Vibrational relaxation of vibrationally and rotationally excited CO molecules by He atoms

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER MARCH 2002 Vibrational relaxation of vibrationally and rotationally excited CO molecules by He atoms Roman V. Krems a) Department of Chemistry, Physical Chemistry, Göteborg University, SE , Göteborg, Sweden Received 9 April 2001; accepted 27 December 2001 This work presents a detailed quantum mechanical study of rovibrationally inelastic He CO collisions in a wide range of translational and internal energies of the collision partners. Fully converged coupled states calculations of rate constants for vibrational relaxation of CO(v 1) by He are found to be in excellent agreement with experimental measurements at temperatures between 35 and 1500 K. The role of rotational energy for vibrational relaxation of CO is investigated and it is illustrated that the CO molecules in the first excited vibrational state can exhibit near-resonant vibrational relaxation when they are initially in high rotational excitation and the collision energy is small. A reduced channel coupled states approach neglecting low vibrational states in the basis set is implemented for calculations of rate constants for vibrational and rotational energy transfer in collisions of vibrationally excited CO molecules with He atoms. It is shown that initial vibrational excitation significantly increases rate constants for vibrationally inelastic collisions but does not affect purely rotational energy transfer American Institute of Physics. DOI: / I. INTRODUCTION Many theoretical and experimental studies of He CO collisions have been performed recently in order to provide data of interest in astrophysics and chemistry, 1,2 test and refine interaction potential energy surfaces PESs, 3 7 and understand mechanisms underlying vibrational and rotational energy transfer in collisions of diatomic molecules possessing large moments of inertia with light atoms The experimental measurements of the vibrational relaxation VR of CO(v 1) have, thus, yielded rate constants over the temperature range K. These results have been used as the reference data in several theoretical investigations of vibrationally inelastic He CO collisions at low temperatures. It has been shown, for example, by Balakrishnan et al. 9 that quantum mechanical close coupling calculations of rate constants for VR performed on the most recent PES 14 at temperatures less than 100 K agree well with experimental data. Reid et al. 6,7 have reported vibrational close couplingcoupled states VCC-CS calculations of rate constants for VR of CO(v 1) below the temperature 300 K. Their results obtained on the same PES have underestimated the experimental values by a factor of 1.6 on average. A large number of closely spaced rovibrational levels in the CO molecule prohibits accurate quantum mechanical calculations at high collision energies and only approximate breathing sphere 15 or rotational infinite-order-sudden RIOS 12,15,16 calculations of rate constants have been performed for the He CO system at temperatures higher than 300 K. Approximate classical approach quantum encounter treatment has also been proposed for vibrationally inelastic He CO collisions. 17 a Present address: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138; electronic mail: rkrems@cfa.harvard.edu Unfortunately, both the experimental measurements and the quantum mechanical calculations at the RIOS level of approximation produce averaged results which provide little, if any, mechanistic insight into the collision dynamics. Understanding dynamics of rovibrational transitions in collisions of rotationally and vibrationally excited diatomic molecules with atoms is important for modern experimental techniques allowing measurements of the state-resolved energy transfer processes. 18 The collisions of excited diatomic molecules with atoms show many interesting features like the resonant vibrational energy transfer 19,20 or the strong competition between vibrational and purely rotational energy transfer. 21 These effects cannot be reproduced by the approximate quantum mechanical calculations neglecting the rotational energy spectrum of diatomic molecules and the classical dynamics calculations do not always provide reliable results. Accurate quantum mechanical calculations of rovibrational transitions in collisions of excited diatomic molecules with atoms should, therefore, assist experimental measurements and development of adequate theoretical models. To the best of our knowledge no quantum mechanical calculations of VR in vibrationally excited CO molecules have been reported. In the present paper we focus on investigation of collisions of vibrationally and rotationally excited CO molecules with He atoms. Rate constants for vibrational relaxation of CO(v 1) by He are computed using the VCC-CS methodology and compared with experimental data in a wide range of temperatures extending to 1500 K. Vibrationally reduced channel coupled states approach using energetically local basis sets of vibrational functions is implemented to obtain rate constants for VR of higher vibrational states and investigate the dependence of the vibrational and rotational energy transfer on the vibrational quantum number. The role of rotational /2002/116(11)/4517/8/$ American Institute of Physics

2 4518 J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Roman V. Krems excitation of CO for vibrational relaxation is investigated in the calculations of cross sections and rate constants for VR of CO(v 1) excited to as high a level as j 40. The details of the calculations are briefly reviewed in Sec. II and the presentation of the results in Sec. III is followed by a summary giving concluding remarks. II. DYNAMICAL CALCULATIONS For scattering calculations of the present work we employ the J-labeled variant of the vibrationally close couplingcoupled states VCC-CS approach initially presented by Pack. 22 This method has been well documented in literature see, e.g., Refs. 23 and 24, and references therein and will, therefore, be described very briefly in the following. An alternative l-labeled coupled states approach originally proposed by McGuire and Kouri 25 and investigated by many researchers 24,26 is known to give better results for orientationally dependent quantities. 24 We have illustrated, however, in the accompanying article 27 that the J-labeled CS approach describes VR in heavy diatomic molecules significantly better. The following notation for quantum numbers is used hereafter: J denotes the total angular momentum for the collision, while j is the rotational angular momentum of the CO molecule. Ĵ z and ĵ z are the operators that give the z component of Ĵ and ĵ, respectively, and denotes the projection of the total angular momentum on the body-fixed quantization axis. The notation v is used to denote the vibrational states of the diatom. The conventional Jacobi coordinates representing the He-CO center of mass separation (R), the interatomic distance in CO (r), and the angle between the vectors corresponding to r and R are used for description of the internal dynamics of the triatomic complex. The caret over a symbol denotes the corresponding operator. The coupled states approximation is applied in the body fixed coordinate system to the centrifugal term in the total Hamiltonian of the triatomic system as follows: 22 Ĵ ĵ 2 Ĵ 2 ĵ 2 2Ĵ z ĵ z. In this expression the terms responsible for coupling between states with different values are omitted while all operators contributing to the diagonal elements of the centrifugal coupling matrix are retained. As a result, the collision problem is parametrized by three constants of motion: total energy E, total angular momentum J and. The partial wave functions are expanded in terms of products of vibrational and rotational functions of the diatomic molecule with expansion coefficients being functions of the interparticle separation (R), J 1 R v, j F J v j R j v r Y j,0. Substitution of the expansion 2 into the stationary Schrödinger equation with the total Hamiltonian of the triatomic system results in a system of coupled second-order differential equations to be solved for fixed values of E, J, and : 1 2 d2 dr 2 k 2 v j J J 1 j j R 2 F J v j R J 2 v j V r,r, v j F v j R. 3 v, j In this equation is the reduced mass of the colliding particles, k v j 2 (E v j ), and v j is the energy of the rovibrational channel v j. The three-dimensional interaction potential V(r,R, ) depending on all Jacobi coordinates is conventionally expanded in a series of Legendre polynomials that allows to evaluate integrals over the angular part of the problem analytically. 28 Integrals over the r variable are evaluated numerically using 50-point Gauss Hermite quadratures. The vibrational wave functions ( j v ) as well as the rovibrational energy levels ( v j ) of the diatomic molecule are determined numerically by solving 1 d 2 j j 1 2 CO dr 2 V CO r 2 CO r 2 j v r v j j v r, where CO denotes the reduced mass of the diatomic and the spectroscopic potential V CO (r) has been taken from Ref. 29 in the form of a Simons Parr Finlan function. The use of this potential gives the energy levels v j very close to the experimental values presented, e.g., in Ref. 30. We note that the vibrational wave functions used in the present work for evaluation of the matrix elements entering Eq. 3 are dependent on the rotational quantum number j. Asymptotic solution of Eq. 3 yields a scattering S matrix from which the cross sections for rovibrational transitions are obtained as v j v j E 2 2 j 1 k v j 2J 1 J max J S v E j;v j 2. In this expression max is limited by min(j,j,j ). The energy dependence of the cross sections can be integrated with the Maxwell Boltzmann weighting function to give rate constant for v j v j transitions, k v j v j T 8k 1/2 bt v j v 0 j E exp E/k B T EdE k b T 2, where k B is the Boltzmann constant and T is the temperature. The experimentally observable rate constant for vibrational relaxation of the v state is obtained by summation of these state-resolved rate constants over all final rotational states and Boltzmann averaging over all initial rotational states as follows: k v v j 2 j 1 exp v j v j 0 /k B T j k v j v j. j 2 j 1 exp v j v j 0 /k B T

3 J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Relaxation of CO molecules 4519 The potential energy surface for the He CO interaction V(r,R, ) has been a subject of several recent studies. 4,14,31 37 In the present work we use the PES by Heijmen and co-workers 14 that includes an explicit dependence on the vibrational coordinate r. This PES was used by Heijmen and co-workers 14 for calculation of the bound state energies and the infrared spectrum of the HeCO complex that were found to be in a good agreement with experimental data. Reid and co-workers 7 and Balakrishnan and co-workers 9 have also employed this PES for their dynamical calculations. Another three-dimensional PES has recently been generated by Kobayashi et al. 4 The dynamical calculations performed on this PES in Ref. 4 have shown that it is of similar quality to the PES by Heijmen and co-workers. 14 All scattering calculations presented in this paper are performed using the program recently developed by the author for accurate and approximate quantum mechanical calculations of cross sections for inelastic collisions of diatomic molecules with atoms. The performance of the code has been tested against literature data and selected results obtained by two alternative programs: MOLSCAT 38 and HIBRIDON. 39 More details concerning the program used in the present work can be found on the web. 40 III. RESULTS In order to compute the experimentally observable rate constants for vibrational relaxation VR of CO(v 1) by He in a wide range of temperatures we perform VCC-CS calculations of cross sections for v 1,j v 0,j transitions at 41 -values and 99 total collision energies in the range cm 1. All calculations are checked to be converged with respect to integration parameters and a total number of 80 partial waves are taken in the sum in Eq. 5 to ensure convergence of cross sections for VR at high energies. Equation 3 is propagated to the maximum grid value R 40 Å. A total number of 55 rotational levels in the ground vibrational state of CO, 52 rotational levels of v 1, 40 rotational levels of v 2, and 23 rotational levels of v 3 are taken in the basis set for the calculations. The computations are performed at every second partial wave and the resulting cross sections are multiplied by 2. We have found 27 that the cross sections for VR of CO molecules at low collision energies are sensitive to high-order Legendre components in the angular expansion of the interaction potential when the J-labeled CS approach and accurate j-dependent vibrational wave functions are used for the calculations. The sensitivity of cross sections for VR of CO to details of quantum calculations has been investigated in the accompanying paper 27 and a total number of 39 terms in the Legendre expansion of the potential are used in the present work. In order to evaluate the low-magnitude terms of the Legendre expansion accurately we use Gauss Legendre quadratures with 50 points. We have repeated calculations with 80 points in the quadratures and the results have not changed. To finally verify the validity of our calculations we have computed selected cross sections with an alternative program MOLSCAT 38 and repeated calculations of the previous literature data for He CO vibrationally inelastic scattering 7,17,27 using the same propagators and integration parameters. In all TABLE I. Rate constants for vibrational relaxation of CO(v 1) by He (cm 3 molecules 1 s 1 ). T K Reference 7 Experiment a Experiment b This work a Reference 12. b Reference 2. test calculations we have obtained values differing from the reference data by not more than 1%. It must be noted, however, that cross sections for VR in He CO collisions at low energies ( 100 cm 1 ) are so small that even slight variations of the basis set, the angular expansion of the interaction potential, or the asymptotic energies of CO may significantly affect quantitative values of the cross sections. In addition, as has been pointed out by Reid and co-workers 10 and Balakrishnan and co-workers, 9 the rate constants for VR in He CO collisions at low temperatures are influenced by scattering resonances. We have done our best to obtain the fully converged results at these low collision energies but the reader and the future investigator should be warned that the quantitative results at temperatures below 70 K must be treated with caution. Table I presents averaged rate constants for VR of CO(v 1) by He in comparison with the previously published data of Reid et al. 7 and experimental measurements. 1,2,12 The main difference of our calculations from those of Reid et al. 7 at low temperatures is in the vibrational basis functions used for evaluation of matrix elements in Eq. 3 and the labeling of the CS approximation. j We use exact vibrational wave functions v depending on the rotational quantum number and the J-labeled CS approach, while Reid et al. have used harmonic vibrational wave functions in the l-labeled CS calculations. Table I illustrates that our rate constants at temperatures below 300 K are closer to experimental values than the data by Reid et al. 7 by about 55% on average. The sensitivity of cross sections for VR in heavy diatomic molecules to the choice of vibrational wave functions and the different parametrization of the CS approximation has been analyzed in detail in the accompanying paper. 27 It can be seen from Table I that our calculations are in close agreement with experimental data over the whole interval of temperatures considered. This agreement in a wide range of temperatures indicates that the PES of Heijmen et al. 14 accurately describes the He CO interaction over large variations of the Jacobi coordinates and the CS

4 4520 J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Roman V. Krems TABLE II. Rate constants for vibrational relaxation of CO(v 1,j) byhe (cm 3 molecules 1 s 1 ). j T 30 K ( ) T 300 K ( ) T 800 K ( ) approximation that we are using is accurate for the present system. Table II shows rate constants for VR of CO(v 1,j) by He as functions of the initial rotational quantum number at three translational temperatures. At low temperature, T 30 K, the rate constant shows a slight increase and then decreases with initial rotational excitation, whereas at higher temperatures, T 300 and 800 K, the rate constants increase monotonously as j increases up to the value 25. The increase of the rate constant is faster at T 300 K than at T 800 K. The interpretation of such behavior of rate constants can be obtained from Fig. 1 where we plot the dependence of cross sections for VR of different v 1,j levels on the translational collision energy. The curves corresponding to different initial j levels show similar dependence in the high energy interval with cross sections for VR of higher rotational states being larger by approximately an energy independent factor. At low energies, however, the cross sections for VR of high rotational states of CO tend to zero much faster than the cross sections for VR of lower j states and the curves cross. More detailed information on the mechanism of VR can be obtained from an analysis of rotational distributions of diatomic molecules after VR. Figure 2 presents diagrams of the rotational distribution of the CO molecule after VR of various v 1,j states at the collision energy 10 cm 1. The resonant rotational levels ( j res )ofv 0, i.e., the levels which are closest in energy to the initial rotational levels of v 1 ( j i ), are also shown on the graphs. The rotational distribution after VR of low j levels is broad and centered in the FIG. 1. Energy dependence of cross sections for vibrational relaxation of CO(v 1,j) by He: full curve j 0, dot dashed curve j 10, dotted curve j 30, broken curve j 40. region of low j values. As the initial rotational excitation increases, the final rotational distribution narrows and shifts toward larger j values, peaking in the case of VR of j 38 close to the resonant rotational level. This type of nearresonant energy transfer has been observed experimentally and in theoretical calculations for the diatomic molecules possessing small moments of inertia H 2, HF, Li 2. 18,41 44 Figure 2 illustrates that molecules with small rotational constants like CO can also exhibit near-resonant VR if they are initially at high levels of rotational excitation. Diagrams of the final rotational distributions after VR of different rotational levels of CO by He at higher collision energies are shown in Figs. 3 and 4. An increase of the collision energy broadens the final rotational distribution supporting, in the case of VR of low rotational states, population of higher rotational levels up to the resonant one. It has been proposed by Schinke and Diercksen 16 to use the RIOS factorization formula for scaling cross sections for VR of v 1,j 0 computed in the VCC-CS formalism to higher initial j values. The RIOS factorization formula is given by v 1,j v 0,j C 2 jj j 000 v 1,j 0 v 0,j, 8 j where C( j, j, j 000) is the Clebsch Gordan coefficient. From Eq. 8 and the unitarity of the Clebsch Gordan coefficients it follows that j v 1,j v 0,j j v 1,j 0 v 0,j, and rate constants in Table II should be independent of the initial rotational quantum number. The RIOS factorization formula has been checked for the He CO system in Ref. 16 at a single collision energy that could not give an adequate account of the applicability of Eq. 8. Table II illustrates that the RIOS factorization formula will produce accurate to within 50% results for the total rate constant for VR of 9

5 J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Relaxation of CO molecules 4521 FIG. 2. Rotational distributions after vibrational relaxation of CO(v 1,j i ) by He. Values of energetically closest rotational levels of v 0 (j res ) below j i are also indicated. The collision energy is 10 cm 1. CO(v 1) by He at low ( 30 K) and high ( 800 K) temperatures. It may not be so at temperatures close to T 300 K. Approximately 25 rotational levels should be considered in summation Eq. 7 in order to obtain the converged value of rate constant at this temperature and it follows from Table II that rate constants for VR of high rotational states of CO obtained from the RIOS factorization formula will underestimate the CS values by as large a factor as 6. This is a consequence of the near-resonant VR of high rotational states of CO that cannot be reproduced within the RIOS approximation. FIG. 3. The same as in Fig. 2 but for the collision energy 307 cm 1. FIG. 4. The same as in Fig. 2 but for the collision energy 1090 cm 1.

6 4522 J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Roman V. Krems TABLE III. Cross sections (Å 2 ) for vibrationally elastic rotational excitation of CO(v, j 0) by He. The numbers in parentheses denote the vibrational states in the basis and N is the total number of rovibrational basis states. Collision energy is 1000 cm 1. Basis N j 0 j 3 j 0 j 5 j 0 j 10 j 0 j v ,5, ,4,5, ,..., ,..., Full CS 0,..., v 10 10, ,10, ,9,10, ,8,9,10, ,..., In order to compute cross sections and rate constants for VR of higher vibrational states of CO than v 1, we perform coupled states calculations using local basis sets of vibrational wave functions that neglect low vibrational states making an apparently small contribution to dynamics of the vibrational state of interest. The partial wave expansion in Eq. 2 is modified as follows: J 1 R v v m j v v n J j F v j R v r Y j,0, 10 where v denotes the vibrational level of interest and the second summation is performed over all rotational states of the given vibrational level. The convergence of the cross sections can now be sought by gradually increasing both the m and n parameters. Tables III and IV illustrate the convergence of cross sections for rotational and vibrational relaxation of high vibrational levels of CO with respect to variation of the lower cutoff parameter n. For all calculations we use 55 rotational states providing they are below the upper cutoff energy that is found from separate convergence tests in each vibrational level. It follows from Table III that the rotational energy transfer is not affected by the presence of lower vibrational states in the basis set and the calculation of rotationally inelastic cross sections can be made using only one vibrational level in expansion 10. Cross sections for state-resolved and total VR of v 5 and v 10 Table IV are well converged in a basis of four vibrational states (v 1,v,v 1,v 2). It may be noticed that convergence of cross sections for VR is slower when CO is initially in v 10, which is a consequence of a closer spacing of vibrational states around v 10. Table V collects room temperature rate constants for vibrational relaxation and purely rotational excitation of various v, j 0 states of CO as functions of the vibrational quantum number. The vibrational energy transfer is significantly enhanced by increasing the v number, whereas rate constants for vibrationally elastic total rotational relaxation are almost not affected by changes of the initial vibrational level. This is in agreement with results previously obtained for vibrational and rotational relaxation of vibrationally excited HF molecules by Ar atoms. 21 Figure 5 shows energy dependencies of cross sections for VR of various v, j 0 levels of CO by He. The curves corresponding to different initial vibrational states are different by an energy independent factor. The significant increase of vibrational relaxation with initial vibrational excitation has also been observed in TABLE IV. Cross sections (Å 2 ) for vibrational relaxation of CO(v, j 0) by He. The numbers in parentheses denote the vibrational states in the basis and N is the total number of rovibrational basis states. Collision energy is 1000 cm 1. Basis N v, j 0 v 1,j 0 v, j 0 v 1,j 10 v, j 0 v 1, j v, j 0 v, j v 5 4,5, ,4,5, ,3,4,5, ,2,3,4,5, Full CS 0,..., v 10 9,10, ,9,10, ,8,9,10, ,7,8,9,10, ,6,7,8,9,10,

7 J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Relaxation of CO molecules 4523 TABLE V. Room temperature rate constants for total vibrational relaxation and total rotational excitation of CO(v, j 0) by He (cm 3 /molecules s 1 ). The notation j is used for the rate constants summed over final j levels. v k v, j 0 v 1, j k v, j 0 v j the quantum mechanical close coupling calculations of vibrationally inelastic transitions in the H H 2 collisions in Ref. 45, where it was shown that the matrix elements coupling adjacent vibrational levels are significant over a wider range of intermolecular separation R, when H 2 is vibrationally excited, due to a greater stretching of the diatomic molecule. It does not explain, however, why the purely rotational energy transfer, at least for the cases of He CO and Ar HF 21 collisions, is not affected by the vibrational excitation. Unfortunately, no systematic experimental investigations of the vibrational and rotational energy transfer in vibrationally excited CO molecules have been reported. It will be interesting to see if experimental measurements will verify the predicted trends. IV. SUMMARY The main results of the present work can be summarized as follows. i Accurate vibrationally close coupling-coupled states calculations of the rate constants for vibrational relaxation of CO(v 1) by He are performed and compared with experimental measurements in a wide range of temperatures extending to 1500 K. The theoretical results are in close agreement with experimental data in the whole range of temperatures considered. FIG. 5. Energy dependence of cross sections for vibrational relaxation of CO(v, j 0) by He: full curve v 1, broken curve v 3, dot dashed curve v 5, dotted curve v 7, short dash curve v 10. ii The role of the initial rotational quantum number for vibrational relaxation of CO(v 1) is investigated and it is shown that the CO molecules can exhibit near-resonant energy transfer when they are initially in high rotational excitation and the collision energy is small. The final rotational distribution broadens as the collision energy increases. The final population of high rotational states of v 0 which are close in energy to the initial rotational state of v 1, however, remains predominantly large after vibrational relaxation of high rotational states of CO even at high collision energies. iii Evidence is given that the rotationally infinite-ordersudden factorization formula will produce accurate results for the total vibrational relaxation rate constants at low ( 30 K) and high ( 800 K) temperatures but may underestimate the coupled states results by a large factor at the intermediate temperatures. iv A reduced channel coupled states approach using energetically local vibrational basis sets is implemented for calculations of rate constants for vibrational and rotational energy transfer in collisions of vibrationally excited CO molecules with He atoms. It is shown that increasing initial vibrational excitation results in a significant increase of vibrational relaxation but does not affect purely rotational energy transfer. ACKNOWLEDGMENTS Many useful discussions with Professor Sture Nordholm and Dr. Nikola Marković and Dr. Alexei Buchachenko are gratefully acknowledged. Thanks are due to Professor van der Avoird for providing a FORTRAN subroutine generating the He CO interaction potential. The work has been supported by the Swedish Research Council. 1 J. C. Stephenson and E. R. Mosburg, Jr., J. Chem. Phys. 60, ; W. H. Green and J. K. Hancock, ibid. 59, R. C. Millikan, J. Chem. Phys. 40, ; D. J. Miller and R. C. Millikan, ibid. 53, E. Bodo, F. A. Gianturco, and F. Paesani, Z. Phys. Chem. Leipzig 214, ; F. A. Gianturco, F. Paesani, M.F. Laranjeira, V. Vassilenko, M. A. Cunha, A. G. Shashkov, and A. F. Zolotukhina, Mol. Phys. 94, ; F. A. Gianturco, N. Sanna, and S. Serna, J. Chem. Phys. 98, R. Kobayashi, R. D. Amos, J. P. Reid, H. M. Quiney, and C. J. S. M. Simpson, Mol. Phys. 98, S. Antonova, A. Lin, A. P. Tsakotellis, and G. C. McBane, J. Chem. Phys. 110, J. P. Reid, C. J. S. M. Simpson, H. M. Quiney, and J. M. Hutson, J. Chem. Phys. 103, J. P. Reid, C. J. S. M. Simpson, and H. M. Quiney, J. Chem. Phys. 107, J. R. Fair and D. J. Nesbitt, J. Chem. Phys. 111, N. Balakrishnan, A. Dalgarno, and R. C. Forrey, J. Chem. Phys. 113, J. P. Reid, C. J. S. M. Simpson, and H. M. Quiney, Chem. Phys. Lett. 246, ; J. P. Reid and C. J. S. M. Simpson, ibid. 280, M. M. Maricq, E. A. Gregory, C. T. Wickham-Jones, D. J. Cartwright, and C. J. S. M. Simpson, Chem. Phys. 75, C. T. Wickham-Jones, H. T. Williams, and C. J. S. M. Simpson, J. Chem. Phys. 87, A. J. Banks and D. C. Clary, J. Chem. Phys. 86, T. G. A. Heijmen, R. Moszynski, P. E. S. Wormer, and A. van der Avoird, J. Chem. Phys. 107, C. T. Wickham-Jones, G. G. Balint-Kurti, and M. M. Novak, Chem. Phys. 117,

8 4524 J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 Roman V. Krems 16 R. Schinke and G. H. F. Diercksen, J. Chem. Phys. 83, N. Markoviv, T. D. Sewell, S. Nordholm, and A. Miklavc, Chem. Phys. 211, B. Stewart, P. D. Magil, and D. E. Pritchard, J. Phys. Chem. 104, A. J. McCaffery and R. J. Marsh, J. Phys. Chem. 104, S. Clare and A. J. McCaffery, J. Phys. B 33, R. V. Krems and S. Nordholm, J. Chem. Phys. 115, R. T Pack, J. Chem. Phys. 60, K. McLenithan and D. Secrest, J. Chem. Phys. 80, V. Khare, D. J. Kouri, and R. T Pack, J. Chem. Phys. 69, P. McGuire and D. J. Kouri, J. Chem. Phys. 60, ; P. McGuire, ibid. 62, See, for example, R. B. Walker and J. C. Light, Chem. Phys. 7, ; S. I. Chu and A. Dalgarno, J. Chem. Phys. 63, , 64, ; B. H. Choi and K. T. Tang, ibid. 62, ; M. H. Alexander and P. McGuire, ibid. 64, ; G.A. Parker and R. T Pack, ibid. 66, ; Y. Shimoni and D. J. Kouri, ibid. 66, ; S.M. Tarr, H. Rabitz, D. E. Fitz, and R. A. Marcus, J. Chem. Phys. 66, ; L. Monchick and S. Green, ibid. 66, ; K. McLenithan and D. Secrest, ibid. 80, R. V. Krems, J. Chem. Phys. 116, , following paper. 28 R. N. Zare, Angular Momentum Wiley, New York, C. G. Diaz, F. M. Fernandez, and E. A. Castro, J. Mol. Struct. 280, H. Telle and U. Telle, J. Mol. Spectrosc. 85, B. Kukawska-Tarnawska, G. Chalasinski, and K. Olszewski, J. Chem. Phys. 101, F. M. Tao, S. Drucker, R. C. Cohen, and W. Klemperer, J. Chem. Phys. 101, B. Kukawska-Tarnawska, G. Chalasinski, and K. Olszewski, J. Chem. Phys. 101, R. Moszynski, T. Korona, P. E. S. Wormer, and A. van der Avoird, J. Chem. Phys. 103, F. A. Gianturco, N. Sanna, and S. Sernamolinera, Mol. Phys. 81, R. J. Leroy, C. Bissonnette, T. H. Wu, A. K. Dham, and W. J. Meath, Faraday Discuss. 97, C. E. Chuaqui, R. J. Leroy, and A. R. W. McKellar, J. Chem. Phys. 101, J. M. Hutson and S. Green, MOLSCAT computer code, version , distributed by collaborative Computational Project No. 6 of the Engineering and Physical Sciences Research Council UK. 39 HIBRIDON is a package of programs for the time-independent quantum treatment of inelastic collisions and photodissociation written by M. H. Alexander, D. E. Manolopoulos, H.-J. Werner, and B. Follmeg with contributions by P. F. Vohralik, D. Lemoine, G. Corey et al roman/abc3d/ 41 B. Stewart, P. D. Magill, T. P. Scott, J. Deronard, and D. E. Pritchard, Phys. Rev. Lett. 60, A. Miklavc, N. Markovic, G. Nyman, V. Harb, and S. Nordholm, J. Chem. Phys. 97, R. V. Krems, N. Markovic, A. A. Buchachenko, and S. Nordholm, J. Chem. Phys. 114, R. V. Krems, A. A. Buchachenko, N. Markovic, and S. Nordholm, Chem. Phys. Lett. 335, N. Balakrishnan, R. C. Forrey, and A. Dalgarno, Chem. Phys. Lett. 280,