Quantum reactive scattering in three dimensions: Using ...morrison/Research/Papers/PKM00.pdfQuantum reactive scattering in three dimensions: Using tangent-sphere coordinates to smoothly

JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 3 15 JULY 2000

Quantum reactive scattering in three dimensions: Using tangent-spherecoordinates to smoothly transform from hypersphericalto Jacobi regions

Gregory A. Parker,a) Mark Keil, and Michael A. Morrisonb)

Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019-0225

Stefano CrocchiantiDepartmento Di Chimica, Universita´ di Perugia, Perugia, Italy

~Received 29 November 1999; accepted 14 March 2000!

Hyperspherical coordinates are well suited for treating rearrangement processes in the stronginteraction region, and several different hyperspherical coordinates have been used successfully forquantum reactive scattering by various research groups. However, it is well known thatasymptotically the appropriate set of coordinates~for a three particle system! are the three sets ofJacobi coordinates. In this paper we show how one can smoothly connect the hypersphericalcoordinates in the rearrangement region to Jacobi coordinates in the nonrearrangement region usingtangent-sphere coordinates. This procedure reduces the computational time required to solve thequantum Schro¨dinger equation and eliminates the need for numerical projection. To illustrate thismethod, we apply it to the F1H2HF1F reaction, comparing reaction probabilities to those fromprevious benchmark calculations based on a conventional formulation. ©2000 American Instituteof Physics.@S0021-9606~00!00622-X#

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I. INTRODUCTION

Rearrangement and exchange processes are importanuclear,1,2 atomic,3–5 and molecular scattering.6–9 Accuratetheoretical treatments of such processes are paramouunderstanding many chemical, atomic, and nuclear phyproblems, including three-body recombination rates for ctrolling limitations in Bose–Einstein condensation,3,4,10

collision-induced dissociation and recombination for acrate treatment of chemical kinetics,11 and studying isotopicanomalies of the upper atmosphere. They are also the kecalculating cross sections for muon and positron scatterin12

and for e2e processes.13 Many numerical methods have beedeveloped to study these processes. Prominent amongare techniques which use hyperspherical coordinates, thevantages of which have been long recognized in mcontexts.7

In wave-function propagation methods for solving tscattering equations for rearrangement collisions, significpractical difficulties arise because different coordinate stems are appropriate to different regions of configuratspace. When reactants are well separated, the set of intcoordinates most suited to their behavior are those ofJacobi system. But when these constituents are in close pimity, hyperspherical coordinates are far better suited toscribe their physics. The point at which the gap betweenhyperspherical and Jacobi regions must be bridged is wthere appear such difficulties as nonphysical couplinglarge hyperradii, the need to use different bases in differsectors of the domain of the hyperradius, the need to pro

a!Electronic mail: [email protected]!Electronic mail: [email protected]

9570021-9606/2000/113(3)/957/14/$17.00

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gate through the same region twice, and loss of unitaritytheSmatrix. In this paper we describe how to use a thirdof orthogonal coordinates—tangent sphere coordinatesbridge this gap via a propagation variable that continuouand smoothly connects hyperspherical and Jacobi regand therefore completely eliminates the need for algebrainumerical matching procedures of any kind. In additionobviating numerical problems, use of tangent sphere coonates results in a savings of CPU time.

In Sec. II we outline the quantal scattering theoryhyperspherical and Jacobi coordinates. We also detail theof tangent sphere coordinates to avoid the need to projecscattering function in hyperspherical coordinates ontocounterpart in Jacobi coordinates. In Sec. III we describstep-by-step implementation of this procedure. Then in SIV we apply this method to the F1H2HF1F reaction, as-sessing its accuracy and value by comparing present reto those from previous benchmark calculations based on cventional formulation. In Sec. V we conclude with a summary of essential points.

II. THEORY

In wave-function propagation scattering methodsgoal is to propagate a single-variable~radial! scattering func-tion through configuration space into the asymptotic regwhere one can extract the scattering~S! matrix by matchingto known analytic boundary conditions. For rearrangemcollisions, such methods must overcome the problem thatgrouping of particles in the exit channel may differ from thin the entrance channel.7 If, for example, the collision of an

© 2000 American Institute of Physics

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958 J. Chem. Phys., Vol. 113, No. 3, 15 July 2000 Parker et al.

atom A and a diatom BC leads to an atom and a diatom, tthree possible arrangements of the constituents A, B, anare possible:

A1BC→A1BC, ~1a!

→B1AC, ~1b!

→C1AB. ~1c!

To denote a particular arrangement of particles, we shallindicest for the atom of massmt andt11 andt12 for thediatom of massmt111mt12 . Thus for the entrance and exchannel arrangements in Eq.~1a!, t5A, t115B, andt125C. Although we shall write in terms of atom-diatomcollisions, the formulation presented here applies equallyrearrangement collisions~1! in any three-particle system. Foexample, in electron–hydrogen atom scattering, the protile electron plays the role of the ‘‘atom’’ A, while the ‘‘diatom’’ is the target hydrogen atom consisting of proton B abound electron C.

A. Jacobi coordinates

The set of internal coordinates that best describeasymptotic behavior of the atom and diatom for a givenrangementt are those of the Jacobi system JSt . For colinearmotion, these coordinates reduce to Cartesian coordinWe use mass-scaled Jacobi coordinates. LetXt denote a po-sition vector from the origin of a space-fixed coordinate stem to particlet. After separation of the center-of-mass mtion, the Jacobi coordinates for the relative motion adefined as

Rt5Xt2mt11Xt111mt12Xt12

mt111mt12, ~2a!

r t5Xt122Xt11 . ~2b!

The vectorr t is the internuclear axis of the diatom, whileRt

is a vector from the center-of-mass of the diatom toatomt.

The corresponding mass-scaled Jacobi coordinateSt

andst are scaled versions ofRt and r t , respectively,14–16

St5dtRt and st5r t

dt. ~3!

The dimensionless scaling factor is defined in terms ofthree-body reduced mass,

m5S mAmBmC

M D 1/2

, ~4!

and the total mass of the systemM5mA1mB1mC, as

dt5Fmt

m S 12mt

M D G1/2

. ~5!

These coordinates for one arrangementt are illustrated inFig. 1.

The scale factors in Eq.~3! change the lengths of thposition vectors but not their orientation. Thus the angleQt

between mass-scaled Jacobi vectorsSt andst is

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Qt5cos21S St•st

StstD5cos21S Rt•r t

Rtr tD . ~6!

Physically,St is the translational coordinate, which corrsponds to the distance from the center-of-mass of the diato the atom, andst is the vibrational coordinate, which corresponds to vibration~or dissociation! of the diatom. TheangleQt corresponds to the orientation of the internucleaxis of the diatom with respect to the translational vectorSt .

The three coordinates (St ,st ,Qt), which we shall callthe internal coordinates, uniquely specify the center-of-mposition of the three particles in the plane defined bysystem. To orient this plane in space, three additioangles—the three Euler angles—are required. Their effeceasily included in expansion bases using Wigner rotatmatrices~see Sec. II F!.17 Our emphasis will be on the internal coordinates. For reactive collisions of the form~1!, thereare three sets of internal Jacobi coordinates.~The Faddeevapproach18,19 uses all three sets simultaneously.!

Mass-scaled Jacobi coordinates are convenient forreasons. First, they yield an especially simple form forkinetic energy operator, viz.15,7

T52\2

2m~¹St

2 1¹st

2 !. ~7a!

When written in terms of the rotational angular momentumjof the diatom and the orbital angular momentumL of theatom about the diatom, this expression assumes a form thuseful for conversion to other coordinate systems,

T52\2

2m

1

StstS ]2

]St2 1

]2

]st2 1

1

St2 Lt

211

stj t2DStst . ~7b!

Second, transformations between mass-scaled Jacobordinates for different arrangementst andt8 are effected bysimple kinematic rotations of the form7

S St8st8

D5R~xt8,t!S St

stD . ~8a!

HereR is a 636 matrix which depends on the skew angxt8,t between arrangementst andt8 as

FIG. 1. Mass-scaled Jacobi coordinates (St ,st ,Qt) for an atom–diatomsystem.

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959J. Chem. Phys., Vol. 113, No. 3, 15 July 2000 Quantum reactive scattering: Coordinates

R~xt8,t!5S cos~xt8,t!I sin~xt8,t!I

2sin~xt8,t!I cos~xt8,t!ID , ~8b!

where I is the 333 unit matrix. The skew angle betweeadjacent arrangement channelst and t8 is the negative ob-tuse angle defined by7

cosxt8,t52m

dtdt8mt9and sinxt8,t52

1

dt8dt, ~8c!

wheremt9 is the mass of the atom in the arrangement otthan the two arrangements that define the skew angle. Tfor xAB , t5A, t85B, andt95C; this angle is shown in Fig2. The skew angles obey the symmetry relationsxt,t50 andxt,t852xt8,t . The sum of all three skew angles must equ2p

(t8

xt8,t522p and (t8

xt,t8512p. ~9!

Note that t, t8, and t9 are cyclic permutations~A,B,C!,~B,C,A!, and~C,B,A!.

B. Hyperspherical coordinates

Jacobi coordinates are not appropriate if the atomdiatom are in close proximity. In this region, the interactioamong the particles are strong and rearrangementsplace. More suitable for this rearrangement region are hyspherical coordinates. A variety of hyperspherical coornates are used in nuclear,1,2 atomic,3–5 and molecularscattering,6–9 and one can easily transform from one hypspherical coordinate system to another using rotational fratransformations, since all use the same hyperradius. Herchoose Delves hyperspherical coordinates,15 which are par-ticularly convenient for describing motion in each arrangment channel. For colinear motion, the Delves hypersphcal coordinates reduce to the usual plane polar coordina6

@In practice, we begin propagating the scattering function

FIG. 2. Hyperspherical coordinatesr and ut in two dimensions and theskew anglexAB between thezt axes for arrangement channelst5A andt5B. Also shown is the hypercircle of radiusr. The masses of the threconstituent particles are taken to be equal, so all three skew angles areto 2p/3.

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Adiabatically-adjusting Principal-axis Hyperspherical~APH!coordinates and transform to Delves coordinates at the oboundary of the hyperspherical region.7#

For convenience, we choose thextzt plane and measurethe angleut from thezt axis, as illustrated in Fig. 2.~Fromthis point on, we assume that we have completed propation in the three-dimensional hyperspherical region andtransformation from APH to Delves hyperspherical coornates.! This angle and the hyperradiusr are related to the JSt

coordinates by

St5r cosut and st5r sinut . ~10!

In three dimensions, the internal rotation angle in HSt coor-dinates isQt ; this angle is common to the JSt and HSt

systems. Unlike the anglesut andQt , the hyperradiusr isindependent of the arrangementt, i.e.,

r5ASA2 1sA

2 5ASB21sB

25ASC21sC

2. ~11!

In principle, the domain of these coordinates is

0<ut<p

2, 0<Qt<p, 0<r,`. ~12a!

In practice, however, the values ofr andut are limited in thenonrearrangement regions, as

0<ut<utmax, 0<Qt<p, 0<r<rmax<rasy,

~12b!

wherermax is large enough to enclose all rearrangement pcesses, andrasy is the hyperradius at which one projects onJacobi channel bases. Later in this section, we show howrasy

can be reduced tormax by matching in tangent-sphere coodinates.~In the rearrangement region, 0<ut,ut

max to allowfor exchange of any two particles.! The valueut

max is theangle at which the atom–diatom interaction potential bcomes repulsive; the hyperradiusrmax is the outer constant-rcontour of the HSt region, beyond which we switch to another coordinate system. In general,ut

max for two adjacentarrangement channels, such ast5A and t85B in Fig. 2, isrelated to the skew angle by

utmax1ut8

max<xt8,t . ~13!

In practice, we desire near equality in this equation, so acent hyperspherical regions touch at the maximum excursof their respective polar angles.

The determination ofrmax requires consideration of thpoint of contact between HSt and JSt regions. In Jacobi co-ordinates surfaces of constant vibrational coordinatest arehalf-cylinders of radiusst , as shown in Fig. 3. The maximum radiusst

max is determined by the vibrational wave function of the diatom in arrangementt: the domain 0<st

<stmax must encompass the range ofst where this function is

nonzero. These wave functions appear in the expansion bfor the system wave function, and it is important that thebe no overlap of basis functions for adjacent arrangemesee Fig. 4. Since the maximum hyperradius for a givenrangement is tangent to the Jacobi cylinders, we must deminermax for each arrangement so as to ensure that adjacylinders do not overlap. So, although strictly speaking

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hyperradii in Eq.~11! are independent of arrangement,practice we obtain three different hyperradii,rAB

max, rBCmax, and

rACmax. For example, for the adjacent regions shown~in the

plane! in Fig. 2 we have

FIG. 3. Jacobi coordinates (St ,st ,Qt) for the three arrangement channeof an atom–diatom system with three equal masses. The translational pgation variable isSt , the vibrational coordinate isst , and the angle be-tweenSt andst is Qt . Surfaces of constantst are half-cylinders of radiusst , and surfaces of constantSt are half-planes normal to thezt axis.

FIG. 4. Hyperspherical coordinates (r,ut ,Qt) for the three arrangemenchannels of an atom–diatom system with three equal masses. The trational propagation variable is the hyperradiusr. The Delves hyperangle isut , andQt is the angle between the Jacobi vectorsSt andst . The maxi-mum values of the anglesut are limited so as to exclude the repulsivpotential region, here illustrated for arrangement channelst5A and t5B.

rABmax5

1

sinxABA~sA

max!212sAmaxsB

maxcosxAB1~sBmax!2. ~14a!

~Here and in other figures, the coordinate labelsxt , yt , andzt do not refer to the positions of the particles. Rather, threpresent three orthogonal coordinates appropriate to therangementt and the region under discussion. Thus, in Jt

coordinateszt5St , xt5st , and yt5Qt .) To obtain asingle outer boundary for all three HSt regions, we choose

rmax[max~rABmax,rBC

max,rACmax!. ~14b!

This choice leads to the lower bound onr in Eqs.~12b!.Equations for transforming between Jacobi and Del

hyperspherical coordinates are given in Table I. Thesethe familiar polar coordinate transformations, which can abe expressed as a conformal map. Volume elements for tsystems, which are required for evaluation of matrix ements, appear in Table II.

Figure 4 shows the Delves HSt coordinates fort5A foran atom–diatom system of three equal masses. Were winclude the full range of HSt coordinates, Eqs.~12a!, thecontours of constantr would be hemispheres of radiusr. Butin order to ensure zero overlap of vibrational wave functioin adjacent regions, we chooseut

max5p/3. ~In HSt coordi-nates, the vibrational distance isr sinut'rut for small polaranglesut .) This choice excludes arrangements in whichthree interparticle distances are large; here the potenV(r,ut ,Qt) is repulsive and the wave function for the sytem can be assumed to be zero.

C. Solving the scattering equation in Jacobi andhyperspherical coordinates

In scattering methods based on the close-couplingproximation, the many-body Schro¨dinger equation for the

FIG. 5. A planar slice of the JSt and HSt systems for the three rearrangement channels of an atom–diatom system with three equal masses.thick solid line indicates a possible trajectory for propagation of the scating function into the asymptotic~Jacobi! region for channelt5B. The gapbetween the hyperspherical and Jacobi regions arises because the hydius is tangent to the Jacobi contour of minimumSt at only one point.Procedures for bridging this gap are discussed in the text.

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TABLE I. Relationships between JSt , HSt , and TSt coordinates. The top entry in each block is the propation variable; the bottom entry is the ‘‘vibrational’’ coordinate.

Region Hyperspherical Tangent sphere Jacobi

Hyperspherical r5r r5Armax2 1wt

2~11vtrmax!2

11vt2wt

2 r5ASt21st

2

ut5ut ut5tan21F wt

vtwt21rmax~11vt

2wt2!G ut5tan21Sst

StD

Tangent sphere vt5r cosut2rmax

r21rmax2 22r rmaxcosut

vt5vt vt5St2rmax

st21~St2rmax!

2

wt5r21rmax

2 22r rmaxcosut

r sinut

wt5wt wt5st

21~St2rmax!2

st

Jacobi St5r cosut St5vtwt

2

11vt2wt

2 1rmax St5St

st5r sinut st5wt

11vt2wt

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atom–diatom system is reduced to a set of coupled sinvariable differential equations by expanding the system wfunction in an appropriate basis.20 The functions in this basisare constructed from complete sets in the relevant coonates. In the JSt region, the basis for expansion of a wafunction in the body-fixed reference frame consists of pructs of vibrational wave functionsfv j

(JS)(st) for the diatom inarrangement t and associated Legendre polynomiaPj

L(cosQt) for the rotational motion of the diatom.@For ex-pansion in the space-fixed frame, the associated Legepolynomials are replaced by coupled angular functioYj l

JM(St ,st), whereJ andM are the quantum numbers for thtotal angular momentumJ5j1L and its projection on thespace-fixedz axis, andj andl are quantum numbers forj andL .20

The resulting single-particle scattering equation mustpropagated from the origin into the asymptotic region whmatching to asymptotic boundary conditions yields the sctering matrix. For rearrangement collisions, this propagatencounters special problems owing to the use of differcoordinate systems in different regions of configuratspace. These problems are evident in Fig. 5, which showplanar slice~colinear plane! through the JSt and HSt coor-dinates for all three arrangements of an equal-mass sys~The corresponding three-dimensional JSt and HSt contoursappear in Figs. 3 and 4.! At the juncture between the JSt andHSt systems, the propagation variable changes fromr to St .For each arrangementt, the hypersphere of radiusrmax istangent to the corresponding Jacobi contour of minimumSt

at only one value of the vibrational coordinate,ut5st50.For any other angleut , there is a gap between the poi(rmax,ut) and the start of the corresponding JSt region,St

(min) . So if one knows the value of the scattering function the surface of the hypersphere atrmax, one knows thevalue at only a single point in the JSt region. Only at thispoint, therefore, can one determine the scattering funccontinuously from the HSt origin r50 to an asymptoticvalue of the corresponding JSt translational coordinateSt .

To bridge the gap between the HSt and JSt regions, twostrategies have been proposed so far.7 ~We shall propose a

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third in Sec. II D.! The most widely used approach ismatch on the surface of the hypersphere of maximradius.7 A circle on this hypersphere is highlighted in Fig.the actual matching surface is obtained by rotating this cuaround each of thezt axes. In practice, rather than transforthe scattering function atrmax to JSt coordinates and propagating further, one simply propagates in HSt coordinatesfrom r50 to a hypersphere of radiusrasy in the asymptoticregion. There one projects the solution to JSt coordinates andmatches to asymptotic boundary conditions in these coonates~see Sec. II G!. Of course, the value ofrasy must ac-commodate all anglesut ; i.e., this value must be in theasymptotic region for all required values ofSt . This require-ment results in a maximum hyperradius large enough thatjust evaluate the JSt scattering function on the region of thsphere where the HSt and JSt systems now overlap, thecross-hatched region in Fig. 6.

Additional difficulties with this first procedure resufrom the dependence of the HSt vibrational basis functions

FIG. 6. One procedure for bridging the gap in Fig. 5. Solutions are matcto asymptotic boundary conditions in the region where the Jacobi cylinoverlaps with this hypersphere~cross-hatched region!.

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on r, a consequence of the limited domain of the ‘‘vibrtional’’ coordinateut in Eq. ~12b!. This dependence resultin a nonphysical channel coupling at large hyperradii trequires the inclusion of basis functions for closed vibtional channels. Worse, it requires subdivision of the domof the hyperradius into sectors, each of which has a differbasis. One must therefore change bases at each sector bary, a requirement that demands considerable memorysignificant CPU time. Strategies for accommodating thismand are sector adiabatic bases,7 diabatic by sector bases,12

and smooth variable discretization.21

The second proposed way to bridge the gap betweenHSt and JSt regions is to match scattering functions on Jcobi planes rather than on the hypersphere. This approillustrated in Fig. 7, entails propagating in HSt coordinatesout to the JSt plane that corresponds to the minimum valof St . On this plane, the HSt scattering function is matcheto regular and irregular solutions that have been propaginward from the asymptotic value ofSt . In addition to theproblems discussed in relation to matching on a hypersphthis alternative suffers from the fact that, as shown in Figthe distance from the origin to the Jacobi matching pladepends on the hypersphere angleut . This dependence requires one to propagate through the cross-hatched retwice, outward in HSt coordinates where multiple hype

FIG. 7. An alternative to the scheme in Fig. 6 for bridging the gap in FigOne propagates the scattering function in HSt coordinates out to the planasurfaces of the Jacobi cylinders, shown by thick lines for all arrangemchannelst. One matches to solutions in Jacobi coordinates in the regwhere hypersphere overlaps these cylinders~cross-hatched region!.

TABLE II. Volume elements and are lengths along the propagation variaq1 for the JSt(q15St), TSt(q15vt), and HSt(q15r) coordinate systems

Region Volume elementPropagatedarc length

Hyperspherical 14 r5 sin2~2ut!dr dut dSt dst

rmax2rmin

Tangent spherewt

4~vtwt21rmax1vt

2wt2rmax!

2

~11vt2wt

2!6 dwt dvt dSt dst wt tan21S wt

2rmaxD

Jacobi St2st

2dSt dst dSt dst St(max)2St

(min)

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spherical sectors cross the boundary and inward in JSt coor-dinates. Moreover, this technique does not guarantee unity of the resultingS matrix; in fact, the extent to which theresultingSmatrix violates unitarity varies with the scatterinenergy. Finally, it is not clear how to adapt this strategyformulations based on variational methods, such as the Kvariational method.22 These difficulties have prompted ouintroduction of a third strategy based on tangent sphereordinates.

D. Tangent sphere coordinates

Tangent sphere coordinates appear in the compendof coordinate systems by Moon and Spencer.23 ~We havereplaced their variableu by 1/wt andv by vt ; we have alsoshifted their coordinatez by rmax.) For each arrangementt,the TSt system is an orthogonal coordinate system whcoordinates (vt ,wt ,Qt) are related to rectangular coordnates by

~st!x5xt5wt

vt2wt

211cosQt , ~15a!

~st!y5yt5wt

vt2wt

211sinQt , ~15b!

St5zt5vtwt

2

vt2wt

2111rmax. ~15c!

Note thatQt , the rotational angle of the diatom, is commoto the TSt , JSt , and HSt systems.

The nature of these coordinates is most clear in(xt ,zt) plane obtained from Eqs.~15! by settingQt50. InFig. 8, curves of constantvt are circles of radius21/2vt

centered atrmax11/2vt ; these circles are tangent to a linparallel to thezt axis atxt5rmax. The contourvt50 is thext axis. Similarly, curves of constantwt are circles that aretangent to thezt axis atzt5rmax. In the limit wt→`, thesecurves approach thezt axis.

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FIG. 8. Tangent sphere coordinates in two dimensions. For each coordvt ~dashed curves! andwt ~solid curves! we show three constant-coordinatcontours: 1~thick curves!, 2, and 3~thin curves!.

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Constant-vt and constant-wt contours for three-dimensional TSt coordinates can be obtained by twirling thcircles in Fig. 8 around thez axis. These coordinates arillustrated for an equal-mass sytem in Fig. 9. Strictly speing the domains of the TSt coordinates are

0,wt,`, and 0,vt,`, and 0<Qt<2p. ~16!

These limits give for contours of constantvt spheres,

xt21yt

21Fzt2S r11

2vtD G2

51

4vt2 , ~17a!

and for contours of constantw toroids about the origin~withno center opening!,

xt21yt

21~zt2r!25wtAxt21yt

2 . ~17b!

But because only the domain@0,p# is physically mean-ingful for the rotation angleQt , we restrict the maximumvalue of this variable top. Hence the resulting surfaces anot closed. Surfaces of constantvt are now hemispheres, ansurfaces of constantwt are half-toroids.~Additional restric-tions must be imposed because of the proximity of the Tt

system to the JSt and HSt systems, as we shall discuss blow.! The coordinate origin is regained by settingwt50 andtaking either limitvt→6`. The limit wt→` is thezt axis,and vt50 corresponds to the half of thextyt plane forwhich yt.0. To ensure that the tangent sphere region Tt

will not overlap the HSt and JSt regions, the following limitsshould be used: 0,wt,st

max, 21/(2rmax),vt<0, and 0<Qt

max,p.

E. Solving the scattering equation in tangent spherecoordinates

The tremendous advantage of TSt coordinates for rear-rangement collisions is that they allow us to introducesingle propagation variable that varies continuously asmoothly from the origin ~in the HSt region! to the

FIG. 9. Tangent sphere coordinates for the three arrangement channelsatom–diatom system with three equal masses. The translational propagvariable isvt , the vibrational coordinate iswt , and the rotation angle abouthezt axis isQt . Surfaces of constantvt are hemispheres, while surfacesconstantwt are half-toroids.

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asymptotic (JSt) region. Figure 10 illustrates such a propgation in the (xt ,zt) plane, and Figs. 11 and 12 show ththree-dimensional, three-arrangement analog. Beginninthe origin, the variabler varies fromr50 to r5rmax, whereit joins smoothly to the TSt variablevt at vt521/2rmax.This variable increases through the TSt region until itsboundary atvt50, where it joins smoothly toSt at St

(min) .From here we can easily propagate to the asymptotic vaof this variable.

Use of TSt coordinates as an intermediary betweenHSt system, which is physically appropriate to the situatiin which the three particles are in close proximity astrongly interacting, and the JSt region, which is appropriateto the asymptotic limit, completely eliminates the needalgebraic or numerical matching procedures such as thdescribed in Sec. II C. Rather, by imposing continuity of t

f anion

FIG. 10. Hyperspherical (HSt), tangent-sphere (TSt), and Jacobi (JSt)coordinates in thexz plane. The thick dashed curves delimit the TSt region.The propagation variable in the HSt and JSt regions arer and St . In theTSt region, the propagation variable isvt ~thick curve!.

FIG. 11. Bridging the gap between the HSt and JSt regions~see Fig. 5! withtangent-sphere coordinates. The TSt region for t5B is delimited by thickcurves. After propagating the scattering function in HSt coordinates out tormax, one transforms this function algebraically to TSt coordinates and con-tinues propagating with respect to the variablevt . At vt50, one performsa second algebraic transformation to JSt coordinates and propagates into thasymptotic region. Note that the arc atrmax in the HSt region is also thecontour of constantvt521/2rmax, and the line of constantSt in the JStsystem is also the contour of constantvt50 in tangent-sphere coordinate

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-.

pace


scattering function and its first derivative at these boundawe can trivially determine theR matrix at the inner boundaryof the TSt region from theR matrix at the outer boundary othe HSt region~see Sec. II G for details!. ~In our implemen-tation, a frame transformation is performed at the ouboundary of the HSt region to transform to the Delves hyperspherical coordinates.! A similarly simple procedure connects theR matrices at the outer TSt boundary and the inneJSt boundary. The propagated are lengths given in Tablcan be used to ensure that the propagation steps are unin each coordinate system.

As noted above, the domains in Eq.~16! are limited inpractice by the upper limitQt5p and by the proximity ofeach TSt to its adjacent HSt and JSt regions. For examplethe TSt translational coordinatevt is bounded as indicated iFig. 10,

21

2rmax<vt<0. ~18!

The JSt vibrational coordinatest is bounded at the maximum valuesmaxt by the properties of the vibrational wavfunctions in the JSt basis, which requires that the coordinawt of the TSt region be limited to the same physical rangThe vibrational coordinate ranges for the TSt and JSt sys-tems must extend into the classically forbidden regionenough that the basis functions for arrangementt are essen-tially zero.

FIG. 12. Jacobi, tangent-sphere, and hyperspherical regions for all arrament channels of an atom–diatom system with three equal masses.curves highlight the boundaries between the HSt and TSt region and be-tween the TSt and JSt regions, both for configurationt5A. Note that therotation angleQt , which is shown in Figs. 3 and 4, is common to all thrcoordinate systems. The domain of the vibrational coordinate, which isthogonal to both the rotational and translational coordinate, extends intoclassically forbidden~repulsive potential! regions far enough that the vibrational basis functions are essentially zero at the highlighted boundaries

s

r

IIrm

.

r

F. Hamiltonians and expansion bases

The kinetic energy operator in JSt coordinates is cus-tomarily expressed using spherical coordinates (St ,St), and(st ,st), where St and st denote the polar and azimuthaangles of their respective vectors. Writing the Laplacian oerators¹St

2 and ¹st

2 in Eq. ~7a! in these coordinates an

introducing the orbital angular momentumL t of atom tabout the center-of-mass of the corresponding diatom,rotational angular momentumj t of the diatom, and the totaangular momentumJ5 j t1L t , the kinetic energy operatobecomes

T52\2

2m

1

stStF ]2

]St2 1

]2

]st2 1

1

St2 Lt

211

st2 j t

2GstSt . ~19!

The system Hamiltonian in JSt coordinates is then

H5T1V~St ,st ,Qt!. ~20!

To transform the kinetic energy operator into HSt or TSt

coordinates, we simply apply the chain rule using the apppriate conformal transformation from Table I. Let us denoby q1 the propagation variable in any of the three systeand byq2 the vibrational coordinate~see Table III!. In termsof these generalized coordinates, we can write a generic ffor this operator that pertains to any of the three systems23

T52\2

2m

q1kq2

h

S~q1 ,q2!s~q1 ,q2! F(i 51

21

Ag

]

]qiS giiAg

]

]qiD

1L2

S2~q1 ,q2!1

j 2

s2~q1 ,q2!G S~q1 ,q2!s~q1 ,q2!

q1kq2

h ,

~21!

where the metric coefficientsgii andAg are given in TableIII. The exponents in the factorsq1

k and q2h are chosen to

eliminate first derivative terms from the resulting kinetic eergy operator in each system; in JSt coordinates, for ex-ample,k5h50, and the kinetic energy operator is~19!. Theinverses ofq1

kq2h multiply the corresponding radial function

in Tables IV and V. The kinetic energy operators in the Ht

system that results from applying Eq.~21! is

ge-ick

r-he

TABLE III. Coordinates and metric coefficients for JSt , TSt , and HStcoordinate systems. The volume elements for the full six-dimensional sis st

2St2Ag; see Table II.

HSt TSt JSt Generalized coordinate

Propagation variable r vt St q1

‘‘Vibrational’’ variable ut wt st q2

g11 1wt

4

~11vt2wt

2!2 1

g22 r2 1

~11vt2wt

2!2 1

Ag rwt

2

~11vt2wt

2!2 1

tena

rom

tin-

-ce-

arendes.tse-

IV,

lts

end-

onsat

ion

h–d

c-

t.

sn-


T52S \2

2m D 2

r5/2sin 2utF ]2

]r2 11

r2

]2

]ut2 1

1

4r2

11

r2 cos2 utLt

211

r2 sin2 utj t2G r5/2sin 2ut

2, ~22!

and that for the TSt system is

T52S \2

2m D 1

wt2

~11vt2wt

2!2

vtwt21rmax~11vt

2wt2!

3H ~11vt2wt

2!2F 1

wt4

]2

]vt2 1

]2

]wt2G

1~11vt

2wt2!2

@vtwt21rmax~11vt

2wt2!#2 Lt

21~11vt

2wt2!2

wt2 j t

2J3wt

2vtwt

21rmax~11vt2wt

2!

~11vt2wt

2!2 . ~23!

The exponents in the scale factors for the HSt coordinatesarek51/2 andh50; those for the TSt coordinates arek50andh51.

In each coordinate system, we solve the Schro¨dingerequation for the scattering function by expanding the syswave function in a basis that is complete in the vibratio

TABLE IV. The Schrodinger equations used to define the vibrational funtions in the space-fixed and body-fixed bases. In the HSt system, differentvibrational bases are used in different sectors, as discussed in the texclarity, however, we suppress the sector index onr.

Region Vibrational Schro¨dinger equation

Hyperspherical H2 \2

2m

1

r2 F ]2

]ut2 1

jt~ jt11!

sin2 utG1Vt~r sinut!2En,j

~HS!~r!J3fv j

(HS)(ut)50

Tangent sphereH2 \2

2m~11vt

2wt2!2F ]2

]wt2 1

jt~ jt11!

wt2 G

1VtS wt

11vt2wt

2D2En,j~TS!~vt!fv j

~TS!~wt!50

Jacobi H 2\2

2m F ]2

]st2 1

j t~ j t11!

st2 G1Vt~st!2En, j

~JS!J fv j~JS!~st!50

TABLE V. Space-fixed expansion of the system wave functionCg0(q1 ,q2)

for HSt , TSt , and JSt coordinate systems. The channel indices for theexpansions areg5(t,n, j ,l ;JMp). The subscript 0 denotes the initial chanel.

Region Space-fixed expansion

Hyperspherical (g8

2

r5/2 sin 2ug8Gg8,g0

~HS!~r!fv8 j 8

~HS!~ug8!Yj 8 l 8

JM~St8 ,st8!

Tangent sphere (g8

~11vg82 wg8

2!2

wg82

~vg8wg82

1rmax1vg82 wg8

2 rmax!Gt8,g0

~TS!~vg8!

3fv8 j 8~TS!

~wt8!Yj 8 l 8JM

~St8 ,st8!

Jacobi (g8

1

St8st8Gg8,g0

~JS!~St8!fv8 j 8

~JS!~st8!Yj 8 l 8

JM~St8 ,st8!

ml

and angular coordinates of that system. For scattering fan initial channelg05(t0 ,n0 , j 0 ,l 0 ;JMp), we denote thewave function byCg0

(q1 ,q2), leaving implicit the constantsof the motionJ, M, and the parity

p5~21! j 1 l . ~24!

All channel indices containJ, M, p, and the arrangemenchannel indext. In addition, space-fixed channels are distguished by the quantum numbersv, j, and l, correspondingto the vibrational Hamiltonian,j t , andL t , respectively. Tospecify channels the body frame, we replacel by the L,which corresponds to projection ofL t on the body-framezt

axis. Thus in the body frame, the initial channel isg0

5(t0 ,n0 , j 0 ,L0 ;JMp).20,7 The general form of the expansion, for any coordinate system in either body-fixed or spafixed reference frames, is

Cg0~q1 ,q2!5(

gGg,g0

~q1!Fg~ q1 ,q2!, ~25!

where$Fg(q1 ,q2)% is the appropriate basis.In each coordinate system, the functions in this basis

products of vibrational wave functions for the diatom aangular functions for either the body- or space-fixed framIn the body-fixed JSt system, for example, the basis consisof products of functionsfv j

(JS)(st) and associated Legendrpolynomials Pj

L(cosQt), which represent rotational eigenstates of the diatom. The JSt vibrational Hamiltonian whoseeigenfunctions appear in this basis is given in Tablealong with those for the HSt and TSt systems.

In the expansion ofCg0(q1 ,q2) in this basis, the body-

fixed JSt radial functionGg,g0

(JS) (St) is multiplied by the scale

factor 1/stSt to cancel the factorq1q25stSt in the JSt ki-netic energy operator~19!. Finally, the expansion is multi-plied by the normalized Wigner rotation matrix17 to take ac-count of the overall orientation of the system. There resuthe JSt expansion of the body-fixed wave function,

Cg0~q1 ,q2!5(

g8DL8,M

J~at8 ,bt8 ,gt8!

1

St8st8

3Gg8,g0

~JS!~St8!fv8 j 8

~JS!~st8!Pj 8

L8~cosQt8!. ~26a!

The Wigner rotation matrix effects rotation into thspace fixed frame and so is not required for the correspoing space-fixed expansion. Rather, the rotational functiPj

L(cosQt) are replaced by coupled angular functions thtake into account rotations of the diatom and its orientatin the space-fixed frame,

Yj lJM~St ,st!5 (

L8,m8C~ j 8l 8J;L8m8M !Yj 8,L8~ st!Yl 8,m8~St!,

~26b!

where we adopt the conventions of Rose for the ClebscGordan coefficients.17 The expansion of the space-fixewave function in the JSt basis is

For

e

mS

e

cin

oot

ona-c-

lerio

,nt

i-

er

n,

als

n

at

e

sl

he


Cg0~q1 ,q2!5(

g8

1

St8st8Gg8,g0

~JS!~St8!

3fv8 j 8~JS!

~st8!Yj 8 l 8JM

~St8 ,st8!. ~26c!

Corresponding expansions ofCg0(q1 ,q2) in HSt and

TSt systems are given in Table V for the space-fixed fraand Table VI for the body-fixed frame. In regard to the Tt

basis, note that the TSt vibrational Hamiltonian in Table IVdepends on the TSt translational coordinatevt in addition tothe vibrational coordinatewt . Hence the TSt vibrationalwave functions depend parametrically onvt . In practice,this features causes no problems. The set$fv j

(TS)(wt)% iscomplete for any vt in the physically relevant rang21/2rmax<vt<0. For example, by choosingvt50 in thisHamiltonian, we can make the vibrational Hamiltonians~andwave functions! in the JSt and TSt systems identical.

G. Transforming the scattering function at boundariesbetween coordinate systems

As discussed in Sec. II E, the use of tangent sphereordinates greatly facilitates propagation of the scatterfunction from the origin to the asymptotic region becausereplaces computationally intensive matching procedures~atvalues of the propagation variable where a change of cdinate system is made! by simple matrix transformations. Aeither the HSt–TSt boundary or at the TSt–JSt boundary,we merely impose continuity of the scattering functiCg0

(q1 ,q2) and its first derivative with respect to the propgation coordinateq1 . This chore is made easier by introduing theR matrix, which is defined~in any coordinate system!by the matrix product

R~q1!5G~q1!@G8~q1!#21. ~27!

Each boundary defines two regions: the one to itsand the one to its right. The continuity conditions concethe propagation variable as it changes between these regWe denote this variable in the left region byq1 and in theright region byq1 . At the HSt–TSt boundary, for examplethe maximum propagation coordinate for the left regioq1

max, is rmax, while the minimum coordinate for the righ

TABLE VI. Body-fixed expansion of the system wave functionCg0(q1 ,q2)

for HSt , TSt , and JSt coordinate systems. The channel indices for theexpansions areg5(t,n, j ,L;JMp). The subscript 0 denotes the initiachannel.

Region Body-fixed expansion

Hyperspherical (g8

2

r5/2 sin 2ut8Gg8,g0

~HS!~r!fv8 j 8

~HS!~ut8!

3Pj 8L8(cosQt8)DL8,M

J (at8 ,bt8 ,gt8)Tangent sphere

(g8

~11nt82 wt8

2!2

wt82

~vt8wt82

1rmax1vt82 wt8

2 rmax!Gg8,g0

~TS!~vt8!

3fv8 j 8(TS)(wt8)Pj 8

L8(cosQt8)DL8,MJ (at8 ,bt8 ,gt8)

Jacobi(g8

1

St8st8Gg8,g0

~JS!~St8!fv8 j 8

~JS!~st8!

3Pj 8L8(cosQt8)DL8,M

J (at8 ,bt8 ,gt8)

e

o-git

r-

ftnns.

,

region, q1max, is vt

(min) . Propagation and vibrational coordnates at both boundaries appear in Table VII.

Imposing continuity of the scattering function at eithboundary yields

Cg0~q1 ,q2!uq15q

1max5Cg0

~ q1 ,q2!u q15q1min. ~28a!

Inserting the generic expansion~25! and using orthogonalityof the basis functions appropriate to the left regioFg(q1 ,q2), yields

Gg,g0~q1

max!5(g8

Gg8,g0~ q1

min!^FguFg8&q15q1max. ~28b!

We can write this result in matrix notation as

G~q1max!5OG~ q1

min!, ~28c!

whereO is the matrix whose elements are overlap integrbetween expansion bases for the adjacent regions.

Similarly, imposing continuity of the scattering functioat a boundary yields

]

]q1Cg0

~q1 ,q2!Uq15q

1max

5]

]q1Cg0

~ q1 ,q2!Uq15q

1min

, ~29a!

5]q1

]q1

]

]q1Cg0

~q1 ,q2!Uq15q

1min

.

~29b!

Note that]q2 /]q1 is zero at any region boundary, becausea boundary unit vectors alongq1 and q1 are parallel. Thequantity ]q1 /]q1 in ~29b! is just the scale factor for thetransformation (q1 ,q2)→(q1 ,q2). Application of the ge-neric expansion ofCg0

(q1 ,q2) and orthogonality of basisfunctions for the left region yields a matrix equation for thfirst derivatives that is analogous to Eq.~28c!,

G8~q1max!5QG8~ q1

min!, ~30!

whereQ is the matrix of scale factors at the boundary.Combining Eqs.~28c! and ~30!, we obtain the desired

transformation equation for theR matrix at a region bound-ary,

R~q1max!5OR~ q1

min!@Q#21. ~31!

eTABLE VII. Values of the propagation and vibrational coordinates at tboundaries between the HSt , TSt , and JSt regions. The third column givesrelationships between the vibrational coordinatesq2 in each region for val-ues ofq1 given in the second column.

Coordinate system propagation coordinate (q1) vibrational coordinate (q2)

HSt–TSt boundary

HSt q1max5rmax ut52 tan21S wt

2rmaxD

TSt q1min5vt

~min!521

2rmaxwt52rmax tanSut

2 DTSt–JSt boundary

TSt vt(max)50 wt5st

JSt St(min)5rmax st5wt

thc

nd

th

l,th

tio

l

ly,

-l

d

i-

y

e-

rossbe-

,-

te.

s an-

ar-


To complete this description we need only determinescale factors at each of the region boundaries. We can fatate implementation of Eq.~31! by judiciously exploiting theaforementioned parametric dependence of the TSt vibra-tional Hamiltonian onvt . If we choosevt521/2rmax inthis Hamiltonian, then the basis functions in the HSt and TSt

regions are identical. The overlap matrixO in ~31! then re-duces to the identity matrix. Moreover, since at this bouary ]q1 /]q15wt

22, the elements of the matrixQ of scalefactors are simply

K FgU 1

wt2UFg8L

q15q1max

5E fv j~HS!~ut!

1

wt2 fv8 j 8

~TS!~wt8!dwt8 . ~32!

Note that for this choice ofvt , the vibrational coordinateswt andut are related by

wt52rmaxtanS ut

2 D , ~33!

so we can evaluate the one-dimensional integral in~32! withrespect to eitherwt or ut .

At the TSt–JSt boundary, we choosevt50 in the TSt

vibrational Hamiltonian. This choice makes the TSt and JStvibrational basis functions identical, and again reducesoverlap matrixO in Eq. ~31! to the unit matrix. The scalefactor at this boundary is again equal towt

22, so the matrixelements ofQ have the same form as those in Eq.~32!. Sinceat vt50, the TSt and JSt vibrational coordinates are equawt5st , we can easily evaluate these matrix elements inJSt system.

H. Calculating the differential cross section from theR matrix

Once the propagation described in the previous sechas reached the asymptotic region, a valueSt

(max) in the JStcoordinate system, we extract theK matrix, with elementsKg,g8, by matching the JSt scattering function to the usuaasymptotic boundary conditions,

Gg,g0

~JS! ~St!5a~St!2b~St!K . ~34!

For open channels, the elements ofa andb are proportionalto the Ricatti–Bessel and Neumann functions, respective24

ag,g05dg,g0

kg1/2St j l~kgSt!, ~35a!

bg,g05dg,g0

kg1/2Stnl~kgSt!, ~35b!

where the channel wavenumberkg is defined by conservation of the total system energyE in terms of the rovibrationaenergiesev, j of the diatom as

E5\2

2mk0

21ev0 , j 05

\2

2mkg

21ev, j . ~36!

For asymptotically closed channels, the matching contions are

Gg,g0

~JS! ~St!5c~St!2d~St!K . ~37!

eili-

-

e

e

n

i-

Defining

kg5 ikg , ~38!

where kg5ukgu, we follow McLenithan and Secrest25 inchoosing

cg,g05dg,g0

kg1/2St j l~ ikgSt!, ~39a!

dg,g05dg,g0

kg1/2Sthl

~1!~ ikgSt!. ~39b!

The quantitiesag,g0are real and closely related to the mod

fied spherical Bessel functionsI l g11/2, which are regular atthe origin. The quantitiesbg,g0

are also real. They are closelrelated to the modified spherical Bessel functionsKl g11/2,which decay to zero exponentially at large distances.24

With these conventions, we can determine theK matrixfrom theR matrix ~27! at St as

K5~Rb82b!21~Ra82a!, ~40!

wherea8 andb8 are the first derivatives ofa andb evaluatedat St .

If the boundary conditions given above are applied bfore the dying closed-channel coefficientsdg,g0

are negli-gible, then the resultingK matrix will contain elements cor-responding to both open and closed channels. The csection, however, depends only on the scattering matrixtween open channels. So only the open–open blockKoo ofthe full K matrix contributes to theS matrix,25

S5~ I1 iKoo!~ I2 iKoo!21. ~41!

From theS matrix, we calculate the transition matrixwith the conventionT5I2S, and thence the scattering amplitude for a transition~in the space-fixed frame!,

g05~t0 ,n0 , j 0 ,l 0 ;JMp!→g5~t,n, j ,l ;JMp!. ~42!

In terms of elements ofT, this amplitude is

f ~kg ,g←k0 ,g0!52p

Akgk0(JM

(l l 0

i l 02 l 11

3C~ j lJ ;m,M2m,M !Yl ,M2m0* ~ k0!

3Yl ,M2m~ kg!Tg,g0

J , ~43!

wherekg5St denotes the scattering angle in the final staFinally, the differential cross section for this transition is

I g0→g~St!5kg

k0u f ~kg ,g←k0 ,g0!u2. ~44!

III. IMPLEMENTATION

To summarize the tangent sphere procedure and aguide to future applications, here we outline the implemetation for reactive collisions of the type shown in Eq.~1a!.Figure 12 combines all three regions and shows the demcation boundaries referred to in this description.

~1! Choose the range of total energiesE over which we re-quire cross sections for comparison to experiment.

ncchnauc

te

f

S

fbencis

ype

haanhe

aeeletaem

tpanu

suon

s

e

.

st

-thhisd

eetheultsevi-ec-of

e

e

tal

rre-is

lu-a-

vi-ouseted

vi-ri-

A

n-

d by

-


~2! Estimate the number of required vibrational basis futions and calculate vibrational wave functions for eaarrangement channel in the asymptotic limit using alytic basis sets consisting of Sturmian functions for Colombic systems or simple harmonic oscillator eigenfuntions for molecular systems. Integrals are evaluausing Gauss–Hermite quadrature designed to conformthe maximum value ofst as determined by the extent othe vibrational basis functions.

~3! Calculate the maximum propagation radius in the Ht

region, rmax, from Eqs. ~14!. Using vibrational wavefunctionsfv j

(HS)(ut) at rmax, we check convergence othe vibrational eigenenergies with respect to the numof basis functions, increasing the number of these futions if necessary. These tests ensure that our basadequate in hyperspherical coordinates atrmax and inJacobi coordinates forrmax<St<`. Since this basis isalso adequate at the HSt–TSt and HSt–JSt boundaries,it is comparably accurate over the entire TSt region,21/2rmax<vt<0.

~4! Generate contour plots of the interaction potential in hperspherical coordinates at several values of the hyradius. From these we estimate the initial value ofr forthe propagation, making sure that all contours at tvalue ofr are much higher than the maximum energywhich we will calculate cross sections. This check esures that for all energies propagation will begin in tclassically forbidden region.

~5! Calculate HSt surface functions and eigenenergiesseveral values ofr nearrmax, using the discrete variablrepresentation, analytic basis set method, or finitement method. Using these we make the estimated sing value ofr precise via the WKB approximation. Ware quite conservative in our choice of this minimuvalue ofr.

~6! Generate surface functions and coupled equations inHSt region, storing on disk matrix elements and overlato be used in the transformations from one sector toother discussed in Sec. II C. This fairly extensive calclation requires numerous convergence checks andstantial disk space. This step, however, is the lastbefore propagation.

~7! Propagate theR matrix in hyperspherical coordinatefrom our initial value ofr to rmax, where we switchfrom APH coordinates to Delves coordinates. We thtransform theR matrix function to TSt coordinates usingEq. ~31! with the simplifications discussions in Sec. II GThis initializes the propagation with respect tovt

through the TSt region. At the outer boundary of thiregion, we again implement the equations of Sec. II Gtransform theT matrix into the JSt region. Furtherpropagation with respect toSt yields the scattering function at a value of this coordinate large enough thatmatching equations of Sec. II H are applicable. Tyields theK matrix, from which we calculate the desirecross sections.

-

---dto

r-is

-r-

tt-

t

-rt-

hes-

-b-e

n

o

e

IV. APPLICATION TO F¿H2 SCATTERING

Since the F1H2HF1F reaction continues to be thfocus of both theoretical and experimental activity wchoose this system to demonstrate the effectiveness oftangent sphere method. In this section we compare rescalculated with the tangent sphere against those from prous calculations in which we used a two-dimensional projtion of asymptotic Jacobi solutions onto a hypersphereradius26–28 rasym59.5a0 . In both calculations we used thpotential energy surface of Brownet al. ~often referred to asthe T5a or Truhlar 5a potential!29 ~see Fig. 13!.

In this reaction two of the particles are identical, and widentify arrangement channels as A5F and B5C5H. Weperformed scattering calculations at 95 values of the toenergy in the range 1.65 eV<E<2.4 eV. Figure 14 showscolinear contours for this system. The dashed contour cosponds to the maximum collision energy, 2.4 eV. As thdiagram shows,rmax must be greater than'6.8a0 to allowfor tunneling in the interior hyperspherical region. We evaated the values ofsmax, the largest Gauss–Hermite quadrture point for each arrangement channel, using the samebrational basis and parameters as in our previcalculations;26–28 these values are given in Table VIII. Thskew angles and scale factors in this table were calculafrom the atomic masses. Using Eq.~14! we obtainedrmax

56.830 . So we could use surface functions from our preous calculations26–28in propagation through the hypersphecal region, we choosermax56.975a0 , a value at which apreviously calculated surface function was available.26–28

Figure 15 shows hyperspherical~APH! contours for a con-stant hyperradius of 6.875a0 . Figure 13 shows that for thisvalue ofrmax the barrier height for scattering from channelto channels B or C is 3.9 eV with a width of 1.6a0 at themaximum collision energy, and the barrier height from chanel B to C is 710 eV with a width of 5.4a0 at the maximum

FIG. 13. Colinear barrier heights for rearrangement processes illustratea two-dimensional ‘‘slice’’ through the T5a potential surfaceV(r56.975,uAPH5p/2,xAPH) of Brown et al. in APH coordinates at a hyperradius of rmax56.975a0 .

thtte

Ht

.ew

omry

l

ceula-

m-m-her

om-ricalses.

s toJa-rdi-

le I.ionsring

Vra-

lesin

re

h-ap-

e totes,he thiss toingas-

e

gy

ion

e ofllyum


collision energy. These values ensure that tunneling intangent-sphere and Jacobi regions is negligible at all scaing energies of interest.

We propagated the coupled-channel equations fromr52.2a0 to r5rmax in the hyperspherical region using APcoordinates. We then applied the unitary transformationtransform theR matrix from APH to Devles coordinatesUsing Eq.~31! to evaluate theR matrix in the tangent-spherregion, we then propagated through that region. Finallyagain used Eq.~31! to evaluate theR matrix in the Jacobiregion, after which we propagated through that region frSt5rmax to 20a0 , at which we applied asymptotic boundaconditions.

Figure 16 shows reaction probabilityPn f←(n i , j i )R (E) for

scattering from initial state (n i , j i)5(0,0) into a final vibra-tional manifold n f ~summed over all final open rotationastates! as a function of the total energyE(eV) for the F1H2

system with total angular momentumJ50. The curves cor-respond to the present tangent sphere calculations anddots to our previous results.26–28 Clearly, results from the

FIG. 14. Potential energy contour plot of the T5a potential energy surfacBrown et al. in APH coordinates at a hyperradius ofrmax56.975a0 . Thewhite areas are classically forbidden regions where the potential energreater than the maximum collision energy of 2.4 eV~solid circle!.

TABLE VIII. Parameters used in the present tangent-sphere calculatand in a benchmark calculations based on a two-dimensional projectioasymptotic Jacobi solutions onto a hypersphere of radiusrasym59.5a0 .

Parameter Channel A Channel B or C

Atomic mass~amu! 18.9984032 1.00782503dt 1.379 1.00xt,t8 2.3309960 1.6211932we (a.u.) 2.0053431022 1.88555731022

r e (a0) 1.40112 1.732517nmax 1 1j max 12 31smax(a0) 2.770 3.352rmax5Smin(a0) 6.975 6.975Smax(a0) 20 20

er-

o

e

the

two calculations are identical to graphical accuracy. Sinthe size of the hyperspherical region in the present calctions is roughly half that in our previous study,26–28 thepresent calculations require roughly a factor of 2 less coputation time. More importantly, the current procedure copletely eliminates the matching procedure required in otmethods.

V. CONCLUSIONS

The tangent-sphere method eliminates the need for cplicated matching procedures such as those in hypersphemethods for treating exchange or rearrangement procesTo accomplish this, we use tangent-sphere coordinatesmoothly propagate from hyperspherical coordinates tocobi coordinates. The relationships between these coonates are shown in Figs. 11 and 12 and collected in TabFor convenience we have gathered most of the key equatrequired to implement this approach in tables: the scatteequations in Table IV, expansion bases in Tables VI andand boundary values of the propagation variable and vibtional coordinate in Table VII. We hope that these tabalong with the step-by-step implementation schemeSec. III will facilitate other applications of tangent-sphecoordinates.

In addition to eliminating the need for numerical matcing between the hyperspherical and Jacobi regions, theproach described herein reduces the maximum distancwhich one must propagate in hyperspherical coordinawhich is now limited to the range of hyperradii over whicrearrangement and/or exchange processes occur. Outsidrearrangement region we use simple Jacobi coordinatepropagate to a sufficiently large distance that all coupland phase contributions are negligible. In addition to incre

of

is

nsof

FIG. 15. Potential energy contour plot of the T5a potential energy surfacBrown et al.29 for the colinear configuration. The white areas are classicaforbidden regions where the potential energy is greater than the maximcollision energy of 2.4 eV. The outer circle is the hyperradiusrasym of ourprevious calculations.26–28 The inner circle is the hyperradiusrmax for cur-rent implementation of the tangent-sphere technique.

h-ro-


FIG. 16. A comparison of F1H2HF1F transitionprobabilities from our previous calculations26–28 ~soliddots!, which uses the two dimension projection tecnique, with results from the current tangent-sphere pcedure~curves!.

sth

cE

v,

-

.

.

in

-

C..

ing significantly the efficiency of hyperspherical methodcomputer programs required by the tangent sphere meare simpler and easier to use.

ACKNOWLEDGMENTS

We acknowledge the support of the National ScienFoundation under Grants No. PHY-9722055 and No. CH9710383.

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