james r. henderson, steven miller and jonathan tennyson- discrete variable representations of...

8/3/2019 James R. Henderson, Steven Miller and Jonathan Tennyson- Discrete Variable Representations of Large-amplitude R

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J . CHEM. SOC. FARADAY TRANS. , 1990, 86(11), 1963-1968 1963

Discrete Variable Representations of Large-amplitude Ro-vibrationalStates in a generalised Coordinate System

James R. Henderson, Steven Miller and Jonathan TennysonDepartment of Physics and As tronomy, University Colleg e London, London WC7E 6BT

W e pre sen t a formulation of t h e triatomic ro-vibrational nuclear motion problemi n a set of gene ral ise d coordi-nates using a discrete variable representation (DVR). These coordinates are expressed in t e r m s of tw o dis-tances and an included angle, and contain many well known coordinate systems. Applications toLiCN, Na, an dH i which illustrate t h e power of the method a re descr ibed . The resulting energ ylevels a n d wavefunctions arediscussed. Our Na, fundamental frequenc iesare found to be i n good agreement with e xp e r im e n t .

1. Introduction

Highly excited small molecular systems are now an area of

much study for both the experimentalist and the theorist.Improved technology in the form of lasers and molecularbeams has allowed spectroscopists to study high-lying ro-vibrational levels. More suitable and efficient methodologiesare helping the theorist, as are of course the great advances incomputer facilities. It is challenging to try to understand thespectroscopy and dynamicsof highly excited systems, and theidea of qua ntu m chaos attract s much interest.

Conventional variational methods, using basis-set expan-sions, now allow the determination of low-lying ro-vibrational states of triatomic molecules, in some cases tospectroscopic accuracy. For a given potential-energy func-tion, these techniques, applied within th e Born-Oppenheimerapproximation, are used to solve the nuclear motion on the

hypersurface. Effectively one constructs the Hamiltonianmatrix in the basis of some truncated set of polynomial (ornumerical) functions and diagonalises this matrix to yield theeigenvectors and eigenvalues. Th e general approac h is varia-tional in that the eigenvalues will converge monotonicallyfrom ab ove t o their exact values as the size of the set of func-tions in each coordinate increases indefinitely.

For relatively high-lying states, however, it has becomeincreasingly apparent that these finite-basis representations(FBRs) begin to become intractable, owing primarily to theextremely large basis sets required for convergence of thehigher levels. This is particularly the case w ith molecules thatcan exhibit very large-amplittide motion in one or more ofthe internal coordinates and display large anisotropy in the

potential-energy surface. These systems are floppy and tendto isomerise, or may encounter other features of thepotential-energy surface. The problem is that the convention-al FBR methods may fail to provide a good representationwhen the molecule is executing very delocalised motions andis accessing highly anharmo nic regions of the p otential.

The past five years have seen BaEiC, Light and co-workersdevelop the discrete variable representation (DV R) as anattempt to overcome the difficulties encountered with theFBR techniques. The notion, essentially, is that the basisexpansion of the FBR can be operated on by an orthogonalDVR transformation matrix to givea representation in coor-dinate space rather than in function space. This part of thetheory relies on standard transformation properties and wasfirst suggested by Harris et a!. The second part of the DVRtheory utilizes the notion of successive diagona lisation an dtruncation of intermediate, reduced-dimensionality Hamilto-nian matrices. This effectively gives one the power to createbasis sets of different types and sizes at different grid pointsover the potential-energy function, thus ensuring a morephysical and realistic representation of the physics, and

winning major computational savings over FBR methods.One is also at liberty in coordinate space to reject quadraturepoints t hat may be energetically inaccessible. Th e procedureswill be explained in detail in section2.

The most popular DVR approach so far, in the ro-vibrational study of floppy triatomic systems, employsJacobi coordinates. The angular coordinate 0 is usuallyexpressed in a DVR whilst the radial coordinates remain inan FBR. (Light and co-workers usually like to use a distrib-uted Gaussian basis in the radial coordinate^.^)It is also pos-sible to discretise more than one and sometimes all of theinternal coordinates.

In this paper we develop the theory for the DVR in a gen-eralised set of body-fixed coordinates, prescribed by param-eters g1 and g2 which has, for example, Jacobi, Radau andbond length-bond angle coordinates as special cases. Thecoordinate system is most general in that it represents anytwo lengths and the included angle (see fig.1). The angularcoordinate has been discretised via a transformation inassociated Legendre polynomials and the radial coordinatesuse an FBR in either Morse oscillator-type functions4 orspherical oscillator^.^ Rotational excitation is accounted forusing the two-step variational approach of Tennyson andSutcIiffe.6

This formalism has been used by us for a series of calcu-lations on flop py triatomic sy stems. In this work we discussthe calcula tions on L E N , H i and Na,. Na, illustrates thepower of the DVR particularly well in that we were able toconverge many energy levels, whereas ou r FBR efforts failedto converge even the lowest-lying states satisfactorily.

Fig. 1. Coordinate system. Ai represents atom i. The coordinates inthe text are given by r l = A,-R, r 2 = A,-P and 0 = A , Q A , .


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1964

2. Theory2.1 The Ham iltonian Operator

DVR techniques have been well reviewed recently by BaEii:and Light.' In this section we present the DVR theory usinga generalised body-fixed Hamiltonian operator, as developedby Sutcli ffe and Te n n y ~ o n .~he DVR in any of the,internalcoordinates is obtained by transforming the respective FBRHamiltonian matrix elements.

First then, before applying the transformation, we presentthe Hamiltonian operator in the FBR. We use the close-coupling approach of Arthurs and Dalgarno,8 and followTennyson and Sutcliffe? Effectively this entails definingangular basis functions of the form

Ij, k ) = 0 , ( e ) D L k ( a * A Y) (1 )where 0 , (e) is an associated Legendre polynomial (assumedto be normalised) in the phase convention of Condon andS h ~ r t l e y , ~nd D L k ( a ,3, y ) is a Wigner rotation matrix in theEuler angles (a , /3,7). In this notation J is the total angularmomentum of the system and k is the projection of J ontothe body-fixed z axis.

On e then lets this exact Hamiltonian o per ato r7 act onIj, k ) , multiplies from the right byv, 'l and integrates overall the angular variables. This results in an effective radialHamiltonian of the form

@ r 1 , r 2 )= @) + Izr) + + @A+ 6 k . k O " k I W l , 2 , 6) Ijk), (2)

where V ( r l , 2 ,6) is the potential-energy function. The kineticenergy operators a re given by

J . CHEM. OC . FA RAD AY T RAN S . , 1990, VOL. 86

direction necessitates the changesr l ++ r2 and pl * 2 in theterms involving the rotational qua ntum numbersJ and k .

This Ha miltonia n oper ator is effective in the set of coordi-nates illustrated in fig.1 . The most general use of these coor-dinates is given by Sutcliffe and T e n n y ~ o n . ~or our use it issufficient to use the p aram eters g1 and g2 given by (see fig.1)

(3)

(4)

The an gular factors used above are defined as

c: = [ J ( J + 1) - (k f ) y (7)

(8)( j - k + l k + k + l ) l"

d j k = [ (2 j + 1X2j + 3 ) 1(9)

The above operators are for the body-fixedz axis embed-ded along the r l direction. Embedding thez axis along the r2

A3 - P A3 - Rg1 =-. I g2=-.

A3 - A2 A 3 - A1

The reduced masses are defined in terms of the parameters g1

(12)

(14)

Note that orthogonal coordinate systems are those forwhich p;; = 0, and hence @) = @A = 0. Such systemsinclude scattering (or Jacobi) and Radau coordinates, andany other (g l, g2)solutions of p;; = 0.

The above close-coupled equations (2) are in an FBR,labelled by j, corresponding to the rotational functionsforming the angular basis for any particularJ an d k . This setof equations can now be transformed to yield a representa-tion la belled by a se t of discrete angles.

and g2(0 < g l , g2 < 1 ) ;p1-' = girn;' + m i ' + 1 - 2)'rn; '

p i 1 = gfrn;' + m i ' + ( I - l)'rn;'.p;; = ( 1 - lX1 - g2)rn;' - 2rn;' - l r n i l (13)

2.2 The Discrete Variable Representation

We would like to switch from amplitudes expressed in theassociated Legendre polynomials to amplitudes representedat Gauss-associated Legendre quadrature points. Dickinsonand Certain" showed that if the functions involved are a setof j,, + 1 classical orthogonal polynomials, then there existsan orthogonal transformation to switch to a representationat jmaX 1 suitably weighted Gauss-polynomial quadraturepoints. Furthermore, from their work one can define the par-ticular transformation required in this case. Ifxkn(= cos O k U )an d a k n re the points and weights of N-point ( a = 1, N )Gauss-associated Legendre quadrature, then the DVR trans-formation can be w ritten

where N j knormalises 0,.

in the FB R t o yieldOn e can now transform the effective Ham iltonian op erator

N + k - 1 N ' + k ' - 1

f i w a r , k U = T!:a.fi(rl, r2)T!aj = k j ' =k c


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J. CHEM. SOC. FARADAY TRANS., 1990, VOL. 86

The new, transformed matrices that are diagonal ink aregiven by

1965

matrices in the 2D calculations or in the final,3D calculation.It is generally considered better to include them in the3Dcalculation as one can then assign a physical meaning to the2D solutions, as eigenvalues of the stretching vibrations(rather like an adiabatic approximation). Essentially, bytaking this option, the2D eigenvalues are independent of thenumber of DVR points a nd faster convergence of finalsolu-tions is achieved.

The full expression for fi in a DVR has been given forcompleteness, and indeed one can go ahead from this andobtain ro-vibrational solutions entirely in aDVR in the gen-eralised coordinates. For vibrational problems( J = 0) it isseen that p R 0 and so one purely solvesAk in the DVRfor k = 0. To account for rotational excitation ( J > 0) on ecould now solvep R lso in a DVR, for all k .

Th e treatmen t tha t we prefer, however," partly because ofthe computational considerations, is to treat the non-Corioliscoupled (Ak only) problem in a DVR, for each k . Take thelowest of these solutions (for eachk ) , back-transform to anFBR and then couple the k s to solve the full Coriolis-coupledHamiltonian, using the solutions of each k calculation,expressed in an FBR, as a basis to solve the finalfi. Ofcourse the advantage here again is that not all of the non-Coriolis solutions are required; one chooses the lowestM .This is a neat way of performing ro-vibrational calculationsutilising both a DVR and the two-step variational approachof Tennyson and Sutcliffe.6

where the sum goes from max(k,k & 1) to min(N + k - 1,

The details of each matrix have been given, particularly asthe non-orthogonal contributions have not been publishedbefore. Note that each of these transformed matrices isformed by effectively the same operation andso all of themcan be computed at the same time.

On e can express eqn (15) in a more com pact notation as

N' + k & 1 - 1).

f i = A " + P " (21)where

Ak = P( r l , rz)Z+ w,,(rl, rz)Lk3

+ C ( - l ywi( rl , r2 )k Pki = 1

ZP R -uo(rl , rz)Qk*- 2 udr,, rz)R(i)k. (23)

In de riving eqn (15) we have assumed the usualDVRapproximation in that the potential function is diagonalacross the grid po ints,i.e.

i = 1

N + k - 1

1 T;.,O"k I W l , r z ,6) l jk > T!a L, l , z I ka).j, ' = k

(24)

This is equivalent to ?V-point Gauss-associated Legendrequadrature." This quad rature approximation means that theDVR method is not strictly variational, owingto the numberof functions being inextricably linked to the number ofpoints. It is observed, however, that the method behavesvariationally (as one would expect) if N is suficiently large.

As a consequence of eqn(24) the potential is now diagonalin the angles, and the only angle-coupling is provided by thetransformed matrices, which are relatively simple to evaluate.

The DVR procedure then is to solve the two-dimensional(2D) Hamiltonian P , contained in t3*, or each different anglea. This can be thought of as finding solutions at various2D'cuts' across the potential-energy surface. The 2D eigen-vectors obtained can then be used as a well adapted basis inwhich to represent the fully coupled problem. The majorbenefit is that not all the solutions of these lower-dimensionalproblems are required to converge the desired states offi.One can select a subset of each of the2D solutions at eachangle according to an energy cut-off criterion, then the finalbasis constructed from these will be of reduced size and henceoffer great comp utational savings.

The elements in P are block-diagonal in angle, with allother terms spanning the wholeAk. Clearly one could eitherinclude the diagonal contributions from the angle-coupling

3. Applications/Results

3.1 Scattering Coordinates: L E N , HZ

We discuss here some recent studies using theDVR in scat-

tering (or Jacobi) coordinates. In termsof fig. 1, these areobtained when point R coincides with atom A, and point Pis at the centre of mass of the diatom A2A,{gl= [ m z /( m 2 m3)], g2 = 0). These orthogonal coordinates are partic-ularly suitable for atom-diatom systems that tend to be'floppy' and isomerise.

Henderson and Tennysonl 2 have recently studied thevibrational ( J = 0) states of the LiCN system using the SCFpotential-energy surface of Esserset a l l 3 with the C N bondfrozen at its equilibrium value. This potential has an absoluteminimum for linear LiNC (0 = 180") and a metastableminimum for LiCN (0 = 00). These isomers are separated bya relatively low barrier.

The rz coordinate, the Li-CN stretch, was treated in an

FBR using M orse oscillator-like functions. TheDVR methodproved to be powerful in that some 900 vibrational stateswere converged (with respect to basis-set truncation). Thiscalculation involved diagonalising a final Ham iltonian m atrixof dimension 1870. The best FBR calculation on thissystem14 converged only the lowest80 levels.

In the high-energy region studied all the states aresuffi-ciently energetic to cross the LiNC/LiCN barrier. Mostvibrational states were found to be irregular and delocalised.However, Henderson and Tennyson found states correspond-ing to LiNC normal modes, LiCN normal modes and free-rotor states where the Li' orbits the CN -, throughout theenergy range studied.

Approximate mode excitation assignments could be madefor these states by examining contour plots of the wavefunc-tion. Indeed, state 900 is found to be an LiNC state with20qua nta of stretching energy in the Li-CN bond! Fig.2shows some of the wavefunction structures for various states.

The large number of levels involved enableda reliable sta-tistical analysis to be made on nearest-neighbour level spac-ings. Fits were made to a Brody parameter,15 for various


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1966 J. CHEM. SOC. FARADAY TRANS., 1990, VOL. 86

7

5

3

7

R 5

3

7

5

3

I , , , , , , , I

(d1 '

1 I

0 .0

t i

0 180 0er

18 0

Fig. 2. Con tour plots of six typical wavefunctions for the LiN C/LiC N system. Approx imate assignm ents are ;(a) tate 180, LiNC normal mode(7,4); (c) state 353, LiCN normal mode(7, 6); (a) state 609, free-rotor (0, 65);( f ) state 900, LiNC normal mode(20,0), where (us, ub) correspondsto (no. of stretching quanta in the Li-CN bond,no. of quanta in the bend). State s250 and 75 0 [(b) and (e)] are irregular and cannotbeassigned. Solid (dashed ) con tours enclose regions where the w avefunction has positive (negative) amp litude. Con tours are d rawn at4, 8, 16, 32and 64% of the maximum am plitude of the wavefunction. The out er dashed co ntou rs represen t the classical turning pointof the potential forthe associated eigenvalue.

ranges of levels, and indicated increases in irregularity withenergy. This finding was consistent with the decreasing

number of states tha t c ould be, assigned even approxim atequantum numbers.

H: is a system that has been studied extensively in recentyears by both experimental ists and theorists . This elec-tronically simple species has proved to beof great interest tothe theoretical nuclear dynamicist and spectroscopist . Theobservation by Carrington and Kennedy,16 of27 OOO t ran-

sitions in a 222 cm - near-dissociat ion region provides muchmotivation in this area. This spectrum remains unassigned

an d largely unexplained.Tennyson and co-workers '7 , 1 8 recently performed a series

of parallel qua ntum and classical2D calculat ions o n H: with0 frozen at 90". Th e two radial coordinates were expressed ina n FBR: Morse-l ike functions in r l and spherical oscil latorfunctions in r 2 .The spherical oscil lators were used t o ca rryr2as the Morse-type functions do no t behave correctly whenr2


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approaches zero, i.e. as the system becomes linear. Thisgeometry is sampled at ca. one third of the H; dissociationenergy.

These calculations studied the structure of the wavefunc-tions for all the bound states of the system, using two differentpotential-energy functions. A variety of rotationally excitedstates was examined. An important result of these calcu-lations was the observation of horseshoe states in quantumcalculations (see fig. 3). Classical studiesIg had suggested tha tthis horseshoe motion was responsible for the underlyingstructure in the Carrington-Kennedy spectrum. This spec-trum, the transitions of which are between quasi-boundstates, can only be fully understood by calculations includingboth vibrational and rotational excitation.

Tennyson and Henderson, using the highly accurate abinitio surface of Meyer et al., performed similar calculationsfor a range of other angles, and this 2D basis was employedfor the full 3D calculations using a DVR in 0.

The lowest 180 vibrational (J = 0) solutions were obtainedand J =

1+

0excitation energies were computed for the

lowest 41 vibrational states. This considerably extended therange of ro-vibrational states studied, and showed the con-vergence difficulties suffered in previous calculations.21

This work identified horseshoe states o y theJirst time in afull 31) quantum-mechanical study. These states may actuallybe thought of as the highly excited bending motion of aquasi-linear molecule. Fig. 3 shows a wavefunction contourplot for a horseshoe state from a 2D calculation, and fig. 4shows the same state in a 3D calculation. The authors arecurrently developing a multi-dimensional DVR program tomake a serious attempt, at the very highest energies, to repro-duce the Carrington-Kennedy spectrum quantum-mechanically.

1967

l-----l

3.2 Radau Coordinates: Na,

The Na, cluster presents a very challenging problem. ThisJahn-Teller system has a very complicated potential-energy

4

7,

2

472

- 0 2Fig. 3. A wavefunction of a J = 0 horseshoe state of H i obtainedfrom a 2D (0 = 90) calculation (see text). This state (number 54)contains 12 quanta of excitation in the h orseshoe mode. Contoursare for 64, 32, 16 and 8% of the maximum amplitude with solid(dashed) curves enclosing regions of p ositive (negative) amplitude.The outer contour gives the classical turning point. The radial co-ordinates are mass weighted so that i l = a r I an d P, = r J a , wherea = (3/4)14.

4

7,

2

I-----..

02 4

720

Fig. 4. Cuts through the 3D wavefunction of state 150 of the J = 0even states containing 12 quanta in the horseshoe mod e. The angle0 is fixed at 90 for this plot. The con tours and radial coordinates areas given in fig. 3.

surface.* Below the conical intersection there are threesymmetry-related absolute minima and three metastableminima, separated by six saddle points. There are a furtherthree low-lying saddle points at linear geometries. The linearsaddle points are a long way from the minima and the

conical intersection resulting in long, narrow and nearly flatportions of the surface.

A number of spectroscopic studies have been performed onNa,. The most intriguing, by Broyer et ~ l . , ~ound a verydense spectrum at relatively low energy. Classicalcalculationsz4 suggested that this behaviour could be associ-ated with classical chaos and in particular with the asym-metric stretching mode which, with relatively small amountsof excitation, links the absolute minima.

We therefore decided to perform a quantum-mechanicalstudy of this system using the single ab initio potential-energysurface of Thompson et a1. Initially we attempted an FBRcalculation. We tried using scattering (gl = 4, gz = 0),bondlength-bond angle (gl = gz = 0) and Radau [gl = gz =

,/3/(1 + ,/3)] coordinates. None of these calculations gavesatisfactory results. The best results were obtained usingRadau coordinates and even then the lowest levels were con-verged to only ca. 0.5 cm- with very poor results for higherlevels, see table 1.

DVR calculations were therefore attempted using scat-tering and Radau coordinates. Again the Radau coordinatesgave the better results. In this case satisfactorily converged(to within 0.1 cm- or better) results were obtained for the70-plus levels lying below the linear saddle points (table 1).These calculations were successful because we were able touse large radial basis sets to obtain converged solutions forthe 2D radial Hamiltonians. The lowest of these solutionsthen gave a very satisfactory basis for the full 3D problem.

Effectively, these calculations mimicked FBR calculationswith in excess of 30000 basis functions! We note that Batik etaLz5 used Radau coordinates for a successful DVR calcu-lation on the vibrational states of water.

Rather surprisingly the very low-lying levels of Na, ,including the fundamental frequencies, are well representedby the harmonic approximation. Table 2 compares our fully


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1968

Table 1. Convergence of J = 0 vibrational calculations on Na,levels with odd symmetry performed in Radau coordinates


usually those of choice owing to the extra computationaleffort required for cross-terms and the mixed derivatives. Insome situations however, they are desirable if not necessary.Sutcliffe and Tennyso n, for exam ple, when working o nHeHD* found that i t was necessary to prescribe the non-orthogonal, geometric, coordinates defined byg1 = 0.5 an dgz = 0.0. The use of Jacobi coordinates led to unconvergedresults owing to theloss of the symm etry in the potential .

Recently the a utho rs have developeda computer programto treat al l three internal coordinates ina DVR using thesame coordinate systems presented here. Whitnell andLight have already employed a 3D-DVR and useda hyper-s phe ri ca l coo rd i nat e sy st em t o s t udy H: I na multidimen-sional DVR the diagonalisat ion-truncation procedure leadst o a rather elegant expression of the problem and offers amost systematic solution strategy. The only problem is decid-ing in which order one should solve for the internal coordi-nates. Also of course, one might ask which internalcoordinates should be in a DVR a nd wh i c h i n an F B R?These questions are unde r investigation.

FBR DVR

level L = 2 1 0 0 L = 2 4 6 0 L = 200 L = 50 0 L = 1800

16

11162126313641

- 953.6-2812.1- 758.8-2716.0-2678.5-2654.6-2619.9- 594.1-2577.8

- 953.9-2813.7- 758.9-2721.8- 681.5-2660.3-2625.8- 603.8-2581.1

- 954.17- 816.34-2762.57- 726.82- 710.06- 671.30- 662.71- 627.08-2618.80

-2954.17- 816.34- 762.57-2726.82- 710.06- 671.30-2662.71-2627.09- 618.81

-2954.17- 816.34-2762.57- 726.82- 710.06- 671.30-2662.71-2627.09-2618.81

The energies of the selected levels are given in cm- relative to disso-ciation. The finite-basis representation (FBR) calculations usedquantum number selec tionz9 o give a final secular matrixof dimen-sion L. hese calculations used upto 71 Legendre polyn omials for0and up to 19 Morse oscillator-like functions in the radial coordi-nates. The discrete variable representation (DVR) selected theL

lowest solutions from a grid of 70 Gauss-Legendre quadraturepoints. The vibrational problem was solved for the35 uniquepoin ts using up to 30 Morse osc illator-like functions in each rad ialcoordinate.

coupled results with Thompsonet al.s harmonic calculat ionsan d experiment? These results show tha t Tho mp sonet al.spotential-energy surface is accurate in the region about theminima. For completeness we also give the tunnell ing spli t-t ing of the levels we calculate. These splittings are all smallfor the fundamentals.

T he J = 0 high-lying solutions of the Na, problems tha twe have obtained show interest ing structure that does notimmediately bear out the predict ionsof the classical calcu-l a t i on~ . ~ t ho rough ana ly s i sof these results as well as cal-culations on rotationally excited states is in progress and fullresu lts will be given elsewhere.

4. ConclusionsWe have presented a discrete variable representation, in oneinternal coordinate 8, of a most generalised coordinatesystem. A summary of recent DV R calculat ions using Jacobiand Radau coordinates has been given. The systems dis-cussed here exemplify different strengthsof the DVR method.For LiCN results were obtained for some 900 vibrat ionalstates. For H high-lying rotationally excited states were

obtained. These calculations also necessitateda clever use ofC,, symmetry in coord ina te space . For Na, , where FBRmethods failed to converge even the lowest states, wemanaged to converge over70 vibrational levels.

Th e generalised coordinates also en compass an infinite setof non-orthogonal coordinate systems which have now beenpresented in a DVR. Non-orthogonal coordinates are not

Table 2. Com parison of Na, band origins ina -

DVR

band harmonic exp tb frequency splitting

V O - 954.2 0.0006v2 58 49.5 54.0 0.0070v3 94 87 92.5 0.02002VZ 105.2 0.0440V 1 142 139 137.8 0.0101

We thank Brian Sutcliffe for many helpful discussions duringthe course of this work. This work was part ial ly supported bythe S.E.R.C. under grants GR /E/98348 an d GR/F/14550.

References12

345678

9

10

111213

141516

171819

20

2122

23

24

2526

272829

Z. BaEiC and J. C. Light, Annu. Rev . Phys . Chem.,1989,40,469.D. 0 . Harris, G. 0. Engerholm and W. Gwinn,J . Chem. Phys.,1965,43, 1515.Z . BaEiC and J. C. Light,J . Chem. Phys., 1986,85,4594.J. Tennyson and B. T. Sutcliffe,J . Chem. Phys., 1982,77,4061.J. Tennyson and B. T. Sutcliffe,J . Mol. Spectrosc., 1983,101, 71.J. Tennyson and B. T. Sutcliffe,Mol. Phys., 1986,58, 1067.B. T. Sutcliffe and J. Tennyson,Int. J . Quantum Chem., in press.A. M. Arthurs and A. Dalgarno, Proc. R. SOC.London, Ser. A ,1960,256,540.E. U. Condon and G. H. Shortley, The Theory of AtomicSpectra (Cambridge University Press, Cam bridge1935).A. S. Dickinson and P. R. Certain,J . Chem. Phys., 1968, 49,4204.J. Tennyson and J. R. Henderson,J . Chem. Phys., 1989,91,3815.J. R. Henderson and J. Tennyson,Mol. Phys.,in press.R. Essers, J. Tennyson and P. E. S . Wormer, Chem. Phys. L ett . ,1982,89, 223.S . C. Farantos andJ. Tennyson,J . Chem. Phys., 1985,82,800.T. A. Brody, Let t. Nuovo Cimento,1987,7,482.A. Carrington and R.A. Kennedy,J . Chem. Phys., 1983,79,43.J. Tennyson,0 . Brass and E. Pollak,J . Chem. Phys., in press.0 . Brass, J. Tennyson and E. P ollak,J . Chem. Phys., in press.M. Berblinger, E. Pollak and Ch. Schlier,J . Chem. Phys., 1988,88,5643.W. Meyer, P. Botschwina and P. G. Burton, J . Chem. Phys.,1986,84,891.R. M. Whitnell andJ. C. Light,J . Chem. Phys., 1989,90, 1774.T. C. Thompson, G. Izmirlian Jr,S. . Lemon, D. G. Truhlarand C. A. Mead,J . Chem. Phys., 1985,82,5597.M. Broyer, G. Delacretaz, G-Q. Ni,R. L. Whettan,J. P. Wolfeand L. Woste,J . Chem. Phys., 1989,90,4620.J. M. Gomez Llorrente and H.S . Taylor, J . Chem. Phys., 1989,91,953.Z . BaEiC, D. Watt and J. C. Light,J . Chem. Phys., 1988,89,947.M. Broyer, G. Delacretaz, P. Labastie,J. P. Wolf and L. Woste,J . Phys. Chem., 1987,91,2626.

S . Miller, J. R. Hen derson andJ . Tennyson, to be published.B. T. Sutcliffe and J . Tennyson,Mol. Phys., 1986,58,1053.J. Tennyson,Comput. Phys. Reps, 1986,4, 1.

The absolute energy of the ground state , relativeto dissociation,isgiven for comparison. Ref.(22). Ref. (26).

Paper 9/054491; Received 21st December, 1989

james r. henderson, steven miller and jonathan tennyson- discrete variable representations of...

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