jonathan tennyson and steven miller- calculating molecular spectra

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Calculating Molecu lar Spectra

Jonathan Tennyson and Steven M il le rDepartment of Physics and Astronomy, University College London, London WC1 E 6 B T

1 IntroductionSpectra, the absorption or emission of light at characteristicwavelengths, have long been known t o provide accurate finger-prints of molecular species. Spectroscopic analysis is standardfor characterizing species and for identifying and quantifyingmolecules in complex mixtures. It also provides in formation onsuch things as temperature and isotopic abundances. We areparticularly interested in these properties as a diagnostic for‘cool’ astronomical bodies. Astronomically, molecular spectraprovide a unique handle on the physical conditions in environ-ments such as giant molecular clouds, planetary atmospheres,and cool stellar atmospheres. As a counter example infra redspectroscopy is widely used in the petroleum indus try to monit oreverything from the oxidation of oils in engines to the compo-sition of exhaust gases after combustion.’

The detailed information contained in the many transitionsthat characterize the spectrum of a molecule is a sensitivereflection of the underlying interactions within that molecule.Thus the rotational energy levels of a molecule are largelygoverned by the molecular geometry: microwave spectroscopy,the study of rotational transitions, has long provided the mostaccurate det ermination of molecular bond lengths. Converselythe vibrational energy levels of a molecule are determined by theease with which the atoms can move relative to each other withinthe molecule: infra red spectroscopy, which covers the wave-lengths of the strongest vibrational transitions, provides, a t leastin principle, detailed and often very accurate information onhow the ato ms in a molecule interact.

The interaction of atoms within a molecule is governed by theelectronic potential energy surface of the system (see Section 2).Potential energy surfaces are important entities within chemistry

Jonathan Tennyson studied Natural Sciences at Kingj . CollegeCambridge ( B . A . 1977). In 1980 he completed a Ph.D. inTheoretical Chem istry at S ussex University under the supervisionof Professor John Murrell. H e then spent two year s at theUniversity of Nijmegen, The Netherlands, as a Royal SocietyWestern Exchange Fellow followed by three years at the S E R CDaresbury Laboratory where he worked on calculationsof photo-ionization and electron molecule scattering. H e moved in 1985 tothe Department of Physics and Astronomy at University CollegeLondon and was promoted to Reader in 1991.

His current research interests include calculating the ro-vibra-tional spectra o f molecules, the consequencesof classical chaos onmolecular spectra, electron (pos itron) m olecule collision calcula-tions, and the calculation of molecular data for astrophysics(including observational studie s).

Jonathan Tennyson is the grea t, great grandson of AIfred LordTennyson.

Steven Miller gained B.Sc. (19 70 ) and Ph.D. (197 5) degreesfro m Southampton Universi ty. Two years atU M I S T as researchfellow and two at Shefle ld University were ollowed by seven yearsworking as a political journalist before he joine d th e MolecularPhysics Group at University College London to work onro -vibrational transitions of triatomic molecules, includingH i .

He has pioneered the use of H i as an astronomical probe o fplanetary atmospheres and ma de the first identijication of H ioutside the solar system (in Supernova 19 87A ) in 1990.

as they determine not only spectra, but also reaction dynamics,transport properties, and interactions in the liquid and solidstate. Alth ough all these properties a re sensitive to the potential,spectroscopy is undoubtedly the most accurat e source of infor-mation on th e potential. Unfortunat ely, except for diatomicmolecules, there is no general method of extracting potentialsfrom spectroscopic data. Th is leads t o a n alternative strategy.Potentials constructed by any means, from first principlesapplication of quan tum mechanics to guess work, are tested byusing them to generate spectra which can be compared withobservation. If the agreement is unsatisfactory the potential canbe adjusted and the calculation repeated.

Recorded spectra, particularly for polyatomic molecules, areoften very complicated. They can contain many thousands oflines each corresponding to transitions between unknown levels.The process of assigning such spectra involves ascribing degreesof vibrational a nd rota tional excitation to b oth the initial a ndfinal levels involved in the transition. Often this is quit e straight-forward once a few assignments have been made but the firstassignments may involve inspired guesswork and c an be greatlyaided by calculations. Similarly predictions of where the transi-tions can be found greatly aid the search for a particular species.

Calculations of molecular spectra are thus important fortesting and developing potential energy surfaces, interpretinglaboratory data, and predicting spectra. However, even for athree-atom molecule, such as water, at room temperature thespectrum can contain a very large number of transitions. Thetotal absorption of light as a function of wavelength, which iscalled the opacity of the system, is an important property ofmany bodies including the Earth’s atmosphere.

A detailed knowledge of the opacity of the atmosphere isrequired t o model the greenhouse effect. The peak of the Sun’sradiation is at optical wavelengths to which the Earth’s atmos-phere is largely transparent. The Earth is cooler which meansthat it re-emits radiation at longer wavelengths - mainly in theinfrared. If this emission passes through the atmosphe re then theear th will lose heat; if the atmosphere abso rbs the emission thenthis heat is retained. This is a fine balance which is dependent onthe component s of gases in the atmosphere. Although CO, iswell known a s a greenhouse gas, many other atmospheric tracespecies such a s methane, Chloro-Fluoro-Carbons (CFCs) andtheir derivatives, actually have a much greater potential t o act asgreenhouse gases. The generation of the many transitionsinvolved in synthesizing opacities for these species is naturallyhandled computationally. We are currently embarking on such aproject focusing on the behaviour of cool stellar atmospheres.

2 PotentialsAs mentioned above, potential energy surfaces form a commonstrand running through practically all areas of physicalchemistry. Actually the very existence of these surfaces relies onan approxi mation, albeit one which usually is valid.

Chemists generally consider molecules to be composed ofelectrons which are light and singly charged, and compoundnuclei which are comparatively heavy (hydrogen is 1836electronmasses) and multiply charged. Given the large mass ratio, it isgenerally assumed that the electrons can relax instantaneouslyto any given nuclear geometry. It is within this framework,known as the Born-Oppenheimer approximation, tha t molecu-

lar potential energy surfaces can be defined. The potentialenergy for a particular nuclear configuration, generally denoted

91



92 CHEMICAL SOCIETY REVIEWS. 1992

I/, is a function only of the relative nuclear coordinates as theelectronic motion is assumed to be fully adjusted to thisgeometry.

Potentials can be calculated using first principles quantummechanics or ab in i tio , by choosing a set of internuclear sepa-rations, solving the resulting Schrodinger equation for the

electronic motion and repeating the procedure until a grid ofpoints has been generated. These points must then be fitted tosome cont inuous function to produce a full surface. Solution ofthe electronic Schrodinger equatio n is a formidable problem inits own right but one which has advanced rapidly with theoreti-cal developments and computer technology. It should be noted,however, that the generation of full ab in i tio potential energysurfaces for polyatomic molecules remains computationallyexpensive.

The traditional method of extracting potentials fromobserved spectra has been vi a force constants. These constantsunderpin a theory of vibrational motion which assumes thenuclei undergo only small amplitude motion. The force con-stants represent the derivatives of the potential at the equili-brium geometry of the molecule and as such give a very high

order representation of the surface at one point rather t han aglobal surface. There have been a number of suggestions as towhich coordinates give the best extrapolation of the potentialaway from equilibrium. But no completely satisfactory solutionexists.

An alternative method of obtaining potentials from experi-ment has been to make some initial guess at a potential functionwhich contains parameters tha t can be optimized by comparisonwith experimental data. This procedure places great emphasison the development of computationally efficient procedures fo rcalculating spectra as this is the slowest step in the process. Themethod was first extensively used for atom-diatom van derWaals complexes such H,-X and HCl-X (where X = He, Ne,Ar, Kr).273 These weakly bound systems have the advantage thatthe relatively stiff diatomic vibrational mode can be neglected.

The other vibrational modes undergo large amplitude motionswhich are poorly represented as an expansion about equili-brium. More recently this method has been applied to chemi-cally bound systems such as ~ a t e r , ~ , ~ ,S,5 and HCN.6

3 Diatomic MoleculesGiven a potential energy curve the calculation of rotation-vibration spectra for diatomic molecules is relatively straightfor-ward. F or these systems the potential energy surface and hencealso the vibrational wavefunction is one dimensional. Thismakes visualization of both these functions straightforward.Similarly the degree of vibrational excitation in a diatomic isgiven by the number of times the wavefunction passes throughzero. These zero amplitude points ar e called nodes.

A number of model potentials have been proposed for diato-mics. These include harmonic, Morse, and Lennard-Jones 6-1 2potentials. The first two of these are illustrated in Figure 1 .

The harmonic potential is important not so much for itsquantitative properties, but because most of the language usedto interpret vibrational spectra is based upon it. Representingvibrational motion as harmonic oscillations is valid if theamplitude of these vibrational motions is small. In this approxi-mation vibrational energy levels are evenly spaced. However,the harmonic model neglects all possibility of molecular disso-ciation. This introduces so called anharmonic effects whichusually cause the spacing between vibrational levels to decrease.Of course vibrational motion becomes increasingly large ampli-tude as dissociation is approached.

The Morse potential, also shown in Figure 1 , contains the

basic properties of most diatomic molecules. It is stronglyrepulsive when the nuclei are close together, has a minimum a tsome intermediate distance and dissociates at long bond length.It is thus capable of representing basic anharmonic effectsalthough for fully quantitative results more sophisticated func-tions are generally required.

Figure I Potential energy curves, V ( R ) , or a diatomic with equilibriumbond length Re and dissociation energy D,. he blue curve is harmonicand the red curve is a Morse potential. Horizontal lines indicatevibrational energy levels. Both potentials have the same curvature(force constant) at R = Re.

There are now many computational techniques for directnumerical integration of one-dimensional second-order differ-ential equat ions of type encountered fo r diatomic systems. Thismeans that, given a potential, the rotation-vibration energylevels of a diatomic system can be routinely obta ined . This is incontrast to polyatomic systems.

4 Coordinate Systems for Polyatomic

If a molecule consists of N nuclei then 3N coordinates arerequired to describe the position of these nuclei in space. The

translational motion of the molecule can be represented by the 3coordinates of its centre-of-mass. Similarly for a non-linearmolecule, rotational motion can be represented by a further 3coordinates. This leaves 3 N - ( 3 N - 5 for a linear molecule)coordinates t o represent the vibrational motion of the molecule.These are usually described as internal coordinates .

For diatomics the bond length of the molecule is also thenatural internal coordinate of the system. Even for triatomicsystems, however, there is no unique choice of internal coordi-nates which gives a good picture for all systems.

Traditionally so called normal coordinates have been used torepresent the motions of polyatomic molecules. These coordi-nates are obtained as the solution of the multi-dimensionalharmonic problem assuming that only harmonic terms areretained in the molecular potential. But these coordinates are

unsatisfactory in many cases. This is because, given enoughinternal energy, all molecules undergo large amplitude vibratio-nal motion which is not well represented within the harmonicapproximation. In particular there is a technical problem thatwith these coordinates it is all too possible to leave the truedomai n of the problem. This is illustrated by Figure 1 where itcan be seen that f or high energies the harmonic potential crossesR = 0 and allows the vibrational wavefunction to sample nega-tive bond lengths.

Normal coordinates have now been largely abandoned foraccurate calculations on small molecules. Instead internal coor-dinates defined in terms of internuclear separations and asso-ciated angles are usually employed. Figure 2 illustrates 3 ofmany such triatomic coordinate systems.

The internuclear coordinates ( r l , r 2 , r 3 ) , Figure 2a, would

appear a good coordinate choice for a number of triatomicsystems. This is particularly so for the molecule Hi, discussedbelow, whose equilibrium geometry is an equilateral triangleand for which it is thus desirable to have coordinates reflectingthe high symmetry of the system. Unfortunately these coordi-nates are inconvenient to work with as their allowed ranges are

M ol ecu es



CALCULATING MOLECULAR SPECTRA-J. TENN YSON AN D S. MILLER 93

I

0r A2

3

Figure 2 Internal coordinates for triatomic systems: (a) bond lengthcoordinates ( r l r 2 . 3) ;(b) bond length ~ bond angle coordinates ( r l ,r 2 , a ); (c) scattering coordinates ( r , R , 0) .

linked by a series of triangulation relationships of the formIr1 -. r3 I < y2 d 1 + r 3 . This makes numerical integration

very difficult and as a consequence these coordi nate s are gener-ally not used.

Bond length-bond angle coordina tes ( r l , r 2 , a), Figure 2b,give a good representation of many triatomics such as H 2 0 ndH 2 S . These coordinates have often been used for these mole-cules. Recently Radau coordinates, originally developed to

study planetary motion about the Sun, have also been used forsuch systems. When the central atom is heavy, these coord inat esare very similar to the bond length-bond angle coordina tes buthave the adva ntage tha t they yield a much simpler kinetic energyoperat or in the nuclear motion Hamiltonian.

Scattering coordinate s Y, R , B) , Figure 2c, like Radau coord i-nates, also give a simple (diagonal) kinetic energy operator.These coordinates got their name from their use to representatom-diatom collisions: the intersection of Y and R being at thediatom centre-of-mass. They are appropriate not only forweakly bound atom cli ato m van der Waals complexes but alsofor a molecule like HCN as the coordinates can link theequilibrium struct ure of the two linear isomers HC N and H NC .

Even if there is no coordina te system in which the vibrationalmotion is separable, it is still usual to talk about a molecule

having 3N - 6 vibrational modes. Like the internal coordinates,how these modes are chosen is not unique. The process ofpartitioning the vibrational energy between the vibrationalmodes, known as assignment, will be discussed below.

There are also many ways of defining the 3 coordinates thatrepresent the overall rotational motion of the system. This isbecause these coord inat e depend on how one fixes (‘embeds’) thex, , and z axes to the molecule. Thus, for example, in thescattering coordinates of Figure 2c the z axis of the system isoften embedded along either Y or R depending on what is mostappropriate for the system.

5 HamiltoniansThe equations of motion for N interacting particles, or their

quant um mechanical equivalent given by the Hamilton ian, caneasily be written down. However, the removal of translationalmotion, combined with coordin ate transformation s to internalcoordinates and a particular axis embedding means the Hamil-tonian itself must be transformed. Each time one defines a newset of internal coordinates or axis embedding it is necessary to

construct a new Hamiltonian for the system. A general methodfor deriving these Hamiltonians has been developed by Sut-cliffe7 but its application is often algebraically messy.

Unfortunate as this may seem, there is a more seriousproblem. Th e very process of defining internal coordinate s andaxis embeddings introduces geometries into the Hamiltonian

where it is badly behaved. For example, these badly behavedgeometries, or singularities, are often encountered when amolecule becomes linear. In this case the system only has 2instead of 3 rotational degrees of freedom and 3 N - instead of3N - vibrational modes. Special care is thus required for anybent molecule which samples linear geometries.

6 The Variational PrincipleIn its simplest form the Raleigh-Ritz Variational Principlestates that for a given Hamiltonian ope rator the energy of anyapproximate wavefunction will always be greater or equal to thelowest (ground state) energy of the system. This theorem canalso be extended to give upper bounds fo r the energy of excitedstates provided th at the trial wavefunctions obey certain simple

constraints.The Variational Principle has been a rock upon which much

of quantum chemistry has been founded. It allows trial wave-functions to be improved systematically with the best functionbeing given unambiguously by the function with the lowestenergy. Particularly powerful in variational calculations arebasis functions. These allow the trial wavefunction to beexpanded as linear combinat ion of suitable functions:

The coefficients, given by the vector c , can be varied until the besttrial wavefunction is obtain ed. This procedure can be written interms of matrices and is thus particularly well suited to computerimplementations. These (‘secular’) matrices are diagonalized to

yield the c’s as eigenvectors and energies as eigenvalues.In the secular matrix method, which is commonly used in allbranches o f quan tum chemistry,* matrix elements are given by

In this expression, Hi s the secular matrix, fi he Hamiltonian forthe system and the 4’s are the basis functions. The integral runsover all space with d7 being the appropriate volume element.Usually at least some of the integrals involved in this expressionhave to be evaluated using numerical quadrature. Provided thematrix elements are real, the secular matrix is symmetric ( i . e .

For vibrational problems suitable basis functions includesolutions of the harmonic oscillator and Morse problems dis-

cussed in Section 3. The basis sets required to give a completerepresentation of the vibrational motion are always infinite.One’s aim is therefore to make a judicious choice of functions sothat only a few are required to give a good representation of t hewavefunction. Convergence is reached when the use of addi-tional functions only leads to an insignificant lowering of theenergy.

For rotational degrees of freedom the situat ion is simpler. Fora given rotational quantum number, J, it is only necessary tohave (2 J + 1) functions to represent the rotat iona l wavefunctionfully. These functions, known as Wigner rotation rn a t r i c e ~ , ~ anthus be used to expand any rotational problem and their use isuniversal.

Fo r triatomics, variational calculations performed using basissets give a computer time requirement which is almost always

determined by the size of the secular matrix that needs diagona-lizing. A technique which has proved highly successful inmolecular vibration-rotation calculations is the application ofthe Variational Principle in several steps. In such procedures aproblem of reduced dimension is solved using a basis setexpansion and the solutions of this problem used as the basis for

H . = H . ).‘ J J:‘



94 CHEMICAL SOCIETY REVIEWS, 1992

a higher dimensional problem. If the reduced problem is wellchosen then only a minority of its solutions are needed to obtainconverged results fo r the full problem. This can lead to massivesavings in computer time. Such methods have allowed thesolution of many problems which would otherwise be beyondthe scope of current computers.

7 Molecular PropertiesThe most obvious results ob tain ed from solving the vibration-rotation problem are the (approximate) wavefunction of thesystem and its associated energy levels. However, quantummechanics tells us that with the wavefunction we can calculateall other knowable properties of the system. Of particularinterest in this context are the intensities of the transitionsbetween the various states of the system.

Most common transitions which involve absorbing or emit-ting light are driven by dipoles. Thus the intensity of a purerotationa l transition is proportional t o the squar e of the perma-nent dipole of the molecule. For vibrational transitions theimportant property is the change in the dipole moment in the

vibrational coord inat e being excited. In the harmonic model ofmolecular vibrations this is simply given by the derivative of thedipole at the equilibrium geometry of the molecule. However,more accu rate calculations require a knowledge of the dipole asa function of the internal coordinate s of the molecule, as well asthe wavefunction of the initial and final state. In fact, because thedipole is a vector, 3 surfaces (2 for planar molecules such astriatomics) are required.

Transition dipoles not only give the strength of individualabsorption or emission features, they can also be used to givefluorescence lifetimes of excited states. (Fluorescence lifetime isthe average length of time an excited state will survive beforedecaying to a lower level by the spontaneous emission of aphoton.)

Dipole surfaces can also be used to give vibrationally resolved

dipole moments. Similarly any other property of the moleculewhich is geometry dependent, such as bond lengths, rotationalcons tant s, or polarizabilities, can be obtained for different statesof the system by using the wavefunction to d o the appropriateaverage.

The wavefunction of the system can always be labelled by thetotal an gular momentum of the system, J , as this is a constan t ofmotion. However it is usual t o label molecules by the number ofquanta of vibrational excitat ion in each vibrational mode of thesystem. Such labelling is often based on a harmonic model of th esystem and is always approximate. The process of attachingthese labels is called assignment. Assigning levels is important a sa wealth of detailed understanding of how systems behave hasbeen developed in terms of these assignments. Fu rthe rmore , asexperiments never record all transitions for a system, assign-

ments are crucial in any comparison between theory andexperiment.Usually low-lying levels can be assigned without difficulty

either by studying energy patte rns o r by using the coefficients inthe basis set expansion. Fo r higher-lying levels making assign-ments can be very much trickier. The density of vibrationalstates increases with energy, making energy differences unreli-able, and individual states may no longer be dominated by asingle basis func tion.

Another technique fo r making assignments, used extensivelyby us, is visual inspection of the wavefunction. This is normallyachieved by making contour plots of the wavefunction or cutsthrough the wavefunction - see Figures 3 and 4. States t hat caneasily be assigned are characterized by nodes in the wavefunc-tion which give a simple grid patte rn; some examples are given in

Figures 3a and 4a. For higher-lying states there may be noassignable nodal structure. This may be because inappropriatecoordinates have been used for preparing the cont our plots orbecause the sta te may be inherently unassignable.

Unassignable states can usually be associated with energyregions for which classical solutions of the same problem are

3

State 5

] !

. -

0 e 180 0 8 180

(b)

State 397 1 State 398I t

3

0 8 180 0 e 180

Figure 3 Contour plots of the wavefunction of 8 vibrational states ofLiCN: (a) low-lying states, (b) highly excited states. The plots are inscattering coordinates with the CN bond length, r , frozen at itsequilibrium value. 0 = 180" for linear LiNC an d 0" for linear L iCN.Solid lines enclose regions where the wavefunction is positive anddashed lines regions of negative amplitude. Nodal planes occur wherethe wavefunction has zero amplitude. The outer dotted curves repre-sent the limits of the classically accessible potential for each state inquantum mechanics the wavefunction can tunnel into this region.

(After J . R. Henderson and J. Tennyson, M o l . Phys., 1990,69, 639.)

chaotic. How classical chaos manifests itself in quanta1 systems

remains a controversial subject and is beyond the scope of thisarticle. However, one property t hat is observed in both mecha-nics is that even above the transition to classical chaos, someassignable solutions are f ound . Figure 4b depicts wavefunctionsof the H i molecular ion whose energies are well above theclassical transition to chaos. For three states the wavefunction



CALCULATING MOLECULAR SPECTRA-J. TENNYSON AN DS. MILLER 95

State 3 I

5 .

State 4

f 3 i

I I I I

State 148

' I' t @( I

I I I I

State 149

1 I II

State 150

I

5

!r 3 !

I I I II

State 151

, - I . , , , I

1 3 5 1 3 5

R R

I

5

r 3

1

5

r 3

1

1 3 5 1 3 5

R R

Figure 4 Con tour plots of the wavefunction8 vibrational states ofH i:(a) low-lying states; (b) highly excited states. The plots are in scatter-ing coordinates with8 frozen at90". The contours are asfor Figure 3 .

(After J. R . Henderson andJ . Tennyson, Chem. Phys. Lett . , 1990, 173,133.)

appears irregular. State number 150 has a clearly defined nodalstructure spread along a half horseshoe shape - and is called ahorseshoe state. O

The energy levels of the system can also be used t o give otherproperties of the system. The most obvious of these is thepartition function, the factor which normalizes the Boltzmanndistribution of molecules into particular states. The partitionfunction is needed if one wants to synthesize spectra as afunction of temperature.

8 Sample Resul t sVariational calculations have been performed on many tri-atomic and some tetratomic molecules. We choose a few of theseto give a flavour of what can be achieved.

8.1 H:: from Jupiter to ChaosPerhaps the most exciting project that we have been involved inconcerns the fundamental molecular ion H i . This seeminglysimple molecule consists of 2 electrons and 3 protons. Itsrelatively simple electronic structure means that potentialenergy surfaces for this molecule can be calculated more accu-rately by using first principles quantum mechanics', than byanalysing experimental data. This situation is probably uniquefor a polyatomic.

H i is rapidly formed when H, is ionized by the exothermicreaction

H, + H:+Hl+ H ( 3 )

As H, is the most abundant molecule in the universe, there aremany astronomical situations where H f s expected, although itremains largely unobserved. Similarly H i is easily formed in thelaboratory using a hydrogen discharge.

H f s an equilateral triangle in its equilibrium geometry.However, because of its light nuclei, H i undergoes large ampli-tude vibrational motion. We have performed a series of variatio-nal calculations on this molecule in an attempt to aid bothlaboratory assignment of the complicated infra red spectra ofthis molecule and its astronomical detection.

These calculations took a new significance following obser-vations of Jupiter by Drossart et aZ.13 These workers werestudying a known hot region near Jupiter's south pole. Inparticular they were looking for very weak transitions due tomolecular hydrogen. They observed these lines but at the sametime saw 28 other, unexpected, transitions which they wereunable to explain. It transpired that a spectrum like this hadbeen observed at the Herzberg Institute of Astrophysics inOttawa but only assigned as a result of these o b s e r ~ a t i o n s . ~ ~Because it was thought that the experimental spectrum wasprobably due t o Hf our calculations were enlisted.

Figure 5 gives a comparison of our calculated spectrum andthat observed by Drossart et aZ. Note that the observed line at4721 cm- ' is due to H,. The agreement is striking. Ourcalculations match the observed lines positions to within about0.02%. There are a number of surprising aspects of this firstextraterrestrial observation of H i . The actual transitions involvejumps of two vibrational quanta. Conventional wisdom is thatsuch transitions should be very weak, but the floppiness of H imakes the two-quanta transitions nearly as strong as the funda-mental or one-quantum transition, which has since also beenobserved in Jupiter. l Another surprising outcome of thisobservation was that the observed spectrum could only bemodelled by H i with a temperature in the region of 1000 K. Thisis a high temperature for a planet whose body is at ab out 200 K.

The observation of H i in Jupiter has stimulated a very activearea of astronomical research studying H i in Jupiter andelsewhere. This will be the subject of another article by us in thisjournal.

While the electronic properties of H i may be fairly simple, itsnuclear dynamics are extremely rich. Attention was focused onthis by the observation of a very unusual spectrum by Carr-ington and co-workers. This experiment prepared H i in adischarge and probed the resulting ions with an infra redlaser. Amass spectrometer was then used to monitor if any protons wereproduced by photodissociation.

Photodissociation is a common technique for investigatingmolecular systems and the resulting spectra are often quasi-continuous with maybe rather lumpy features. The spectrumobtained by Carrington and co-workers was extraordinary fortwo reasons. The energy of the laser was only a small fraction ofthat required to dissociate the ground state of the Hf and the



96 CHEM ICAL SOCIETY REVIEWS.1993

Figure 5 Observed (blue) and simulated (red) emission spectrum fromJupiter's southern polar region. The labels identify the rotationallevels of H i involved: R(J) is a J + 1 t transition.

(Reproduced by permission fromChem. B r. , 1990,26, 1069.)

spectrum, taken over a small range of wavelengths, containedover 26 000 narrows lines.

This spectrum, occurring as it does right at the dissociationlimit of the molecule, presents a formidable challenge to theory.Further analysis of the experiments showed that the protonsgenerally left the molecule with more energy than they got fromthe laser. This could only be rationalized by placing both theinitial and final state of the system above the dissociation limit ofthe molecule. Such quasibound states, physicists call them shaperesonances, a re well known alt hough such a profusion of them isunusual. They result from rotat iona l excitation of the moleculewhich causes the potential t o be distorted by a hump or barrierto dissociation. The molecule can get trapped behind this hump,but because of quantum mechanical tunnelling, the state onlyremains trapped for a finite time. The Uncertainty Principlemeans that states which are trapped only for a short time havelarge uncertainty in their energy and appear a s broad lines in any

spectrum. All the observed Hf lines are nar row suggesting thatthey are associated with long-lived states.The dissociation spectrum of Hf as observed has little struc-

ture. However, if a spectrum is synthesized by broadening theobserved lines, then the spectrum collapses to 4 egularly spacedfeatures. This intriguing piece of information has been a chal-lenge to theoreti cians. Classical analysis of the H i systemsuggests that the molecule is almost totally chaotic once themolecule has chance to become linear. Such geometries becomeaccessible at energies about one third of the way to dissociation.In chaotic systems it is usual for some solutions ('trajectories') tobehave in a regular o r quasiperiodic fashion. Classical mechani-cians argue that the structure in the H i spectrum is caused bysome underlying regular motion with the many individualtransitions being a reflection of the chaotic nature of the system.

This explanation leaves two questions. What is the underlyingregular motion and how does the fact that Hf obeys quantumand not classical mechanics alter this picture?

One proposal for the regular motion is that the atoms undergoa 'horseshoe' motion. O In scattering coordinates, see Figure 2c,this motion can be described in terms of the coupled motions of r

and R with 8 fixed at 90". A s R goes towards zero, r becomes largeas the other two atoms move apart to allow the central atomthrough. As R moves away from zero, r decreases again. Plottingthis motion as a function of r against R , where R has the range- co + + co, ives a horseshoe shape.

Quantum mechanical calculations on these high energyregions are very difficult, not least because the density of states

rises rapidly with energy. However modern supercomputers a ndthe adapt ation of the variational procedures described above touse methods based on finite elements rath er than basis functionshave allowed such calculations to be attempted. So, for example,a quan tal calculation has recently estimated the position of everybound vibrational stat e of H i. Figure 4 hows wavefunctionsproduced by these calculations. The states of H i are plotted inscattering coordinates with 8 fixed at 90".In these plots R has therange 0 - + co. State 150has a (half) horseshoe-like shape and isregular in the sense that nodes can easily be counted. The otherstates in Figure 4b, which are typical of many states in the highenergy region, have irregular structures.

Although there is clear evidence for horseshoe states in thequantum mechanical calculations, their exact role in the H idissociation spectrum remains controversial. So far the quantal

calculations have not been sophisticated enough to generateactual spectra: in particular no-one has managed to study theeffect of rota tion al excitation on the high-lying states. A decadeafter the first infra re d photodissociation spectrum of H i wasrecorded there is still clearly some way to go before it is fullyunderstood.

8.2 Van der Waals Complexes: Ar-N,Van der Waals bonding is the weak attr action which results fromcharge clouds adjusting to each others instantaneous fluctua-tions. It is the weakest of the chemical bonds and thus van derWaals molecules are only very weakly bound.

Van der Waals complexes are thus aggregates of stable, oftenclosed shell, molecules. The most studied triatomic van der

Waals systems are the complexes formed between the Noblegases (He, Ne, Ar . . . .) and diatomics such as H,, HF , HC1, andN,. These systems have been prototypical in the development ofvariational methods. This is because their flat potential energysurfaces mean tha t the concept of vibrational motion as a (small)displacement from an equilibrium geometry is unhelpful. Inter-



CALCULAT ING MOLECULA R SPECTRA-J. TENNYSO N AN D S. MILLER 97

nal coordinates, particularly scattering coordinates, have thusbeen used for some time for van der Waals complexes.

As mentioned earlier, the wealth of experimental data oncertain van der Waals systems has meant that empirical poten-tial energy surfaces have been derived for a n umber of complexesby performing cycles of calculation which involve guessing the

surface, calculating the transition frequencies predicted by thesurface, comparing with experiment, and repeating the pro-cedure until a satisfactory surface is p r ~ d u c e d . ~ ecently anattempt has been to use a similar procedure t o generate a dipolesurface for the Ar-N, complex.'*

For Ar-N, the experimental spectrum was obta ined in theinfra red by using wavelengths in the region of the N, vibrationalfundamen tal. ' As excitation of the vibrational mode of isolatedN, is forbidden, this experiment is sensitive to those N,'s whichare part of a complex. Furthermore the transitions do not occurexactly at the frequency of the forbidden N, tran siti ons becausevibrational modes of the van der Waals complex can also beexcited. The spectrum thus has several features, all of which aresuperimposed on a br oad conti nuous background due to pres-sure broadening. Pressure broadening is the result of using

relatively high pressures to produce the van der Waals com-plexes meaning that the molecules in the experimental cell can nolonger be considered as isolated.

The experiments on Ar-N, were performed at liquid nitrogentemperature s of about 77 K. For a van der Waals complex this ishot and means that for Ar-N, all levels are thermally occupied.The result is tha t instead of consisting of a series of discrete lines,the experimental spectrum comprises a number of features eachcontaining many transitions. Indeed the calculations found thatthe strongest individual transitions were ten times weaker thanthe features in the experiment. It was thus necessary to considersome 15 000 transitions in synthesizing the spectrum. These wereobta ined by calculating all vibration-rotation states of thesystem which lie below the dissociation limit of the complex.

Figure 6 compares the calculated and experimental spectra for

the Ar-N, complex. The agreement is not spectacular but thecomparison cont ains a wealth of information about the poten-tial used for the calculations and the interpretation of theexperiment. For instance, the broad shoulder in the calculatedspectrum 7 cm- from the N, fundamental frequency (taken atthe origin on the figure) is almost certainly a genuine featurewhich was lost in the process of removing the background due topressure broadening from the observed spectrum. Conversely,the misalignment of the peaks mar ked S(0) is due to the potentialenergy surface used in the calculation. It would appear thepotential f or the Ar to move about the N, was not flat enough inthe low-energy region. It will also be noticed th at the observedspectrum has a much greater extent than the calculated one. This

Figure 6 Observed (blue) and simulated (red) infra red absorbtionspectrum of the Ar-N, van der Wa ds complex. Both spectra are for atemperature of 77 K and a density of 1.7 amagat.

(After Garcia Allyon et al., M o l . Phys. , 1990, 71, 1043.)

is because the calculations only considered truly bound stat es ofthe system, whereas the higher frequency features are due toquasibound states trapped behind rotational humps in thepotential.

8.3 Potential Energy Barriers: Isomerization in LiCNMany molecules are found to have more than one stablestructure. These different structures are called isomers a nd ingeneral become more common as the number of atoms in amolecule increases since this also increases the number ofcandidate structures. Some triatomic systems exist as differentisomers. For example hydrogen cyanide is found with two linearstructures. Indeed, an unresolved astrophysical problem is whyHNC is almost as abundant in interstellar clouds as its muchmore stable HCN isomer.

Quantum chemical calculations have predicted that bothlinear forms of LiCN are also stable .20 However, in this case theLiNC isocyanide is predicted to be the more stable form. Thishas been confirmed by microwave experiments., ' AlthoughHC N has been the subject of many variational calculations ( e g .

ref. 6), LiCN is actually easier to work on. This is because inHC N the H-CN stretching mode is at a similar frequency to theC-N stre tch. The heavier Li atom means that the high fre-quency C-N mode is approximately decoupled from the othervibrational modes in the system. Furthermore, the theoreticalbarrier between the two linear isomers of LiCN is small, only3500 cm- ' (0.4 eV) or less than 5 quant a of the Li-CN stretch.As the LiCN is heavily ionic, the potential can be thought of asbeing Li + orbiting C N- with only a secondary sensitivity to theactual orientation of the CN -.

LiCN has thus been the subject of a series of calculations all ofwhich have frozen the CN motion. Within this model, the 6Y hstate of the system is already in the region of the barrier. Asrecent calculatio ns22 have obtained wavefunctions for the low-est 900 states of the system, the behaviour of the system below

the barrier, in the region of the barrier, and well above thebarrier can be studied.

Plots of low-lying and high-lying states of LiCN are given inFigure 3. The lower states can all be easily assigned by theirnodal pat terns. However, as more energy is put into the Li-CNbending motion, the states become increasingly irregular. It isthe bending motion which samples the isomerization barrier.The potential in this region is of course highly anha rmon ic andthus the assignments, based as they are on a harmonic picture ofmolecular vibrations, rapidly break down.

Above the barrier t o isomerization an increasing proport ionof states (more than 90% above state 400) have a highly irregularappearance and cannot be assigned. However, in this regionthere are series of state s with clear nodal patt erns. Perhaps themost pronounced of these series is the overtones of the isocya-

nide Li-CN stretch. These states go throughout the regionstudied. Thus st ate 400, depicted in Figure 3b, has 12 quanta ofstretch.

An interesting and topical quest ion is how the many irregularstates in a system such as LiCN would manifest themselves inobserved spectra. Calculations of the conventional absorptio nspectrum of the molecule in its ground vibrational states showfew unusual features. This is because the spectra are domi natedby excitations of the Li-NC stretching mode. The ionic natureof the molecule means that this mode has a large dipole whichleads to very intense transitions. Similar behaviour has beenobserved experimentally for HCN.

- A T ~ak-aiing-pi-cxuredT-ihe'.bekvioakuT ne ekcfteti $ratesof LiCN is obtained by considering their fluorescence life-t i m e ~ . , ~ his is illustrated in Figure 7 which has an interesting

structure: the assigned ('LiNC regular') states fall into a numberof series. One series, for which the lifetime decreases withexcitation, all have no stretching excitation and increasingqua nta of bending energy. Conversely the other series are for 1,2, 3, and 4 quanta of stretching excitation respectively, with thelifetime getting shorter as the degree of stretching excitation



98 CHEMICAL SOCIETY REVIEWS, 1992

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Figure 7 Fluorescence lifetime of excited vibrational s tates of LiNC withJ = 0. Ei is the energy of each state relative to the LiNC ground states.Assignments as ‘regular’ or ‘chaotic’ were made by analysing contou rplots of the wavefunctions such as those given in Figure 3.

(Reproduced by permission from Chern. Phys, 1986, 104, 399.)

increases. Within each series, increasing the bending excitationleads to slightly longer lifetimes.

All this can be understood by remembering that because of itsionic nature the transitions in LiCN a re very stro ng for stretch-ing states. Thus any excited state which can do so decays rapidlyby losing a quantu m of stretch. Conversely decaying by losingone or more qua nta of bending excitation is slow. However, asthe molecule is excited there is an increase in bend-stretchmixing. This means that the higher bending states take on asmall but increasing stretch character which shortens theirlifetime whereas increasing the bending character of the stretchexcited states lengthens their lifetime.

As the states become more excited the bend-stretch interac-

tion becomes stronger. This leads to a host of states that havedistorted wavefunctions which defy assignment. It also leads to asituation where all states have rather similar fluorescence life-times as represented by the clump of unassigned (‘chaotic’)states in Figure 7.

Analysis of classical calculations on LiCN shows that bendingexcitation is the main cause of chaot ic behaviour in the system.24Indeed regular (‘quasiperiodic’) stretching states can be foundover a wide energy range, possibly all the way to dissociation,provided that the bending excitation is kept low. Similar analysisof the quantal results24 show again that regular stretching statescan be found over an extensive energy range, see state 400 ofFigure 4b fo r example.

A direct quantitative comparison of classical and quantalresults is given by Figure 8. In this figure the energy is appor-

tioned to either stretch or bending. The resulting state/trajectoryis then assigned as either regular/quasiperiodic or irregular/chaotic and the app ropr iate entry made. It is clear that there is awide measure of agreement between the predictions in the twomechanics although the onset of irregular quan tum states occursat a slightly higher threshold energy. This behaviour has beencalled ‘quantum sluggishness’.

9 Conc lus ionsIn this article we have given a brief review of how one performsvibration-rotation calculations on small molecules. We havetried to explain why these calculations are impo rtan t and to givethe scope of problems that can be addressed with these calcula-tions by the discussion of some sample results. It will be noted

that all the problems discussed are for triatomic molecules. Intwo of these cases the calculations involved finding all the boundstates of the system - all the vibrational states of HS and all thevibrational and rota tional states of Ar-N,. These and mostother variational calculations on triatomics can now be per-formed so accurately that the major source of error is the

Figure 8 Comparison of the quasiperiodic (blank) - chaotic classicaldomain (hatched in blue) with ‘regular’ - ‘chaotic’ quantum states(red) for LiNC. Classical results were obtained by starting 50 trajec-tories with the energy indicated in the stretch and bend coordinates.Quanta1 results were obtained by partitioning energy between stretch-ing and bending modes. This cannot be done for most ‘chaotic’ stateswhich therefore do not appear on the figure.

(After J . Tennyson and S . C. Farantos, Chem. Phys., 1985,93,237.)

potential energy surface used for the calculation. This is still truefor the most accurate vibration-rotation calculations available,viz. our own on Hi, which reproduce a range of experimental

data with an error of about 1 part in 5000.The advent of variational vibration-rotation calculations has

really opened the way for serious and systematic theoreticalanalysis of the highly-excited states triatomic systems. Suchstates, as implied by the discussion of the H i and LiCNproblems, ar e often found a t energies where classically one findschaos. What the exact consequences of this are for quantummechanical wavefunctions or indeed spectroscopy is still amatt er for considerable speculation. At least theoreticians nowhave the necessary tools in their armoury to tackle suchproblems.

Our discussion has concentrated almost exclusively on tri-atomic molecules. This does not imply that larger molecules arewithout int erest. Quite the con trar y, these systems are extremelychallenging. A number of variational calculat ions on tetratomic

systems have now been performed. The computational aspectsof the tetratomic problem are rather different than fortriatomics.

Although it has not been considered here, in order to form thematrix elements necessary to construct the secular matrix, i t isnecessary to integrate over the coordinates of the problem. Ifany arbitrary potential function is to be used, this integrationmust be done numerically in 3N - 6 dimensions. It turns outthat for triatomic systems ( N = 3) this integration is much lesscomputationally demanding than the later step in the calcula-tion of diagonalizing the secular matrix. Fo r tetratomic system( N = 4), 6D numerical integration must be performed. As thenumber of integration points increases roughly as M 3 N - 6 ,vena modest number of points in each dimension such as M = 10leads to a thousandfold increase in computer time requirements.

It is clear that any serious advance in the area of calculatingspectra for larger molecules using variational procedures mustsomehow first break this integration bottleneck.

Acknowledgements. We thank our various collaborators, par-ticularly James Henderson and Brian Sutcliffe, from whose



CALCULAT ING MOLECULA R SPECTRA-J. TENNYSO N AN D S . MILLER 99

work we have quoted freely. We also gratefully acknowledgefunding from the Science and Engineering Research Council,the EEC, the British Council, the Research Corporation Trust,and NATO.

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J. Birnie, Spectrosc. World, 199 1, 3(3), 12.R. J. Le Roy and J. S . Carley, Adv. Chem. Phys. , 1980,42, 353.J. M. Hutson, Annu. Rev. Phys. Ch em., 1990,41, 123.P. Jensen, J . Mol . Spectrosc., 1989, 133,438.E. Kauppi and L. Halonen, J . Phys. Chem., 1990,94, 5779.S . Carter, N . C. Handy, and I . M. Mills, Proc. R . Soc.London, Ser. A ,1990,332,309.B. T. Sutcliffe, n ‘Methods in C omputati onal Chemistry’, Vol. 5 , ed.S. Wilson, Plenum, (in press).R. McWeeny, ‘Methods of Molecular Quantum Mechanics’, 2ndedition, Academic, London, 1989, Chapte r 2.R. N. Zare, ‘Angular Momentum’, Wiley, 1990.J. M. Gomez Llorente a nd E. Pollak, J. Chem. Phys. , 1988,88,1195;1989,90, 5406.J. Tennyson, S . Miller, and J. R. Henderson, in ‘Methods inComputationa l Chemistry’, Vol. 5 , ed. S . Wilson, Plenum, (in press).

12 W. Meyer, P. Botschwina, and P. G. Burton, J . Chem. Phys. , 1986,84, 891.

13 P. Drossart, J. -P. Maillard, J. Caldwell, S. J. Kim, J. K. G. Watson,W. A. Majewski, J. Tennyson, S. Miller, S . Atreya, J. Clarke, J. H.Waite Jr., and R. Wagener, Nature (London), 1989,340, 539.

14 W. A. Majewski, P. A. Feldman, J. K. G. Watson, S . Miller, and J .Tennyson, Astrophys. J. , 1989,343, L51.

15 S. Miller, R. D. Joseph, and J. Tennyson, Astrophys. J. , 1990, 360,L55.

16 A. Carrington, J. Buttenshaw, and R . A . Kennedy, Mol. Phys. , 1982,45,753; A. Carrington an d R. A. Kennedy, J . Chem. Phys. , 1984,81,91.

17 J . R. Henderson and J. Tennyson, Chem. Phys. Lett . ,1990,173,133.18 A. Garcia Allyon, J. Santamaria , S. Miller, and J. Tennyson, M o l .

19 A. R. W. McKellar, J . Chem. Phys. , 1988,88,4190.20 R. Essers, J. Tennyson, and P. E. S. Wormer, Chem. Phys. Lelt.,

21 J. J. van Vaals, W. L. Meerts, and A . Dymanus, Chem. Phys. , 1984,

22 J. R . Henderson and J. Tennyson, M o l . Phys., 1990,69,639.23 J. Tennyson, G. Brocks, and S . C. Farantos, Chem. Phys. , 1986,104,

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jonathan tennyson and steven miller- calculating molecular spectra

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