jonathan tennyson, steven miller and brian t. sutcliffe- beyond ro-vibrational separation

8/3/2019 Jonathan Tennyson, Steven Miller and Brian T. Sutcliffe- Beyond Ro-vibrational Separation

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J . Chem. Soc., Furuday Trans. 2, 1988, 84(9), 1295-1303

Beyond Ro-vibrational SeparationJonathan Tennyson* and Steven M iller

Department of Physics and Astronom y, University College London, Gow er Street,London WClE 6B TBrian T. Sutcliffe

Department of Chemistry, University of York, Heslington, York YO1 5 D DResults are presented for nuclear motion calculations on D: H z D + and thevan der Waals compiex ArCO in rotationally excited states. These calcula-tions are performed using a two-step variational procedure which allowslarge ro-vibrational interactions (Coriolis couplings) to be treated accurately.The difficulty of assigning states in systems such as HzD+where the Coriolisinteractions are large is illustrated and the limitations of effective Hamil-tonians derived from perturbation theory discussed.

Conventional wisdom, as built up over many years by spectroscopists, is that the nuclearmotion of most molecules can be well understood in terms of small-amplitude vibrationsand near-rigid rotations of the nuclei about some fixed point on the potential-energysurface. Theoretically these ideas are underpinned by the ro-vibrational Hamiltoniandue to Eckartl and Watson,' and by the perturbative analysis of molecular spectra.

Although rotations and vibrations are often treated separately, it is well known thattheir interaction, through so-called Coriolis forces, can be significant. However, formost spectroscopic purposes, it is normally considered sufficient to consider Coriolisinteractions in isolated cases, usually involving accidental or near degeneracy.

In the increasingly lively area of molecular vibrations, to which this meeting is atestimony, theoreticians have,. until very recently, been content to study problems inwhich ro-vibrational interactions were small or even completely neglected.374This stateof affairs was encouraged by the difficulty in performing calculations on polyatomicmolecules with*anything but the lowest levels of total angular momentum, J.

Recently, a two-step variational procedure was proposed5 as a way round thiscomputational bottleneck. This and a similar method6 have now been implementedwith a variety of vibrational procedufes.'-9 Fully coupled calculations on moleculeswith J > 5 are now available for H2D+,5HT,") CH;," ArCO" and H20.699

Experimental studies have also been performed on spectral regions where the usualperturbative Hamiltonians may no longer be valid." Recent studies on f~ rm a ld eh y d e '~and acetylene" have probed regions where ro-vibrational couplings appear to havedestroyed completely the structure of the spectrum.

In this paper we discuss the problems that arise when one begins to explore theregion where rotational and vibrational motion is inextricably coupled. We will do thiswith the aid of theoretical calculations on the H;D+ and ArCO molecules. These providea suitable contrast as the large rotational constants of H,'and, especially, its asymmetrictop isotopomers lead to strong ro-vibrational coupling, whereas Ar-CO is typical ofthe van der Waals complexes for which the Coriolis interactions have often beenignored.3q4

MethodThe methods used to study ro-vibrational coupling effects have to be capable of treatingrelatively high values of the total angular momentum quantum number, J. Because of

1295


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1296 Beyond Ro-vibrational Separationthe linear increase in size of the secular problem with J , conventional one-step variationalprocedures have never been applied to problems with J > 4. The basic idea behind thetwo-step procedures that have been used is that the Hamiltonian of the problem can bepartitioned thus

f i = k v i - k v R + V ( Q ) (1)where the kinetic energy operator has been divided between a vibration-only operatorand a vibration-rotation operator which is null for J = O . The potential, V , dependspurely on the internal coordinates, Q, of the system.

Although there is no unique form for the Hamiltonian of the nuclear motion problem,any body-fixed Hamiltonian canke represented in this way. The idea behind the two-stepvariational approach is that if K V K s only trsated approximately then it is possible todecouple the resulting vibrational problem. The solutions of the decoupled problemsare then used as a basis set to expand the full problem. Clearly if all functions fromthe first step are retained in the second then the basis has merely been transformed.However, the savings come because in the second step not all functions are usuallyrequired to obtain satisfactory convergence and the resulting secular matrix has asimplified structure.

Exactly how this procedure is implemented depends on the nature of the Hamiltonian,i.e. on the choice of internal coordinates and the embedding of the axes. Chen et al.( C M W ) 6 used solutions of the J = O problem as expansion functions for their fullproblem. Tennyson and Sutcliffe (TS)' used solutions of problems for which only theoff-diagonal Coriolis interactions were neglected. In this case k, the projection of Jalong the body-fixed z-axis, is a good quantum number. TS employed a Hamiltonianfor which there is a range of different methods of defining the body-fixed z-axis.16 Thisflexibility is useful as the better the approximation that k is a good quantum number,the fewer the number of functions that will be required for the second step of thecalculation.

The method of CMW has the advantage that only one secular problem for the firststep need be solved for all J of interest. The TS method requires the solution of J + 1such problems for each J ; however, i t incorporates the centrifugal distortion term intothe first step, producing a basis better adapted to the full problem. As calculations inthe very high J regime where centrifugal distortion effects are large are yet to beperformed with the C M W method, i t is not possible to say definitely which method isthe best fo r studying the ro-vibrational coupling. We note that Carter and Handy' haverecently followed TS in that they included the diagonal part of the centrifugal distortionterm in their first-step Hamiltonian.

A recent refinement of the TS method17 is to select basis functions for the secondvariational step according to the energy of the solutions of the first. This gives enhancedconvergence over the usual procedure of using the same number of functions for eachk. This procedure cannot be implemented with the C M W method.

CalculationsIn this section we present the results for two-step calculations on two contrasting triatomicsystems. First, we analyse rotationally excited states of isotopomers of Hl. Thesemolecular ions have been much studiedI8 because of their fundamental nature and therichness of their spectra. Secondly, we consider the Ar-CO van der Waals complex,which is typical of systems for which the ro-vibrational coupling between the levels hasoften been neglected. All the calculations were performed in scattering c ~ o r d i n a t e s ' ~using the program suite SELECT,RIATOM and ROTLEV.")

Table 1 compares results of calculations on D; with previous calculations18q21ndexperiment" for levels with J d 4. As the molecule has D3,,symmetry, special quantum


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Table 1. Calculated rotational term values for D; ( in cm-')ground state V I

level exptl theory theoryJ K s ref. (22) this work ref. (18) ref. (21) this work ref. (18)1 01 12 02 12 23 03 13 23 3 +13 3 -14 04 14 24 3 -14 3 +14 4

~~

43.6 132.32130.59119.3785.62260.47

249.35215.91159.87159.87432.56421.57388.53333.17333.16255.02

43.6032.32130.55119.3485.61

260.40249.29215.86159.83159.83432.4442 1.46388.44333.09333.08254.9 1ban d origin

43.5032.24130.26119.0785.42

259.81248.72215.38159.47159.48420.5 1332.34332.33

43.7032.38

130.84119.6085.80261.06

249.91216.39160.21160.21

42.8 131.72128.20117.1883.99

255.72244.79211.91156.78156.78424.69413.89381.41326.96326.95250.03

2301.25

42.74'3 1.67128.00116.9983.87255.3 1

244.40211.59156.54156.57413.23326.46326.452301

F ( J , G, U = -1) F ( J , G, U = +1)level exp tl theo ry exptl theory

J G s ref. (22) this work ref. (18) ref. (22) this work ref. (18)1 01 01 11 22 02 02 12 22 32 33 03 03 13 23 33 33 44 04 04 14 24 34 34 44 5

+1-1

+1-1

+1-1+I-1

+1-1+1- 1

+1-1

49.7353.40

133.63144.55121.33

258.90280.50247.46212.52

424.86460.35413.88380.57323.67323.71

49.7053.37

133.56144.50121.27

258.78280.42247.352 12.43

424.67460.23413.70380.42323.53323.58

band origin

49.5853.29

133.33144.29121.06

258.34280.09246.952 12.09

424.0 1459.67379.83

43.9112.67

133.42100.3845.65

268.77233.94177.04177.1799.03

448.50412.05354.32354.65274.06

1834.67

43.8912.67

133.38100.3645.6545.67

268.69233.88176.98177.1198.96

448.3841 2.94354.22354.55273.93172.551835.06

43.8212.69

133.19100.2445.6245.70

268.38233.61176.77177.0198.01

412.47

273.781831.07


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1298 Beyond Ro-vibrational SeparationTable 2. C ompar i son of ab initio energies ( in cm- ' ) wi th fitted values fo r H2D+with J = 11"

level ab initio fitted A ass ignment123456789

1011121314151617

2348.93002.43438.73682.34099.74253.24664.34788.45028.35311.15375.55589.95759.15896.25969.76179.96189.2

2355.03005.83430.7367 1.74097.84290.44669.44776.45059.05326.55349.05599.55725.85909.95976.76135.6

6.13.4

-8.0-10.6

-1.937.2

5.1-12.0

30.714.6

-26.59.6

-33.313.77.0

-44.3

v g = 1.00vo = 1.00vo = 1.00vo = 1.00vo = 1.00

v(J= 1 oo

v1 = 1.00v2 = 0.20, v3 = 0.80

~2 = 0.87, ~3 = 0.13

~2 = 0.14, ~3 = 0.86v 2= 0.81, v 3= 0.19

v 2 = 0.72, v3 = 0.18v 2 = 0.38, v3= 0.62V? = 0.7 1 , ~3 = 0.29

1'1 = 1.00(2vd

v2 = 0.40, v3 = 0.60

numbers are required to characterise the rotational levels. Those used here (J, K, s forvo and v l , nd J, G, U, s for v.1 are due to Watson." The present calculations employedthe accurate potential of Meyer et al. (MBB)24and reproduce the observed levels closely.Further details of these calculations, including fits to an empirically motivated perturba-tive Hamiltonian due to Watson23can be found in ref. (25). These calculations havebeen extended to consider the low-lying rotational levels of H l hot bands.26

In tables 2 and 3 we present results for the rotationally excited states of H 2 D +withJ = 11 and 15, respectively. Again the calculations used the MBB potential. To obtainresults converged to ca. 0.1 cm-* it was necessary to include an average of 300 solutionsof the first variational step for each k in the second step of the calculation. This gavesecular problems of dimension 3600 for J = 11 and 4800 for J = 15. The exact numberof functions with a particular k varied between 232 and 340 because the functions werechosen using the energy-ordering ~ r i t e r i on . ' ~hese results form part of a comprehensivestudy of the highly rotationaly excited states of H 2 D + , full details of which will bepublished elsewhere.

Only levels corresponding to the totally symmetric ( e e ) representation are presentedin tables 2 and 3 . The levels are grouped as manifolds corresponding to the number ofrotational levels with this symmetry belonging to the ground and first two ( v 2 and v3)vibrational states.

The levels designated fitted in tables 2 and 3 were not obtained by fitting the datapresented here. Instead they are the levels given using the parameters fitted to theeffective Hamiltonian of Foster et al.*' by Miller and Tennyson.25 These parameterswere obtained by least-squares fits to the calculated levels of H2D' with J G 4 . OnlyCoriolis interactions between the v2 and v3 vibrational states were allowed for in thisfit. Hence the assignments, which were made by matching fitted and calculated energylevels, can only be to pure vo or v l states, or to mixed v 2 / v j states.

The method of assignment by matching a b inirio and fitted energy levels appears tobe reliable for the J = 11 results. Here the differences, A, though quite sizeable, areconsiderably smaller than the spacings between the levels. This is not true for the J = 15levels, whose assignments must thus be treated with caution.


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J. Tennyson, S, Miller and B. T. Sutclife 1299Table 3. Comparison o f ab initio energies ( in c m - ' ) with fitted values fo r H,D'with J = 15"

level ab initio fitted A ass ignment123456789

1011121314151617181920212223

4121.84992.95633.65888.26063.46332.76554.36818.56895.77077.67351.97449.27651.67717.37878.57970.68133.78143.78205.68292.28519.68593.68649.2

4161.75014.55630.35981.16083 O6275.66508.468.06.97023.37139.17245.97476.97626.27834.87980.08000.38214.88215.98366.58620.88706.7

39.921.6-3.398.719.6

-57.1-45.9-11.6127.661.527.7

117.5101.529.781.1

- 106.0-25.4

10.374.327.257.5

vo = 1.00vo = 1.00vo = 1.00v, = 1.00

v 2= 0.89, v3 = 0.11vo = 1-00

v 2 = 0.12, 15= 0.88vo = 1.00

v 2 = 0.59, v3 = 0.41v1 = 1.00v2 = 0.40, v3 = 0.60v, = 1.00v2 = 0.21 , v3 = 0.79v2 = 0.54, v3 = 0.46v 2= 0.41, v3 = 0.59

Vl = 1.00

(24vo = 1.00( v z + v,)?v1 = 1.00

v 2= 0.46, v3 = 0.54

v2 = 0.59, v3 = 0.41

u2=0.83, v3 = 0.17

Table 4. Lowes t 15 levels of the Ar-CO van de r Waals c o mp le x with J = 5' ( f requenc ies a rerelative to dissoc ia t ion of the c o mp le x , s e e text fo r a discussion of th e ass ignments)~~

frequencies/cm-' assignmentlevel no Coriolis full k i k 1

123456789

101112131415

-82.5-80.0-72.9-68.5-62.0-61.9-59.1-55.6-51.9-51.6-48.8-48.1-47.1-42.7-42.6

-82.6-80.0-73.0-68.5-62.0-62.0-59.0-55.7-51.9-51.6-48.8-48.2-47.2-43.4-42.1

0 11 I2 10 21 21 20 31 30 42 20 52 34 10 60 6

1.00I .oo1 oo0.990.41 3 1 0.900.89 3 1 -0.420.980.990.990.990.990.991 oo0.76 1 4 -0.62

-0.55 1 4 0.74


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1300 Beyond Ro-vibrational SeparationTable 5. Lowest 20 levels of the Ar-CO van der Waals complex with J = 10' (frequencies are

relative to dissociation of the complex, see text for a discussion of the assignments)frequencies/cm-' assignment

level no Coriolis full k i k 1123456789

1011121314151617181920

-77.0-74.3-67.3-63.4-56.6- 6.3-54.0-50.5-46.8-46.2-44.1-42.9-41.5-38.1-37.8-36.2-32.1-3 1.6-31.5-30.3

-77.1-74.5-67.5-63.4-57.0-56.6-53.8-51.0-46.8-46.3-43.9-43.3-41.8-39.3-37.0-35.7-32.7-32.3-31.8-29.8

0 11 12 10 21 23 10 31 30 42 20 52 34 10 61 40 61 61 63 30 7

~ ~~ ~~~~ ~~ ~~~ ~

0.990.980.980.980.900.970.950.960.950.940.960.960.980.77 1 4 0.540.65 1 5 0.56

-0.51 1 5 0.73-0.59 3 2 0.67

0.67 3 2 0.640.960.95

Tables 4 and 5 present results for calculations on the Ar-CO van der Waals complexusing the potential-energy surface of Mirsky.** The calculations, which kept the COfrozen at its equilibrium bondlength, were performed as part of a comparison of thequantum and classical behaviour of this weakly bound system. '* These calculationsshow that the highest truly bound rotational state of ArCO is for J = 35 .

In tables 4 and 5, the levels of the complex are given relative to dissociation intofree Ar and CO. The deepest point on the potential, which corresponds to a bentgeometry with the Ar nearer the 0, is bound by 105 cm-'. The J = 0 ground state isbound by 84.6 cm-'. The projection, k, of J along the Ar-CO coordinate is nearly agood quantum number for this system. Thus only very few functions from the firstvariational step, 10 for each k, are needed to obtain converged energy levels. The resultspresented here were actually obtained by including the lowest 80 solutions of the firstvariational step for each k. The extra functions were included to guarantee convergenceof the highest states and because, for this system, the size of the second step of thecalculation had little influence on the computer time used.The Ar-CO levels in tables 4 and 5 are assigned according to the dominant basisfunctions in the second step of the calculation. The ordering of the levels within a givenk manifold is given by i. Thus, within a particular k manifold, i = 1 corresponds to thelowest state, i = 2 the first excited state etc. We note that for J = O , i = 2 is the firstbending excitation with a frequency of 14.3cm-' and i = 3 is the Ar-CO stretchingfundamental which has a frequency of 24.2 cm-' for this potential.For Ar-CO the states are dominated by basis functions with low k. Inspection ofthe no Coriolis calculations shows that the complex has no bound states with k a 7 .


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J, Tennyson, S. Miller and B. T. Sutclife 1301

DiscussionFor low levels of the total angular mom entum there can be no d ou bt that the potentialof Meyer et al.24 nd o ur ro-vibrational m etho ds comb ine to give results of high accuracy.This accuracy is un pre ced ente d in the first-principles calculation of vibration-rotationlevels of polyatom ic systems. Fo r high er rotational levels, where, fo r exam ple, rotationalman ifolds of m any vibrationa l states overlap, th e situation is less clear. This is du e inpart to the absence of observed spectra in this region, although we note that Majewskiet al.29have recently extende d their H; da ta u p to J = 10. However, there remainconsiderable difficulties in interpreting, an d he nce assigning, spectra in the high-J region.From inspection of the levels in tables 2 an d 3, it is clear that these levels are notwell reprod uced by the parameters obtained by fit ting to the lo w -J levels of the samesystem. The standard deviation of the original fi ts were 0.004, 0.01 a nd 0.008 cm-' forvo, v2 plus v3 a n d v,, espectively. These should be co mp ared with s tand ard deviat ionsof 6.5, 31 an d 1 1 cm-' for the J = 1 1 da ta and 45, 84 a n d 42 cm-' for J = 15.These large differences cann ot be ex plain ed solely in terms of the increased imp ort-an ce of centrifugal distortion. Th e original fits inclu ded all centrifugal distortion termsu p to fou rth or de r as well as making so me allowance for sixth-order effects. It is clearthat Coriolis coupling effects also have a large part to play. J = 1 1 is the lowest valueof J for which the rotat ional manifold of th e vibrat ional grou nd state begins to ov erlapthat of a vibrational fundamental. By J = 15 the grou nd-state manifold overlaps thoseof al l the fundamentals .In order to persist with the fit t ing and interpretation of spectra in this region viaeffective Ha m iltonia ns based on perturb at ion theory i t would be necessary to conside rall these ov erlap ping vibrational states simultaneously. While this is technically possiblefor the fund ame ntals , i t unfortunately does no t provide a real solut ion. The reason forthis can be fou nd in the increasing ov erlap of the rotational m anifolds w ith vibrationalexcitation. Th us for J = 15, the v3 rotat ional manifold would ap pea r to be overlap pedfrom above by the rotat ional manifolds of v l , 2 v 2 , v2+v3 an d 2v3 in addi t ion to thestates below it . These complications can only become worse with further rotational orvibrational excitation. This means tha t there is no o bvious poin t where the effect ofhigher vibrational levels can easily be neglected.In view of these difficulties o ne is prom pted to ask 'why attem pt to fit the sp ectrumat al l? ' O ne could ab and on th e interpretation of the levels an d merely stack them onan energy-ordering basis. However, in doing this on e loses the pow er to predict theproperties of spectra of the system. In this situation o ne can no long er estimate whichtransitions are likely to be intense and therefore promising candidates for observationusing propensity rules. Instead one is reduced to having to calculate every transitionintensity explicitly. We ar e currently doin g this for the h igh -J ( J = 10-30) region ofH 2D + n the ho pe that it will reveal insight in to the st ructure of the spectrum.By contrast to the H l systems, van de r Waals complexes such as ArC O ap pea r wellbehaved. O ur tabula ted results for this system show th at even for values of the totalangu lar mom entum considerably larger than tho se previously consid ered, the neglectof off- diag on al Coriolis couplings gives a very good approximation to the exact ro-vibrational energies (although this approximation is known to be poor for transitionintensities3*). However, neglecting these coupling terms can lead to errors when neardegeneracies lead to r eso nan ce interactions. This is i l lustrated by levels 14 and 15 fo rt h e J = 5 c alcu la tio n. ~In A rCO a m ore com plicated situation ap pe ars to arise at higher levels of vibrationalexcitation. Th ere is an increase of interactions between neighbo uring states. Theseinteractions often involve several states, see for example levels 14-16 of the J = 10


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1302 Beyond Ro-vibrational Separationcaiculation. This behaviour is probably not so much due to large amounts of rotationalexcitation but has more to do with behaviour of the underlying vibrational states. Evenfor J = O the vibrational levels in the near dissociation region of the complex form acomplicated manifold. In this region the assignment of quantum numbers, in terms ofAr-CO stretching and bending quanta, is no longer possible. It appears that theselevels are sensitive to the perturbation caused by rotational excitation which results inconsiderable recoupling in the vibrational wavefunction.

ConclusionIn this work we have focussed on two situations where Corioiis interactions cause asignificant perturbation of the spectrum. The first is when the rotational manifoldsbecome larger than the vibrational spacings, causing a general overlap of the levels.For small molecules, this is most likely to be important in systems with large rotationalconstants. The second situation occurs when the underlying vibrations are stronglycoupled and are thus sensitive to being perturbed by ro-vibrational couplings. Thissituation will be found in regions where the potential is significantly anharmonic. It isthis that is being observed in the ro-vibrational spectra recently labelled c h a o t i ~ . ' ~ , ' ~

There is a third situation which can lead to strong recoupling in the rotationalmanifold. This occurs when a bent molecule becomes linear. This situation has recentlybeen explored by Carter and Handy,' who found for water that even for J = 7 therewas strong interaction between levels with different k. They also found that to reproducethe observed spectrum of this system it was necessary to adapt the usual force-fieldrepresentation of the potential-energy surface to enforce the correct behaviour as themolecule becomes linear.

With the exception of the potential due to Schinke et d.," ll the potential-energysurfaces recently developed for the HT system concentrate on the region in the immediatevicinity of the equilibrium geometry. These potentials d o not behave correctly as themolecule approaches linearity, nor, perhaps more crucially, do they allow the systemto dissociate correctly. As there remain many unanswered questions about the dissoci-ation spectra of this system,32 this is a serious deficiency.We thank the S.E.R.C. for the provision of computer facilities and funding for S . M .References

1 C. Eckar t , Phys. Rev., 1935, 47 , 552.2 J . K. G. Watson, Mol. Phys., 1968. 15, 479.3 S. L. Holmgren , M . Waldman and W. Klempere r , J. Chem. Phys. , 1978, 69, 1661.4 J . T e n n y s o n a n d A. van d er Avoird , J. Chem . Phys., 1982, 77 , 5664.5 J. T e n n y s o n a n d 9. T. Sutcliffe, Mol. Phys., 1986, 5 8 , 1067.6 C - L . C h e n , 9. Maes s en and M . Wolfsberg, J. Chem. Phys. , 1985, 83, 1795.7 V. Spirko , P. J ens en , P. R. Bunker and A. C e j c h a n , J. Mol. Spectrosc., 1985, 112, 183.8 V. Spirko , P. J ens en and P. R. Bunker , J. Mol. Specfrosc., 1986, 115, 269.9 S. C a r t e r a n d N . C . H a n d y , J. Chem. Phys. , in press.10 P. J e n s e n a n d V. Spirko , J. Mol. Specfrosc., 1986, 118, 208.11 9. T. Su tc li ff e an d J . Tennys on , J. Che m. SOC.,Furaduy Trans. 2, 1987, 83, 1663.12 S. C . F a r a n t o s a n d J. Tennyson, to be publ ished .13 9. Lemoine, M. Bogey an d J. L . Des tombes , Ch em . Phys. Letr. , 1985, 117, 532.14 H. L. Dai, C. L. Ko rpa , J . L. Kinsey an d R. W. Field, J. Chem . Phys., 1985, 8 2 , 1688.15 R. L. Sun dberg , E. Abram son, J. L . Kinsey an d R. W. Field , J. Chem . Phys., 1985, 83, 466.16 9. T. Sutcliffe and J . T e n n y s o n , Mol. Phys., 1986, 5 8 , 1053.17 9. T. Sutcliffe, S. Mil le r and J . T e n n y s o n , Cornput. Phys. Commun. , 1987, s ubmi t t ed .18 J. Tennys on and 9. T. Sutcliffe, J. Chem. Soc., Furuduy Trans. 2, 1986, 8 2 , 1151.19 J. Tennys on a nd 9 . T . Su tc li ff e , J . Chem. Phys. , 1982, 77, 4061; 1983, 79, 43.20 J . Tennyson, Cornput. Phys. Com mun ., 1986, 42, 257.


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J. Tennyson, S. Miller and B. T. Sutclife 130321 G. D. Ca rne y , Chem . Phys., 1980, 54 , 163.22 J. K. G . Watson, S. C. Fos t er a n d A. R. W. McKellar, Can. J. Phys., 1987, 65, 38.23 J. K. G . Watson, J. Mol. Spectrosc., 1984, 103, 350.24 W. Meyer, P. Botschwina and P. G. Burton, J. Chem . Phys., 1986, 84 , 891.25 S. Miller and J. Tennyson, J. Mol. Specrrosc., 1987, in press.26 S, Mil le r and J . Tennyson, J. Mol. Specrrosc., in press.27 S. C. Fos te r , A. R. W. McKellar, I. R. Peterkin, J. K. G. Watson, F. S. Pan, M. W. Crofton, R. S. Altman28 K. Mirsky, C h e m . Phys., 1980,40, 45; J . Manz and K. Mirsky, Chem. Phys., 1980, 40, 457.29 W. A. Majewski , M. D. Marshal l , A. R. W. McKellar, J. W. C. J o h n s a n d J. K. G. Watson, J. Mol .Spectrosc., 1987, 112, in press.30 J. Tennyson, G. Brocks and S. C. Farantos, Chem. Phys., 1986, 104, 399.31 R. Sc hinke , M. Dupui s a n d W. A. Lester Jr , J . Chem . Phys., 1980, 7 2 , 3909.32 A. Carr ington and R. A. Kennedy, J. Chem. Phys., 1984, 81, 91.

a n d T. O k a , J. Chem . Phys., 1985, 84, 91.

Paper 7/ 1528; Received 17th August, 1987

jonathan tennyson, steven miller and brian t. sutcliffe- beyond ro-vibrational separation

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