James R. Henderson, Hoanh A. Lam and Jonathan Tennyson- Highly Excited Vibrational States of the KCN...

8/3/2019 James R. Henderson, Hoanh A. Lam and Jonathan Tennyson- Highly Excited Vibrational States of the KCN Molecule

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J. CHEM. SOC. FARADAY TRANS., 1992, 88(22), 3287-3293 3287Highly Excited Vibrational Statesof the KCN MoleculeJames R. Henderson, Hoanh A. Lam and Jonathan TennysonDepartment of Physics and Astronomy, University College London, London, W C lE 6BT UK

Vibrational (J= 0) states for the KCN molecule are calculated in Jacobi coordinates, employing a discretevariable representation (DVR) for the angular interna l coordinate. The power of the DVR method is once againillustrated in that some 800 vibrational (J= 0) states are converged for a two-dimensional potential-energysurface. The energy region studied is that where the classical dynamics of the system are known to be chaotic.Most of the states are found to be irregular, although there are classes which appear to be regular and can beassigned effective quantum numbers corresponding to excitation in particula r 'sepa rable' modes of the system.Phenomenological aspects of the wavefunctions are discovered via graphical analysis, in particular many lin-early localised states are identified where the potential-energy surface actually has a saddle point. The sta-tistical behaviour of the level spacings is also investigated. Comparison with a similar study on LiCN (J. R.Henderson and J. Tennyson, Mol. Phys.,1990,69,639) is made.

In recent years experimental advances have initiated interestin the high-lying rotation-vibration states of small molecularsystems. The problems involved in the rather demandingtheoretical investigations have been alleviated somewhat bythe recent introduction of novel techniques. A most usefulmethod for very high-lying states is the discrete variable rep-resentation (DVR) originally proposed by Harris, Engerholmand Gwinn.' The DVR has been extensively developed byBaEii: and LightY2 nd has seen further development and useby Tennyson, Henderson and co-~orkers.~-~he methodhas proved to be very useful in dealing with small molecularsystems which can execute highly anharmonic and largeamplitude motions, and which may have a potential-energysurface with more than one minimum. Such systems areusually regarded as being 'floppy'. The alkali-metal cyanidesL E N , NaCN and KCN are known t o be of this type.The dynamics of KC N are known from both the liquid7and ~rystalline'*~hases to be particularly interesting andunusual. This is because of the facile transition between theCN group being rigid and undergoing free rotation.

In the gas phase, KCN is to our knowledge unique in thatclassical calculations predict the onset of classical chaosbelow the quantum ground state of the system." These cal-culations were also unusual in that the coupling between theK-CN bending and stretching modes was so strong that noquasiperiodic trajectories were found above the classical tran-sition to chaos.",' ' This behaviour should be contrastedwith that of LiCN in which the Li-CN bend and stretchshow much weaker coupling. In this case, the transition toclassical chaos is at much higher energy and quasiperiodictrajectories can be found all the way to dissociation.12 Thisbehaviour is also reflected in the quantal states of thissy ~t em .~ .' Conversely, limited quantal calculations onRbCN suggest that it will display behaviour similar to that ofKCN.I3

In the gas phase, the observed spectrum of KCN has beenunderstood only in the microwave region.l4l' Initial studiesconcluded that the KCN system has a linear equilibriumgeometryt4 as in LiNC and HCN, although now it is knownto have a more or less T-shaped structure."

Theoretical investigations on the rotation-vibration behav-iour of the KCN molecule have been made by severalworkers,".' 6-18 but only at low energies, encompassing thefirst 35 states. The main reason for this is that the classicaldescription for this system predicts a highly chaotic behav-iour at very low energy indeed, and it is then usual for thecorresponding quantum studies to prove difficult.

The main aim of this study is to provide further details ofthe interesting dynamics of the KCN system. Comparisonsare made with our study of the LiNC-LiCN ~ y s t e m . ~CNand LiCN are distinctly different because their equilibriumgeometries are not the same (LiCN and LiNC are both linearminima on the potential), and, as noted above, the angular-radial coupling in KCN is very much stronger than in LiCN.The difference in mass between the potassium and the lithiumsuggests investigation into their relative (semi-)classicaldescriptions.

The following section gives a description of the potential-energy surface used, the calculations performed and anaccount of the method. The Discussion section presents ourresults as tabulations of energies and contour plots of variouswavefunctions. The final sections then give a discussion of theinteresting and phenomenological aspects of the results andconclusions.

Calculations and ResultsThe potential-energy surface used in this work is that due toWormer and T en n y ~o n ,~ ~nd is shown in the form of acontour plot in Fig. 1. This potential was calculated at theSCF level for a fixed value of the CN bond length of2.186 a, .t The SCF calculations used an extended basis of 98GTOs: (1 s,6pY2d/6s,3p,2d)on C and N, and (2Os,15p,ls/l Is,9p,ld). While vibrational frequencies are not usually rep-resented accurately by the SCF approximation, this model isappropriate for the highly ionic KCN molecule, where theinteraction is between closed-shell CN - and K + fragments.As can be seen, this potential predicts the T-shapedminimum, in reasonable agreement with e~ pe ri me nt .' ~

The calculations presented here were made in 'scattering'or ' acobi' coordinates. These coordinates are given by rCNfixed at 2.186 a,, R , the distance of K to the CN centre ofmass, and 8, he angle between rCN and R . Linear KNC willthen be located at 180" and linear KCN at 0". t can be seenfrom the potential-surface plot that the equilibrium configu-ration is given by 8 x 105" and R - a0 , with saddle pointsto either side at the linear configuration.

It would be well within the capabilities of our method toinclude the motion of the CN bond; however, there is no fullthree-dimensional potential available for this system. This isnot surprising since any electronic structure calculation

t 1 a, = 5.29177 x lo-" m.


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32887.00

6.50

6.00

5.50ct

5.00

4.50

4.006/degrees

Fig. 1 Contour plot of the potential-energy surface for KCN. Thevertical axis represents R, the K-CN centre of mass distance, and thehorizontal axis represents the angular coordinate 6. Note that thepotential minimum occurs approximately at R = 5.0a, and 8 = 105"indicating that this system has a T-shaped equilibrium geometry. Thecontours are marked with their appropriate energies in E, (1 E, x4.35975 x J)

Table 1 Details of the radial basis functions used

TS" 5.13 0.001 18 0.173this work 5.50 o.Oo094 0.178r e , w eand D, are give in au." Ref. (18).

J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88hoping to treat the CN stretching coordinate reliably wouldbe forced to use a level of correlation which would make theinclusion of a K atom in the calculation prohibitively expen-sive.

We would argue that freezing the CN bond is reasonableon physical grounds. Wormer and Te n n y ~ o n ' ~erformed anumber of calculations testing the effect on their optimizedCN- bond length of placing a K+ on at several geometries.They found that K CN behaved like an atom-diatom van derWaals complex in that the coupling between the 'intra-molecular' CN- coordinate and the other 'intermolecular'coordinates was weak. Furthermore, the CN- has a high fun-damental stretching frequency relative to the other modeswhich will act t o weaken the coupling further.

In this work, the angle 8 is represented in a DVR, with theremaining coordinate, R , being treated in the more conven-tional finite basis representation (FBR). This particularhybrid method, in conjunction with these coordinates, hasproved successful on a number of other molecules, includingL EN 3 and HCN,20,21where angular-radial aniso tropy isknown to be'high. H3+,4 SiH z,2 2 Na, and H,O 23 havealso been studied successfully using our program DVR1D.24

The solution algorithm then is first, to make the angularcoordinate discrete in point space, whilst solving the radialcoordinate in the function space of the FBR. This means thatfor each discrete angle one diagonalises a one-dimensional(1D) Hamiltonian in the coordinate R . A selection criterion isthen imposed and the N lowest 1D radial solutions areretained. These N eigenvectors are then used as a basis inwhich to couple the angular coordinate and solve the fullproblem. The angular DVR points were defined by making adirect transformation of the Hamiltonian from a Legendrepolynomial basis based on Gaussian quadrature. The radialbasis uses the Morse-type functions of Tennyson and Sut-cliffe. The functions are parametrised by three parametersI, , D, and o, which were optimised variationally at thebeginning of the investigations. The previous values used byTennyson and Sutcliffe" were optimised to be appropria tefor the low-lying levels. We took these and re-optimised themto be suitable for describing the high-lying levels, makingsure that they could still give an adequate representation forthe lower levels also. Both sets are shown in Table 1.

The final calculation converged the lowest 80 0 vibrationalstates for J = 0 to within 6 cm-'. The first 40 0 levels areactually converged to within 0.1 cm-'. This required 46radial functions in R (NR= 45 in the code), with 53 integra-tion points and 72 DVR points in the angular coordinate 8( N o= 72). The lowest 1800 intermediate solutions were selec-ted, to construct the final Hamiltonian matrix, from a

Table 2 Convergence of the KCN band origins as a function of parameters used in the calculationsNR 56 56 40 45 50 56 56 56Ne 62 72 90 90 90 80 80 80ERA, 13866 13866 14042 13950 13866 12613 13866 15231L 1455 1661 1870 1870 1870 1578 1700 1800level400 4494.2 4494.2 4494.2 4494.2 4494.2 4494.2 4494.2 4494.2500 5089.2 5089.2 5089.4 5089.4 5089.4 5089.2 5089.1 5089.1600 5646.6 5646.6 5647.4 5644.3 5643.9 5646.5 5646.0 5645.9650 5905.7 5905.7 5908.4 5904.8 5904.3 5905.7 5904.0 5903.4700 6162.2 6162.5 6185.3 6163.4 6157.5 6162.5 6158.5 6157.6750 6425.1 6426.0 6448.7 643 1.6 6429.1 6426.0 6411.5 6409.2800 670 1.4 6701.5 6748.8 6726.2 6717.5 6701.4 666 1.4 6656.2

NR gives the number of Morse oscillator-like functions used for the r2 coordinate. Ne gives the number of discrete points used in the angularcoordinate, 6. ERAY gives the cut-off energy for solutions of the 1D radial problem in cm-' relative to the KNC minimum of the potentialyielding a final Hamiltonian matrix of dimension L. All band origins are given in cm- ' elative to the KNC ground state. Comparison of levelsbelow level 500 showed them all to be converged to within 0.1cm-' by the calculations presented here.


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J. CHEM. SOC. FARADAY TRANS. , 1992, VOL. 88Table 3 Assignments for regular states localised about the K CN minimum

3289

assignmentlevelno. frequency/cm - ' VS '

12345678910

1 11314151718192023

0.0116.1217.7294.3314.5379.5420.4448.3494.7530.2574.5623.6661.0688.9739.2781.3788.7838.6892.0

0000000001100010002?

0123?345?566?77?8998?10lo?lo?

assignmentlevelno. frequency/cm - ' vs '242526272830313334353839404346545968

920.6949.6973.01008.61017.71068.11082.11126.61 139.71173.61235.21251.61261.81329.41465.31508.11606.81724.3

l?l?0000200010100111

8?lo?121 1l l ?138?12?1313?1314141614151616

States are assigned by inspection of the wavefunction (see Fig. 2) quanta of the K-C N stretch, vs , and bend, v b States for which the nodalstructures are greatly distorted are denoted by ' ?'.

maximum of 46 x 72 = 3312. This selection criterion isequivalent to selecting all those solutions that lie below38929 cm-'. Table 2 illustrates the convergence of a selec-tion of levels as a function of the various parameters.We find it very useful to make contour plots of the wave-functions. The main advantage of this technique is that itimmediately allows an identification of excitation in a partic-ular mode, coordinate or direction, and hence can allow(tentative) assignment of quantum numbers. In this context,we refer to a particular state as being irregular if no approx-imate quantum numbers can be assigned in this way. We alsochoose to plot, with an outer dotted contour, the classicalturning point for the particular state being plotted. It is thenseen that an irregular state will, in general, fill up the whole

state 38state24 1

0 180 0B/degrees eldegrees 180Fig. 2 Contour plots of four typical KCN states localised about theequilibrium. The axes are as in Fig. 1 . Contour lines are drawn at 4,8, 16, 32 and 64% of the maximum amplitude of the wavefunction.Those w hich are solid represent positive amplitude, whilst the dashedrepresent negative amplitude. The dashed outer contour representsthe classical turning point for that particular energy

of the coordina te space available to it. This method was usedby Tennyson and Farantos for the lower-lying states ofKCN," and by Henderson, Tennyson and co-workers for thehigh-lying states of other For instance, thisgraphical analysis helped to confirm the existence of thequantum analogue of the so-called 'horseshoe' periodic orbi tof H3+predicted by classical mechanic^.^

As the equilibrium geometry of KCN is approximately T-shaped, one may expect a number of states to exist about thisequilibrium. We find ca. 70 such states, all lying below thehigher of the two linear saddle points, of which we are abletentatively to assign ca . 36. These are tabulated in Table 3,with contour plots of some of them shown in Fig. 2.

Above the two saddle points the majority of the statesobserved in this work are irregular in nature. Fig. 3 shows aselection of these. As the KCN molecule is not linear, onemay not necessarily expect to find linear bound states as

0$ 6

4

' o 180eldegrees O/degrees

0Fig. 3 Contour plots of four typical irregular, or delocalised states.Contours as in Fig. 2


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3290 J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88

0$ 6

4

0$ 6

4111111111111111111110 180 0 - 180eldegrees eldegreesFig. 4 Contour plots of four typical saddle-point states localised atthe linear KNC geometry of the molecule; contours as in Fig. 2

these geometries are actually saddle points on the potential-energy surface. We do observe these states, however. One canactually think of them as localised isomeric transition states.Contour plots of some of these states are shown in Fig. 4 and5 . Tabulations of their energies, and approximate assign-ments, are given in Tables 4 and 5 . We believe that this is anovel observation of a series of quantum bound states that

02 6

0 180 0 180eldegrees OldegreesFig. 5 Contour plots of four typical saddle-point states localised atthe linear KCN geometry of the molecule; contours as in Fig. 2

are not localised about a potential-energy minimum.The possibility of KCN free-rotor states, where the K +

'freely rotates' with high orbital angular momentum aboutthe CN- ion, was first discussed by Clementi et Suchstates are found to be very pronounced in KCN, existingright across the energy range considered here. A selection ofthese very distinctive wavefunction patterns are shown in Fig.

Table 4 Assignments for linear KCN statesassignmentlevel

no. frequency/cm- v s '818284858687109110112113139140143148171

20521523924024 1258277278296312316317339349355356360

1920.021932.851954.231968.801980.021987.092237.462255.292270.542273.152544.8 12548.262573.342628.802834.683120.28321 1.063392.763400.673405.713531.533675.793677.023808.033911.193949.649955.794102.464165.744210.984215.634233.13

00000011112222344555566677777888

02?3?2?03?000l ?00l ?10017100000110012010

assignmentlevelno. frequency/cm- v s '36 139840 5438440446449486490492538588590623626629638642643684698700702705709716728736741764788792

4242.294480.234519.044725.514741.704770.994784.725009.695027.705043.735302.425574.055588.345764.425776.965795.195843.415861.255862.786081.576147.546157.666169.876185.676206.026241.056302.906336.156366.396483.836600.5 366 18.08

898101010101111111213121314141414141415151515141515?1415151615

0010003?2?4?2?03?23?15 ?1104?6?000203?3?2?12?3

States are assigned by inspection of the wavefunction (see Fig. 5 ) quanta of the K-CN stretch, v, , and bend, v,, . States for which the nodalstructures are greatly distorted are denoed by ' '.


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J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88Table 5 Assignments for linear KNC states

3291

assignmentlevelno. frequency/cm- vs '16293647699210111812114915115918118318421822022 125725929829930433 1338383384427428444

729.831043.641182.991404.781744.8 12037.792 150.452329.072353.472633.622650.922725.792919.892934.082944.663225.393238.283243.433524.073539.833820.213824.313861.744049.004100.624388.534390.354664.124670.244764.43

122345566777788999101011111112121313141414

l ?03?000300004000000000010000300

assignmentlevelno. frequency/cm- vs '47247447652252357 1582594618619620622670672674695701708715722723720734770775776775783793

4939.764946.824955.675214.97522 1.975487.345549.365607.685742.775746.145452.405759.316012.496020.026024.236128.876168.616203.346234.496276.846279.696311.146332.846514.556537.746539.746540.186563.066630.74

15151416161713171818?18?1819191919?19182020201819212121212019

3?02?10012?2?0l ?023?3?24?3l ?104?2102104

States are assigned by inspection of the wavefunction (see Fig. 4) quanta of the K-CN stretch, v,, and bend, vb . States for which the nodalstructures are greatly distorted are denoted by ' '.6. Free-rotor states are also found to exist with a quantum ortwo of stretching excitation. Their assignments and energiesare given in Table 6.Fig. 7 shows a family of states exhibiting a totally unpre-dicted type of state, not observed in the LiCN s t ~ d y . ~hiswill be discussed in the next section.A useful technique for analysing quanta1 energy levelswhose classical analogue is chaotic is the plotting of nearest-neighbour level spacing distributions. It is said that for aselection of levels that behave regularly, the nearest neigh-bour spacings will tend to a Poisson distribution,26 and for

8 - 7 1

$6cr

4I I I I I . 1 1

s t a t e d 11 state671 {02 6

180 180eldegrees O/degreesFig. 6 Contour plots of four typical free-rotor states; contours as inFig. 2

irregular levels to a Wigner di~tribution.~~ig. 8 shows fourhistograms, one for each equal window on the 800-staterange. As can be seen, the general trend is increasinglyWigner-like as the energy increases.

DiscussionA large amount of qualitative information is obtained fromthe graphical analysis, in particular the nodal structure of thewavefunctions. For the first time, we have been able to inves-

L I 1 I I 1 I I I l I I 1 1 1 1 I I I 10 180eldegreesldegrees

Fig. 7 Contour plots of four typical 'hingeing' states; contours asin Fig. 2


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3292 J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88Table 6 Assignments for the free-rotor states

assignmentlevelno. frequency/cm- v , 'b80889611712015016016117017719921023 1250265268280293307

318319332335344347351365377379380393

1903.832005.222094.0923 16.102341.122645.292732.333747.262818.692886.873070.3931 52.893326.463470.363584.393603.193694.473788.883883.0139 62.9 23963.504055.184076.664132.744153.324 174.894273.074347.78437 1.683473.1 16680.23

01000 .00001111010000000001000001

192222262632?32303231?3435?37?4141?4142?4546454744484846495050?515150?

assignmentlevelno. frequency/cm- vs 'b39740941342343044144746348249549951653453655155457359360561262563265 167 169 1713732754773789799

4473.624546.564575.024643.274675.294745.264776.96487 8.48498 1.765058.345084.715185.585287.505290.885277.875391.625495.705599.675670.375703.525776.225807.725907.99601 5.936117.206224.156325.25643 1.396533.996604.356649.08

000103?00O?2?00004?000022000003?0021

52505348?5454?555657545859546060?616263586464656667686969?70?70?7073

States are assigned by inspection of the wavefunction (see Fig. 6) quanta of the K-CN stretch, v , , and bend, vb . States for which the nodalstructures are greatly distorted are denoted by ' ?'.

tigate the behaviour of this system above the two saddlepoints a t linearity.Table 7 shows previously calculated (performed in FBR)

levels of KCN compared with the low-lying ones of thisstudy. Note that the previously calculated levels all lie belowthe two saddle points. This illustrates nicely the fact thatFBR-based methods have been adequate in the past fordealing with these fairly low-energy, mode-localised motionsbut that the DVR now allows one to investigate the moredelocalised behaviour at very much higher energies where thesystem may be classically chaotic or new types of quasi-periodic motion may arise.

Families of regular states are identified which are assign-able in terms of zeroth-order Hamiltonians in various modes,

0.20S 0.15B2 0.10e>

0.05

2 0.15m 0.10n& 0.05

.--0

I I 1 I I 1levels 1-200 I levels 200-400t 1

t 1

0.000 10 20 30 10 20 30 40level spacing/cm-' level spacing/cm-'

Fig. 8 Nearest-neighbour level spacing distributions for fourwindows, each of 200 states

some of which are also observed in the LiCN m~lecule.~Note that the number of 'localised equilibrium' states thatare unassignable is rather large compared to that for L E N ;this could be anticipated because of the unusually low onsetof classical chaos in KCN combined with the strong modecoupling." In fact we note that these 'harmonic normalmode' states centred about equilibrium do not exist abovestate 68, whereas in LiCN-LiNC they appear at very highenergies, and are almost certainly stable right the way to dis-sociation.I2 Furthermore, in KC N these states seem to bedominated by bending motions, whilst in LiCN the bends

Table 7 Comparison of some of the energies computed in this studywith those of previous studiesstate this work ref. (16) ref. (17) ref. (11)

23456789101520253035

116.12 17.7294.3314.5379.5420.4448.3494.7530.2688.9838.6949.61067.71173.6

116.1217.7294.33 14.6380.0421.2451.8499.7533.4-----

116.1217.8294.3314.6379.6420.5448.4494.9530.4-----

---314.6----530.2688.9838.6949.61067.71173.9

Energies are given relative to the ground state which is -38861.4cm-'.


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J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88 3293become damped at fairly low energy and very highly excitedstretches predominate.

The free-rotor states are found once again t o have absencesin their assignments. If one plots the energy of the free rotoragainst l( l + 1) (where 1 is the approximate assignment) thenthe result ought be approximately a straight line of gradientB, where B is the rotational constant for that motion. We finda disagreement by a factor of 2 between our gradient and anapproximate by hand value for B.28 n contrast, we havefound agreement to within 10% for LiCN.28 In both mol-ecules, the free-rotor paths do not appear necessarily to bearthe same structure. In particular, it seems that the free-rotororbits for KCN may actually be more affected by thepotential-energy barrier, whereas the LiCN orbits tend toprefer to stay to the minimum-energy path. Classical trajec-tory calculations would be of use here.

The existence of the transition states, for each of thelinear geometries, is somewhat surprising and rather inter-esting. One could say that these states are actually linear,metastable isomers of KCN. Classically, one could envisage astability on a potential-energy local maximum so long as themotion is precisely normal to the tangent at that point. Arecent study by Gomez Llorente et has investigatedsimilar behaviour. Their classical and quantum studies on theLiCN system have lead them to discuss these states as saddle-point resonances. We note that a similar, but mechanicallydifferent, phenmenon referred to as vibrational bonding hasbeen discussed by P01lak.~

Those plots presented in Fig. 7 represent a fascinating setof a few bound states which we shall refer to as hingemotions. This can be viewed as a highly excited rotation ofthe K + about the C N- ion, centred at the N-end of the ion.It can be seen that this is not a full rotation as in the free-rotor states, but rather it is a restricted rotor and bouncesback from the repulsive wall at somewhere between 6 = 60and 80.So these states could actually be regarded as highlyexcited bending states of a quasilinear molecule, and indeedthat quasilinear molecule could well be represented by thesaddle-point resonances discussed above. This idea of arestricted rotor-highly excited bend against a potential wallhas also been considered by Rohlfing in studies on the C,molecule.31 Note that states 245 and 417 also seem to have adegree of stretching excitation.

ConclusionsWe have been able to compute 800 accurate bound states forthe non-linear 2D KCN system. In comparison to LiCN,some remarkable, rather unexpected, differences in the natureand dynamics of the vibrational states exist. Also it appears,in accordance with classical predictions, that the onset ofchaos is much sooner and stronger in KCN. The majority ofstates are certainly irregular and delocalised. The predicted

regular states corresponding to model Hamiltonians wereobserved, along with the identification of two more inter-esting types of motion. The identification of the hingeingmotion and the transition states suggests that classical inves-tigations would be of use.

As yet, no classical studies have discovered any periodicorbits for this system, unlike LiCN. We feel that this studymay encourage and assist further calculations.We thank Florentino Borondo and Rosa Benito for helpfuldiscussions. J.R.H. acknowledges the SERC for a Fellowship.

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D. 0.Harris, G. 0.Engerholm and W. Gwinn, J. Chem. Phys.,1965,43, 1515.Z . BaEiiC and J. C. Light, Annu. Rev. Phys. Chem., 1989,40,469.J. R. Henderson and J. Tennyson, Mol. Phys., 1989,69,639.J. Tennyson and J. R. Henderson, J. Chem. Phys., 1989,91,3815.J. R. Henderson, S. Miller and J. Tennyson, J. Chem. SOC.,Faraday Trans., 1990,86,1963.J. R. Henderson and J. Tennyson, Chem. Phys. Lett., 1990, 173,133.S. Miller and J. H. R. Clarke, J. Chem. SOC., Faraday Trans. 2,1974, 74, 160.H. T. Stokes and R. D. Swinney, Phys. Rev. B, 1985,31,7153.P. Bourson and P. Durand, Ferroelectrics, 1991, 124, 339.J. Tennyson and S. C. Farantos, Chem. Phys. Lett., 1984, 109,160.S. C. Farantos and J. Tennyson, J. Chem. Phys., 1985,82,800.J. Tennyson and S. C. Farantos, Chem. Phys., 1985,93,237.G.Brocks, Chem. Phys., 1987,116,33.P. Kuijpers, T. Torring and A. Dymanus, Chem. Phys. Lett.,1976,42,423.T. Torring, J. P. Bekooy, W. L. Meerts, J. Hoeft, E. Tiemannand A. Dymanus, J. Chem. Phys., 1980,73,4875.J. Tennyson and B. T. Sutcliffe, Mol. Phys., 1982,46,97.J. Tennyson and A. van der Avoird, J. Chem. Phys., 1982, 76,5107.J. Tennyson and B. T. Sutcliffe,J. Chem. Phys., 1982,77,4061.P. E. S. Wormer and J. Tennyson, J. Chem. Phys., 1981, 75,1245.Z. BaEii and J. C. Light, J. Chem. Phys., 1987,86,3085.M. Mladenovic and Z. BaEii,J. Chem. Phys., 1990,93, 3039.C. R. Le Sueur and J. Tennyson, to be published.J. A. Fernley, S. Miller and J. Tennyson, J. Mol. Spectrosc., 1991,150,597.J. R. Henderson and J. Tennyson, Comput. Phys. Commun., to besubmitted.E. Clementi, H. Kistenmacher and H. Popkie, J. Chem. Phys.,1972,57,5137.M. V. Berry and M. Tabor , Proc. R. SOC.,London, Ser. A, 1977,P. Pechukas, Phys. Rev. Lett., 1983,51,943.F. Borondo and R. Benito, personal communication.J. M. Gomez Llorente, F. Borondo, N. Berenguer and R . M.Benito, Chem. Phys. Lett ., 1992, 192,430.E. Pollak, Comm. At . Mol. Phys., 1985,15, 73.E. A. Rohlfing,J. Chem. Phys., 1989,91,4531.

356,375.

Paper 2/03266J; Received 22nd June, 1992

James R. Henderson, Hoanh A. Lam and Jonathan Tennyson- Highly Excited Vibrational States of the KCN...

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