brian t. sutcliffe and jonathan tennyson- variational methods for the calculation of rovibrational...

8/3/2019 Brian T. Sutcliffe and Jonathan Tennyson- Variational Methods for the Calculation of Rovibrational Energy Levels of

1/12

J . Chem. SOC.,Furuduy Trans. 2, 1987,83(9), 1663-1674

Variational Methods for the Calculation of RovibrationalEnergy Levels of Small MoleculesBrian T. Sutcliffe

Chemistry Department, University of York, Heslington, York YO 1 5 D DJonathan Tennyson

Department of Physics and Astronomy, University College London, Gower Street,London WCIE 6BT

Variational rovibrational calculations performed on the molecules H,O andCH; are discussed with a view to pin-pointing the best solution strategy foreach system. While all the methods discussed appeared to be reliable forthe low-lying levels of water, CHf has proved a more testing system. Arecently proposed method for calculating highly rotationally excited statesis applied to the I = 10 levels of CH;. Comparison of these calculationswith those of Carter and Handy suggests that at this level of rotationalexcitation it will be necessary to consider full rovibronic coupling effects inthe characterisation of any spectra of this floppy system.

It is nearly twenty years since W atson derived w hat is now w idely accepted as th esimplest and most elegant quantum-mechanical form of Eckarts3 classical vibration-rotation Ha m iltonian . W atsons Ha m iltonian differed by on e term (th e so-called Watsonterm) from tha t wh ich had become commonly used o n exper imen ta l g r o u n d ~ . ~ha tWatsons Hamil tonian is correct is now beyond doubt and i t has been widely used.How ever, as was noted by Sayvetz in 1939, this formalism nee ds to be ad ap ted fo r thecase of large am plitude vibrational m otion , and, as sh ow n by Watson himself,6 it is notapp ropria te for l inear molecules fo r which the W atson term becom es singular.Although a variety of H am iltonian s have been used for variation al calculations,2 inmany ways the derivation by Watson marks the start of variational calculations on thevibration-rotation spec tra of polyatomic m olecules. In the past two decade s severalsignificant steps have been made on the path to accurate variational calculations onpolyatomic systems. As Handy has been involved in many of these stages, this wouldappear to be an appropria te time an d p lace to survey past work an d poin t to what theimmediate future would appear to hold in the increasingly rich area of bound statenuc lear motion calculations. In discussing previous ca lculatio ns we will not attem pt tobe comprehensive, but will instead focus on tw o co ntrast ing tr iatomic molecular systems,water and CH;. Between them these systems hav e form ed the testing ground for manyideas on how best to calcula te the rovibrational spec tra predica ted by a given potential.In the development of ab initio bound-state nuclear motion techniques, water hasoften been used as the prototype, near-rigid molecule for which the Eckart conditionsare undou bted ly valid. It has a high barrier to linearity, ca . 11 000 cm-,- a n d its fewelectrons have m ad e it the subject of many state-of-the-art electronic stru cture calcula-tions.- Expe rimental da ta on its low-lying rovibrational states ar e also widely avail-

In contrast , CH; is a Renner-Teller system w hose grou nd ele ctron ic state has abar rie r t o l inea ri ty of le ss t han 1 0 O O ~ m - . ~ ~his means that the system is not onlya n d h av e y ie ld ed se ve ra l ( se m i- ) e m pir ic al po te ntia l- en er gy ~ u r f a c e s . ~ * ~ * - ~ ~

1663


2/12

1664 Variational Rovibrational Calculation on Small Moleculeshighly anh arm on ic with a large-amplitude ben ding m otion, but also displays non-Born-Oppenheimer effects due to coupling with the vibrational levels of the other Renner-Tel ler component .26 As yet there is no recorded spectrum of CH Z, al though the ion isknow n to be an importan t const i tuent of som e interstel lar

TheoryWithin the Born-Oppenheimer approximation, the Hamil tonian for the motion of thenuclei in an N-atomic molecule is given by

h 2 3 N 12 i = l mikful,=-- v:+ V ( q )

where the potent ial depends only on q, the 3N - internal coordinates of the system.This H amil tonian suffers from the st rong disadvantage that i t st il l contains the cont inuou sspectrum given by the centre of mass motion of the whole system. There is also nodistinction between motions that are usually associated with vibrations and thosecommGnly ascribed to rotations. These prob lem s, besides making direct calculationwith Iffullumbersome, also lead to considerable difficulties in the interpretation ofresults.The translational part of the Hamiltonian can be separated off by defining thecentre-of-mass motion coordinate1 1x=-c- -x . I , M=Crn, .M i mi 1

Of the remaining 3N - degrees of freedom , 3 (o r 2 for a l inear system) c an be associatedwith the overall rotation. Th ese rotationa l coordina tes can be defined in terms of theEuler angles re quired to ro tate the laboratory fixed axes into the fram e of the m olecule,whose remaining 3N - ( 3 N - ) internal coordinates carry the vibrational motion.Fprmally one is free to choose any coordinate set and system of axes and t ransformHfullnto the ap prop riate body-fixed form.An ex am ple of such an em bedding are the tw o condit ions proposed by Eckart3 whichare the basis of Watson's Hamiltonian'

(3fiw s expressed in normal coordinates, Qk , which are mass-weighted displacementsfrom some fixed (equi l ibrium) geometry. In the above, ra re the Cartesian com pon entsof the Coriolis terms, II, are the components of the total angular momentum withrespect to the body-fixed axes and the matrix p is closely related to the inverse inertiatensor. A more detailed discussion of these terms can be foun d in ref. ( 1 ) .A procedure for embedding an arbi t rary set of internal coordinates and axes hasbeen given by Sutcliffe.** Application of this procedure is not without its problems.The a lgebra is usually lengthy and complicated, resulting in at least one acknowledgedfailurez9 an d oth er incorrect results. Recently, however, Handy3' has ad ap ted Sutcliff 'sprocedure so that the a lgebraic language REDUCE can be used to perform the t ransforma-t ion. Additional ly, som e care must be taken in choo sing the internal coordinates, asthe domain for which the resulting body-fixed Hamiltonian is well behaved may notencompass the physical domain of the problem."Recently, we have used Sutcliffe's proc edu re to derive a body-fixed H am iltonian fortriatomic systems in terms of the distance between atoms 2 a n d 3, r , , he distance, r 2 ,f rom a n arb i t ra ry poin t on r l to atom 1 , and the angle, 8, between rl a n d r2.32 Once


3/12

B. T . Sutclifle and J. Tennyson 1665integration has been performed over the rotational coordinates, whose motion is carriedby rotation matrices IJk), the Hamiltonian for the internal coordinates can be written

( 4 )i - @ ) + Z y + $ l ) + @ST - V R V R + v ( r l , r 2 , e,with

The form given above is appropriate for the z-axis embedded along rl . The embeddingalong r2 is obtained simply by making the exchanges r , e 2 and p ,- 2 , In using theoperator in this form any integral over the internal coordinates must be performed withthe volume element r f r ; sin 8 dr , dr2 d6. The range of rj is (0,a) nd that of 8 (0,T ) .The coefficients Csk are the usual step-up and -down coefficients

c;k = [J(J+ 1 ) - ( k f ) ]1 '2 (9)and the reduced masses are given by

p ; l = my'+ my'p;z '=g(rn; l+m;l) -m; lpZ1= m;'+g2rnT1+ (1-g)'m,'

withg = ( r2- / r2 ( 1 1)

whece v is the distance from atom 2 to the point a t which r2 cuts r l .HSTgeneralises a number of previously known Hamiltonians. In particular, whenr2 links atoms 1 and 3 ( g = l ) , the resulting bond length-bond angle Hamiltonian is inthe form of Lai and H a g ~ t r o m . ~ ~ - ' ~hen r2 cuts r , at the ce$re of diatomic mass[ g = m2/(m,+ m3)], he resulting Hamiltonian, in which p;; = K y )= k?; = 0, is wellknown from atom-diatom scattering c a l ~ u I a t i o n s . ~ ~ * ~ ~discussion of other coordinatechoices with this Hamiltonian can be found in ref. (38) . HsT also provides a choice ofaxis embedding as the z-axis can lie along either rl or r,; a further generalisation whichallows the z-axis to be embedded along any line between these has just been proposed,39but difficulties remain over its application.


4/12

1666 Variational Rovibrational Calculation on Small MoleculesUntil recently, fully coupled, variational rovibrational calculations were limited toproblems with low angular momentum, J s 4 . This restriction was caused by the rapidincrease in the size of the secular matrix with J. However, recently the use of a

two-variational step has been proposed4' as a means to obviate this problem. Essentiallythis involves using the solutions of a Hamiltonian in the form

to solve the full problem. The saving arises because not all solutions of eqn (12) arerequired to give converged solutions of eqn (4), and because the resulting secular matrixhas a sparse structure which can be utilised computationally.40 Chen et aL4' havedeveloped a similar procedure based on Watson's Hamiltonian.Calculations on the Water MoleculeStandard Variational TreatmentsThe earliest fully variational calculation on the low-lying rovibrational levels of thewater molecule, indeed on any triatomic, was by Bucknell and Handy.42 The calculationswere performed on a number of surfaces using a Hamiltonian derived by Handy andco~orkers ,4~hich was later44 shown to be equivalent to, but more cumbersome than,Watson's Hamiltonian of eqn (1).Another notable early variational calculation was performed by Lai33using the bondangle-bond length coordinates discussed above. Unfortunately this work, though widelyreferred to [see ref. (24) and (41) for example], was never published. However, Lai'swork does appear to be the first fully variational vibrational calculation which did notrely on the Eckart conditions and the concept of an equilibrium geometry.The early work based on the Eckart approach by Bucknell and Handy was consoli-dated in two papers by Whitehead and Handy (WH),44745 oth of which consideredwater. These works presented and applied an efficient variational algorithm whichemploys a basis set of harmonic oscillator functions (weighted Hermite polynomials)and constructs the matrix elements using Gauss-Hermite quadrature. This method isstill the one of choice for solving the Watson Hamiltonian for non-linear molecules [seefor example ref. (8 ) , (15), (41), (46) and (47)] and provides the benchmark againstwhich alternative variational techniques must be calibrated.Calculations on the water molecules similar to those of WH were also performed inthe mid-seventies by Carney et al.24(CCL) and Sorbie and Murrell.' Both these worksused the potential derived from a perturbation theory based fit to experiment due toHoy et a1.22(HMS) as a starting point to providing improved potentials. Table 1compares the results of a number of calculations on this potential.The WH method essentially settled the argument over how to perform variationalcalculations on the low-lying levels of a fairly rigid molecule such as water (see laterfor a discussion of less orthodox solution strategies). This method was used by Sextonand Handy'' in an extensive a6 initio study on the fundamentals of water. More recentlyChen et ~ 1 . ~ 'CMW) have used the WH method to solve the J = 0 problem as the firststep in a two-step variational study of the rotational states of water with J S 10.In order for a direct comparison of variational rovibrational methods to be meaning-ful, it is necessary for the methods to be applied to the same potential-energy surface.In the case of water such a comparison is difficult because of the variety of surfacesthat have been studied. The most frequently used potential is that of HMS; table 1compares calculations performed on this surface. The most obvious conclusion to bedrawn from table 1 is that all theoretical results are in good agreement, with the majordifferences still attributable to small changes in the potential. For example the two mostextensive calculations on the same surface, by Lai33and CCL,24 agree to within 1 cm-'


5/12

B. T. Sutclife and J. Tennyson 1667Table 1. Com parison of variational calculations on the H M S potential: ( a )potential expansionin displacement coordinates and ( b ) potential expan sions in SPF coordinate^.^^'^^

( a ) ( b )Lai CCL W H b BQ CCL CC C M W exptl

v 1 v2 v3 ref.: 31 24 44 49 24 48 41 200 0 00 1 00 2 01 0 00 0 10 3 01 1 00 1 10 4 01 2 00 2 12 0 01 0 10 0 2

46521597315937173821468352995403616868956950741975207626

46521597315937173821468453005403617668466950742075217626

46521598 15973160 31603717 37163821 382053005402684269 5 1

46391596315636593758467952375335616967776877721 172617453

46391595315436593758467752365334615967756875721072597452

1596 15953156 31523659 36573759 375646675235533 1613667756872720172507445All frequencies in cm- are relative to the (0, 0,O) ground state. Corrected results reported asa personal communication by Huber, whose o wn calculations agree with WHs to within 1 cm-.

for the lowest eight states. Th erea fter Lais results are variationally slightly sup erior,du e to the use of a relatively large basis. Murrell a nd Sorbies calcu lation8 o n th e HM Spotential is not included in table 1 because they ap pea red to ha ve an incorrect sign inone of the potential coefficient^.^^ The resul ts of Cropek and Carney4 ( C C ) , whichused the bon d length-bond angle coordinates of Lai , are variational ly the best for thesecond version of the potential.Th e sensitivity of the results to the po tential is i l lustrated by co m parin g the resultslisted u nd er headings ( a ) n d ( 6 ) of table 1 . Those in co lumn ( a ) sed th e Taylor seriesderived by HMS in its original displacement form, while column (6 ) gives resultsobtain ed using the sam e coefficients but with Simons-Parr-Finlan (SP F) coordinate^.^'The results computed with the latter version of the HMS potent ial are clearly muchcloser to experiment. Rec ent calculations by CM W 4 suggest that the best force field istha t du e to Lai a nd H a g ~ t r o m . ~ ~Table 2 surveys a6 initio calculat ions of the fund am ental vibrat ions of water. Thetable conta ins a numb er of interesting features. The first is the great improvem ent overthe last dec ade or so in the quality of electronic s tructu re calculatio ns tha t it has becomepossible to do. While it is apparent that the accuracy of this step in the calculation isof para m oun t importance in obtaining good resul ts , analysis of the results of K raem eret al. 2 (KRS) i l lustrate the importance of including good nuclear m otion calculat ions.Their publ ished fundamentals , based upon harmonic frequencies with anharmoniccorrections, suggest that their results are fairly poor with two fundamental frequenciesmore than 30 cm- in erro r. How ever, variational calculations p erfo rm ed on their surfaceby Sexton and H andy (S H ) suggest that the KR S surfa ce actually gives the best cur renta6 initio values for the fun dam ental frequency of water.When com paring the errors in the fundam entals , it should be remem bered that eventhe best of the variational electronic stru cture calcula tions on w ater give energies manythousands of wavenumbers above the t rue energy. This means that the calculation ofa surface well above, but parallel to, the actual surface is being attempted. In this


6/12

1668 Variational Rovibrational Calculation on Small MoleculesTable 2. Comparison of ab initio calculations of the vibrational fundamentals of the watermolecule'

fundamental/cm-'workers ref. type of surface year V l v2 v3

BHW HHKDBSP'KSHKRS

SHexptl

RE S

K R S ~

424510911

1412121520

SCFSDCISDQSDQMBPTCASSCFM R D C IMRDCIMRDCI

197419761976197819791982198219821984

1728162916331649161016451647161316231595

4045372036893619370236913662365436673657

41393 80737983753378937943787375737523756

a The year and type refer to the data of and method used for solving the electronic structureproblem. Vibrational analysis of Romanowski and Bowman."ibrational analysis by KSH.I4Vibrational analysis by SH.I5context it is interesting to note that although the properties of the various electronicstructure methods with respect to dissociation has been the subject of much debate, itis the non-dissociating bending coordinate whose fundamental has proved the mostdifficult to calculate accurately.Another source of error in these calculations is the ability analytically to continue(or fit) the calculated grid of points to produce a potential-energy surface. Typicalfitting errors are ca. 2 ~ m - ' , ' ~hich is significant if highly accurate results are de~ i r ed . '~WH side-stepped this problem by performing electronic structure calculations at thegrid of points required for their Gauss-Hermite quadrature, but for reasons discussedlater in the section on CH;, this method has not found general favour.The development of sophisticated procedures for finding (analytic) derivative^^^suggests that one might be able substantially to reduce the number of grid points requiredto characterise an ab initio potential if these points were supplemented with the additionalinformation available from derivative calculations. However, this leads to two problemswhich so far remain unresolved. How should this reduced grid be distributed formaximum efficiency'' and what relative weighting should be given to energy points asagainst derivatives?An alternative strategy for using these derivative techniques has been pioneered byCaw and Handy.5s Their approach is to calculate directly, from the derivatives of themolecule at its equilibrium geometry, the constants commonly used by spectroscopiststo represent their data . This method, which of course is not variational , relies on thevalidity of the perturbed harmonic oscillator-rigid rotor model from which the para-meters are derived. The approach has shown that high-order parameters (force constantsetc.) can be calculated for molecules such as water with surprising accuracy usingrelatively crude theoretical techniques.Alternative MethodologiesBesides the standard variational techniques discussed above, the nuclear motions of thewater molecule have been the subject of a large number of other calculations. If weare to say anything about the future of a6 initio rovibrational spectroscopy it is interestingalso to consider some of these. The method with, perhaps, the longest pedigree as faras rovibrational calculations on water are concerned is the (non-)rigid bender method


7/12

B. T. Sutclife and J. Tennyson 1669as em ployed by Bunker a n d c o - ~ o r k e r s . ~ ' ~ ~ * ~ ~his technique d oes a full t reatment forthe bending a nd ro tat ional coordinates, with a perturbative treatment of the more rigidstretches. Perhap s the mos t interesting recent result of this work is the conclusion byBeardsworth et ~ 1 . ~ ~hat the standard spectroscopic approach of using perturbat iontheory to trea t the effects of anh arm onicity , Coriolis co uplin g and centrifugal distortionwill be 'unsatisfactory very often'. This em ph asi ses the desirability of using variationalapproaches to t reat the nuclear motion problem.An a ppro ach in direct contrast to the non-rigid ben der m ethod has been employedby Lawton a nd C hild57 o study the highly excited stretching states of water. Their localm odes analysis gave qu ant um , semi-classical a nd classical results into the region (a bov e30 000 cm -') where classical calculations fou nd many c hao tic trajectories. If the barrierto linearity in water is only ca. 1 1 000 cm -', analysis of oth er triatomics suggests5' thatthe onset of chaos would be considerably lowered by inc ludin g the bend ing motion.Semi-classical studies which include the bending mode are rare, a notable exceptionbeing the work of Colwell and Handy." W ork using local m odes for water, studyingfor exam ple th e effects of high ro tatio nal excitation,6' is still co ntin uin g.A number of author^^^^^*,^' have used an S C F app roac h to calculate the vibrat ionallevels of th e water molecule. As in electronic stru ctu re calculations, it is necessary tofollow th is ste p with a configuration intera ction calculation if accu rate reuslts are to beachieved. Romanowski and Bowman5* compared S C F CI ca lcu la t ions wi th an approxi -mation which m ade an adiab at ic separat ion between the st retching a nd bending modes.We note, however, that they omit ted the Watson term from the Hamil tonian andconsidered n o more than thre e qua nta of vibrational excitation. Even a t this relativelylow level of excitation some of the levels were found to be up to 100cm-' too low inthe adiab atic model. Th is is in line with calcu lations which suggest that such adiab aticsepa rations are of l imited accuracy for highly excited ben ding m odes.62 Recent analysisby Jo h ns o n a n d R e i r ~ h a r d t ~ ~ ' i 'uggests that the reverse adiabatic approximation showsmore promise for water in highly excited states.Burden an d Q ~ i n e y ~ ~BQ ) uti lised the S C F C I method , but with a numerical SC Fprocedure , on the H M S water potential. Their resul ts com pare favourably with theother variational trea tme nts of this potential s how n in table 1 , suggesting that this SCFCI procedure can indeed give a compact and accurate representat ion of the nuclearmotion problem. We note, however, that BQ's resul ts for the H C N pro blem were lesssatisfactory. In a seco nd work, Burden64 used hyperspherical coo rdina tes to treatproblems ranging from e l to water and ArH Cl using the same algorithm. High accuracywas not aimed for in these calculations.Finaily, mention m ust be m ade of the recent use by H andy,65 following a proposalby Coke r and Watts ,66of the quan tum M onte Ca rlo method for a vibrat ional calculat ionon water. He de mon strates that this procedure, which is not variat ional but rather aimsat a direct solution of the nuclear motion Schro dinger equat ion, can be ma de viable forcalculating zero-point energies. How ever, at this t ime it is uncle ar how the method canbe ada pte d to deal routinely with vibrational excitation a nd h ence transition frequencies.Calculations on the CH; RadicalIn the absen ce of an observed spec trum of the CH,' radic al, the discussion of this systemwill c onc entra te solely o n a b initio calculations. Th e original calculation s were performedby Bartholomae et al.(j7 (BMS) who performed electronic CI calculat ions at the gridpoints requ ired for Ga uss- He rm ite integration alon g their vibrational coordinates. Theyused this surfa ce for vibrational calculations p erfo rm ed with Watson's Ham iltonian,

-i. The reader is referred to this work fo r a more comple te l i s t of non-variat ional vibrat ional calculat ionson water.


8/12

1670 Variational Rovibrational Calculation on Small Moleculeseqn (1). It was soon apparent that while the equilibrium structure of CH, is bent, thelarge-am pli tude bending m ode samples l inear geometries. BMS suppre ssed the result ingsingularities in the Watson term by weighting the vibrational wavefunctions so as no tto sam ple l inear geometries.The chal lenge of this system was taken up by Carter and Handy25(H C ) who usedWatsons isom orphic Ham il tonian6 for l inear molecules. In ord er to perform thesecalcu lations H C refitted BMSs surface to the for m of Sorbie an d M urrell.8 This i llustratesa drawback of performing a6 initio calcu lations at a set of grid po ints (a s originallyproposed by W hitehead and Han dy45) rom the point of view of com paring calculations.Although performing calcu lations only a t a set of grid points implicitly assumes a fit tothe surface, namely in terms of the orthogonal polynomials used in the quadratureprocess, that fit is not an explicit one. Thu s there is no potential-energy surface thatcan be used for a comparat ive calculat ion and the vibrat ional calculat ions cannot beunam biguously reproduced by an al ternative metho d. This has mea nt that this methodof eliminating the explicit representation of the potential has fallen from favour.Th e H C potential was utilized by us (TS)68 to perform fully cou ple d rovibrationalca lcu la t ions on CH; us ing sca tter ing coord ina tes . These ca lcu la t ions a l ~ o ~ e x pl o re dheefficiency of neglecting Coriolis interactions in the two em bed dings of H,, discussedabove. In turn, these calculat ions were used by C arter et (CHS) to t5st theircalculations, which used a bond length-bond angle Hamil tonian similar to Hsr, butwith the z-axis placed symmetrically along the bisector of the H C H angle. Furthercalculat ions by us (ST)32 onfirmed that the bon d length-bond angle coordinates givea m uch m ore efficient representat ion fo r this system than the scat tering coordinates usedpreviously by us. Finally, Ca rter an d Ha nd y (HC 2)26 augm ented BMSs electroniccalculat ions on the ground state with calculat ions on the fi rst exci ted state and usedthese for variat ional non-B orn-O ppenheim er calculat ions which included (in some partsapproxim ately) the vibrat ional coupling between the two surfaces.Table 3 summarizes the results of the various vibrational ( J = 0) calculations. It isapparent from this table that most of the resul ts show a fair measure of agreement .Besides the H C2 vibronic results, the exce ption to this are the results of BMS. Th eircalculat ions appear superior, in a variational sense, to the others because of the lowerabso lute energies they obtained . How ever, i t would app ear that this improvement is anartifact of the W hitehead-H andy m e t h ~ d ~ ~ , ~ ~nd in par t icu lar the W atson H amil ton ian .This uses an embedding which is valid in only the small amplitude limit, but can failfor larger ampli tude m otion as the coordinates can leave the t rue dom ain of the problem .In part icular, i t is possible for the bending coordinate to pass through l inear geometriesand to sample regions which should be related to the t rue problem by a rotat ion andnot a vibration. This difficulty was even enc oun tered for the low-lying states of waterby Chen et aL4 It probably gives as severe a restriction on the utility of the Wkitehead-Ha ndy m ethod as does the singularity for l inear geometries en countered in H , .I t was jus t thi s p roblem which prompted C arter and Han dy to trea t CHZ ( and C H 2)as a quasi-l inear molecule.2s As the com parison of results shows, this method givesreliable results, but suffers from the disadvantage that the bending coordinate is beingrepresented by a basis set expanded about the incorrect equilibrium structure. Theresulting inefficiency m akes this met ho d unattractive for oth er quasi-linear systems andto our knowledge it has not been used subsequently.

Similarly, ou r (TS ) scattering c oord inate ca lculation s gave a reliable representationof the lowest 1 1 vibrational states of CH,, but the system is not naturally representedas a collision complex between C + an d H Z . This mean t that a relat ively large num berof basis functions (432 for each symmetry) were req uired to converge the higher states.Although this basis set is larger than those used in the corresponding calculations, i tt We note tha t a t e rm - ( h 2 / m , j cos 8 ( d / d R , dRz )was omi t ted f rom fiv in this reference.


9/12

B. T. Sutclife and J. TennysonTable 3. Comparison of vibrational calculations a on CH;

1671

B M S HC TS CHS ST HC2v l v 2 v3 ref.: 67 25 68 69 32 260 oo 0 -70519 .6 -70398 .7 -0 2O 0 908 7180 4O 00 6' 01 oo 0 2935 29990 oo 1 3162 327 11 2O 00 2O 10 8O 01 4O 00 4* 10 l o o 0

-70400.67 18.3161 1.02770.72998.83270.63678.63966.841 17.44551.14839.45598.5

-70 400.6718.31610.92770.22998.83270.63678.43966.741 15.44550.14839.25587.6

-70 400.6718.41611.42772.72998.93270.73678.63966.9

938.12021.83293.62998.83268.93898.14181.74717.14952.85240.0

a Th e vibrational band origins, in cm-', are relative to the (0, Oo , 0) ground state. The labelling isas for a linear triatomic; for further details see text.should be noted that in o u r methodology7* diago nalisa tion is the rate-limiting step, du ein part to reduction in integration times achieved by taking adva ntage of the propertiesof orthogonal polynomial expansion^.^^ In cont ras t , the speed of the then currentalgorithms of Carter a nd H andy w as very m uch d etermined by integrat ion t imes.72 Wenote, however, that Ca rter an d Ha ndy have since implem ented a n algorithm35 basedupon the optimised choice of integration points suggested by Schwenke and Truhlar7'in which diagonalisation is also the rate-limiting step.T he CH: vibrat ional calculat ions label led C H S and ST both used the same Hamil-tonian (HSTn its bond length-bond angle form) but differed in their choice of basisfunct ions and integrat ion procedures. In part icular, several of the angular integralswhich CHS evaluated using Gauss-Legendre integrat ion were shown to have analyt icform s by ST. However, the sm all differences between th e predicted ba nd origins isalmost certainly attributable to the smaller basis set used by ST, which contained atleast 40% fewer funct ions than any of the o ther calculat ions presented in table 3, withthe exception of t he anomalous B M S calculations.Finally, the colum n label led HC 2 presents the results of Carte r and Handy's Renner-Teller calculations. It is clear from these results that fo r CH,f the ben ding m ode is veryseverely pertu rbed by n on-B orn- Op pen heim er effects, even at very low levels of excita-tion. This calculation c onsis tuted the first fully mo de coup led vibron ic calculation a ndwill almost certainly be a prelude to similar investigations of other systems with morethan one low-lying electronic state.All the calculations discussed abo ve cons idere d only low-lying rotatio nal levels withJ S 2 . However, CH,f is a l ight molecule and thus does not need much rotat ionalexcitation for the separa t ion between rotat ional and vibrat ional levels to becom e comp ar-able. Table 4 gives the results of a two-step variational calculation on the J = 10 levelsof CH,', a calculat ion that w ould require a secular matrix of dim ension 7382 if it wereperformed in a single variational step. As can be seen, convergence to better than0.1 cm-' can be achieved using a matrix of dim ensio n only 880.The rapidi ty of this convergence, which is muc h bet ter than foun d in H 2D +,40s duein part to the fact that k, the project ion of the angular m om entum along the body-fixedz-axis, which was placed alon g on e C-H bo nd in these calculation s, is a nearly goodquantum number. This can be seen from the pairs of nearly degenerate levels withdiffering parity . Levels 3 , 4 a n d 6 with e parity are nearly degenerate with levels 2, 3


10/12

1672 Variational Rovibrational Calculation on Small MoleculesTable 4. Low-lying levels of CH; with J = 10 calculated using a two-step variational procedureand bond length-bond angle coordinates.a

J = 10 J = 1 6level N = 40 80 120 160 100

12345678910

801.912889.9521133.4661423.3031512.0061769.9051822.3972160.9652227.8902422.275

801.858889.8201133.3461423.2211511.7931769.8341821.7 192160.9072226.636242 1.267

801.856889.8151133.3581423.2031511.7601769.8301821.6822 160.9042226.564242 1.242

801355889.8 151133.3571423.20215 11.7591769.8291821.6802 160.9032226.5552421.222

937.8461131.8611423.2891769.7701872.1472 160.8912232.7132588.8192658.0852968.003

Frequencies, in cm-, are relative to the J = O ground state.an d 4 of the f s tack and correspond to k s of ca. 2, 3 an d 4 respectively. States with kequal to 0 occur only for the e stack and thus levels 1 , 5 an d 10 can b e assigned as thelowest members of the rotat ional manifold of the (0, Oo, 0), (0,2, 0) a n d (0,4, 0)vibrational states. As the J = 10 rotat ional manifold for each of these states has 11 ean d 10f members, this m eans there is a large degree of o verlap between the manifoldsbelonging t o different vibra tional states. As HC2s calculat ions show ed that the bendingexcitations of CHT w ere strongly pertur bed by vibro nic interactions, a com plete determi-nat ion of these levels would require a fully coupled rovibronic treatment.

ConclusionsThe calculat ions considered above on water and CH; suggest that , for t r iatomics a tleast, reliable variational methods are now available for obtaining rovibrationalwa vefu nction s. T his is certainly tru e for relatively low degrees of vibratio nal excitationa n d new r n e t h o d ~ ~ ~ ave opened an exciting new world of high rotational excitationto acc urate calculation. How ever, there is stil l much work to be don e on highly excitedvibrational states. For exam ple, the observation of ca. 27 000 lines in a small region ofthe near dissociat ing spe ctrum of HT by Carrington an d Ken nedy74has highlighted theinadequacy o f o ur knowledge in this spectral region and provided a tantal ising targetfor the theoretician to aim at. So far only very approximate quantum-mechanicalca lcu la t ions have been possib le on th is p r ~ b l e m . ~Some promising work in this direction is now becoming available, mostly on sim-plified or m odel systems. Am ongst these the use of the discrete variable tech niqu es byL ight a n d c o - ~ o r k e r s ~w hich have alrea dy been used for the low-lying levels of water51),the at tem pt to stabi lise highly excited states outside the variat ional app roa ch by C han get d. and the use of dynamical basis sets as suggested by Blanco and Heller7 perhapsshow the most p romise for the fu ture. Car ter and Handy79have recent ly dem onstrated,again using CH;, how to cons ider higher exci ted states, in the fashion of Cha ng et a ~ , ~ and still obtain variational results.References

1 J. K. G. Watson , Mol. Phys., 1968, 15, 479.2 G. D. Carney, L. L. S prande l and C. W. Kern , Ado. Chem. Phps., 1978, 37, 305.


11/12

B. T. Sutclifle and J. Tennyson 16733 C. Eckar t , Phys. Rev., 1935, 47, 552.4 R. Wertheimer, Mol. Phys., 1974, 27, 1673.5 A. Sayvetz, J . Chem. Phys., 1939, 6, 383.6 J . K. G. Watson, Mol. Phys., 1970, 19, 465.7 P. R. Bunker and J . M. R. S tone , J. Mol. Spectrosc., 1972, 41, 310.8 K. S. S orb ie and J. N. Murre l l , Mol. Phys., 1975, 29, 1387.9 P. Hennig, W. P. Kram er , G. H. F. Diercksen and G. Strey, Theoret. Chim. Act a, 1978, 47, 233.10 B. J. Rosenberg, W. C. Ermle: and I. Shavit t , J. Chem. Phys., 1976, 65, 4072.11 R. J. Bartlett, I . Shavi tt and G . D. Purvis 111, J. Chem . Phys., 1979, 71, 281.12 W. P. Kraemer, B. 0. R o o s a n d P . E. M. Siegbahn, Chem . Phys., 1982, 69, 305.13 P. Saxe, H. F. Schaefer 111 a n d N. C. Handy , Chem. Phys. Lett . , 1981, 79, 202.14 P. J. Knowles, G. J . S ex ton and N. C. Handy , Chem . Phys., 1982, 48, 157.15 G. J . Sexton and N. C. Handy , Mol. Phys., 1984,51, 1321.16 W. S. Benedict , N . Gai l a r and E. K. Plyler, J. Chem . Phys., 1956, 24, 1139.17 G. Steenbeckeliers and J. Bellet , J. Mol. Spectrosc., 1973, 45, 10; J. Bellet , W. J . Laffertyy and

18 L. A. Pugh and K. N. Rao , J. Mo l. Spectrosc., 1973, 47, 403.19 J . M. Flaud and C. Carny-Peyret , Mol. Phys., 1973, 26, 811; J. Mol. Specrrosc., 1974, 51, 142; J. M .F laud , C . Cam y-P eyre t an d J . P. Maillard, Mol. Phys., 1976, 32, 499; C . Camy-Peyret, J. M. Flaud, J.P. Mail lard and G. Guelachvi l i , Mol. Phys., 1977, 33, 1641.20 J . S. Garing and R. A. McClatchey, Appl . Opt . , 1973, 12, 2525; C . C. Ferri s0 and C. B. Ludwig,

J. Chem. Phys., 1964,41, 1668.21 D. F . Smith Jr and J . Overend , Spectrochim. Acta, Part A , 1972, 28, 471.22 A. R. Hoy, I. M. Mills and G. Strey, Mol. Phys., 1972, 24, 1265.23 A. R. Hoy and P. R. Bunker, J. Mol. Spectrosc., 1974, 52, 439; 1979, 74, 1.24 G. D . C a r n e y, L. A. Cur t i s s and R . S. Langhoff, J. Mol. Spectrosc., 1976, 61, 371.25 S. Car t e r and N. C. Handy , J. Mol. Spectrosc., 1982, 95, 9.26 S. Car t e r and N . C. Handy , Mol. Phys., 1984, 52, 1367.27 G. Pineau des Forets, D. R. Flower, T. W. Hartquis t and A. Dalgarno, M o n . N a t . R. Astr. Soc., 1986,28 B. T. Sutcliffe, in Current Aspects o Quantum Chemistry, e d . R. C a r b o (St ud ies in Theoretical Chemistry,29 S. Car t e r and N. C. Handy , Mol. Phys., 1984, 53, 1033.30 N . C. Handy , Mol. Phys., 1987, 61, 207.31 B. T. Sutcliffe, Mol. Phys., 1983, 48, 561.32 B. T. Sutcliffe and J . Tenn yson , Mol. Phys., 1986, 58, 1053.33 E. C. K. Lai, Maste rs Thesis (India na Universi ty, 1975).34 S. Car t e r and N . C. Handy , Mol. Phys., 1982, 47, 1445.35 S. Car t e r and N. C. Handy , Mol. Phys., 1986, 57, 175.36 J. Tennyson and B. T. Sutcliffe, J. Chem. Phys., 1982, 77, 4061.37 G. Brocks, A. van der Avoird, B. T. Sutcliffe and J. Tennyson, Mol. Phys., 1983, 50, 1025.38 B. T. Sutcliffe and J. Tennyson, in Molecules in Physics, Chem istry and Biology, ed . I. Csizmadia an d39 B. T. Sutcliffe, to be publ ished.40 J . Tenny son and B. T. Sutcliffe, Mol. Phys., 1986, 58, 1067.41 C-L. Chen, B. Maessen an d M. Wolfsberg , J . Chem . Phys., 1985, 83, 1795.42 M . G . Bucknell and N. C . H a n d y , Mol. Phys., 1974, 28, 777.43 M . G. Rucknell, N . C. H a n d y a n d S. F. Boys, Mol. Phys., 1974, 28, 759.44 R. J . Whi tehead an d N. C . H a n d y , J. Mol. Spectrosc., 1975, 55, 356.45 R. J. Whitehead and N . C. Handy , J. Mol. Spectrosc., 1976, 59, 459.46 N . C. Handy and S . Car t e r , Che m. Phys. Lett . , 1981, 79, 118.47 B. Maessen and M. Wolfsberg, J. Chem . Phys., 1984, 60, 4651; B. Maessen, M . Wolfsberg and L. B.48 D. Cropek and G. D. Carney , J. Chem. Phys., 1984, 80, 4280.49 F. R. Burden and H. M. Quiney, Mol. Phys., 1984, 53, 917.50 J. M . Simons, R. G. P a r r a n d J. M. Finlan, J. Chem . Phys., 1973, 59, 3229.51 D. Huber , Mol. Phys., 1986, 59, 1007; Int. J. Qua ntum C h em. , 1985, 28, 245.52 H . Romanowski and J . M. Bowman, Chem. Phys. Lett . , 1984, 110, 235.53 J. Tennyson and B. T. Sutcliffe, J. C h e m . SOC.,Faraday Trans. 2, 1986, 82, 1151.54 Geometrical Derivatives of Energy Surfaces and Molecular Properties, ed. P. Jorgensen and J. Simons55 J. F. C a w a n d N . C. Handy , Chem . Phys. Lef t . ,1985,121,321; 1986, 128,182; J . F. G a w , Y. Yamaguchi ,56 R. Beardsworth, P. R. Bunker, P. Jensen and W. P. Kraemer, J. Mol. Spectrosc., 1986, 118, 50.57 R. T. Lawton and M . S. Chi ld , Mol. Phys., 1981,44,709.

G. Steenbeckeliers, J. Mo l. Spectrosc., 1973, 47, 388.

24, 801.Elsevier, Amsterdam, 1982) vol. 21, p. 99.

J. Maruan i (Reidel , Dordrecht, 1987).

Hard ing , J. Chem. Phys., 1985, 89, 3324.

(Reidel , Dordrecht , 1986).H. F. Schaefer 111 an d N . C. Handy , J. Chem . Phys., 1986,85, 5132.


12/12

1674 Variational Rovibrational Calculation on Small Molecules58 J. Tennyson and S . C. Farantos, Chem. Phys., 1985, 93, 237.59 S. M. Colwell and N . C. Handy, Mol. Phys., 1978, 35, 1183.60 R. B. Shirts, J. Chem. Phys., 1986, 85, 4949; W. B. Clodius and R. B. Shirts, J. Chem. Phys., 1986, 84,61 T. C. Thompson and D. G. Truhlar, J. Chem. Phys., 1982, 77, 3031.62 S. C. Farantos and J. Tennyson, J. Chem. Phys., 1986, 84, 6210.63 B. R. Johnson and W. P. Reinhardt, J. Chem. Phys., 1986, 85 , 4538.64 F. R. Burden, J. Phys. B, 1986, 19, 1397.65 N . C. Handy , Commun. Phys. Chem., 1987, 1, 1.66 D. F. Coker and R. 0. Watts, Mol. Phys., 1986, 58, 1113.67 R. Bartholomae, D. Martin and B. T. Sutcliffe, J. Mol. Spectrosc., 1982, 87, 367.68 J. Tennyson and B. T. Sutcliffe, J. Mol. Spectrosc., 1983, 101, 71.69 S. Carter , N . C. Handy and B. T. Sutcliffe, Mol. Phys., 1983, 49, 745.70 J. Tennyson, Comput. Phys. Rep., 1986, 4, 1.71 J. Tennyson, Comput. Phys. Commun., 1985, 38, 39.72 S. Carte r, personal comm unication, 1983.73 D. W. Schwenke and D. G. Truhlar, Comput. Phys. Commun., 1984,34, 57.74 A. Carrington and R. A. Kennedy, J. Chem. Phys., 1984, 81, 91.75 R. Pfeiffer and M. S . Child , Mol. Phys., 1987, 60, 1367.76 J. C. Light, I. P. Hamilton and J. V . Lill, J. Chem. Phys., 1985, 82, 1400; Z. B a c k a n d J . C . Light,77 J . Chang, N. Moiseyev and R. E. Wyatt, J. Chem. Phys., 1986, 84, 4997.78 M . Blanco an d E. J. Heller, J. Chem. Phys., 1985, 83, 1149.79 S. Carter and N. C . Handy , Comput. Phys. Commun., 1987, 44, 1.

6224; R. B. Shirts and K . M . Holbrook, Int. J. Quantum Chem., 1987, 85 , 6210.

J. Chem. Phys., 1986, 85, 4594; 1987, 86, 3065.

Paper 7/111; Received 22nd January, 1987

brian t. sutcliffe and jonathan tennyson- variational methods for the calculation of rovibrational...

Documents