Jonathan Tennyson- Why Calculate the Spectra of Small Molecules?

8/3/2019 Jonathan Tennyson- Why Calculate the Spectra of Small Molecules?

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J. C H E M. SOC. FARADAY TRANS., 1992, 88(22), 3271-3279 3271FARADAY RESEARCH ARTICLE

Why Calculate the Spectra of Small Molecules?Jonathan TennysonDepartment of Physics and Astronomy, University College London, London WCI E 6BT,UK

Calculations on the bound nuclear motion states of molecules are now making a contribution to many areas ofphysical science including astrophysics, planetary atmospheres and chemical kinetics. This article discussesthe various uses made of ro-vibrational calculations on small, part icularly triatomic, molecules. A non-technicaltheoretical overv iew compares the three main methods for performing these calculations: the traditional pertur-bation theory approach, basis set methods using the variational principle and a finite-element approach, thediscrete variable representation (DVR). The reasons for maintain ing parallel basis set and DVR programs aregiven. Applications of ro-vibrational calculations to the area of traditional high-resolution spectroscopy, i.e.low-lying states, are discussed. These include proving and developing potential-energy surfaces, predict ing andassigning spectra, calculating transition intensities and generating data for the calculation of thermodynamicand emissivity parameters. The links between highly excited ro-vibrational states, reaction dynamics andquantum chaology are discussed and the importance for improved calculations in this energy region outlined.

1. IntroductionHigh-resolution spectroscopy has produced a wealth of dataon small and not so small molecules. Transitions are oftenrecorded to an accuracy of eight or nine figures. This preci-sion is impossible for the theoretician to match, particularlyone working ab initio. Nonetheless the first-principles calcu-lation of spectra of small molecules has been a major growtharea.

The initial motivation for calculating transition frequencieswas often as a test of theoretical methods against observedresults. The hope was that if calculations could be madeaccurate enough then useful predictions could be made for asyet unobserved transitions. As theoretical methods haveimproved a number of other reasons for performing these cal-culations have come to the fore. Some of these allow theoryand experiment to combine to obtain the maximum informa-tion from observed spectra. Other, more creative, uses of cal-culations aim to exploit the computers ability to generateand handle very large data sets to extend studies into areaswell beyond those covered by traditional molecular spectros-copy. Furthermore, while spectra may often be observed withvery high precision, the nature of the transitions involvedmay often be far from obvious. Calculations, in which effectsfrom various contributions may be analysed individually,have proved useful in unravelling many complicated spectra.

The development of first-principles methods of treatingnuclear motion has made possible a wide variety of applica-tions. My group at University College London (UCL) havebeen involved in studies ranging from the spectrum of super-nova 1987A to quantum chaos and from planetary auroraeto equilibrium constants for chemical reactions. These andother applications are described in this review.

The article is organised as follows. Section 2 gives a brieftheoretical overview which will aim at giving a flavour ratherthan a rigorous exposition of how calculations are performed.Section 3 discusses results obtained for the low-lying states ofsmall molecules. Section 4 considers the high-lying rotation-vibration states of molecules including those near disso-ciation.

2. Theoretical Overview2.1The low-lying states of chemically bound molecules areusually considered to undergo small-amplitude vibrational

Perturbation Theory: The Traditional Approach

motion about some equilibrium geometry with their rota-tional motion approximated by the rotations of a rigid body.The interaction between vibrational and rotational motionuia Coriolis forces is neglected. Within this simple frameworkvibrational wavefunctions are products of harmonicoscillators and rotational wavefunctions are easily obtainedas the solution of the rigid rotor model with appropriatemoments of inertia.2 Of course these models do not give anaccurate picture but they provide most of the language uponwhich the labelling schemes (assignments) of modern infra-red and microwave spectroscopy are based.For quantitative treatments the harmonic oscillator-rigidrotor model is improved by use of perturbation theory.Spectra are parametrized using force constants for thevibrational motion (see ref. 3 for example) and rotationalcon~tan ts .~he force constants are the coefficients of aTaylor expansion about equilibrium while the rotational con-stants are the coefficients of a power series in the rotationalangular momentum and its projection. Both series containinformation about the underlying potential of the system.

Thus the force constants, for example, give values for thederivatives of the potential at equilibrium. As such they givea high-order representation of the potential at a single point.A completely satisfactory scheme for extrapolating thesederivatives away from equilibrium to yield values of thepotential at any arbitrary molecular geometry has yet to befound. Thus, although the ro-vibrational states of a moleculeare very sensitive to the potential of the system, force con-stant methods do not yield potentials in a form that is appro-priate for other studies which rely on a knowledge of thepotential far away from equilibrium.The theory underpinning the harmonic oscillator-rigidrotor model is based on the use of the Eckart conditions5which maximize the separation between vibrational and rota-tional motion. Eckart proposed these conditions for a clas-sical system and the simplest quanta1 Hamiltonianincorporating them has been given by Watson for both bent6and linear molecules.

2.2 Variational Principle: Basis Set MethodsIt has long been recognised that perturbation theory is inade-quate for treating molecules with large-amplitude vibrationalmotion. Van der Waals complexes and chemically bound


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3272 J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88molecules containing hydrogen are sufficiently floppy for per-turbative methods to be inappropriate;8 at higher excitationenergies, all systems must undergo large-amplitude vibra-tional motion in one or more modes prior to dissociation.

Diatomic molecules have only one vibrational mode andhence only one internal coordinate. This means that for agiven potential, V(R), he Schrodinger equation can be solvedaccurately by direct numerical integration.' Early attempts totreat large-amplitude motion in polyatomic molecules identi-fied a single large-amplitude mode which could be treated ina similar fashion to a diatomic molecule, with the othermodes kept (near) rigid."

For triatomic (and to a lesser extent tetratomic) molecules,ro-vibrational energy levels and wavefunctions are nowusually obtained using basis set expansions and the varia-tional principle. For example, energy levels and wavefunc-tions for Watson's Hamil tonians have been obtained byWhitehead and and ot he r~ .' ~- '' These calcu-lations use products of harmonic oscillators (Hermitepolynomials) as a basis to represent the vibrational motionsand Gauss-Hermite quadra ture to evaluate the matrix ele-ments of the Hamiltonian, particularly over the potential.

Watson's Hamiltonian has generally fallen out of favourfor variational calculations. This is because of the difficulty ofgoing smoothly from bent to linear systems as illustrated by aseries of calculations on the quasi-linear CH; mol-ecule. 3716-1 ' he preferred solution to these problems is touse internal coordinates which are defined in terms of inter-atomic distances and related geometric parameters. Onegeneral scheme for defining internal coordinates for triatomicmolecules is given in Fig. 1.

For any set of internal coordinates, Q, t is necessary toderive the appropriate Hamiltonian. How this is done is dis-cussed in a recent review by Sutcliffe.20 However, we shouldnote that the process of transforming from 3 N particle coor-dinates to three translational, three rotational and 3 N -vibrational (internal) coordinates inevitably introduces singu-larities into the Hamiltonian. This means that not all physi-cally desirable coordinate choices are actually usable. TheHamiltonian implied by the coordinates of Fig. 1 is given bySutcliffe and Tennyson,21 who also discuss these problems insome detail.

Given a coordinate system, Q, nd a Hamiltonian the nextstep in the variational approach is to choose basis functionsto represent the motions of the nuclei. Usually vibrationalmotion is represented as products of suitable one-dimensionfunctions, P(Q) ,so that the wavefunction of vibrational state ican be written

YAQI Q2 3 Q3 , - .) = C Cf , k, I , ... f'AQi)PAQ#'i(Q3)..j , k, , ...(1)

where the expansion coefficients c' are obtained by diagonal-king the secular matrix constructed in terms of the basisfunctions. This method has been labelled the finite basis rep-resentation (FBR).23

Suitable basis functions include harmonic oscillators(Hermite polynomials), Morse oscillators (Laguerre poly-nomials) and related functions, and free rotor functions(Legendre polynomials). In the case of the Morse and harmo-nic oscillators these functions contain parameters that can beoptimised using the variational principle to obtain compactone-dimensional basis sets.For a given basis set it is necessary to integrate over allcoordinates to form the secular matrix. While some integralsmay be evaluated analytically, integration over arbitrarypotential functions requires numerical quadrature. A verysatisfactory way of doing this is to note that all the basis

Fig. 1 General triatomic internal coordinate system.,' Ai representsatom i. The coordinates in the text are given by r l =A,-R, r , =A,-P and 8 = A , Q A , . The Hamiltonian is defined by geometricalparameters, see ref. 21A 3 - P . A3 - R

91 =-A , - A , ' g2=pA3 - A1functions mentioned above can be expressed in terms oforthogonal polynomials. The integrals can be evaluatednumerically using the appropriate Gaussian quadraturescheme24 or each function.

For triatomic molecules the computational bottleneck invariational calculations is diagonalisation of the secular(Hamiltonian) matrix. It is for this reason that care must bechosen to use appropriate, compact, basis sets. However,operations on matrices, including diagonalisation, are veryefficiently handled by computers with vector processing capa-bility and calculations with several thousand basis functionsare now routinely performed. In this context it should benoted that the matrices being diagonalised are small in com-parison with those used routinely ab initio electronic struc-ture calculations using configuration interaction (CI).However, in contrast to CI calculations one is usually inter-ested in tens or hundreds of the solutions. Furthermore,although the matrices being diagonalised may be ~ p a r s e , ~ ' . ~ ~they are usually much less sparse and less diagonally domi-nant than the CI ones.Rotational motion is carried in these calculations byWigner rotation matrices, D i , M ( a ,8, y). Unlike the vibra-tional basis functions discussed above, these form a finite set:for a given value of the rotational angular momentumquantum number, J , it is necessary to include only the 25 + 1functions with - < k < +J , where k is the projection of Jon the molecular z axis.

Experience has shown that the best way of solving thecoupled rotation-vibration problem is in two Inour method, the first step involves solving J + 1 vibrationalproblems, one for each value of I k ( . The lowest solutions ofthese problems are selected as a basis for the fully coupledproblem.28 This method is YO effective that rotational levelscan even be found at the dissociation limit for chemicallybound molecule^;^' the two-step procedure has effectivelysolved the rotational problem for small molecules.

2.3Variational calculations using basis sets have been very suc-cessful in predicting spectra for a wide variety of triatomicsand some larger molecules. However usually in these calcu-lations only a small proportion (ca. 5 % ) of the solutionsobtained by diagonalising the secular matrix are of any sig-nificance. This means that basis set calculations normallyperform poorly when a large number of energy levels arerequired.

Finite Elements: he Discrete V ariable Representation


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J . CHEM. SOC. FARADAY TRANS., 1992, VOL. 88 3273In the last few years a number of finite-element methods

have been proposed for solving the nuclear motion Schrod-inger equat ion. The most widely used of these is the discretevariable representation (DVR) which was originally suggestedin the sixties22 but has since been extensively developed byLight and co-workers for ro-vibrational problems. A com-prehensive review of the DVR has been given by BaEik andLight.2

Although the DVR method is not strictly variational, it hasstrong formal3' and practical links with the basis set methodsdiscussed above. This is because the formulation of theproblem in a DVR first requires, at least in principle, the con-struction of the secular matrix in terms of appropriate(polynomial) basis functions. This matrix is then transformedto a grid of points determined by the appropriate Gaussianquadrature scheme for each function.

Superficially this transformation achieves nothing as theDVR and FBR matrices are i~omorphic.~' owever, in theDVR it is easy to define a hierarchy of problems which canbe diagonalised and the lowest solutions selected and used toexpand the next problem. In this fashion final Hamiltonianmatrices are constructed with a very high informationcontent-up to half the solutions of the final matrix may bephysically ~ignificant.~An illustrated discussion of thisdiagonalisation and truncation procedure, which is similar tothe two-step method for rotationally excited states discussedabove, is given by Light e t ~ 1 . ~ ~

2.4 DVR us. FB RGiven that the DVR typically yields many more physicallysignificant solutions for a final matrix of given size (and hencefor a given amount of computer time) one might expect i t tobe universally the method of choice. However, the FBR andDVR methods have rather different numerical characteristicswhich means that it is the policy of the UCL group to main-tain parallel and complementary FBR and DVR codes forgiven problems.In the DVR method the quadrature points also serve as theexpansion points for the wavefunction. This means thatnumerical integration and basis set size are inextricablylinked so that the only way of improving the accuracy ofintegrals which have to be determined numerically is byincreasing the number of points used to represent the wave-function and hence the size of the problem. The pitfalls in thisapproach have recently been extensively analysed for aproblem where DVR calculations appeared to give convergedresults which violated the variational p r i n ~ i p l e . ~ ~ - ~ ~In the FBR method, quadrature schemes may be selectedsimply to give integrals to the required precision with noother consequences for the calculation. This means that aconverged FBR calculation is inherently more accurate thanthe corresponding DVR one. FBR calculations will thereforecontinue to be used when very accurate results are requiredfor low-lying states. This is the most common situation whencalculations are performed for comparison with conventionalhigh-resolution spectroscopy.

Furthermore, the coupling of quadrature and matrix sizemeans that it is generally impossible to perform a small DVRcalculation as the numerical quadratures become too unreli-able and the results lose all significance. It is, in principle,possible to circumvent this problem by transforming matrixelements determined numerically using the appropriate FBR(as has been done in other circumstance^^^*^^), but there islittle advantage in doing this rather than a small FBR calcu-lation. Small calculations are important for developingappropriate basis functions,36 the derivation of spectro-scopically determined potentials (see below) and other tests.

D V R calculations on rotationally excited systems havenow been performed by a number ofHowever, because of the large bases used for the vibrationaldegrees of freedom in these calculations, they tend to be com-putationally expensive. This again suggests that FBR calcu-lations will continue to be used for studies involving manyrotational levels especially at low levels of vibrational excita-tion.

3. Low-lying States3.1 Proving Potential-energy SurfacesMuch of chemical physics is concerned with properties whichare determined by inter- and intra-molecular potentials.High-resolution spectroscopy in particular is very sensitive tothe details of potential-energy surfaces. As spectra containmuch information about these surfaces, it would seem naturalto use these data to construct accurate surfaces which can beused for further spectroscopic studies or in other contexts.However, only for the simple one-dimensional curves govern-ing the nuclear motion of diatomic systems is it possible togo directly from observed energy levels to the potential.For polyatomic molecules potential-energy surfaces mustbe constructed by some other means. This surface can thenbe used to predict known ro-vibrational transition fre-quencies for comparison with experiment. From this com-parison the accuracy of the surface can be determined, atleast in the region to which the observed data are sensitive.

The traditional way of fitting the experimental data toforce constants and rotational constants using perturbationtheory has been discussed above. These approaches do notreally yield true global potentials of the system and are grad-ually being superseded by other methods.

The improvement in electronic structure calculationsmeans that at least for small, triatomic and tetratomic, mol-ecules containing light atoms it is possible to calculatepotential-energy surfaces ab initio. Although in few cases canthese surfaces hope to meet the demanding standards of accu-racy produced by high-resolution spectroscopy, they can beused for predictive purposes and as the starting point forfurther refinements.Actually electronic structure calculations yield values forthe potential only at a grid of points. To obtain the fullpotential it is necessary to interpolate analytically betweenthese points. This is usually done by means of least-squaresfitting a surface of suitable functional form and with adjust-able parameters. This step in the calculation remains some-thing of a black art (see for example ref. 15). An appropriatefunctional form should be capable of ( a ) displaying (near)harmonic behaviour about the equilibrium geometry, (b )reflecting the (permutation) symmetry of the molecule, ( c )going smoothly to dissociation products and (d) showingsaddle points at appropriate places, for example lineargeometries for a non-linear molecule. Functions used to fitpotentials in the region of equilibrium rarely satisfy (d)andnone that I know of satisfy ( c ) .Alternatively, potentials suchas the Sorbie-Murrell which dissociate correctlyusually have difficulty giving the required accuracy aboutequilibrium. This problem is becoming a barrier to the devel-opment of high-quality global potentials with at least onevery accurate ab initio calculation remaining unfitted becauseof difficulties finding a suitable functional form.42

Alternatively it is possible to construct spectroscopicallydetermined surfaces by guessing a suitable potential, calcu-lating its ro-vibrational levels, comparing them with experi-ment, adjusting the parameters in the potential and repeatingthe procedure until the calculations reproduce experiment.


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

I

This method has been used extensively for Van der Waalsmolecules43 and is now becoming a significant source ofpotentials for chemically bound triatomics.

Fig. 2 summarizes these two procedures. In both cases thecalculation of ro-vibrational levels is a key step in the com-parison with experiment.One system for which many potential-energy surfaces havebeen constructed is water. The better ab initio surfaces repro-duce the fundamental frequencies to within a few wavenum-bers (see ref. 44 for a survey). Conversely there are fourspectroscopically determined surfaces available all of whichreproduced the 63 observed band origins of water lying up to22000 cm- with a standard deviation of ca. 10 cm-. Table1 summarizes the results of a systematic test of these surfacesby Fernley et ~ 2 1 . ~ ~nd shows that the surface due to J e n ~ e n ~ ~gives reliable results for a large range of levels.

electronicstructurecalculationfu nct o nalI form

L it optimizeparametersenergysurface

i-

Fig. 2 Scheme showing how potential-energy surfaces are con-structed and tested against observation. The left-hand side depictsthe first-principles route and the right-hand side spectroscopicallydetermined potentials. Hybrids between the two methods are alsopossible

Table 1 Statistical comparison of four spectroscopically determinedwater potential-energy surfaces for the 63 observed vibrational bandorigins and rotational term values for the lowest 10 vibrationalstates4 [given are the mean (observed - calculated) error and stan-dard deviation (a),both in cm- 3band origin term values

potentiala mean a mean 0CH 4.1 11.5 0.10 0.35HC - 1.2 7.2 - .59 1.20J - .1 6.4 0.01 0.14KH 0.2 10.3 - .05 0.37

The potentials are: CH, S. Carter and N. C. Handy, J. Chem. Phys.,1987, 87 , 4294; HC, L. Halonen and T. Carrington Jr., J. Chem.Phys. , 1988, 88, 4171; J, P. Jensen, J. M o l . Spectrosc., 1989, 133,438;KH, E. Kauppi and L. Halonon, J. Phys. Chem., 1990,94,5779.

3.2 Predicting and Assigning SpectraIn spectroscopy it is possible to observe large numbers oftransitions to high accuracy without knowing what it is thatis being measured. Theory is rarely able to compete with thisaccuracy but can often help with the understanding.

The usual procedure by which spectra are understood iscalled assignment. Both levels in a transition are assignedvibrational and rotat ional quantum numbers. In fact most ofthese quantum numbers are only approximate, a theme I willreturn to in section 4. It is often true that once a few tran-sitions have been assigned patterns emerge which allow theassignment of many others.A molecule in which theory has played and is continuingto play a particularly strong role is H Z . This unusual tri-angular ion is believed to be the cornerstone of most gas-phase reaction cycles in the interstellar m e d i ~ m . ~ ~ . ~ ~et itwas only in 1980 tha t the first HZ spectrum was observed:Oka49 observed infrared absorptions of the degeneratebending fundamental v2 following pioneering theoretical cal-culations by Carney and Porter.

Since then a series of calculations by Miller andTennyson have lead to the assignment of the bands 2v2trecently, following the predictions of Miller et a1.,55 theforbidden v 1 + v 2 + 2 band has been ~bserved.~hiswork also reported the first observations of transitions in theinfrared-forbidden breathing fundamental, v , . Table 2 com-pares the observed transition frequencies with those predictedfrom theory for this band.

The importance of the H; work was shown by the seren-dipitous observation of its 2v, -+ 0 emission spectrum fromthe southern polar regions of Jupiter.59This observation hasled to a flurry of observational activity and the assignment ofHZ emission spectra in both supernova 1987a6 andUranus.6

0,52 2v2+- v 2 , 5 3 v 1 + v 2 t- v 2 5 3 and 3v2 ( l = 1)+ 0.54 Most

3.3 Calculating Transition IntensitiesIt is often said tha t a spectrum acts like a fingerprint, giving aunique characterization of a particular species. While this istrue, most applications of spectroscopy actually aim at morethan a simple diagnosis of the presence of a particular mol-ecule. Quantitative information is also required.A ro-vibrational spectrum contains considerable physicalinformation about the system under study. For systems in

Table 2 Observed and predicted transition frequencies of theinfrared-inactive stretching fundamental, v 1 , of H i the transitionsare observable due to intensity stealing from nearby transitions ofthe infrared-active bending fundamental, v ttransition

J ,K+ , Kwavenumber/cm -

O ~ S ? calc.7 3+ 6,26,5t 27,6+ 6,35 3 +-4,26.6+ 5,37,7t .46,5+ 6,26,6+ 6,37,7+ 7,4

3282.3083202.1693 144.4543 120.1993066.5613026.1542709.4792569.7262454.41 7

~~ ~

3282.273202.153144.333 120.243066.553025.962709.612569.852454.39a Quantum numbers J,K denote the total angular momentum and itsprojection along the HZ symmetry axis5


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J. CHE M . SOC. FARADAY TRANS., 1992, VOL. 88 3275thermodynamic equilibrium, spectra can give the absolutequantity of the species being observed and the temperature,as well as information on isotopic and spin isomer abun-dance if appropriate. In non-equilibrium conditions, a spec-trum can give the amount of the species in each levelabsorbing/emitting radiation. However, to extract any of thisinformation it is necessary to know the strength of the indi-vidual transitions observed.While many ro-vibrational transition intensities have beenmeasured,62 certain set-ups make the measurement of absol-ute transition intensities nearly impossible. This is true, forexample, of non-equilibrium observations such as those madeof cold plasmas, flames or transient species. In this case thecalculation of transition intensities may be vital for any usefulapplied spectroscopy even though the actual transitionsinvolved have been well characterised in the laboratory.

H3f provides a prototype of this approach. The extensivelaboratory measurements have all been made in plasmas forwhich it is very difficult to determine the occupancy of indi-vidual levels of the ion. Thus the experiments yield littleinformation on the transition intensities. However, these canbe calculated from first principles63 and extensive lists havebeen compiled."Fig. 3 shows a recent infrared emission spectrum of Uranustaken with a resolution of 0.0031 pm.61 It is easy to assignthe 11 most prominent features of this spectrum to lines orblends of known transitions in the fundamental bendingmode of H i , v , . However, use of the transition intensitiesalso yields a column density, p , a temperature, T and anortho fraction, f,. T and n are important for modellingauroral processes in Uranus-in particular they are sensitivemonitors of both the energy deposition and emission in theionosphere. f, is important as values near 3, as observed,suggest that the H3f has thermalised before emitting. This isbecause H3f formed from cold H, might be expected to haveanf, value nearer $.61 The ro-vibrational spectrum of H i isnow routinely being used as a monitor of auroral activity inthe Jovian planet^.^^,^'3.4 Generating Thermodynamic Da taThe development of reliable and fairly automatic programsfor calculating ro-vibrational energy levels and spectra66means that it is possible to compute comprehensive sets ofenergy levels or transitions. These in turn can be used to gen-erate thermodynamic or emissivity data of interest.

3.0

' 2.5P

5"E 2.0z 1.5stc

l . l . . . . , . , . . I . . . . l . . . l.3.90 3.95 4.00 4.05

WPmFig. 3 Emission spectrum from Uranus (April lst , 1992) recordedby Trafton et ~ l . , ~ loints, and fit to the spectrum using the H: dataof Kao et ~ l . , ' ~urve. A telescope resolution of 0.0031 pm wasassumed in the fit. H: parameters: T = 740 K, f, = 0.51,p = 6.5 x 10'' m2

For example, sums over energy levels, E i , give partitionfunctions, z, as a function of temperature, T:z =1 2Ji+ l)gi exp(- i / k , T) (2)

where gi is the nuclear spin degeneracy factor. Note that thisformulation makes no separation between rotational andvibrational motion and hence z is the combined partitionfunction for these motions. Besides their use for calculatingtemperature-dependent spectra, partition functions can beused to derive equilibrium constants as a function of tem-perature.

I

Consider the reactionA + B + C + D (3)

Then the temperature-dependent equilibrium constant, K (T),is given by the formulaz;:zz i z;K =- xp(- U / k , T) (4)

where U, he heat of the reaction at absolute zero, is given bythe difference between the zero-point energies of the productsand reactants measured on a common (absolute) energy scale.Note that z' differs from z defined above in that it alsoincludes the translational partition functions which can beincorporated as a simple mass factor for reactions which con-serve the number of particles6'

Isotope-exchange reactions are important for isotopeseparation schemes, for the formation of exotic molecules (e.g.ref. 68) and astrophysically. The thermodynamics of thesereactions can be characterised by using the ro-vibrationallevels of the species involved.Thus, for example, the deuterium fractionation reactionsHz + D + H zD + + H ; U = 509K ( 5 )

andH3f + HD + H2D' + H,; U = - 39.5K ( 6 )

are both exothermic by amounts significant for temperaturesof 100 K or less. They are thus very important in the coolinterstellar medium where they can lead to an H,D+/H:ratio several orders of magnitude greater than the naturalabundance of D.69 Fig. 4 shows the temperature dependenceof the equilibrium constants of the two reaction^.^'These data are important input for models such as that ofMillar et ~ 1 . ~ ~

10

LUJ- 0 5

~ I 1 I

\00 500 1000 1500

TIKFig. 4 Calculated equilibrium constants, K, as a function oftemperature" for reactions H: + D +H,Df + H (solid curve) andH: + HD+ H,D+ + H, (dashed curve)


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3216 J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88Knowledge of the total absorption of a medium, often called

the opacity, as a function of molecular composition, wave-length and temperature is important for modelling planetaryand cool stellar atmospheres. On Earth it is vital for under-standing the greenhouse effect . Traditionally such absorp-tion profiles have been constructed from databases oftransitions individually measured in the laboratory, of whichHITRAN62 is a much used example.

Even for a triatomic molecule, such as water, depending onthe temperature of interest, it may be necessary to knowbetween lo4 and lo7 transitions in order to synthesize anaccurate opacity function. Generating this large amount ofdata experimentally is tedious and expensive. Converselycomputers are well suited to the repetitive nature of suchtasks. So far, datasets of all the possible ro-vibrational tran-sitions of a molecule have only been generated for simplesystems.71 However, a number of groups, including the oneat UCL, are working on extending these methods to mol-ecules of general atmospheric and astrophysical interest.In fact, at UCL we have a very extensive, but far fromexhaustive, dataset of H3+ transition frequencies and lineintensities. Part of an earlier version of this dataset has beenpublished,58 but one of the problems of this work is present-ing the large quantities of data involved in a form accessibleto other workers. So far our attempts a t guessing which por-tions of this dataset are the most important to publish havenot proved very accurate !

4. High-lying States4.1 SpectroscopyandReaction DynamicsMuch of high-resolution spectroscopy is concerned with tran-sitions between low-lying levels of molecules, but this is notalways the case. Laser-based techniques, such as stimulatedemission pumping, allow high-lying levels of molecules to beprobed. Besides the intrinsic interest in these levels, it is inthis region that spectrosocopy links up with other areas ofchemical physics and in particular the study of reactiondynamics.

Unimolecular reactions in the form of isomerisation shouldbe amenable to study using ro-vibrational techniques, pro-vided the reaction occurs on a single electronic potential-energy surface. A prototype system of this form which hasbeen much studied is LiCN.72-79This molecule has a linearisocyanide structure80.81 and is predicted to also have ametastable linear cyanide isomer.81

A study by Henderson and Tennyson using the DVR tech-nique and a model which froze the CN bond obtained 900vibrational wavefunctions for These results coveredan energy region up to four times the barrier to isomerisationin the LiCN system. The calculations found vibrational stateslocalised in both LiNC and LiCN minima, some of whichpersisted throughout the entire energy region. Above thebarrier to isomerisation an increasing proportion of the statesare completely delocalised. Of these states, a few are free-rotor like (or p ~ l y t o p i c ~ ~ )n appearance, but most arehighly irregular in nature, meaning that no sensible assign-ment of vibrational quantum numbers could be attempted.The study on LiCN parallels similar but less complete 3Dstudies on isomerisation in HCN.84*85 ibrational wavefunc-tions for highly excited states of HCN have been used to esti-mate the relative formation rates of HCN and HNCS6 in anattempt to explain the anomalously high population of HN Cobserved in the interstellar medium.

A decade ago Carrington et reported an infraredphotodissociation spectrum of H3+ showing a very largenumber of narrow transitions between what proved to be

quasibound (or predissociating) ro-vibrational states of theion.88 This spectrum, which remains largely unexplained andcompletely ~nassigned,~~as provided a major challenge totheory.

Initially theoreticians focussed on the classical behaviour ofthe H; nuclear motion problem. This showed that whilemuch of the H; phase space in the near dissociation regionwas chaotic, quasiperiodic trajectories were found by anumber of workers (e.g. ref. 90). H i is, however, a highlyquantal system.

Henderson and co-workers performed a series of calcu-lations using a DVR in and three33*34+91imensionsobtaining approximate energies and wavefunctions for allthe vibrational bound states of H;. This work has takentheory into the realms of the dissociation for chemicallybound polyatomic systems although there is still some way togo before it yields a full explanation of the predissociationspectrum of Carrington et al .

An interesting comparison between (semi-)classical andquantal calculations on H i can be obtained by comparingthe density of states predicted by the two methods. The func-tion is important for partition functions and for statisticalmodels of chemical reaction. Fig. 5 shows a comparison ofsemi-classical and quantal estimates of the J = 0 density ofH3+ states based on the work of Berblinger et ~ 1 . ~ ~s can beseen, the agreement is excellent, suggesting that direct inte-gration of phase space gives a reliable method of obtainingdensities of states for highly excited molecules.

4.2 Quantum ChaologyThe subject of chaos in classical systems has aroused muchrecent research and popular interest. A particular branch ofthis work involves the behaviour of conservative (i.e.constantenergy), non-linear Hamiltonian systems.93 Anharmonicallycoupled oscillators are a particular example of such a systemand as such their general properties have received muchattention.

More controversial is the behaviour of systems which areknown to be classically chaotic under quantum mechanics(e.g. ref. 94). The phrase quantum ~ha olog y ~ as beencoined to describe this area of research.

Molecular vibrations give some of the simplest and mosteasily observed examples of anharmonically coupled quantaloscillators. Indeed a number of observed molecular spectra

1400

1000 Izb 60012004001 /

0 5 10 15 20 25 30 35 40/lo3 cm-

Fig. 5 Number, N, ( E) , of vibrational ( J = 0) states of the H imolecular ion lying below a particular energy, E. The solid line givesquan tal predictionsg1 and the dashed line gives semi-classical resultsobtained by Monte Carlo integration of the volume of phase space.92For ease of comparison doubly degenerate vibrational stateshave been counted twice. The dissociation energy of H l is ca.35 OO cm-


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J. CHEM. SOC. FARADAY TRANS. , 1992, VOL. 88 3277have been pronounced 'chaotic' on the basis of the behaviourwith respect to some indicator of 'quantum chaos'. Smallmolecules so studied include HCN,96 CH 20H ,97 C2H, ,98NO, ,9 9 SiH,"' and Na3 I o 1This is an area where quantum-mechanical calculations areprobably more reliable and easier to interpret than experi-ment. Direct comparison between experiment and classicalcalculations relies on a series of assumptions about suchthings as the accuracy of the potential-energy surface, theBorn-Oppenheimer approximation, neglect of nuclear andelectronic angular momentum couplings and assumptionsabout unobserved transitions. In conventional spectroscopicstudies, these effects can often be ignored or allowance madefor them using perturbation theory. However, chaotic trajec-tories are unstable with respect to perturbations and it is thusdimcult to decide a priori how a seemingly small perturbationmay affect a spectrum.Conversely it is possible to set up and solve, at least inprinciple, a well defined model using quantum mechanics.For molecular vibrations this same model can usually bestudied classically ( e . g . ref. 75 , 76). Direct unambiguousquantum-classical comparison is then possible. If theseresults also agree with observation, then the model is a validone. If not, it may be possible to improve the model used forboth the quantal and classical calculations.As the wavefunction of a system contains all the informa-tion knowable about that system, it is possible to interrogatecalculated wavefunctions using several different methods.Table 3 gives the results of such an analysis for the LiCNcalculations of Henderson and Tennysongl discussed in theprevious section. The two parameters compared are the dis-tribution of the spacings between neighbouring vibrationalenergy levels of the system, as parametrized by the Brodyparameter 4 , and the proportion of states in the same energyregion which could not be assigned approximate quantumnumbers by visual inspection of plots of the vibrational wave-function, u. The Brody distribution is a generalization whichgives a Poisson distribution, the usual distribution displayedby the spacing between neighbouring levels of a regularsystem, for 4 = 0, and a Wigner distribution, the distributionof eigenvalues obtained by diagonalising matrices of randomnumbers, for 4 = 1. In practice the parameter q is least-squares fitted to the nearest-neighbour distribution (see ref.102). It can be seen here that there is a high degree of corre-lation between the two measures: one based on energy levelsand the other directly on the wavefunctions.The one major conclusion of the various comparisons ofclassical and quantal calculations concerns 'scarring' of thewavefunction. This is the observation that the wavefunctiontends to collect amplitude disproportionately about quasi-periodic classical orbits. This is the basis for the explanationTable 3 Level spacing distribution Brody parameter,'" q, for thelevels of LiCN ca lculated by Henderson and Tennyson'l

levels energy range/cm - S 4 fY U1-30

31-10010 1-300201-400301-500401-600501-70060 1-80070 1-900

0-22402286-40904090-7 1505768-83587 150-94278350- 104459427-11388

10445-1 224711388-13086

77.526.015.313.011.410.59.89.08.6

0.1540.5880.7850.7980.7640.7940.8590.8301.049

0.0340.0300.0120.0150.0170.01 30.010.0120.01

7Y o4481858891939495

The energy range of each bin is given relative to the LiNC groundvibrational state. The average level spacing, s, n cm-' and the stan-dard deviation, 0, of the fit in units of probability as well as thepercentage of unassigned states, u, in each fit are also given.

of coarse-grained regularity in a number of otherwise irregu-lar spe ~tra.~' . ' '~ndeed scarring of the quantal wavefunctionhas actually lead to the identification of otherwise unnoticedstable orbit^.^'^,'^^

5. ConclusionIn 1978 Carney, Sprandel and Kern'" published a muchquoted review on the state of rotation-vibration calculationsfor chemically bound small (triatomic) molecules. This workgave a comprehensive summary of variational techniques inuse at the time. Its major concerns were (a) whether the(Eckart) Hamiltonians used by different groups were actuallythe same and (b ) whether the different solution strategiesemployed actually gave the same answers. All except onework quoted used normal coordinates, and all studies con-centrated on the vibrational fundamentals and, perhaps, alow level of rotational excitation.

During the 1980s geometrically defined internal coordi-nates have largely superseded normal coordinates and anumber of benchmarked internal coordinates codes havebecome generally available (e .g . ref. 106). Methods have beendeveloped for dealing with very high degrees of ~ i b r a t i o n a l ~ ~and rotationalt7 excitation.The first principles calculation of molecular spectra usingthe variational principle has thus become a mature techniquecapable of making useful contributions to a number of areasof physical science. In this review I have tried to i llustrate therange of problems that can be tackled using these techniques.

Perhaps the most compelling illustration of this progresscan be seen from the role of first-principles ro-vibrational cal-culations in the development of accurate potential-energysurfaces. A major driving force for the development of thesetechniques was the progress in electronic structure calcu-lations and hence the availability of good ab initio potential-energy surfaces. Ro-vibrational calculations were required tomake comparisons between t,hese surfaces and high-resolution spectroscopy. Now it is becoming a standard pro-cedure to use variational ro-vibrational calculations to derivepotential-energy surfaces directly from the observed spectra.I would like to thank my co-workers from whose work I havequoted freely here. In particular I must thank Brian Sutcliffeof the University of York for a long and fruitful collabo-ration, and the members of my group at UCL, in particularSteven Miller, James Henderson and Ruth Le Sueur. Thework discussed in this article has been supported by anumber of SERC grants and has received support from theEEC, NATO, the British Council, the Research CorporationTrust and NASA, all of whom are gratefully acknowledged.

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J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88Kinsey, J. Chem. Phys., 1985,83,453.99 S. L. Coy, K. K. Lehmann and F. C. De Lucia, J. Chem. Phys.,1986,85,4297.100 J. W. Thoman Jr, J. I. Steinfield, R. I. McKay and A. E. W.Knight, J. Chem. Phys., 1987,86,5909.101 J. M. Gomez Llorrente and H. S. Taylor, J . Chem. Phys., 1989,91,953.102 E. Haller, H. Koppel and L. S. Cederbaum, Phys. Rev. Let t . ,1984,52,1665.

3279103 J. Tennyson, 0. Brass and E. Pollak, J. Chem. Phys., 1990, 92,3005.104 J. R. Henderson, H. A. Lam and J. Tennyson, J. Chem. Soc.,Faraday Trans., 1992,88,3281.105 G. D. Carney, L. L. Sprandel and C. W. Kern, Adu. Chem.Phys. , 1978,37, 305.106 S. Carter, P. Rosmus, N. C. Handy, S. Miller, J. Tennyson andB. T. Sutcliffe, Comput. Phys. Commun., 1989,55, 71.

Paper 2/03063B; Receioed 10th June, 1992

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