C. Ruth Le. Sueur, James R. Henderson and Jonathan Tennyson- Gateway states and bath states in the...

8/3/2019 C. Ruth Le. Sueur, James R. Henderson and Jonathan Tennyson- Gateway states and bath states in the vibrational

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Volume 206, number $6 CHEMICAL PHYSICS LETTERS I May 1993

Gateway states and bath states in the vibrational spectrum of HcC. Ruth Le. Sueur, James R. Henderson * and Jonathan TennysonDepartm ent of Physics &Astronomy, UniversityCollegeLondon. Cower Street, London W CIE 6Br UK

Received 7 November 1992; n final form 1 March 1993

Vibrational band intensities are obtained from discrete variable representation calculations of eigenstates for Hf. Transitionslinkinghighly excited states to the ground state show large variations in intensities, with gateway states corresponding to highlyexcited bending motion (horseshoes) which leak intensity into the nearby bath states. It is suggested that these gateway statesmay be observable. Implications of these results forthe interpretation ofthe infrared predissociation spectra of H:are discussed.

1. IntroductionThere is a great deal of interest in Hz : after an ini-

tial dearth of assigned transitions, experiments andobservations have lately come thick and fast. To alarge extent, the paucity of assigned lines arose fromthe fact that there were no good indications as towhere to look for the lines. Recently, however, theavailability of an extremely high-accuracy ab initioelectronic potential energy surface due to Meyer,Botschwina and Burton (MBB) [ 11, has enabled theposition of transitions to be pinpointed with greataccuracy. The latest and most accurate calculationusing this surface, for example, boasts that all thebound vibrational band levels are converged to within2 cm- [21,Furthermore, this surface was also usedfor ab initio calculations of rovibrational levels whichenabled the assignment of a spectrum emanatingfrom Jupiter [ 3 1.

Carrington and Kennedy observed a very complexspectrum of Hz, with over 27000 lines in 220 cm-.This spectrum, which results from transitions be-tween highly excited metastable rovibrational states,has so far eluded explanation, implying perhaps thatthe molecule at these energies may be completelychaotic in its motions. However, by convoluting thestronger lines to obtain a pseudo coarse-grainedspectrum, they observed four broad bands, at Current address: ULCC, 20 Guilford Street, London WClN1DZ, UK.

1033.62, 978.45, 928.02 and 875.65 cm-, indicat-ing that there is at least some regular behaviour un-derlying this spectrum.

Over the years since that spectrum was first ob-served, there have been many attempts to explainthese four bands, At first Carrington and co-workersnoted the fact that they correlated well with N= 5+- 3rotational transitions in Hz with 0, I,2 and 3 quantaof stretch, but nothing more came of this suggestion[4,5]. The currently favoured explanation is that thefour broad bands are due to rotational transitions inthe R branch of highly excited vibrational states. Ithas been noted that all the vibrational levels ofH: have roughly the same rotational constants ofapproximately 25 cm- I, which would imply a spac-ing between consecutive rotational transitions ofroughly 50 cm- [ 61. This is in reasonable agree-ment with the observed spacings. The only questionremaining is what states are long lived enough to beobserved. The most likely candidate states are thehorseshoe states [71 in which the complex under-goes what are in effect large amplitude bending mo-tions of a quasi-linear molecule.

Although there have been many calculations ofH: energy levels (see, for example, refs. [8- 131 forcalculations extending above the barrier to linear-ity), calculations of transition intensities are verymuch rarer. The assignment of the Jupiter spectrumwas one case where transition intensities were in-valuable, enabling a spectrum to be generated in ex-tremely good agreement with the observed data [31.

0009-2614/93/$ 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved. 429


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Volume206,number 5,6 CHEMICAL HYSICS E?TERS 7 May 1993of polynomials g,(x) which are orthogonal over adifferent metric W(x) ,

the products [w(x) / W(x) Ifn(x)fm(x) must all beexpressible as linear combinations of a finite numberof products g,,(x)g,( x) (and hence as polynomialsof order n t m). If that is true then the x, must bethe zeros of the polynomial gM( x) whose order M isgreater than that of any polynomial gn( x) involvedin the linear combinations.2.2. Eckart axes

It has been shown previously [ 141 that in order tocalculate vibrational band intensities from varia-tional wavefunctions, it is necessary to refer the di-pole surface to the Eckart coordinate system [ 191.This coordinate system is defined as one in whichthe position vectors of the atoms obey the followingrelationships:

(7)where r: denotes the position vector of atom i atequilibrium. The first of these merely places the or-igin of the coordinate system at the centre of mass.The latter ensures that the angular momentum in thiscoordinate system is approximately constant. Clearlythese relationships will hold in any of an infinite setof coordinate systems related by a fixed rotationabout the centre of mass, and so referring dipoles tothe Eckart coordinate system is simple. One simplyplaces the Eckart axes parallel to the original internalaxes at equilibrium, and then calculates how muchthe E&art axes rotate with respect to the original axesat each position away from equilibrium [ 141. Thistechnique is particularly suited to DVR calculationswhere all integrations are done using quadrature.

It must be noted that the Eckart conditions werederived by assuming that the molecule undergoes

only small deviations from equilibrium, and so thistechnique may not be appropriate for looking at theintensities of transitions between large amplitudemodes. Nevertheless, tests performed earlier [ 141show that, certainly for transitions involving low-lying states, the calculations are reasonably accurate.One good test is to observe when the symmetry-basedselection rules break down. The results presented inthis paper show that the A coefftcients for forbidden(A-A) transitions are typically several orders ofmagnitude lower than those for allowed transitions,at least in the range where symmetry assignments arefirm.

3. CalculationsThe eigenstates of H3+ used in this Letter were pro-

duced earlier [21, using the DVR3D program [201and implementation A [211 of the MBB potential[ 11. The calculation was done in scattering coordi-nates, with internal coordinates rl , the H-H dis-tance, r2, the scattering length (the distance from themidpoint of r, to the third atom), and 6 he anglebetween I, and r,. The full symmetry of Hz was notused, but the Hamiltonian was split into two blocksaccording to whether even or odd combinations ofv( 6) and v( ~-6) were taken. The coordinates weretreated in the order 8+r,-+r2, with 32, 36 and 40points respectively. The Hamiltonian was truncatedbetween treatment of each coordinate using an en-ergy criterion (so that intermediate solutions withvery high energies were ignored), with a final matrixsize of 6500. The resulting eigenstates are orthogonalto within the precision of the machine [i.e. to 13decimal places)

Since it is possible for this system to become linearat relatively low energies (about 12000 cm- abovethe vibrational ground state), it was necessary to usefunctions for the scattering length, r, , which behavedcorrectly at r,, which behaved correctly at r,=O.Spherical oscillator functions of the type21/2j?4N,,,+,,2 exp( -~x)x(~+)~~L~+~~~(x) ,

x=/!lrBr:, (8)were therefore used, where LE+12(x) is an associ-ated Laguerre polynomial. Henderson et al. [21

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Volume 206, number 5,6 CHEMICAL PHYSICS LETTERS 7 May 1993Table 1A comparison of A coefficients in s-t. The top row indicates who did the work and the second row gives the dipole surface used. Vibrs-tionsl states are assigned as (II,, vi)Transition

(0,0)+(0, 1)(0,0)+(o,22)(0,0)+(1, 1)(o,O)+(l,Oo)(0, 0) + (0,2)(0, 00)+-(2, 0)(0, l)+(O, 2)

Frequency CP [ 151 DMT [ 161(cm-) CP [ 151 MBB [l]2521.3 124.0 I28.84991.4 174.0 144.65553.7 1.5 0.33118.3 forbidden4777.0 forbidden6262.0 forbidden2476.1 253.0 256.0

This workMBB128.9ll28.9149.4J154.7

0.4/0.51.2x 1o-S1.2x 10-37.6x lo-

255.71256.8

MBB (corrected)12a.afl28.8149.3/149.3

0.4/0.41.8X lo-2.6x 1O-91.2x 10-l)

255.4f255.4

ref. [22]120.5/120.5161.51167.5

16.6/16.65.2 x IO-3.8x10-96.8 x lo-

218.6f218.6

Table 2Assignments of the upper levels of the bright transitions observed in fig. INo. of quanta inhorseshoe mode

State No.(even/odd)

Frequency (cm-)(even/odd)

A coefficient to ground state (even/odd) (s-l)MBB ref. [22]

1 211 2521/2521 128.8/128.82 5/2 4997/4997 149.31149.33 a/4 7003/7003 16.5/16.54 14/a 9108/9108 6.5516.555 21/13 10853/10853 2. 17/2.176 28118 12294112294 0.67fO.677 38/25 13681/13681 0.56/0.568 50133 15103/15103 0.29fO.299 66/47 16712/16712 0.13/0.13

10 86164 18432/18431 6.0x 10-/6.0x lo-11 115187 20238/20238 2.5x 10-*/2.5x IO-*12 148/115 22115/22115 1.1x10-~/1.1x10-~13 1931155 24034/24033 7.7x 10-3/7.7x 1W14 2441198 25912125912 3.3x 10-/3.3X10-15 308/256 27803/27803 1.8x 1O-3/1.8x 10-l16 383/324 29707/29707 8.1~10-~/8.5~10-~17 4701403 31551/31548 8.6X 10-4/8.3x lo-18 567/491 33301/33297 3.5x 10-4/5.3x 10-419 682/597 35036/35026 2.3x 1O-4/3.5x 1W20 819/722 36757/36737 8.5x 1O-5/9.7xlO-5

120.5/120.5167.5/167.531.9f37.912.6/12.65215.21.411.41.6/1.60.96/0.960.23/0.230.14/0.146.4x 1O-2/6.3x1O-22.0x 10-2/2.0x 10-Z1.4x10-~/1.4x10-*5.6x 1O-3/5.7~1O-33.7x 10-/3.7x10-31.3x10-3/1.4x10-31.3x10-/1.2x10-4.7x 10-4/7.3x10-3.0x 10-4/4.7x lo-1.1 x10-4/1.3x10-

found that in order to obtain the best convergence,Q!needs to be 0 and I for the even and odd blocksrespectively. The metrics for these two sets of func-tions are e -xx12 and e-Xx3f2 neither of which canbe represented as a linear function of the other, andso a mutually satisfactory quadrature scheme mustbe found using the theory outlined above. The sim-plest method is to search for a quadrature schemebased on associated Laguerre polynomials LY, (whichhave a metric eBXxY). Using the theory outlined in

the previous section it is clear that x-~ and x3/2-Ymust be whole powers of x and hence that y- f mustbe zero or a negative integer. However, since asso-ciated Laguerre polynomials are only defined in termsof y> - 1, y can only by + f. We have chosen to usey= f The value ofM should be chosen to ensure thatall the orthonormality integrals implicit in the DVRcalculations are still evaluated correctly under thenew quadrature scheme. In this case, this means us-ing at least one more point than was used in the orig-

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Volume 206, number 5,6 CHEMICAL PHYSICS LETTERS 7 May 1993

inal DVR calculations. We have tested the stability the Jensen dipole surface is less good: A coefftcientsof the new quadrature scheme by doubling the num- for the latter are significantly larger than the valuesber of points used. The results were exactly the same, used in the figures (see table 2). Nevertheless, theindicating that it is well-behaved. striking structure shown in the figures remains.

Once a satisfactory quadrature scheme had beendevised, A coefficients were calculated using the DVRwavefunctions and the dipole surfaces of Meyer,Botschwina and Burton (MBB [ 11) and of Jensenand Spirko [22]. Table 1 presents the results fortransitions between low-lying states, and comparesthem with calculations of some other workers. Onlythe calculations by Camey and Porter (CP [ 15 ] ) didnot use Eckart axes (at least explicitly), but it is clearthat this approximation does not significantly affectthe results. The calculations by Dinelli et al. (DMT[ 161) used the same MBB surfaces as were used toproduce the results in column 5, but the wavefunc-tions were produced,using the FBR (finite basis rep-resentation). The sets of two numbers in columns 5,6 and 7 result from the symmetry breaking of ourcalculations - if the upper state is of E symmetry,then two separate A coefficients may be calculated.This demonstrates a problem with the publishedMBB dipole surface as it stands: it does not have thecorrect symmetry for H,f , resulting in the disagree-ment between the sets of coefficients quoted in col-umn 5. Once the dipole surface had been corrected[23 1. by replacing equation ( 12 ) in the MBB paperby[d,, &I= n&L,&~~m+k, I

4. ResultsIn tig. 1 the A coefficients for transitions to the

ground state ar e plotted as a function of band origin.The A coefficients are plotted on a log scale in orderthat the relatively low coefficients for the high-lyingstates may be seen. It is clear from this figure that theA coefficients drop off much more slowly than isusual for other molecules, which goes part way to ex-plaining why there has been so much success latelyin observing up to Au=3 transitions (see, for ex-ample, ref. [ 24 ] ) . The A coefficients which drop offapproximately in a log fashion with frequency aredue to transitions from E states, which are symmetryallowed. Transitions from A states, which are sym-metry forbidden, contribute to the gradually risingscar of A coefficients. It would appear that around30000 cm- the A and E states gradually become in-distinguishable from each other in the main. The in-teresting feature to note from this graph is the reg-ular pattern of transitions with higher than averagebrightness, spaced about 1800 cm- apart. Inspec-tion of the bright states shows that each peak is as-sociated with a horseshoe state. Only the horseshoeswith 14-20 quanta show clearly in this figure, but it

X[cosk+, (-l)k-sink$], (9)the agreement is much better as demonstrated by theresults in column 6.

The results presented in section 4 have all beencalculated using the MBB dipole surface and DVRwavefunctions calculated from a Hamiltonian of di-mension 6500. Comparisons with results obtainedfrom the Jensen dipole surface (see table 1, column7 and table 2) and with results from DVR wave-functions calculated from a Hamiltonian of dimen-sion 6000 show that the salient features of figs. 1 and2 remain. For example the root-mean-square per-centage difference, once the very small A coefIicients( c lo-*) have been excluded, between the 6000 and6500 results is 11%. Absolute agreement between re-sults calculated using the MBB dipole surface and

Fig. 1.Einstein A weffkients for transitions to the ground vibra-tional state, plotted as a function of upper state band origin inwavenumbers. Note the log scale used for the A coefficients.

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Volume 206, number 5,6 CHEMICAL PHYSICS LETTERS 7 May 1993is possible to pick out a full sequence of horseshoestates (see fig. 3 and table 2)) extending all the wayfrom below linearity (where they may be assignedconventionally) to above dissociation, merely bysearching for A coefficients that are larger thanaverage.

Fig. 2 shows similar plots to that in fig. 1, but us-ing other low-lying states as the lower state in thetransition. It is interesting to see that the variationsin brightness which showed so strongly in fig. 1 grad-ually decrease as more quanta of stretch are put intothe lower state (figs. 2a, 2c and 2d), until by the timethat there are three quanta of stretch in the lowerstate, it has all but disappeared. When the lower stateis of E symmetry, as in fig. 2b, all transitions aresymmetry allowed, and hence the scar due to thesymmetry forbidden A++A transitions apparent inthe other plots is missing in this plot.It would thus appear that the strong transitions are

to high bending overtones. These are already knownto be strong for the second to fourth overtones [ lo].Exciting the stretch turns these transitions into dif-ference transitions and greatly reduces their strength.

Because of vibrational angular momentum, thereare several horseshoe states with similar modelstructures even in an unperturbed model. Not all ofthese have E symmetry. Furthermore, the couplingto the bath also causes the horseshoe scar to be spreadover several states. naming a particular state as thehorseshoe state is therefore largely a subjective mat-ter. The states plotted in fig. 3 are actually the E stateswith the strongest horseshoe character. As can be seenfrom fig. 4, the presence of a horseshoe scar corre-lates well with a high A coefficient.

Wavenu mber /cm- Wavenu mbsr /cm4Fig. 2. Einstein A coefficients for transitions to low-lying vibrational states: (a) to the (1, 0) state; (b) to the (0, 1) state; (c) to the(2,@) state and (d) to the (3,O) state. Note the log scale used for the A coefficients. The horizontal scale is again the band origin ofthe upper state in wavenumbers, making comparisons between the various graphs slightly easier.434


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Volume 206, number 5,6 CHEMICAL PHYSICS LETTERS 7 May 1993

Horreshoe 6

k LO 2023&4cm-

Hormwboe 24807.4cm~ Hormahoe 4@WUkm Horrerhw 6lOtl5S&&

Horseshoe 71366lD.9CUl~

Horsmhoe 6i51oa.icm+ Hormahoe 9167lZlcm~ Homeshoe 10la43l.lkm~

Horwahor l2 Hormahoo 131 Eorwrhoe 141

r2/aoFig. 3. Contour plots of the upper (even) states of the bright transit&s observed in fig. 1 in scattering coordinates, with 6 fixed at 90(plots of cdd states in this configuration re meaningless ince hey are defined to be zero). The band origin of each state is given. Notethat the classical turning point allows the possibility of linear configurations (rz=O) from horseshoe 5 onwards. On reflecting throughthe r, = 0 ordinate, the horseshoe epithet becomes plain.

s, , I

Fig. 4. Even eigenstates in the immediate vicinity of the hor-seshoe with 16 quanta in the bend, plotted in scattering coordi-nates with 0 fixed at 90 (plots of odd states in this configurationare meaningless since they are defined to be zero). The A ax!&cient for the transition to the ground state is also given. States383 and 385 are brighter than states 384 and 386 and have hor-seshoe character (state 383 is given in fig. 3 and table 2 as thehorseshoe).

5. ConclusionsAlthough the potential and dipole surfaces used in

this study are not absolutely accurate at energies neardissociation (in particular the MBB potential isknown to dissociate to too high a dissociation limit,and the dipole surface behaves incorrectly at ge-ometries where all three atoms are well separated),we feel that the features we have observed are trueto the nature of HT. In other words, we believe thatthere is a highly harmonic series of bright levels re-sulting from bending motion of the quasi-linear mol-ecule, which provide a gateway in intensity to thesurrounding bath of (normally) dark states. We areconfident that given enough sensitivity, these tran-sitions could be observed.

AcknowledgementThis work was produced on a IBM RS/6000 work-

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Volume 206, number 5,6 CHEMICAL PHYSICS LETTERS 7 May 1993station and the Convex C38400 at the University ofLondon Computer Centre. We thank N.G. Fultonfor providing the program with which the contourplots in figs. 2 and 4 were produced. We are gratefulto the SERC for funding this research through grantsGR/GO1874 and GR/H41744.

References[1 ] W. Meyer, P. Botschwina and P.G. Burton, J. Chem. Phys.84 (1986) 891.[2] J.R. Henderson, J. Tennyson and B.T. Sutcliffe, J. Chem.Phys. ( 1993), in press.[3]P. Drossart, J.-P. Maillard, J. Caldwell, S.J. Kim, J.K.G.Watson, M.A. Majewski, J. Tennyson,S. Miller,S. Atreya,J. Clarke,J.H. Waite Jr. andR. Wagener,Nature 340 (1989)539.[4] A. Canington and RA. Kennedy, J. Chem. Phys. 81 (1984)91.[ 5)A. Canington, LR. &Nab and Y.D. West, J. Cbem. Phys.98 (1993) 1073.[6] J. Tennyson, 0. Brass and E. Pollak, I. Chem. Phys. 92(1990) 3005.[7] J.M. Goma Llorente and E. Pollak, J. Chem. Phys. 90(1989) 5406.[8] R.M. Whitnell and J.C. Light, J. Chem. Phys. 90 (1989)

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[ 91 I. Tennyson and J.R. Henderson, J. Chem. Phys. 91 ( 1989)3815.

[ IO] S. Carter and W. Meyer, J. Chem. Phys. 93 (1990) 8902.[ 1 ] J.R. Henderson and J. Tennyson, Chem. Phys. Letters 173(1990) 133.[ 121P.N. Dayand D.G. Truhlar, J. Chem. Phys. 95 (1991) 6615.[131Z. BatiCand J.Z.H. Zhang, Chem. Phys. Letters 184 (1991)513.[141C.R. L-eSueur, S. Miller, J. Tennyson and B.T. Sutcliffe.Mol. Phys. 76 (1992) 1147.[IS] G.D. Camey and R.N. Porter, J. Chem. Phys. 65 (1976)3547.[161B.M. Dinelli, S. Miller and J. Tennyson, J. Mol. Spectry.153 (1992) 718.[171A.S. Dickinson and P.R. Certain, J. Chem. Phys. 49 (1968)4209.[181R.N. Zare, Angular momentum (Wiley-Interscience, NewYork, 1988).[191E.B. Wilson Jr., J.C. Decius and PC. Cross, Molecularvibrations (Dover, New York, 1980).[20] J.R. Henderson, C.R. Le Sueur and J. Tennyson, Comput.Phys. Commun. (1993), in press.[ 21 M.J. Bramley, J.R. Henderson, J. Tennyson and B.T.Sutcliffe, J. Chem. Phys., submitted for publication.

(221 P. Jensen and V. spirko, J. Mol. Spectry. 118 ( 1986) 208.[231 W. Meyer, private communication.[241 S.S. Lee, B.F. Ventrudo, D.T. Cassidy, T. Oka, S. Miller andJ. Tennyson, J. Mol. Spectry. 145 (1991) 222.

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