Volume 89. number 6 CHLMICAL LEITERS 9 July 1981

DISCRETEVARlABLE REPRESENTATIONS AND SUDDEN MODELS IN *

J.V. LILL, G.A. PARKER * and J.C. l7re JarrresFranc/t insrihrre and The Depwrmcnr ofC7temtsrry. Tire llm~crsrry of Chrcago. Otrcago. l7lrrrors60637. USA

Received 26 September 1981;m fin11 form 29 May 1982

An c\act fOrmhSm In which rhe scarrcnng problem may be descnbcd by smsor coupled cqumons hbclcd CIIIW bb bans iuncltons or quadrature pomts ISpresented USCof each frame and the srnrplyculuatcd unitary wmsformatlon which connects them resulis III an cfliclcnt procedure ror pcrrormrnpqu~nrum scxrcrrn~ ca~cubr~ons TWO ~ppro~mac~~~ arc compxcd wrh ihe IOS.

1. Introduction “ergenvalue-like” expressions, rcspcctnely. In each case the potential is represented by the potcntud Quantum-mechamcal scattering calculations are function Itself evahrated at a set of pomts. most often performed in the close-coupled representa- Whjle these models have been shown to be cffcc- tion (CCR) in which the internal degrees of freedom tive III many problems,there are numerousambiguities are expanded in an appropnate set of basis functions in their apphcation,especially wtth regardto the resulting in a set of coupled diiierentral equations UI choice of constants. Further, some models possess the scattering distance R [ 1,2] _The method is exact formal difficulties such as loss of reversal sym- IO w&in a truncation error and convergence is ob- metry, non-physical coupling, and non-conservation tained by increasing the size of the basis and hence the of energy and momentum [ 1S-191. In fact, it has number of coupled equations (NJ While considerable never been demonstrated that sudden models fit into progress has been madein the developmentof efficient any exactframework for solutionof the scattering algonthmsfor the solunonof theseequations [3-71, problem. as the number of equations is increased the computa- This latter point IS the reason for the use of the tron ume becomes proportronal to N3 due to the ma- term “model” rather than “approximation” ut the trLxwork involved [8] _The primarycause of this in- present work. Ihe purpose of thrs paper is to intro- crease is the prohieration of rotational states which duce a new exact scattering formalism which includes become available at typical scattering energies. sudden-type equations in a welt-defied mathematical In recentyears a numberof decoupledmodels for framework. In the following section the discrete vari- the quantum scattering problem have been proposed able and fmite basis represcntarions (DVR and FBR) [9-131 (for a review, see ref. [ 141). The centrifugal are developed for a simple two-dimensional problem. sudden (CS), energy sudden (ES), and infmite order -t-he former variable-labeled representation is shown sudden (10s) models achieve their decoupling by re- to be appropriate for smallscattering distances while placement of orbital, rotational, or both orbital and the latter function-labeledframe best dcscrrbcs the rotational kinetic energy operators with constant system at large scattering distances. Next the theory is applied to a standard rigid rotor problem. Finally * Tins material is based upon work supported by the Nationti Science Foundauon under Grant WE-7906896. a bnef dtscusstonwill be given. * Present address- Department of Physics and Astronomy, Umversw OlOklshoma. Norman. Oklahoma 73019. USA

0 009-26 14/82/0000-0000/S 02.75 0 1982 North-Holland 483 VOlUIllC 89. IlUIllbti 6 CHEMICALPHI SICS LETTCRS 9 July 1981

2. DVR and FBR quadrature points and weights, respectively, the ortho- gonality and completeness relations [(3a) and (3b] be- In order to mrroduce the DVR and FBR consider come 111ssinlplc two-dimensional scattering problem H(R,x) X *T(R,s) = E*T(R._v) where the hamllroman IS:

M(R._\) = -d’/d@ + II(I) + l’(R.x) (14

and (6b) Itm I’(R,.u) _ 0 . (lb) R-= In the languageof numerical analysis the internal degree Here R IS the scattenng coordm.rte and s 1s the single of freedom is now described by a drscrete (as opposed internal vanable. The internal hamdtonian I+) posses- to a continuous) orthogonal system [ZO]. ses 3 complete set of ortltonornl3l eigenfunctions. The points and wrights so described are uruque, not only providmg the discrete orthogonality conditron (6a). lr (x j I$, (s) = E, I?, (s) (3 but also minimizing the dtscrete rms error when quadn- wlrh orrhonormabty and complerencss relatrons. ture approximations to the mtegrak (SC) are performed [2 I j. Such relations hold for any set of orthogonal I dv p, (s) ,s, (A ) = q . (33) polynomtats. Eq. (6b) IS really just the Christoffel-Dsrboux iden- ury in drsguise (see eq. 8.418 m rei. [Zl]). It is clear 2 ~~(-\.)~~(_~‘)=6(s-x’). (3b) k=l that eqs. (6a) and (6b) define an orthogonal transforma- uon- Here 5: IS the Kronecker delta and 6(s - _I-‘)1s the [T]; 3 ~&) w;” . Drrac delta functron. the range ofs need not be specl- (74 fied. TTT=I=TTT. (7b)

TIE f&r has been nored in an independent (and quite different) derivation [%?I concerned with the efficient computation of matrix elements of unusual potenttal tW.-~)I, = $, tf@)I:.~~W. (9 functions 1231. What is believed to be completely novel in the pres- ~hcrc “I” mdcws the “imtr;il” , results ent approach is the use of the transformation (7a) to in the truncated (A’-drmennonal) CCR equations. connect dual representations of the scattering problem. To this end it 1s convenient to defme the Ndimension- [d’l/dR’+61-e]f(R)=V(R)f(R), (W aJ diagonal matrix of the potential evaluated at the where* quadrature points:

[Ejj = s; E, (sb) [U(R)]i =:sf V(R,x,). (8) and Now in analogy with the partial the DVR equations may be written as [v(R)l; q sdv q(.4 t’(R,x)~+,(.r). (54 [d’l/dR’+El -TETT]fDVR(R)=U(R)fDVR(R) The DVR and FBR squattons may now be developed @a) in analogy to the parttal differential and CCR equations by supposrngthere esists an A-point gaussian quadrature and in analogy with the CCR equations, the FBR such that the first M orthogonality relations (3a) are equations may be written as- reproduced esactly. Letting {I,) and {w,} denote the

484 Volume 89. number 6 CHEMICAL PHYSICS LETTERS 9 July 1982

[d’l/d@ tEi - E]fTBR(R)=fTu(R)~JrBR(R) sudden-type equations and provrdes an approach to (W corrections to them. The diagonal poruon of the DVR equations 1s m fact Just a particular set of IOS Not2 in particular that equations with a particular s2t of “constants” dic- tated by the transformation of the internal hamil- c;‘= Jbr+M-+&)=E; Jd-W,(x)q(x) toman. Taking these equahons as definining a zero. order problem, on2 may perform a Series of calcula- tions of varying degrees of accuracy by mcludmg the off-diagonal terms by various perturbation schemes. Similarly the &agonal portion of the FBR cqua- tions defines the so-calkd distorted solutron. so rhe Internal hamdtoman is the same in the CCR Thx may similarly be improved through the use of and FBR. Further note that perturbation theory. The DVR and FBR solutlons may then be connected exactly at some value of the scattenng coordinate R, by the dscrete variable transformation (7a). The optunal choice of R, involves dctermimng is just the usual potential mauL! element exept that which frams is more diagonally dommant. At small now the mner product is the gaussian quadrature in- scattering distances where the potential dominates stead of the integrauon as m (5~). Obviously as the the colliaon the DVR is more diagonally dominant, number of quadrature points and basis functions go at larger scattermg distances where centrifugal effects to infinity, the DVR and FBR results converge (m dommate, the FBR IS more dragonally dominant. A the mean) to the exact soluuon. convenient measure of this dominance is the magni- Since the dimensionality of the problem is un- tude of the difference between the largest and sm3Uest changed the advantages of rhe DVR-FBR analysis dagonal matrix clzmcnts. the frame with the larger ax not Immedmrely obvious. First note that the splirtlng is more ct~agonally dommant smce the hrgesr DVRprovides anexact numerlcal method in which andsmallest diagonal elements arc closer to thecorrt- the potential matrix Elements are merely given by spondmg eigenvalues. the potential Itself evaluated at the set of quadrarure Finally rhe exact scattering boundary conditions points Since the internal hamiltonian T&TT is m- may be appliedm the FBR exactly as in the CCR dependent of R (this follows from the separation of variables UI the original partial differentral equation) bm fFBR =;?a [I(R) - U(R)S] , (10) it need only be calculated once. Thus there is much R-- less JJz work done in constructmg the interaction UI where I(R) and O(R) are the appropnate mcommg the DVRand this frame shouldalways be preferable and outgoing , and S is the scattering matrix. III exact calculanons. In a quasi-adiabatic approach the elgenvalues obtained by diagonrdization of the DVR and FBR interactions are exactly equivalent 3. EmpIe since they drffer by only a unitary transformation; rhe eigenvalues so obtained approach those of the As an example a 16channel rigid rotor problem cal- CCR m the limit of infmite dunensionahty. Further, culated by Tsien and Pack [24] was chosen. This IS a it may be shown that under a broad set of circum- relatirely high energy system for whxh the “matrix stances,the errors in the eigenvaluesdue to the re- tiagonahzatlon” sudden model of Tsienand Pack [9] placement of the mtegration by the quadrature are is known to be quite accurate. of the same order as those mtroduced by fruncatlon The DVR for this problem IS defined in the body- of the basis. fixed (helicity) frame of Pack [x]; the states for a to- Secondly - and more importantly - the DVR- tal angular momentum J and space-futed projection FBR analysis IS an exact method which contams AI,are labeled by the body-fned projection K and the Volunw 89. number 6 CHEFllCAL PHYSICS LCI-TCRS 9 July 19E’, - - Tsblp I Tsienand Packwith] = I= 0. These are slightly more Pomt, and rvc~~l~tsfor the I‘UR-DVR trzuWormatton cosxd, accurate than the results withy= 0 and T= 6 quoted IS the cosmrl 0i the atom-dxitom snflc by those authors. The third set of transition probabili- ties are obtamed by integration of the diagonal por- !i cos Au UG ~-- tions of the DVR and FBR equations and matchmg at 0 -0 9602899 0.~0~4571 R = 8.76 bohr with the discrete variable transformation 0 -0.7966663 0 4J-17611 0 -0 525532.t 0 6’7-1133 [T]: = (1 -cos~?~~)-‘~~~~P~(cosx~)w~‘~. (11) 0 -0 18353J6 0 1153616 Here (1 - co~~_~~)~~~~~P~(cosx,)is the polynoml I -0 8997580 0.0630323 1 -0.6771863 0.7972808 part of the normahzed associated Legendre function, 1 -0 3631175 0 6015008 and cos s, and o, are the points and weights of table 2 -0 8196460 0 049345-t 1. In this diagonal calculation the couphng results from 1 -0.3106016 0 3233837 tile phase built up in the DVR region and the discrete 1 -0 198677-1 0 6939376 variable transformation itself. 3 -0 7198343 0 0658866 ! -0.385270-l 0433635 Finally the fourth set of transition probabdities 4 -0 5974051 0 1138061 listed are obtamed by mclusion of Ihe ofi-diagonal cou- -t -0 2076225 0 6888923 phng in the DVR and FBR regions to fist order in per- 5 -0 -1-172136 0.1841603 turbation theory and again matching at R = 8.76 bohr 6 -0 2581989 0 6819847 -..-- with the discrete variable transformation (11). In the language of the variable interval-variable step integra- Gauss-assoclarcd Legendre pomts as shown m cable 1. nor. this is a twomrerval calculation (one DVR and

Because the dlarom is symmernc and the potential one FBR interval) with many steps in each interval. contams only PO and P2 terms, the ‘xnck” ofusmg In more conventional language, the IOS-hke equations only neg~r~ve points make all lmphed quadratures of are corrected to fiist order in the DVRinternal and basts functions over the potential equivalent to the the distorted wave equations are corrected to fist or- corresponding intcgratlon. The DVR and FBR are thus der in the FBRinterval. The discrete variable transfor- each equtvalent to the CCR for this rype of problem. mation, being a transformation between Hdbert spaces, Sclecred rest&s of several calculations performed IS independent of the accuracy of the solutions ob- wtth the VIVSintegrator are presented in table 2 The tamed m either region. first trims1~1on probabtity quoted is ihe exact result To aold exponential growth in the closed channels, obtamed by mtegrarion of either the CCR, FBR. or only tie diagonal portlon of the DVR equations were DRV equations. differences between the results ob- mtegrated in the nonclasstcal region; one step per in-

tained in the various frames are due purely to numen- terval was taken up to the point where alJ channels cal roundoff on the computer and do not appear in were open IO stabilize the solutions. the numbers quoted. The second ser of Root-mean-square (rms) errors are hsted in table 3. probabditresxc obtamedusing the IOSmodel of Ftnally the variation in rms error of the perturbed cal-

Selcctcd lrxwuon problblhucs. Hcrc DDVR-DTBR and PDVR-PI‘BR refer to the dlsgond and perturbed DVR-TBR caJcula- lions. rcspccu~cl) The numbers m pxrnrhcscs arc rhc poncrs of WI by ~h~uchthe value hsrcd should bc multlphcd All rrsul~s haw been rounded to rhc ncxcst I X IOmJ

Dbgonal elrmcnts

1.1 0,6 2,8 A.10 6.12 7.6 1,s 6,lO =.,-I

c\xt 4319(-l) 3.837(-l) 3 933(-l) 6 729(-l) 4 814(-l) 5.314(-l) 7.391(-l) 3.872(- 1) 10s 4.585(-l) 3.37@(-1) 3.160(-l) 6.026(-l) 1987(-l) S-078(- 1) 6.993 (- 1) 4 046(-l) DDVR-DTBR 8 015(-l) 7 187(-l) 6 756(-l) 8 166(-l) 8 115(-l) 8.108(-l) 8.799(-l) 7.539(-l) PDVR-PTBR 4.462(-l) 3.929(-l) 3 933(-l) 6 673(-l) 4.932(-l) 5 457(-l) 7.441(-l) 3.950(-l)

486 Volume 89. number 6 CHEMICAL PHYSICS LETTERS 9 July 1982

Table Z (conrinued)

I.1 -136 6.8 4.4 6.6 -1.2 6.4 6.2 6.0

ciact 5.138(-l) 7 585(-l) 5273(-l) 7.361(-l) 3.950(-l) 6.968(-l) 6 364(-l) 6 644(- 1) IOS 5248(-I) 7 309(-l) 5.419(-l) 7 159(-l) 3.951(-l) 6.867(-l) 6 173(-l) 6.501(-l) DDVR-DTBR 8.3X(-l) 8.775(- 1) 8.166(-l) 8599(-l) 7.491(-l) 8.656(-l) 8 %-1(-l) 8519(-l) PDVR-PI-BR 5298(-l) 7.623(-l) 5 381(-l) 7 387(-l) 4 026(-l) 7 033(-I) 6.399(-l) 6 685(-I)

Elements for innal/, I= 0.6

liilall. I 06 28 4.10 6.12 1.6 4-8 6.10 2.4

e\acr 4.439(- 1) Z 051(-l) 148(-Z) I.?(-3) 1231(-l) l.JO(-2) 7(-4) l.548(- 1) IOS 4 585(-l) 1.997(-l) 2.55(-2) 1.6(-3) 1.180(-l) 135(-Z) 8(-J) I 559(-l) DDVR-DTBR 8.015(-l) 7 06(-Z) 9 5(-j) 2 5 (-3) 4.18(-2) 5.0(-3) 1.3(-3) 5 I’(2) PDVR-PTBR 1162(-1) 2 026(- 1) 2.30(-Z) J 3(-3) J.‘3’(--1) 1.32(-Z) 7(-J) 1.5X(-11

iinal], I -1.6 6.8 -r;l 6.6 A.1 6A 6.1 6.0

C\PCI 1.17(-l) 6(-J) 1.11(-a) 6(-J) 110(-Z) 5(-j) 5(--t) 4(-J) IOS 1.09(-Z) 6(-J) 101(-2) 6(-J) I 27(-Z) 5(-J) 5(-4) 4(-4) DDVR-DFBR 3.1(-3) 1 O(-3) 3 8(-3) 9(-j) -I 7(-3) 8(-J) 7(-J) 6(-J) PDVR-PI-BR 1.10(-Z) 6(-J) 105(-Z) 6(-J) 1.32(-Z) 5(-4) 5(-4) 4(--I)

Elemcnrs for iruJn1~. I = 2.8

iilnll. I 0.6 2.8 -1.10 6,12 2.6 -1.8 6.10 2.4

lX3CI 2051(-l) 3 837(-l) 1626(-l) 3 19(-2) 5 53(-Z) 2 08(-Z) -I at-31 3 OO(-2) IOS 1.997(-l) 3 730(- 1) Z 702(-J) 1.17(-Z) 5 50(-Z) 2 10(-l) 5 3(-3) z 75(-Z) DDVR-DI-BR 7 06 (-2) 7 187(-l) 1.183(-l) 9.6(-3) 3 86(-Z) 166(-Z) ‘.O(-3) 1.40(-L) PDVR-PTBR 1.026(-l) 3.929(-l) Z 574(-l) 3.10(-Z) 5 601-Z) ,.08(-Z) -!A(-3) 2 81(-Z)

rmaI].l 4.6 6.8 4.4 6.6 4.2 6.4 6.2 690

C\3r’l 2.8(-3) 7(--u I.?(-3) l t-41 I.-l(-3) 0(-J) OH 0(--u 10s 3.1(-3) B(-4) 1.3(-3) l(-4) I ‘(-3) w--0 W-4) w--u DDVR-DI-BR 3 5(-3) 6(-4) z 4(-3) 7(-J) 4.1(-3) 1(-J) 3(-4) 3(-J) PDVR-PFBR 3.0(-3) 7(-4) 1.4(-3) I(-J) 1.3(-3) 0(-J) O(-4) O(-4)

Elcmcnls ior maal~. I= 4.10

rill. 1 0.6 2.9 4.10 6.12 1.6 -V3 6.10 2-l

C\XL Z 48(-Z) 1.626(-l) 3.533(-l) 2 875(-l) l&(-2) 1.16(-Z) 6.7(-3) 19(-3) 10s 2 55(-Z) 2.701(-l) 3 260(-l) 3455(-l) 108(-Z) 1.12(-2) 7 6(-3) l.B(-3) DDVR-DTBR 9.5(-3) 1183(-l) 6.756(-l) l.j-lO(-1) 5.6(-3) 1.61(-Z) 9 l(-3) 6.1(-3) PDVR-PI-BR Z 30(-Z) 2.574(-l) 3.9-13(-l) 2 971(-l) 1 03(-Z) 1.21(-Z) 7.6(-3) l.S(-3)

finali, 1 4,6 698 4.4 696 4,2 6.4 6.2 6.0

exact B(--1) 2(-4) 1f-4) W-4) l (--1) W-J) 0(-4) W-4) 10s 9(-4) 3(-4) 1(-4) 0(-4) 0(-J) 0(-4) 0(-4) W-4) DDVR-Dt-BR l-8(-3) 1 I(-3) 4(--I) 3(-4) 1.2(-3) 2(-J) 2(-4) 2(-4) PDVR-PI BR 8(--t) 3(-B) 1(-4) W-4) l(-4) W-4) 0(-j) W-4)

487 Volume 89. number 6 CHEMICAL PHYSICS LEl-iERS 9 July 1981

Table3 Why then is the (uncoupled) IOS solution more rms erroIs of the various dppro\lm.mofls krc rmsD. rmsO accurate than the &agonal DVR-FBR? The essential and rms arc 111cdlqonal. ofi-dla&onaland loral rms error. rc- di&rence between the IOS and DVR-FBR equations SpCcurcl) - is in the asymptotic boundary conditions. In the 10s model “scattering-rype” boundary conditions are ap Calcukmon mlSD rmso rms --. phed in the variable-labeled frame and these boundary 105 00317 0.0067 0 0101 conditions are not equivalent to the exact scattering dlqonal DVR-I DR 0 2741 0 0388 0 0763 boundary conditions employed in the DVR-FBR pLrrurbrd D\ R-l UR 0 0089 0.0015 0.0026 analysis. It should be added that the non-physical bounh conditions of thr IOS model are requlrad TabIt .l to make the mathematical problem well posed; if the Vxmrlon oi rmj errors !\nh R,. tlcrc R, IS ~hc scarrcrmg co- IOS boundary conditions were not dictated by the ordm3re 31 H hlch the sir uch from DVR to FUR IS made. e IS choice of “e~genvalue-like” constants the transitron rhc IWO oi the dliicrcncc oi the lugcst and rmAxr m~crac- probabdities would keep changing with the endpoint WXIm31n\ clcn~~nls m the I‘UR IO Ihat m Ihc DVR. and rms IS rhc rmc. error Ibr the pcrturbarcd DVR-FBR calculation for of integration as R -+m. However this means that the lbc tivcn R 5 sudden model IS. strictly speaking, a scattering prob- -- - lem which is mathematically distinct from the exact Rj -c rms ___--- problem. And there is no guarantee that performing 8 36 0 869 0 0013 an approximate calculation of the exact problem (as 846 0901 0 0025 in the DVR-FBR calculation) WIUyield better results 8 56 0 93.1 0 0029 than an exact calculation of some model problem - 8 66 0 967 0 0030 8 76 1001 0 0016 such as the IOS model - for some specific problem. 8 86 I 016 0 0023 The great benefit of usingthe DVR-FBRformd- 8.96 1071 0 0025 km is that one is guaranteedof convergingto the exact result. This could be done with the dlabatic culatlon wnh R,, as well as the ratios of the sphttings VIVS mtegrator by simply takmg more mtervals in of the DVR and FBR interactIon matrix diagonal ele- the DVR and FBR regtons. ments are shown in table 4. Ftnally it is interesting to note one final connec- tlon between 10s models and the DVR-FBR theory. The “matrix diagonalization” sudden used here uses a 4. Discussion transformationwhich diagonalizes the potentialmatrix for all R. This transformationis in fact equivalentto the The values of A. and COJ x,, in re5le 1 label the discrete variable transformation (11). In homonuclear DVR: in solving the DVR equations one is in effect diatomic scattering when there are potential terms working m a miked parkd-coupled ordinary frame, higher than P1, or in heteronuclear diatomic scattering with the appropriate dlscretization gven by the qua- when there are potential teTms greater than PII no such dnrurr. Solution of rhc equauons U-I rhe DVR would diagonalizing transrbrmarlon exists. However Secresr appear preferable for R < 8.76 bohr as is indicated in [II] ha s s h own that in the limit of an infinite basis one table 4, past thts point the FBR is more diagonally can diagonalize the interaction matrix by transforming dominant. Table 4 also indicates only shght differ- the CCR equations back to partial differential form. ences in the perturbed DVR-FBR results for reason- Similarly the discrete variable transformation (11) will able choices oT R,. diagonal&e the FBR interaction matrix for aUR as the Table 3 clearly indtcates the superiority of the per- dimensionsgo to infinity. The DVR-FBR analysis is turbed DVR-FBR results. The dtagonal DVR-FBR thus a synthesis of the matrix diagonahzation and trans- results are very poor and constderation of table 2 re- formational approaches to the IOS equations included veals the diagonal DVR-FBR transitton probabilittes in w exact mathematical framework. It provides a to be much too Aastic. Simply a change of frame is unique representation in which the potential matrix is not enough IO provide the coupling required by this just a diagonal matrix of the potential itself evaluated problem. at 3 specific set of points. 488 Volume 89. number 6 CHELIICAL PHYSICS LETTERS 9 July 1982

[ 121 P hl&uur and D.J. Kourr. J Chcm. Phjs. 60 (1974) 1388. (131 V. Ehax. J. Chem Phls 68 (1978) 4631. [ 11 \V H. hllllcr. cd . D~namris oi molecular coll~aons, Vols [ 141 D J. Kouri. in. Atom-molecule colliuon rhcory, cd A, B (Plenum Press, New York, 1976). R.B Bcrnsrcin (Plenum Press, New York. 1979) 171 R.B. Bernstein. cd., Atom-molcculc coilwon thcor) 1151 V Kharc and D J. Eourr. J. Chrm Phys. 72 (1980) (Plenum Press. New York, 1979). 2317 [ 31 ~lgor~lhms and Compurcr Codes for Atomic and hlolcc- 1161 R B. Walker and J C Light. Chum. Phys 7 (1975) 84 ulx Quantum Scarrenng Theory, NRCC Procerdmgs No. 1171 GA Parher and R T Pack. J. Chcm Plr) s 66 (1977) 5. LBL-9501. Vcls 1.2 (1979, 1980). 2850, I?] G A. Parher. T C. Schmalz and J.C Lghr, J. Chum. [IS] V. lihc and D J. Eourl. J Chcm Phys 69 (1978) Phys. 73 (1980) 1757. 49 16. 151 G A Parher, B R Johnson and J C. Lrght.CIwn. Phys. [ 191 J.M. IJopmJn and K T. Lee, J. Chem Phbs 71(1980) Leirers 73 (1980) 571. 5071. [6] R B. Walker. E B. Srcchcl and J.C Lrghr. J Chcm. Pbs [ZOl C Dalqurst and A. BJorck. Numcrul methods. L~JIIS. 69 (1978) 3518. N Anderson (Prenr~ce Hall. Lngletrood Chiis. 197-l) [7] J V. Ldl, TG. SchmaL~ and J C. Light. Imbeddcd hlarn\ 1211 T.B. HddebrJnd, lnrroducuon 10 numcrrcai anal>sa Green’s I unctions m Atomic and hlolccular Scarrcrmg (Prentrcc Hall, [email protected] Chffs. 197-l) Thcorv. IO be oubhshed. 1221 A S. Dickinson and P.R. Ccrrs~n, J. Chcm. Phys 19 181 H. R&z, UI bynarnics ofmolecubr colhons. Vol A. (1968) 4205. cd H II. Cldlcr (Plenum Press, NC\\ Yorli. 1976) ch 2 113J D 0. Hansen. C C Enprholm and W 0. Gwnn. J (91 J P Tsren and R.T Pack, Chcm. Phys. Lc~tcrs 6 (1970) 54 Chcm Phys. 43 (1965) 1515 [ 101 G A Parker and R T Pack. J. Chem Phys 68 (1978) [Xl T.P Tncn Jnd R T Pack.Chem Ph)s Lcllcrs6 (1971) 1585 579. [ 111 D Sccrcst, J. Chem Phys. 62 (1973) 710. 1251 RT Pack. J. Chcm. Phls 60 (1974) 633.