-Ames. Shown are the ac­tual computation times and theoretical times determined as t;_1 (N;lN;_1 )2, wheret;_1 is the actual computation time for the previous case, and N; the number of coupled channels for case i. It is immediately apparent that the scaling is actu­ally slower than N 2

, although it will ultimately approach this limit as the number of channels becomes larger. These re­sults are encouraging as to the ability of wave packet meth­ods to treat large numbers of channels in gas phase atom­diatom scattering.

In Table II and Figs. 1-4 we present more detailed re­sults for the 969 channel case. The physical parameter pR ~//, which measures the relative size of the diatom mo­ment of inertia, we take to be

pR ~// = 0.0001,

which corresponds to a heavy diatom. The total angular mo-

TABLE I. Time scaling with number of coupled channels.

Execution time N-scaled time Channel No. (cpu s) (cpu/s)"

25 85 64 212 557

144 568 1073 256 1700 1795 969 20400 24357

a N-scaled time is the predicted execution time given by t,_ I (N,IN,_ 1)2,

where (Nil N, _ I ) is the ratio of current channel number to previous one and t, _ I is the previous execution time. All times are for the CRA Y2 supercomputer at NASA-AMES.

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Y. Sun and D. J. Kouri: Atom-rigid rotor scattering 2961

TABLE II. Transition probabilities at a total energy" of 1000.

j IS(jI .... jolo) 12 j IS( jl .... jolo) 12

0 30 0.177(0) 8 30 0.324( - 3) 2 28 0.193(0) 8 32 0.366( - 3) 2 30 0.177(0) 8 34 0.360( - 3) 2 32 0.204(0) 8 36 0.318( - 3) 4 26 0.432( - I) 8 38 0.332( - 3) 4 28 0.384( - I) 10 20 O.S12( - 5) 4 30 0.440( -1) 10 22 0.47l( - 5) 4 32 O.4Sl( - 1) 10 24 0.S80( - 5) 4 34 0.SI9( - 1) 10 26 0.778( - 5) 6 24 0.388( - 2) 10 28 O.IOS( -4) 6 26 0.338( - 2) 10 20 0.136( - 4) 6 28 0.413( - 2) 10 32 0.16l( -4) 6 30 O.SOO( - 2) 10 34 0.170( -4) 6 32 0.S3S( - 2) 10 36 0.157(-4) 6 34 O.S04( - 2) 10 38 0.130( - 4) 6 36 0.SS6( - 2) 10 40 0.12S( - 4) 8 22 0.188( - 3) 12 18 0.119( - 6) 8 24 0.160( - 3) 12 30 0.9SI( -7) 8 26 0.198( - 3) 12 22 O.l1S( - 6) 8 28 0.2S9( - 3) 12 24 O.ISS( - 6)

• This corresponds to the average energy of the wave packet. Total J is 30, jo=O.

mentum J was set equal to 30, and the initial jo was set equal to zero. Consequently, only even parity states arise, since initially 10 == 30, and therefore jo + 10 is even. The maximum rotor state jmax was taken to be 76. The larger probability results at the medium energy are shown in Table II, and the complete energy dependence is shown in Figs. 1-4. It is in­teresting to note the behavior of the transition probabilities function of final I for given final j. For j> 0, the probabilities decrease as I decreases down to IJ - jl, within some cases, an oscillation. The value for the lowest value of I( = IJ - jl) invariably goes up. This value of I corresponds to the situa-

0.22

/,"":(0,30>-(2,32) N

0.20 -.. -t 0 0.18 (II) . 0 -i 0.18 0

___ (0,30)-(0,30) 0.14

tion where the orbital angular momentum I and rotor angu­lar momentum are as aligned as possible, and it results in the lowest centrifugal potential. It is readily seen from the re­sults that at low energy, the dominant transition becomes this lowest I value (I = IJ - jl ). This corresponds to the 1-dominant coupling regime as discussed by DePristo and Alexander, II and it reflects the dominance of the long-range attractive part of the potential at lower energies.

As the energy increases, the dominant transition is even­tually the one for the largest value of I ( = J + j), which cor­responds to motion such that the I andj vectors are opposite­ly oriented. One expects that as the collision energy increases, the short-range repulsive part of the potential will become more important, and this is reflected in the decreas­ing importance of the I-dominant transition I = IJ - jl. However, it is intriguing that at the higher energies, it is the maximum 1= J + jwhich is the dominant transition, for all final j. In addition, we note that for final I> 6, the probabili­ties oscillate as a function of final I. Finally, we again note the upturn of the probability for 1= IJ - jl, all j, at the higher energies. This type dependence on the orbital angular mo­mentum is reminiscent of the behavior of rotational rain­bows,6 which are well known to result in atom-diatom scat­tering due to the short-range repulsive part of the potential. It is also very similar to the magnetic rainbows predicted 12

and observed 13 in molecule-corrugated surface collisions. Indeed, the possible occurrence of orbital rainbows is

plausible since the angle r can be thought of equally well as relating to the molecular orientation with respect to a z axis along the scattering vector R, or the orientation of the scat­tering vector R with respect to a z axis along the molecular axis. As is the case with rotational rainbows,S a convenient approximate framework for analysis is the lOS approxima­tion. 14 The relevant amplitUdes SJ(j/ljolo) in the lOS can be expressed as

FIG. 1. Transition probabilities for J = 30, for initial (j,/) equal to (0,30), to final states (0,30) and (2,/), asafunctionofener-gy in units of If /2p,R !.. .

0.12~--~------~~------~--------L-------~--------~ 900. 950. 1000. 1050. 1100. 1150.

SCATTERING ENERGY

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2962 Y. Sun and D. J. Kouri: Atom-rigid rotor scattering

0.050

0.045

0.040

0.035

0.030

1000. 1050.

SCATTERING ENERGY

SJ(j1 Uo/o) = p7-1-~, J iR dr yftM*(Rr)

X e2i-'1/(rl yt.t: (Rr), (28)

where the total angular momentum state yftM(Rr) is4

A A

yftM(Rr) = L (1M -).j)·IJM)YIM_A (R) lj.tCr), A

(29)

the angle r is defined by

cosr=r·R. (30)

"h (r) is the phase shift for scattering of the atom by the diatom when the relative angle r is fixed, and I is the CS parameter. 14,15 For simplicity, we shall employ the so-called

1100. 1150.

"final-I labeling," so that

1= I.

FlO. 2. Transition probabilities for J = 30, forinitial (j,/) equal to (0,30), to final states (4,/), as a function of energy in units of"z /2p,R !..

(31) A

Now in evaluating the integrals over R,t. in Eq. (28), it is possible either to orient the z axis along R or r. The former corresponds to what is used in the analysis ofEq. (29) when rotational rainbows are of interest. 5,16 However, the latter is equally valid, and the result is

y-!r(Rr) = (2j + 1)112 J 41T

XL (/j.tjOIJj.t)D~M(SXt/J)YlPo(r,O), (32) Po

where now r is the polar angle of the scattering vector in the new coordinate system. The S-matrix element then becomes'

0.007~--~------~-------r------~--------r---~~'-'

N

Q

G O.OOS

b (I)

o r 0.004 en

0.003

(0,30)-(8,24)

(0,30)-(8,28)/

0.002L---L-------~-------L------~--------~------~ 800. 8S0. 1000. 1050. 1100. 1150.

SCATTERING ENERGY

J. Chern. Phys., Vol. 89, No.5, 1 September 1988

FlO. 3. Transition probabilities for J = 30, for initial (j,/) equal to (0,30), to final states (6,/), as a function of energy in units of"z /2p,R !. .

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Y. Sun and D. J. Kouri: Atom-rigid rotor scattering 2963

0.0005

N 0.0004

1 (0,30)-(1,21)

o CW)

S t,

0.0001

(0,30>-(8,38) (0,30>-(1,31)

FIG. 4. Transition probabilities for J = 30, for initial (j,/) equal to (0,30), to final states (8,/), as a function of ener­gy in units of -rr /2p,R !..

800. 850. 1000. 1050. 1100. 1150.

SCATTERING ENERGY

SJ(j/ljo/o) = i-lo

(211" ds (211" dtfJ (11" dxsinX (11" dr sin r[(2j+ 1)(2jo+ 1)]112 L (If.LjOIJf.L)(lof.LojoOIJf.Lo) 41T Jo Jo Jo Jo 1',1-'0

XD"{:M(SXtfJ)D~(SXtfJ) Yt (r,O)exp [2i1]1 (r) ] Ylol-'o (r,O). (33)

The integrals over the D J matrices are easily done yielding

SJ(j/ljo/o)

[(2j+ 1)(2jo+ 1)]1/2 ,1-10 = I (2J + 1)

XL (/f.L jOIJf.L) (I Of.L joO I Jf.L ) I'

X i1l" dr sin rP/1' (cos r)exp[2i1]/(r)]PloIJ (cos r),

(34)

where PII' are normalized associated Legendre polynomials. Now we recall that within the lOS, SJ(j1 Uo/o) satisfy a scaling relation, 14.17 so that it suffices to consider 10 equal to zero (which also implies that f.L == 0). Then we have

. ·1

SJ(j1 IJO) = [(2j + 1)/(2J + 1)] 1/2 _1_ (lOjOIJO) Ii

xi1l" drsinrP/(cosr)exp[2i1]l(r)].

(35)

We apply a semiclassical approximation to PI (cos r) so that

1 PI(cosr) = (ei[(/+1I2)Y+1I"/4J

i~21T sin r _ e - i((/+ 112)y+ 1I"/4J). (36)

Then the result is

'1-1 SJ(j1 IJO) = [(2j + 1)/(2J + 1)] 1/2 _I - (lOjOIJO)

2,fi

X (11" dr sin 1/2 r[ i[2'1I(Y) + (/ + 112)y+ 11"/4] Jo

-it -2'1I(Y) + (/+ 112)Y+1I"/4]] -e . (37)

We evaluate this, as usual, by stationary phase to obtain ./- 112

SJ(j/IJO) = [(2j + 1 )/(2J + 1) P/2 _I -

Ii

[ a21](r)]-1I2

X (lOjOIJO) L ~sin rv 2 f v a Yv

- i[ - 2'1/(Yv) + (/ + 112)yv+ 11"/4]] -e , (38) where r v are determined by the stationary phase conditions

2 a1]/(r) = + (I +.!.) (39) ar - 2

for integer I. The rainbow results whenever

a21]/(r) I = 0 (40) ay2 Y=Yv

is also satisfied for r equal to rv' The quantity 2(~rlar) is the usual classical excitation function L(r) and the rain­bows occur at any extrema in this function. It is interesting to note that the analysis of a rotational rainbow in the j

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2964 Y. Sun and D. J. Kouri: Atom-rigid rotor scattering

dependence of the SJ(j/ljo/o) is entirely analogous to th~ above, except that the Z axis would have been taken along R rather than r, and the conditions ofEq. (39) would have had I replaced by j.s,16 ThefactthatEqs. (39) and (40) are well known to be satisfied in the case of rotational rainbows sug­gests strongly that orbital rainbows also occur . We can carry through a simple approximate evaluation of the phase shift and concommittant rainbow condition by assuming that the scattering is described by a rigid body whose shape is the locus of classical turning points Ro ( r) at ellergy E (and radi­al kinetic energy filkJI2m). Then the phase shift would be approximately

11/(r) = - kjRo(y)· (41)

We note that the dependence of 11/ (r) on I arises from the centrifugal potential, and on j through the fact that, in gen­eral, the ES parameter] is a function of j. Now by Eqs. (40) and (41), the rainbow condition is then

d:~o = O. (42)

We conclude then that the orbital rainbow reflects an inflec­tion point in the locus of turning points as a function of relative orientation of the projectile and target diatom. We again note the fact that this analysis is parallel to that of Schinke and of Korsch and Schinke for rotational rain­bows. 16

IV. SUMMARY

In this paper, we have considered the CCWP-J method recently introduced for atom-diatom gas phase scattering. We have shown that a very convenient initial packet can be constructed which reduces the number of distinct packets that must be propagated in order to obtain scattering out of a particular initial jo, mo state, while still considering a well­defined total angular momentum J. The N 2

, or slower, scal­ing of the labor involved in such calculations was verified by considering a series of increasingly more difficult problems, with the largest consisting of 969 coupled channels. The nu­merical results showed interesting variations in the transi­tion probabilities as a function both of energy and of final orbital angular momentum. The low energy results reflect the I-dominant behavior, II while at higher energies, one sees scattering which is dominated by the short-range repulsive anisotropy. The detailed variation of the results with I at higher energy was rationalized by suggesting that orbital an­gular momentum rainbows were being observed. An lOS approximation analysis, similar to what is used for rota­tional rainbows,s.16 was employed. Combined with a rigid molecule collision model, the analysis indicated that the or­bital rainbow hypothesis is a reasonable explanation of the observed behavior.

Further improvements of the wave packet method for gas phase atom-diatom scattering are under study, includ­ing modifications that should lead to a method which scales

as 18 N IN rather than N 2• In the case of the 969 channel

calculation, such a method would result in a reduction in the computational labor by a factor close to 30. It may also be possible to develop a procedure which would scale as N In N

(by carrying out FFfs in all variables), in which case an additional factor of 4 reduction in labor would result. The prognosis for such procedures is bright.

ACKNOWLEDGMENTS

This research was supported in part by National Science Foundation Grant No. CHES6-00363 and by R. A. Welch Foundation Grant No. E-60S. The calculations were done on the NASA-Ames Research Center CRA Y2 (Navier) UD­

der the support of the National Science Foundation. The authors acknowledge helpful discussions of the results re­ported herein with Professor M. H. Alexander and Professor H. Mayne.

Iy. Sun, R. C. Mowrey, and D. J. Kouri, J. Chem. Phys. 87, 339 (1987). 2W. A. Lester, Jr. and R. B. Bernstein, J. Chem. Phys. 48, 4896 (1968). 3B. R. Johnson, D. Secrest, W. A. Lester, Jr., and R. B. Bernstein, Chem. Phys. Lett. I, 396 (1967).

4A. M. Arthurs and A. Dalgarno, Proc. R. Soc. London Ser. A 256, 540 (1960).

sR. Schinke, J. Chem. Phys. 76, 2352 (1982); see also R. Schinke and J. M. Bowman, in Molecular Collision Dynamics, edited by J. M. Bowman (Springer, Berlin, 1982), and references therein.

c.see any standard text on quantum mechanical collision theory; for exam­ple, M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964); R. D. Levine, Quantum Mechanics of Molecular Rate Pr0-cesses (Oxford, London, 1969).

7H. Tal-Ezer and R. Kosloff, J. Chem. Phys. 81, 3967 (1984). 8D. KosloffandR. Kosloff,J. Comput. Phys. 52, 35 (1983); R. Kosloffand D. Kosloff, J. Chem. Phys. 79, 1823 (1983).

9See also R. Bisseling and R. Kosloff, J. Comput. Phys. 59, 136 (1985). lOA similar, though not identical, quantity has been introduced earlier in

studies of the coupled states (CS) approximation; see Y. Shimoni and D. J. Kouri, J. Chern. Phys. 66, 2841 (1977).

"A. E. DePristo and M. H. Alexander, J. Chem. Phys. 64, 3009 (1976); Chem. Phys. Lett. 44, 214 (1976); J. Phys. B 9,2713 (1976).

12D. J. Kouri and R. B. Gerber, Isr. J. Chem. 22, 321 (1982); T. R. Proctor, D. J. Kouri, and R. B. Gerber, J. Chem. Phys. 80, 3845 (1984).

1:Iof. R. Proctor and D. J. Kouri, Chem. Phys. Lett. 106, 175 (1984); in Dynamics on Surfaces, edited by B. Pullman, J. Jortner, A. Nitzan, and R. B. Gerber (Reidel, Dordrecht, 1984), pp. 89-102; H. R. Mayne, C.-Y. Kuan, and R. J. Wolf, Chem. Phys. Lett. (in press).

l"There are many references. As a few examples, see T. P. Tsien, G. A. Parker, and R. T Pack, J. Chern. Phys. 59, 5373 (1973); R. T Pack, ibid. 60, 633 (1974); D. Secrest, ibid. 62, 710 (1975); R. Goldflam, D. J. Kouri, and S. Green, ibid. 67, 5661 (1977). A few papers containing early contributions to this approach include S. I. Drozdov, JETP I, 591 (1956); D. M. Chase, Phys. Rev. 104, 838 (1956); K.AlderandA. Winther, Mat. Fys. Medd. Dan. Vid. Selsk. 32, 1 (1960); K. H. Kramer and R. B. Bern­stein, J. Chern. Phys. 40, 200 ( 1964); C. F. Curtiss, ibid. 49, 1952 ( 1968). For a detailed discussion, see the review by D. J. Kouri, in Atom-Molecule Collision Theory: A Guide for the Experimentalist, edited by R. B. Bern­stein (Plenum, New York, 1979).

"See also Y. Shimoni and D. J. Kouri, J. Chern. Phys. 66, 2841 (1977); G. A. Parker and R. T Pack, ibid. 66, 2850 (1977); V. Khare, D. J. Kouri, and D. K. Hoffman, ibid. 74, 2275, 2656 (1981); 76, 4493 (1982).

l<;gee also R. Schinke, Chem. Phys. 34, 65 (1978); J. M. Bowman, Chem. Phys. Lett. 62, 309 (1979); H. J. Korsch and R. Schinke, J. Chern. Phys. 73,1222 (1980); 75, 3850 (1981).

17R. Goldftam, S. Green, and D. J. Kouri, J. Chem. Phys. 67, 4149,5661 (1977); V. Khare, ibid. 68, 4631 (1979); S.1. Chuand A. Dalgarno, Proc. R. Soc. London Ser. A 342, 191 (1975); A~ E. DePristo, S. D. Augustin, R. Ramaswamy, and H. Rabitz, J. Chem. Phys. 71, 850 (1979); D. K. Hoffman, C. Chan, and D. J. Kouri, Chem. Phys. 42, 1 (1979); C. Chan, J. W. Evans, and D. K. Hoffman, J. Chern. Phys. 75, 722 (1981); R. B. Gerber, L. H. Beard, and D. J. Kouri, ibid. 74, 4709 (1981); L. H. Beard, D. J. Kouri, and D. K. Hoffman, ibid. 76, 3623 (1982); L. Eno, B. Chang, and H. Rabitz, ibid. 80,1201,2598 (1984); L. Eno, ibid. 82,1063 (1985); 84,4401 (1986).

'"Yo Sun, R. C. Mowrey, and D. J. Kouri (unpublished).

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