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Discrete Variable Representations and Sudden Models in Quantum Volume 89. number 6 CHLMICAL PHYSICS LEITERS 9 July 1981 DISCRETEVARlABLE REPRESENTATIONS AND SUDDEN MODELS IN QUANTUM SCATTERING THEORY * J.V. LILL, G.A. PARKER * and J.C. LIGHT 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 time 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” quantum state, 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 differential equation 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 wave 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 should always be preferable and outgoing waves, 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.
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