摘要:

FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 55 1973 Molecular Beam Scattering THE FARADAY DIVISION CHEMICAL SOCIETY LONDONFARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 55 1973 Molecular Beam Scattering THE FARADAY DIVISION CHEMICAL SOCIETY LONDONA GENERAL DISCUSSION ON Molecular Beam Scattering 16th-18th April 1973 A GENERAL DISCUSSION on Molecular Beam Scattering was held in the Chemistry Department, University College, London on the 16th, 17th and 18th April 1973. The President, Prof. Sir George Porter, F.R.S., was in the Chair at the Opening Session and 151 members and others were present. Among the overseas visitors were : Prof. V. Aquilanti, Italy Dr. D. Bassi, Italy Dr. A. Ben-Shaul, Israel Prof. R. D. Bernstein, U.S.A. Prof. P. R. Brooks, U.S.A. Dr.P. W. Brumer, Israel Dr. U. Buck, Germany Prof. D. L. Bunker, U.S.A. Prof. Byung Chan Eu, Canada Dr. W. H. Cramer, U.S.A. Dr. H. W. Cruse, U.S.A. Dr. P. J. Dagdigian, U.S.A. Dr. A. M. G. Ding, Canada Dr. R. Dueren, Germany Dr. J. J. Everdij, The Netherlands Dr. J. B. Fenn, Germany Prof. D. D. Fitts, U.S.A. Prof. K. F. Freed, France Prof. J. H. Futrell, U.S.A. Dr. E. Gersing, Germany Prof. T. F. George, U.S.A. Prof. R. G. Gordon, U.S.A. Dr. A. H. M. Habets, The Netherlands Prof. D. Ham, U.S.A. Mr. W-D. Held, Sweden Prof. D. R. Herschbach, U.S.A. Dr. H. Heydtmann, Germany Dr. G. Hunter, Canada Dr. G. M. Kendall, Germany Dr. B. Lantzsch, Germany Prof. Y. T. Lee, U.S.A. Dr. G. D. Lempert, Israel Dr. J. Leonhardt, East Germany Prof. R. D. Levine, Israel Dr. C. A.Linse, The Netherlands Dr. H. Loesch, Germany Prof. G. Liuti, Italy Prof. R. A. Marcus, U.S.A. Prof. J. D. McDonald, U.S.A. Prof. M. Menzinger, Canada Prof. W. H. Miller, U.S.A. Dr. A. M. C . Moutinho, Portugal Dr. E. E. Muschlitz, Jr., U.S.A. Dr. B. Neidhart, Germany Dr. S. E. Nielsen, Denmark Dr. C. Nyeland, Denmark Dr. J. Ogilvie, Canada Prof. J. M. Parson, U.S.A. Prof. Dr. H. Pauly, Germany Mr. D. S. Perry, Canada Prof. J. C. Polanyi, Canada Dr. L. L. Poulsen, Denmark Prof. S. A. Rice, U.S.A. Mr. G. Rotzoll, Germany Dr. A. M. Rulis, The Netherlands Mr. J. L. Schreiber, Canada Dr. K. Shobatake, U.S.A. Dr. J. J. Sloan, Canada Prof. Dr. J. P. Toennies, Germany Dr. F. Torello, Italy Dr. J. C. Tully, U.S.A. Mr. J. M. M. Van Deventer, The Prof. G. G. Volpi, Italy Netherlands0 The Chemical Society and Contributors 1973 Printed in Great Britain at the University Press, AberdeenCONTENTS I.THEORETICAL 9 22 30 34 45 51 59 80 93 I00 I13 129 145 Introduction-The Theoretical Approach by R. A. Marcus Rational Selection of Methods for Molecular Scattering Calculations by R. G. Gordon Franck-Condon Transitions in Multi-curve Crossing Processes by M. S. Child Semiclassical Theory for Collisions involving Complexes (Compoimd State Resonances) nnd for Bound State Systems by R. A. Marcus Partial Averaging in Classical S-Matrix Theory: Vibrational Excitation of H2 by He by W. H. Miller and A. W. Raczkowski Multidimensional Canonical Integrals for the Asymptotic Evaluation of the S-Matrix in Semiclassical Collision Theory by J.N. L. Connor GENERAL DrscussroN-Dr. G. D. Bag, Dr. H. Fremerey, Prof. J. P. Toen- nies, Dr. G. G. Baht-Kurti, Dr. B. R. Johnson, Dr. M. D. Pattengill, Prof. J. C . Polanyi, Prof. R. A. Marcus, Dr. R. G. Gilbert, Prof. T. F. George, Dr. S. Bosanac, Dr. J. P. Simons, Prof. K. F. Freed, Prof. W. H. Miller, Mt. J. L. Schreiber, Prof. R. D. Levine, Prof. J. N. Murrell, Dr. J. N. L. Coiinor, Mr. Y-W. Lin, Prof. K. Morokuma. Collision Complex Dynamics in Alkali Halide Exchange Reactions by P. Brumer and M. Karplus Alkali-Methyl Iodide Reactions by D. L. Bunker and E. A. Goring-Simpson Energy Disposal and Energy Requirements for Elementary Chemical Reaction by R. D. Levine and R. B. Bernstein GENERAL DIscussIoN-prof. D. R. Herschbach, Dr. S. M. Lin, Dr. R. Grice, Dr.P. Brumer, Prof. S. A. Rice, Prof. R. A. Marcus, Mr. D. S. Y. Hsu, Prof. W. H. Miller, Dr. J. N. L. Connor, Prof. R. B. Bernstein, Dr. R. A. LaBudde, Dr. P. J. Kuntz, Prof. D. L. Bunker, Prof. R. M. Harris, Dr. A. Ben-Shaul, Prof. J. C. Polanyi, Mr. J. L. Schreiber, Dr. J. J. Sloan, Prof. R. D. Levine, Prof. M. Menzinger, Mr. D. J. Wren, Mr. D. S. Perry, Dr. C. Woodrow Wilson Jr. 11. ELASTIC SCATTERING Introduction by J. P. Toennies Central-$eld Intermolecular Potentials from the Diflerential Elastic Scattering of H,(D,) by Other Molecules by A. Kuppermann, R. J. Gordon and M. J. Coggiola 56 CONTENTS 158 The Scattering of Metastable Mercury Atoms by T. A. Davidson, M. A. D. Fluendy and K. P. Lawley 167 Scattering Experiments with Fast Hydrogen Atoms: Velocity Dependence of the Integral Elastic Cross Section with the Rare Gases in the Energy Range 0.01-1.00 eV by R.W. Bickes Jr., B. Lantzsch, J. P. Toennies and K. Walaschewski 179 Characterization of the Intermolecular Potential Well from Elastic Molecular Beam Scattering Data by D. D. Fitts and M. Liao Law 185 GENERAL DIscussioN-Dr. U. Buck, Dr. H. 0. Hoppe, Dr. F. Huisken, Prof. H. Pauly, Dr. R. Gengenbach, Prof. J. P. Toennies, Dr. W. Welz, Dr. G. Wolf, Prof. V. Aquilanti, Prof. G. Liuti, Dr. F. Vecchio-Cattivi, Prof. G. G. Volpi, Dr. K. P. Lawley, Dr. J. F. Ogilvie, Mr. R. W. Davis. 111. INELASTIC SCATTERING 191 Introduction-Molecular Beam Studies of Inelastic Collision Processes by H. Pauly 203 Measurement of Diferential Cross Sections for Indiuidual Rotational Quantum Transitions in the Scattering of Li+ by H2 at Ec.,,.= 0.6 eV by H. E. van den Bergh, M. Faubel and J. P. Toennies 21 I Experimental Inelastic Diflerential Cross Sections for Electronic Excitation in Atom-Atom and A tom-Molecule Collisions b y E. Gersing, H. Pauly, E. Schadlich and M. Vonderschen 221 GENERAL DrscussIoN-Prof. R. D. Levine, Prof. R. B. Bernstein, Dr. R. A. LaBudde, Dr. R. Bottner, Dr. U. Ross, Prof. J. P. Toennies, Mr. D. L. King, Dr. €3. J. Loesch, Prof. D. R. Herschbach, Dr. A. M. G. Ding, Prof. J. C. Polanyi, Dr. G. M. Kendall, Mr. R. A. Larsen, Dr. J. R. Krenos, Prof. V. Aquilanti. IV. REACTIVE SCATTERING 233 Introduction by D. R. Herschbach 252 Eflect of Changing Reagent Energy on Reaction Probability and Product Energy- Distribution by A.M . G. Ding, L. J. Kirsch, D. S. Perry, J. C. Polanyi and J. L. Schreiber 277 Crossed-Beam Reactions of Barium with Hydrogen Halides: Measurement of Internal State Distributions by Laser-induced Fluorescence by H. W. Cruse, P. J. Dagdigian and R. N. Zare 293 Translational Energy Dependence of Product Energy and Angular Distribution for the K+CH,I Reaction by R. B. Bernstein and A. M. Rulis 299 Scattering of K Atoms from Oriented CF31 Reaction at Both " Ends" by P. R. BrooksCONTENTS 7 307 GENERAL DrscussroN-Prof. R. D. Levine, Dr. A. Ben-Shaul, Prof. C. A. Parr, Prof. J. C. Polanyi, Dr. W-H. Wong, Prof. D. C . Tardy, Mr. D. J. Douglas, Dr. J. J. Sloan, Dr. P. J. Dagdigian, Prof. M. Menzinger, Mr. D. J. Wren, Prof. R. A. Marcus, Prof. D. 0. Ham, Dr. W.S. Struve, Dr. J. R. Krenos, Dr. D. L. McFadden, Prof. D. R. Herschbach, Dr. A. M. C. Moutinho, Mr. P. E. McNamee, Dr. K. Lacman, Dr. G. Marcelin, Dr. R. P. Brooks. 320 Reacti2;e Scattering of Alkali Dimers: Alkali Atom- Dimer Exchange Reac- tions by J . C. Whitehead and R. Grice 33 I Facile Four-Centre Exchange Reactions by D. L. King and D. R. Herschbach 344 Substitution Reactions of Fluorine Atoms with Unsaturated Hydrocarbons: Crossed Molecular Beam Studies of Unimolecular Decomposition by J. M. Parson, K. Shobotake, Y. T. Lee and S. A. Rice 357 Reactive Scattering of Methyl Radicals: CH3 -k ICl, IBr, I, by C . F. Carter, M. R. Levy and R. Grice 369 GENERAL DIscussroN-Dr. D. W. Davies, Mr. G. del Conde, Dr. S. M. Lin, Dr. R. Grice, Prof. J. D. McDonald, Prof. J. C. Polanyi, Mr. J. L. Schreiber, Dr. J. C. Whitehead, Mr. D. A. Dixon, Mr. D. L. King, Prof. D. R. Hersch- bach, Prof. J. D. McDonald, Mr. J. T. Cheung, Prof. R. A. Marcus, Dr. J. M. Parson, Dr. K. Shobatake, Prof. Y. T. Lee, Prof. S. A. Rice, Mr. C . F. Carter, Mr. M. R. Levy, Dr. K. B. Woodall, Dr. Y. C. Wong, Dr. D. D. Parrish, Dr. G. Hunter. 389 Summarizing Remarks by J. C . Polanyi 410 AUTHOR INDEX

ISSN:0301-7249    

DOI:10.1039/DC9735500001

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

GENERAL DISCUSSIONS OF THE FARAD14Y SOCIETY Date 1907 1 YO7 1910 1911 1912 1913 1913 1913 1914 1914 191s 1916 1916 1917 1917 A917 I918 1918 1918 1918 1919 1919 1920 1920 1920 1920 1921 1921 1921 1921 1922 1922 1923 1923 1923 1923 1923 1924 1924 1924 1924 1924 1925 1925 1926 1926 1927 1927 1927 Subject 3smotic Pressure Hydrates in Solution The Constitution of Water High Temperature Work Magnetic Properties of Alloys Colloids and their Viscosity The Corrosion of Xron and Steel The: Passivity of Metals Optical Rotary Power The Hardening of Metals The Transformation of Pure lron Methods and Appliances for the Attainment of High Temperatures in a Refractory Materials Training and Work of the Chemical Engineer Osmotic Pressure Pyrometcrs and Pyrometry The Setting of Cements and Plasters Electrical Furnaces Co-ordination of Scientific Publication The Occlusion of Gases by Metals The Present Position of the Theory of Ionization Tha Examination of Materials by X-Rays The Microscope : Its Design, Construction and Applications Basic Slags : Their Production and Utilization in Agriculture Physics and Chemistry of Colloids Electrodeposition and Electroplating Capillarity The Failure of Metals under Internal and Prolonged Stress Physico-Chemical Problems Relating to the Soil Catalysis with special reference to Newer Theories of Chemical Action Some Properties of Powders with special reference to Grading by The Generation and Utilization of Cold Alloys Resistant to Corrosion The Physical Chemistry of the Photographic Process The Electronic Theory of Valency Electrode Reactions and Equilibria Atmospheric Corrosion.First Report Investigation on Oppau Ammonium Sulphate-Nitrate Fluxes and Slags in Metal Melting and Working Physical and Physico-Chemical Problems relating to Textile Fibres The Physical Chemistry of Igneous Rock Formation Base Exchange in Soils The Physical Chemistry of Steel-Making Processts Photochemical Reactions in Liquids and Gases Explosive Reactions in Gaseous Media Physical Phenomena at Interfaces, with special reference to Molecular Atmospheric Corrosion. Second Report The Theory of Strong Electro\ytcs Cohesion and Related Problems Laboratory Eautriation Orientation Volume Trans. 3 3 6 7 8 9 9 9 10 10 11 12 12 13 13 13 14 14 14 14 15 15 16 16 16 16 17 17 17 17 18 18 19 19 19 19 19 20 20 20 20 20 21 21 22 22 23 23 24GENERAL DISCUSSIONS OF THE PARADAY SOCIE7V Dare 1928 1929 1929 1929 1930 1930 1931 1932 1932 1933 1933 1934 1934 1935 1935 1936 1936 1937 1937 1938 1938 1939 1939 1940 1941 1941 1942 1943 1944 1945 1945 1946 1946 1947 1947 1947 1947 1948 1948 1949 1949 I943 1950 1950 1950 1950 1951 1951 1952 1952 1952 1953 1953 1954 1954 Subject Homogeneour Catalyia Atmospheric Conosioo of Metals.Third Ropon Molecular Spcara and Molecular Structure Colloid Sciena Appliad to Biology Photochemical P r m Tho Adsorption of Garas by Solida The Colloid hpoctn of Taxtile M a W b Liquid Cryrtala and Anisotropic Melta Free Radicals Dipob Moments Colloidal Eloctroly tar Tho Structure of Metallic Coatings, Films and Surfaoor The Phenomena of Polymerization and C o n d d o n Dkpane Systems in Ga~cs : Dust, Smoke and Fog Structure and Moiecular Forcts in (a) Pure Liquid& and (b) Solutions The Properties and Punction~ of Mernbraam, Natural and Artificial Rcaction Kinetics Chemical Reactions Iovolviop Solid6 Lumiwscanca Hydrocarbon Chemistry The Elsctrical Double Layer (owing to the outbreak of war the meeting Tho Hydrogen Bond The Oil-Water loterface The Mechanism and Chamkal Kinetics of Organic Raactiotu in Liquid The Structure and Reactions of Rubbcr M o b of Drug Action Molacular Weight and Molecular Weight Disbbution in High PolymCrL (Joint Meeting with the Plastica Group, Society of Chemical Industry) Tho Application of Infra-red Spectra to Cbemical Problems Oxidation Dielcctrica Swelling and Shrinking Elactrodo Processu Tho Labile M o k u k Surface Chemistry.(Jointly with the Socittt dc Chimie Physique at -8td StrUctUre and cbcmical Constitution optical Rotatory Power was abaodoaad, but tbc papen wert printed in thc Transactlow) SyStOmS Bordeaux.) Published by Butterworths Scientific PubkatioM, Ltd. Volume 24 25 25 25 26 26 27 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 35 36 37 37 38 33 40 41 42 42 A 42 b Disc. 1 2 Colloidal Electrolytes and Solutions The Interaction of Water and Porous Materials The Physical Chemistry of Procees Metallurgy Cryrtal Growth Lipo-Proteins Chromatographic Analysis Hcteropncous Catnlysis Physiw-chemical Propertic8 and Bchaviour of Nucltar Acids Spectroscopy and Molecular Structure and Optical Methods Electrical Double Layor Hydrocarbons The Size and Sbape Factor in Colloidal Systems Radiatioo Chemistry The Physical Chemistry of Proteins The Reactivity of Free Radicals Thc Equilibrium Properties of Solutions of Non-Electrolytes The Physical Chemistry of Dyeing and Tanning The Study of Fast Reactions Coagulation and Flooculatioo vmtigating Call Structure Tram.43 Disc. 3 4 3 6 3 8 Trans. 46 Disc. 9 Trans. 47 Disc. 10 11 12 13 14 15 16 17 18 of 10-GENERAL DISCUSSIONS OF THE FARADAY SOClETY Date 1955 1955 1956 1956 1957 1958 1957 1958 1959 1959 1960 1960 1961 1961 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 subjw Volume Microwave and Radio-Frequency Spectroscopy 19 Physical Chemistry of Enymcs 20 Membrane Phenomena 21 Physical Chemistry of Proctsscs at High Prcaaura 22 M0kcula.r Mechanism of Rate Procuua in Solids 23 Interactions in Ionic Solutions 24 Configurations and Interactions of Macromoldcs and Liquid Crystah 25 Ions of the Transition Eltmenta 24 Energy Transfer with special reference to Biologkal Systunr 27 crystal h p d ~ ~ t i ~ ~ and the Chemical Reactivity of Solids 28 Oxidation-Reduction Reactions in 10- Solvents 29 The Physical Chemistry of Aerosols 30 Radiation Ef€” in Inorganic Solids 31 The Structure and Properties of Ionic Melts 32 Inelastic Collisions of Atoms and Simple Molcculcs 33 High Resolution Nuclear Magnetic Resonance 34 Tha Structure of Electronically-Excited Species in the Gas-Phase 35 Fundamental Processes in Radiation Chemistry 36 Chemical Reactions in the Atmosphere 37 Dislocations in Solids 38 The Kinetics of Proton Transfer Procassas 39 Intermolecular Form 40 The Role of the Adsorbed State in Heterogeneous Catalysis 41 Colloid Stability in Aqueous and Non-Aqueous Media 42 Thc Structure and Properties of Liquids 43 Molecular Dynamics of the Chemical Reactions of Gasas 44 Electrode Reactions of Organic Compounds 45 Oxidat ion 46 Bonding in Metallo-Organic Compounds 47 Motions in Molecular Crystals 48 Polymer Solutions 49 The Vitreous State 50 Electrical Conduction in Organic Solids 51 Surface Chemistry of Oxides 52 Reactions of Small Molecules in Excited States 53 The Photoelectron Spectroscopy of Molecules 54 Molecular Beam Scattering 55 Homogeneous Catalysis with Special Reference to Hydrogenation and For current availability of Discussion volumes, see back cover.

ISSN:0301-7249    

DOI:10.1039/DC973550X003

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

GENERAL DISCUSSIONS OF THE FARADAY SOClETY Date 1955 1955 1956 1956 1957 1958 1957 1958 1959 1959 1960 1960 1961 1961 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 subjw Volume Microwave and Radio-Frequency Spectroscopy 19 Physical Chemistry of Enymcs 20 Membrane Phenomena 21 Physical Chemistry of Proctsscs at High Prcaaura 22 M0kcula.r Mechanism of Rate Procuua in Solids 23 Interactions in Ionic Solutions 24 Configurations and Interactions of Macromoldcs and Liquid Crystah 25 Ions of the Transition Eltmenta 24 Energy Transfer with special reference to Biologkal Systunr 27 crystal h p d ~ ~ t i ~ ~ and the Chemical Reactivity of Solids 28 Oxidation-Reduction Reactions in 10- Solvents 29 The Physical Chemistry of Aerosols 30 Radiation Ef€” in Inorganic Solids 31 The Structure and Properties of Ionic Melts 32 Inelastic Collisions of Atoms and Simple Molcculcs 33 High Resolution Nuclear Magnetic Resonance 34 Tha Structure of Electronically-Excited Species in the Gas-Phase 35 Fundamental Processes in Radiation Chemistry 36 Chemical Reactions in the Atmosphere 37 Dislocations in Solids 38 The Kinetics of Proton Transfer Procassas 39 Intermolecular Form 40 The Role of the Adsorbed State in Heterogeneous Catalysis 41 Colloid Stability in Aqueous and Non-Aqueous Media 42 Thc Structure and Properties of Liquids 43 Molecular Dynamics of the Chemical Reactions of Gasas 44 Electrode Reactions of Organic Compounds 45 Oxidat ion 46 Bonding in Metallo-Organic Compounds 47 Motions in Molecular Crystals 48 Polymer Solutions 49 The Vitreous State 50 Electrical Conduction in Organic Solids 51 Surface Chemistry of Oxides 52 Reactions of Small Molecules in Excited States 53 The Photoelectron Spectroscopy of Molecules 54 Molecular Beam Scattering 55 Homogeneous Catalysis with Special Reference to Hydrogenation and For current availability of Discussion volumes, see back cover.GENERAL DISCUSSIONS OF THE FARADAY SOClETY Date 1955 1955 1956 1956 1957 1958 1957 1958 1959 1959 1960 1960 1961 1961 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 subjw Volume Microwave and Radio-Frequency Spectroscopy 19 Physical Chemistry of Enymcs 20 Membrane Phenomena 21 Physical Chemistry of Proctsscs at High Prcaaura 22 M0kcula.r Mechanism of Rate Procuua in Solids 23 Interactions in Ionic Solutions 24 Configurations and Interactions of Macromoldcs and Liquid Crystah 25 Ions of the Transition Eltmenta 24 Energy Transfer with special reference to Biologkal Systunr 27 crystal h p d ~ ~ t i ~ ~ and the Chemical Reactivity of Solids 28 Oxidation-Reduction Reactions in 10- Solvents 29 The Physical Chemistry of Aerosols 30 Radiation Ef€” in Inorganic Solids 31 The Structure and Properties of Ionic Melts 32 Inelastic Collisions of Atoms and Simple Molcculcs 33 High Resolution Nuclear Magnetic Resonance 34 Tha Structure of Electronically-Excited Species in the Gas-Phase 35 Fundamental Processes in Radiation Chemistry 36 Chemical Reactions in the Atmosphere 37 Dislocations in Solids 38 The Kinetics of Proton Transfer Procassas 39 Intermolecular Form 40 The Role of the Adsorbed State in Heterogeneous Catalysis 41 Colloid Stability in Aqueous and Non-Aqueous Media 42 Thc Structure and Properties of Liquids 43 Molecular Dynamics of the Chemical Reactions of Gasas 44 Electrode Reactions of Organic Compounds 45 Oxidat ion 46 Bonding in Metallo-Organic Compounds 47 Motions in Molecular Crystals 48 Polymer Solutions 49 The Vitreous State 50 Electrical Conduction in Organic Solids 51 Surface Chemistry of Oxides 52 Reactions of Small Molecules in Excited States 53 The Photoelectron Spectroscopy of Molecules 54 Molecular Beam Scattering 55 Homogeneous Catalysis with Special Reference to Hydrogenation and For current availability of Discussion volumes, see back cover.

ISSN:0301-7249    

DOI:10.1039/DC97355BX006

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

1. THEORETICAL The Theoretical Approach BY R. A. MARCUS Department of Chemistry, University of Illinois, Urbana, Illinois 6 180 1 Received 4th June, 1973 1. A QUESTION There are a number of areas in chemical kinetics where generalizations have been helpful in interpreting and correlating a large body of experimental data in gas phase or solution reactions. I am reminded here of Bronsted’s relation between rate constants and equilibrium constants,l Eyring’s and Evans and Polanyi’s work on transition state theory,2 Rice, Ramsperger and Kassel’s work on unimolecular reaction^,^ later augmented to RRKM,4 treatments of the curve crossing problem^,^ Hammett’s ap relation and acidity function, and the subsequent equations they stimulated,6 theories of three-body recombination of atoms and of electron transfers in solution * and at electrode^,^ simple BEBO calculations on activation energies,” the Woodward-Hoffman rules and their implications for activation energies, Breit-Wigner and later treatments of resonances,1 models for ion-molecule reac- t i o n ~ , ~ ~ to name a few.In the case of inelastic non-reactive collisions one would include the SSH theory,14 distorted wave theory for some systerns,l5 the Anderson theory of spectral line broadening l6 and its later extensions.17 The interested observer, as well as the seasoned practitioner, might well ask which of these generalizations of analytical thought apply to current problems of molecular dynamics, what new ones have been developed, or what experimental generalizations are there, if any, which literally cry out for a theoretical answer.He might ask, too, whether the present field is sufficiently different from the previous ones that the approximate analytical theory will be literally swept under by a Spartan-like phalanx of exact classid trajectories and their semiclassical and quantum mechanical counter- parts, with much analytical thought going into this army. We shall not attempt to answer all of these questions here, but shall summarize instead some of the trends which appear to be developing in the field. Calculations in the area are diverse, and some classification would be useful. A possible scheme for dynamical calculations is proposed in this introductory paper. 2. INTERACTIONS AND SURFACES Perhaps it would be well to begin our inquiry with this chemical topic.Hersch- bach and his colleagues in some of their recent studies have obtained or inferred information about shapes, linear versus nonlinear, of activated complexes (i.e., of the * This research was supported by a grant from the National Science Foundation at the University of Illinois. 910 THE THEORETICAL APPROACH " reaction geometry ") and have correlated the results with Walsh's rules based on molecular orbital theory? The work of Grice and coworkers in this Discussion also treats this problem for methyl radical reactions. In their paper, Herschbach et al. comment on implications of the Woodward-Hoffman rules for four-centre reaction activation energies involving electron rearrangement, and the relation to their study.Studies of infra-red chemiluminescence, notably by Polanyi and coworkers, including the paper presented at this symposium, have revealed much on the disposi- tion of energy in exothermic reactions. In turn such information has been correlated with characterization of the relevant potential energy surfaces as early downhill, late downhill, and mixed energy release. The purely theoretical quantitative calculations include the many LEPS surfaces, as well as the modest surprising-in-its simplicity BEBO method.1° The latter, with its use of non-kinetic data, has yielded reasonably good results for activation energies of a class of reactions (one bond broken, one formed), give or take a few kcal/mol. The need for a knowledge of potential energy surfaces having the right shape is well- known, of course, to dynamicists, and several ab initio surfaces for reactions or ine- lastic collisions have been calculated, such as H + H2, F + H2 and Li+ + H2.20 Bernstein and Rulis, in their Discussion paper, have drawn our attention to the many facts which must be satisfied by any reputable K+ CHJ surface.The rise and the subsequent rapid drop of reaction cross-section with increasing initial translational energy, and its possible relation to curve crossing, was considered earlier.21 3. MOTION ON THE SURFACES The theoretical treatments for motion on these potential energy surfaces are now many, as recent reviews amply confirm. These theories can be classified as (1) dynamical, (2) statistical, (3) statistical-dynamical, a term coined elsewhere,22 and (4) numerical or " exact ".We shall explore some of the current trends in these areas, beginning with a brief recall of some relevant history of the pre-beam era. (i) EARLIER THEORY The earlier theory of reactive collisions had a flavour different from that of inelastic collisions, both because of the difference in data available and in complexity of theory : prior to the 1950's essentially all the data on reactive collisions were of a highly statistical, ensemble-averaged nature (rate constants of systems in averaged initial states) while much of the inelastic collision data offered dynamics fairly directly in the form of vibrational relaxation in sound dispersion experiments. Thus, a statistical treatment of the former and a partly dynamical treatment of the latter was needed and responded to.Again, the complexity of the dynamics of reactive collisions, arising mainly from the fact that the coordinates of the reactants do not conveniently describe in a simple way the motion of the products, made a statistical treatment a matter of necessity in the 1920's to the 1950's. Even today, for most of the reaction rate data in chemistry in solution and the gas phase, statistical treatment is a matter of necessity. While the chemical kinetics data lent themselves to statistical theory, it is refreshing to recall that many of the dynamical concepts in use today were considered in the earlier years, including the concept of adiabaticity for the internal motions, and the role of excess energy in stimulating vibrational e~citation.~~ To be sure, in the 1960's and onwards these concepts have been further developed and made more quantitative.In the case of inelastic collisions, the dynamicai grounds were laid in the early 1930's ' ' 9 24 and fashioned into a practical tool, SSH, in the 195O's.l4R. A. MARCUS 11 (ii) CURRENT DYNAMICAL PICTURE For purposes of examining current trends 2 5 in dynamical calculations, I would suggest a classification such as in table 1 : TABLE 1 .-A CLASSIFICATION OF DYNAMICAL CALCULATIONS Dynamical Inelastic and Reactive Collision Theories I I Exact (Q, C, SC) I Approximate (Q, C, SC) I I I Static Adiabatic (in each valley) (natural collision coordinates) I I I I Distorted Wave Classical Path I I I “ Distorted Wave ” Classical Path i 1 I I Sudden DIPR Half I Collision Spectator Stripping The non-italicized portion refers to inelastic collisions.All parts of the table, including the italicized ones, refer to reactive collisions. The abbreviations Q, C and SC denote quantum, classical and semiclassical, since the approximations can be formulated for each of them. Quantum is used in the sense that at least some of the internal coordinates of the system are treated quantum mechanically. Classical indicates that all degrees of freedom satisfy Hamilton’s equations of motion. Semi- classical is used in the sense adopted in the papers by Miller, Connor and Marcus in this Discussion. A principal distinction in table 1 is the “ static ” versus “ adiabatic ” approxima- tion. The label “ static ” or “ adiabatic ” is one used for brevity in table 1 and is intended to signify the zero’th order calculation plus a higher order one needed for calculation of change of rotational-vibrational state.(Hence, for example, the “ adiabatic ” in table 1 includes the usual zero’th order adiabatic calculation plus a nonadiabatic correction.) We recall the difference in the zero’th order calculations for the static and adiabatic models as follows. In calculations some reaction coordinate s is first selected, typically the radial coordinate in the case of inelastic collisions. The static model has, for the zero’th order s-motion, an effective potential obtained by averaging the interaction potential V over the unperturbed initial internal state.* In the adiabatic model, this effective potential is obtained instead by averaging V over the ZucalZy adjusted internal state at any s.That adjusted internal state is obtained beforehand by solving the equation for the internal motion at each s. In the case of a reactive collision, s is some curvilinear coordinate, leading smoothly from reactants to products in the adiabatic case. * The importance, at least in some instances, of using an averaged V instead of one which ignores the dependence of Y on internal coordinates, has been noted.2612 THE THEORETICAL APPROACH However, in the static case, s would be the radial coordinate of the reactants (products) when used for the unperturbed wave function of the reactants (products), with no possibility of having a smooth progress from one to the other. I have extended the customary meaning of the terms " distorted wave " and " classical path " for use in table 1, in that " distorted wave " in table 1 indicates first or higher order perturbation theory, be it Q, C or SC; classical path in table 1 indicates a zero'th-order classical treatment of the s-motion and any order Q, C or SC treatment of the internal motions.* Conventionally, distorted wave was typically only used for a first or higher order perturbation of the quantum mechanical static case. The classical path appro~imation,~~ so-called by workers in the line broadening field 16* l7 and sometimes called semiclassical 28a by other researchers, was usually reserved for the case where the relative motion is treated classically (and even approxi- mated by straight-line trajectories in some instances), and where the internal motion is treated quantum mechanically.However, the basic idea underlying " distorted wave " and " classical path " can be applied to the C and SC cases also, and so an extended definition is useful. The " exponential approximation ",25 not listed sep- arately in table 1, is one method for achieving a high order, in certain respects, of perturbation. The derivative nature of the remaining approximations in table 1 will be recog- nized : when the static " classical path " case is further approximated by neglecting energy differences of the internal states (and hence classically setting the frequencies of internal motion equal to zero) one obtains a " sudden " approximati~n.~~ Spectator stripping 13b* 30 could be regarded as a particular case of the latter approximation in which the initial internal energy is neglected.Use of a " classical path " model in the exit channel of a reaction, together with specified initial conditions for motion in that channel, leads instead to the DIPR approximation 31 (bimolecular) or to the half- collision approximation 32 (unimolecular dissociation). (One incidental application of table 1, by the nature of its emphasis, is the use of approximate C calculations, readily made and tested, to make predictions about the accuracy of Q and SC cal- culations for the same approximation.) For the Q case, examples exist for many of the items in table 1. They include " exact " calculations for several three-dimensional inelastic collision^.^ and for one- dimensional collinear reactive collision^.^ There are no three-dimensional Q calculations of comparable accuracy for reactive systems with smooth potentials, though less accurate two- and three-dimensional reactive calculations exist. 33-3 There are static Q calculations, both with distorted wave and with classical path for inelastic colli~ions.~~ Adiabatic Q calculations have been made for collinear inelastic collisions using the distorted wave 36a or classical path appro~irnation,~~~ and for reactive collisions using distorted wave,37- 38 or classical path.36* 3 8 9 39 For the C case, the exact calculations are well-represented in this symposium and, of course, currently constitute the main link between molecular beam and infra-red cherniluminescence data and molecular properties.Examples of the C static classical path approximation are available for collisional rotational-translational 40 and vibrational-translational energy 41 42 transfer. The adiabatic classical path approxi- mation (including nonadiabatic calculations, as emphasized earlier) has been used for the C collinear inelastic 43 and reactive problems.44 There are C examples of the spectator stripping,l 3b* 30 DIPR and half-collision models.32 * Thus, for the C case the distorted wave and classical path approximations differ only in that the latter uses a zeroth order approximation for the s-motion, while the former may use higher order approximations.R. A. MARCUS 13 For the SC case there are now many exact calculations (and some approximate ones).Several Discussion papers either refer to or make use of the calculations. We comment on SC theory later. The adiabatic type calculations are much fewer than the static due to the greater complexity of the former. On the other hand the static calculations are sometimes quite inaccurate, because of considerable distortions of the internal motion during the collision. In the collinear vibration-translation problem, that static C calculation, for the case where a light atom is sandwiched between two heavy ones, yielded an error of as much as a factor of 4000 in the calculated energy transfer,42 whereas an adiabatic C calculation involved only a 10 % error.43 The static C calculation for rotation-translation energy transfer appeared to become poorer when the reduced moment of inertia (1/p2, where Q is a Lennard-Jones distance parameter) became small.40a The physical reason for the breakdown of the static calculation is probably the same in the two cases above : the motion of relatively light masses or molecules with a small moment of inertia is strongly perturbed in the usual collisions and is not well- represented by approximations which largely neglect these strong perturbations in zero’th order.For a different reason, namely the discontinuity involved in any “ reactant channel ” versus “ product channel calculation ”, the Q static distorted wave calculation was apparently poor for the one case tested (H + H2).45 Again, the first order static rotational-translational calculation can be inaccurate at low impact parameters b(b/o< 1) 40b and either higher order or a hard body 40b (but exact) might be used there.At high translational energies the adiabatic approximation itself cannot be as good as at lower energies, because the internal state of the system does not have time to adjust to this fast s-motion. The adiabatic model (with nonadiabatic corrections included) for collinear reactive collisions has been treated with the aid of natural collision coordinates in the C classical path appro~imation,~~ in the Q classical path approximation 38 and in the distorted wave approxirnati~n.~~ Both the curvature of the reaction path and the vibration frequency along the path contributed to the change in quantum number (classical action variable).38* 44 The results gave quite good agreement with exact trajectory results for the H + H2 reaction.44 However, the model should break down at large energy transfers : the usual classical path approximation is not a self-consistent one, since it doesn’t allow for energy loss of the s-coordinate and so even allows an infinite build-up of energy in the oscillator (it is a “ forced oscillator ” 28b problem).TABLE 2.-sUMMARY OF THEORETICAL CONTRIBUTIONS* reactive inelastic Dynamical : Exact quantum I - Exact classical C13 A6, A7, D15 Exact semiclassical A3, A4, A5 A3 Distorted wave - - Classical path - - Curve-crossing A2 A6 Stat istical : - Statist i d - D y namical - A6, A8, D20, D21 * The entries in this table refer to papers presented at this Discussion and are here identified by the surname of the first author : A2-Child ; A3-Marcus ; AGMiller ; AS-Connor ; A6-Brumer ; A7-Bunker ; A8-Bernstein ; C13-van den Bergh ; D15-Ding ; D20-Herschbach ; D21-Lee.14 THE THEORETICAL APPROACH Application of the model is made in the paper by Zare and coworkers to their results on vibrational energy distribution in this Discussion.A number of calculations have been performed with hard-body type potentials (dumbells, spheres, etc.,) and might typically be labelled as " exact " though with these idealized potential^.^^ There is also, among the " exact '' quantum mechanical calculations, a recent interesting development using averaged-over-rn-states potentials to reduce the number of channels in an otherwise exact quantum cal~ulation.~~ All of the previous discussion concerned motion on one potential energy surface or curve, but the problem of curve crossing is an important one for chemistry.8 * 47* 48 The case of multiple curve crossings is considered by Child in his Discussion paper. The theoretical contributions in the present symposium can be roughly categorized as in table 2. Exact calculations are seen to predominate. A prescription for choosing between the exact and several of the approximate ones is suggested for inelastic collisions by Gordon in his Discussion paper. For reactive collisions, on the other hand, we have already noted the difficulty in finding good approximate dynamical theories. There are a number of semiclassical contributions in table 2, and several pertinent aspects of exact semiclassical theory are considered in the next section.(iii) s E M I c L A s s I c A L T HE o R Y The ubiquitousness of exact classical calculations for reactive collisions and the current virtual absence of exact three dimensional quantum calculations suggest that " exact " semiclassical calculations 49-51 may be helpful for reactive (or indeed for inelastic) collisions. The main quantum effects expected for collisions are (1) quantum mechanical interferen~es,~~ (2) penetration of classically forbidden ~egions,~ and (3) quasi-bound state effects.54 The first of these is well-known in elastic collisions (rainbows, supernumeraries, superimposed oscillations on them) and also in curve crossing The interferences " wash out " when the phenomenon is state-dependent and the results are averaged over the states,50- 5 5 but can be preserved in suficiently state-selected experiments.The second phenomenon, which can also be called n-dimensional tunnelling, does not " wash out ". It is of particular importance for any given state being formed either in a threshold region for formation of that state or in what we might term a " twilight region ", where the formation of this once important state has almost ceased (in a plot of probability of formation versus some parameter such as energy). Thus, the tunnelling is not limited to regions where the given state is just becoming energetically-allowed. In the threshold and twilight regions, one may expect semi- classically a significant contribution from complex-valued trajectories, and so the latter should not be neglected there.The third phenomenon is treated semiclassically in the Discussion paper by the present author. Previous exact semiclassical calculations for nonseparable systems were concenrned only with direct collision but in this latest paper the quasi-bound state trajectories were calculated and matched to direct collision traject- ories to yield the S-matrix elements. The resulting quantum effects would be ob- served for quasi-bound complexes with sufficiently widely-spaced states. The elusive problem of semiclassical eigenvalues for bound state systems is also treated in this paper. An example of quasi-bound states for separable systems arises in the curve crossing problem for diatomic molecules.48b A fourth quantum effect also exists : the probability distribution function ofR.A. MARCUS 15 coordinates and momenta in the lowest quantum state(s) of an oscillator differs significantly from the classical value. The exact semiclassical theory for direct collisions adopted in several papers of this Discrlssion was developed during the past three years using both Feynmann propagator 50 and wave function appro ache^.^^ Calculations on a number of syst- ems have been performed, and many problems have been investigated. They include the following : (a) inelastic collisions: numerical calculations for vibration-translation energy transfer in one dimension,49 9 50 rotation-tran~lation,~~* rotation-vibration- translation,50* 51 quasi-bound states 51 ; (b) reactive collisions : ~ n e - , ~ O * ~ ~ * 56*57 two- and three-dimen~ions,~~* multidimensional potential energy surface- crossings so ; (c) other and related topics : derivation~,~~~ 5 0 ranging from intuitive to the more rigorous ; uniform approximation^,^^-^^* 59 complex-valued trajector- i e ~ , ~ ' * 6o selection rules,49 Wigner 39 coefficients,5o* 51 Wigner 6-j coefficient^,^' spectral line br~adening,~~ canonical perturbation theory,49 partial averaging,50 and " exact '' bound state eigenvalues.51 The paper by Connor in this Discussion considers the problem of a uniform approximation for multidimensional semiclassical integrals, a problem which arises in atom-diatomic collisions when attention is focused on formation of a given vibrational- rotational state, or in collisions involving polyatomic molecules when, as is typically the case with widely-spaced levels, a particular state of the several vibrations is needed.The paper by Miller and Raczkewski describes the important method of partial averaging 50 of semiclassical results and its use in calculation of cross-sections for inelastic collisions. 4. STATISTICAL THEORIES The most commonly used statistical theory for reaction rate constants is, of course, activated complex theory. Here, as is well-known, one assumes a quasi-equilibrium between reactants and systems crossing a particular hypersurface, " the activated complex ", calculates the probability of finding the system near the hypersurface, per unit length of reaction coordinate, multiplies the latter by the local velocity along that coordinate, integrates over all velocities, and sums over all states of the activated complexes.Classical trajectories have been used, mainly by Karplus and co-workers, to test the activated complex theory of bimolecular reaction^.^^'^^ All tests apart from the first have compared the classical trajectory data with the classical form of activated complex theory (as they should), instead of with the quantum form. Comparison was made with the microcanonical form of activated complex theory,62 which in turn is related by a Laplace transform to the usual theory. The agreement between exact and trajectory values of the reactive flux was good in the region of thermal interest. Breakdown in this comparison of bimolecular reaction rates occurred at the higher translation energies, because of reflection near the curved portion of the reaction path.The reflection led to a reduced rate both when the activated complex was in the initial channel (highly exothermic reaction) and so was recrossed to reform reactants, and when the activated complex was in the product channel (highly endothermic reactions) and so was never reached. The quantum form of activated complex theory (ACT) has also been tested, though only for collinear systems, by comparing with the results of exact numerical solution of the Schrodinger equation for the H+H, reaction.64 The exact results show more tunnelling in the threshold for reaction. It will be interesting to see how much of this difference occurs in three-dimensional calculations.16 THE THEORETICAL APPROACH Unimolecular reaction rate theory in its RRKM form has been useful in interpret- ing rate constant versus pressure data, using the full number of degrees of freedom.65 It has received support from trajectory calculations,66 for molecules having lifetimes longer than an estimated intramolecular relaxation time - 10-l1 s.A comparison for another type of reaction, a four-centre reaction, appears in a paper by Brumer and Karplus in this Discussion, where it was found convenient to divide the trajectories into those for short- and long-lived complexes. The comparison with RRKM is made for the latter. An example of breakdown has recently been found for molecules with lifetimes shorter than - 10-l2 s in the reaction involving the formation of the higher energy intermediate, CF2-CF-CF-CF2 * followed by elimination of a CF2 from either ring.67 It was estimated from pressure effect data on the vibrationally-hot intermediate that the time needed for randomiza- tion of energy among the two rings, i.e., for random emission of CF2 from either ring, was 5 s.67 In general, in other systems, one would expect an even greater intramolecular relaxation time at sufficiently low energies and correspondingly low anharmonicities. Activated complex theory in its usual form is essentially devoid of dynamics. For example, it needs dynamics only in an infinitesimal interval (s, s+&), and these dynamics are trivial.(When the curvilinear reaction coordinate is treated quantum mechanically, one uses dynamics over a somewhat larger s-interval, e.g., in the WKB approximation, to avoid conflict with the uncertainty principle.) The lack of extensive dynamics suffices for calculation of canonical or micro- canonical rate constants and the lack of need of dynamics is, in a sense, one of the strengths of activated complex theory.It is also its greatest weakness. For example, apart from the cited case of the loose activated complex, activated complex theory cannot be used to predict the dependence of reaction probability on initial state and on initial relative translational velocity ; nor can it predict the relative formation rate of products in given final states and with given final velocities. One can adapt activated complex theory to make such calculations, by imposing added approximations. Measurements of final states or velocities, therefore, do not test activated complex theory, and hence do not test RRKM, but rather ACT (or RRKM) plus added assumptions. This point appears later in a discussion on the most interesting data of Y. T.Lee et aZ. in this Discussion.* Sometimes this " strength " or " weakness " of activated complex theory has been statistical vibrational adiabaticity 7 0 ) have been used to understand activated complex theory under certain conditions, ACT is both more general and less powerful than VA. Any breakdown of VA or SVA does not automatically constitute a breakdown for ACT. Thus, the fact that an exothermic reaction yields products in highly excited vibrational states, even though the reactants were in low vibrational states, constitutes a breakdown of VA and SVA but not necessarily of ACT.In fact, such reactions frequently have an activated complex which is almost loose and their reaction * A new development in the field, involving the direct observation of the internal energy distri- butions of these products of molecular beam systems, by measuring their infra-red chemiluminescence, has been devised by McDonald and coworkers.68 It promises to provide much needed information, nicely complementing the measurements of Y. T. Lee et a!. misunderstood in a different way : while vibrational adiabaticity 2309 3 7 9 69 (orR. A. MARCUS 17 rate constants can be calculated by ACT using an essentially loose complex. The activated complex for the reaction rate does not necessarily, in fact usually does not, coincide with a “ transition state ” region where marked dynamical excitation or de- excitation occurs.The phase space theory of reaction^,^' unlike activated complex theory, aims at predicting final states of reaction products without using properties of the potential energy surface for intermediate configurations. Because of the increased density of final rotational-orbital states with increasing energy, the common phenomenon of an inversion in the hal vibrational state population requires an at least partly-dynamical, rather than a purely phase space, argument. The reaction cross-reaction, like the rate constant, depends on the dependence of the potential energy surface on internal coordinates. In the case of the loose activated complex, this dependence is simple and the cross-section can be calculated by phase space theory, as well as by ACT.For other complexes, a more detailed (statistical-dynamical) theory is needed. Never- the less, the idea of counting phase space states of products, like the idea of counting activated complex states, is a significant one for future modifications, and has had a very stimulating effect on the field. We return to the statistical-dynamical problem in the next section. Recently, in a most interesting development 72-74 Levine, Bernstein and coworkers have found that the distribution of vibrational states of products of some A+BC reactions can be presented by P,,(E- Ev)Pv(Ev), where Pro@- Ev) is the density of rotational-orbital states and for a rotational plus translational energy of E- E, and is proportional to (E-Ev)+ for this system.Pv(Ev) was found empirically to be exp( - AVEv), where AV is a constant. (A vibrational population inversion corresponds to a negative 4). Such an exponential dependence can be derived on statistical grounds * if one allows the coordinate to take on unlimited (or almost unlimited) values of E,, but such a situation presumably does not apply here and a different explanation must be sought. The phenomenology has been extended to a conditional rotational di~tribution.’~ The general development is discussed by Levine and Bernstein in their Discussion paper, both for product and angular distributions. 5. STATISTICAL-DYNAMICAL THEORY A statistical-dynamical theory is one which would employ approximate dynamics for one or more of the degrees of freedom and use a statistical approximation for the remainder.For example, in a treatment of vibrational distribution of the products it might use analytic or trajectory calculations for collinear collisions to calculate an ab initio Pv(Ev) in the preceding section (but caution is needed) and use the statistical Pro(E- E,) for the remaining coordinates. One simple example, not appropriate to these highly exothermic or endothermic reactions but more appropriate to some thennoneutral reactions without marked reaction path curvature has been given. It uses a vibrationally adiabatic approximation for the vibration. The remaining coordinates are treated statistically. An integral equation for the reaction cross- section was then solved. The results for the reaction cross-section were in reasonable agreement with trajectory data, without introducing my unknown parameters.22 An analogous procedure could be employed for other models for calculating Pv(E,). A classical path quantum calculation for collinear collisions has been used to relate (E,) to reaction path curvature.38 (Cf. also ref. (44), where the phase-averaged * e.g., in ref. (73) it was derived by maximizing an entropy -kC Pv In Pv subject to a condition of n a preassigned <Ev>, i.e., C EvPv. The derivation placed no upper limit on Ev. n18 THE THEORETICAL APPROACH result is (E").) Thus, in a sense the P,,(E-E,)P,(E,) in ref. (72) and (74) is a form of a statistical-dynamical theory, but only in some average way, since the form of P,(E,) used 7 2 9 74 differed from that estimated in approximate dynamical calcula- t i o n ~ .~ ~ We have already commented on the shortcomings of the usual classical path forced-oscillator problem for treating large excitations. An example of a statistical-dynamical theory which relates RRKM theory for energy distributions in the activated complex to energy distributions of the products has been given by Herschbach and coworkers,75 who treated the dynamics for the loose activated complex. Their result is the same, they note, as would be obtained from phase space theory. RRKM theory and phase space theory have been shown 76 to be equivalent for the case of a loose activated complex (and zero potential energy barrier). This result is not unexpected, since phase space theory 71 does not use any detailed properties of the surface on one hand, and since the dynamics from loose activated complex to products are simple, on the other.As noted earlier, with other systems other models need to be considered and can be developed in order to attempt to deduce the properties of the activated compIex from those of the products and vice versa. 6. SUMMARY The extensiveness of references in recent reviews 25 reveals both the enthusiastic activity in this field and the sobering fact that much remains to be done, even in the field of inelastic collisions. Fully collinear inelastic collisions are reasonably well- understood analytically, and substantial progress has been made in the three-dimen- sional ones as well, at least when the interactions are not too strong.Suggestions range from the purely dynamical to a partial use of dynamics in one regime and a use of statistical in another.25 The situation for reactive collisions is qualitatively fairly well-understood for collinear collisions. There has been some progress quantitatively, both for the near- adiabatic and, to some extent, depending on one's willingness to accept the cited classical path estimate, in the more nonadiabatic regime as well. For the fully-three dimensional collisions there is again some qualitative understanding and, in the case of rather special models (e.g., spectator stripping ; hard bodies), quantitative analyt- ical insight. The progress is slowed, in comparison with the collinear case, by the absence of diagrams similar to the highly useful and much-studied skewed-axes plots of potential energy contours.(To paraphrase a quite different remark of Herschbach at an earlier Discussion, the trouble with three-dimensional systems is that they have two dimensions too many!) For this reason, phenomenology such as that discussed by Levine and Bernstein for reactive collisions is a most welcome one, with its use of some statistical insight. The possibility, too, of extending activated comples theory with the addition of dynamical elements, i.e., of having statistical-dynamical theories is already with us. Two examples have been given in the previous section and more and better ones will undoubtedly be developed. In the realm of exact calculations, exact classical theory remains the most potent method for treating experimental data on simple chemical reactions.Moreover, semiclassical theory may become a significant method for including the quantum effects absent in the purely classical calculations. While individual S-matrix elements are sometimes in error by a factor of two, the averaging that occurs when a reaction cross-section is calculated from S-matrix elements might reduce the error further, to the extent that some of the errors in the elements may be random.R. A. MARCUS 19 The present symposium touches on many of these and related questions, and At the very least, represents one more step toward answering some of the questions. we all enjoy it. J. N. Bronsted and K. Pedersen, 2. phys. Chem., 1924, A108, 185, cf. P. R. 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Magnetic Resonance, 1968, 3, 1. D. R. Herschbach, in Proceedings of the Conference on Potential Energy Surfaces, ed. W. A. Lester, Jr. (IBM Research Laboratory, San Jose, Calif., 1971); J. D. McDonald, P. R. Le- Breton, Y. T. Lee and D. R. Herschbach, J. Chem. Phys., 1972,56, 769. l9 cf. T. Carrington and J. C. Polanyi in MTP International Review of Science, Phys. Chem. Sec. I,20 THE THEORETICAL APPROACH Vol. 9, Chemical Kinetics, ed. J. C. Polanyi (Butterworths, London, 1972), p. 135 ff. ; J. C. Polanyi, Acct. Chem. Res., 1972, 5, 161. 2o e.g., I. Shavitt, R. M. Stevens, F. L. Minn and M. Karplus, J. Chem. Phys., 1968, 48, 2700; S. V. O’Neil, P. K. Pearson, H. F. Schaefer and C. F. Bender, J. Chem. Phys., 1973,58,1126 ; W.A. Lester, Jr., J. Chem. Phys., 1971,54, 3171. 21 R. A. Budde, P. J. Kuntz, R. B. Bernstein and R. D. Levine, Chem. Phys. Letters, 1973, 19,7. 22 R. A. Marcus, J. Chem. Phys., 1966,45,2630; 1967, 46,959. 23 e.g., (a) J. 0. Hirschfelder and E. Wigner, J. Chem. Phys., 1939,7,616; (b) M. G. Evans and M. Polanyi, Trans. Farahy Soc., 1939,35, 178. 24 cf. also 0. K. Rice, J. Amer. Chem. Soc., 1932,544558 ; L. D. Landau and E. Teller, Phys. Z. Sowjetunion, 1936, 10, 34 ; C. Zener, Phys. Rev., 1931,28, 277. 25 For excellent recent reviews see R. D. Levine, in MTP International Review of Science, Phys. Chem. Ser. I, Vol. 1, Theoretical Chemistry, ed. W. B. Brown (Butterworths, London, 1972), p. 229 ff. ; D. Secrest, Ann. Rev. Phys. Chem., 1973,W; T. F. George and J.Ross, Ann. Rev. Phys. Chem., 1973,24. 26 F. H. Mia, J. Chem. Phys., 1964, 40,523 ; R. E. Roberts, J. Chem. Phys., 1968,49,2880. 27 cf. R. D. Levine, ref. (25), pp. 240-241, for detailed references. 28 (a) cf. R. J. Cross, Jr., J. Chem. Phys., 1968,48,4838, M. D. Pattengill, C. F. Curtiss and R. B. Bernstein, J. Chem. Phys., 1971,54,2197 ; F. E. Heidrich, K. R. Wilson and D. Rapp, J. Chem. Phys., 1971, 54, 3885; R. E. Roberts and B. R. Johnson, J. Chem. Phys., 1970,53,463; (b) E. H. Kerner, Canad. J, Phys., 1958, 36, 371. 29 K. Alder and A. Winther, Danske videns. Selskab. Mat. Fys. Medd., 1960, 32, No. 8 ; R. W. Fenstermaker and R. B. Bernstein, J. Chem. Phys., 1968,47,4417, and refwenoes cited therein. 30 cf. R. E. Minturn, S. Datz and R. L. Becker, J.Chem. Phys., 1966, 44,1149; cf. J. L. Kinsey, ref. (19), pp. 203-204. 31 P. J. Kuntz, M. H. Mok and J. C. Polanyi, J. Chem. Phys., 1969,50,4623 ; P. J. Kuntz, Trans. Faruduy Sac., 1970, 66, 2980. 32 K. E. Holdy, L. C. KIotz and K. R. Wilson, J. Chem. Phys., 1970,52,4588 ; F. E. Heidrich, K. R. Wilson and D. Rapp, J. Chem. Phys., 1971,54,3885. 33 M . Karplus and K. Tang, Disc. Faraday SOC., 1967,44, 56. 34 R. P. Saxon and J. C. Light, J. Chem. Phys., 1971,55,455. 35 G. Wolken and M. Karplus in Physics of Electronic and Atomic Collisions, eds. T. R. Gover and 36 (a) E. Thiele and R. Katz, J. Chem. Phys., 1971,55, 3195 ; (6) W. A. Cady, Ph.0. Dissertation 37 G. L. Hofacker, Z. Naturforsch., 1963,18a, 607 ; R. A. Marcus, J. Chem. Phys., 1966,45,4493. 38 G. L. Hofacker and R.D. Levine, Chem. Phys. Letters, 1971, 9, 617 ; R. D. Levine, Chem. 39 R. D. Levine and B. R. Johnson, Chem, Phys. Letters, 1971,8,501. 40 (a) A. 0. Cohen and R. A. Marcus, J. Chem. Phys., 1968,49,4509 ; J. Chem. Phys., 1970,52, 41 L. D. Landau and E. Teller, ref. (24) ; K. Takayangi, Progr. Theor. Phys. Suppl. (Kyoto), 1963, 42 J. D. Kelley and M. Wolfsberg, J. Chem. Phys., 1966, 44, 324 ; 1970, 53, 2967. 43 M. Attermeyer and R. A. Marcus, J. Chem. Phys., 1970,52, 393. 44 R. A. Marcus, J. Chem. Phys., 1966,45,4500; S. F. Wu and R. A. Marcus, J. Chem. Phys.. 45 e.g., cf. ref. (33) and (39, and the factor of 50 difference. 46 H. A. Rabitz, J. Chem. Phys., 1972, 57, 1718 ; 1973, 58, 3975 ; R. Conn and H. Rabitz, J. Chem. Phys., in press. 47 T. Carrington and D.Garvin, ref. (146), Chap. 3. 48 e.g., in collisions or predissociations involving electronic changes : (a) R. E. Olson and F. T. Smith, Phys. Rev. A, 1971, 3, 1607 ; R. E. Olson, Phys. Rev. A, 1972,5, 2094, and references cited therein to these “ Stueckelberg oscillations ” ; (6) D. S. Ramsay and M. S. Child, MoI. Phys., 1971,22,263 ; C. E. Caplan and M. S. Child, Mol. Phys., 1972,23, 249 ; M. S. Child, Mol. Phys., 1972,23,469 ; and references cited therein. 49 R. A. Marcus, Chem. Phys. Letters, 1970, 7, 525 ; J. Chem. Phys., 1971, 54, 3965 ; J. N. L. Connor and R. A. Marcus, J. Chem. Phys., 1971,55, 5636 ; W. H. Wong and R. A. Marcus, J. Chem. Phys., 1971, 55, 5663; 1972, 56, 311 ; 1972,56, 3548; J. Stine and R. A. Marcus, Chem. Phys. Letters, 1972, 15, 536; R.A. Marcus, J. Chem. Phys., 1972, 57,4903. W. H. Miller, J. Chem. Phys., 1970, 53, 1949 ; 1970, 53, 3578 ; Chem. Phys. Letters, 1970, 7, 431 ; J. Chem. Phys., 1971, 54, 5386; W. H. Miller, Ace. Chem. Res., 1971, 4, 161; C. C. F. J. deHeer (VII ICPEAC, North-Holland Publishing Co., Amsterdam, 1972), p. 302. (University of Illinois, 1972). Phys. Letters, 1971, 10, 510. 3140; (6) R. A. LaBudde, J. Chem. Phys., 1972,57, 582, and references cited therein. 25, 1. cf. reviewed by D. Rapp and T. Kassal, Chem. Rev., 1969,69, 61. 1970,53,4026.R. A. MARCUS 21 Rankin and W. H. Miller, J. Chem. Phys., 1971, 55, 3150; W. H. Miller and T. F. George, J. Chem. Phys., 1972,56,5668 ; T. F. George and W. H. Miller, J. Chem. Phys., 1972,57,5722 ; J. D. Doll and W. H. Miller, J.Chem. Phys., 1972, 57, 5019 ; J. D. Doll, T. F. George and W. H. Miller, J. Chem. Phys., 1973, 58, 1343. 5 1 D. Fitz and R. A. Marcus, J. Chem. Phys., 1973, 59; A. Turfa and R. A. Marcus (to be submitted) ; H. R. Kreek and R. A. Marcus (to be submitted) ; R. Ellis and R. A. Marcus (to be submitted) ; J. Stine and R. A. Marcus (to be submitted) ; J. Stine and R. A. Marcus, J. Chem. Phys., 1973, 59, cf. Discussion remarks; W. Eastes and R. A. Marcus (to be submitted), cf. Discussion remarks. 5 2 See related semiclassical studies of P. Pechukas, Phys. Rev., 1969, 181, 166, 174 ; R. D. Levine and B. R. Johnson, Chem. Phys. Letters, 1970, 7 , 404; 1971, 8, 501 ; I. C. Percival and D. Richards, J. Phys. B, 1970, 3, 315, 1035. 53 e.g., L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1958).54 e.g., R. D. Levine, Acct. Chem. Res., 1970, 3, 273 ; D. A. Micha, Acct. Chem. Res., 1973, 6, 138, and references cited in these articles. 5 5 L. D. Landau, ref. (5). 56 J. M. Bowman and A. Kuppermann, Chem. Phys. Letters, 1973, 19, 166. 57 S. F. Wu and R. D. Levine, private communication. '* J. C. Light and R. P. Saxon, private communication. 59 J. N. L. Connor, Mol. Phys., 1972, 23,717 ; and submitted for publication. 6o J. Stine and R. A. Marcus, ref. (49); T. F. George and W. H. Miller, ref. (50); J. D. Doll, 61 M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Phys., 1965, 43, 3259. 62 R. A. Marcus, J. Chem. Phys., 1966, 45,2138. 63 K. Morokuma, B. C. Eu and M. Karplus, J. Chem. Phys., 1969,51, 5193 ; K. Morokuma and M. Karplus, J. Chem. Phys., 1971,5563 ; G . W. Koeppl and M. Karplus, J. Chem. Phys., 1971, 55, 4667. These papers also contain niicrocanonical activated complex theory for collisions on a line. 64 E. M. Mortensen, J. Chem. Phys., 1958,48,4029 ; D. G. Truhlar and A. Kuppermann, J. Chem. Phys., 1972, 56, 2232, and references cited therein; D. J. Diestler, D. G. Truhlar and A. Kuppermann, Chem. Phys. Letters, 1972, 13, 1. 6 5 For a recent review see D. W. Setser in MTP International Review of Science, Phys. Chem. Ser. I, Vof. 9, Chemical Kinetics, ed. J . C. Polan9 (Butterworths, London, 1972), Chap. 1 ; L. D. Spicer and B. S. Rabinovitch, Ann. Rev. Phys. Chem., 1970, 21, 349 ; E. A. Hardwidge, B. S. Rabinovitch and R. C. Ireton, J. Chem. Phys., 1973, 58, 340. 66 D. L. Bunker, J. Chem. Phys., 1962,37,393 ; D. L. Bunker and M. Pattengill, J. Chem. Phys., 1968,48,772. 67 J. D. Rynbrandt and B. S. Rabinovitch, J. Chem. Phys., 1971,54,2275, and private communica- tion. J. D. McDonald, J. Chem. Phys. (to be submitted), and Discussion remarks. T. F. George and W. H. Miller, ref. (50). 69 R. A. Marcus, J. Chem. Phys., 1965, 43, 1598. 'O R. A. Marcus, Disc. Faraday SOC., 1967, 44,7. 71 P. Pechukas and J. C. Light, J. Chem. Phys., 1965, 42, 3281 ; J. C. Light, Disc. Faraday SOC., 7 2 A. Ben-Shaul, R. D. Levine and R. B. Bernstein, Chem. Phys. Letters, 1972, 15, 160 ; J. Chem. 73 G. L. Hofacker and R. D. Levine, Chem. Phys. Letters, 1972, 15, 165. 74 R. D. Levine, B. R. Johnson and R. B. Bernstein, Chem. Phys. Letters, 1973, 19, 1. 7 5 S. A. Safron, N. D. Weinstein, D. R. Herschbach, Chem. Phys. Letters, 1972, 12, 564. '' F. H. Mies, J. Chem. Phys., 1969,51,798 ; S . A. Safron, Ph. D. Dissertation (Harvard University, 1967,44, 14; cf. J. C. Keck, J. Chem. Phys., 1958, 29,410. Phys., 1972, 57, 5427. 1969) (cited in ref. (75)).

ISSN:0301-7249    

DOI:10.1039/DC9735500009

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

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.

ISSN:0301-7249    

DOI:10.1039/DC9735500022

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

Franck-Condon Transitions in Multi-curve Crossing Processes BY M. S. CHILD Department of Theoretical Chemistry, 1, South Parks Road, Oxford Received 24th January, 1973 The high energy solution of the multi-curve crossing problem involving many vibronic levels is shown to reduce to that of a single curve crossing modulated by Franck-Condon factors. A quanti- tative validity criterion is derived. While the dynamical problem associated with a single crossing between potential curves is essentially solved,’-’ certain molecular processes may be described in a multi-curve-crossing form, and here the situation is less satisfactory. Important examples involve interaction between covalent, M +XY, and ionised, M+ +XY-, species, with each vibronic state being represented by a different curve, as shown below.\ FIG. 1 Theoretical difficulties then arise from overlap between adjacent crossing regions, because a given crossing has a dynamic width,4 given in the linear approxiniation Vi(r) = Vo-Fi(r- R), Ar,, = [hu/21Fi - F’1]f, where u is the velocity at the crossing point, R. This width may be seen either as the range within which the “ classical ” momentum uncertainty at distance Ar from the crossing point, cannot be accommodated within the uncertainty principle, or, from a mathematical viewpoint as the region outside which rapid oscillations in the equations of motion 30M. S. CHILD 31 damp out any contribution to the transition probability. The possibility of such overlap clearly restricts the use of formulae based on the accumulation of single crossing probabilties lo-ll to the low velocity region.At the same time, eqn (2) shows that while Ar increases with v it does so less rapidly than the velocity itself. Hence the interaction time, Ar/u decreases as v-* and must in the limit fall to a value small compared with the time period of the internal motion. Under these conditions we might expect to revert to a single electronic level crossing model modulated by Franck-Condon populations in the final vibrational states.’ The purposes of this note are first to justify this expectation and secondly to provide quantitative conditions on its validity. We assume for this purpose a model based on unperturbed internal states xln(p) and xzm(p) with the energies E l , and Ezn in channels 1 and 2 respectively, in which the electronic interaction term, V1 2(r), is independent of the internal coordinates p (this is an essential pre-condition of any Franck-Condon result). Furthermore, the translational velocity is assumed sufficiently high to allow the use of a time dependent classical trajectory formulation,’ the relation between time and distance being of the form r = ro+u(tl.(4) The resulting equations for the amplitude coefficients cln(t) therefore become where the elements define the (orthogonal) internal state overlap, or Franck-Condon amplitude, matrix. Clearly a first-order perturbation solution, valid for small VI2(r), subject to the bound- ary conditions C l n ’ ( - 00) = 6,”*, - a) = 0, (7) = T 2 1 S n n 9 (8) will always yield a solution of the desired Franck-Condon form where in this case with t , used to denote the time at the crossing point.channel 2 to obtain new coefficients The theory in the more general case relies on an orthogonal transformation in a 2 n ( t ) = C S m n ’ ~ 2 m ’ ( t ) , (10) nr32 MULTICURVE CROSSING PROCESSES in terms of which by the orthogonality of S c,m(t) = C Srn'adt). (11) n' Note the use of the label n to denote the channel one level to which the transformed channel two states are coupled. Substitution in (4) now yields m These equations are, apart from the terms in Xnn.9 appropriate to a single curve cros- sing between (14) (15) where W i is the energy at the crossing point, the final forms in (14) and (15) being justified by the linear approximations (1) and (4). Furthermore the terms in Xnnt in eqn (12) may be recognised as those responsible for spontaneous decay of the non-stationary states represented by (11).Hence we wish in physical terms to find conditions under which the time constant for this spontaneous decay is large compared with the as yet unspecified interaction time, to. The argument is facilitated by the substitutions W&) = E,,+V,(t) E! W;-F,o(t-',) W2n(t) = E,,+V,(t) 21 W",F&-t,) and the appropriately averaged curve and the introduction of a new dimensionless variable z = ( t - t,)/to. Eqn (12) then take the forms . dbln - 'm) exp [Iia(z)]bzn(z) h dz ' db2n dz I2 I - - I-=- '''' 2 ( t ) exp [ - ia(z)]bIn(z) + whereM . S . CHILD 33 The onset of rapid oscillations, la(z)l> n, leading to widespread destructive inter- ference on integrating (1 8) may therefore be set at the points z = * 1, by defining the interaction time as The final term in (18) therfore provides a contribution to b2,, of order twice the coefficient ( t o X n n r / A ) when (18) is integrated over the interaction zone, - 1 <z< 1.The condition for neglect of this term, for moderate values of [toV12(t)/?i] is therefore to = [27EA/Up71-F214. (20) Cases for which (toVI2/h) < 1 are seen to be covered by eqn (9). Hence, if the condition (21) is satisfied, the solution of (18) is equivalent to that for a single curve crossing between Wln(r) and the appropriate mean curve wzn(r) ; and the resulting transition amplitude, given in the Landau-Zener approximation by (22) is readily converted, byt the use of (1 1) and (16) into a set of Franck-Condon trans- ition amplitudes Finally, by virtue of the closure relation ITi"z'l = I~z,(.o)l = exp { - ~Cv,z(t,)12~~ul~1 - & I 1 9 I K m t = ICAW)I = ISmnTC;lI.(23) C Snmsmn = 1 7 m P , , =c ITnm12 = lT'1"1I2. the total probability of scattering into channel two is given m The validity of (23) and (25) does not however rest on the Landau-Zener approxi- mation. The overall conclusion is that a single curve crossing, Franck-Condon, model is justified at velocities sufficiently high that (21) is satisfied, provided that the asymptotic is calculated from the correct near vertical electron affinity The validity criterion (21) is seen to depend on the ratio of the interaction time, to, to the time period, (E2m-E2n)/lZ, of the internal motion, and also on the extent of overlap between the covalent and ionic vibrational states. E(X2) = E(X)+D(X,)-D(X,)+Eln--~n. (27) l L. D. Landau, Phys. Z . Sow. Union, 1932,2,46. C. Zener, Proc. Roy. SOC. A , 1932, 137,696. E. C. G. Stuckelberg, Helu. Phys. Acta, 1932, 5, 369. D. R. Bates, Proc. Roy. SOC. A, 1960, 257, 22. V. K. Bykhovskii, E. E. Nikitin and M. Ya. Ovchinnikova, J. Expt. Theor. Phys., 1965,20,500. G. V. Dubrovskii, J. Expt. Theor. Phys., 1964,19, 591. ' M. S . Child, Mol. Phys., 1971,20, 171. * E. E. Nikitin, Opt. Spektr, (trans.), 1961, 11, 246. H. Hartmann, Chemische Elementarprozesse (Springer, Berlin, 1968), pp. 43-77. V. I. Osherov, Zhur. Exp. Teor. Fiz., 1965,49, 1157. Yu. N. Demkov, Dokl. Akad. Nauk. S.S.S.R., 1966,166,1076. G. M. Kendall and R. Grice, Mol. Phys., 1972,24, 1373. l 3 J. B. Delos, W. R. Thorson and S. K. Knudson, Phys. Reu. A, 1971, 6, 709. 55-B

ISSN:0301-7249    

DOI:10.1039/DC9735500030

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

Semiclassical Theory for Collisions Involving Complexes (Compound State Resonances) and for Bound State Systems t BY R. A. MARCUS Department of Chemistry, University of Illinois, Urbana, Illinois 61 801 Received 23rd January, 1973 Semiclassical theory for bound states is discussed and a method is described for calculating the eigenvalues for systems not permitting separation of variables. Trajectory data are supplemented by interpolation to connect open ends of quasi-periodic trajectories. The method is also applied to quasi-bound states. Previously, semiclassical S-matrix theory has focused on " direct " reactions. Processes involv- ing complexes (compound state resonances) are treated in the present paper and an expression is derived for the S-matrix. Use is made of the above analysis of quasi-bound states and of trajectories connecting those states with open channels.The result deduced for the S-matrix has the expected factorization property, and expressions are given for computing the quantities involved. Some extensions and applications will be described in later papers. An implication for classical trajectory calculations of complexes is noted. 1. INTRODUCTION A semiclassical S-matrix theory for " direct " inelastic and reactive collisions has been developed in recent papers.l* (Ref. (3) contains related studies.) On the other hand, many collisions and other processes involve short- or long-lived vibra- tionally-excited intermediates 4* : unimolecular reactions, molecular beam reac- tions involving complexes,6 other bimolecular reactions (e.g., possible at a thre~hold),~ and intramolecular energy transfer in general.In the present paper we consider the dynamics of coupling between open channels and quasi-bound states (" compound state resonances "9), and formulate a semi- classical S-matrix theory for such collisions. The theory of direct collisions is first summarized in Section 2. A method utilizing classical mechanical trajectory data is proposed in Section 3 for calculation of eigenvalues of bound states. It is non- perturbative and is directed toward systems for which one cannot separate variables. It is adapted in Section 4 to quasi-bound states. The principal result for quasi-bound states is given by eqn (4.9), (4.12), (4.16) and (4.17). A method for calculating the quantities is described. One-dimensional (" shape ") resonances, such as those occurring in orbiting, have earlier been treated semiclassically.o-l 2. SEMICLASSICAL THEORY OF DIRECT COLLISIONS The semiclassical wave function $+(q, nE) is a function of the coordinates, denoted For direct inelastic (2.1) -f This research was supported by a grant from the National Science Foundation at the University collectively by q, the quantum numbers n, and the total energy E. collisions, this wave function at large separation distances R is $+(q, nE) = (qlnE+} = [A' exp{iF2'} + A exp{i(F2 -3n}] exp{iZnn/2}, of Illinois. 34R. A . MARCUS 35 where the term with superscript i describes the incoming particles, and the second term is for the outgoing ones.* The A’s and F’s are functions of q, n and E ; - n/2 is the usual phase loss for a reflection ; Znn/2 is present by convention, where In is the orbital quantum number contributing to n.Units of A = 1 are used. F2 in (2.1) is a classical mechanical generating function l3 and serves to make a canonical transformation of variables from q, p to momenta in nE and to conjugate coordinates). Illuminating discussions of the relation between classical canonical transformations (e.g., embodied in Fz(q, nE)) and quantum mechanical unitary transformations (e.g., embodied in (qlnE+) have been given. 14-1 8’; for a partial wave, when the q’s consist of a radial coordinate R and angle coordinates w, is given by (2.2). [The value when the q’s are conventional coordinates appears later in (4.5).] Fi(wR, nE) = -knR+2nnw, (2.2) The w’s are canonically conjugate to 2nn and are frequently employed for the internal motions.‘.2* 17-19 Their properties are very convenient and have been summarized in Part 1V.l nw is an abbreviation for I=njwj, the summation being over the various internal degrees of freedom (including orbital, if any). F2(wR, nE) is the phase integral calculated from the classical trajectory passing through the cited q [the point (w, R)], beginning at the given n and E and hence at some initial point (wo, R,) : R W RO wo F,(wR, nE) = -k,Ro+2nnwo+ 1 pRdR+2n1 i i d w , (2.3) where Z is the instantaneous n along the path. vation 1 s 20-22 or by normalization to a 6(n - n’)[2n6(E- E’)] it is the determinant t The amplitude A in semiclassical solutions can be evaluated either by flux conser- I5-l6 When normalized to A = ]PF2/dadql* (2-4) = on-+, A = v-+jawo/awl+, (2.5) where the a’s are E and (since h = 1) 2nn.One finds from (2.2)-(2.4) that where u, and li denote the magnitude of the initial and final radial velocity for the trajectory specified by the final point (wR) and the initial momenta present in nE. The semiclassical S-matrix Smn can be defined via the expression for t+h+ at large R : v;*(exp (2nim w})[8,, exp { -i(k,R --+Zmn)] - $+( wR, nE) = m S,, exp {i(kmR-3Zmn))]. (2.6) Comparison of (2.1) and (2.6) yields (Part 11, ref. (1)) .1 S,,, = 1 18 wo/a wl*(exp iA) dw (to be stationary phased), 0 using the fact that only the stationary phase points contribute (whence ZJ = Un,).’ given by A is (2.8) A = F2( wR, nE) - Fi(wR, mE) ++(1, + 1, + 1)~. * The wave function given by eqn (2.1)-(2.6) is for a volume element dq.If the volume element is gt dg, e.g., R2 dR dw,’ then (2.1) and (2.6) are multiplied by g-$, e.g., R-I. t This is equivalent in (2.5) to the normalization used in ref. (l), where each A was normalized to unit radial flux at a given R.36 RESONANCES A N D BOUND STATES A stationary phase evaluation of (2.7) yields evaluated at fi = m. The summation is over stationary phase points wt, i.e., over the one or several traject- ories leading from the initial momenta in nE to the final momenta in mE. (The Riemann sheet discussion in Section 3 and 4 provides an understanding of the fact that several trajectories lead from nE to mE, namely one per sheet.) F4 is a gener- atorI3 far a canonical transformation from nE to mE, and was denoted by At, apart from the z-terms, in ref.(1). F,(mE, nE) = - w d i = - J q dp. (2.10) A uniform approximation of (2.7) can be made. It usually involves Airy functions and has wider validity than the stationary phase value, (2.9). 9 The amplitude A in (2.5) becomes infinite on certain surfaces ((‘caustics ”). This infinity in A is detected by the intersection there of neighbouring trajectories. (E.g., see later, “ surfaces ’’ AB, BC, CD and DA in fig. 1) : the resulting vanishing of the “cross-sectional area” between the trajectories causes the amplitude A to become infinite to “ conserve probability flux.” In such cases one can still usually use (2.7) to obtain a stationary phase or uniform approximation for Sm,, since the stationary phase points are themselves usually nonsingular.To have a useful integral expression for S,, one can change the coordinate repre- sentation to one which is sometimes singularity-free, by a canonical transformation (Part III of ref. (1)) from wR to Wz, where the i7 are constant and z is a time variable. This representation leads to the same result as (2.7), but with dw amd dw replaced by a@ and dK, and with A having an added term 2n(ii-m)@. The uniform and station- ary phase values of the new expression agree exactly with those of (2.7)-(2.9), but the new integral is now sometimes also of particular use when asymptotic evaluation methods become poor. The form of this integral expression had been predicted by intuitive arguments.3. SEMICLASSICAL TREATMENT OF EIGENVALUE PROBLEMS To obtain the phase F2(q, n) of the wave function for a bound state at a point q, one may integrate along a trajectory, as in (2.3). Except in the case of degeneracy, accidental or intrinsic, this trajectory does not close on itself, i.e., is not periodic. Moser 23 and Arnold 23 have proved an important theorem for celestial mechanics and thereby for the present nonlinear mechanics. Under certain conditions for systems not permitting (or permitting) separation of variables, the motion is quasi- periodic (multiply-periodic) rather than ergodic. That is, the p’s amd q’s can be re- presented as functions of time by Fourier series, e.g., Z aml,. .mN exp(i Z 2nm,vit), where the m’s denote the integers from -co to co and where the coefficients a decrease exponentially with (Irn,l+ .. . +lmN[).23 The contrast between ergodic and quasi- periodic is seen in fig. 1 : the former would occupy the whole space within the line of constant energy, while the other would be more confined spatially. In a nondegenerate system, N is the number of degrees of freedom. The vi are the frequencies of the true angle variables wf for this problem, i.e., those canonically conjugate to the actions, N m l . . .mN i = 1R . A . MARCUS 37 2nnI, the classical counterparts of the quantum numbers. Hamilton’s equations of motion yield (3.1) d(21tnl)/dt = -aH(n)/awi = 0, dw,/dt = ~ H ( ~ I ) / ~ ( ~ z T Z ~ ) = v,(n). Quasi-periodicity implies that there is a canonical transformation from (q, p) to (w, 2nn) and hence a generating function F,(q, n) * for this transformation. In turn, F,(q, n) defines a congruence of trajectories, directed along VF,.Later, F2 will be the phase in a semiclassical wave function, each “ surface ” of constant phase serving as a wave front and the trajectories serving as rays along the normal to the front. FIG. 1 .-“ Box-like ” orbit formed by a simple trajectory in a bound or quasi-bound state. (The actual figure was made for two uncoupled oscillators, using conventional coordinates.) Caustics are AB, BC, CD, DA. The elliptical curve is a constant energy curve, with energy equal to the total energy. In the nondegenerate situation depicted in fig. 1 a single trajectory, for the case of two degrees of freedom, generates four congruences of rays, corresponding to the four possible algebraic signs of the two components of the momentum p, as seen in fig.1 and as emphasized in fig. 2 and (in Appendix) fig. 4. The corresponding VF, (= p) FIG. 2.-Congruences of rays present in fig. 1, each corresponding to a branch (Riemann sheet) of the function p(q). then has four branches (Riemann sheets).20 [When there are N coordinates instead of 2 there are 2N branches.] The rays in fig. 1 do not cross when they are on the same sheet, except at the boundaries, i.e., at the caustics AB, BC, CD, DA. * The symbol F2(q, n) is shorthand notation for F2(q, 2 4 . An analogous remark applies to F2 in Section 2.38 RESONANCES AND BOUND STATES The semiclassical wave function is a linear combination of the solutions, A(q) exp(iF2(q, n)], one per sheet ; F2 is Jp dq.In the several-dimensional wave diffrac- tion literature, the formula for connecting such asymptotic expansion terms is found by assuming a local separation of variables near the caustic and solving the local problem exactly.24 The local solution usually involves the Airy function and its derivative. When this procedure is applied to the present problem in the vicinity of caustics AB and AD, near point A, one obtains (3.2) for the case of two vibrations. (It is easily generalized to N-coordinates, and then has 2N terms.) Each sheet of F2 is described by a Roman numeral : (3.3) (3.4) a denotes 2nn1 and 2nn2 ; Cis a normalization constant. The pre-exponential factors, which are absolute values of the determinants, are equal in the region near A.One sees from (3.4) that congruence II(1V) is related to I(II1) by time-reversal. If one similarly obtains a local solution in the vicinity of D, and uses it to obtain the connection formula.relating the ( A exp iF2)'s, the resulting t,b is similar to (3.2), but with A replaced by D and with a different arrangement of the n/2's, namely 0, 0, -7r/2, n/2 for branches I to IV. Since @ is single-valued these two solutions can differ at most by a multiplicative constant. A further analysis (Appendix) then establishes (3.5). A similar comparison of (3.2) with the t,b resulting from a local solution near B yields (3.6) : 2n(n,+3) =fc, Pdq = /;Pldq+ lP P' Plrrdq (3.6) where 0, 0', P, P' are arbitrary points on their respective caustics, as in fig. 2.It may be emphasized for this derivation that the local separation of variables near a caustic is used only to obtain the connection formulae of the ( A exp iFJs. The global nature of the ( A exp iF2)'s as solutions, for points not near caustics, is the principal tenet (asymptotic expansion) of semiclassical theory. Eqn (3.5) and (3.6) have been obtained earlier by a different argument 2o : each ( A exp iF2) term was assumed to be single-valued and A was allowed to vary on passage through a caustic. (This appears to be a type of phase integral argument.) The present discussion and that in Section 4 avoids a phase integral approximation, though the latter frequently suffices. F2 satisfies the Hamilton-Jacobi partial differential equation and the method of characteristics (classical trajectories) is a convenient method for solving it.However, that method is not the only one. Thus, it should be emphasized that the Jpdq's need not be along dynamical trajectories, that is, along a ray congruence. In fact, in the case of a nondegenerate system the motion is not periodic and so one cannot compute the C1 and C2 path integrals merely by integration along classical trajectories.R. A . MARCUS 39 Instead, one can use an " interpolation '' of the exact trajectory data to close the ends of an open ended path. For example, the jpdq data obtained from the single trajectory can be represented by a Fourier series. One may then use that series to join two ends of a trajectory and so compute $p dq along the two independent paths.Fourier series, suitably chosen to avoid the " small divisor '' problem when necessary, have been extensively used in the astronomical l i t e r a t ~ r e . ~ ~ Fig. 1 describes an orbit which has been termed " box-like " in the computer-simulated stellar dynamics literature.26 Other classes of orbits exist (" shell-like," " tube-like "),26 readily understood physically, and can be analogously treated. The need for supplementing trajectory data by interpolation for calculation of $p dq has been missed, incidentally, by a number of who accordingly but incorrectly insisted on periodic dynamical trajectories. (The classical Feynman propagator, with its usual dynamical associations, was used as a starting point.) Two other approaches should also be recalled : perturbation theory 28 and a proposed mapping of the nonseparable problem onto a separable one.2 Quasi-periodicity implies incidentally, that there exists a canonical transformation for a mapping of the nonseparable one onto a separable one, since the problem in (3.1) has become separable.- ' =+ & 4. SEMICLASSICAL THEORY OF COMPOUND-STATE RESONANCES The description of the quasi-bound (q-b) state, like the bound state, involves caustics. The two states differ, in that the q-b state is connected via (real or complex) trajectories to other states, e.g., to open collision channels. We consider, by way of example, the situation depicted in fig. 3, where AB, BC, DC and DA are caustics bounding the q-b state. The semiclassical wave function for collisions involving compound-state resonances is constructed below so as to satisfy the boundary conditions at R = 03, and in the vicinity of the various caustics.I@ -~ IB,40 RESONANCES A N D BOUND STATES As in Section 3 a solution based on local separation of variables near A, to find the connection formula, yields (3.2). Comparison with a solution determined near D yields (3.5). A local separation of variables is made in a segment near caustic EB, followed by use of the one dimensional solution for the coordinate normal to EB and the two turning-point one dimensional solution l 1 for the curvilinear coordinate parallel to EB. (The two turning points occur on EF and on BC.) When the latter is fitted to the solution (3.2) one obtains the desired connection formula, whence a standing wave solution (q(rt2E) for the q-b state InzE) is found to be <qIn2E) = 231d2Fi/dqda()(exp {iG') + exp (iG"))+ where 21j82F',"/aqdalt(exp (iG"') + exp (iG")) (4.1) GI = F\ +a,, - &bS - &T, G" = 3'; - 6, + &5s + $71 (4.2) GIv = Fiv - a,., + $+s GI11 = FIII 2 + 6,s - 3 4 s , F: = J' p,dq (y = I , .. ., IV). E (4.3) The pr's are related as in (3.4), and so F" = -Pi and Piv= -F!. Thus, (4.1) is a standing wave solution. a,, and c $ ~ are independent of n and m and are discussed later. s denotes n,E. Eqn (4.1) contains two radially ingoing terms (branches I1 and IV) whose sum is denoted by (qln2Bi), and two radially outgoing terms (branches I and 111), whose sum is denoted by (qlnzE9, so that (qln2E> is the sum of these.Each sum is separately normalized to 6(nz-n2)[2n6(E-E)] (and hence to unit radial flux). . We turn now to the $+(q,nE) satisfying the appropriate boundary conditions for a collision. The incident term in (2.1), denoted by (qlnE'), is given by (4.4) using conventional coordinates, (qlnE') = 2-*[la2Pk1'/i3qi3a,J3 exp (iFk"++in) + where 1d2~$1v/aqda,l* exp (iF$IV - tin}] exp (i~"n/2}, (4.4) (4.5) FLY = s' py dq-k,R ( y = 11, IV). BPI B,, is the vibrational turning point in state nE at the given R (fig. 3). In branch I1 the vibrational momentum points toward and in branch IV away from B,,. Similarly, the wavefunction (qlnEf) for an outgoing wave is identical with (4.4)-(4.5) but with I1 and IV replaced by 111 and I, respectively. The unperturbed wave function is the sum (qlnE') +(qlnEf).To construct the desired wave function $+(q, nE) analogous to (2.1) with (4.4) as the incident term, it is necessary to follow the two congruences of rays in (4.4) during the collision until they have become outgoing rays at large R. To do this we match an ingoing ray(s) of (q1.E') with one(s) of (qln,Ei) by finding the stationary phase vaIue of J<nzEilq> dq'(q1nE'). (The q' indicates integration at a fixed large R.) The stationary phasing serves to match a p(q) in InE'} with one in InsEi) and so provide a smooth trajectory to the q-b state. The outgoing ray(s) emerge as <qlnzE'>. Thus, the rays lead from the initial state nE to a congruence in the q-b state. Then, from another congruence in the q-b state they go out to some q at large R.Upon summing the contributions from all q-b states and including the contribution (if any) from anyR. A. MARCUS 41 direct collision trajectories not involved in the caustics of fig. 3 ((qlnE+),) one obtains $+(q, nE) (qlnE+) = (qlnE'> + (qlnE+)d+ (qln2Ef><n2EilnEi>, (4.6) n2 where the first term is the incident term, as in (2.1), and (n,E1lnEi) denotes the stationary phase value of l(n,Eilq) dq' (qlnEi). Comparison of eqn (2.6) and (4.6) shows that -Smn = (mE'lnE+)d + (mEfln2Ef)(n2EilnEi),I (4.7) n2 where (rnEfln2Ef) is obtained by stationary phasing J(rnEfJq> dq'(qln2E?. The latter also serves to match an outgoing ray(s) in <qlnzEf) with one(s) in (qlml?). Eqn (4.1)-(4.5) yield (after some manipulation related to that in eqn (7) of ref. (1 5)) (n2E'lnE') = P n s exp W r s - - + + s - + 4 } 9 (4.8) Pns = 3 c Iiaw;la4+ exp (i[~1;+3(ln+3)n]>, (4.9) where y = 1I.W and F i = P,,dq-k,R = - q dp, ( y = 11, IV), (4.10) where wY, is canonically conjugate to 2nn2, being equal to aFqYp(2~n~), and the inte- gration over py in (4.10) is over the path from B,, to E.s 1- s (mE'InE') = where P i s = and Fj; = The a,, in eqn (4.8) and p,, dq- kmR = - q dp, (7 = I, 111). (4.13) Jt J (4.1 1) is given by and, when EF and BC are real caustics, (4.14) (4.15) C3 is a contour encircling BC and EF. & a function of OS given in ref. (1 l), is close to zero unless the system is near the top of the barrier between BC and EF. 0, is given by a related equation l 1 when BC and EF are complex caustics. The exponential, exp{ 1, in (4.8) can be written as (exp( } - 1)+ 1 and, it can be shown, that stationary phase value of X&pns has the same form as (2.11).It constitutes contribution from the direct collision trajectory involving reflection from caustic EF. If the sum of this contribution and of S:,, is written as Szn, eqn (4.7), (4.8) and (4.1 1) yield (4.16) nz Snl" = - C' flAsPnsKexP W r s - i4.D - 11,42 RESONANCES AND BOUND STATES where the sum over s denotes the sum over n2 (at the given E). If the exponent is expanded about E-E, one obtains l1 S m n = G'm - C P~sPnsrsI(E-E,+ir,/2), (4.17) where the E, are the E's for which nl is an integer (when +s w 0) and where r, is given by eqn (24) of ref. (1 1). When exp( - OJ 4 1, we have (4.18) where vl, is the frequency i3E/i3(271n1) for the mode corresponding to n1 in the q-b state s.iOs can be calculated as the difference of the $p dq overa path (from any nE to any mE) which encircles caustic EF and one over a path which encircles both EF and BC. (The ends of the paths at large R are joined analytically, and care is taken to compare paths on the same ingoing branch y.) To apply eqn (4.16) or (4.17) in a form utilizing exact trajectories it is necessary to obtain the F2 for the q-b state, as deduced from Section 3. From caustic BC one then calculates trajectories leading to the open channels. The usual analytic argu- ments at large R then yield the integrals in /Ins and #&.* S L P w v,sexp{ - OS), 5. DISCUSSION Eqn (4.16) has the appropriate factorization pr~perty,~ one which reflects the " loss of memory " of" indirect " collisions occurring in the q-b state.It also: has the appropriate time-reversal symmetry of S,,, since Bt is a time-reversed j?. When state In2E) couples strongly with some state, the #I for that state will be close to and can be replaced by unity or evaluated by an integral expression (suitably expressed in terms of W's) used to obtain (4.8). The small P's can still be evaluated by the semiclassical expression (4.8) and (4.11). When the collision system is purely elastic, the sum over s reduces to a single term pf& (= l), and the formulation reduces to that l1 for elastic collisions, as it should. One implication of (4.17) for purely classical calculations involving complexes may be noted. When the relevant I'Gs are large, one may anticipate that classical tra- jectories will reasonably well reproduce the quantum dynamical behaviour.When, however, a relevant rs is small one should, at least, exclude from such calculations certain trajectories-those for which an individual vibration associated with the r, has, for any significant time, an energy substantially less than its zero-point energy. Otherwise the calculations could not approximate in this respect the corresponding quantum mechanical behaviour. Some extensions and applications, computational and perturbative, will be des- cribed elsewhere.§ As in the case of direct collision trajectories some partial averag- ing 30 should serve to isolate aspects of the collision which are predominantly quantum mechanical.* E.g., to calculate Pns the analytic expression for py(q) in state nE at large R is used to calculate - jq dp, from Bn to a point on the trajectory leading to BC. The pr(q) data for that trajectory are then employed, and the q-b py(q) data are used to obtain the contribution to reach B. Subtraction of i0J2 then yields the value to reach E. J Eqn (4.3)-(4.13) refer, for simplicity, to a vibration-translation problem, but are readily extended to include rotational and orbital problems by including appropriate generating functions, functions which disappear in the second halves of (4.10) and (4.13). Certain +7r terms, arising from passage through vibrational turning-points, are also omitted for brevity.R. A . MARCUS 43 APPENDIX DERIVATION OF EQN (3.5) A N D (3.6) The path from A to D on any branch y can be deformed to lie along the caustic AD.The component of p normal to AD is zero, and so (fig. 2) pr = prv = --pn = --pru on AD. The integral lpydq from A to D for y = I or IV thus equals that from D to A for y = I1 or 111. This fact is used in the comparison of the solution (3.2) and the corresponding one originating from D to show that the magnitude of the above integral equals n(n2+3), whence f p dq equals 27t(n2+$) if the cyclic path C2 lies along AD. To prove (3.5) one may suitably deform the above path, and to do this it is necessary to remove the multivaluedness of p. While p (and hence VF2) is a function of q, one may introduce a Riemann surface on which it is single-valued.20 As usual the pair of sheets which have the same p at a branch cut (caustic) are joined.When one does this at all caustics, one obtains for the Riemann surface a torus,’O as in fig. 4, where the torus is an open “cylinder”, whose rims are the “ellipse” passing through C and D and the one passing through A and B. A I FIG. 4.-Riemann surface for the system in fig. 1 and 2. The two lines joining D to C coincide spatially, as do the two joining A and B. The congruences in fig. 1 and 2 are indicated, as are the cyclic paths C2 and C1 in eqn (3.5) and (3.6). The momentum p(q) is now a (single-valued) analytic function of q on this surface, and so one may deform the contour C2 for f p dq, which originally went from A to D and back to A along AD, to be any other equivalent cyclic Cz path as indicated in fig.4. Mathematically, this path is equivalent to the second half of (3.5). Eqn (3.6) is obtained similarly. In three dimensions, points A and D lie at the corner of a cube, instead of a square, and AD lies along an edge. Once again all pr’)~ are equal in magnitude on AD, since the com- ponents of p normal to the edge vanish. Thus, the preceding argument can be generalized. For an N-dimensional system, it leads to p dq equal to 2n(ni+3), for the N topologically independent paths Cj. Fig. 2 is, essentially, a deformed square. An equilateral triangle has also been treated, by a different method.20 $ R. A. Marcus, Chem. Phys. Letters, 1970, 7, 525; J. Chem. Phys., 1971, 54, 3965; J. N. L. Connor and R. A. Marcus, J. Chem. Phys., 1971, 55,5636 (Part 11) ; R.A. Marcus, J. Chem. Ptrys., 1972,56,311 (Part 111), 3548 ; J. Chem. Phys., 1972,57,4903 (Part V) ; W. H. Wong and44 RESONANCES AND BOUND STATES R. A. Marcus, J. Chem. Phys., 1971,35,5663 ; J . Stine and R. A. Marcus, Chem. Phys. Letters, 1972,15,536 (Part IV). W. H. Miller, J. Chem. Phys., 1970,53, 1949; 1970,53, 3578; 1970,54,5386; Chem. Phys. Letters, 1970,7,431 ; Acc. Chem. Res., 1971,4,161 ; C. C. Rankin and W. H. Miller, J. Chem. Phys., 1971,55,3150; W. H. Miller and T. F. George, J. Chem. Phys., 1972,56,5668, 5722; J. D. Doll and W. H. Miller, J. Chem. Phys., 1972, 57, 5019. For related studies see P. Pechukas, Phys. Rev., 1969, 181, 166, 174 ; R. D. Levine and B. R. Johnson, Chem. Phys. Letters, 1970, 7, 404; 1971, 8, 501 ; I. C. Percival and D. Richards, J.Phys. B., 1970,3, 315, 1035. e.g., R. D. Levine, Acc. Chem. Res., 1970,3,273 ; 0. K. Rice, Phys. Reo., 1930,35,1538,1551; J. Phys. Chem., 1961, 65, 1588; J. Chem. Phys., 1972, 55,439; D. A. Micha, Chem. Phys. Letters, 1967,1, 139 ; T. F. George and J. Ross, J. Chem. Phys., 1972, 56, 5786 ; R. A. van Santen, J. Chem. Phys., 1972, 57, 5418. F. H. Mies and M. Krauss, J. Chem. Phys., 1966,45,4455 ; F. H. Mies, J. Chem. Phys., 1969, 51,787,798 ; 0. K. Rice, ref. 4. W. B. Miller, S. A. Safron and D. R. Herschbach, Disc. Farday Soc., 1967, 44, 108 ; J. I). McDonald, J. Cheung and D. R. Herschbach, unpublished; J. M. Parson, P. E. Siska and Y. T. Lee, J. Chem. Phys., 1972, 56,4658; Y. T. Lee et al., this Discussion. R. D. Levine and S.-F. Wu, Chem. Phys. Letters, 1971,11, 557; Mol.Phys., 1971, 22, 881. * K. E. Rieckhoff, E. R. Menzel and E. M. Voigt, Phys. Rev. Letters, 1972,28,261; E. R. Menzel, R. E. Rieckhoff and E. M. Voigt, Chem. Phys. Letters, 1972, 13,604. H . Feshbach, Ann. Phys., 1958,5,357 ; 1962,19,287 ; 1967,43,410 ; J. R. Taylor, Scattering Theory (John Wiley and Sons, New York, 1972), Chap. 20. l o M. V. Berry, Proc. Phys. Soc., 1966,%%, 285 ; A. S. Dickinson, Mol. Phys., 1970, 18,441 ; K. W. Ford, D. L. Hill, M. Wakano and J. A. Wheeler, Ann. Phys., 1959, 4,239. I' J. N. L. Connor, Mol. Phys., 1968, 15, 621 and references cited therein. l2 J. N. L. Connor, Mol. Phys., 1968,15, 37; 1970, 19, 65; Chem. Phys. Letters, 1969,4,419. l3 H. Goldstein, Classical Mechanics (Addison Wesley, Reading Mass., 1950), Chap. 8, 9. l4 V. A. Fock, Vest. Leningrad Univ. Ser. Fiz. Khim., 1959,16,67. (Technical Translations, 1960, 4, NO. 60-17464). R. Schiller, Phys. Reu., 1962,125,1109. l6 W. H. Miller, J. Chem. Phys., 1970,53, 1949. l7 R. A. Marcus, J. Chem. Phys., 1966,45,4500 ; 1968,49,2617 ; A. 0. Cohen and R. A. Marcus, J. Chem. Phys., 1968,49,4509; S.-F. Wu and R. A. Marcus, J. Chem. Phys., 1970, 53,4026. l 8 M. Born, Mechanics of the Atom (Ungar, New York, 1960). l9 J. Light, J. Chem. Phys., 1962,36, 1016. 2o J. B. Keller, Ann. Phys., 1958,4, 180; J . B. Keller and S. I. Rubinow, Ann. Phys., 1960,9, 24. J. H. Van Vleck, Proc. Natl. Acad. Sci., 1928, 14, 178. 22 V. P. Maslov, Zhur. Vychisl. Mat. :Mat. Fiz., 1961, 1, 113, 638. (Engl. translation: USSR Compt. Math.-Math. Phys., 1962, 1, 123, 744). *'e.g., see C. L. Siege1 and J. K. Moser, Lectures on Celestial Mechanics (Springer-Verlag, Berlin, 1971), 526, $36 and references cited therein ; J. K. Moser, Mem. Amer. Math. Soc., 1968, No. 81, 1. 24 Yu A. Kravtosov, Izuest-Vuzov Radiofizika, 1964, 7, 664 [Technical Translations 14, No. 12, p. B3, #'IT-65-30131(1965) ; D. Ludwig, Comm. Pure Appl. Math., 1966,19,215 ; 1967,20,103 ; R. M. Lewis, N. Bleistein and D. Ludwig, Comm. Pure Appl. Math., 1967,20,295 ; M. V. Berry, Sci. Progr.,l969,57,43 ; S. V. Khudyakov, Sou. Phys. JETP, 1969,29,507. c.g., Y. Hagihara, CeZestzhI Mechanics (M.I.T. Press, Cambridge, Mass., 1972), Vol. 2, Part 1 a d 2. 26e. g., G. Contopoulos, Int. Astron. Union Symp. No. 25, The Zlzeory of Orbits in the Solar System and in Stellar Dynamics, ed. G. Contopoulos (Academic Press, New York, 1966), p. 3 . 27 M. Gutzwiller, J. Math. Phys., 1971, 12, 343; W. H. Miller, J. Chem. Phys., 1972, 56, 38 ; K. F. Freed, J. Chem. Phys., 1972,56,692. '* J. Doll and W. H. Miller, J. Chem. Phys., 1972, 57, 4428. 29 P. Pechukas, J. Chem. Phys., 1972,57, 5577. 30J. Doll and W. H. Miller, ref. (2).

ISSN:0301-7249    

DOI:10.1039/DC9735500034

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

Partial Averaging in Classical S-Matrix Theory Vibrational Excitation of H2 by He BY W. H. MILLER AND A. W. RACZKOWSKI Inorganic Materials Research Division, Lawrence Berkeley Laboratory and Depart- ment of Chemistry, University of California, Berkeley, California 94720 Received 28th December, 1972 Within the framework of a general semiclassical theory that combines exact classical dynamics and quantum superposition it is shown how a certain averaging procedure allows one to treat some degrees of freedom in a strictly classical sense while others are quantized semiclassically. This enormously simplifies the application of the theory to three-dimensional collision systems and also leads to an interesting formal structure in the theory: the quantum-like degrees of freedom are quantized semiclassically via use of double-ended boundary conditions, while the unquantized classical-like degrees of freedom enter only through a phase space average over their initial coordin- ates and momenta.Preliminary results for vibrational excitation of H2 by He are presented and compared with available quantum mechanical calculations. 1. INTRODUCTION The last several years have seen an increasing use of classical trajectory calcula- tions in describing inelastic and reactive molecular collisions.l# The advantage of classical approaches is that the equations of motion can always be solved (at least numerically) without the necessity of introducing any dynamical approximations, whereas this is generally not the case for a quantum description. The shortcoming of a classical theory is, of course, that real molecules obey quantum rather than classical mechanics.The object of our research in recent years has been to show how exact classical dynamics (i.e., numerically computed trajectories) can be used as input to a general semiclassical theory.3* The principal physical idea is a natural extension of the Ford and Wheeler treatment of potential scattering, namely that one uses a quantum mechanical formulation of the scattering problem (so as to incorporate quantum superposition of probability amplitudes), but evaluates the dynamical parameters of the theory within the classical limit. For a collision between species which possess internal degrees of freedom, the result of this semiclassical theory is a prescription for how one uses classical mechanics to construct the classical-limit of S-matrix elements, the “classical S-matrix ”, which are simply the probability amplitudes for transitions between specific quantum states of the collision partners.In a number of examples it has been seen that this combination of “ classical dynamics plus quantum super- position ” accurately describes the quantum effects in molecular collisions. One of the most practically important aspects of classical S-matrix theory is the ability to analytically continue classical mechanics in such a way as to describe cZnssicaZZy forbidden processes, i.e., those which do not take place via ordinary classical dynamics ; Section 2 discusses such processes in detail. This paper describes a feature of the theory that is particularly useful when some of the internal degrees of freedom are very classical-like, but others are highly quantized.This is common in an atom-diatom collision, for example, where there is typically a large number of 4546 CLASSICAL S-MATRIX THEORY rotational states that are strongly coupled and may thus be treated by strictly classical methods, but only a few vibrational states are involved so that this degree of freedom is highly quantum-like. The " partial averaging " approach allows one to use a strictly classical Monte Carlo treatment of the classical-like degrees of freedom while the quantum-like degrees of freedom are quantized semiclassically-all without introducing any dynamical approximations into the theory. Preliminary results for vibrational excitation of H2 by He are presented in Section 3.2. SUMMARY OF THE THEORY A. CLASSICALLY ALLOWED TRANSITIONS All formulae in this paper will be written explicitly for nonreactive collisions of atom A and diatomic molecule BC. The cross section for collisional excitation of BC from initial vibration-rotation state (nl, j , ) to final state (n2, j2), summed and averaged over the m-components of the rotational states of BC, is given by where Sn2j2z2,nljlll(J, E) is the S-matrix element for the A+BC collision system ; El is the initial collision energy, kf = 2pE1/A2, E is the total energy, J is the total angular momentum, and 2 is the orbital angular momentum for relative motion of A and BC. The classical limit of a particular S-matrix element is given by where 4(n2j2Z2, nljlZl) is the classical action along the classical trajectory that is determined by the indicated double-ended boundary conditions (units are used such that PZ = 1); specifically, where (R,P) are the translational coordinates and momenta for radial motion of A and BC, and (qn, n) (qj,j), (qz, I ) are the action-angle variables for these degrees of freedom ; the initial and final values, (nljlZ,) and (n2 j2Z2), are required semiclassically to be integers.The pre-exponential factor in eqn (2) is the Jacobian relating the final values (n2j2Z2) to the conjugate initial values (qnl, qjl, qJ which lead to these specific final values ; i.e., with (nljlZl) fixed, one varies (qnlqjlqll) to cause (n2jJ2) to take on their desired final integer values.Typically, however, there is more than one classical trajectory that satisfies these double-ended boundary conditions; eqn (2) is then a sum of similar terms, one for each such trajectory. In forming the square modulus of the S-matrix element as it appears in eqn (1) interference terms thus result. In the co-linear A+BC collision these interference effects are quite prominent (and are accurately described by classical S-matrix theory), but it has been noted that the sums that occur in eqn (1) diminish their effect for a three dimensional A + BC system ; i.e., the interference terms are quenched. If the interference terms are neglected, then it is easy to see that eqn (1) and (2) giveW . H . MILLER A N D A . W . RACZKOWSKI 47 where it is assumed that enough integer values of J, ZI, and 1, contribute to justify replacing the sums over them by integrals.If many integer values of n, and j , are accessible from the initial state nljl, then it is also convenient to average eqn (4) over a quantum number increment about n2 and j , : where the limits of the j , and n, integral are the integers plus and minus 3. In eqn (9, one can now change integration variables from (n2j212) to (qnlq,lqzl) ; this elimin- ates the Jacobian from the integrand, giving J' d(qn1/2n) J' d(q j1/2n> J' d(q,1/2n)(l) (2.6) where the limits of the integral over qnl,qjl, and qzl are values such that the final values of n and j are in the increment (n, +$, n2 -3) and ( j , +*, j , -+), respectively. The simplest way to evaluate eqn (2.6) is to sweep qnl, qjl, and qzl through their complete domains (0,2n), putting the outcome of each trajectory into a quantum number " box " labeled by the closest integer value of the final values of n and j ; this is essentially what is done in a standard Monte-Carlo calculation for this type of quantity.If interference eflects are neglected, therefore, classical S-matrix theory for classically allowed processes reduces to standard Monte-Carlo methods. As an aside regarding the above strictly classical expression, it is actually more consistent if one also averages the cross section over a quantum number width of the initial quantum numbers n, and j , ; this gives the form of a complete phase space average over initial conditions ; the limits of the nl andj, integrals are the initial integer values plus and minus -5.This latter expres- sion, which treats initial and final states on an equal footing, satisfies microscopic reversi bility . B . CLASSICALLY FORBIDDEN TRANSITIONS In some cases there may be no classical trajectories (at the given energy) that con- nect the specific initial and final states (nljl) and (n&) ; the transition is then said to be classicdy forbidden which in practice simply means that the process is " weak ", i.e., has a small transition probability. Vibrationally inelastic transitions in low energy collisions of light diatomics with atoms are usually such processes. Other important examples of classically forbidden processes are tunnelling in reactive systems that have activation barriers and electronic transitions between different adiabatic electronic states .Although there are no ordinary classical trajectories that contribute to these48 CLASS~CAL S-MATRIX THEORY processes, it is in general possible to analytically continue classical mechanics and h d complex-vdued trajectories that do so. This can actually be accomplished by inte- grating the equations of motion with complex initial conditions and with a complex time variable. Along such complex-valued trajectories the action integral 4 is complex, so that the S-matrix element in eqa (2.2) has an exponential damping factor, exp( - Im4) ; classically forbidden processes are thus a generalization of the concept of tunneling in one-dimensional systems. Just as for classically allowed processes, there may be several different classical trajectories (complex-valued oms) that contribute to the specific S-matrix element in eqn (2.2) ; because of the sums that appear in eqn (2.l), however, it is still reasonable in most cases to disregard interference between these different trajectories. Further- more, even though the nl j , -tn& transition is classically forbidden for n, # n,, it will typically be true that many different j2 values have comparable probability. In such cases it is thus possible to average over a quantum number width ofj, as in Section 2A (replacing sums by integrals) but not far n2. Changing from integration over final values to integration over initial values as in Section 2A, thus gives where PnZcnl is essentially a one dimensional-like vibrational transition probability that depends parametrically on the initial conditions of the other degrees of freedom and is calculated by holding the initial conditions (Zljl, qllqjl) constant while qnl is varied to make n, equal to the desired integer value.The important practical advantage in this " partial averaging " scheme is that one must deal with double-ended boundary conditions (through a root search procedure) only for the vibrational degree of freedom, the one that is being quantized semiclassically, with the orbital and rotational degrees of freedom entering only through a phase space average over their initial conditions. The four dimensional integral in eqn (2.7) can now be evaluated by Monte Carlo methods. Furthermore, one can obtain all partial cross sections-i.e., the distri- bution in final rotational quantum number j2 and/or the distribution in scattering angle (the differential cross section)-in the usual Monte Carlo fashion by assigning the numerical value of the integrand in eqn (2.7) to the appropriate " box " labeled by j2 and scattering angle. In summary, the only approximations involved in eqn (2.7) and (2.8) beyond classical S-matrix theory itself are (i) neglect of interference between different traject- ories that lead to the same final values of Z2 and j,, and (ii) repIacement of sums over integer values of Z2 and j , by integrals. As has bem noted, the interference terms would essentially average to zero even if they were included, and one only needs a few integer values of l2 and j , to justify replacing the sums by integrals.The import- ant practical advantage of this partial averaging approach is that double-ended boundary conditions (and the related root search) are required only for the quantized degrees of freedom (i.e., vibration), while the other (unquantized) degrees of freedom enter only through a phase space average over their initial conditions.W. H , MILLER AND A . W. RACZKOWSKI 49 3. RESULTS FOR He+H2 COLLISIONS Calculations based on eqn (2.7) and (2.8) are being carried out for the He+H, collision system and its isotopic variants. The interaction potential is that of Gordon and Secrest,6 and the H2 potential is the accurate fit of Waech and Bernstein to the Kolos-Wolniewicz potential, Fig. 1 shows our preliminary results for the total 0- 1 vibrationally inelastic cross section as a function of the initial collision energy ; i.e., the quantity shown is 0 1 +oo(E,) = C 0 1 j z + O d E l ) * (3.1) i 2 Although these results are not final, they should be the correct semiclassical values to within at least a factor of 2.At 5 eV collision energy the O-, 1 transition is still classically forbidden ; i.e., with El = 5 eV, n, = j , = 0, there are no values of J, ZI, qzl, qjl for which a real-valued classical trajectory leads to n2 = 1. El lev FIG. 1.-The cross section for excitation of Hz (by collision with He) from its ground state (nl = jl = 0) to its first excited vibrational state (n2 = l), summed over all final rotational states as a function of initial collision energy ; the quantity shown is defined by eqn (3.1) and (2.1).The arrow indicates the energetic threshold for this transition. FIG. a t - \ 1 $ 4 1 0 1 \ \ -I - 0 2 4 G B j 2 2.-The distribution in final rotational states j 2 that accompany the O - t l vibrational in He + H2 collisions, u1 j z + O O , as defined by eqn (2.1). The initial collision energy is excitation 3 eV.50 CLASSICAL S-MATRIX THEORY Fremerey and Toennies have recently carried out coupled channel (i.e., quantum mechanical) calculations for this system and find a value of - 1.0 x A2 for the total 0+1 cross section at Eo = 1.09 eV. Within the uncertainty in our preliminary semiclassical results, therefore, there is excellent agreement with this quantum mechanical value. Fig. 2 shows the quantity ~ l j z + O O as a function ofj, at El = 3 eV collision energy ; i.e., this is the distribution in final rotational state that accompanies vibrational excitation from the ground state.Although j , = 0 is the single most probable final rotational state, there is a significant amount of rotational excitation which accom- panies the O+ 1 vibrational excitation. Further calculations for this system are in progress and more details of the calcula- tional procedure will be presented in a later report. This work has been supported by the National Science Foundation and the U.S. Atomic Energy Commission. Appreciation is also expressed to Dr. C. Sloan for helpful discussions. D. L. Bunker, Meth. Comp. Phys., 1971, 10,287. J. C. Polanyi, Acc. Chem. Res., 1972, 5, 161. (a) W. H. Miller, J. Chem. Phys., 1970, 53, 1949 ; (b) 1970, 53, 3578 ; (c) Chem. Phys. Letters, 1970,7,431; (d) J . Chem. Phys., 1971,54,5386 ; (e) C. C. Rankin and W. H. Miller, J. Chem. Phys., 1971,55,3150 ; (f) W. H. Miller and T. F. George, 1972,56,5668 ; (9) T. F. George and W. H. Miller, J. Chem. Phys., 1972,56,5722 ; 1972,57,2458 ; (h) J. D. Doll and W. H. Miller, J. Chem. Phys., 1972,57,5019. See also the related sequence of papers by Marcus et a!., (a) R. A. Marcus, Chem. Phys. Letters, 1970,7,525 ; (b) R. A. Marcus, J. Chem. Phys., 1971,54,3965 ; (c) J. N. L. Connor and R. A. Marcus, J. Chem. Phys., 1971,55, 5636 ; (d) W. H. Wong and R. A. Marcus, J. Chem. Phys., 1971,55,5636 ; (e) R. A. Marcus, J. Chem. Phys., 1972,56,311; (f) R. A. Marcus, J. Chem. Phys., 1972,56,3548 ; (9) J. Stine and R. A. Marcus, Chem. Phys. Letters, 1972,15, 536 ; (h) R. A. Marcus, J. Chem. Phys., 1972,57,4903. K. W. Ford and J. A. Wheeler, Ann. Phys. (N.Y.), 1959,7,259,287. M. D. Gordon and D. Secrest, J . Chem. Phys., 1970,52, 120. H. Fremerey and J. P. Toennies, private communication. ’ T. G. Waech and R. B. Bernstein, J. Chem. Phys., 1967,46,4905.

ISSN:0301-7249    

DOI:10.1039/DC9735500045

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年代:1973

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Multidimensional Canonical Integrals for the Asymptotic Evaluation of the S-Matrix in Semiclassical Collision Theory BY J. N. L. CONNOR Dept. of Chemistry, University of Manchester, Manchester M13 9PL Receiced 29th December, 1972 The uniform asymptotic evaluation of multidimensional integrals for the S-matrix in semiclassical collision theory is considered. A concrete example of a non-separable two-dimensional integral with four coalescing saddle points is chosen since it exhibits many of the features of more general cases. It is shown how a uniform asymptotic approximation can be obtained in terms of a non- separable two-dimensional canonical integral and its derivatives. This non-separable two-dimen- sional canonical integral plays a similar role to the Airy integral in one-dimensional integrals with two coalescing saddle points.The uniform approximation is obtained by applying to the two- dimensional case the asymptotic techniques introduced by Chester et al. for one-dimensional integrals. An exact series representation is obtained for the canonical integral by means of complex variable techniques. The series representation can be used to evaluate the canonical integral for small to moderate values of its arguments, whilst for large values of its arguments existing asymptotic tech- niques may be used. 1 . INTRODUCTION An important goal in the semiclassical theory of inelastic and reactive molecular collisions is the derivation of integral representations for the S-rnatri~.’-~ The integral representations may often be evaluated by asymptotic techniques, in which certain canonical integrals play an important role.Consider, for example, the one-dimensional integral 11 = @x) exp ; x)ldx, (1.1) which arises in the semiclassical analysis of the non reactive collinear atom + diatom collision problem.’b* lC* 2c* 2d The main contribution to the integral (1.1) arises from the saddle points of f(a; x), that is, the points at which af(a; x)/dx = 0. Physically, each saddle point is associated with a real or complex classical trajectory, and the parameter a corresponds to the final quantum number of the transition (the initial quantum number is fixed by the initial conditions). The position of the saddle points depends on the parameter a and for a certain value of a, two saddle points may coincide.For this situation, the asymptotic method of Chester et al.’ provides a uniform approximation for the integral (1.1). The first two terms of this uniform asymptotic expansion were derived by the author and Marcus,2C* and in the case where f”(a ; xl)> 0 and f”(a ; x,) < O when the saddle points x1 and x2 are real, are given by52 MULTIDIMENSIONAL CANONICAL INTEGRALS The subscripts indicate that g(x) andf"(a ; x ) are to be evaluated at the appropriate saddle point, and Ai( -5) and Ai'( -[) are the Airy function and its derivative respec- t i ~ e l y . ~ The formula (1.2), used in conjunction with real or complex trajectories,2g accurately reproduces the exact results of Secrest and Johnson8 over a range of ;s loo to = 10-l'. In the one-dimensional example described above, the Airy integral is the canonical integral, and the usefulness of eqn (1.2) and (1.3) is dependent on the fact that the properties of the Airy integral are known and its values are tab~lated.~ When the S-matrix is represented by a multidimensional integral, the derivation of uniform asymptotic expansions is more difficult.For certain special cases the Airy integral is again the canonical integral, for example, when the phase is separable 2h or in the case of two nearly coincident saddle points in n-dimensions.6 In general, however, an n-dimensional integral requires an n-dimensional canonical integral for its uniform asymptotic evaluation. Very little work has been done on this problem of obtaining uniform asymptotic expansions or of investigating the properties of multidimensional canonical integral^.^" This present paper is devoted to this problem.A concrete example of a two-dimensional integral with four nearly coin- cident saddle points is studied. This example is typical of more general ones. Uniform approximations were introduced into the semiclassical theory of elastic scattering by Berry.g In Section 2 it is shown how a uniform asymptotic approximation for this two- dimensional integral with four nearly coincident saddle points may be obtained in terms of the non-separable canonical integral aJ (1.4) 1 w, 5, q) = - (27c)2 1; oo du du exp [i(3u3 + i u 3 + 5u + 5u + quo)], -a3 and its derivatives. This is achieved by applying to the two-dimensional integral the techniques introduced by Chester et aL5 for one-dimensional integrals.An exact series representation for the canonical integral (1.4) is obtained in Section 3 with the help of complex variable techniques. The series can be written as the sum of products of derivatives of the Airy function. For small to moderate values of the parameters c, c, and q, the series can be summed numerically to obtain an exact value of U((, 5, q). For large values of these parameters, existing asymptotic techniques (e.g., the ordinary saddle point or stationary phase method) can be used. Finally, Section 4 discusses the general n-dimensional case and points out some of the difficulties that are encountered as the number of dimensions increases. 2. UNIFORM ASYMPTOTIC EVALUATION OF A TWO-DIMENSIONAL INTEGRAL WITH FOUR NEARLY COINCIDENT SADDLE POINTS The integral under consideration is J - c o J - a ~ The functionf(x, y ) depends on two parameters a = (a1, u2), but they have not been indicated explicitly in order to simplify the notation.For an atom+rigid rotator collision, these two parameters correspond to the final rotational quantum number and final relative orbital angular momentum quantum number.'" It is assumed that the integral (2.1) possesses four saddle points (xi, y l ) with i = 1, 2, 3, 4, which for a certain value of the parameters coalesce or come close together at (xo, yo).J . N. L. CONNOR 53 2.1 NON-UNIFORM ASYMPTOTIC APPROXIMATIONS Suppose that in the vicinity of the point (xo,yo) where the four saddle points coalesce or come close together, f ( x , y ) and g(x, y ) have the Taylor expansions Ax, Y ) = f(x0, Yo) +fx(xo, Yaw- xo) +fv(xo, Yo)(Y -Yo) +fxY(XO, Yo>(x-xo)(Y -Yo) + 4fucx(xo, YO>(X - xol3 + 4fyyY(xo, Y O X Y (2.2) and g(x, Y ) (2.3) Then it is clear that in this approximation the integral (2.1) can be expressed in terms of U (eqn (1.4)) and its derivatives Us, Uc and U,,.This is achieved by a change of origin and normalisation of the highest order terms in the expansions (2.2) and (2.3). This approximation is, however, only valid when all four saddle points are close together. Note that eqn (2.2) is not the only Taylor series expansion that is possible for four coalescing saddle points. For each of the remaining cases, a uniform asymptotic approximation expressed in terms of a different canonical integral can be obtained by applying the techniques described in this paper.On the other hand, when the saddle points coalesce in pairs but with the pairs well separated, the asymptotic approximation to the integral (2.1) is d x o , Yo) +gx(xo, Yo)(=--Xd +gy(xo, Yo)(Y -Yo) + gxAx0, Yo>@ - Xo)(Y -Yo). where and 4x61 = f 2 - f 1 , 4 x t 3 = f 4 - j . 3 - The saddle points have been labelled so that ( x , , y , ) coalesces with ( x 2 , y 2 ) and (x3, y 3 ) coalesces with (x4, y4). When 9W*)( - xij) is replaced by its asymptotic values of zero or unity, eqn (2.4) becomes 9. A t I2 - 2xi C 2 exp (if;.), the ordinary saddle point or stationary phase resu1t,lo and where the sum is over all contributing saddle points. The asymptotic expansion (2.6) is valid when all four saddle points are well separated.The Taylor expansions (2.2) and (2.3) are valid when all four saddle points lie close together, and eqn (2.4) is valid when the saddle points coalesce in pairs, but with the pairs well separated. The problem now is to obtain a unifarm asymptotic expansion which reduces to the non-uniform results of eqn (2.2)-(2.6) in the appro- priate limits. This is done in the next subsection.54 MULTIDIMENSIONAL CANONICAL INTEGRALS 2.2 UNIFORM ASYMPTOTIC APPROXIMATION A uniform asymptotic approximation is obtained for the integral (2.1) by applying to the two-dimensional case the procedure used by Chester et aL5 for one-dimensional integrals. We map f ( x , y ) onto a cubic polynomial in the variables u and v : f(x, y ) = ~ ~ ~ + ~ v ~ + ~ u + ~ u + q u v + A .(2.7) Notice that eqn (2.7) is an exact transformation and not the first few terms of a Taylor series expansion (as in eqn (2.2)). The transformation (2.7) is one-to-one if the saddle points on either side of (2.7) correspond The saddle points (ui, v,) are to be labelled so that their distribution in the (u, u) plane corresponds as closely as possible with the distribution of saddle points in the (x, y) plane. The saddle points are determined by the solutions of the equations and Except in the separable case where q- = 0, the solutions u1 = u,((, r, q) and vi = ul([, r, q-) are too complicated in form to be written down explicitly, but this is of no consequence for the development of the theory given below. u4 + 2[u2 + q3u +p + t q 2 = 0, u4 + 2tV2 + q3v + t2 +[q2 = 0.If the saddle points (2.8) are substituted into eqn (2.7), there results f(xi, yi) = 31413 + + I : + C U ~ + t ~ i + quivi + A, i = 1,2, 3,4. (2-9) Equation (2.9) represents four non-linear equations in the unknowns (, 5, q and A. These equations can be solved (in principle at least) for 5, e, q and A in terms of the saddle points off(x, y). The integral (2.1) now becomes I 2 = 1 du 1 O0 dvg(u, v)J(u, v ) exp [i(4u3 + 3v3 + [u + t v + quv + A)], (2.10) where J(u, v ) is the Jacobian of the transformation (2.7). It is not difficult to show by means of the chain rule that at a saddle point the Jacobian satisfies the equation -03 (fuufvv - f 3 u i , vi> = J2(ui, U i U x x j y y - .L$Xxi, Yi), 4uivi - q2 = J2(ui, VXfxxfyy-.f~)(xi, Yi).or with the help of eqn (2.7), (2.11) (2.12) The first four terms of the asymptotic expansion for I2 are obtained by writing g(u, v)J(u, u) = p+qu+rv+suu. If the saddle points (2.8) are inserted into eqn (2.12), there results g(ui, vi)J(Ui, ~ r ) = P+qui+rvi +SUiUiy i = 1, 2, 3,4, (2.13) which represent four hear equations and these can be solved (by Cramer's Rule for example) for the unknowns p , q, r and s. If the expansion (2.12) is substituted into the integral (2. lo), it follows that I, - (271)2(p U - iqUc - ir U, - isU,,} exp (iA), (2.14)J . N . L . CONNOR 55 where is the canonical integral defined by eqn (1.4). Eqn (2.14) is the uniform asymptotic expansion for four nearly coincident saddle points that we have been seeking. The quantities p , q, r and s are found from eqn (2.13) and 5, c, q and A from eqn (2.9).It is not too difficult to verify that the uniform asymptotic approximation (2.14) reduces to the non-uniform results (2.4)-(2.6). This is done by replacing U and its derivatives by their appropriate asymptotic expansions (found by application of eqn (2.4) and (2.6) to the canonical integral (1.4)), and using eqn (2.11) and (2.13). As this is a straightforward calculation, details of it will be omitted. Iff@, y ) and g(x, y ) are expanded in the Taylor series (2.2) and (2.3), this allows [, 5, q-, and A in eqn (2.7) and p , q, r and s in eqn (2.12) to be identified in terms of the Taylor coefficients. It is then readily shown that the uniform asymptotic approxi- mation (2.14) reduces to the non-uniform result contained in eqn (2.2) and (2.3) for four nearly coincident saddle points. u = w, 5, v ) 3.SERIES REPRESENTATION FOR THE CANONICAL INTEGRAL In the previous section it was shown how a uniform asymptotic approximation (eqn (2.14)) for a two-dimensional integral with four coalescing saddle points could be expressed in terms of the canonical integral (1.4) and its derivatives. The problem now is to evaluate the canonical integral. When the saddle points are well separated or coalesce in pairs with the pairs well separated, the asymptotic formulae (2.4) and (2.6) can be applied to the canonical integral. When however, the saddle points lie close together, corresponding to the Taylor expansion (2.2), exact values of the canonical integral are required.In this section an exact series representation for the canonical integral is obtained with the help of complex variable techniques similar to those used in obtaining a series representation for the Airy integral. l1 The canonical integral (1.4) is first transformed into the form O3 1: du 1: dv exp [i f(u, -v)] + 1 " m, r, v ) = (-{Io du 1 dv exp [if(., v)l+ 0 1: du 1: dv exp [if( - u, v)] + 1: du 103 dv exp [if(-., -v)]), (3.1) 0 where now f(u, v) denotes f(u, v) = +u3 + 4v3 + 5u + c v 4- quv. Consider in the first term of eqn (3.1), the integral over u. Since the integrand contains no singularities, a contour integral along the closed contour O+R+R exp (in/6)+0 will be zero by Cauchy's theorem.12 The contribution along the arc R to R exp (in/6) can be shown to tend to zero as R+oo by Jordan's Lemma.If a similar transformation is applied to the integral over v, the first term in eqn (3.1) becomes = exp (in/3)G([ exp (i2n/3), 5 exp (i2n/3), q exp (i5n/6)), where G(a, p, r) = -2 1; d x J r dyexp [-3x3-~y3+~x+By+yxy]. (3.2) ( 2 456 MULTIDIMENSIONAL CANONICAL INTEGRALS A similar analysis can be applied to the remaining terms in eqn (3.1) except that where the integrand is of the form exp (-i3u3), the closed contour O+R+ R exp (-in/6)+0 is used instead. The canonical integral becomes u(C, 5 , q ) = exp (i71/3)G(5 exp (i2n/3), t exp (i2n/3), 11 exp (iSn/6))+ G(C exp (i2n/3), t exp (- 2?1/3), - iq) + G(C exp ( - i2n/3), t exp (i2n/3), - iq) + exp (-in/3)G(C: exp (-i21t/3), < exp (-i2n/3), q exp (in/6)). (3.3) In the integral (3.2) the cross term can now be expanded in a power series and the series integrated term by term to give 1 " ...n where K(a, n) = exp (ax - +x3)xn dx.(3.5) 1: If eax in eqn (3.5) is also expanded in a power series and integrated term by term, it is found that If the expansions (3.4) and (3.6) are substituted into eqn (3.3), then By recognizing that the series (3.7) can be written in the form O3 q" d"Ai([) d"Ai(C) U((, 5, q) = c i-" - - ___ . n=O n! d r dt" Eqn (3.7) and (3.8) are the series representations of the canonical integral we have been seeking. Numerical summation of these series can be used to obtain exact values of the canonical integral for small to moderate values of c, and q. Because of the relation Ai"(r) = cAi(5) the series (3.8) can be written in terms of the Airy function and its first derivative alone, with higher derivatives absent.The result (3.8) can also be derived from eqn (3.3) and (3.4) be recognizing that K(a, n) is the nth derivative (to within a constant factor) of the integral 1 J" exp (ax-+x3) dx, 2zi which is defined and whose properties are discussed by Jeffreys and Jeffreys. AnJ . N. L. CONNOR 57 alternative derivation of the results (3.7) and (3.8) can be given using convergence factors,13 a method which avoids the complex variable techniques used in the present paper. It can be checked that the series converges for all q, C, and f. 4. DISCUSSION The uniform asymptotic approximation (2.14) is derived from the mapping (2.7) which in turn is suggested by the Taylor series expansion (2.2).The analysis is not restricted to this special case however. When a different Taylor series expansion for four coalescing saddle points is valid, similar techniques can be applied to obtain a uniform asymptotic approximation in terms of a different canonical integral. It is also clear that the techniques described in this paper can be applied to higher dimensional integrals. As the number of dimensions increases, however, so do the number of possible canonical integrals. It is necessary to examine the Taylor expan- sion in the neighbourhood of those points where a number of saddle points coalesce to ensure that the correct canonical form is chosen. If an incorrect choice is made, the remainder term in the asymptotic expansion may become of the same order of magnitude as that of the leading terms, or worse, the remainder term may overwhelm the leading terms.The latter can easily happen, for example in the case of a classically inaccessible transition when the contribution from the leading terms is exponentially small. l (a) W. H. Miller, J. Chem. Phys., 1970, 53, 1949 ; (b) 1970, 53, 3588 ; (c) Chem. Phys. Letters, 1970,7,431; (d) Acc. Chem. Res., 1971, 4, 161 ; (e) J. Chem. Phys., 1971,54,5386 ; (f) C. C Rankin and W. H. Miller, J. Chem. Phys., 1971, 55, 3150 ; (g) W. H. Miller, J. Chem. Phys., 1972,56, 38 ; (h) 1972, 56,745 ; (i) W. H. Miller and T. F. George, J. Chem. Phys., 1972, 56, 5637 ; ( j ) 1972,56, 5668 ; (k) T. F. George and W. H. Miller, J. Chem. Phys., 1972,56,5722. ; ( I ) 1972, 57, 2458; J.D. Doll and W. H. Miller, J. Chem. Phys., 1972 57, 5019. (a) R. A. Marcus, Chem. Phys. Letters, 1970, 7, 525 ; (b) J. Chem. Phys., 1971, 54, 3965 ; (c) J. N. L. Connor and R. A. Marcus, J. Chem. Phys., 1971,55,5636 ; (d) W. H. Wong and R. A. Marcus, J. Chem. Phys., 1971, 55, 5663 ; (e) R. A. Marcus, J. Chem. Phys., 1972, 56, 311 ; (f) 1972,56,3548 ; (g) J. Stine and R. A. Marcus, Chem. Phys. Letters, 1972,15,536 ; (h) R. A Marcus, J. Chem. Phys., 1972,57,4903. The asymptotic formula derived in this reference for a two dimensional integral with four coalescing saddle points (and a similar one given by Miller in ref. (l(e))) is only valid for a separable f(x,y) = f'(x)+f&) when all four saddle points are close together. A detailed discussion is given in ref.(13). (a) R. D. Levine and B. R. Johnson, Chem. Phys. Letters, 1970, 7, 404; (6) 1971, 8, 501 ; (c) R. D. Levine, Mol. Phys., 1971,22,497 ; (d) J. Chem. Phys., 1972,56,1633. (a) P. Pechukas, Phys. Rev., 1969,181, 166, 174; (b) J. C. Y. Chen and K. M. Watson, Phys. Rev., 1969,188,236 ; (c) R. E. Olson and F. T. Smith, Phys. Rev. A, 1971,3,1607 ; (d)B. C. Eu, J. Chem. Phys., 1972, 57, 2531 ; (e) M. D. Pattengill, C. F. Curtiss and R. B. Bernstein, J. Chem. Phys., 1971, 54, 2197; cf) D. R. Bates, Comment Atom Mol. Phys., 1971, 3, 23; (g) D. Richards, J. Phys. B., 1972, 5, L58; (h) K. F. Freed, J. Chem. Phys., 1972, 56, 692; (i) P. Pechukas and J. P. Davis, J. Chem. Phys., 1972, 56,4970; (j) R. A. La Budde, Chem. Phys. Letters, 1972, 13, 154 ; (k) M. A. Wartell and R. J. Cross, J. Chem. Phys., 1971,55,4983. ( I ) J. B. Delos and W. R. Thorson, Phys. Rev. A, 1972, 6,720. (a) C. Chester, B. Friedman and F. Ursell, Proc. Camb. Phil. Soc., 1957,53,599 ; (6) B. Friedman J. Soc. Ind. Appl. Math., 1959, 7, 280; (c) F. Ursell, Proc. Camb. Phil. SOC., 1965, 61, 113; (d) 1972,72,49. J. N. L. Connor, Mol. Phys., 1973, 25, 181. H. A. Antosiewicz, Natl. Bur. Std. (US.) Appl. Math. Ser., 1964, 55, 446$ edited by M. Abramowitz and I. A. Stegun. (a) M. V. Berry, Proc. Phys. Soc., 1966, 89,479 ; (b) J. Phys. B., 1969, 2, 381 ; (c) M. V. Berry and K. E. Mount, Rep. Prog. Phys., 1972,35, 315. lo (a) B. Nagel, Arkiv Fys., 1964,27,181 ; (b) M. Born and E. Wolf, Principles of Optics (Pergamon London, 1965, 3rd revised edn.) Appendix 111, pp. 753-754; (c) N. Chako, J. Inst. Math. * D. Secrest and B. R. Johnson, J. Chem. Phys., 1966,45,4556.58 MULTIDIMENSIONAL CANONICAL INTEGRALS Applics., 1965, 1, 372 ; ( d ) R. M. Lewis, Asymptotic Solutions oj’Diferentia1 Equations and their Applications, (1964, Publication No. 13 of the Mathematics Research Center, U.S. Army, University of Wisconsin, edited by C. H. Wilcox, Wiley), pp. 104-106 ; (e) D. S. Jones, General- ized Functions, (McGraw-Hill, N.Y., 1966), p. 344. l 1 H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge Univ. Press, 3rd Edn., 1956), p. 508. l 2 V. I. Smirnov, A Course ofHigher Mathematics, Vol. 111, Part 2, (Pergamon, London, 1964), Chap IV. l 3 J . N. L. Connor, Mol. Phys., 1973, in press.

ISSN:0301-7249    

DOI:10.1039/DC9735500051

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年代:1973

数据来源: RSC

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GENERAL DISCUSSION Dr. G. D. Barg, Dr. H. Fremerey and Prof. J. P. Toennies (Gottingen) said: To examine the validity of classical mechanics in calculating differential inelastic cross sections we have compared the results of classical trajectory and close coupling calculations of rotational excitation of He-H, ( j = 0) at E,, = 1.09 eV. The po- tential hypersurface used included the repulsive potential model of Krauss and Mies l with the Gordon and Secrest parameters,2 to which was added the long range aniso- tropic potential of Victor and Dalgarn~.~ A Morse vibrotor potential was used to describe the H2 molecule. 2000 trajectories were used in the Monte Carlo calculation and a n = 0, j = 0, 2,4 and n = 1, j = 0, 2,4 basis set (18 channels) was used in the close coupling calculation.In the case of the j = 0-2 differential cross section, the close coupling calculation shows a gradual drop off from a maximum of da/dw E 6 x lo-' t f 2 sr-' at 8,, = 0" to 8 x t f 2 sr-l at 180". The classical cross section is zero for 8,, < 30" and then rises up sharply to the quantum mechanical value. Agreement is good at larger angles, except at 180" where the classical cross section drops off somewhat. For the j = 0+4 cross section a similar behaviour is observed. The classical cross section vanishes below 90" and good agreement is achieved at larger angles. This comparison suggests that whereas the classical torque in the weak potential well ( E - 1 meV) is apparently too small to produce a transition, the wave mechanical uncertainty makes such a transition possible.This picture is consistent with the case of Li+-H2 (E-0.25 eV) where the agreement between quantum and classical mech- anics is good down to small angles (8,, - 5"). Dr. G. G.Balint-Kurti (University ofBristoZ) and Dr. B.R. Johnson (Ohio State Univer- sity) said : We would like to comment on a method of extending the range of validity of the distorted wave Born (DWB) approximation beyond that mentioned by Gordon. The conservation of particle flux in nonrelativistic quantum mechanics requires that the S matrix be unitary. In most perturbation approximations such as the Born and DWB approximations this property of unitarity is destroyed. A unitary matrix S may be written in the form S = (1) where A is a Hermitian matrix. The Born and DWB approximations both expand S directly in a perturbation series, using the strength of the potential as a perturbation parameter.If, instead, however, the matrix A is expanded in a perturbation series, then the resulting S matrix will always be unitary, so long as the approximate A matrix is Hermitian.'-' It can further be shown, by comparing the Born series for the S matrix with the expansion of elA, that evaluating A to first order only in the poten- tial is equivalent to including a part of all orders in the Born series.8 Thus the M. Krauss and F. H. Mies, J. Chem. Phys., 1965,42,2703. M. D. Gordon and D. Secrest, J. Chem. Phys., 1970,52, 120. G. A. Victor and A. Dalgarno, J. Chem. Phys., 1970,53, 1316. Comment in this Discussion by G. M. Kendall and J. P. Toennies.R. D. Levine and G. G. Baht-Kurti, Chem. Phys. Letters, 1970, 6, 101. G. G. Balint-Kurti and R. D. Levine, Chem. fhys. Letters, 1970, 7 , 107. ' R. D. Levine, MoZ. Phys., 1971,22,497. R. D. Levine, J. Phys. By 1969, 2, 839, also G. G. Balint-Kurti, to be published. 5960 GENERAL DISCUSSION exponential Born and exponetftial distorted waue approximations may be regarded as partial resummations of the Born and distorted wave Born series respectively. As an example let us consider the case of atom-rigid rotor collision. The potential is taken to be of the form where R is the distance from the atom to the centre of mass of the diatomic and 0 is the angle between the axis of the diatomic and the line joining the atom to the centre of mass of the diatomic.Following Arthurs and Dalgarno the total wavefunction is expanded in terms of angular functions which are eigenfunctions of the total angular momentum. The problem is reduced in this manner to a set of coupled, second order differential equations in R. In the exponential distorted wave approximation these equations are uncoupled in some manner, the uncoupled equations are solved for the distorted waves and these are then used to evaluate the perturbation expressions for the A matrix elements. In the present case we use the distortion decoupling procedure to define the distortion potential (i.e., the distorting potential is taken to be the diagon- al part of V ) . The S matrix can then be written in the form : roo The functions qEl(RP) are the total angular momentum eigenfunctions defined in ref. (9, and J, j and I are the quantum numbers corresponding to the total, the rotational and the orbital angular momentum respectively.The distorted wave &(R) is the solution of the differential equation : which is regular at the origin and becomes a sine wave at large distances &(R) - sin (kjR-ln/2+Sjl) R+CO ( 5 ) The approximation outlined in eqn (3)-(5) is the exponential distorted wave distortion (EDWD) approximation which has previously been examined. It has also been shown that an identical perturbation expression arises if the A matrix is expanded in a double perturbation series and evaluated to lowest order in both the A. M. Arthurs and A. Dalgarno, Proc. Roy. SOC. A, 1960,256,540. R. D. Levine, B. R. Johnson, J. T.Muckerman and R. B. Bernstein, J. Chern. Phys., 1965,49,56. R. D. Levine and G. G. Balint-Kurti, Chern. Phys. Letters. 1970, 6, 101. G. G. Balint-Kurti unpublished, work ; see also R. D. Levine and B. R. Johnson, Chem. Phys. Letters, 1970, 7, 404.GENERAL DISCUSSION 61 potential and in I?. lt is therefore, completely in keeping with the EDWD approxi- mation to use a semi-classical JWKB approximation to evaluate the distorted waves x$(R). This additional approximation reduces the task of evaluating the functions x ~ ~ ( R ) to that of performing a quadrature, and greatly simplifies the computation of the A matrix elements. To evaluate the JWKB wavefunction we use in essence the prescription given by Bernstein : x;~(R) = 0 k j "0.629 271 +a458 7452 = [ g ] - 0.104 878z3 - 0.038 229z4] = Fc-* sin (kjReff+n/4) where 3 z = e) kj(R- Ro), and Ro is the classical turning point of the motion. k-; = The phase shift is given aj, = in the semi-classical approximation - l < z < l < -1 (6) z > l dR'.by 1) dR'. (7) The use of the JWKB approximation to evaluate the distorted waves (eqn (6)) together with eqn (3) to calculate the S matrix is called the exponential semi-classical distorted wave distortion (ESCDWD) approximation. The quadratures were per- formed by a simple trapezoid rule integration. All of the quadratures, including the evaluation of the A matrix elements (eqn 3) were performed simultaneously. The potential energy parameters chosen were suitable for describing an Ar + TlF collision. As the potential, however, contains no odd Legendre polynomials it is inore appropriate to a homonuclear diatomic.One of the distinguishing character- istics of the model potential is the large short range anisotropy (al2 = 0.6). It is on account of this large anisotropy that the exponential Born approximation gives poor results for this case. Fig. 1 compares the exact and various approximate opacities for t h e j = O+j' = 2 first order allowed transition, at an energy of E/E = 1.7, as a function of the orbital angular momentum. The calculations include 9 coupled channels and the exact calculations were performed using the amplitude density m e t h ~ d . ~ As may be seen from fig. 1 , the distorted wave results (SCDWD) start to deviate appreciably from the exact ones for I values below 54.The exponentiated (ESCDWD) and exact results are in much better agreement over the whole range of 2 examined. The present model R. B. Bernstein, A h . Cliern. Pliys., 1966, 10, 75. * The potential energy parameters used were : B = 2p&R;/fi2 = 1200, &h/& = 0.01 where Eth is the lowest threshold energy ( j = 0+2), a12 = 0.6 and a6 = 0.2. G. G. Balint-Kurti and R. D. Levine, Chem. Phys. Letters, 1970, 7, 107 D. Secrest and B. R. Johnson, J. Chem. Phys., 1966,45,4556 : B. R. Johnson and D. Secrest, J. Cliem. Phys., 1968, 48, 4682.62 GENERAL DISCUSSION I I I 1 30 40 5 0 6 0 70 I FIG. 1.-Exact (dots) and approximate opacities for j = 0+2 transition calculated using 9 coupled channels. I FIG. 2.-Exact (dots) and approximate opacities for j = 0->4 transition calculated using 9 coupled channels.GENERAL DISCUSSION 63 potential, with its large anisotropy parameters, forms a specially severe test for any perturbation approximation.It is seen, however, that even at low I values the ESCDWD approximation gives reasonable values for the opacity. The EBDWD approximation fails completely due to the omission of distortion effects in this method. 5 0- 40- s J. 2 30- !% + - 20- 4 4 e-4 10- 6 0 . b I I:2 1:4 I :6 l:8 2.0 El€ FIG. 3.-Variation of the P44(2t0) opacity with energy. Fig. 2 shows the opacities for the j = 0-jf = 4 first order forbidd n transition, and fig. 3 shows the effect of varying the energy on the j = 0-j’ = 2 opacity. In both these cases the ESCDWD method yields reasonable results. In general it seems true to say that by the use of the exponential method the range of validity of a perturbation approximation, such as the Born or distorted waue Born, may be significantly extended with relatively little extra computational effort.We thank Prof. R. D. Levine for suggesting much of this work. Dr. M. D. Pattengill and Prof. J. C. Polanyi (University of Toronto) (contributed): In his introductory remarks, Marcus referred to a number of simple models for chemical reaction-the impulsive model, the DIPR model, and so on. There exists a category of model (a sub-category in Marcus’s scheme) which can be loosely designated as “ retreat-coordinate models ”, since these models lay their major stress on the forces that operate as the products separate.2 Models of this type tend to be based on the concept of a forced oscillator.The new chemical bond is regarded as, in some sense, existing at the outset. The repulsion between the products forces some vibration into G. G. Baht-Kurti and R. D. Levine, Chem. Phys. Letters, 1970, 7, 107. J. C. Polanyi and J. L. Schreiber, in Physical Chemistry-An Adcanced Treatise, Vol. VT, Kinetics of Gas Reactions, eds. H. Eyring, W. Jost and D. Henderson (Academic Press, New York, 1973) Chap. 9.64 GENERAL DISC U SSION this new bond ; the balance of the reaction energy goes into product translation and rotation. A feature of the chemical process that has been omitted from the simple forced oscillator models is the fact that the new bond is in the process of being formed while the product repulsion is being released.This is, in fact, a distinctive feature of chemical reaction, as compared with energy-transfer between an atom and a stable molecule. We have been developing a classical model of the retreat-coordinate variety which explicitly includes the effect of ‘‘ tightening ” in the new bond, AB, concurrently with the release of repulsion along the coordinate of separation, BC. The model (termed FOTO, for Forced Oscillation of a Tightening Oscillator) in its present form assumes that reaction proceeds collinearly through a selected intermediate configuration A--B.C. The dynamics of energy-release are followed in the retreat from this configuration. The extended A--B bond is treated as an harmonic oscillator of diminishing equilibrium separation and increasing force constant, being “ forced ” by a B-C repulsion of constant magnitude and finite duration.As the repulsion is released, A- -B evolves from a fractional bond to a normal one. Several assumptions,2 along with the use of empirical relationships (Pauling’s bond-order relationships, and Badger’s rule), suffice to parametrise the initial A- -B.C configuration and the required time dependences in terms of a single input parameter, namely the initial fractional bond-order of A- -B. The model has been applied to ten reactions for which some experimental data and trajectory results exist. The correspondence is encouraging. The model is sufficiently complete to embody recognisable analogues of “ attractive ”, “ mixed ” and “ repulsive ” energy relea~e.~.Prof. R. A. Marcus (University of Illinois) said: The agreement of the results of Balint-Kurti and Johnson with the exact values is most encouraging. I believe that for their system, the reduced moment of inertia I/pa2(in the notation of ref. (5)) is still in the region where the adiabatic distortion of the rotor is not too serious, judging from the results in ref. (5). It might be interesting to compare ESCDWD with exact results for a system with a substantially smaller I/pa2 (e.g., HX + M or H2 + M), where the “ static ” classical approximation seemed to show some signs of breaking down,5 although the latter results were quite incomplete and should be extended. Dr. R. G. Gilbert (University of Sydney) and Prof. T. F. George (University of Rochester) said : We have investigated the exponential Born distorted wave approxi- mation (EBDW) for collinear H+H2(u = O)+H2(u = O)+H at energies where only the ground vibrational state of H, is open? The S-matrix in the distorted wave approximation (DW) is given as S = I -iB, where Bll = BZ2 are the DW elements of B for elastic scattering and BI2 = B,, are for reaction.The EBDW form of S is given as S = exp(i6,) exp( - i6’) exp(i6,), where 8‘ is the off-diagonal part of B and A simple quantum model which stresses the strengthening of the new bond is that of G. L. Hofacker and R. D. Levhe, Chem. Phys. Letters, 1971, 9, G17. cf. F. E. Heidrich, K. R. Wilson and D. Rapp, J. Chem. Phys., 1971,54, 3885. J. C. Polanyi and J. L. Schreiber, in Physicuf Chemistry-An Advanced Treatise, Vol.VI, Kinetics of Gas Reactions, eds. H. Eyring, W. Jost and D. Henderson (Adademic Press, New York, 1973) Chap. 9. For a full account see M. D. Pattengill and J. C. Polanyi, Chem. Phys., 1974, in press. A. 0. Cohen and R. A. Marcus, J. Chem. Phys., 1970,52,3140. R. G. Gilbert and T. F. George, Chem. Phys. Letters, 1973.GENERAL DISCUSSION 65 2&, with elements 2a1 = 2d2 = 26, is the diagonal part of 6.l. Expanding the matrix exponentials in terms of the projection operators of the matrices, we find the elements of S to be Sll = = cos(Bzl) exp(2i8) and S12 = SZl = -i(sin(Bz1) exp(2i6)). The EBDW probability of reaction is then lS2,12 = sin2(Bzl). We obtain distortion surfaces for reactants and products by fitting the lowest adiabatic Porter- Karplus surface to a Morse curve at the local minimum (at fixed separation of H from the centre of mass of H& with the constraint that this curve support at least " 0.4 0.5 0.6 energy (eV) FIG.1 .-The EBDW probability (0) for collinear H+ H2(u = O)+H2(u = 0)+ H as a function of total energy and the probabilities from the coupled-channel calculations of Diestler (D) and Wu and Levine (WL). A are the results of the Condon approximation applied to the DW S-matrix. one vibrational state. Wave functions in B2 are evaluated in the perturbed-station- ary-state approximation. The results (0) are shown in fig. 1 in comparison with the coupled-channel calculations of Diestler (D) on the Porter-Karplus surface. (A are the results for Szl = -iBzl (DW) with the Condon approximation applied to Bzl : B21 is assumed to be proportional to the nuclear overlap integral between reactant and product wave functions.) The EBDW S is unitary, whereas the DW S yields reaction probabilities as high as four.6 However, the EBDW probability drops off too soon near a total energy of 0.6 eV, which is most likely due to the first vibrational state becoming locally accessible.and Wu and Levine (WL) Dr. S. Bosanac (Bristol University) said: Without loss of generality, we can take the potential of a three-particle system to be of the form G. G. Balint-Kurti and R. D. Levine, Chem. Phys. Letters, 1970, 7, 107. R. D. Levine, Mul. Phys., 1971, 22,497. R. N. Porter and M. Karplus, J. Chem. Phys., 1964,40, 1105. D. J. Diestler, J. Chem. Phys., 1971, 54, 4547. S.-F.Wu and R. D. Levine, Mol. Phys., 1971,22,881. R. B. Walker and R. E. Wyatt, Chem. Phys. Letters, 1972 16, 52. 55-c66 GENERAL DISCUSSION For multichannel scattering, (C collides with (AB)) this form reduces to the we11 known (2) where Y, R, and 8 are the standard coordinates. We now discuss multichannel scattering. The standard way of solving the Schrodinger equation with the potential (2) is to reduce the six dimensional differ- ential equation to only one dimension by expanding the wave function in the complete set of solutions for Vl(r). The members of the complete set we designate by $"(r) where n stands for quantum numbers defining the solutions. After neglecting the contributions from the continuum states, we get a set of M coupled differential equations in one variable only, the coupling being the matrix elements of V4(r, R, cos 8) between the eigenfunctions &(r).One way of solving the set of the equations is to uncouple them and solve M separate two-body problems. The solutions we designate by $A@) and $i(R), where again n refers to the quantum numbers and the indices 1 and 2 designate the solutions with different asymptotic behaviour for large R. A solution of the multichannel equations is now given in the form of an integral equation vf = vl(r)+ v4(r, R, cos e) $(R) = $'(R)+ dR'lc(R, R')$(R') (3) 0 where and K(R, R') is the kernel of the integral equation, given by is a diagonal matrix of the boundary conditions imposed on $(I?), and $2(R)$1(R')V;(R'); R 2 R' $'(R)$2(R')V;(R'); R < R'. K(R, R') = The prime on V4(R') denotes the potential matrix with no diagonal elements and C is a constant diagonal matrix.Iterating the integral equation once, we get a distorted wave Born solution to the wave equation. However, the S-matrix from this solution is not unitary, and we make it such by exponentiating the Born approximation Let us discuss the validity of such a method. We do this by developing the exponential function in the power series S - 1 +LJ * dRf$'(R')V:(R')$'(R') + c o (6) + J* dR'$ '(R')V:( R')$ dR"$' (R") V i ( R")$' ( R") c.2 0 and compare the result with the iterated solution of the integral eqn (3) S , - l+f dR'$'(R')V&(R')$'(R')+ c o dR"$l( R") V;( R")$'( R") + dR"$2( R")Vi(R")IC/ (R"). (7)GENERAL DISCUSSlON 67 The third term in the series for S could be treated as an estimate of the exact The estimate is good, provided value given by the third and fourth terms in (7).Yk(R’) is small. In other words, the quantity $ = jJic(R, R’) ic(R‘, R) dR’ dR (8) is a measure of validity for the exponential distorted wave approximation. However, in the case of reactive scattering, Yk(R’) is not in general small. If we neglect the term V3(rac) in (l), (in the case of reactive scattering A+ (BC)+C + (AB)) the potential assumes the form v’(r, R , cos 0) = Vl(r)+ V4(r, R , cos O)+V2(JR2+a2r2-2arK cos 0) (9) - Vi(r)+ V:(i., R)+ V,”(JR2 +a2r2)+P,(cos O)(V:(r, R)+ V$(r, R)). The term Vi(r, X) which enters in (9) and contributes to $ (8) cannot be taken as small because the expansion in P,(cos 0) of a two-body potential is poorly convergent.The difficulty could be resolved by solving the complete three-body problem for the potential and then using the exponential distorted wave approximation. This, however, would involve solving the Fadeev equations. v’’ = vl (rAB) + V 2 ( r B C ) + v3(rAC> (10) Dr. J. P. Simons (University of Birmingham) said : The successful use of a Franck- Condon modulation method for treating the multi-curve crossing problem discussed by Child has been demonstrated in a semi-classical “ impulsive half-collision ” calculation for energy transfer from Hg(63Po) into vibrational levels in CO or NO. V 0 2 4 6 B 1 0 42 14 16 18 0-3 0 0.12 I I I I I I I 1 I 1 i I 1 0 1 2 3 4 5 6 7 8 9 V FIG. 1.-Vibrational distributions in the quenching of Hg(63Po) by CO and NO.Experimental data : filled circles, CO, open circles, NO, (ref. (5)). Calculated distributions : broken lines, (assum- ing an exponential repulsive potential curve and a decrease in bond order of 0.6 in C-0 and N-0, ref. (2)). Intersystem crossing from a linear triplet Hg-CO or Hg-NO collision complex transfers the system onto a steeply repulsive part of the singlet potential surface and the subsequent recoil forces the molecular species into oscillation. If the equilibrium M. Child, this Discussion. J . P. Simons and P. W. Taker, Mol. Phys., 1973.68 GENERAL DISCUSSION C-0 or N-0 bond lengths remained constant throughout the entire process the recoil would force the molecules from an initial vibrational state i = 0 into a range of final statesf, the relative populations of which, would follow a Poisson distribution.'.If their bond lengths were extended at the instant of intersystem crossing a range of initial states i> 0 would be populated ; it was assumed (cf. ref. (3)) that their relative populations were governed by the appropriate Franck-Condon factors for the radia- tionless transition. The distribution over final states was obtained by calculating the set of probabilities Pi,, weighting each one by the corresponding Franck-Condon factor S& and then summing over the range of initial states i. Very good agreement with the experimental data was obtained assuming a decrease in bond orders of 0.6 in the triplet complexes of both CO and (see fig. 1). Prof. K. F. Freed (University of Chicago) said: I would first like to comment on Marcus' discussion concerning the contribution of the quasi-periodic classical orbits to the semi-classical bound states.Below is a non-rigourous proof which illustrates the the nature of the quasi-periodic classical orbits which can contribute to the quantum mechanical bound states and which demonstrates the existence of these types of orbits at all the classically allowed energies. (Of course, the quantization conditions are not necessarily satisfied at all these energies.) Following Gutzwiller and others,6* let $i(q) and Ei be the eigenfunctions and eigenvalues of a multi-dimensional Hamiltonian, and consider the sum over states where E is a complex parameter. As is well known, the poles of G in the E-plane are the eigenvalues and the residues at these poles provide the eigenfunctions.If we are just interested in the eigenvalues, then it is possible to focus attention on the spectral function 5 * G(E) dqG(q, q; E ) f C ( E - E i ) - ' , s 1 which has poles at the eigenvalues corresponding to the bound states. Gutzwiller and others 6* investigated the semi classical description of the bound states of multidimensional nonseparable systems by introducing the semi-classical approximations G,,(q, q; E) into (2). Upon evaluating Jdq in (2) by stationary phase, Gutzwiller '9 was lead to the conclusion that only the periodic classical orbits provide contributions to the semi-classical approximation to (2), and hence to the quantum bound states. (see below) on the full G,,(q, q' ; E ) showed that the periodic or multiply periodic orbits could contribute to the semi- classical quantization of bound states.Since the multiply periodic orbits can have incommensurate frequencies, this class includes the quasi-periodic orbits which Marcus emphasizes in his paper as yielding the semi-classical bound states. Our previous work K. E. Holdy, L. C. Klotz and K. R. Wilson, J. Chem. Phys., 1970,52,4588. R. D. Levine and R. B. Bernstein, Chem. Phys. Letters, 1972, 15, 1. M. Child, this Discussion. G. Karl, P. Kruus and J. C. Polanyi, J. Chem. Phys., 1967, 46,224 ; G. Karl, P. Kruus, J. C. Polanyi and 1. W. M. Smith, J. Chem. Phys., 1967,46,244. M. C. Gutzwiller, J. Math. Phys., 1967,8, 1979; 1969, 10, 1004; 1970,11,1971 ; 1971, 12,343. W. H. Miller, J. Chem. Phys., 1972, 56, 38.' K. F. Freed, J . Chem. Phys., 1972, 56, 692.GENERAL DISCUSSION 69 Let 6 g be a small volume in configuration space which is centred at point q. Consider the coarse grained average of G over the region 5q, Provided 5q is small and q is not near a mode of some $i(q), the contribution from the ith state in (1) to (3) will approximately be $i(q)$T(q)(E-Ei)-l. So, for a given q, we might lose a few states from the coarse graining in (3), but by varying q all should be found as poles in Ga,(q; E). At this juncture, we introduce the well-known semi- classical approximation 9 G,,(q', q"; E ) = (ih)- ID(q', q"; E)J' exp [iA(q', q"; E)/h - ivn/2], (4) classical paths where the sum in (4) is over all the classical trajectories leading from q' to q" with energy E , A is the classical action 4 q ' , qtt; E ) = l: p(zi ; ElD (5) ph; E ) is the classical momentum for the system at q with energy E, lZ2A(q', q"; E ) a2A(q', q"; E)l - _ _ _ ~ ~ ~ and v is the number of times the classical path encounters a caustic.into (3) and express the ldq' and jdq" as the limit of a summation to yield [G,,(q; E)ISc = lim Aq-+u { ( i h ) - ' ( d q ~ - ~ C C ID($, q"; E)I* x Substitute (4) ni3 x classical Aq',Aq'' paths exp [iA(qf, q"; E)A/-ivn/2]Aq'Aq".). (7) Eqn (7) has contributions from all the trajectories of energy E which start in the region 6q about q and end in that same region. By Poincark's theorem, this class of traject- ories is infinitely dense, since all trajectories starting in the region of phase space 6q and any allowed momentum must return to this region infinitely often in the course of time.As noted previously, this must lead to destructive interference unless the infinite number of different paths in the sum have actions which bear some relationship to each other, i.e., the paths must be parts of periodic or multiply periodic orbits. For 6q small and for trajectories which have the same number of traversals n through the region 6q, D,(q', q" ; E ) is slowly varying provided q is not near a caustic as we now assume. Thus, the only differences between such contributions can be phase differ- ences which arise from the action differences Consider now the difference in the contributions of various terms to (7). = s ( q f , qrt; E ) . 6, +;--"(qt, i3A q"; E ) . 62 3- .. . w aq" l M. C. Gutzwiller, J. Math. Phys., 1967,8, 1979; 1969,10, 1004; 1970, 11, 1971 ; 1971,12, 343. P. Pechukas, Phys. Rev., 1969, 181, 166, 171.70 GENERAL DISCUSSION As 6q is small, terms in (6)2 can be ignored in (8), and and 6, can be replaced by dq, a typical radius vector of 6q, in order to obtain an order of magnitude estimate. Thus, we have where are the final and initial momenta, respectively, for trajectories of energy E going from q‘ to q”. Thus, the phase difference between ‘‘ neighbouring ” classical trajectories in (7) is approximately 6p*6q/h. So, for cases in which the paths starting in and returning to 6p6g will all contribute to (7) and not destruct- ively interfere, i.e., all trajectories in the “ quantum box ” Gp*Sq<A contribute. The classical trajectories satisfying (1 1) are generally distinct from those which contribute to the normalization (6) for a periodic trajectory that traverses 6q, as periodic trajectories with energy E need not pass through a 6q (for some q) that the quasi-periodic trajectories do.In the limit 6q-+O, the operator (1/6q)J dq’ becomes Jdq’d(q-q’), where 6 is the Dirac delta function, and (12) Hence, all the quasi-periodic trajectories passing through q and satisfying (1 1) contri- bute to the semi-classical bound states so long as they satisfy the auxiliary quantization conditions. Another way to obtain the result (11) is to note that in the stationary phase evaluation of the integral in the semi-classical approximation to (2), the phase need only be stationary to order ti as this is “ effectively ” zero in the phase classical limit.I would also like to mention some work that John Laing and I have been doing to investigate the general semi-classical limit of multichannel scattering. We are interested in cases in which certain degrees of freedom must be treated quantum mechanically, while others may be treated semi-classically as in WKB approximations. Such cases arise, for example, in photochemical dynamics of polyatomic molecules where the molecule can rattle around on more than one nested potential energy surface before dissociation. A special case of this problem is to investigate the purely classical limit, if any exists, of motion on several potential surfaces when transitions may occur over very wide regions and not just along the intersections of the surfaces.2 I shall not go into the details, but instead just note that the time-dependent Schrodinger equation for multichannel scattering can be represented in a Feynman path integral from involving only the particles’ masses, the potential surfaces and the matrix elements connecting these surfaces.This removes a deficiency of Pechukas’ approach which required transition-matrices for arbitrary motions of the non- P. Pechukas, Phys. Rev., 1969, 181, 166, 174. J. C. Tully and R. K. Preston, J. Chem. Phys., 1971,55,462 ; W. H. Miller and T. F. George, J. Chem. Phys., 1972, 56, 5637. R. P. Feyntnan and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). 6p. sqpi < O(1) (1 1 ) 6q GSqh ; E)+G(q, q ; E).GENERAL DISCUSSION 71 quanta1 degrees of freedom.In our case, what was an action integral in the single channel case now becomes a matrix, corresponding to a classical mechanics with internal degrees of freedom. However, this action matrix is nonlocal in time and can only be represented as a Magnus type expansion.' Consequently, the classical equations of motion have forces which depend on the whole trajectory of the particle. We are currently investigating the applicability of this formalism in the natural approximation that the Magnus expansion is truncated in lowest orders. Prof. R. A. Marcus (University of Illinois) said: I should like to report on some recent results on the semi-classical calculation of eigenvalues and resonances by Dr. Walter Eastes of this laboratory, using a method outlined in my paper at this Discussion.Eastes considered the collinear problem of a harmonic oscillator interacting with an atom via a Morse potential and calculated the bound states and the resonances quantum mechanically (numerically). The semiclassical results obtained thus far are encouraging. For example, when the quantum mechanical ground state (0,O) energy was - 1.806, and a value of zero was assigned to one quantum number, the following values were found for the other semiclassical quantum number n for various energies : n E n E - 0.064 - 1.83 + 0.01 6 - 1.80 - 0.01 1 - 1.81 + 0.043 - 1.79 - 0.0004 - 1.806 One sees that the trajectory giving the correct n (namely zero) serves to locate the eigenvalue of - 1.806.(At each energy several trajectories were used to find the ones giving the constant values of n listed in the above table.) Extension to larger inter- actions is being made and the results will be submitted for publication elsewhere. In response to a question posed by Freed regarding Gutzwiller's belief that periodic trajectories are needed to calculate eigenvalues, the arguments in my paper (and really those of Keller given earlier) show that such trajectories are not needed. None of the trajectories used to construct the above table were periodic. Prof. W, H. Miller (University of California) said: Regarding the point made in the last paragraph of Section A of Miller and Raczkowski's paper, S. M. Hornstein and I have carried out such a calculation for the collinear H + H2 reaction on the Porter- Karplus potential surface; the interest was to see if this symmetrical averaging procedure could describe the energy region near the classical threshold for reaction more accurately than the " quasi-classical ", or unsymmetrically averaged reaction probability.For the collinear H+H, system let x(nl, q1 ; E ) be the function such that x(n1, 41 ; E ) = 1 if the trajectory with the indicated initial conditions [(n,, ql) for the action-angle variables of the vibrational degree offreedom, and R1 = large, PI = - {2p[E-&(n,)])* for the translational degree of freedom] is reactive, and zero otherwise. It is easy to show, then, that the usual Monte Carlo, or " quasi-classical '' reaction probability is given by Punoym(E) = ( 2 W 1; dq,x(O, 41; El, (1) P.Pecliukas and J . C . Light, J . Cbem. Phys., 1966, 44, 3897.72 GENERAL DISCUSSION whereas the symmetrically averaged quantity is (For energies E in the threshold region all final quantum numbers correspond to the n2 = 0 " box ".) Thus the unsymmetrical procedure quantizes the initial vibrational state but not the final one, whereas the symmetrical procedure quantizes neither ; the classical S-matrix approach quantizes both initial and final vibrational states (via the boundary conditions on the trajectories). Fig. 1 shows the results : the solid line is the classical S-matrix result,' which is essentially the same as the exact quantum mechanical result, the dotted line is the unsymmetrical reaction probability [eqn (l)], and the dashed line the symmetrically averaged reaction probability [eqn (2)].(The abscissa is the nominal initial transla- tional energy, Eo = E - e(O).) The symmetrically averaged reaction probability, which would be preferred on any a priori theoretical grounds, is actually seen to be poorer than the unsymmetrical averaging procedure; it is difficult to give any solid theoretical reason why this should be so, and it may in fact be fortuitous. 0.4 - h Y .C( - 2 0.3- 0.12 0.14 0.16 0.18 0.20 0.22 0.24 collision energy/eV FIG. 1. The moral of this example presumably is that if there are so few quantum states involved that it matters how trajectories are assigned to " boxes ", then the only reliable way of using trajectories is via the semiclassical theory, which quantizes the appropriate quantum numbers initially and finally.Mr. J. L. Schreiber (Uniuersity of Toronto) (communicated) : Miller and Raczkow- ski have observed that cross sections calculated by the currently popular classical trajectory method do not satisfy a microscopic reversibility relation, due to the dis- symetric way of handling initial and final conditions. They have proposed a further averaging over initial internal state indices (n) as a " fix " for this situation ; however, the resulting cross sections still do not satisfy a microscopic reversibility relation. In the spirit of the quasi-quantum boundary conditions applied to the trajectories, it seems desirable for the resulting cross sections to satisfy a quantum-like algebraic microscopic reversibility relation.The well-known microscopic reversi bility relation between forward and reverse T. F. George and W. H. Miller, J . Cliem. Phys., 1972,57, 2458.GENERAL DISCUSSION 73 cross sections (averaged over degenerate levels of the initial state, and summed or integrated over the degenerate levels of the final state) 6, may be written where p, p' are the initial and final momenta, and gn is the degeneracy (or density of degenerate states) associated with the internal state index n (which may be a collection of indices, in which case integration over n is a multiple integration over all indices). For a given transition ni+n; the quantitiesp andp' are related by conservation of total energy. Cross sections calculated by the conventional classical trajectory method involves grouping the (continuous) n' values in a certain range n; - 3 to n;+ +, and associating these with the quantum transition into the state n;: Typically, pr and nf are fixed, and the cross section 0, for the ni+n; transition is the integral of all n' values in this range.Since the total energy is constant there is an implicit integration over p' as well n;+3 a, c,(pini -+ p;n;) = 1 dn' 5 dp' 6(E(p', n')-E(p>, n;))xo(pini -+ p'n') (2) where E(p, n) is the total energy of the system with the indicated internal state and momentum. It is clear that the forward and reverse nc's do not satisfy a simple (algebraic) microscopic relation of the form (1). n f -t 0 In order to obtain the simple relation, we define a new quantity, 0, 1 n i + f co Cc(pini -+ p;n;) = dn gn dp p2 x S(E(P, n>-E(pi, ni>)ac(pn + pin>)* PI gni J n i - * J* f, = E,gni s"'" n i - $ This average over the classical manifold of states associated with the quantum state p i n , is at a fixed total energy.Doing the integral overp and denoting ET = p:/2p, the (fixed) nominal translational energy, we get dn gn(ET+E(ni)-E(n)} x cc(ETn -+ Ein') (3) where ET is the quantity in curly brackets in the integrand, the translational energy required by conservation of total energy, and E(n) is the internal energy of the state n. This form for the classical cross section is easily applied, and results in a simple relation of the form (1) between forward and reverse cross sections so defined. a,(pini -+ p' n' Prof. R. D. Levine (Jerusalem) (communicated): The conclusions of this paper support earlier expectations by Miller In particular, one can show that for a certain class of properties, namely those which have a classical analogue, the loss of interference upon averaging is expected on a rigorous basis and is not an approximation.and by myself.3* Prof. J. N. Murrell (Sussex University) and Dr. S . Bosanac (Bristol University) said : We have been applying the classical S-matrix theory of Miller and Marcus and The classical cross sections u are continuous functions of their arguments, and as such satisfy this relation for a differential element of phase space, but not generally for an entire " correspondence principle " interval. * W. H. Miller, J. Chern. Phys., 1971, 54, 5386. R. D. Levine, Abstracts of Papers, VII ICPEAC (North-Holland, Amsterdam, 1971) p.912. R. D. Levine, J. Chern. Phys., 1972, 56, 1633.74 GENERAL DISCUSSION co-workers to the atom-diatom vibrational relaxation over a strongly attractive potential surface. The potential used as a model for the co-linear triatomic system A-B-A was based upon the spectroscopic properties of the ground state of C 0 2 and consisted of a CO Morse potential and a repulsive 0-0 Hulthkn potential. This combination produced a potential with an activation barrier to reaction of 2 eV and a well depth of 5.4 eV relative to CO +O. From our provisional results we wish to make some comments about the difficulties encountered in this type of calculation. 4 3 (c.2 B 0 0 I 2 3 4 5 6 FIG. 1. 4i Fig. 1 shows the results of trajectories obtained at an initial total energy of 1.3 eV greater than the activation barrier.Full lines represent reactive collisions and dotted lines non reactive. As was shown by Rankin and Miller for the H+C12 linear collision [a potential with no stable triatomic] the trajectories may be divided into " direct " and " complex ". In the figure we have only shown those that are direct, and in the regions of qi [the initial phase of the diatomic] not covered by the direct trajectories one obtains an apparently random set of final vibrational quantum numbers, nf, both reactive and non reactive. It is the problem of the separation of the direct and complex regions that we wish to comment upon. Given complete accuracy in the integration procedure all trajectories would be found to lie on continuous segments of nf[qi) curves.In practice it is possible to distinguish direct and complex regions according to the range of qi spanned by these segments. For example in the case ni = 0 there is one reactive segment that spreads from qi = 4.30 to qi = 2n + 1.16, and two shorter non reactive segments. A scan of Aq = 0.025 over the whole range, with an integration procedure that left the total W. H. Miller, J. Chem. Phys., 1971, 54, 5386.GENERAL DISCUSSION 75 energy accurate to better than 1 x eV, showed no other continuous regions with a width greater than 0.2, all other regions were therefore designated as complex. Because of the time reversibility of the classical trajectories, or because of the unitarity of the S-matrix, there must be connections between the continuous regions for different values of ni.For example, the region labelled A in n, = 0 shows trans- itions are possible over the specified scan of qi to nf = 0-4. It follows that there must be a nonreactive continuous region in each of ni = 1-4 in which the transition to nf = 0 is allowed. These are indicated by the letter A in each scan. We note however, again because the S-matrix must be unitary, that the slope [dnf/dqi] must be the same for n, = 0, n, = 4 as for ni = 4, nf = 0. Given that the continuous curves are approximately parabolic, it follows that the parabola must become steeper as ni increases. In other words one cannot make an absolute judgement on whether a region of trajectories is direct or complex based on the range of Aq over which a continuous curve is found.The region A in the scan ni = 4 must be designated direct even though the curve is continuous only from 6.02 to 6.10. The figure gives a labelling to each of the short continuous regions which shows their correspondence for different ni. If one takes region D for example this appears in ni = 2 but as it does not allow for nf = 1 or 0 there is no such region in ni = 1 or 0. In D the transition 2+3 occurs at four values of q1 so that there will be two curves D1 and D, in ni = 3 at which the transitions 3-2 are allowed. However, the transition 2+4 in D only occurs at two values of qi so that although there are two curves D, and DZ, in only one of these [Dl] is the transition 4-2 allowed. Prof. R. A. Marcus (University of IZZinois) said: The paper of Connor on the evaluation of multi-dimensional semi-classical integrals is very interesting.In most instances the mapping of the phase of the integrand onto quadratic or cubic functions is quite adequate for obtaining reasonably accurate results. However, we have noticed a number of instances where the phase along one coordinate is so slowly- varying that a cubic or quadratic mapping is inadequate, even in one dimension. TRANSITION PROBABILITY exact Airy Bessel uniform parameters (a, p, E ) transition quantum uniform (Stine) 0.114, &, 3.8 1-1 1 .00 4.7 1 .00 0.300, 3, 3 1-1 0.98 1.46 0.97 0.300, 3, 6 1-1 0.22 0.22 0.23 0.300, 3, 10 0-1 0.22 0.21 0.21 0.300, 3, 4 0-1 0.11 0.11 0.10 0.1 14, 3, 3 1-0 7.1 x 10-4 7 . 6 ~ 10-4 6 .8 ~ Recently, therefore, J. Stine in our laboratory has developed a new uniform approximation, involving a mapping of the phase onto a sinusoidal function and an appropriate mapping of the pre-exponential fact0r.l He has obtained encouraging results, a few of which are listed in the following table. The a, p and E refer to the parameters in the Secrest-Johnson problem of a collinear collision of a diatomic molecule with an atom (exponential repulsive interaction potential). Complex- valued trajectories While Bessel functions have been used before in semiclassical perturbtion calculations, they J. Stine and R. A. Marcus, J. Chem. Phys., 1973, 59. J. Stine and R. A. Marcus, Chem. Phys. Letters, 1972, 15, 536 ; W. H. Miller and T. F. George, J. Chem. Phys., 1972, 56, 5668, 5772.were used for the last two rows in the table.76 GENERAL DISCUSSION apparently haven’t been used to construct uniform approximations. Dr. Kreek is now applying the method to the two-dimensional integral arising in collisional rotational-translational energy transfer problems, and we hope to have results shortly. Dr. J. N. L. Connor (University of Manchester) said: I have an addition to make to my paper.l The integral for which a uniform asymptotic approximation is derived is a special case of the n-dimensional integral The position of the saddle points (classical trajectories) depends on the set of para- meters a. No concrete example of such an integral in semiclassical collision theory was given. However, an example can be found in the paper by Marcus,2 specifically eqn (2.7) where A = F,(wR, nE)-Fi(wR, mE)+3(ln+Zm+ 1 ) ~ .The correspondence between the two integrals is as follows. For a fixed energy E, the set of parameters a is equal to m = (rn,}, the set of final quantum numbers for the collision. When the energy E is allowed to vary as well, there is then an additional parameter on which the position of the saddle points (classical trajectories) depends. Mr. Y-W. Lin, Prof. T. F. George and Prof. I(. Morokuma (University of Rochester) said : The analytic continuation of classical mechanics for the description of classically forbidden processes, as discussed in the paper by Miller and Raczkowski, involves the analytic continuation of potential energy surfaces for complex values of nuclear coordinates. Such analytic continuation is necessary for a semiclassical description of electronic transitions in molecular collisions between adiabatic surfaces of the same symmetry, if the description is carried out strictly in the adiabatic representation.We have applied the semi-classical theory of Miller and G e ~ r g e , ~ which has been developed in the adiabatic representation, to the collinear rearrangement H+ + D2 + HDf + D, as proceeding through a transition between the two lowest singlet, adiabatic surfaces of HDZ. A trajectory started asymptotically on surface 1 with vibrational quantum number nl as as action variable for D2 propagates on the same surface in real time steps until AE, the absolute energy difference between the two surfaces, reaches a local minimum. Allowing time steps to become complex, we continue the trajectory to the nearest (complex) point of intersection, where the trajectory switches continuously to surface 2.Hence we assume the transition to be localized at the point of intersection. Our method can be viewed as an extension of the method of Stueckelberg for a single nuclear degree of freedom to several nuclear degrees of freedom. Unlike his method, however, ours must deal with many transitions due to the many local minima in AE. Fig. 1 shows the complex r-motion of a trajectory from nl = 0 on surface 1 to n2w0 on surface 2, which has propagated on surface 1 into the product valley, where r = tr+iri is the complex internuclear distance of HDf (strictly speaking, t is the internuclear distance of HD when the system is on surface 1, J.N. L. Connor, this Discussion. R. A. Marcus, this Discussion. W. H. Miller and T. F. George, J. Chem. Phys., 1972,56, 5637. E. C. G. Stueckelberg, Helv. Phys. Acta, 1932, 5, 369.GENERAL DISCUSSION 77 and of HD+ when the system is on surface 2). The four branch points, corresponding to the zeroes of AE(r, R), and their extended branch cuts are drawn for R = (6.9+ 0.03i) Bohr, where R is the complex distance from the D in HDf to the atom D. The r-motion in fig. 1 is in the vicinity of R = (6.9 +O.O3i) Bohr. AE goes through a minimum when r, goes through a turning point. Near this minimum the trajectory 0.3. 0.2. 0.1 - 2 5 0.0. i" -0.1 . -0.2. -0.3. R = (6.9+0.03i) Bohr I / - 3 t 3.0 I I- FIG. 1.-The complex r-motion of HD+, D for real time steps in the vicinity of R = (6.9+0.03i)BohrY where Y = r,+ ki is the complex internuclear distance of HDf and R is the complex distance from the D in HD+ to the atom D.The four branch points and their extended branch cuts for the function AE(r, R) are drawn for R = (6.9+0.3i)Bohr. can proceed in complex time steps to either of the two nearest branch points to switch continuously to surface 2, and it need only cross the branch cut without actually passing through the point itself. This is illustrated by the thin line crossing the cut and continuing on surface 2 (dotted line). Hence there are two different branching procedures for continuing on surface 2 from this local minimum. Likewise there are two branching procedures for remaining on surface 1 : crossing no branch cuts or crossing both branch cuts.In this manner we see that many trajectories with different branching procedures can end in n2 = 0 on surface 2. If A ( j ) is the complex action along such a trajectory j , the classical limit of the S-matrix element for the transition from n, = 0 on surface 1 to n2 = 0 on surface 2 is given by a sum over j of terms each proportional to exp(iA/(j)/A). Preliminary calculations show the probability of this transition to be of the order of The analytically continued surfaces used in this calculation are obtained through the insertion of the appropriate complex values of r and R into the elements of the electronic Hamiltonian matrix, where the elements are derived as functions of real r and R for HD; in the diatomics-in-molecules method used by Tully and Preston.' This method of analytic continuation is subject to error, especially when the selected real points of Y and R used in the drivation of these function elements are not close to J.C. Tully and R. K. Preston, J . Chern. Phys., 1971, 55, 562. for an initial relative translational energy of 3 eV.78 GENERAL DISCUSSION each other and when the imaginary parts of Y and R are large. The most powerful means of analytic continuation involves the complete diagonalization of the electronic Hamiltonian, He,, for complex values of r and R , and we are currently carrying this out for the HD; system. As a test case we have diagonalized He, for the one-electron system HeH++, focusing on the potential energy curves, E36 and E4,, corresponding to the 30 and 40 molecular ion states. Due to the Wigner noncrossing rule,' AE(R) = E,,(R) -E,,(R) can never be zero if R is complex. The state wave functions which diagonalize He, are expressed as linear combinations of atomic orbitals on both the helium and hydrogen nuclei (LCAO MO), where the atomic orbitals are Gaussian orbitals with four s type on hydrogen and eight s type and four p type on helium. + 4 + 5 + 4 + 3 + 2 - 2 - 3 - 4 -5 - 6 0 1 2 3 4 5 6 AEr/10-3 Hartree FIG. 2.-The complex energy difference AE(R) = E4,(R)-E3,(R) = 4Er+iAEi for HeH++ as a function of complex internuclear distance R = R,+iRj. The thick solid line is for real R, i.e., Ri = 0. Since our LCAO MO calculations are in excellent agreement with the exact results of Bates and Carson for real R, we do not include d type orbitals. The electronic Hamiltonim matrix is complex and non-Hermitian, so that the resulting eigenvectors, i.e., states of HeH++, are members of a biorthogonal basis. AE(R) has a minimum at R = 3.6440 Bohr (for real R) of 0.002 94 Hartree, and AE(R) = 0 (i.e., JAEJ < for R = (3.643 408 8k0.037 475 326i) Bohr. For complex R we can write AE = J . von Neumann and E. Wigner,. Z . Phys., 1929, 30,467. S. F. Boys, Proc. Roy. SOC. A , 1950,200,542. D. R. Bates and T. R. Carson, Proc. Roy. SOC. A, 1956,234,207.GENERAL DISCUSSION 79 AE,+iAEi and R = R,+iRi, and fig. 2 shows AE for lines of constant R, and of constant Ri. Since AE is an analytic function of R except at the branch point where AE = 0, the mapping is conformal so that each line of constant R, crosses each line of constant Ri perpendicularly. Since the electronic Hamiltonian and the atomic orbitals are real functions of R, AE is a real function of R, so that AE(R*) = {AE(R)) * and we need show only lines of positive Ri. The lines of constant Ri are not sym- metric about the line AE, = 0, which is anticipated since AE for real R is not sym- metric about the line corresponding to the real part of R for which AE = 0. Our results are quite encouraging, and the task of finding surfaces for HD; for complex values of nuclear coordinates does not seem very difficult.

ISSN:0301-7249    

DOI:10.1039/DC9735500059

出版商:RSC    

年代:1973

数据来源: RSC