Rational Selection of Methods For Molecular Scattering Calculations" BY ROY G. GORDON Dept. of Chemistry, Narvard University, Cambridge, Massachusetts, U.S. A. Received 26th January, 1973 A critical discussion is given of some of the more useful and accurate methods for the calculation of cross sections for various types of molecular collisions. Quantum mechanical, classical and semiclassical methods are considered. Criteria are summarized for the feasibility of various calcula- tions and for the accuracy of the results. A flow chart is formulated, which uses these criteria to select, for given molecules and types of experiments, the easiest calculational method which yields accurate results. Examples of this selection process are given, drawn mainly from recent calculations of inelastic scattering.1 . INTRODUCTION Scattering theory is the link between intermolecular forces and the various experi- mmts with molecular beams, gases, etc., which depend on collisions between molecules. This link is used in both directions: in the theoretical approach the intermolecular forces are used to predict the outcome of experiments. In the empirical approach, experimental results are inverted or analyzed to obtain informa- tion about the intermolecular potential. For most molecular scattering phenomena, it is usually assumed that non- relativistic quantum mechanics provides an accurate description. Therefore, one might expect the field of molecular collision phenomena to be nicely unified by the application of non-relativistic quantum-mechanical scattering theory.Instead, one finds that a bewildering variety of methods, approximations, techniques, formula- tions and reformulations are used to treat molecular collisions. One might be tempted to blame this multitude of approaches on the conceit of the many theoreticians who have worked in this area, each developing his own point of view. In fact, this variety is more nearly due to the following two circumstances : (i). Exact quantum mechanical scattering calculations are not yet feasible for all types of molecular collisions. Therefore some types of approximations are necessary to treat the quantum mechanically intractable cases. (ii) The very richness and variety of mole- cular scattering processes require that a number of different approximation methods be used in different situations.We believe that suitable methods have in fact been developed to treat successfully almost all types of molecular collisions. The question thus arises : how do we select the most appropriate method for a given problem? In Section 2 we discuss some criteria for choosing between methods. In Section 3 we propose an explicit algorithm for selecting the best available method for a given collision process, and for a given set of experiments measuring that process. Then we apply this algorithm to a number of examples, mainly from inelastic scattering. It is hoped that these examples will * Work supported in part by the National Science Foundation. 22R. G . GORDON 23 illustrate the way in which one should choose between methods, and the kind of information such a choice requires.In addition, the examples described in Section 3 are all chosen to represent real cases for which calculations have been completed, or are in progress. Thus, they provide a guide to some recent applications of each of the methods discussed, and the reader himself can evaluate the state of the art in applications of each method. 2. CRITERIA FOR CHOOSING A N APPROPRIATE SCATTERING THEORY In order to make a rational selection of a scattering theory to apply to a specific problem, we must formulate criteria upon which this choice is to be based. It seems to us that there are three main considerations : 1. FEASIBILITY It is necessary that the method be applicable, in a practical sense, to the problem of interest.Difficulties may occur at various stages: analytic difficulties (e.g., in evaluating matrix elements, or in transforming coordinate systems) ; exceeding memory size or running time of computers; difficulties in averaging and analysis of results into a form to compare with experiments. 2. ACCURACY The results must be sufficiently accurate to interpret the experiments of interest. In a complete quantum-mechanical calculation, this accuracy can be verified by convergence tests within the calculation. In classical, or other approximate methods, accuracy and reliability generally must be judged by experience with test comparisons with complete quantum-mechanical calculations. The numerical stability of the method must also be considered. 3. EASE OF CALCULATION When more than one method meets the above criteria of feasibility and accuracy, one has the luxury of choosing the easiest of the possible methods.Some considera- tions in the " ease " of calculation might include the following : if the evaluation of the interaction potential is difficult (as it is likely to be in any realistic case), one would prefer the method which requires the smallest number of values of the potential. Other considerations might be the complexity and cost of the computer calculations, and the availability of well-documented and reliable computer programs. Next, we must discuss the specific methods of calculation which we shall recom- mend, in the light of the three criteria discussed above. A. The feasibility of a full quantum scattering calculation depends mostly upon the number (N,) of internal states which are coupled together by the interaction potential, during the strongest part of the collision.The most efficient quantum scattering method currently available is based on piece-wise analytic solution to model potentials which approximate the true potential to any prescribed degree of accuracy.2 While one can program this method to work with whatever size computer is available (using disc storage if necessary), the number of disc accesses becomes rather large unless the computer memory is large enough to store at least eight N, by N, matrices (8 NZ numbers). Up to about 100 N: multiplications and additions are required to construct a single scattering matrix. These storage and timing restrictions typically restrict feasible calculations to N, about 100 or less.Quantum scattering (" close coupling ").'24 MOLECULAR SCATTERING CALCULATIONS The accuracy of the quantum scattering results is limited mainly by the number of internal states included (close-coupling approximation). Therefore, one must check that the predictions of interest converge as one increases the number of internal states. The accuracy of the radial integration can be set at any pre-determined value. The method was constructed to be numerically stable, and in practice not more than two digits are lost in round-off error, even in calculations involving millions of arithmetic operations. As for ease of calculation, only a small number (say 30) of radial integration points are required, so that not too many evaluations of the potential are necessary. A complete computer program for quantum-mechanical elastic and inelastic scattering is a~ailable.~ The quantum theory of reactive scattering is not as highly developed as for inelastic scattering.No generally applicable algorithm has yet been perfected, particularly for three-dimensional reactions. However, many promising approaches are being explored. B. Quantum scattering calculations are sometimes made using the distorted wave Born appro~imation.~ Such calculations have the advantage of almost always being feasible numerically. For simple cases, one can also obtain some results analyti~ally.~ However, the accuracy of the results is generally poor for most molecular collisions. A necessary condition for the results to be accurate is that all the calculated transition probabilities be small compared to unity.However, this is not a sufficient condition, since small transition probabilities can result from fortuitous cancellation of large negative and positive contributions to the perturba- tion integrals. One can test for this possibility by checking whether the sum of all the perturbation integrals remains small as we build them up by adding on contri- butions from the various radial intervals. This provides both a necessary and sufficient condition for the validity of perturbation theory. C . Classical mechanics provides an approximate description of scattering, which has the important advantage of almost always being feasible to carry out. Only three circumstances occasionally make it difficult to obtain results with classical scattering theory : (1) There may be points at which the coordinates chosen for integration become singular or undefined.6 If a trajectory approaches one of these points, the numerical integration may break down.Such difficulties may be avoided by changing co- ordinate systems. (2) If some coordinates change much more rapidly than others, the equations become difficult to integrate numerically. These difficulties may be reduced by using action-angle coordinates for the rapidly varying coordinates,’ and by using a very stable and accurate integration technique, such as Runge-Kutta. (3) Some trajectories in both inelastic * and reactive collisions are long and compli- cated, corresponding to resonances or long-lived collision complexes.Unless one really needs to know the details of such collisions, it is probably best to use a statistical theory to describe the distribution of results for these collisions. The accuracy of classical calculations is usually adequate when the experiments of interest average over at least several quantum states. If, however, no classical trajectories connect the initial and final states of motion, the classical prediction is a vanishing cross section or rate constant for that process. The correct quantum- mechanical prediction may, however, be a small but non-zero rate for such a “ classi- cally forbidden ” process. “ Tunnelling ” through a potential barrier is a simple example. The connection formulas in the WKB method may be viewed as providing a complex-valued trajectory which does link the “ classically forbidden ” states.Jn the WKB treatment, the probability for passing through this complex trajectory,R. G. GORDON 25 is related to the exponential of the imaginary part of the classical action function accumulated along the complex path. Recently, this treatment has been generalized to inelastic and reactive scattering.'O The main difficulty at present in applying this method, is finding the actual complex trajectories in a numerically stable way. Several approaches have been suggested, and this is an active field of current research. One should note that the method appears also to require that the interaction potential be an analytic function of all its coordinates, so that it, too, can be analytically continued.Whether a continuation method can be applied to a potential defined by a table of numerical values and some interpolation formulae, is not clear at present. When classical mechanics is applied to experiments involving only one or two quantum states, the results are generally less accurate than for the cases involving averages over many quantum states. However, even simple correspondence principle arguments, assigning classical results to the quantum state of nearest angular momentum, predict line-broadening cross sections to an accuracy comparable to the experimental uncertainty.8* 11 Moreover, by including interference effects between different trajectories,l one can make fairly accurate predictions for elastic,' vibrationally l3 and rotationally l4 inelastic, and reactive lS scattering.This is a very useful approach, which will certainly be used more in future calculations, to improve the accuracy of classical predictions. D. Another approach to scattering calculations uses a quantum-mechanical description of the internal states, but classical mechanics for the translational motion. This " classical path " method has been popular in line-shape calculations.16 It is almost always feasible to carry out such calculations in the perturbation approxima- tion for the internal states.16 Only recently have practical methods been developed to perform non-perturbative calculations in this approach. ' To get accurate results from this approach, it is necessary that the collisional changes in the internal energy be small compared to the translational energy.Then one can accurately assume a common translation path for all coupled internal states. In the usual applications of this method, one does not include interference effects between different classical paths, so that translational quantum effects, including total elastic cross sections, are not predicted. If the perturbation approximation is also used, accuracy can be guaranteed only when the sum of the transition probabilities remains small throughout the collision. These classical path calculations are relatively easy to carry out, and analytic results are available in the straight-line path, perturbation limit.18 Thus when the approximations are valid, this classical path approach should be used.3 . AN ALGORITHM FOR CHOOSING AN APPROPRIATE SCATTERING THEORY Using the criteria discussed above, we wish to select the easiest method of calcula- tion which is both feasible to apply to the molecules of interest, and whose results are sufficiently accurate to describe the relevant experimental results. We have found it convenient to organize this selection process into a flow chart, which is given in fig. 1. Starting at the top, one makes a sequence of decisions based upon the criteria for feasibility and accuracy. Decisions about the relative ease of different methods are not made explicitly; they are implicit in the organization of the flow chart. When one's path in the flow chart reaches a box with no lines going out from it, and double underlines at its bottom, one has arrived at the most suitable method.26 __.--- Let NE be the maximum number of internal states or basis functions which are coupled during collision. Do about 8N: numbers fit into your computer's memory, and can you afford about 100N2 multiplications on your computer. per S matrix? MOLECULAR SCATTER I N G C A LC U I. A TI 0 N S no YCS Do all the experiments you are interpreting average over more than about 10 internal states? I Accept these quantum I scattering results. no Y e no can-( no Do real classical trajcctories connect the initial and final quantum states of interest? ' trajectories which connect the quantum states of interest? r Compute your results from analytically Compute your rcsults using these real continued classical mechanics : trajectories, plus a correspondence (complex) trajectories (ref.(10)) principle, if necessary (ref. ( 1 1)) L 1 t I W I 1 1 Do any of the experiments of interest have angular resolution sufficient to rcsolve oscilla- tions due to quantum interference or to observc the total elastic cross scction? Yes1 & Use a fixed classical path, independent of internal states, and perturbation theory on the intcrnal stales (ref. (16)). Are a11 the transition probabilities C' I I z...l at all times during the collision? 8 v Try a calculation using the Distorted Wave Born Approximation (ref. (4)). Are all the transition probabilities 2' I zj I ' ~ ~ ~ ~ 1 at all radii during collision? Are the changcs in internal energy small compared to the translational no ' energy ? I T Compute " classical " S-matrices (ref. (I?)), with interferences betwcen different trajectories.no ( i ' Accept the results of this classical path, yes quantum internal statcs calculation. - Use a fixed classical path, independent of internal states, with an exact, non- perturbative treatment of internal states. (ref. (17)). Do these rcsults converge as internal states 3rc added?R . G . GORDON 27 In some cases, one’s decision at some point may be conditional on a variable in the problem. For example, transition probabilities may be small compared to unity for large orbital angular momenta, but not for small ones. In such cases, one should follow both branches of the decision, and arrive at two different methods, one for each range of the variable.In a few such cases, both branches may later rejoin, and only one method is recommended after all. In more difficult cases, as many as three different methods have been found to be necessary for different ranges of the variables. We first follow the flow chart for the simple case of elastic scattering of structure- less atoms. The number of internal states, N,, is one, and quantum scattering calculations are feasible and recommended, for even the smallest modern computer. The Numerov method has often been used for such calculation^,'^ but the recent method based on analytic approximations by Airy functions obtains the same results with many fewer evaluations of the potential function. The WKB approxi- mation also requires a relatively small number of function evaluations, but its accuracy is limited, whereas the piecewise analytic method can obtain results to any preset, desired accuracy.Next we consider rotationally inelastic scattering of H2 with He. At room temperature, the maximum rotational angular momentum state which is significantly populated is j,,, = 4. Thus, we estimate N, = (jmax/2+ 1), = 9, including all the nz-states. The data storage SN,” is less than lOOOnumbers, only a small addition to the quantum scattering program code (about 100 k bytes). Assuming a multiply time of 1 ps., 100 N: is less than 0.1 s computer time per S matrix. Thus the quantum scattering calculations are quite practical, and have been carried out for more than a dozen different potential surfaces.2o The results are in good agreement with mole- cular beam results, sound absorption, and line shapes in light scattering and n.m.r.Because of the wide spacing of the rotational levels, and the relatively weak angle- dependent potential, these results converge very quickly as j,,, increases, and j,,, = 4 is adequate for all the experiments at temperatures up to 300 K. For collisions of H2 with atoms at higher energies, both vibrational and rotational excitation occurs. At 1 eV, about 50 channels are open. For a complete quantum scattering calculation, we estimate data storage at SN: 2: 20000 single precision words, and computer time of 12 s per S matrix (again assuming a 1 { i s multiply time). Convergence is obtained with the addition of a few closed channels, and such calculations are feasible, and have recently been carried out for H, + He,21 and H2 + Li+.22 For vibrational and rotational relaxation of D, at 1 eV, about 140 channels are open, so the quantum scattering estimates are about 160 000 numbers in data storage, and about 5 min computing time per S matrix, or 2 s per initial condition.While such calculations are feasible on a large computer, they might be too expensive. Then, if one is averaging over rotational states to find vibrational transition probabilities, the flow chart suggests classical trajectories. However, the vibrational coupling is so weak that no real trajectories connect different vibrational states, so complex trajectories must be calculated to find the vibrational transition pr~babilities.,~ One should note, however, that if one wants to find all the individual rotation- vibrational transition probabilities, the quantum calculation, at 2 s, per initial condition, uses less computer time than the complex trajectory calculation, which requires about 2 s per complex trajectory, and a search of several complex trajectories for each initial condition.If we consider the collisions of two molecules (rather than atom+molecule, as above), the number of coupled channels is approximately the square of the number of Examples of all these cases have been found.28 MOLECULAR SCATTERING CALCULATIONS accessible internal states of either molecule separately. Thus for rotational excitation of two hydrogen molecules near room temperature, N, x (jmaX/2 + 1)4 = 81 for jmax = 4, and quantum calculations are feasible. However, for vi bration-rotation transitions at 1 eV, 50 internal states for each molecules correspond to N, = 2500 channels, and exact quantum calculations are not feasible.If we want individual transition probabilities for this case, the flow chart brings us to try the distorted wave Born approximation, which is feasible and accurate for this case. Next we consider some more difficult cases, in which several methods are recom- mended for different parts of the calculation. For rotational excitation of HCI by Ar at room temperature, the maximum rotational angular momentum quantum number coupled during collision is about 12. The maximum number of coupled j, m states is Nc = omax + l)(jmax + 2)/2 = 91, since HCI is a heteronuclear molecule, and thus all states of the same total parity are coupled.With 91 channels, the quantum scattering calculations are feasible, but rather expensive. A further compli- cation of the quantum calculations for this case, is the fact that many bound states of HCl+Ar exist, which will lead to many resonances in the scattering, and thus difficult energy averaging the cross sections. Thus we explore the alternative methods with the flow chart. For interpreting infra-red line-widths, we average over the 2j+1 m-states. For an initial j greater than 5, we thus average over enough m states so that the classical method, plus the correspondence principle, is adequate for these cases. For the low-j lines, we observe that in the absence of differential cross section measurements, we do not require a " high resolution " quantum calcula- tion.The rotational energy changes, for the low j states, are small compared to the typical translational energies, so the fixed classical path approximation is valid. For collisions at large impact parameter, the classical path-perturbation theory results are of acceptable accuracy. However, for small impact parameter cases, the perturbation theory fails. To select a method for the remaining cases we note that the maximum number of coupled initial states up to j = 5 is N, = G+ l)(j+Z)/ 2 = 21. The storage estimates for a non-perturbative classical path calculation are thus 91(91+2 x 21) 21 21 O00 numbers, and computer time 50(91)2(91 +21) x loA6 s =46 s per S matrix. This classical path method is thus feasible for the remaining initial conditions, and has been used to calculate infra-red and n.m.r.line shapes for this system. For a heavier system, such as N20 + Ar, a calculation8 of rotational transitions and microwave or infra-red line widths would follow the same course through the flow chart, as that followed above in detail for HCl+Ar. However, at the last stage (low j, small b collisions), the number of coupled states would probably be too large for the non-perturbative, fixed classical path calculation to be practical. Then one should calculate " classical S matrices " including interference between trajec- tories, to cover these remaining collisions. 4. CONCLUSION The theory of molecular scattering has now been developed to the point that scattering calculations can be made with an accuracy sufficient for comparison with current experiments.Thus any discrepancy between theory and experiment should be traced to an inadequate knowledge of the interaction potentials, or to experimental errors, rather than to approximations in the collision dynamics. This tighter coupling of theory and experiment should permit a much more fruitful utilization of the results of molecular beam scattering.R. G . GORDON 29 For reviews of recent quantum scattering methods, see Methods in Contputational Physics, ed. B. Alder et al. (Academic Press, New York, 1971), vol. 10. R. G. Gordon, ref. (l), chap. 2, p. 81. Program No. 187, Quantum Chemistry Program Exchange, Chemistry Dept., Indiana Univer- sity, Bloomington, Indiana 474.01, U.S.A.4see, for example, L. S. Rodberg and R. M. Thaler, Introduction to the Quantum Theory of Scattering (Academic Press, New York, 1967), chap. 12. G. Starkschall and R. G. Gordon, to be published. R. J. Cross, Jr. and D. R. Herschbach, J. Chem. Phys., 1965, 43, 3530. ' A. 0. Cohen and R. G. Gordon, to be published. R. Pearson and R. G. Gordon, to be published. P. W. Brumer and M. Karplus, to be published. l o (a) W. H. Miller and T. F. George, J. Chem. Phys., 1972,56,5668 ; (b) 1972,56,5722 ; (c) J. D. Doll and W. H. Miller, J. Chem. Phys., 1972, 57, 5019; (d) R. A. Marcus, H. R. Kreek and J. R. Stine, Farahy Disc. Chem. Soc., 1973,55, 34. R. G. Gordon, J. Chem. Phys., 1966,44,3083; R. G. Gordon and R. P. McGinnis, J. Cheni. Phys., 1971, 55, 4898 ; D. I. Bunker, ref. (1) chap. 7. l 2 (a) K. W. Ford and J. A. Wheeler, Ann. Phys., 1959, 7,259 ; (b) W. H. Miller, J. Chern. Phys., 1970, 53, 1949 ; (c) 1970, 53, 3578 ; ( d ) Chem. Phys. Letters, 1970, 7, 431 ; (e) R. A. Marcus, J. Chem. Phys., 1972, 57, 4903 and references therein. l 3 W. H. Miller, Chem. Phys. Letters, 1970, 7 , 431. l4 W. H. Miller, J. Chem. Phys., 1971, 54, 5386. Is C. C. Rankin and W. H. Miller, J. Chem. Phys., 1971,55,3150. l6 P. W. Anderson, Phys. Reo., 1949,76,647 ; for a recent review, see G. Birnbaum, Ado. Chem. l 8 R. J. Cross and R. G. Gordon, J . Chem. Phys., 1966,45,3571. "J. W. Cooley, Math. Computation, 1961, 15, 363. 2o R. Shafer and R. 6. Gordon, 1973,58, 5422. 21 W. Eastes and D. Secrest, J. Chem. Phys., 1972, 56, 640. 2 2 H. van den Bergh, R.-David, M. Fraubel, H. Fremerey and J. P. Toennies, Furaday Disc. 23 W. H. Miller, Faraday Disc. Chem. SOC., 1973, 55,45. Phys., 1967, 12,487. W. Neilsen and R. G. Gordon, J. Chem. Phys., 1973,58. Chem. SOC., 1973, 55, 203; W. A. Lester, to be published.