James R. Henderson and Jonathan Tennyson- All the vibrational bound states of H3^+

8/3/2019 James R. Henderson and Jonathan Tennyson- All the vibrational bound states of H3^+

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Volume 1 73, number 2,3 CHEMICAL PHYSICS LETTERS 5 October I990

All the vibrational bound states of H3+James R. Henderson and Jonathan TennysonDepurtm ent ofPhys ics and Astronomy, Univ ersity College London, Gow er Street, London W ClE 6BT. U K

Received 8 August 199 0

Vibrational calculations are presented for three H $ potential energy surfaces using a discrete variable representation in all threeinternal coordinates. These calculations converge ali the J= 0 bound states of Hz to within IDcm- giving at least 88 1 states foreach potential. The wavefunctions of these stateshave been analysed in an attempt to find assignable or spatially localised statesof the system. The significance of this work to the unassigned near-dissociation spectra of H$ is discussed.

The unassigned, near-dissociation, infrared spec-tra of Hz recorded by Carrington and Kennedy [ 1,2has given rise to much interest and speculation. Thegenerally accepted explanation of these extremely richspectra is that high angular mom entum states arebeing observed with the final state undergoing ro-tational predissociation [ 3-61. However coarse-grained versions of these spectra display regu lar fea-tures which have been attributed to underlying rcg-ular features in the classical motions of the system.An example of this motion is the horseshoe pe-riodic orbit of Gomez Llorrente and Polk& [ 7 1.

Several limited quantum-mechanical calculations[8-l 1) have suggested that the horseshoe-likestates are indeed found. All these studies have beenperformed or analysed in such a fashion that otherfamilies of regular states could we ll have been over-looked. As yet no fully coupled 3D quantum-me-chanical calculations have been performed up to thedissociation limit. Indeed, until recently it wouldhave been u nthinkable to sugge st such a calculation.Wh at is required is nearly a thousand states of a clas-sically chaotic, three-mode system. However, recentdevelopments of theoretical techniques for treatinglarge-amplitude vibrational motion have made suchcalculations possible.

The most significant new development has beenthe use of the discrete variable representation (DV R)by B aGC , Light and co-workers [ 121 for vibrationalproblems. This finite element technique relies onsuccessive diagonalisations and truncations to build

up a final Ham iltonian matrix w ith a very high in-formation content. Thus Whitnell and Light [ 13 1used hyperspherical coordinates and a DVR in allthree internal coordinates to obtain the lowest 100states of H: . We [9 ] used Jacobi or scattering co-ordinates with a DVR in the angular coordinate toobtain the low est 180 vibrational states which ex-tend about half way to dissociation. The accuracy ofthese results has recently been confirmed by Canerand Meyer [ 141 who used hyperspherical coordi-nates and a basis set contracted using a diagonali-sation and truncation procedure to calculate the sam elevels.

In this work we use a new implementation of theDVR method in scattering coordinates which trans-forms the entire internal coordinate H amiltonian intoa DVR. This method combines the efficiency of thepolynomial basis sets that we have developed over anumb er of years to treat m-vibrational problems [ 15with the power of the DVR to give us estimates, towithin 10 cm-, of the positions of all the vibra-tional bound states of the H3f molecule.

In this work we used three HZ potentials. The po-tential energy surface due to M eyer, Botschw ina andBurton (MBB ) [ 161 is of near spectroscopic accu-racy at low energies [ 171. However, although thissurface is well behaved up to dissociation [ 141, itdoes not actually dissociate correctly. The less ac-curate ab initio p otential due to Sch inke, Dup uis andLester (SDL) [ 18 was designed for scattering cal-culations and thus dissociating problems. However,

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Volume 173, number 2,3 CHEMICAL PHYSICS LETTERS 5 October 1990in the course of this work, we found that actually ifone considers a coordinate where all three H atomsmove ap art symm etrically this potential becom es ill-behaved below the H+-H2 dissociation limit. Thisproblem was circumvented by forcing an artificiallyhigh energy in the region of poor behaviour. The di-atoms-in-molecules (DIM) potential [ 191 has beenused extensively for (semi)-classical calculations. Itis well behaved over the entire energy range of con-cern to us but, at low energy, is significantly less ac-curate than the ab initio potentials. Although wepresent results for all three potentials, we have con-centrated on the M BB potential becau se the quan -tum-mechanical calculations cited above [ 9,13,14]all used versions of this potential.

possible final Hamiltonian matrix. This is done bytreating the coordinate with the smallest density ofstates last. In the present problem, the highest den-sity of states is in the dissociating r2 coordinate andthe lowest in the 6 oordinate. The coordinates weretherefore treated in the order r2+rl+0.Both radial basis sets contain parameters whichcan be adjusted to give the variationally best func-tions for a given problem. The rad ial basis functionswere optimised using a previously written [ lo] 2D(8 frozen) program w orking in a basis set rather thanDVR. These tests ensured that sufficient functionswere included to represent all the bound states of theproblems for several values of 8. Details of the op-timised radial functions are given in table 1.DVR theory as applied to vibrating triatomics has

been thoroughly reviewed by BaEiC nd Light [ 121and w ill not be repeated here. In our imp lementationwe have used scattering or Jacobi coordinates: rl isthe H-H separation; r2 is the distance from the centreof the H2 fragment to the third H atom ; 8 is the ang lebetween rl and r2_We note that unlike hyperspher-ical coordinates, this coordinate system does not re-flect the full symmetry of Hz.

In each coordinate we used orthogonal polyno-mials to carry the motion. Thus the rl motions werecarried by M orse-oscillator-like functions [ 201, ther2 motions by spherical oscillator functions [ 2 11, asthese behave correctly at the r2=0 limit, and Lc-gendre polynomials to represent the angular motion.Both the radial functions can be related to Laguerrepolynomials. For each coordinate it is thus possibleto define a Gaussian quad rature which is then usedas the basis of the DVR transformation [ 121.

As the size of the final 3D H amiltonian was notactually dependent on the number of radial gridpoints used (because 19 s the last coordinate in-cluded), we could be generous in our provision ofradial points. Convergence is thus ensured providedthe final step includes sufficient contracted radialpoints at each angular grid point. The final Hamil-tonian was constructed by selecting the N radial con-tractions with the lowest eigenenergies for the 2Dsubproblems.

The remaining technical problem is to determinein which order the coordinates should be treated inthe successive diagonalisation a nd truncation pro-cedure. T his question h as been discussed by Light etal. [22]; the objective is to minimize the CPU time(and m emory) required by obtaining the smallest

Our final results were computed using a grid of 32Gauss-Legendre quadrature points for the 0 coor-dinate. Because of symmetry only 16 of the pointsneed be explicitly con sidered [ 91. Tests showed thatincreasing this number actually degraded our finalresults as, for fixed N, less radial functions per anglecould be considered. Calculations which increasedthe number of angular points using a fixed energyselection criterion for the radial functions showed thecalculations to be stable with respect to increasingthe size of the angular grid, The uncontracted DVRgrids used here are equivalent to a basis set expan-sion comprising 23040 functions for each symmetry.

Table 2 shows the convergence of a selection oflevels as a function of dimension, N, of the final

Table IDetails of the radial basis functions used. r., w . and 0, are given in atomic units. n is the num ber of DVR points in each coordinateCoordinate Oscillator n r, w, &r1 Morse-like [20 1 36 3.16 0.11085 0.0060r2 spherical [21 1 40 _ 0.009s

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Volume 173, number 2,3 CHEM ICAL PHYS ICS LETI-ERS 5 October 1990Table 2Convergence of a selection of even-symmetry H z vibrational band origins as a function of dimension, N, of the final Hamiltonian. Allvalues are given in cm- relative to the Hf ground state at 4363 .5 cm- above the minimum of the surface. These results were obtainedfor the M BB potential func tion

Level N2100 2400 2700 3000 3300

1 2521.28 2521.28 2521.28 2521.28 2521.2851 15202.9 15202.8 15202.7 15202.7 15202.6

151 22265.8 22264.3 22263.6 22263.3 22262.3251 26170.1 26164.8 26162.1 26160.3 26158.3351 29014.3 29008.6 29004.6 29002.8 29001.1451 3 1239.8 31226.9 31218.5 31212.5 31209.7551 33104.6 33082.7 33070.7 33063.6 33059.1651 34733.1 34688.6 34668.0 34657.6 34652.2701 35443.4 35368.2 35341.3 35332.0 35323.8

Hamiltonian matrix. Inspection of the table suggeststhat the top levels are converged to within 10 cm-(note that even level 701 is actually above the dis-sociation energy of the MBB potential [ 231). Thelower levels are considerably better converged thanthis. It is possible to g et an indepen dent view of theconvergence by comparing the energies of the evenand odd symmetry calculations. About half the J= 0states of H$ should be of degenerate E symmetry. Inthis case one even and one o dd eigenvalue should bedegenerate. Experience [ 91 has sh own u s that thesplitting between these levels (the odd level is nearlyalways of lower energy) gives a good m easure of theconvergence.

The energies of our lowest 180 states are in goodagreement with previous studies [ 9,141. They rep-resent a systematic improvem ent on our previouswork [ 93 where we estimated that the highest levelswere only converged to 10 cm- . Carter an d M eyer[ 141, conversely, claim convergence for all their lev-els to 0.2 cm-. Detailed com parison w ith their re-sults suggest while their states of A, symm etry maywell be accurate to this am ount, their states of A2an dE symm etries are often as much as 6 cm- higherthan ours.

To estimate the total number of bound states weobtain it is necessary to assign all the E levels, something that becomes increasingly difficult near thedissociation limit where the splitting of the E statesbecomes greater than the mean spacing betweenstates. Alternatively, at high energy, it is possible to

make a statistical assum ption about the symmetry ofthe states calculated. We assumed that two-thirds ofthe odd states were E, the remaining being AZ.Withthis assumption the total number of states below aparticular energy is given by the number of evenstates plus one-third the numb er of odd states. Table3 compares this approximation with the actual num-bers for the lowest 400 M BB states which have beenassigned. The lowest 180 were assigned by compar-ison with Carter and Meyer [ 141, the remainder bylooking for degeneracies in the even and odd cal-culations. This latter method is likely to overesti-mate the total num ber of E states slightly, and henceTable 3The number of hound states supported by each of the three sur-faces for a range of energies. The energies (column 1) are givenrelative to the H $ ground state for each surface. Do is the disso-ciation energy [23]: 35035.2 cm- for MBB, 37810.6 cm- forDIM, and 34424.5 cm- for the SDL surfaceEnergy MBB(cm-) (assigned)10000 1914000 4418000 9522000 17426000 30 328000 39730000 -32000 -34000 _DO

MBB DIM SDL

21 22 2047 52 47

100 106 100183 19 2 18431 4 31 7 31 5401 398 40450 2 49 5 51 0624 607 63 7781 739 79288 1 1071 82 8

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Volume 173, number 2,3 CHEMICAL PHYSICS LETTE RS 5 October 1 990underestimate the total number of states, becausepairs of A, a nd A2 states which happen to be closein energy will be assigned as E.Table 3 show s that the statistical approximation isa reasonably good one. Using it we predict that theMBB potential supports 881 states. As these calcu-lations are essentially variational in nature (see ref.[ 121 for a fuller discussion on whether the D VRmethod is variational), this num ber gives a lowerbound on the actual number of bound states. How-ever our tests strongly sugge st that it is unlikely thatwe are missing more than 10 states. The statisticalapproximation may add a further error of 10 states,giving a total maximum error of 20 states.

Table 3 also presents results for the SDL and DIMpotentials. The SDL results are very sim ilar to thoseobtained using the MBB potential; this potential

supports a slightly higher density of states at higherenergies but a sm aller total numb er of states becauseof its lower dissociation energy. At low energies theDIM potential supports more states than the othertwo, but the density drops near dissociation. TheDIM potential overestimates the dissociation energyof H3+ and this results in the po tential supportingover 1000 bound vibrational states. Although we havenot done a d etailed convergence analysis on the DIMpotential we suspect that the DIM calculations arenot quite so well converged as the others. Previousstudies [ lo,11 ] have show n the DIM p otential to beclassically more chaotic (strongly coupled) than theMB B potential an d hence harder to obtain con-verged results for.In previous studies [ 9- 11 ,241,contour plots of thewavefunction have proved very useful as a method

5

r1 3

1

Fig. 1. Cuts through the wav efunction of four even Hz vibrational states calculated using the MBB potential with 0= 90 . C ontours arefor 64%, 32%, 16%and 8% of the maxim um am plitude w ith solid (dashed) curves enclosing regions of positive (negative) amplitude.The outer contour gives the classical turning point. The ban d origins of the states are 333 1I, 33317,33326 and 33335 cm-. The radii1coordinates are given in atomic units (a,,).136


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Volume 173, number 2,3 CHEMICAL PHYSICS LETTERS 5 October 19 90

I state 328

I1 3 5

r2

Fig. 2. Cuts through the wavefunctions of four even H$ vibrational states calculated using the M BB potential with B = 90 . Contours a sin fig. 1. The bandoriginsofthe statesare28014,28055,28091 and28138 cm-.

of obtaining insight into the nature of individualstates. However analysing many hundreds of 3Dstates with 2D plots is a formidable un dertaking. Sofar we have only looked at the even states and at theDV R point nearest 13~90 ; this corresponds to theprevious analysis of Hz wavefunctions. This partic-ular cut through the wavefunction is appropriate forfinding horseshoe states. These horseshoe statesare perhap s better though t of as highly excited bend-ing states of a quasilinear molecule. We find suchstates extending in a single progression all the wayfrom the ben ding fundam ental (ark) to dissociation- and indeed above it; we found a 19~~state justabove the dissociation limit of the MBB potential.

Figs. 1 and 2 illustrate typical contour p lots of thewavefunctions of the high-lying states. Note that be-cause we are confined to the coordinate range rz > 0,only half horseshoes app ear in the plots. Fig. 1showstwo even states with 18~~ (numbers 566 and 569)

and their immediate neighbours. Fig. 2 shows evenstates (numbers 327 and 328) with 1425 as well asa quantum of excitation in the transverse stretchingmode (Y,). Again neighbouring states are includedfor comparison. The high-lying states that we assignto horseshoe modes have a large gathering of am-plitude in the vicinity of the horseshoe periodic or-bit, or in other words are scarred by it. However, noneof the assignments we make in the high-energy re-gion are particularly clear cut. This is in contrast tothe intermediate energy horseshoe states analysedpreviously which show ed very clear nodal structures191.The search for other regular features in the vibra-tional w avefunctions of this system is underway. Thiswill be done both by plotting different cuts throughthe wavefunctions and also by transforming thewavefunctions to coordinates which better displayother classical periodic orbits.

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Volume 173, number 2,3 CHEMICAL PHYSICS LETTE RS 5 October 1990In summary, we have developed a 3D DVRmethod in scattering coordinates with which we ob-

tain converged vibrational wavefunctions for H3+upto dissociation. This is a major advance in our abil-ity to treat th is challenging and dynam ically rich sys-tem. W e believe that a proper qua ntal treatment ofthe predissociating states of H,+ observed by Car-rington and co-workers [ 1,2] is near.

Finally we note that, co mpared for example to thecost of constructing 3D ab initio potential energysurfaces, these calculations are computationally fairlymod est. For example a calculation with N= 2400 took2 min to perform all the steps up to and includingconstructing the final Hamiltonian matrix and 22 minto diagon al&e this m atrix, all timing s are for a sing leprocessor of a Gray-XMP. Our decision to stop thecalculation at N=3 300 was solely determined bymemory constraints.

We would like to than k the staff of the Universityof London Computer Centre for their help in per-forming these calculations. This work was supportedby SERC grant GR/F/ 14550.

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[ 81 J.M. Gomez Llorente, J. Z akrewski, H.S. Taylor and KC.KuIander, J. Chem. Phys. 89 (1988) 5959; 90 (19 89) 1505.[9] J. Tennyson and J.R. Henderson, J. Chem. Phys. 91 ( 1989)

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72 (1980) 3909.[ 19) Ch. Schlier and U. Vix, Chem. Phys. 95 (198 5) 401.I201 J. Tennyson and B.T . Sutcliffe, J. Chem . Phys. 7 7 ( 1982)

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71 .(221 J.C. Light, R.M. Wh itnell, T.J. Pac k and SE. Choi, in:Supercomputer algorithms for reactivity, dynamics andkinetics OF mall molecules, ed. A. L agan& NA TO ASI seriesC 277 (Kluwer, Dordrecht, 1989) p. 187.

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James R. Henderson and Jonathan Tennyson- All the vibrational bound states of H3^+

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