B.T. Sutcliffe and J. Tennyson- The Construction and Fitting of Molecular Potential Energy Surfaces...

8/3/2019 B.T. Sutcliffe and J. Tennyson- The Construction and Fitting of Molecular Potential Energy Surfaces and Their Use in

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The Construction and Fitting of MolecularPotential Energy Surfaces and Their Use in

Vibration-rotation CalculationsB. T. SUTCLIFFE

Department of Chemistry, Un iversity of York, Heslington, York YO1 5DD, EnglandJ. TENNYSON

Department of Physics and Astronomy, University College London, Cow er Stree t , LondonWCIE B T, England

AbstractTh e state-of-the-art in non-empirical calculation s of potential energy surface s for small molec ulesis discussed, as is the present position with respect to the analytic fitting of such surfaces. Theresults of some nonempirical vibration-rotation calculations performed on such analytic surfacesare presented and compared with experimental results. An attempt is made to assess the extentto which present met hod s of electronic structur e calculation and present analytic fit ting m ethodsare adeq uate to produce surfaces for the interpretation of high-resolution spectral data.

IntroductionIn the past decade computational quantum chemistry has developed, as com-puters have developed, to the position where it is possible, almost as a matter

of routine, to obtain very accurate and extensive potential energy surfaces fromelectronic structure calculations for systems consisting of 3 or 4 nuclei and upto about 10 electrons. There have also been parallel developments in the ratherless accurate calculation of such surfaces for larger systems with more nucleiand more electrons. It is, however, the first sort of calculation on which interestwill be centered in this paper, for it is these calculations that offer the bestbasis for nonempirical calculations of whole-molecule wave functions; that is,calculations of not only electronic structure but of nuclear motion behaviortoo. The interest in such calculations lies not only in the role that they play ininterpreting the rotation-vibration spectra of molecules, but also in the role thatthey might play in understanding chemical reactions. Thus, if one could findmethods of dealing with very highly excited vibration and rotational states,where the molecule is on the verge of dissociation, then one would be ap-proaching from below the elusive region of chemical reaction, which is ap-proached from above by scattering and collision theory.

The work described in this article represents only a modest step toward the

International Journal of Quan tum Chemistry: Quantum Chemistry Symposium 20,507-520 (1986)0 1986 by John Wiley & Sons, Inc. CCC 0360-8832/86/010507-4$04.00


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508 SUTCLIFFE AND TENNYSONgoal of describing the reactive d om ain , but it doe s illustrate what ca n be doneby purely nonempirical mea ns to d escribe quite highly excited vibration androtat ion states of small systems.

The Theoretical BasisThe complete non relat ivist ic molecular Hamiltonian expressed in a labo-

(1)where the sub scr ipts e and n designate the electrons and nuclei, respectively,and th e symbols a re otherwise conventional.In electronic structure calculations it is usual simply to ignore R,, to t reatV,, a s a classical q uan ti ty, and t o try to obtain solutions to th e problem specified

Aelec Re + ven+ Pee (2)as func tions of the nuclear geometry. T hu s if there ar e N ( > 2 ) nuclei in theproblem, th e electronic energy E(R) associa ted with is a function of 3N - 6internal variables (denoted collectively a s R). These var iables m ust be cho senso that fieIecs invariant to translation and rigid rotation of the system, butotherwise may b e ch osen a t will. V,,, considered as a classical quantity, canalso be exp ressed in terms of these internal variables, a nd th e sum of E(R) andV,,(R) is the potential energy surface (or function) for the system . As is wellknown, if there is a minimum in this surfac e at some geom etry R,,, then theenergy at this minimum is generally, to a very good approximation, the truetotal energy of the bound system with equilibrium geometry R,,.In the standa rd formalism fo r moving beyond the electronic s tructure cal-culation, i t is assum ed that the w hole-system wa ve function can b e writ ten asa sum of products :

ratory-fixed fram e has th e formA = R, + K e + v,, + ven+ vee,

by

* = lz X n @ n , (3 )nwhere @ is an electronic wav e function exp ressed as a function of e lectronicvariables and of nuclear geom etry, and xn is a nuclear motion function.Clearly, th e si tuation is not quite s traightforward here, b ecau se one m ust beable to express xn in (3 ) in terms of the chosen internal variables used in @,and six more variables, three of which describe the translation of the systemand three i ts rigid rotat ions, thus covering the m otion of the whole system . Toput this an oth er way, if xn is to oc cur as a solution to a n effective problemarising from (1 ) after electro nic motion ha s been integrated ou t, then it isessential tha t the Ham iltonian (1 ) be reex pressed in terms of the chosen internalcoo rdinates, with three co ordina tes to describe translations an d three m ore todescribe r igid rota t ions of the sy stem as a whole.This is a technically formidable task whose solution was first convincingly


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MOLECULAR POTENTIAL ENERGY SURFACES 509tackled by Eckart [ l ] for the case where the stable system has a very deepminimum at some unique equilibrium geometry, &,and where at that minimum,the other electronic states in the expansion (3) are, energetically, well separatedfrom the state of primary interest. In this case careful analysis shows that, ifthe nuclei exhibit only small-amplitude motions, it is sufficient to consider justone electronic state, and the influence of the electrons on the nuclear motioncan be incorporated via the potential energy function alone. This is usuallycalled the Born-Oppenheimer approximation.

The conditions given by Eckart [ l ] may be used to express the Hamiltonianin terms of three translations, three rotation and 3N - 6 translation- androtation-free internal coordinates, and after a great deal of algebra, it may beshown that the effective Hamiltonian for nuclear motion (ignoring the trans-lational term, which may be factorized off) is:

This form was first given by Watson 121, and there a detailed explanation ofthe terms can be found. It is sufficient to note here that the internal coordinatesQk are usually normal coordinates, the La are angular momentum operators forthe whole system (and depend only on the angular variables), the .rr, are thecoriolis coupling operators, and the matrix p is a matrix closely related to theinverse of the moment-of-inertia tensor for the problem. The term V ( Q ) s thepotential energy function expressed in terms of the coordinates Q k , which arechosen such that at the equilibrium geometry the Qk are all zero.

For small amplitude motions the Qk are small, the ranegligible, and theintertia tensor well approximated by its equilibrium value. The potential V ( Q )can be expanded in a Taylor series about the minimum, so that the secondorder

where F is the so-called force-constant matrix, whose elements can be deter-mined by computing the second derivatives of V ( Q ) and evaluating them atQ = 0.Thus, for sufficiently small-amplitude motion (4) can be well approx-imated as the sum of rigid-rotor plus simple-harmonic oscillator Hamiltonians.The solution of t h e problem in this approximation is of the form

where @,,(Q) s a product of harmonic-oscillator functions and IJk) are thesymmetric-top eigenfunctions with arbitrary quantization along the space-fixedaxis.Now if the whole sequence of approximations that lead to a solution of theform (6) are adequate, then the calculation of whole-molecule properties, al-


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510 SUTCLIFFE AN D TENNYSONthough tricky, need actually involve only a relatively few electronic structurecalculations, in fact just enough to locate the minimum of the potential energy-surface and to get the second derivatives at the minimum. The advantages thatflow from this are clear. A good nonempirical calculation of an electronic energyis a formidably expensive and lengthy undertaking, even with the best tech-niques that are now available, so the fewer such energies that are needed thebetter. Furthermore, in this approach it is not necessary to have any particularview about the shape of the potential energy surface near the minimum, forpurely numerical methods can be used to calculate the first and second deriv-atives.

It was early discovered however that, unfortunately, this simple proceduredid not in general lead to very accurate results, and the deficiency was identifiedas lying in the approximation for the potential ( 5 ) and the consequent approx-imation in the nuclear motion wave function, (6).

To transcend the approximation (5) is formally easy. It is simply necessaryto determine V(Q)over a suitable range, to extend the trial wave function tobe a sum of vibration functions (possibly J and k dependent), and to use thelinear variation theorem. In practice, however, two difficulties arise. In thefirst place it is feasible to calculate V only at a set of points; it is not clear atthe outset of the calculation how these points should be chosen. Neither is itclear how the function should be continued (or fitted) between the calculatedpoints. Furthermore, because of the expense and labor of such calculations, itis imprudent to engage in indiscriminate calculation. It is also, in general,difficult to calculate matrix elements of the Hamiltonian between the functionsthat form the linear-variation basis. Even if a suitable choice of basis is madeso that analytic expressions are obtainable for the kinetic energy integrals, itis not easy to avoid numerical integration for the matrix elements involving Vand the elements of p, nd 3N - 6 dimensional quadrature is a difficult andtime-consuming business.

Pioneering attempts in this area were made, for triatomic systems, by White-head and Handy [3,4] and Carney and Porter [ 5 , 6 ] . Whitehead and Handyargued that it was advantageous to avoid explicit fitting of the surface, butrather to decide on a quadrature scheme in advance and simply to evaluate thepotential at the required quadrature points. In their work they found the normalcoordinates by solving the usual problem with a nonempirical F matrix, choseas basis functions products of simple-harmonic oscillator functions, and usedGauss-Hermite quadrature. The course of action that they advocated is equiv-alent to assuming that the potential energy surface can be fitted as a sum ofproducts of Hermite polynomials. It can thus be regarded as a special case ofa more general technique in which the potential energy surface is fitted byproducts of general orthogonal polynomials, a technique used explicitly in, forexample, Bartholomae et al. [7].

Carney and Porter, on the other hand, fitted the potential directly, adoptinga somewhat more physically motivated approach to describing i t , recognizing


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MOLECULAR POTENTIAL ENERGY SURFACES 511in their functional form both the nature of the bonds involved and their dis-sociation behavior. They did this by modifying a fitting technique introducedfor diatomics and linear triatomics by Simons, Parr, and Finlan (SPF) [81. (Asimilarly physically motivated approach to fitting the potential was also sug-gested, a little later, by Sorbie and Murrell [9]). he matrix elements of thepotential were evaluated by means of a generalized three-dimensional gaussianquadrature scheme adapted to the form of the potential and the chosen basisfunctions.It was apparent from the early calculations that the results obtained weresensitive not only to the quality of the ab-initio calculation itself but also tothe method (implicit or explicit) of fitting the potential energy. It is of coursepossible to quantify the defects in a fit by calculating the root-mean-squareerror, but this is only a measure of how well the fit matches the calculatedpoints. It is obviously at least equally important to know how well a fit inter-polates between the calculated points, and this can be done only by using thefitting function to estimate uncalculated values and comparing the estimate withactual calculations. The expense of actually calculating fresh points is generallyso great that this is seldom done, and all calculated points are deployed inmaking the fit, so that most fitted surfaces are, in a sense, untested. It is thusnot unusual, as will be seen, for distinctly different results to be obtained innuclear-motion calculations that are nominally on the same potential energysurface but in which different fits have been made of the potential.Besides the problems discussed above, associated with the potential energyfunctions and the evaluation of matrix elements, there is another problem thatarises from the actual Hamiltonian for nuclear motion expressed in body fixedcoordinates. It is the case (see, for example, Ref. 10) that the separation ofrotational motion from the other motions always results in a Hamiltonian con-taining terms that are incipiently singular. In the Eckart Hamiltonian (4) p issuch a term, for it is the inverse of a matrix that can become singular forsufficiently large amplitude vibrations: for example, when a bent triatomicbecomes almost linear. The underlying reason for this kind of singularity is theimpossibility of parameterizing the proper rotation group, S 0 ( 3 ) , in such a waythat the parameterization function is analytic everywhere. In the present con-text that means that in a nonrigid system it is not always possible to defineproperly the Euler angles that determine the angular momentum operators.The possibility of such a difficulty was anticipated early by Sayvetz [ l l ] ,who suggested ways in which the Eckart Hamiltonian could be modified toavoid the more usual problems, such as free internal rotation and vibration tolinear or planar configurations. This seminal work has led to many develop-ments in the theory of what are now called semi- or nonrigid molecules (see,for example, Refs. 12 and 13), but in the work to be described here anotherpath has been taken with a view to coping simultaneously with the incipientsingularities in highly-excited triatomics and dealing with the problems of po-tential fitting and matrix element evaluation in an efficient way.


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512 S U T C L I F F E A N D T E N N Y S O NCalculations on Triatomic Systems

It is clear that in describing highly excited nuclear motion states of triatomics,it is not a good approximation to assume that the molecule executes only smallamplitude vibrations about an equilibrium configuration; nor is it sensible toassume that the molecule rotates approximately as a rigid body. The states ofinterest may, for example, be described better in terms of the collision processof an atom with a diatomic, and thus in terms of the standard scattering co-ordinates as used, for example, by Arthurs and Dalgarno [14]. Such coordinatesare appropriate for van der Waals complexes, where the potential in the regionof interest can be fitted in terms of functions of such scattering coordinates.

It is equally the case, however, that in certain energy regions the moleculeis perhaps better described in terms of a more traditional bond-length, bond-angle system of coordinates. It is clearly something of a problem to obtain aHamiltonian that describes the various possibilities in a coherent and uniformway without making any assumptions about the rigidity of the system.

However, the authors have recently shown [151 how a suitable Hamiltonianmay be constructed that allows for at least some of the possibilities. For thisHamiltonian it is assumed that the three laboratory-fixed coordinates xiof thenuclei are transformed to a space-fixed set of two translation-free coordinates,ti, by the transformations

3t; = c xjv,j=

with(7)

From (7) it is seen that tl is the,bond-length vector from particle 2 to particle3. By choosing g = 1 in (7), t2 becomes the bond-length vector from particle2 to particle 1, and by choosing g = m Z ( m 2+ r n 3 ) - ' , t2 becomes the vectorfrom the center of mass of the diatomic system 2-3 to the particle 1, so that inthis case the t l and t2 are the space-fixed scattering coordinates. Clearly otherchoices are possible by varying g.The body-fixed system is defined by an orthogonal transformation C suchthat

CTtj = r l[:] ,CTt2= r2 [sii1 (9)where r l and r2 are the lengths of t , and t2, respectively. 8 s the angle betweentl and t2 with range 0-T, and the embedded coordinate system is chosen to beright handed. An alternative embedding is obtained by swapping tl - 2andr l - 2. cos 8


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M O L E C U L A R P O T E N T I A L E N E R G Y S U R F A C E S 513Th e sche me outl ined in [16] may be used to c onstruc t the body-fixed Ham -

iltonian. This Hamiltonian is then allowed to operate on the manifold of ro-tational functions IJk) and the results multiplied from the right by (Jk' l . Onintegrating out o ve r the rotational variables, th e following effective form con-taining internal coordinates only is obtained(10)i = kv) Kv' + kvk + kvk + V

with

l a 72 a aKV ) = S k ' k2 A(- 6- - ine-ar, ar, sine ae ae( 1 1 )

sine -- --PI2 arlar2 r l r2 sin0 a0 aeO S Y a . a )

a2Kv' = 8 k ' k fif_ (- COSe-l a l a 1( l ar2 r2 ar, r l r2+ s ine - - + -- + -) 3) (12)

1 I+ k2cosec28(3 - -The above form i s for t l chosen as defining the z axis of the body-fixedsystem. T he form for z chosen along t2 is obtained by the exchange r l CJ r2 ,p l ++ p2.Th e kro neck er deltas show th e couplings within the rotat ional man-ifold.In (1 1)-(14)

p;' = m;' + m y 1p ~ ' g(m;' + mq l ) - m;'p;' = m;' + g2m;' + ( 1 - g)2m;'.

andCA = (J(J + 1) - k(k _t 1))1'2 (15)I t should be n oted that if scattering co ordina tes are ch osen, pfi' is zero , andhence the opera tor s a2)nd a2k re null operators . The Hamiltonian thusreduces to the s tandard one used b y the au thors in the previous work [17]. If


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514 SUTCLIFFE AND TENNYSONthe bond-length bond-angle coordinates are chosen, the Hamiltonian is that ofHagstrom and Lai [18]. The general Hamiltonian takes a volume element ofintegration +$sin 8drldr2d8.

For systems in which this coordinate representation is appropriate, the po-tential may be expanded as

where P A s a standard Legendre polynomial. The V , may be obtained exactlyfrom this assumed form by (A + 1)-point Gauss-Legendre quadrature over 8for any choice of rl,r2.A computer program to do this is available [19]. Withthis form then the explicit fitting problem is reduced to fitting the V A ( r I , r 2 ) .

It is seen that, as expected, the Hamiltonian (10) also has incipient singu-larities arising possibly from the rT 2 terms and from the terms in cosec28as 8tends to 0 and T.That the effect of these singularities can be mitigated withoutabandoning any physically significant information is shown in detail in [151.Essentially what is done is to choose a basis of products of the form

where the ejk are standard normalized associated Legendre functions definedin the convention of Condon and Shortley [20] and the ~ k jre chosen asproducts of central-field functions (for example Morse or spherical oscillatorfunctions). If the effective operator derived above is applied to functions ofthe type (17), it can be seen that the cosec28 terms in KVA cancel with thosethat arise from the operation of the derivative terms in KV.) and KV) upon theejk.The incipient divergences are thus removed, and the results of the operationmay be multiplied by elk,and integrated over 8 to yield an effective operatorthat operates only on a basis of radial functions. In constructing this radialoperator it is convenient to incorporate the radial portion of the jacobian intothe operator to put it in what is often called a manifestly hermitian form.This process is equivalent to choosing radial functions of the formr ; r ; l@&(rl , r2), and the resulting operator determines the @ ik j .The volumeelement of integration thus becomes sinOd8dr,dr2,nd the form of the kineticenergy operator is


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M O L E C U L A R P O T E N T I A L E N E R G Y S U R F A C E S 515

The potential in the form (17) leads to term s of the forma k k v j , k ( r 1 7 r 2 ) (22)

where v j y k are jus t sums over A of products of Gaunt coefficients and theV h ( r I r r 2 )see, e.g. Ref. 15). The auxilliary quantities in the kinetic energyope rators are defined a s

(23)(24)(25)

an d are easily see n to be simply related to Clebsch-Gordon coefficients. Thekron ecke r deltas show th e coupling within the complete angular manifold.The Hamiltonian as given above is not divergent over a suitable basis ofradial functions an d can a ccom mo date the m ost extrem e of bending m otions.It is therefore e minently suitable for describing highly excited vibrational an drotat ional s tate s . The c om puter program [191 mentioned a bov e can be used toperform co mp lete calculations with this Ham iltonian in its scattering coo rdinateform.

d j k = [(i- k + 1) (i + k + 1)/(2j + 1)(2j + 3)]a j k = [(i+ k + 1)(i + k + 2)/(2j + 1)(2j + 3)]2b j k = [(i- k)G - k - 1)/(4~ -

Results and DiscussionI t is app ropriate f irst to put in context what c an b e expected in the w ay ofabsolute ac cu racy from v ibration-rotation ca lculations on triatomics. T he verybest potential energy surface calculations that have been done on the simplestof all triatomics, namely H: , yield an energy at the minimum that is between

150 an d 200 cm -I higher than the absolute energy minimum found in the bestone point calculation. This one point calculation is that of Mentsch and An-derson [21], which was performed using a random walk method that yieldsresults that ar e, arguably, e xac t. It would be extremely difficult to constru ct asurface from points so calculated ho we ver, for the errors in suc h calculat ions


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516 SUTCLIFFE AND TENNYSONdiffer from point to point. It would also be very expensive even if the errorproblem could be obviated. However, as will be seen, modern high-accuracyC I calculations seem able to get the shape of the potential energy surface correct,that is to have about the same error at all the points of interest. This meansthat predicted rotation-vibration spectra turn out to be rather accurate becausethe relative positions of the energy levels are well predicted, although it mustbe admitted that the finer details of the rotational spectrum do appear to berather sensitive to quite small changes in the predicted geometry at the potentialminimum.Within the context outlined above, the kind of accuracy that can be expectedfrom the fundamental vibrational calculations(.I0) are exhibited in Table I,where results arising from various fits to the same potential energy surface andto different surfaces forH; are compared with each other and with experiment.Although the agreement between the calculated and observed results is goodfor all the calculations, the differences between calculations on the same sur-face, but that correspond to different fits, are somewhat disconcerting. It isvery difficult indeed to see a way out of this problem other than by increasingthe density of calculated potential energy surface points so that all points aresufficiently close for a standard interpolation scheme to be used between anypair of them. In some ways that would be to return to the kind of scheme firstadvocated by Whitehead and Handy 151. To use such a dense set would amel-iorate, but not obviate, the difficulty of deciding in advance just where thepoints on the potential energy surface that are to be calculated should lie. Inmost current calculations the number that can be afforded is distributed almostby art across the surface with a view to the kind of fitting technique that is tobe used. If a dense set could be afforded, it would be possible to start from asparse, perhaps randomly distributed, set to interpolate between pairs of pointsand then to calculate at the interpolated points and so continue until the in-terpolation was shown to be essentially exact. This sort of task is ideally suited

T A B L E. Com parison of ab initio calculations of the band origins in c m - ' of H;.H;

Electro nic Calcu lation Fit Used Vibrational Calc ulation YI v2Burton et al [30] SPF" [30]SPF" [30]SPF [25]SPF [25]Morse [32]Morse 1321SPF [34]

Morse [ 2 5 ]

Meyer et al [32]Dykstra and Swope [33]

Observed [35]

~~~

Burton et al (301Tennyson & Sutcliffe [31]Martire & Burton [25]Tennyson & Sutcliffe [3 I]Meyer et a1 [321Meyer et al (321Carney et al (341Martire & Burton 1251

~~ 3189318831763 1753178318631803185-

25092508251925182519252 I252225222521

aThis corresponds to the SPF+ f i t of Table 4 of Ref. 30 .


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MOLECULAR POTENTIAL ENERGY SURFACES 51 7to parallel computation, and as parallel vector-processing computers develop,this course of action wiU perhaps become a possibility at least for triatomics.Currently, however, it must be recognized that this side of nuclear motioncalculations is in a rather unsatisfactory state.

We turn now to the sort of accuracy that can be achieved on any assumedpotential energy surface. In the calculations performed by the authors, ana-lytical forms w r e chosen for the radial (central-field) functions, usually Morseoscillator-like functions. It is of course possible to choose these functions bydirect numerical integration of an artificial one-dimensional problem (see, forexample, Ref. 22), but such a course of action means that the kinetic energyintegrals must be calculated numerically, and a very dense multi-dimensionalquadrature must be done over the potential. It is not yet clear that the extraeffort involved is worthwhile. The angular expansion functions were taken upto a prespecifiedj value depending on the range of A required for the potentialexpansion in (16) and on the k values possible with a givenj. For highj valuesthis means a very extensive set ofj, and recently Carter and Handy [23] havesuggested an ingenious method whereby suitable angular functions may bedetermined self-consistently, which will perhaps prove helpful in shorteningsuch expansions.Because the method of calculating solutions in the present approach is vari-ational, its accuracy can be assessed by examining the convergence of thecalculations under increase in basis size. In Table I1 are shown the results fora selection of vibrational states (J = 0) of Dj+ taken from recent work by theauthors [24]. The results are calculated using the scattering coordinate form ofthe Hamiltonian (where r represents the diatomic coordinate and R the coor-dinate from the diatomic center of mass to the atom). The radial basis chosen

T A B L E1. Vibrational band origins, in cm - I , for D; calculated using the BVDH potential ofMartire and Burton [25]. (a) Simple product ba sis, (b ) perturbation selecte d b asis.Perturbation Selected

(N 800)Quantum Simple Product Basis Bas isNumbers (m c 7 , n c 9 , j c 19)VI Y, / Symmetry j even j od d j even j odd0 I I E'I 0 0 E'0 2 0 A,'0 2 2 E'I I I E'2 0 0 A,'3 0 I E'3 0 3 A , '3 0 3 Az'I 2 0 A , 'I 2 2 E'Zero point energy

I83 I .072297.493523.243647.434052.654549.895208.175400.255704.285791.203102.230

1831.07 183 1.07 1831.072297.493523.233647.40 3647.40 3647.404052.60 4052.6 4052.604549.885206.71 5207.17 5206.715399.285470.59 5468.395703.905788.69 5788.713 102.230


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518 SUTCLIFFE AND TENNYSONis one of products of the Morse-like oscillators Hm(r)Hn(Z?),nd the potentialis that of Martire and Burton [25].The results on the left of the table were calculated using all possible productsof a truncated set of central-field functions, while the results on the right werecalculated using basis functions selected by a first-order perturbation technique.The agreement between the levels is excellent and suggests a high degree ofconvergence. However as can be seen, a very large number of expansionfunctions (800 in each case) are needed to get reasonably accurate energies inthe J = 0 case, and clearly to deal with high J states is going to prove a verydifficult problem, for the size of the secular problem increases as 25 + 1. Theproblem here is essentially one of immediate-access computer storage size, forthe actual time taken by nuclear motion calculations is negligible comparedwith the time taken to calculate an electronic energy.

Recently, however, the authors [26] have developed a secondary variationalmethod with which it has proved possible to get up to J = 20 for the variousisotopomers of H: .The method is similar to that proposed by Chen et al. [27]but differs in that the base Hamiltonian has a dependence on J but is diagonalin k. In this way the base problem remains tractable but also allows for shiftsin geometry caused by centrifugal terms in the potential which, as experiencehas shown, are large for high J states. Table 111shows some results taken fromthis work fo r J = 10states of H2D+on the potential surface of Schinke, Dupuis,and Lester [28] that allow the convergence of the proposed method to beassessed. It is interesting to observe that the lower levels do not necessarilyconverge most quickly. This is because the Coriolis terms for states with lowk are considerably stronger than for those with high k, as would be expectedfrom the form of the Hamiltonian. It is seen, too, that the convergence atN = 160is about 0.1 cm- ',which is a satisfactory level of accuracy. However,for larger J at N = 160 the convergence is less good, and with present machinesthis makes J = 20 about the highest level for which sensible results can beobtained. The difficulty is that, unlike conventional CI electronic structure,where one usually needs a few of the lowest roots, here one needs U + 1roots for each vibrational state of interest, so that a single-root iterative dia-

TABLEll. Convergence of some levels of H 2D + with J = IW (j ven) with increasingprojection of J along R , taken from the no-Coriolis calculations.vibrational basis , N. Energies, in cm - ', are all relative to Eo = -32962.24. Values of k , the

Level number 1 2 3 7 8 9k = 0 2 4 I 0 3No Coriolis 2928.04 3007.67 3234.84 4993.32 5177.72 5202.43N = 40 1982.24 2570.61 2937.70 4191.53 4406.52 4621.4280 1977.14 2568.93 2937.12 4191.37 4392.96 4608.68

120 1976.66 2568.77 2937.05 4191.35 4391.01 4606.93I40 1976.56 2568.75 2937.04 4191.35 4390.68 4606.71160 1976.53 2568.74 2937.04 4191.34 4390.55 4606.64


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MOLECULAR POTENTIAL ENERGY SURFACES 519gonalizer is not very efficient. However, since the matrices involved are rathersparse, it may be possible to increase the value of J that can be considered,and work in this area is in active progress. It should be noted that at high J itis almost impossible to maintain the idea of vibration-rotation separation, forthe rotational splittings become larger than the vibrational splittings. This causesno problems in the computational scheme outlined here, but it does causeproblems of identification and selection for comparison of calculated resultswith experiments that are currently being performed (see, for example, Ref29).

ConclusionsThe work described in this paper does seem to offer a way forward in the

treatment of highly excited rotation-vibration states of triatomic molecules, butdespite its success, there remain problems. The most important of these is thatof the accurate representation of the electronic potential energy surface, andto that problem there seems to be no easy solution. It seems that only thecalculation of more points will suffice, and such many-point calculations willbecome routinely possible only with the further development of parallel pro-cessing computers. There is also the problem of solving the large secular prob-lems that arise in this work for many roots. In this area there is much activedevelopment, and it does not seem unlikely that, with modern numerical tech-niques dealing with sparse matrices and modern vector-processors with in-creased store size, it might be possible to consider states with J -100 in thenear future.The overall picture is then a hopeful one, and the theory of nuclear motionin triatomic systems seems on the verge of a substantial advance.

Bibliography[ I ] C . F. Eckart , Phys. Rev. 45, 552 (1935).[2] J. K. G. Watson, Mol. Phys. 15, 470 (1968).[3] R. J . Whitehead and N. C. Hand y, J . Mol. Spectrosc. 55, 356 (1975).[4] R. J . Whitehead and N . C. Handy, J. Mol. Spectrosc. 59, 459 (1976).[5] G. D. C arney and R.N. Por ter , J . Chem. Phys. 60,4251 (1974).[6] G . D . Carney and R.N. Por ter , J. Chem. Phys. 65, 3547 (1976).[7] R. Bartholomae, D. Martin, and B. T. Sutcliffe, J. Mol. Spectrosc. 87, 367 (1981).[8] G . Simons, R. Parr, and J. M. Finlan, J . Chem. Phys. 59, 3229 (1973).[9] K . S. Sorbie and J . N. Murrell, Mol. Phys. 24, 1387 (1975).

[lo] B. T. Sutcliffe, Mol. Phys. 48, 561 (1983).[ I l l A. Sayvetz, J . Chem. Phys. 6, 383 (1939).[I21 P. R. Bunker, Molecular Symmetry and Spectroscopy, (Academic, N ew York, 1979).[13] G . S. Ezra , Symmetry Properties of Molecules, Lec ture N otes in Chem istry (Springer-Verlag,[14] A. M. A rthurs and A. Dalgarno, Proc. Roy. SOC. ondon Ser. A 256, 340 (1W).[I51 B. T. Sutcliffe and J. Ten nyso n, Mol. Phys. (in press).[I61 B. T. Sutcliffe, in Current Aspects of Quantum Chemistry, Studies in Theoretical Chem istryR. Ca rbo, E d. (Elsevier , Amsterdam, 1982). Vol. 21, p. 99.[17] J . Tennyson an d B. T . Sutcliffe, J. Chem. Phys. 77,4061 (1982).

Berlin, 1982) Vol. 18.


14/14

520 S U T C L I F F E A N D T E N N Y S O N[I81 E. K. C. Lai, M asters Thesis, Indiana University (1975).[I91 J . Tennyso n, Com put. Phys. Commun. 32, 109 (1984); and (in press).[20] E. U. Condon and G. H. Shortley, The Theory o f A f o m i c Specrra, (Cambridge University[21] F. Mentsch and J . Anderson, J . Chem. Phys. 74, 6307 (1984).[22] D. Cropek and G. D. Carney, J . Chem. Phys. 80, 4280 (1984).[23] S. Carter and N . C. H andy, Mol. Phys. 57, 175 (1986).[24] J . Tennyson and B. T. Sutcliffe, J . Chem. SOC.Faraday Tran s. 11 (in press).[25] B. Martire and P. G. Burton, Chem. Phys. Lett. 121, 495 (1985).[26] J . Ten nyso n and B. T . Sutcliffe, Mol. Phys. (in press).[27] C-L. Chen, B. Maessen, and M. Wolfsberg, J . Chem. Phys. 83, 175 (1985).[28] R. Schinke, M. Dupuis, and W. A. Lester, Jr. , J . Chem. Phys. 72, 3909 (1982).[29] B. Lemoine, M. Bogey, and J . L. Destombes, Chem. Phys. Letts. 117, 532 (1985).[30] P. G. Burton, E. von Nagy-Felsobuki, G. Doherty, and M. Hamilton, Mol. Phys. 55, 527[31] J. Ten nyso n and B. T . Sutcliffe, Mol. Phys. 56, 1175 (1985).[32] W. Meyer, P. Botschwina, and P. Burton, J . Chem.,Phys. 84, 891 (1986).[33] C . E. Dykstra and W. C. Swope , J. Chem. Phys. 72, 3909 (1982).[34] C. D . Carney and S. Adler-Golden, and D. C. Lesseki, J. Chem . Phys. 84, 3921 (1986).[35] J. K. G . Watson, S. C. Foster , A. R.W. cKeller, P. Bernath, T. Amano, F. S. Pan, M. W.

Press, Cambridge, (1935).

(1985).

Crofton, R. S. Altman, and T. O ka, Can. J . Phys. 62 , 1875 (1984).Received April 30, 1986.

B.T. Sutcliffe and J. Tennyson- The Construction and Fitting of Molecular Potential Energy Surfaces...

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