Semi-Classical Methods for Vibrational Energy Levels of Triatomic Molecules BY N. C. HANDY, S. M. COLWELL AND W. H. MILLER* University Chemical Laboratory, Lensfield Road, Cambridge CB2 1 EW Received 29th April, 1976 The method recently proposed by Chapman, Garrett and Miller for semi-classical eigenvalues of non-separable systems has been applied to the triatomics SOz and H20. The Hamiltonian is expressed in normal coordinates, using potentials V = V(Arl, Ar2, A@. The energy levels are compared with corresponding quantum mechanical energy levels. For SO2, the fundamental fre- quencies differ by at most 0.1 cm-l, and for H20 they differ by at most 1.6 crn-l. 1. INTRODUCTION Semi-classical methods have been used extensively in recent years to treat atomic and molecular scattering problems,1*2 but only most recently has attention turned to the problem of determining semi-classical eigenvalues for molecular systems.The goal is to generalise the Bohr-Sommerfeld, or WKB, quantum condition,? to multidimensional systems, which are not separable and thus cannot be quantised as individual one-dimensional systems. Eqn (1) has been of immense utility in analys- ing vibrational/rotational spectra of diatomic molecules and determining their inter- atomic potential functions, and it is anticipated that its multidimensional extension would be of similar use for small polyatomic molecules. The multidimensional eigenvalue problem is actually the oldest problem in “ semi-classical mechanics ”, being the central problem of the old quantum theory.The only truly new approach to it in recent years has been Gutzwiller’s periodic orbit theory (with the important modification of Miller), but although interesting and illuminating in a number of ways it is now clear that the quantum condition obtained by periodic orbit theory has dynamical approximations, beyond the semi-classical approximation itself, inherent in it. The quantization procedure of the old quantum theory3 is first to construct the total (classical) Hamiltonian as a function of the complete set of “good” action variables (or adiabatic invariants) which are the constants of the motion of the system, and then to require them to be integers (or perhaps half-integers). To carry out this construction requires the solution of a Hamilton-Jacobi equation for the non-separable system, and can thus be carried out analytically only within the frame- work of perturbation t h e ~ r y .~ Recently, however, Chapman, Garrett and Miller6 (CGM) have shown how the * J. S. Guggenheim Memorial Fellow on sabbatical leave from Department of Chemistry, t Atomic units are used, unless otherwise stated. University of California, Berkeley.30 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS Hamilton-Jacobi equation in action-angle variables, which appears in Born’s3 formulation, can be cast in a form that permits efficient numerical (i.e., non-pertuba- tive) solution. Application6 of the CGM procedure to a two-dimensional anharmonic oscillator model problem has shown the semi-classical eigenvalues so obtained to be in good agreement with the exact quantum mechanical ones, even when the non- separable coupling is quite strong.Other ways of constructing a non-separable Hamiltonian as a function of the “good” action variables have been given by Marcus and Percival* and their co-workers; this work, based on Keller’sg forniula- tion, deals with the Hamilton-Jacobi equation in Cartesian variables. This paper reports application of the CGM approach to determination of the vibrational eigenvalues of SO2 and H20 ; this is the first non-perturbative calculation of semi-classical eigenvalues for real polyatomic molecules. Since a triatomic molecule in its centre-of-mass coordinate system with zero total angular momentum has three degrees of freedom, these present applications involve one degree of freedom more than the model problem treated originally by CGM6 and Marcus7 and co- workers.Section 2 first summarises the CGM approach and the modifications of the original procedure that have been introduced in the present work. The form of the classical Hamiltonian for a non-rotatory, non-linear triatomic molecule is discussed in Section 3, and Sections 4 and 5 present the results of calculations for SO2 and H20. Comparison with available quantum mechanical eigenvalues is in general quite good, the same level of accuracy typically observed for one-dimensional semi-classical eigenvalues. Section 6 summarises and discusses prospects for future developments and outstanding problems. 2. THE CGM METHOD Because CGM has been recently published, we shall here only summarise their The total (classical) Hamiltonian is written in the form theory, and not give the philosophical reasoning behind the approach.2=Xo+X1 (2) where the reference Hamiltonian So is separable. Formulae are simplest if X0 is taken to be harmonic (but this is not necessary), which we now do, where t is the number of degrees of freedom, and where unit mass has been assumed (i.e., mass-weighted coordinates are used). Action-angle variables 3 ~ 1 0 n, q are ob- tained for this Hamiltonian using an Fl type generatori1 Fl(q, x), where Fl(q, x) = - ~ $ w i x r Z tan qr. i The transformation relations are (4) These relations give = JZcosq, p i = - V 2 i n i ~ sin qi.HANDY, S . M. COLWELL AND W. H. MILLER 31 N. C . On replacing n by n and the relations (6) + 3, in accordance with eqn (l), Xo takes the form t z o = 2 + w, i= 1 become xi = J- cos qi In semi-classical theory, eqn (3, for integer ni, gives the energy levels for this separable system. On the introduction of Z1, the action angle variables It, q are no longer good action angle variables i.e., 3 is no longer a function of n only.CGM, following propose using an F2 type generator F2(N7 q) for a canonical transformation from the n, q variables to the good action-angle variables N, Q for Z. For an F2 type generator, the transformation relations are CGM and Born argue, for independent reasons, that the appropriate form for F2(N, q) is t where G(N, q) is a periodic function in q of period 2n. These authors, therefore, propose the form G(N, q) = i 2' B k exp(i k.q) (1 1) k where the prime on the summation implies that k = 0 is excluded.Here we find it preferable to use the equivalent form In eqn (1 l), kl, k2, . . . take both positive and negative values, but in eqn (12), they are all positive (or zero). The notation in eqn (12) implies that all possible terms such as sin klql cos k2q2 cos k3q3 . . . sin k,q, occur in the summation with different coefficients A. The Hamilton-Jacobi equation to determine G(N, q) is Xl will usually be given as a function of p and x, but use of the relations (8) and (9) will express it as a function of Nand q. From relations (9) and (10) we have and thus the Hamilton-Jacobi equation is32 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS In CGM, the coefficients &, which are implicitly a function of N, were determined by equating to zero all the Fourier components (except k = 0) of eqn (14).In practice this means multiplying by e'ik.q, and then integrating 1' .. . [' dq. The procedure is to impose the quantisation condition by setting N to be a given set of integers corresponding to the energy level required. The Fourier components of (14) then give a set of non-hear equations for &, which can be written Bk =fk(B)* (15) The & will have different values for each set of N. Eqn (15) have to be solved by an iterative procedure, and because of the difficult form of S1, the integrations Jdq referred to above have to be performed numerically. Once the eqn (15) are solved the energy E(N) is obtained by integrating eqn (14) Jdq: Clearly the ultimate test of the success of the theory is that the energy E(N) should have " converged " using only a small number of terms in the expansion, eqn (1 1).The use of the alternative expansion eqn (12) leads to similar non-linear simul- taneous equations for the A k A k =fk(A)- (17) These are obtained by multiplying the Hamilton-Jacobi eqn (13) by one of the functions sin k,q, sin k2q2 sin k3q3 {cos k,q,){cos k,q,)(cos k3qJ * {:: ::> I' and then integrating dq, using numerical methods. The algebra required to obtain eqn (17) is a little more complex than that to obtain eqn (15) but it is quite straightforward. The energy is again obtained from eqn (1 1). CGM propose two methods for solving the non-linear simultaneous eqn (17): (i) If eqn (17) are written in the form * = f(x) (18) and x, denotes the Z'th estimate for x in an iteration procedure, the ( I + 1)th estimate is obtained by the direct substitution method XI -k 1 =f(XJ.(19) In practice it was, in certain cases, found more efficient to use some variants of eqn (19) such as Xl + 1= 3x1 + f(x1)l (i) The Newton-Raphson approach. Let % + 1 = XI + A* Determine A from the solution of the linear simultaneous equationsN . C . HANDY, S . M . COLWELL AND W. H . MILLER 33 In CGM, formula (20) is given in more detail, and in particular it is shown that to evaluate a f s ( n , q) is required. Because in our calculations Zl had a complicated form when expressed in terms of n, q, these derivatives were evaluated numerically, and because eqn (20) is only approximate, it does not matter if a%',/an is only approxi- mate.Thus we write ax, ' an Y (21) a s 1 -- - z l ( n + E ) - Zl(n) an E for a suitable E, which we found to be 0.01. that the straightforward estimate is x = 0. of the action-angle variables It, q has the symmetry In both approaches above an initial estimate for x is required, and it is clear In all the problems discussed here, the perturbation Z1 when expressed in terms Zl(n,q,, . . . qi . . . qi) = Xl(n, 2n - qr, 2n - q2, . . ., 2n - qr) (22) i.e., the perturbation is even with respect to inversion in 4 = (n, n, . . ., n). In the general three dimensional case, the 8 expansion functions 2; klql 2; k2q2 Ei"," k3q3 split into two groups of four: (i) Those even under inversion, viz. cos klq, cos k2q2 cos k3q3 cos klq, sin k2q2 sin k3q3 sin klql cos k2q2 sin k3q3 sin klq, sin k2q2 cos k3q3 and (ii) Those odd under inversion, viz.(23) cos klq, cos k2q2 sin k3q3 cos klql sin k2q2 cos k3q3 sin klql cos k2q2 cos k3q3 sin klql sin k2q2 sin k3q3. It can be shown that the odd terms in G are determined only by the even terms in the perturbation, and similarly the even terms G are determined only by the odd terms in the perturbation. As the perturbations considered here have no odd terms, G will contain no even terms, i.e., only the teriiis in group (ii) above will occur in the expansion for G. This symmetry has therefore halved the number of expansion functions for G, and is the main reason why the expansion (12) is preferred to the expansion (11). The other reason is that in all the calculations reported here, all the coefficients A k obtained from the solution of eqn (17) are real.To summarise, the approach is clear, the main question being whether the method will converge for real systems. In the next section we introduce suitable Hamiltonians for triatomic systems for applications of the method. 3. THE CLASSICAL HAMILTONIAN The ideal form of Hamiltoiiian for this theory is one for which its main part Z,, is separable, and in particular when Z0 has the form (3). Such a Hamiltonian is immediately available for vibrating systems when expressed in terms of normal co- ordinates Ql, 02, & and the corresponding momenta pl, p*, p3 (from here on we34 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS shall refer explicitly to triatomic systems, and further limit our discussion to non- linear triatomics).The form of this classical Hamiltonian is given by Wilson, Decius and Cross,12 and for systems with zero angular momentum, it has the form 3 %P 2 = +(FIZ + 4 2 + Ps2) + *z n,,p@cp -t v (24) where n is the so-called vibrational angular momentum The lbi,k are transformation coefficients from mass-weighted Cartesian displacements to the normal coordinates ok: Qk = 2 Li,kmi’(rai - rli) 2 ri = (rli, rzi, r3i) is the position of the i’th nucleus in molecule fixed coordinates, ri0 is the corresponding equilibrium position and mi is the mass of the i’th nucleus; p is closely related to the inverse of the moment of inertia matrix. Detailed informa- tion may be found in ref. (12). The form of the potential V used in these calculations was In H,O for example, Ar12 is the change in the OH1 bond length from equilibrium, Ar13 is the change in the OH, bond length and A0 is the change in the HOH angle from its equilibrium value.The normal coordinates Qk are found by expanding (27) in Cartesian displacement coordinates up to quadratic powers. If this truncated Yis denoted V,, then in normal coordinates it has the form v, = +(w,2Q12 + w2’Q2’ + w32Q32). (28) To connect with the theory of the previous section, we write and 2 0 = *(Pi2 + PZ2 + P3,) + ~ ( u : Q ~ ~ + a?&’ + w ~ ~ Q ~ ~ ) (29) 2 1 = 2 +Z&3ng + v - Yo. (30) G P Zo has the exact form of eqn (3) and, from the mechanics of the system, it is considered that Xl represents a small perturbation.The explicit forms of the potential V, eqn (27)’ can be found in ref. (13). Calcula- tions were performed on SO2 and H,O. The potentials are (i) SO,: The potential of Kutchitsu and Morino14 given in table 5 of ref. (13). The fundamental frequencies wl, co,, co3 for this potential are 1171, 525 and 1378 cm-l. (ii) H,O: The potential of Hoy, Mills and Strey” given in table 1 of ref. (13). The fundamental frequencies are 3832, 1648 and 3942 cm-l. These two triatomic systems have a further symmetry, which when expressed in terms of normal coordinates is (3 1) mQll 02’ QJ = WQl, Q2l 4 3 ) .N. C. HANDY, S. M. COLWELL A N D W. H. MILLER 35 When this symmetry is described in the action-angle variables n, q, it can be written Jfm q1, q 2 , 43) == q1, q 2 , - q 3 ) (32) where q3 is the angle variable corresponding to e3.This symmetry means that solutions of eqn (17) will give A k as zero for all terms with k3 odd, provided accurate integration methods are employed. Such terms were therefore omitted from the expansion (12). In the next section, results of this semi-classical approach will be compared with the quantum mechanical results. The latter were obtained by Whitehead and Handy l3 when solving the quantum mechanical secular equations where !Ds are appropriate vibrational expansions, written as products of Hermite polynomials of the normal coordinates. It is recalled that 3 a=l (34) ZQM = &Q - ‘ 2 8 Pa.* as first derived by Watson.16 On the right-hand side of eqn (34),% is given by eqn (24), with pi replaced by -ia/a&,.Further details on the form and the number of expansion functions required for convergence are given in ref. (13). 4. SEMI-CLASSICAL ENERGY LEVELS FOR SO2 Before giving details of the calculations and results on SOz, it is necessary to specify the selection of the integration points used to evaluate the integrals occurring in eqns (16) and (17). It appears ideal to use integration points which force the sym- metries arising from eqns (22) and (32). This means that the number of integration points must be even. Furthermore if they are equally spaced then Fourier functions are integrated exactly. Thus M points were used in each of the dimensions ql, q2, q3, with 111 even, and distributed as 3 2M- --)2n. 1 2M’ 2M” ’ 2M (35) It will be recalled that M equally spaced points will exactly integrate sin kq and cos kq fork = 0, I, 2 .. . M - 1 from 0 to 2n. The expansion functions used in G, eqn (12) were all functions occurring in (23), compatible with k3 even and 0 < ki < K, = 1, 2, 3 for some K. Initial tests were performed on the lowest (000) state of SOz. (This notation implies that the good action variables N,, N2, N3 were each set equal to zero.) The results are given in table 1 , where various numbers of integration points and expansion TABLE 1.-sEMI-CLASSICAL VIBRATIONAL ENERGIES E FOR THE (000) STATE OF s02, USING VARIOUS NUMBERS OF INTEGRATION POINTS M AND EXPANSION FUNCTIONS (23), WITH ki < K. E 1crn-l M K 1533.52 4 1 1529.33 6 2 1529.30 8 2 1529.12 8 3 1529.13 10 3 1529.12 10 436 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS functions were used.These results show that effective convergence on this ground state is obtained using the expansion functions for G given in eqn (12) with ki 6 3, and with 8 integration points in each q-dimension. All results on SO2 were obtainable using the direct substitution method for solving eqn (17), each converging in at most 20 iterations. In table 2, results for the vibrational energies for various low-lying states of SO2 TABLE 2.-sEMI-CLASSICAL VIBRATIONAL ENERGIES E, rN Cm-l, FOR VARIOUS LOW-LYING STATES WITH kr < K. OF SO2, USING VARIOUS NUMBERS OF INTEGRATION POINTS M AND EXPANSION FUNCTIONS (23), M = 6 , K = 2 M = 8 , K = 1529.33 1529.12 2045.71 2045.36 2556.56 2555.86 268 6.74 2685.1 1 2889.39 2889.07 * 3060.30 * 3 558.49 4050.08 * 3 M = 1 0 , K = 4 1529.12 2045.35 2555.77 2685.06 2889.06 * * * * Not computed.are presented, also with various numbers of integration points and expansion func- tions. No difficulties were encountered in any of the calculations, but it is doubtful whether sufficient expansion functions and points have been included for the higher bending states 030, 040 and 050. In table 3, the semi-classical energies are compared with the quantum mechanical energies taken from ref. (5), table 7 (for brevity we shall let SC and QM refer to semi- classical and quantum-mechanical results respectively). It will be noted that in the absolute energies the difference between the QM and SC results is about 0.5 cm-l. TABLE 3 .-A COMPARISON OF " SEMI-CLASSICAL " (sc) ENERGIES AND QUANTUM-MECHANICAL (QM) ENERGIES FOR LOW-LYING VIBRATIONAL STATES OF SO2 absolute energylcm-' frequency/cm-l state sc QMb 6" sc QMb (000) 1529.12 1529.60 7.7 - - ( 1 0 2685.1 1 2685.63 22.8 1 155.99 1 156.03 (010) 2045.36 2045.81 16.3 516.24 516.21 (020) 2555.86 2556.21 30.5 1026.74 1026.61 (001) 2889.07 2889.53 25.6 1359.95 1359.93 a 6 is the difference between the zero'th order energy, eqn (3), and the exact SC energy.b ref. (13). A similarity between these results and those obtained by other workers on more idealized systems6g7 can be noticed if we consider the quantity where E(QM) and E(SC) are the QM and SC energies in table 3, and Eo is the zero'th- order energy given by eqn (7). It is seen that p decreases from 7.5 for the (000) state down to 2.0 for the (020) state.This order of magnitude, and its decrease with increasing energy are similar to the results on other systems. In coiiclusion, then, for SO2, the semi-classical approach appears to give resultsN. C. HANDY, S . M. COLWELL AND W. H. MILLER 37 which are virtually indistinguishable from the quantum mechanical results and, which, when one has become accustomed to performing such calculations, are as easy to obtain. 5. SEMI-CLASSICAL ENERGY LEVELS FOR H20 Water is a very much more testing system for the applicability of semi-classical methods because of the increased effect of the perturbation arising from the lightness of the protons. Similar tables to those presented for SO2 are presented for H20. In table 4 low- TABLE d.-SEMI-CLASSICAL VIBRATIONAL ENERGIES E, IN Cm-', FOR VARIOUS LOW-LYING STATES OF H20, USING VARIOUS NUMBERS OF INTEGRATION POINTS M AND EXPANSION FUNCTIONS (23) WITH ki < K.state (000) 4 648.16" 4 645.1 1 a 4 645.46" (010) 6 248.35" 6 242.55" 6 242.67" (020) 7 815.94" 7 804.80" * (loo) 8 378.46ab 8 360.90ab 8 358.13 (001) 8 466.5gb 8 466.Bb * M = 6, K = 2 M = 8, K = 3 M = 10, K = 4 a Non-linear simultaneous eqn (17) solved by successive substitution. b Eqn (17) solved by Newton-Raphson method. * Not computed. lying semi-classical energies are given for various numbers of points and expansion functions. However, this time it was not found possible to solve eqn (17) by the direct substitution method for the (001) energy, and the Newton-Raphson method had to be employed.No difficulties were then encountered (this also applies to other states of H20 not published here). The results are much more sensitive to the number of expansion functions- indeed the results indicate that K = 4 is probably insufficient for complete converg- ence (but the same was true for the QM calculation, because a bigger secular matrix had to be diagonalised to obtain full convergence). In table 5, the results for M = 8, K = 3 are compared with the quantum mechan- ical results. Here the QM energies lie above the SC energies by 6 to 10 cm-l. The frequencies differ by up to 2 cm-'. The quantity ,u defined in eqn (36) varies from TABLE 5.-A COMPARISON OF "SEMI-CLASSICAL" (Sc) ENERGIES AND QUANTUM-MECHANICAL (QM) ENERGIES FOR LOW-LYING VIBRATIONAL STATES OF H20 absolute energy/cm- frequency/cm- state sc Q M b 6" sc Q M b (010) 6 242.55 6 249.33 117 1 597.44 1 597.35 - - ( O W 4 645.1 1 4 651.98 65 (020) 7 804.80 7 811.53 198 3 159.69 3 159.55 8 360.90 8 369.29 178 3 715.79 3 717.36 (001) 8 466.58 8 472.75 185 3 821.47 3 820.77 a 6 is the difference between the zero'th order energy, eqn (3) and the exact SC energy.b Taken essentially from ref. (13). However the results in ref. (13) contain a minor error because the 004 Term in U, in our programs, had an incorrect sign. The results have been recalculcated, with insignificant effect on any of the conclusions reached in ref. (13). They now completely agree with results published by G. D. Carney, L. A. Curtiss, and S. R. Langhoff (to be published in J. Mol. Spectr.).38 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS 10 for the (000) energy down to 3.5 for the (020) energy, therefore being much in agreement with the results for SO2. The overall results for this difficult system remain satisfactory, although computa- tionally considerable difficulty was encountered solving eqn (17), until the Newton- Raphson method was employed, use of which makes the method time consuming, 6.CONCLUDING REMARKS The purpose of this paper has been to see if the method proposed by Chapman, Garrett and Miller for obtaining semi-classical eigenvalues of multidimensional systems is applicable to determining the vibrational energy levels of triatomic mole- cules with realistic potential functions. The results indicate that this is the case, with the accuracy of the semi-classical eigenvalues (as compared to the quantum mechanical ones) being about the same as for the one-dimensional WKB quantum condition.The practicability of the method, i.e., the ease of calculation compared to straight quantum mechanics, depends primarily on whether the direct substitution iteration procedure converges or not-as for all the states of SO, investigated, but not for all those of H,O-then the calculation is as easy as for these low vibrational states as a quantum mechanical one, and would be expected to be easier for more highly excited states. If it is necessary to use the Newton-Raphson iteration proced- ure to solve eqn (17), then one must deal with matrices of order of the number of Fourier coefficients.Since the number of Fourier coefficients roughly corresponds to the number of basis functions in a quantum calculation, one does not expect the semi-classical calculation in this case to be easier than a quantum mechanical one. Since the semi-classical procedure obtains individual eigenvalues directly, it might be especially useful in situations where particular excited states, rather than them all, are of interest. Particularly difficult to deal with semi-classically is the case that the reference system has low order resonances or degeneracy, i.e., the case that for some small integers (kl, k2, k3). It is clear from the form of the CGM equations,6 for example, that the direct substitution iteration procedure cannot be used in this situation. The Newton-Raphson iteration is still possible, however, and has been successfully applied l7 in some cases to obtain semi-classical eigenvalues for the degenerate version (q = co2) of the two dimensional model problem considered in ref.(6) and (7); the semi-classical eigenvalues so obtained, though, are not in as close agreement with the quantum mechanical ones as for the non-degenerate cases. Born’s3 way of dealing with degeneracy is to transform the zero’th order problem into one for which there are no such resonances; in general, however, it appears that this is difficult to do in practice. Miller18 has recently suggested an expansion for the generating function F(q, N ) , different from Born’s, that formally alleviates the difficulty with the degenerate case, but the numerical procedure it suggests does not appear promising.A completely satisfactory way of dealing with the degenerate (or nearly degenerate) case thus does not seem to be at hand; research on these problems is continuing in this 1ab0ratory.l~ There is, however, one type of degeneracy which can be successfully overcome using semi-classical methods : namely transverse degenerate vibrations of a linear molecule. A transformation from appropriate Cartesian to polar coordinates over- comes the difficulty, and use of the normal coordinate Hamiltonian for linear mole- cules makes the problem then ideally suited to methods used in the present paper. k@l+ k2% + k3w = 0 (37)N. C. HANDY, S . M . COLWELL AND W. H. MILLER 39 Therefore, the case of degeneracy which is most often met in practice can be overcome and semi-classical methods may be u w l . S.C.M. thanks the S.R.C. for financial support. M. V. Berry, and K. E. Mount, Rept. Prog. Phys., 1972, 35, 315. W. H. Miller, Adv. Chem. Phys., 1974, 25, 69; 1975,30, 77. M. Born, The Mechanics of the Atonz (1924), (Ungar, New York, 1960). M. C. Gutzwiller, J. Math. Phys., 1971, 12, 343. W. H. Miller, J. Chem. Phys., 1975, 63, 996. S. Chapman, B. C. Garrett and W. H. Miller, J. Clwm. Phys., 1976, 64, 502. W. Eastes and R. A. Marcus, J. Chenz. Phys., 1974, 61, 4301; D. W. Noid and R. A. Marcus, J. Chem. Phys., 1975,62,2119. I. C. Percival, J. Phys., 1974, A7, 794; 1. C. Percival and N. Pomphrey, Mol. Phys., 1976, 31, 917. J. B. Keller, Ann. Phys. (N.Y.), 1958, 4, 180; J. B. Keller and S. I. Rubinow, Ann. Phys. (N.Y.), 1960, 9, 24. lo H. Goldstein, Classical Mechanics (Addison- Wesley, Reading, Mass. , 1950), pp. 288-307. l1 Ref. (lo), pp. 237-247. l2 E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations (McGraw-Hill, New York, l3 R. J. Whitehead and N. C. Handy, J. Mol. Spectr., 1975, 55, 356. l4 K. Kuchitsu and Y . Morino, J. Chem. SOC. Japan, 1965, 38, 814. l5 A. R. Hoy, I. M. Mills and G. Strey, Mol. Phys., 1972, 24, 1265. l6 J. K. G. Watson, Mol. Phys., 1968, 15, 479. l7 N. C. Handy and W. H. Miller, unpublished results. l 8 W. H. Miller, J. Chem. Phys., 1976, 64, 2880 l9 K. S. Sorbie, to be published. 1955).