Volume 23, number 4 CHE,CIICAL PHYSICS LETTERS 1.5 December 1973

ELASTIC AND INELASTICANGULAR DISTRIBUTIONS IN THE

i,-CONSERVING COUPLED STATES APPROXIMATION FOR MOLECULkR COLLISIONS

Paul McGUlRE

bras-Planck-histitlrt fti Str~nlLrngsforsclr~rt,g. 34 Gtirtir~gen. Gcrrrlorr~

Received 29 August 1973

An approximation scheme is described which allows the decoupling ot’ molecular rotational j and I angular mo- menta in molecular collisions. \Vith a particular choice or the interaction potential, the potential matri\: couples only the moleculnr states of the system and in particular UIOS in which the ;.asis projection ofi ih conserved. Test calcu- lations on He + H2 for the elastic j = 0 - 0 and rotation;llly inelastic j = 0 4 2 diffcrcntinl cross sections arc prcsentcd in the energy range 0.1 to 0.9 eV. These rcwlts are compnrcd with the full coupled-channel cross .sections and arc found to be cstremcly accurate.

In the study of rotational excitation in molecular collisions an extremely large number of scattering channels arises due to the coupling of the molecular rotationj and the orbital angular momentum I of the system. In the to- tal angular momentum representation where J=i + I, the rotational state j couples all of the orbital angular n!o- mentum states I given by IJ- jl G I GJ + j. This coupling scheme is then repeated for each vibrational level of the

system, thus considerably magnifying the size of the problem. Several applications have been made [I] with this

full coupled-channel formalism on simple systems and for the lower rotational and vibrational transitions. Rabitz [2] has recently introduced the idea of an effective potential which essentially is an average of the interaction PO-

tential over the rotational angular momentum projections nzi. Since this effective potential is angle independent,

the decoupling ofj and I is achieved. We have recently suggested [3] an alternate decoupling scheme and demon- strated its exceptional accuracy in predicting elastic and rotationally inelastic integral cross sections.

The purpose of this letter is to show that this alternative method also provides an accurate description of the

dynamics through calculation of the angular distributions. We consider the atom-diatom collision problem for which the potential has the general form

V(R,r)= ~(~)v~(R,~)Y,~(;)y;~(~), (1)

where r is the diatom internuclear distance, R is the center of mass coor’dinate of the system, and YAP is the usual

spherical hartionic. When we consider a particular interaction in which v depends only on the angles associated ” with r^ and on the coordinates r and R, eq. (1) reduces to

The matrix element connecting the diatom vibration-rotation states rr,i, mi is

‘Mmj’ “,zi’m;)=f,(~)1’*~d1~~(1)~~(R,1)l*r(T)Jdi qki(e,@) YAo(B)Yi,,,,;(O,@), -‘,’ (3) . . : ., :

. ‘. : .; ; &.. : ., .,’ .: ” ;.. ., .:

.. : .; I._.. : . . ,_ _. ...,, ‘. :.. ‘, ,. T .:_ :, .: ,: ,,;: ;. ,._ ,i ,_,, ..

.., :: ;, _’ :: :. .,., :.: :_. :’ _,_, ., :- ::. .. :_;- ,’ :., ..,. :’ : : ,,’ : 1; .,,;.. . -. _ _,: : ‘. I ,._, _. .-.:.::+_.,:

,. : . :. : ._ .: ., ,,. .,I:., ..: .: ;- I, -- . ..*., .: ‘.< ;. ., ;:>I; :

Volume 23 ~ num her 4 CHEMICAL PHYSICS LETTERS 15 December 1973

Table 1 Table 2 The elastic (i= 0 + 0) differential cross section o) for He + Ha at 0.1 cv

The elastic tj 7 0 --* 0) differential cross section a) for He + Ha at 0.9 eV

scattering angle 0 (degrees) Coupledchanncl Coupled-state

SMtering angle 0 (degrees) Coupledshannel Coupled-state

0 0.146(4) 15 0.183(2) 30 0.580(1) 45 0.314(l) 60 O-207( 1) 7.5 0.164(l) 90 0.132(l)

105 0.115(l) i20 0.102(l) 13.5 0.953 150 0.891 165 0.860 180 0.822

0.147(4) 0.183(2) 0.581(l) 0.315(l) 0.208(l) 0.164(l) 0.132(l) 0.115(!) 0.102(l) 0.951 0.889 0.858 0.820

0 0.845 (4) 0.845 (4) 15 0.134(2) 0.134(2) 30 0.361 (I) 0.358(l) 45 0.169(l) 0.167(l) 60 0.960 0.935 75 0.611 0.591 90 0.414 0.398

10.5 0.298 0.286 120 0.227 0.219 135 0.183 0.177 150 0.157 0.152 165 0.145 0.140 180 0.150 0.146

o) Number in parentheses is the power of 10 by which the en- try should be multiplied.

a) Number in parentheses is the power of 10 by which the en- try should be multiplied.

where II, and q,,!,. arc the vibrational and rotational eigenfunctions, respectively. Denoting the first integral by v,$$ (R) and wrrtrhg the second in terms of 3j symbols [4], eq. (3) becomes

hjr?$l V Ir~j”nr,?~ = (-1)“‘i c I,,‘:! (R) [(2j +I)(Z?i’+l)] h

This matrix element depends upon the rotational projection mi and is non-zero only for mi = m,!. Because of the

choice of the potential interaction, eq. (2), the coupling is independent of the angles d and thusj and I will decou-

ple. Physica!ly, then, we are looking at the situation in which the colliding atom is restrained to lie a!ong the quart-

tization axis of the diatom throughout the collision. For rotational transitions, the projectionj, along this axis is

preserved, although the magnitude and orientation ofj may change. This same coupling matrix element has re- cently and independently been employed in a semiclassical approach to molecular collisions [5]. It should be em-

phasized that we maintain a full quantum mechanical description. Test calculations with this approximation for the integral elastic (j = 0 + 0) and rotational inelastic (j = 0 + 2)

cross sections in He + H, collisions have already been reported [3] and are found to be very accurate. The follow- ing calculations demonstrate that the differential cross sections for these transitions are also accurately repro- duced. The Krauss-Mies [6] surface has been employed, II = rr’ =O in all cases and we have considered the energy range 0.1 to 0.9 eV. A three-state molecular basisj = 0,2,4 was used at all energies except at 0.1 eV where only j= 0,2 were included. This three-state basis requires the solution of 9 coupled-channel equations in the usual total angular momen turn representation, but only 3 in the present coupled-state approach.

Tables 1 and 2 give the elastic differential cross sections at 15 degree intervals for 0.1 and O-9 eV. The coupled-

state results at 0.1 eV give the best agreement to the coupled-channel result in this energy range while the 0.9 eV results represent the largest deviation. Clearly,all the deviations are small enough to be considkd entirely negligi- b/e. The differences that do occur come where the elastic differential cross section is comparatively QTEIU (i-e., the backward direction). That the agreement is better at the lowest energy is explainable by the fact that it is here that the inelasticity is the smallest.

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Volume 23, number 4 CHEMICAL PHYSICS LEl-l-ERS 15 December 1973

Fig. 1. The rotationally inelastic (j = 0 * 2) differential cross sction as n function of scattering angle 0 at the collision ener- gies E = 0.1 nnd 0.3 cV. The solid lines are the coupled-state results and the dashed lines are the exact coupled-channel re-

sults. Calculations at 0.3 eV ax with the molecular state basis j = 0,2,4, while at 0.1 eV only j = 0. 2 are included.

1 I I I I

30 60 so 120 150 I80 .e

Fig. 2. The rotationally inelastic (i q 0 + 2) differential cross section as a function of scatteri% angle 0 at the collision ener- gy E = 0.9 eV. The solid line is the coupled-state result and the dashed line is the exact coupledchannel result. The calcu- lations were made with the molecular slate basisj = 0,2,4.

Figs. 1 and 2 give the comparisons between the coupled-state (solid line) and the coupled-channel (dashed line) differential cross sections for the j = 0 + 2 transition at 0.1, 0.3 and 0.9 eV. Deviations occur only in the forward direction where the coupled-state results are below the coupled-channel ones at 0.1 eV, nearly equivalent at 0.3 eV and slightly above at 0.9 eV. The number and positions of the oscillations inside 15 degrees for this cross sec- tion are reproduced with the coupled-state approach but the oscillations have a greater magnitude. The undula- tions in the coupled-channel curve in this region at 0.9 eV have been left out only to present a cleaner figure. This cross section goes to a value of 0.32 A/sr at B = 0. As in the elastic case, the largest deviation from the exact cou- pled-channel result occurs where the inelastic differential cross section hz its smallest values. This is important from the experimental point of view since it is the high intensity (high cross section) region of the angular distri- bution which will be measured first.

Comparisons of differential cross sections are the most sensitive tests possible. Overall, the results shown in ta- bles 1 and 2 and in figs. 1 and 2 suggest that the I;-conserving coupled-state approach is an excellent one. ‘I’be sav- ings in computation time is particularly remarkable, For rotational states in general, a computation requiringN coupled channels will require only N1i2 coupled states. Without the coupling of orbital angular momentum, the computation of the cross sections is also greatly simplified. As an illustration, the 3-state calculation at 0.,3 eV re- quires 60 partial -waves for the convergence of all cross sections and took 36 seconds on the Univac 1 IQ8 comput- eriTbis time includes the computation of all the partial, integral, and differential cross sections.

‘. “.

: 577 : .,.:. .: .‘. ; : _,. :’ ., ‘. . . ‘I. ,.,. ,. ., . .

,‘. ., ., .I,.’ . . :,: ._., .._ ,, ‘,-I .., . . . ;-

_:,...i . . . . _.__‘_ ::: ._-_._.. ..c -’ :.,: . . ,. .:.. . . _ ..’ :. ,: ‘.._’ . . . :. :’ .. ... ,.., ‘,. _, _.,. ‘-

.:, ..,. ‘. -. . j, ..‘_,-.. :,,. : . __. :. ,_. ,.,‘,A,. ,,‘I- I, :’ ::,. .,- -. .::y:,.__

Volume 23. numkr 4 CHEhllCAL PHYSICS LETTERS IS December 1973

We are currently considering higher collision energies, other systems, and the effects of closed channels in this approximation. A complete description of the approach and applications are forthcoming.

The author wishes to thank Professor D-l. Kouri for helpful discussions and to acknowledge the kind !lospital- ily uf Professor J.P. Toennics and the Mau-Plan&-lnstilut fiir Strbmungsforschung.

1 II D. Sccrrsl. Theory of Rolaliorul and Vibraliorul Energy Transfer, lo bc published; T.1:. Gwrp and I. Ross, in: Annul reviews of physical chcmisrry, Vol. 14 (Annual Rcvicws, Palo Alto, 1973).

(1 II. Rabirr.J. Chcm. Phyr 57(197?J 1718; C. Zsrur and Ii. Rnbilz. 10 be publidlcd.

131 P. M~Girc and DJ. Korlri, lo bc publislwl. 141 A.R. Edmonds Angular momcnlum in quanrurn mcchnnics (I’rinceion Univ. Press, Princeron, 1957).

151 W.H. Neilrn and RX. Gordon, 1. Chcm. Phys. 5X I 1973) 4131.4149. 161 35. Sraurr and F.11. Vies. J.Chcm. Phys. 42 ( IYhSl 2703.

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