GENERAL DISCUSSION Dr. D. S. Marynick and Dr. D. A. Dixon (Harvard University) said: We have per- formed calculations on inversion barriers for the AH3 molecules PH3, SH3 +, ASH, and SeH3+ with a minimum basis set (MBS) of Slater orbitals. The valence shell expo- nents and geometry were fully optimized and the results are given in table 1. TABLE 1 A PH3 calculations property MBS PDZ-SCF PDZ-CI Alkcal mo1-I 39.8 36.77 36.55 1.444 1.403 1.41 3 93.9" 95.3 1 O 93.72" angle R'IA" 1.405 1.370 1.377 RIA B AH3 MBS calculations property PH3 AsH3 SH3 -I- SeH3+ 1.444 1.528 1.374 1.465 Alkcal rno1-I 39.8 43.8 22.5 35.4 RIA angle R'/& 1.405 1 A67 1.343 1.444 93.9 93.6 100.13 95.06 (I Bond length for planar AH3 species with D3* symmetries. All bond angles are 120". In order to calibrate these results, we have also performed calculations on the in- version barrier for PH3 using a polarized double zeta (PDZ) basis set plus configura- tion interaction (CI).Our initial basis was the double zeta (DZ) basis of Roetti and C1ementi.l We then optimized the exponents for the hydrogen 1s and 2s orbitals at the DZ level. A 2p function (3 components) on H was added to this basis as was a 3d orbital (5 components) on P and the latter exponent was optimized for this basis. Geometry optimization at the PDZ-CI level was done for the planar and pyramidal geometries. Our CI included all single and double excitations from the valence shell to all virtual orbitals. The calculated geometry of PH3 is in good agreement with expximent for which R is 1.42 A and the bond angle is ~93.2O.~ The barrier at the SCF level is 36.8 kcal mol-l, which is very close to the value of 36.7 kcal mol-1 obtained by Lehn and M ~ n s c h , ~ who usedalargegaussian basis without CI.The barrier obtained by Ahlrichs et aL4 is slightly higher (38 kcal mol-I). We note that our SCF energy of -342.4869 au is 0.03 au lower than the lowest previously reported energy. The CI correction is ---0.157 au for PH3 in both forms and we find essentially no correction to the barrier height (a lowering of 0.2 kcal mol-I). Ahlrichs et aL4 find a significant 3 kcal mole' decrease in the barrier when using their CEPA C. Roetti and E. Clementi, J. Chem. Phys., 1974, 60, 4725. M. H. Sirvetz and R. E. Weston, Jr., J. Chem. Phys., 1953, 21, 898. J. M. Lehn and B.Mimsch, Mol. Pliys., 1972, 23, 91. K. Ahlrichs, F. Keil, H. Lischka, W. Kutzelnigg and V. Staeinmler, J. Chnii. Plzys., 1975,63,455.48 GENERAL DISCUSSION correction, which we feel is too large. We note here that our final SCF-CI energy is -342.6437 au for PH,. We find a significantly lower barrier than Pettke and Whit- ten1 found (40.1 kcal mol-l) using a smaller basis and a partial CI. Comparison of the barrier for PH3 obtained from the best calculation with the MBS barrier shows that the MBS barrier is -10% too large. However, we feel that our relative barrier heights for the AH3 molecules we have studied are qualitatively correct and we note that they are significantly larger than observed for first row atoms such as NH,, where the barrier is -6 kcal mol-1.2 We are now doing large basis cal- culations with some CI on the remaining three AH3 molecules to accurately determine their inversion barriers.This is especially useful for the AH, molecules with large barriers as these barriers can not now be determined by experiment. Prof. W. Kutzelnigg (Boclzum) said: I would like to stress the complementarity of theory and experiment. There are properties which are directly (which does not necessarily mean : easily) accessible from quantum chemical calculations while their extraction from experimental data poses problems of principle. An example is the equilibrium (re) geometry of a molecule, i.e., the geometry for which the electronic energy has its minimum. (The equilibrium geometry is a well-defined quantity only if the Born-Oppenheimer or at least the adiabatic approximation is applicable.) The theoretical calculation of the r,-geometry requires an additional step, namely the averaging over zero-point vibrations.Froin experiment, rz (or some other average geometry) is directly accessible and the extrapolation to re is a rather complicated pro- cedure. Provided that theory and experiment are of comparable accuracy (which at present is possible for some small molecules) one has to conclude that theory is the method of choice for the determination of re and experiment the one for rz. Refined quantum chemical ab-initio methods including correlation effects, like the coupled electron pair approximation (CEPA), yield re values for XH bonds that are accurate3 to within 0.003 A whereas the differences between re and rz are of the order of 0.02 A.Something similar holds for harmonic force constants. There is in principle no difficulty whatsoever to get them from quantum chemical calculations as the second derivatives of the energy at the equilibrium geometry. (In an analogous way higher order force constants can be calculated.) The harmonic vibration frequencies are easily derived from the harmonic force field. A theoretical calculation of the observ- able, " anharmonic " vibration frequencies, or rather of the vibronic energy levels, requires an additional computational step either using the higher order force constants or via a direct quantum mechanical treatment of the vibrational problem with the full calculated surface. On the other hand, experiment furnishes the anharmonic vibra- tion frequencies directly whereas their '' harmonization " as well as the extraction of the force field are nontrivial problems, especially when the number of equations is smaller than the number of unknowns.We conclude that quantum chemistry is the method of choice for the determination of harmonic force constants. In fact CEPA type calculations yield CH force with an accuracy of -l%, but so far overestimate force constants of double bonds by -10%. Significant progress is expected within the next few years. It should also be mentioned that, in view of the accuracy possible at present with both theory and experiment, one has to realize that, e.g., experimental rotational barriers (if they are really static barriers and not just activation energies) refer to an J.D. Petke and J. L. Whitten, J. C/zenz. Phys., 1973, 59, 4855. R. M. Stevens, J. Chern. Phys., 1974, 61,2086. W. Meyer and P. Rosmus, J . Cliem. Pliys., 1975, 63, 2356.GENERAL DISCUSSION 49 average over fast vibrations, but calculated ones to a lowest-energy path. The same is true for many other properties. Prof. D. H. Whiffen (Newcastle upon Tyne) said: Kuchitsu has referred to r, quan- tities which are essentially those discussed by Lucas as vibrational expectation values. These structures require only harmonic force fields, in addition to the primary ob- servations, and are not difficult to evaluate with observations from most of the regular techniques, that is rotational spectroscopy, diffraction methods or n.m.r.in liquid crystals. Such structures are slightly dependent on molecular isotopic species, but nevertheless form a useful basis for intercomparison of different techniques which should give the same answer to high accuracy as discussed by Lucas. Prof. I<. Kuchitsu (Tokyo) said: I agree essentially with Whiffen: (a) The r, nuclear positions constitute a well-defined structure at least for a semi-rigid molec~le.~*~ (I discussed previously4 the problem remarked upon by LucasS that " vibrationally averaged distances and angles '' do not obey the rules of geometry.) (b) In principle, the same structure can be derived from different lines of experiment. (c) Since each experimental method measures a different function of parameters related to molecular geometry, the effective vibrational averages in general differ from one another, and a correction has to be made to derive the r, structure.(d) The corrections are essentially " harmonic ", however, and only an approximate estimate of anharmonicity is suffi- cient to make the correction (except in a nonrigid case). On the other hand, as I pointed out in the text, the systematic uncertainty in the r, structure derived from any experimental method can be much larger than the random experimental error. One has to be very cautious, particularly when the r, structure is derived from a combination of various isotopic rotational constants. If neglected, the very small isotopic dependence of the r, nuclear positions can be magnified several orders and cause a large systematic uncertainty in the r, structure.This systematic uncertainty may be overlooked unless one of the following steps is taken: (a) deter- mination or estimation of the re or rm structure by use of rotational constants of vibrationally excited states and various isotopic species, (b) use of other experimental techniques such as gas electron diffraction, and (c) use of a reasonable model of the anharmonic potential function to estimate isotopic displacements of the r, nuclear positions. Dr. N. C. Handy (with DP. K. Sorbie) (Cambridge) said: I wish briefly to introduce an alternative approach to the method presented in our paper. It is based on the work of Marcus and co-workers, but has some simplifications which make it much more practical. In these problems, for most energies and initial conditions, the motion in phase- space is quasi periodic. The exception occurs near dissociation when the trajectories in phase-space become ergodic.In this brief note, we shall not discuss the latter, although such trajectories do occur with our particular potential at high energies. A quasi-periodic trajectory will close to any accuracy after a sufficient time. We denote such a trajectory I'. From the equation following eqn (13), we have N. J. D. Lucizs, Mol. Phys., 1972,23,825. T. Oka, J. Pliys. SOC. Japan, 1960,15,2274. e.g., K. Kuchitsu, J. Chem. Phys., 1968, 49, 4456. K. Kuchitsu, Bull. Chent. SOC. Japan, 1971, 44, 96. N. J. D. Lucas, Mol. Phys., 1971,22,147; 1972,23, 825.50 GENERAL DISCUSSION From eqn (1 l), G is a periodic function, and hence the last term in eqn (36) is zero.If, during I‘, qt increases by 2nMi, where Mi is an integer, we have dq, = Ni2nM,. (37) The alternative method is therefore to adjust the energy and initial conditions until eqn (37) are obeyed, with Ni = integer + 3. Hamilton’s equations are integrated, step by step, in the variables n, q, accumulating at each step the 1.h.s. of eqn (37), until the trajectory closes to a specified accuracy. We have applied the method to the two dimensional system = *(Pi2 + PZ2) + $(dx12 3- m22x22) - &Xi2& (39) with mi2 = 0.9, 0122 = 1.6 and E = 0.08. The semiclassical eigenvalues Esc determined by this method for certain values (N1,N2) are given in table 6. Also in the table are the quantum eigenvalues E,, TABLE 6.-A COMPARISON OF SEMICLASSICAL ENERGIES, Esc (DETERMINED BY THE “ TRAJEC- TORY ” METHOD), AND QUANTUM MECHANICAL ENERGIES, EQM, FOR THE MODEL HAMILTONIAN EQN (39) eigenvalue no.1 2 53 54 55 56 59 100 115 133 Esc 1.1051 2.3673 15.1710 15.2149 15.28 18 15.4952 15.8819 20.6208 22.2606 23.8654 EQM 1.1058 2.3679 15.1692 15.21 75 15.2834 15.4959 15.8854 20.6382 22.2738 23.9963 Eo 1.1068 2.3717 15.653 15.653 15.653 15.653 16.825 21.345 23.242 25.140 obtained by diagonalising a 290 x 290 secular matrix, using harmonic oscillator functions as a basis. The escape energy for this system is 25.3125; it has approx- imately 150 bound states. Because the higher quantum values have not fully converged, it is probable that the corresponding semiclassical eigenvalues E,, are more accurate.Also in the table are given the zero’th order eigenvalues E,. We are now applying this method to the potential surfaces of triatomic molecules. Prof. J. N. Murrell (Sussex) said: Although the semiclassical approach to poly- atomic vibrational energy levels has introduced some interesting physical concepts it has not yet led to a direct method of determining polyatomic potentials analogous to the RKR method for diatomics. Until such a procedure is found one must solve the “ inverse problem ” of determining a potential from spectroscopic frequencies by the indirect method of adopting a functional form of the potential with parameters ai, calculating eigenvalues for different parameter sets, and selecting values of ai which give the best agreement between calculated and observed frequencies. This is the same procedure that must be used for a quantum mechanical approach to the prob- lem.In view of the possible limitations in the type of potential amenable to semi- classical calculations it is likely that the quantum mechanical solution of the “ inverse problem ”, using perturbation or variation methods, will be the computationally simpler approach unless a direct method becomes available.GENERAL DISCUSSION 51 Dr. M. Shapiro ( Weizmann Institute of Science) and Dr. G. G. Baht-Kurti (Bristol University) said: In view of the interest expressed at this conference, and, in particular, in the papers of Mills and of Kandy, Colwell and Miller, in the calculation of vibrational-rotational energy levels for triatomic and polyatomic molecules, we would like to present the basic ideas of a new quantum mechanical method for performing such calculations.The method was first proposed in an article, by one of us (M.S.), published in 1972.l We are presently implementing the method in a form suitable for the calculation of vibrational-rotational energy levels of triatomic molecules. We expect that the method will prove competitive in both accuracy and computational speed with other available methods. As a simple example consider the calculation of the vibration levels of a one-dimensional oscillator whose potential is of the qualitative form illustrated in fig. 1. The Schro- We will attempt here only to illustrate the basic ideas behind the method. R FIG. 1 dinger equation, whose solution yields the bound state energies and wavefunctions for negative energies is : At positive energies, above the dissociation limit the wavefunctions are continuum wavefunctions and the energies (cb) span a continuous spectrum.In order to find the bound state energies in which we are interested we construct a fictitious scattering problem, involving two additional fictitious " electronic " states or channels of the system. The qualitative form of the potential curves of these addi- tional states is illustrated in fig. 2. I n the fictitious problem the three " electronic " states are coupled to each other through a non-physical potential energy matrix of the form : M. Shapilo, J. Clieni. Pitys., 1972, 56, 2582; see espccially section 111.52 GENERAL DISCUSSION I >r m W c W L \ .\ \\ / scattering \ '\\I enerqy R FIG. 2 We will be interested in calculating the probability of a transition or the T matrix element, between states 3 and 2. We can see from the form of the potential energy matrix that these states are not directly coupled to each other. The only mechanism which will permit a transition between them involves excitation from state 3 to 1, fol- lowed by de-excitation from 1 to 2. This is illustrated below 1 The coupled Schrodinger equations corresponding to our fictitious scattering problem are : J. The expression for the T matrix element for a transition from state 3 to 2 can be shown to be1: M. Shapiro, J. Chem. Phy~., 1972, 56,2582; see especially section 111.GENERAL DISCUSSION 53 where x2- and x3+ are the solutions to the homogeneous equations obtained from eqn (3) by setting V13 = VZ1 = 0.The first term in the above corresponds to a sum- mation over the bound states of V,, and the second term to an integration over its continuum states. The important thing to note about the expression for T2t3 is that the first term " blows up " whenever E = eb, i.e., whenever the energy becomes equal to a bound state energy. At the bound state energies T263 has aJirst order pole. Because we know the analytic behaviour of T2+3 in the vicinity of the pole the exact position of the bound state is very easy to 1ocate.l The method is readily generalised to the case of a triatomic molecule. In the case of water, for example, we treat the system as an oxygen atom scattering off H2.The T matrix element we need is easily calculated using any of the several available scatter- ing programs. We are presently applying the method to the calculation of the vibrational-rotational states of water. Some preliminary calculations have been com- pleted and we hope to publish a detailed description of the theory together with some results in the near future. Prof. R. N. Dixon (Bristol University) said: In their paper Handy, Colwell and Miller have exploited the power of semi-classical methods for obtaining the density of states for a system with an assumed potential function, as have a number of other recent authors. In many instances in spectroscopy the essential problem is the reverse, that is, to derive a potential function from a known spectrum of transitions.It is well known2 that in this inverse problem a knowledge of the energy levels is not sufficient, and that additional data such as vibration-rotation interaction constants are necessary for the derivation of a unique potential function. In electronic transitions band intensities provide such additional data through the use of the Franck-Condon principle. The intensity distribution has long been used with a very simple semi-classical deconvolution for deriving repulsive potential func- tions for diatomic molecule^.^ I should like to encourage Handy, Miller and their co- workers to extend their theories to include the semi-classical formulation of expecta- tion values for such properties as rotational coupling constants and transition probabilities. Such a development would prove invaluable in the interpretation of the spectra of molecules such as NO, or SO, for which the Born-Oppenheimer separation is a poor first approximation.Thus in NOz most recent work has attempted to give a detailed classification of individual energy levels, but has not been able to give any detailed interpretation of the derived energy level manif~ld.~ In this case the visible absorption spectrum involves at least three strongly coupled electronic states. The dimensionality of these problems is such that I am sure that only semi-classical methods are likely to make progress towards the goal of deriving satisfactory potential functions for the various interceding states. Prof. R. J. Le Roy (Waterloo) said: The paper of Handy et aZ.,5 and much of the discussion thereof, was concerned with the problem of determining a reliable multi- dimensional analogue of the first-order WKB quantum condition which has been so M.Shapiro, J. Chern. Phys., 1972, 56,2582; see especially section 111. I. M. Mills, in Molecular Spectroscopy: Modern Research, Vol. I, ed. K. N. Rao (Academic Press, N.Y. 1972) and in Theoretical Chemistry, Vol. 1, ed. R. N. Dixon (Specialist Periodical Reports of the Chemical Society, 1974). G. Herzberg, Spectra of Diatomic Molecules (D. Van Nostrand, N.Y., 1950), p. 392. C. G. Stevens, M. W. Swagel, R. Wallace and R. N. Zare, Chem. Phys. Letters, 1973, 18,465; T. Tanaka, R. W. Field and D. 0. Harris, J. Mol. Spectr., 1975,56, 188. N. C. Handy, S. M. Colwell and W. H. Miller, this Discussion.54 GENERAL DISCUSSION successfully applied to one-dimensional problems. It, therefore, seems appropriate to bring attention to two problems associated with the use of semiclassical quantization conditions in one dimension, as they may also affect multi-dimensional problems.The first is simply the fact that the accuracy of results obtained using the first-order WKB approximation may sometimes be inadequate, particularly for a species with a small reduced mass. For example, we found that the exact (numerical) quantum mechanical eigenvalues of a first-order RKR potential curve (the result of an inversion procedure based on the first-order WKB quantum condition) for the ground state of HF differed by up to several cm-l from the experimental data on which the RKR curve was based.Moreover, Kirschner and Watson showed that second-order WKB procedures may be required for calculating spectroscopically accurate potential curves and centrifugal distortion constants for large reduced mass species like CO. While my preceding remark shows that a “ better than first-order WKB ” treat- ment is often required in one-dimensional eigenvalue problems, my second point is the warning that the second- and fourth-order WKB quantum conditions almost always have non-physical singularities at potential asymptotes. The general WKB quantum condition may be written as v + + = %\R2 LE - Y(r)]* dR + A,(E) + A,(E) + . . . 7di R1 where the i-th order contribution A,(E) is an integral of the potential V(R) and its derivatives on the interval between the two classical turning points at energy E, R1 and R2.To any given order i, the allowed WKB eigenvalues are the values of E for which the sum of terms on the right hand side of eqn (l), truncated after A,(E), is precisely half integer. However, we have recently found2 that for a potential which varies as an inverse power of distance as it approaches an asymptote at energy D, V(R) N D - C,,/Rn, if the power n > 2, then As(E) cc -(D - E)-‘n-2’’2n A,(E) cc (D - E)-3(n--2)’a. Thus, at an asymptote of this type, the third-order quantum condition predicts that (v + 5) -+ - while the fifth-order criterion predicts that (v + 5) --f + a. Since the former is absurd and the latter is incorrect, it is clear that these higher-order quantum conditions cannot be used in this region. In the one-dimensional case, this instability of the higher-order quantization condition only manifests itself at energies extremely close to the potential asymptote, but its importance in multi-dimensional problems is not known.Prof. D. H. Whiffen (Newcastle upon T’ne) said: The work of Handy and others has a long term goal which is to use spectroscopic observations to delineate the poten- tial surface employing some technique such as a least squares adjustment to obtain optimal values of the parameters which describe the surface. This step will require a weighting scheme for the observations used as input. Beaton and Tukey3 have raised the question as to whether the traditional least squares scheme with weights chosen, S. M. Kirschner and J.K. G. Watson, J. Mol. Spectr., 1974, 51, 321. A. E. Beaton and J. W. Tukey, Critical Evaluation of ChemicaZ and Physicnl Structural Informa- tion, ed. D. R. Lide and M. A. Paul. Nat. Acad. Sci. 1974, p. 15. * S. M. Kirschner and R. J. Le Roy, 1976, to be published.GENERAL DISCUSSION 55 somewhat subjectively, after inspection of experimental errors, is always appropriate. The appropriate choice of weights is especially difficult when the observations are of different natures or even with different dimensions and units as wave-numbers for vibrational levels, expressed in cm-l, and frequencies for rotational &values, expressed in MHz, from microwave observations. Indeed for the latter the errors of computa- tion from a given potential surface still exceed the experimental uncertainties which make observational errors unsuitable for suggesting a weight matrix.Recently I have experimented with a scheme, which differs from that of Beaton and Tukey, and has so far proved valuable, although it has not yet been tested for problems outside the area of fitting potential surfaces. Traditionally the diagonal weight matrix element, Wii, for the ith observation is put equal to l/ai2 where a, is the typical error expected for the ith observation. The new scheme replaces this by Wii = 1/[(AcJ2 + (Aei)2]. Here is the difference between the ith observation and its calculated value at the current values of the parameters to be determined. Aei is the expected error in the calculation of the ith observation determined in the usual way from the Jacobian matrix and the variance-covariance matrix for the parameters.This latter involves a preliminary weight matrix based on chosen ci, which should be appropriate in absolute value as well as relative values. (Aei)2 is thus the diagonal element of the variance- covariance matrix for the observations. In principle the influence of the initially chosen ci could be removed by repeated iteration at this stage using the revised Wit to deduce improved (AeJ2 but it is doubtful if this is profitable although such a process would not be very expensive in time; it would have the merit of making the scheme truly objective but has not yet been undertaken. Apart from the choice of W, standard non-linear least squares procedures are used. In miscellaneous use, especially in the fitting of potential surfaces, the results from such a scheme are found to agree with those of a sensible weighting scheme within their mutual standard deviations.One merit, its objectivity has been referred to above, but in practice an equal merit at least comes from a treatment of rogue results. These are incorrect features arising from any cause including copying errors, mis- assignment or truncation errors of any nature which affect special individual calculated values in a specially sensitive way. The existence of (Aei)2 in the denominator of W,, downweights such rogues whereas the standard least squares tries very hard indeed to fit these quantities at the expense of worsening the fit elsewhere. Trials with de- liberate errors in input observations with the new scheme always gave values of parameters close to those most acceptable and suggested that the poor fit to the deliberately erroneous information was exceptional.The same input with standard least squares gives a poor overall fit which does not indicate why this should be. It is interesting that in current work on the potential surface of carbonyl sulphide, the lower vibrational levels are more accurately calculated than the higher levels, even though this bias was not introduced via the a,; such a bias was deliberately introduced in a subjective manner in previous work,l since the truncation errors of the calculation made this pattern appropriate. Prof. I. C. Percival (London) said: It is nice to see Handy, Colwell and Miller applying semiclassical methods to real molecules, going beyond the earlier work on models and bringing the theory into contact with experiment.I will discuss the application of semiclassical methods to the forward problem of determining energy levels from a potential surface and also the inverse problem of determining a potential surface from an energy spectrum. * A. Foord, J. G. Smith and D. H. Whiffen, Mol. Phys., 1975,29,1685.56 GENERAL DISCUSSION In either case it is important to distinguish the principles and the methods. For one degree of freedom, the Bohr-Sommerfeld quantization rule is the semiclassical principle upon which the RKR method is based. For a semiclassical theory of systems with many degrees of freedom, where the hamiltonian may not be separable, the first requirement is to generalize the principle, that is Bohr-Sommerfeld quantization.Two distinct generalizations are that of Einstein, Brillouin and Keller (EBK) and that of Gutzwiller, the latter being based on closed classical trajectories. The first is right and the second is wrong, as was implied by Marcus1 at a Discussion of the Faraday Society, Three methods of obtaining energy levels from potential energy surfaces are based on EBK quantization. The method of Marcus and his collaborators uses stepwise integration of trajectories, which are not necessarily closed. The method of Miller and his collaborators is based on the solution of the Hamilton-Jacobi equation for the action function. The methods I have used with Pomphrey at Queen Mary College are based on a variational principle.Since all the methods are based on EBK quantiza- tion the results obtained from them should in principle be the same apart from the inevitable numerical errors. By contrast the Gutzwiller method generally gives different results. Semiclassical quantization of bound states is discussed in a forthcoming review.2 So much for the forward problem. Now for the inverse problem, the question of the extension of RKR to polyatomic molecules. The inverse problem is tough, so it is convenient to take it in two stages. 1. A method for the solution of model non-separable problems. 2. The generalization of the method to the vibration-rotation spectrum of polyatomic molecules. Consider the first. In the case of a diatomic molecule, the vibrational spectrum alone does not provide a unique potential, and the same applies to polyatomic molecules, and to simple models.This problem of uniqueness is overcome by putting an additional constraint on the potential; that it should be a polynomial in th, p s q uares of the co-ordinates. The example chosen was a particle of unit mass moving in the potential V(x, y ) = a1dc2 + a20x4 + aoly2 + a11x2y2 + a12x2y4 + cxoZy4 + a21x4~2 + a22x4y4, with a10 = 0.5, a01 = 0.25, a20 = 0.0015, all = -0.003, aO2 = 0.001, and the remaining coefficients zero. 21 energy levels were calculated (semiclassically) and then the semiclassical inversion method was used on these levels to derive all eight " unknown " coefficients of the above potential form. All values were obtained correctly with an error of less than 2 x The time of computation was less than 16 s on a CDC 6600. The method used resembles a semiclassical version of the quanta1 method of Foord et al.and Wl~iffen.~ It differs in three respects 1. It is semiclassical. 2. It uses a classical analogue of the Hellmann-Feynmann theorem to accelerate the least-squares procedure. R. A. Marcus, Faraday Disc. Chem. SOC., 1973,55,71. I. C. Percival, Adv. Chent. Phys., 1977. A. Foord, J. G. Smith and D. H. Whiffen, Mol. Phys., 1975,29,1685; D. H. Whiffen, Mol. Phys., 1976,31,989.GENERAL DISCUSSION 57 3. It is in a much more primitive form. Work is in progress on stage 2, an extension of the method to real triatomic molecules. Dr. J. N. L. Connor and Dr. W. Jakubetz (University of Manchester) said: We would like to mention an extension of the simple WKB quantization formula that has recently been used to calculate complex angular momentum eigen~alues.l-~ These eigenvalues are required in Regge Pole theories of elastic scattering phenomena.*-’ Consider the radial Schrodinger equation with a potential of the Lennard-Jones type.Imposing an “ outgoing wave only ” boundary condition leads to the following semiclassical quantization formula for the Regge Poles I,. where n = 0, 1,2, . . . and a(2,) and e(lJ are the complex turning points. Results using this formula are compared with exact quantum ones in the table below.3 A Lennard-Jones (12,6) potential was used with collision parameters of exact quantum semi-classical I2 Re I,, Im l,, Re I,, Im I,, 0 1 80.01 2 21.219 1 80.01 5 21.21s 1 179.239 24.035 179.242 24.034 10 175.074 50.561 175.076 50.560 20 176.187 79.644 176.189 79.643 75 210.626 194.336 2 10.628 194.335 E = 4.0 x 10-21J, Y, = 4.0 x 10-lOm, rn = 4.377 x 10-23g and E = 2.0 x 10-20J.These parameters correspond approximately to the elastic scattering of K by HBr. Agreement between the semiclassical and exact results is seen to be very good. Dr. M. Tabor (University of Bristol) said: Miller has stated that “ there are ‘ good ’ action variables associated with a saddle point region of a potential surface just as there are those associated with a minimum ”. This result has been obtained by making one (in the case of a two dimensional system) of the classical frequencies of motion complex. This is another example of analytic continuation of a classical equation, in this case the Hamilton-Jacobi equation, still yielding quantum mechanic- ally useful results.Solutions of the Hamilton-Jacobi equation in real space imply the existence of so called “ invariant tori ” [see e.g., Appendix 26 of Arnol’d & Avez (1968)].8 These are n-dimensional regions of phase space imbedded in the 2n - 1 dimensional energy shell and contain an infinite family of classical orbits, i.e., the orbits are not isolated. It is not yet known if this holds for complex frequencies. If it does, then a delicate point arises concerning the connection between the current theory and the earlier one N. Dombey and R. H. Jones, J. Math. Phys., 1968, 9, 986. J. B. Delos and C. E. Carlson, Phys.Rev. A., 1975,11, 210. J. N. L. Connor, W. Jakubetz and C . V. Sukumar, J. Phys. B. Atom. Mol. Phys., 1976,9, 1783. J. N. L. Connor, Chenz. Soc. Rev., 1976, 5, 125. C. V. Sukumar and J. N. Bardsley, J. Phys. B. Atom. Mol. Phys., 1975, 8, 568. C. V. Sukumar, S. L. Lin and J. N. Bardsley, J. Phys. B. Atom Mol. Phys., 1975, 8, 577. J. N. L. Connor and W. Jakubetz, Chern. Phys. Letters, 1975, 36, 29. V. I. Arnol’d and A. Avez, Ergodic Problems of Classicnl Mechanics (W. A. Benjamin, N.Y., 1968).58 GENERAL DISCUSSION based on Gutmiller periodic orbit theory. The latter is now known to be incorrect (see remarks at this session by Percival) and we have shown that this is because in the case of a torus of orbits, i.e., when action variables exist, the method of stationary phase as employed by Gutzwiller (and Miller) cannot be applied since the classical orbits are not It thus remains to be seen whether periodic orbit theory and the action variable theory of reaction rates are in fact mutually exclusive or that the former, as claimed, merely represents an approximate version of the latter. The semiclassical eigenvalue method used by Handy, Colwell and Miller is also based on the existence of invariant tori. For systems of more than two degrees of freedom (the minimum dimensionality for nonseparability) there will not only be the obvious problem of increased computational difficulty but also problems related to the more complicated structure of phase space. For two degrees of freedom the two dimensional tori divide the three dimensional energy shell. For higher dimensionality we no longer have this simple topological result and gaps between tori can connect up widely differing regions of phase space. This suggests the possibility of increased con- vergence difficulties in the above method. It should also be mentioned that if the gaps between tori are of the order tid (d degrees of freedom) further problems arise due to the existence of the ‘‘ Irregular Spectrum ” predicted by Per~ival.~ M. V. Berry and kf. Tabor, Proc. Roy. Soc., A, 1976,349,101. M. V. Berry and M. Tabor, J. Phys. A., 1976, in press. I. C. Percival, J. Phys. B., 1973, 6, 1229.