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OFFICERS AND COUNCIL OF THE FARADAY DIVISION 199596 President Prof. H. M. Frey (Reading) Vice-Presidents who have held office as President Prof. A. D. Buckingham (Cambridge) Prof. N. Sheppard (Norwich) Prof, R. H. Ottewill (Bristol) Prof. J. P. Simons (Oxford) Prof. R. Parsons (Southampton) Vice- Presiden ts Prof. A. Carrington (Southampton) Prof. M. W. Roberts (Cardiff) Prof. M.A. Chesters (Nottingham) Prof. I. W. M. Smith (Birmingham) Prof. R. N. Dixon (Bristol) Prof. F. S. Stone (Bath) Prof. M. J. Pilling (Leeds) Prof. M. N. R. Ashfold Ordinary Members Dr. D. C. Clary (Cambridge) Prof. A. J. Stace (Sussex) Prof. P. W. Fowler (Exeter) Prof. Sir John Meurig Thomas (London) Prof. R. K. Harris (Durham) Prof. R. P. Townsend (Port Sunlight) Dr. S.L. Price (London) Prof. A. Zecchina (Turin) Prof. S. K. Scott (Leeds) Honorary Secretary Prof M. J. Pilling (Leeds) Honorary Treasurer Prof. F. S. Stone (Bath) Secretary Mrs. A. C. Bennett Faraday Editorial Board Prof. M. N. R. Ashfold (Bristol) (Chairman) Prof. A. R. Hillman (Leicester) Prof. J. A. Beswick (Paris) Dr. J. Holzwarth (Berlin) Dr. D. C. Clary (Cambridge) Prof. D. Langevin (Bordeaux) Dr. L. R. Fisher (Bristol) Prof. S. K. Scott (Leeds) Prof. 6. E. Hayden (Southampton) Dr. R. K. Thomas (Oxford) Prof. J. S. Higgins (London) Scientific Editor: Prof. A. R. Hillman Managing Editor: Dr. R. A. Whitelock Editorial Production Coordinator: Mrs. S. Shah Technical Editors: Dr. J. S. Humphrey, Miss C. J. Nerney Tho Faraday Division of the Royal Society of Chemistry, previously The Faraday Society, founded in 1903 to promote the study of Sciences lying between Chemistry, Physics and Biology Faraday Discussions (ISSN 0301-7249) is published triannually by the Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 4WF, England.1995 Annual subscription rate ECf207.00, Rest of World f217.00, USA S380.00,including air-speeded delivery, Canada f217 + GST. Change of address and orders with payment in advance to: The Royal Society of Chemistry, Turpin Distribution Services Ltd., Blackhorse Road, Letchworth, Herts SG6 IHN, UK. NB Turpin Distribution Services Ltd., is wholly owned by the Royal Society of Chemistry. Customers should make payments by cheque in sterling payable on a UK clearing bank or in US dollars payable on a US clearing bank. Air freight and mailing in the USA by Publications Expediting Inc., 200 Meacham Avenue, Elmont, NY 1103. Periodicals postage paid at Jamaica, NY 11431. USA Postmaster: send address changes to Faraday Discussions, Publications Expediting Inc., 200 Meacham Avenue, Elmont, NY 11003. All other despatches outside the UK by Bulk Airmail within Europe, Accelerated Surface Post outside Europe. PRINTED IN THE UK. c 0The Royal Society of Chemistry 1995. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form, or by any means, electronic, mechanical, photographic, recording, or otherwise, without prior permission of the publishers.

ISSN:1359-6640    

DOI:10.1039/FD99502FX001

出版商:RSC    

年代:1995

数据来源: RSC

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General Discussions of the Faraday Society/Faraday Discussions of the Chemical Society Date Subject Volume 1907 Osmotic Pressure Trans. 3* 1907 Hydrates in Solution 3* 19 10 The Constitution of Water 6* 1911 High Temperature Work 7* 1912 Magnetic Properties of Alloys 8* 1913 Colloids and their Viscosity 9* 1913 The Corrosion of Iron and Steel 9* 1913 The Passivity of Metals 9* 1914 Optical Rotary Power 10* 1914 The Hardening of Metals 10* 1915 The Transformation of Pure Iron 11* 1916 Methods and Appliances for the Attainment of High Temperatures in a Laboratory 12* 1916 Refractory Materials 12* 1917 Training and Work of the Chemical Engineer 13* 1917 Osmotic Pressure 13* 1917 Pyrometers and Pyrometry 13* 1918 The Setting of Cements and Plasters 14* 1918 Electric Furnaces 14* 1918 Co-ordination of Scientific Publication 14* 1918 'The Occlusion of Gases by Metals 14* 1919 The Present Position of the Theory of Ionization 15* 1919 The Examination of Materials by X-Rays 15* 1920 The Microscope: Its Design, Construction and Applications 16* 1920 Basic Slags : Their Production and Utilization in Agriculture 16* 1920 Physics and Chemistry of Colloids 16* 1920 Electrodeposition and Electroplating 16* 1921 Capillarity 17* 1921 The Failure of Metals under Internal and Prolonged Stress 17* 1921 Physico-Chemical Problems Relating to the Soil 17* 1921 Catalysis with special reference to Newer Theories of Chemical Action 17* 1922 Some Properties of Powders with special reference to Grading by Elutriation 18* 1922 The Generation and Utilization of Cold 18* 1923 Alloys Resistant to Corrosion 19* 1923 The Physical Chemistry of the Photographic Process 19* 1923 The Electronic Theory of Vaiency 19* 1923 Electrode Reactions and Equilibria 19* 1923 Atmospheric Corrosion.First Report 19* 1924 Investigation on Oppau Ammonium Sulphate-Nitrate 20* 1924 Fluxes and Slags in Metal Melting and Working 20* 1924 Physical and Physico-Chemical Problems relating to Textile Fibres 20* 1924 The Physical Chemistry of Igneous Rock Formation 20* 1924 Base Exchange in Soils 20* 1925 The Physical Chemistry of Steel-Making Processes 21* 1925 Photochemical Reactions of Liquids and Gases 21* 1926 Explosive Reactions in Gaseous Media 22* 1926 Physical Phenomena at Interfaces, with special reference to Molecular Orientation 22* 1927 Atmospheric Corrosion, Second Report 23* 1927 The Theory of Strong Electrolytes 23* 1927 Cohesion and Related Problems 24* 1928 Homogeneous Catalysis 24* 1929 Crystal Structure and Chemical Constitution 25* 1929 Atmospheric Corrosion of Metals, Third Report 25* 1929 Molecular Spectra and Molecular Structure 26* 1930 Colloid Science Applied to Biology 26* Faraday Discussions Date Subject Volume 193 1 Photochemical Processes 27* 1932 The Adsorption of Gases by Solids 28* 1932 The Colloid Aspect of Textile Materials 29 1933 Liquid Crystals and Anisotropic Melts 29* 1933 Free Radicals 30* 1934 Dipole Moments 30* 1934 Colloidal Electrolytes 31* 1935 The Structure of Metallic Coatings, Films and Surfaces 31" 1935 The Phenomena of Polymerization and Condensation 32* 1936 Disperse Systems in Gases: Dust, Smoke and Fog 32" 1936 Structure and Molecular Forces in (a) Pure Liquids, and (b)Solutions 33* 1937 The Properties and Function of Membranes, Natural and Artificial 33* 1937 Reaction Kinetics 34* 1938 Chemical Reactions Involving Solids 34* 1938 Luminescence 35* 1939 Hydrocarbon Chemistry 35* 1939 The Electrical Double Layer (owing to the outbreak of the war the meeting was abandoned, but the papers were printed in the Transactions) 35* 1940 The Hydrogen Bond 36* 1941 The Oil-Water Interface 37* 1941 The Mechanism and Chemical Kinetics of Organic Reactions in Liquid 37* Systems1942 The Structure and Reactions of Rubber 38 1943 Modes of Drug Action 39* 1944 Molecular Weight and Molecular Weight Distribution in High Polymers (Joint Meeting with the Plastics Group, Society of Chemical Industry) 40* 1945 The Application of Infra-red Spectra to Chemical Problems 41* 1945 Oxidation 42* 1946 Dielectrics 42 A* 1946 Swelling and Shrinking 42 B* 1947 Electrode Processes Disc.1* 1947 The Labile Molecule 2 1947 Surface Chemistry (Jointly with the Socititti de Chimie Physique at Bordeaux Published by Butterworths Scientific Publications Ltd 1947 Colloidal Electrolytes and Solutions Trans. 43* 1948 The Interaction of Water and Porous Materials Disc. 3 1948 The Physical Chemistry of Process Metallurgy 4* 1949 Crystal Growth 5* 1949 Lipo-pro teins 6 1949 Chromatographic Analysis 7* 1950 Heterogeneous Catalysis 8* 1950 Physico-chemical Properties and Behaviour of Nuclear Acids Trans.46" 1950 Spectroscopy and Molecular Structure and Optical Methods of Investigating Cell Structure Disc. 9" 1950 Electrical Double Layer Trans. 47* 1951 Hydrocarbons Disc. 10* 1951 The Size and Shape Factor in Colloidal Systems 11* 1952 Radiation Chemistry 12* 1952 The Physical Chemistry of Proteins 13 1952 The Reactivity of Free Radicals 14 1953 The Equilibrium Properties of Solutions on Non-electrolytes 15* 1953 The Physical Chemistry of Dyeing and Tanning 16* 1954 The Study of Fast Reactions 17* 1954 Coagulation and Flocculation 18* 1955 Microwave and Radio-frequency Spectroscopy 19 1955 Physical Chemistry of Enzymes 20 1956 Membrane Phenomena 21 1956 Physical Chemistry of Processes at High Pressures 22 1957 Molecular Mechanism of Rate Processes in Solids 23 1957 Interactions in Ionic Solutions 24 1958 Configurations and Interactions of Macromolecules and Liquid Crystals 25 1958 Ions of the Transition Elements 26 1959 Energy Transfer with special reference to Biological Systems 27 1959 Crystal Imperfections and the Chemical Reactivity of Solids 28 1960 Oxidation-Reduction Reactions in Ionizing Solvents 29 1960 The Physical Chemistry of Aerosols 30 1961 Radiation Effects in Inorganic Solids 31 1961 The Structure and Properties of Ionic Melts 32 1962 Inelastic Collisions of Atoms and Simple Molecules 33* 1962 High Resolution Nuclear Magnetic Resonance 34 1963 The Structure of Electronically Excited Species in the Gas Phase 35 Faraday Discussions Date 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 1980 1981 1981 1982 1982 1983 1983 1984 1984 1985 1985 1986 1986 1987 1987 1988 1988 1989 1989 1990 1990 1991 1991 1992 1992 1993 1993 1994 1994 1994 1995 1995 Subject Fundamental Processes in Radiation Chemistry Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Absorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Oxidation Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces PrecipitationPotential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Ion-Solvent Interactions Colloid Stability Structures and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids PhotoelectrochemistryHigh Resolution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Electron and Proton Transfer Intramolecular Kinetics Concentrated Colloidal Dispersions Interfacial Kinetics in Solution Radicals in Condensed Phases Polymer Liquid Crystals Physical Interactions and Energy Exchange at the Gas-Solid Interface Lipid Vesicles and Membranes Dynamics of Molecular Photofragmentation Brownian Motion Dynamics of Elementary Gas-phase Reactions Solvation Spectroscopy at Low Temperatures Catalysis by Well Characterised Materials Charge Transfer in Polymeric Systems Structure of Surfaces and Interfaces as studied using Synchrotron Radiation Colloidal Dispersions Structure and Dynamics of Reactive Transition States The Chemistry and Physics of Small Metallic Particles Structure and Activity of Enzymes The Liquid/Solid Interface at High Resolution Crystal Growth Dynamics at the Gas/Solid Interface Structure and Dynamics of Van der Waals Complexes Polymers at Surfaces and Interfaces Vibrational Optical Activity: From Fundamentals to Biological Applications Atmospheric Chemistry : Measurements, Mechanisms and Models Gels Volume 36 37 38 39 40 41* 42* 43 44 45 46 47 48 49* 50* 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65* 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88* 89 90 91 92 93 94 95 96 97 98 99 100 101 * Not available; for current information on prices etc., of available volumes.please contact the Marketing Ofjicer,Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB 4WF,stating whether or not you are a member of the Society.

ISSN:1359-6640    

DOI:10.1039/FD995020X003

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

FU~U~UY 1995,102, 17-29D~SCUSS., Towards Quantum Mechanical Characterization of the Dissociation Dynamics of Ketene Sean C. Smith Department of Chemistry, University of Queensland, Brisbane, Qld 4072, Australia A combination of numerical techniques designed for efficient computations with very large basis sets which enables the spectral analysis of transitional-mode eigenstates in the dissociation of ketene is described. At the heart of our methods is a pseudo-spectral algorithm for the action of the transition- al mode Hamiltonian (describing the rocking/bending motion of the disso- ciating fragments) on a state vector. This allows the multiplication of the state vector by the Hamiltonian matrix without explicitly storing the matrix, thus enabling very large basis sets to be managed.With the radial separa- tion between the fragments frozen, transitional-mode eigenstates at energies close to threshold, where the adiabatic channels are moderately well separated, are readily computed with the Lanczos iterative technique. At higher energies the spectrum rapidly becomes extremely dense and con-vergence of the Lanczos algorithm is very slow. However, combinations of eigenfunctions with a bandwidth of a few wavenumbers or better can be obtained by shifted inverse iteration. Illustrative results from computations using the Klippenstein-Marcus model interaction potential for ketene are presented. 1. Introduction In recent years, quantum dynamical calculations for molecular systems with increasing dimensionality have become possible through increasing hardware performance and the development of numerical methods which facilitate the handling of very large basis sets (see, for example, Ref.1-14). New developments have been made in the study of low- energy van der Waals clusters,2*10~' '-' highly vibrationally excited molecules,18-20 and scattering systems, both rea~tive~*~*~-~ and inelastic.21 IJnimolecular dissociation reactions provide a fascinating and challenging field for the study of reaction dynamics. They provide an experimental advantage in that laser- induced dissociation of jet-cooled molecules enables very precise definition of the total energy and the total angular momentum of the dissociating ~ystem.~~-~~ This helps to resolve congested spectra, but also makes theoretical modelling easier since fixing the total angular momentum J equal to zero reduces the dimensionality of the problem and yet may provide a reasonable representation of what is happening in the experiment, since the jet-cooled molecules have a very low rotational temperature.Full-dimensional modelling of the photodissociation of triatomic species,' 2-'4 in conjunction with high- precision experiment^'^.^^ has revealed much detail about the dynamics of direct disso- ciation processes occurring on repulsive potential surfaces. The dissociation of a highly vibrationally excited molecule on a potential surface which supports a deep well presents many, as yet unsolved, challenges for dynamical modelling.Not the least of these chal- lenges is the generation of a potential surface of sufficiently high quality to provide meaningful comparison with experiment.29730 Recent work by Klippenstein et aL3 on 17 Dissociation Dynamics of Ketene the title molecule, ketene, represents a significant advance in this direction, and has enabled enhanced accuracy and reliability in the computation of absolute micro- canonical rate coefficients for the dissociation process. Distinct, in the Born-Oppenheimer sense, from the question of computing a reliable potential surface is the question of what to do with it once it has been obtained. One option, which for full-dimensional studies is perhaps still the only feasible one, is clas- sical dynamical simulation of the dissociation process.Monte Carlo sampling of initial conditions followed by time-dependent solution of the equations of is now a well established methodology. Quantum mechanical studies are limited in terms of the number of dimensions that can be managed, particularly in view of the high excitation energies involved, which implies that for each dimension a large number of basis states is required. Reduced-dimensional quantum studies can often provide a rea-listic picture of the dynamics when it is possible to identify certain degrees of freedom which are effectively decoupled on the timescales involved from those modes which are to be explicitly incorporated in the scattering computation. This possibility has been exploited to particular effect in the modeling of van der Waals ~ l ~ ~ t e r wherein the internal vibrations of the monomers are often considered separable from the intermolecular van der Waals modes.A reduced-dimensional model can be formulated which corresponds to the physical picture of two rigid rotors tumbling about each other under the influence of a weakly attractive potential. This is a physically realistic and computationally tractable model, and a number of papers presented at the recent Faraday Discussion on van der Waals clusters are representative of the type of model- ling which is currently feasible using variational methods. 36 Diffusion Monte Carlo rep- resents an altogether different approach which can be used to compute low-lying This technique, although slower to converge, has reduced memory require- ments in comparison with variational methods, and has been used recently to compute low-lying states for the water trimer.16 Unimolecular dissociation proceeding on a potential surface where there is no barrier to the reverse recombination is an important class of reactions where an approx- imate separation of modes much analogous to the case of van der Waals clusters may be usefully exploited in order to study the dynamics in the exit channel region.In this case, the internal vibrations of the dissociating fragments are commonly assumed to be adia- batically separated from the reaction coordinate (the radial separation) and the so-called transitional modes, i.e. the fragment rotations and the orbiting rotation.There is a sig- nificant amount of indirect evidence which supports this viewpoint. (1) ab initio opti-mization of structures and frequencies along the reaction coordinate : taking the dissociation of ketene in its ground (singlet) electronic state as an example, ab initio electronic structure calculations3 reveal that the optimized geometries an internal vibrational frequencies of the fragments relax essentially to those of the completely separated methylene and carbon monoxide species by the time the carbon-carbon bond length has stretched to ca. 2 A. At this stage, however, there is still substantial anisot- ropy associated with the rotations of the fragments within the ‘activated complex’. An interpretation which suggests itself is that the internal vibrations of the fragments are, by and large, impervious to the proximity of the other fragment from about this separa- tion onwards, although the rotations of the fragments may still be strongly coupled.(2) statistical modelling of product state distributions : the formalism for incorporating adia- batic separation within the framework of statistical models has been well established through the statistical adiabatic channel model (SACM) of Quack and Tr~e.~’ Marcus and Klippen~tein~*-~~ successfully utilized an adiabatic separation model in order to interpret the observation that the product vibrational modes receive a larger-than-sta- tistical proportion of the available energy in the dissociation of NCN043 and ketene.44 As with the SACM, they assumed vibrational adiabaticity; however, in contrast to the SACM, their model assumed that the fragment rotations are strongly coupled with each S.C. Smith other and with the reaction Coordinate, and hence that the disposal of energy amongst rotations and translations would be statistical. Our computer code has been developed with the target molecule of ketene in mind. In an accompanying paper (Moore, Introductory Lecture), some of the latest experimen- tall developments in regard to ketene will be summarized. By way of background to some of the theoretical observations which we make below, it is germane to summarize briefly the experimental observations in Professor Moore’s laboratory, and the questions arising therefrom, which provided some of the original motivation for this work.Moore and co-workers observed22 that the product rotational state distributions of the singlet methylene fragment appear to be statistical at energies close to threshold (100-200 cm-’ excess), but that at higher excess energies the distributions become increasingly cold relative to those what would be predicted by phase space theory (PST). Contrast-ingly, the rotational state distributions of carbon monoxide do not appear to be signifi- cantly cold. These observations are in marked contrast with the situation in NCNO dissociation, where the product rotational state distributions measured by Wittig and co-~orkers~~are statistical. What is happening with ketene? Why does the lighter frag- ment come out rotationally cold whereas the heavier fragment does not? Clearly, CH2 has a greater rotational state spacing and a shorter rotational period than CO.Is some sort of adiabatic hierarchy operating, such that CH, behaves in a partially adiabatic manner whereas CO exchanges energy with the reaction coordinate in a more or less statistical manner? How would one demonstrate this from a quantum or classical dynamical standpoint? These questions have relations with the issue of searching for partial adiabatic invariants, which is an active area of research in the field of chaotic dynarni~s.~~-~~In our work, we have taken as an ansatz the approximation of adiabatic separation of the internal fragment vibrations from the reaction coordinate and the transitional modes. We therefore formulate a Hamiltonian for two rigid rotors inter- acting via a potential with arbitrary anisotropy, and seek to develop the requisite com- putational technology for probing the quantum dynamics associated with the transitional modes as they proceed from the transition state, which governs the absolute microcanonical rate coefficients, out towards the asymptotic region where the product state distributions can be analysed.The expressions ‘in the exit channel region’ and ‘from the transition state out’ may justifiably be regarded as somewhat vague prescrip- tions for the region where adiabaticity of the fragment internal vibrations holds; unfor- tunately, for want of more precise information, that is the way things are at the present time.In Section 2 below, we summarize the main features of the pseudo-spectral algorithm that we have developed4’ for operating with the exact transitional-mode (TM) Hamilto- nian on an arbitrary state vector. We then discuss in Section 3 some representative results from computations of individual TM channel eigenvalues and eigenfunctions at low energies where the spectrum is discrete and can be well resolved. In Section 4 we discuss the issues encountered when attempting to compute energy-resolved quantities (such as narrow-bandwidth combinations of eigenfunctions) at higher energies, where the transitional-mode state densities are so large that one is for all intents and purposes dealing with a continuum.Results are presented for the shifted inverse iteration which we have developed for this purpose. 2. Pseudo-Spectral Action of the Transitional-Mode Hamiltonian The choice of coordinate system and representation are central features of almost any quantum mechanical computation. For a gaseous cluster consisting of two monomers (or, in the present context, a loose transition state consisting of two dissociation fragments), two different types of body-fixed coordinate systems are commonly used. The first is appropriate when the two monomers are of roughly similar mass, and defines Dissociation Dynamics of Ketene the body fixed axes (x, y, z) such that the z axis is oriented along the line joining the two monomer centres of mass.35 A second coordinate system, which defines the (x, y, z) axes as being the principal axes of the heavier fragment” is useful when the monomers are of greatly disparate mass, such as clusters of atoms with substituted arornatics.l0*l For ketene dissociation, the first coordinate system is most appropriate.The choice of representation is influenced by a number of factors. Principal among these are the diagonal dominance of the resulting Hamiltonian matrix, the ease of evalu- ation of the matrix elements, and whether or not one has to store the matrix explicitly. The first of these factors has an important effect on the performance of any iterative numerical algorithms utilizing the Hamiltonian, and we shall discuss this further below. The second and third factors become increasingly important as the size of the basis set increases.For the illustrative calculations that we present below, we have used a stan- dard basis set of ca. 2 x lo5 basis functions. This is not unusually large, but already the explicit storage of the Hamiltonian matrix is out of the question. Pseudo-spectral methods have been developed by many workers for application to molecular dynamics problems over the past ten year^.^'-'' The principal advantage of the approach is that one is able to avoid explicit storage of the Hamiltonian matrix by taking advantage of the fact that the Hamiltonian is the sum of kinetic and potential energy operators, which are diagonal (or, for more complex kinetic energy operators, nearly diagonal) in momen- tum and position representations, respectively.So, for example, if the state vector is expressed in a discrete variable representation (DVR), then the potential energy may be approximated as diagonal and can be applied simply by direct multiplication, The kinetic energy is not diagonal, but one can transform the state vector to a momentum representation (taking advantage of fast Fourier transforms where appropriate) wherein the kinetic energy operator is trivially applied, followed by reverse transformation to the DVR. In certain cases, one is able to avoid explicitly transforming the state vector by analytically evaluating the kinetic energy matrix elements (e.g. Ref. 1 and 5). The key feature is that one effectively multiplies a state vector by a Hamiltonian matrix without needing to store that matrix or even needing to evaluate its matrix elements explicitly in any single representation.In this and another recent paper,49 our working basis set consists of body-fixed Wigner functions, in terms of which the kinetic energy operator is nearly diagonal and so may be trivially applied. It is similar to that devised by Brocks et a1.,35 but in contrast to theirs the present basis is not Clebsch-Gordan coupled. This has an advantage in speeding up the transformation to the DVR without introducing much additional com- plexity into the kinetic energy matrix. The action of the potential operator is effected through transforming the state vector to a DVR in the four dimensional angular sub- space defined by the internal angles of the system, i.e.the body-fixed 0 and x Euler angles for CH, and the body-fixed 0 and 4 angles for CO. Details of this formulation of the TM Hamiltonian have been presented in Ref. 49 (see also Ref. 57). For completeness, we include a brief summary in the appendix. 3. Low-lying States The consequence of not explicitly storing the Hamiltonian matrix is, of course, that one cannot use standard techniques for matrix diagonalization. Since all we have is the action of the Hamiltonian on a state vector, not the matrix itself, we must resort to iterative methods which utilize the effect of matrix multiplication on some initial seed vector to extract eigenvalues and eigenvectors of the matrix itself.The Lanczos tech- nique can be usefully applied to converge matrix eigenvalues and eigenvectors iteratively in the low-density regions of a spectrum. We have demonstrated its ability to converge low-lying channel eigenvalues and eigenfunctions for the ketene problem in Ref. 49. For those and the present illustrative computations, we have chosen the analytic model for S. C. Smith the interaction potential between the CH, and CO fragments used by Klippenstein and Marcus.42 The interaction potential is written as the sum of an effective bonding poten- tial and non-bonding interactions, q'nt= v, + L where the bonding potential is given by an effective Varshni potential, modified by simple angular anisotropy factors,42 and the non-bonding potential is described by two- body Lennard-Jones interactions between all pairs of atoms except the carbon-carbon pair.Our calculations have been carried out for zero total angular momentum, hence J = M = R = 0. The upper limits on the remaining indices of our Wigner basis func- tions were set atjmax(CHz) = 15, k,,,(CH,) = 10,jmax(CO)= 25, R,,,(CO) = 10, leading to a basis of ca. 2 x lo5 functions. These limits were arrived at by testing for con- vergence of the lowest energy state at the smallest centre-of-mass separation considered, 3.0 A. In Table 1, expectation values for the energies and corresponding dispersions are presented for the five lowest-energy eigenfunctions computed at a centre-of-mass separa- tion R = 4.0 A, together with the number of Lanczos iterations required to achieve these.Table 2 contains corresponding data for the five lowest-energy eigenfunctions computed at R = 3.0 A. A comparison of the data in the two tables reveals the fact that with decreasing separation of the fragments, convergence of the channel eigenvalues rapidly becomes much more difficult to achieve. This is a reflection of the rapidly increasing anisotropy of the potential surface with decreasing separation, which causes the Hamiltonian matrix in the Wigner function representation to become much less diagonally dominant and hence degrades the performance of the iterative Lanczos scheme. Hence, as one might expect, the Wigner function basis is most appropriate at larger separations.In order to accelerate convergence at smaller separations, one might seek an alternative basis which is better adapted to the large anisotropies which occur as the fragment rotations turn into bends. Some options for pursuing this direction are considered in the conclusion. An interesting property of the low-energy channel eigenfunctions computed with this interaction potential is illustrated in Fig. 1. The projection norm of the five lowest- energy eigenfunctions is plotted for projection onto states with fixed values ofj,, and also for projection onto states with fixed values ofj,,,. At this separation, the model Table 1 Five lowest-energy states, computed at R = 4.0 8, energy/cm- dispersion/cm-' no.of L~ ~ ~~ ~~~~~~~~~ anczos iterations -70.3272 3.55 x 10-l0 300 -53.8334 8.0 x lo-'' 500 -52.4966 4.8 x 10-9 500 -46.8082 1.7 x 600 -45.5128 3.5 x 10-9 700 Table 2 Five lowest-energy states, computed at R = 3.0 A energy/cm-dispersion/cm-' no. of Lanczos iterations -2679.93 1.2 x 10-7 1600 -2446.72 6.3 x 10-5 2000 -2408.53 4.4 x 10-3 2000 -2379.66 1.5 x 10-3 2000 -2294.30 1.7 2000 Dissociation Dynamics of Ketene n oaE2 06 -0 04 0-202 a 0 h z EO8 06 04 0*p02 a 0 E0 oa 06 % 04 u 0 '7 02 aE! - F 1: U9 04 0 E!.-a 02 - 0 -08 € 06 2 04 0 Y.E02 - 0 0 1 2 3 4 5 0 1 2 3 4 5 j(C0) J(CW Fig. 1PI-0jection norm of the five lowest-energy eigenfunctions (computed at R = 4.0 A) as a function of thej quantum number for CO (left half) and for CH, (right half).The projection norm sums to one for the CO projection and for the CH, projection separately. potential has a minimum that is ca. 102 cm-' below the threshold for dissociation, and the highest barrier to rotation is ca. 4400 crn-'. The eigenfunctions clearly involve a much greater amount of mixing of CO free rotor states than for CH,. This is what one might expect intuitively on the basis of perturbation theory, given the greater free-rotor state spacing of methylene. It suggests the possibility that the angular momentum quantum numbers of the methylene fragment are more likely to function as partial adia-batic invariants than those of CO, at least in the later stages of the dissociation process.Nikitin et al. have demonstrated for simplified interaction potentials that there is a charge in nature of the adiabatic channel eigenfunctions from perturbed free-rotor states at larger separations, through a transition region, to pseudo-harmonic-oscillator states at small separations (Ref. 59, and references therein). The results of Fig. 1 are indicative that in the case of ketene, where the fragments have quite different rotational constants, qualitatively different transitional behaviour might be expected for CH, as compared with CO. Hence, the concept of perturbed free-rotor states for which the angular momentum quantum numbers function as adiabatic invariants may be a useful one for describing the dynamics of CH, at larger separations, but not very useful for CO.S. C.Smith 3 2.5 h -2'E -2g 1.5 .4m k12 SEj2 0.5-0 -0.5 0 500 1000 1500 2000 number of Lanczos iterations Fig. 2 Log(dispersion) as a function of the number of Lanczos iterations for the approximate Lanczos eigenvectors. Lines are as in Fig. 5. (-* -), -70.32 cm-'. Despite the suggestive nature of the projection data represented in Fig. 1 for the low-lying channel eigenfunctions, it is necesary to explore a range of energies several hundred wavenumbers above threshold, since it is in this region that a marked deviation of the product CH, rotational state distributions from statistical predictions becomes most apparent. This introduces a number of additional difficulties which we address in the next section.4. Computationof Energy-resolved Functions in the Dense Spectral Region The rovibrational density of states for the transitional modes rises rapidly with energy, such that a (classical) trace of the density operator in phase space by Monte Carlo methods is often a reliable method for evaluating statistical rate coefficients for energies not too close to threshold (see, e.g. Ref. 42,60 and 61). Although it would be satisfying to extract and analyse individual eigenfunctions out of the quasi-continuum, an equally useful and more achievable objective is to develop a stable spectral filter algorithm which will enable us to compute narrow bandwidth combinations of eigenfunctions. There are at least two good reasons for seeking such an algorithm. First, one could use the energy-resolved functions to carry out short-time propagations and analyse the results in search of partial adiabatic invariants. Such propagations can be carried out in the TM (i.e.the angular) subspace at specified separations in order to understand in which regions adiabatic invariants exist and can be usefully invoked to explain the reac- tion dynamics. Secondly, with incorporation of the radial coordinate into the explicit basis set (thereby adding one more degree of freedom to the computation), such a filter algorithm will be a central component of the computation of outgoing scattering waves which provide a direct simulation of the product state distributions.'The simplest way to try to compute a narrow-bandwidth combination of eigen- functions with an expectation value for the energy close to a nominated value E would be to use the Lanczos algorithm. After n iterations, the Lanczos algorithm generates a tridiagonal representation T, of the Hamiltonian matrix HTM in terms of an orthogonal basis in the Krylov subspace of degree n. The nature of the algorithm is such that the eigenvectors of T, corresponding to eigenvalues in the sparse region of the spectrum coverge into the true eigenvectors of HTM most quickly. The eigenvectors of T, corre-sponding to eigenvalues in the dense spectral region converge much more slowly with increasing n, but nevertheless constitute combinations of true eigenfunctions.The rele- vant question is: how quickly does the bandwidth of these approximate eigenvectors decrease to within an acceptable limit? Fig. 2 shows the decreasing dispersion of a Dissociation Dynamics of Ketene 1 0.8 E 0.6 0.2 0 0 50 100 150 200 250 300 number of MINRES steps Fig. 3 Residual norm as a function of the number of MINRES steps, i.e. the number of Lanczos recursions within the MINRES algorithm for the action of the Green operator, with a shift energy of 1000 cm-I and an initial seed vector which is the unit vector 1. number of Lanczos eigenvectors, computed at R = 4.0 A, as a function of n. The disper- sion is computed in the usual fashion as the residual norm, ~j = ($; I (HTM -ql2I $Y>"~ (2) In eqn. (2), 4is the expectation value associated with the approximate eigenvector $7, = ($7 1 HTM I$;> (3) For each number of iterations, n, the dispersions plotted are those for the eigenvectors with energy 4 closest to -70.32 (the zero-point energy), 0 (i.e.the asymptotic threshold energy), 500, 1000 and 2000 cm-l. The rate of reduction in the dispersion at higher energies is unacceptably slow, and a better method is required. Nevertheless, the rate of convergence of the standard Lanczos eigenvectors is a useful yardstick against which to measure other algorithms. Shifted inverse iteration is a standard means of computing sucessively improving approximations to an eigenvector close to a specified energy E.The scheme involves repeated action with the Green operator, until the desired level of convergence is reached. Acting with the Green operator as indicated in eqn. (4) requires the solution of the linear system of equations (HTM-Ebn+1 = xn (5) There exist a number of iterative methods for the solution of large linear systems such as eqn. (5),most of which seek a solution for x, + in Krylov space, so that after i iterations the approximate solution xk+ may be expressed as 7xi+ 1~ span(xn7 (HTM -E)xn 9 (HTM -E)2~n-* * 3 (HTM -W1xnl (6) The conjugate gradient method is perhaps the most efficient, but is prone to instability when the matrix is indefinite, as is the case with the shifted Hamiltonian in eqn. (5). For indefinite Hermitian matrices, the minimum residual algorithm (MINRES) of Paige and Saunded2 is a more stable alternative.MINRES generates a Lanczos sequence as an orthogonal basis for the Krylov space and determines xL+ according to the criterion of minimal residual norm. 2500 2000 'E $ 1500 MG ;1000 rd2 500 0 0 10 20 30 40 number of inverse iteration steps Fig. 4 Mean energy of iteratively improved vectors in the inverse iteration algorithm as a function of the number of inverse iteration steps. The line patterns correspond to those in Fig. 5. We examine initially the convergence properties of the MINRES algorithm for the first inverse iteration step of eqn. (4). Fig. 3 shows the decrease in the residual norm for the approximate solution xi as a function of the number of steps i in the MINRES loop, i.e.the number of Lanczos basis vectors computed for the expansion of x;. Convergence of the action of the Green operator is disappointingly slow. This ought perhaps not to be surprising when we realise that the sequence of Lanczos vectors generated in the MINRES scheme as a basis for expanding the solution x1 is identical to that which would be obtained by performing a normal Lanczos diagonalization calculation with xo as the seed vector: incorporation of the shift does not alter the Lanczos vectors. Since the shift energy E is in a part of the spectrum which is quasi-continuous, the desired solution x1 to the linear system would be a very narrow bandwidth combination of eigenvectors about E.Hence, we cannot expect the approximate solution xi for the action of the Green operator to converge significantly faster than the approximate eigen- vectors of the Lanczos diagonalization in the same part of the spectrum. The fact that it is dificult to converge the action of the Green operator precisely does not, of course, prevent the shifted inverse iteration scheme from converging. As long as the partial convergence of the action of the Green operator is sufficient to gener- ate x,, with a reduced dispersion in comparison with x,, the inverse iteration scheme /-r 72Ee 1.5 .+ c .c!$l .a G 0.5 9 0 -0.5 0 10 20 30 40 number of inverse iteration steps Fig. 5 Log(dispersion) for iteratively improved vectors in the inverse iteration algorithm as a func-tion of the number of inverse iteration steps.(-. -), 0;(---), 500; (-----), 1000; (-), 2000 cm-l. Dissociation Dynamics of Ketene will continue to converge. Results which indicate typical performance of the inverse iteration scheme are plotted in Fig. 4 and 5 for four different shift energies, 0, 500, 1000 and 2000 cm-'. Fig. 4 shows the convergence of the mean energies, and Fig. 5 the reduction in the dispersions, as a function of the number of inverse iteration steps. For each of the four energies, the iterative scheme started from the same initial seed vector xo (the unit vector l),and the number of MINRES steps was held fixed at 50. Hence, the calculations for Fig.4 and 5 contained essentially the same total number of matrix multiplications as the Lanczos calculations which produced the results of Fig. 2. There is a certain amount of flexibility in the choice of number of MINRES steps us. the number of inverse iteration steps. For instance, one can achieve almost as good a result using as few as 10 MINRES steps and a proportionately greater number of inverse iteration steps. Conclusions We have reported progress in the development of computational algorithms for explor- ing the quantum dynamics of dissociation reactions occurring on potential surfaces with no barrier to the recombination process. The Lanczos algorithm in conjunction with pseudo-spectral action of the Hamiltonian matrix provides a powerful tool for the com- putation of channel eigenvalues in the discrete region of the spectrum.Analysis of low- lying channel eigenfunctions computed for the dissociation of singlet ketene indicates that the angular momentum quantum numbers for the methylene fragment may behave as partial adiabatic invariants to a much greater extent than those of carbon monoxide at centre-of-mass separations of 4.0 and greater. Confirmation or otherwise of this suggestion will await an ab initio potential surface for the ketene transitional modes covering the region from the variational transition states out to the non-interacting products, coupled with further dynamical calculations. The methods we have discussed facilitate computations with large basis sets.Our illustrative calculations have been carried out with a basis of ca. 2 x lo5 functions; however, basis sets well in excess of one million functions will not present serious prob- lems. Hence, the incorporation of the radial coordinate explicitly into the dynamical calculations is an achievable objective. Notwithstanding the luxury of being able to use large basis sets, it should be noted that the Wigner function basis which we have used for these calculations is a primitive one. The comparison of Tables 1 and 2 indicates clearly that performance of the algorithm suffers as the potential becomes highly aniso- tropic. This suggests that the efficiency of the algorithm at smaller separations might be substantially enhanced, and the size of the basis set correspondingly reduced, if the pseudo-spectral scheme is generalised to use a basis set that is adapted to the large anisotropies.An approach of this type has been successfully implemented by Leforestier co-workers recently to compute six-dimensional eigenfunctions of the H,O, molec~le.~ We are currently investigating the application of this technique within the present context. The success of shifted inverse iteration as a spectral filtering algorithm, generating narrow-bandwidth functions with energy dispersions of a few wavenumbers or better, opens up the possibility of energy-resolved quantum dynamical calculations over an energy range of several hundreds of wavenumbers above the threshold for ketene disso- ciation, which is sufficient to address the experimental observations of non-statistical rotational product state distributions.This work has been carried out with the support of a University of Queensland New Staff Grant (94/NSRG086). We wish to thank Dr Claude Leforestier and Professor C. Bradley Moore for many helpful discussions. S. C. Smith Appendix +:,=jA The TM Hamiltonian is written as the sum of the rotational kinetic energy_for each fragment KfFO' (F = A, B)>the rotational kinetic energy for the orbital rotation Kerb, and an interaction potential ynl AT, = kr' + KT'+ Rorb + cnt (7) We use body-fixed coordinates with two-angle embedding,35 in terms of which @" has the standard form kp' = + B,JT2 + C,JT2] = +[A, + BF)y2+ (2CF-A, -B,)J?~ + +(AF-BFbY+PI2)] (8) and kerb has the form In eqn.(8), the rotational constanis (wifh units of energy) and angular momentum oper- ators have their usual meanings. Jand j in_eqn. (9) represent, respectively, the magnitude of the total angular momentum vect_or J, and the magnitude of the sum of the two fragment angular momentum vectors J = JA + cf,, so that J++ [ + f (10) .fzand j, are the corresponding body-fixed z-axis projections, and j, and the raising and lowering operators. The working representation is a direct-product basis set consisting of body-fixed Wigner functions 9 7I JMQA kAj, k, 0,) = A-1D CJL*(a,P, O)~g;)kfA(4A OA, xA)D@~~~(#R XJ (1 1) where OF, 4, and X,are the body-fixed Euler angles for fragment F, JV is the nqrmal- izing factor for the three functions, M is the quantum number for projection_of J onto the (spatially fixed) 2 axis and i2 is the quantum number associated with the Jz operator for projection of J^ onto the (body-fixed) z axis.By definition, QZA= R -QB. Since J and M are good quantum numbers, they are suppressed for brevity. The basis set of eqn. (1 1) differs from that of Brocks et in that-we do not make use of Clebsch-Gordan (C-G) coupling t? obtain eigenfunctions of thej operator. As with their basis, evaluation of the action of KYt and RF' yields the standard form (nYa kl,jb kk 0; 1 RY' + I?;' 1 RjAkAjuk,3R,) = 'Wf2 'j2A 'jkjn dQ&l,[Gkgk, hjA kAkA + 'kAkA hj, k Ak~1 (l 2, where hjFki.kF = $'ki.kF[(BF cF)jF(jF + 1) + (2A, -BF -CF)ki] + f~(BF-c,)ci.kF ci.kFf 1 (13) and C$ = [j(j + 1) -k(k _+ l)J1'2 (14) eqn.(13) follows the prolate top convention, such that the A principal axis of a fragment is regarded as its fragment-fixed z-axis. The actions of Jz, J, , J, and J, on the basis functions of eqn. (11) are evaluated in a manner analogous to that which Brocks et al. Dissociation Dynamics of Ketene used for their basis set,35 since both their basis and the present basis are (J,R)-resolved. The action ofjQ and of;* on the present basis set may be evaluated with the use of the following identities J3-2-]A + 1% + 2.!]: I?! .$-k + (15) ? ?A % .l+=J+ +j+ (16) The matrix elements are then readily shown to be The rotational kinetic energy matrix summarized by eqn.(12)-(14) and (17) is very sparse and hence may be stored, or generated as required. The potential operator is applied by a change of representation from the Wigner basis to a discrete variable representation (DVR) in the internal anguiar subspace of the system. For the general case of two non-linear molecukar fragments, qntdepends on five of the body-fixed Euler angles of the two fragments. vntis independent of the external Euler angles a and p which orient the body-fixed z-axis, and, by symmetry of the rota- tion of the whole system about the_ z-axis, Fntis effectively independent of one of the fragpent $ angles. Assuming that qnthas been evaluated explicitly as a function of 4B, , 8, , xB), only the indexes (jA, kA , j, , kB ,aB)i.e.qnt= Knt(8A, xA, ~j~ in the state vector need be transformed. Hence, in general a five-dimensional transformation is involved, and for the case of ketene this is a four-dimensional transformation. The procedure for carrying out this transformation has been described by Lef~restier.~~ After multiplica- tion by the potential, back transformation to the Wigner basis then completes the oper- ation of acting with Knt on the state vector. References 1 S. C. Althorpe and D. C. Clary, J. Chem. Phys., 1995,102,4390. 2 M. J. Weida, J. M. Sperhac, D. J. Nesbitt and J. M. Hutson, J. Chem. Phys., 1994,101,8351. 3 U. Manthe, T. Seidemann and W. H. Miller, J. Chem. Phys., 1994,101,4759. 4 M. S. Reeves, D. C. Chatfield and D.G. Truhlar, J. Chem. Phys., 1993,99,2739. 5 J. Antikainen, R. Friesner and C. Leforestier, J. Chem. Phys., 1995, 102, 1270. 6 D. C. Clary and G. C. 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Light, J. Chem. Phys., 1986,85,4594. 21 M. H. Alexander and P. J. Dagdigian, J. Chem. Phys., 1994,101,7468. 22 I. Garcia-Moreno, E. R. Lovejoy and C. B. Moore, J. Chem. Phys., 1994,100,8890. S. C.Smith 23 H. F. Davis and Y. T. Lee, J. Phys. Chem., 1992,96,5681. 24 C. X. W. Qian, A. Ogai, J. Brandon, Y. Y. Bai and H. Reisler, J. Phys. Chem., 1991,956763.25 X. Luo and T.R. Rizzo, J. Chem. Phys., 1992,96,5129. 26 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994,101,3618; 3634. 27 T.Baumert, J. L. Herek and A. H. Zewail, J. Chem. Phys., 1993,99,4430. 28 A. Ticktin, A. E. Bruno, U. Briihlmann and J. R. Huber, Chem. Phys., 1988,120, 155. 29 L. B. Hardin, R. Guadagnini and G. C. Schatz, J. Phys. Chem., 1993,97, 5472. 30 J. Ischtwan and M. A. Collins, J. Chem. Phys., 1994,100,8080. 31 S. J. Klippenstein, A. L. L. East and W. D. Allen, J. Chem. Phys., 1994, 101, 9198. 32 T. D. Sewell, H.W. Schranz, D. L. Thompson and L. M. Raff, J. Chem. Phys., 1991,95,8089. 33 W. L. Hase and Y. J. Cho, J. Chem. Phys., 1993,98,8626. 34 N. Markovic, G. Nyman and S. Nordholm, Chem. Phys.Lett., 1989, 159,435. 35 G. Brocks, A. van der Avoird, B. T. Sutcliffe and J. Tennyson, Mol. Phys., 1983,50, 1025. 36 Faraday Discuss., 1994,97. 37 H. Sun and R. 0.Watts, J. Chem. Phys., 1990,92,603. 38 R. Wong and S. C. Smith, work in progress. 39 M. Quack and J. Troe, in Theoretical Chemistry, Advances and Perspectives, ed. D. Henderson, Aca- demic Press, New York, vol. 6B, 1981. 40 R. A. Marcus, Chem. Phys. Lett., 1988, 144,208. 41 S. J. Klippenstein and R. A. Marcus, J. Chem. Phys., 1988,89,4761. 42 S. J. Klippenstein and R. A. Marcus, J. Chem. Phys., 1989,91, 2280. 43 C. X. W. Qian, M. Noble, I. Nadler, H. Reisler and C. Wittig, J. Chem. Phys., 1985,83, 5573. 44 I-C. Chen, W. H. Green and C. B. Moore, J. Chem. Phys., 1988,89,314. 45 0.Hahn and H.S. Taylor, J. Chem. Phys., 1992, %, 5915. 46 R. S. Dumont and P. Brumer, J. Chem. Phys., 1987,87,6437. 47 N. De Leon, J. Chem. Phys., 1992,96,285. 48 M. E. Kellman, J. Chem. Phys., 1990,93,6630. 49 S. C. Smith, Chem. Phys. Lett., 1995, 243, 359. 50 G. Brocks and K. van Koeven, Mol. Phys., 1988,63,999, 51 M. D. Feit, J. A. Fleck Jr., and A. Steiger, J. Comp. Phys., 1982,47,412. 52 D. Kosloff and R. Kosloff, J. Comp. Phys., 1983, 52, 35. 53 R. C. Mowrey, Y. Sun and D. J. Kouri, J. Chem. Phys., 1989,91,6519. 54 C. E. Dateo and H. Metiu, J. Chem. Phys., 1991,95,7392. 55 D. Lemoine, J. Chem. Phys., 1994,101, 10526. 56 M. J. Bramley, J. W. Tromp, T. Carrington Jr., and G. C. Corey, J. Chem. Phys., 1994, 100,6175. 57 C. Leforestier, J. Chem. Phys., 1994, 101, 7357. 58 D. T. Colbert and W. H. Miller, J. Chem. Phys., 1992, %, 1982. 59 E. E. Nikitin, J. Troe and V. G. Ushakov, J. Chem. Phys., 1995,102,4101. 60 D. M. Wardlaw and R. A. Marcus, J. Chem. Phys., 1985,83,3462. 61 S. C. Smith, J. Phys. Chem., 1993,97,7034. 62 C. C. Paige and M. A. Saunders, SIAMJ. Numer. Anal., 1975,12,617. Paper 5/06195D; Received 19th September, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950200017

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss., 1995, 102, 3 1-52 Rotation-Vibration State-resolved Unimolecular Dynamics of Highly Excited CH30 (2*E) Part 3.t4tate-specific Dissociation Rates from Spectroscopic Line Profiles and Time-resolved Measurements Stefan Dertinger, Alfred Geers, Jurgen Kappert, Jorg Wiebrecht and Friedrich Temps* Max-Planck-lnstitut fur Stromungsforschung, Bunsenstrasse 10,37073 Giittingen, Germuny Vibration-rotation quantum-state resolved measurements of the unimolecular dissociation rates of highly vibrationally excited CH30 (82E) have been performed over a wide rang: of excitation energies (7000 d E/cm-' < 10000). Single excited CH30 (X) quantum states were prepared using the method of stimulated emission pumping (SEP). State-specific decay constants were determined from direct time-resolved measurements using laser-induced fluorescence detection (LIF) of the excited states and from SEP line profiles measured at higher resolution.In very narrow energy windows, the measured decay constants were found to vary statistically by up to two orders of magnitude. These state-specific fluctuations are in con- trast with the traditional picture from unimolecular rate theory (e.g. RRKM theory). The fluctuations were analysed statistically. The average decay rates were found to increase with increasing molecular excitation energy. This general trend could be nicely described by an RRKM model on average. Indications for small deviations were observed at high energies. Viewed in connection with related data on the kinetics of intramolecular vibrational energy randomization (IVR) processes, these deviations may reflect the inherent limitations of statistical theory at high energies where dissociation and IVR compete.Direct experimental studies of the unimolecular dynamics of highly vibrationally excited molecules are of importance for our understanding of elementary chemical processes from unimolecular bond fission, elimination and isomerization reactions to complex- forming bimolecular reactions. Great interest exists especially in fully vibration-rotation quantum-state resolved measurements of the unimolecular dissociation rates of selected prototypical molecules in their ground electronic state with precisely known overall excitation energy E, angular momentum J, rovibronic symmetry r, etc., over a wide range of excitation energies.Considered in close conjunction with complementing inves- tigations of the ensuing intramolecular vibrational dynamics in the highly energized molecules, especially intramolecular vibrational energy randomization (IVR), single quantum-state resolved studies constitute critical tests of the fundamental assumptions and predictions of the theories which are commonly used for modelling unimolecular reactions and which have gained importance for predicting the rates and product dis- tributions in numerous reaction systems.'- t Part 2, Ref. 11. 31 State-spec$c Dissociation Rates of CH30 (2,E) The CH30 molecule in its 2,E ground electronic state constitutes a nearly unique, prototypical system for studying the unimolecular dynamics of highly excited molecules from different points of intere~t.~ Using stimulated emission pumping (SEP) spectro-sc~py,~specific highly excited vibration-rotation levels of the molecule can be probed, ranging from the bottom of its potential-energy well to energies considerably higher than the H-H,CO bond-dissociation energy.The data which are obtained add to an unprecedented understanding of the unimolecular kinetics of small- and medium-sized polyatomic molecules at a state-resolved le_vel. Highly vibrationally excited CH30 (X) molecules can, in principle, undergo two unimolecular reactions, dissociation to H + H,CO or isomerization to CH,OH; CH,O (x)-+H + H,CO; A,Hg = +6950 cm-' (1) CH30 (2)-+ CH,OH; ArHg = -3500 cm-' (2) Both reactions require significant activation energies.However, the classical threshold energy for the dissociation reaction is at Eo(l)= 8500 cm-'. This value could be derived from a separate measurement' of the activation energy for the H-D isotope exchange reaction D + H,CO -+ H + HDCO, which proceeds via an intermediate CH,DCO* complex, taking into account all corrections for zero-point energy, tunnelling, reverse dissociation of the CH2DCO* intermediate, etc. On the other hand, according to the available ab initio quantum-chemical calculations, the threshold energy for the isomer- ization reaction is in the range E,(2) = 10000-12000 cm-', i.e. several loo0 cm-' The potential barrier for dissociation to H, + HCO would be much higher again.Hence, decay to H + H2C0 is the lowest energy reaction channel which can be opened. In preceding papers,"." w,e have reported a study of the spectroscopy of highly vibrationally excited CH30 (X) and information on the intramolecular vibrational dynamizs encoded in the spectra. SEP spectra of CH30 (2)were obtained exploiting the 'A,-X ,E electronic transition of the molecule. The difference in the structures of the molecule in the two electronic states results in excellent Franck-Condon (FC) overlap for excitation of highly excited 8 levels. Measured low-resolution SEP spectra were found to exhibit a characteristic, remark- ably robust, vibrational structure that persists up to very high energies in the disso- ciative quasi-continuum of short-lived metastable states (resonance states) of the molecule above the H-H,CO bond-dissociation energy on the X state potential surface.In a zero-order picture which ignores possible mode-to-mode couplings, the main features in the low-resolution spectrum fit to the excited x CO stretching vibration (v,) states with v3 = 1-10. These optically _bright levels define FC windows for fully quantum-state resolved studies over a wide X energy range. High-resol_ution SEP scans over the FC windows showed that above a rather sharp threshold at E(X) z 5000 cm-', the coarse vibrational features correspond to clumps of large numbers (ca. 15-100) of narrowly spaced single CH30 (8)vibration-rotation eigenstates.The complex level structures reflect the extensive mixing between the optically bright CO stretching states, which can be accessed directly in zero order, and the large number of optically dark states in the vicinity, which cannot be reached directly because of, e.g., FC, symmetry or angular momentum restrictions. These SEP spectra encode quantitatively detailed infor- mation on the short-time intramolecular vibrational dynamics at the high molecular excitation energies of interest for the unimolecular bond-fission mechanism. The observed coarse vibrational features in the low-resolution SEP spectra reflect a short-time regular (periodic) vibrational motion along the CO stretching coordinate on a timescale reflected by the widths of the corresponding spectral features. The high- resolution results, on the other hatd, demonstrate the strong vibration-rotation coup-lings between the highly excited X states, and thus the statistical properties which Dertinyer et al.-30 -Ig 20-m0 \7 $10-a 0-Fig. 1 Energy diagram of the CH,O system and experimental approach become effective on a picosecond timescale. The single vibratizn-rotation eigenstates are unassignable in a traditional sense. The density of states at X excitation energies of ca. 6000 cm -approaches the full calculated symmetry-sorted rovibronic state density with assumption of complete breakdown of the K quantum number, i.e., strong mixing of the K-rotor with the vibrational degrees of freedom.The nearest-neighbour level spacing distributions are close to the so-called Wigner distribution, a frequently used criterium for chaotic vibrational dynamics.' 2,13 Exploiting the Fourier relationship between the frequency and time domain, the rate and extent of IVR processes in the highly excited molecules were determined quantitatively. Two tiers were observed, with characteristic IVR times of ca. 0.25 and 2-3 ps, respectively. For a small number of highly excited states, hints were found for the presence of some residual mode-specific influences; these indications, however, pertain essentially only to a class of extreme motion states which contain all excitation energy concentrated in the CO stretching vibration in certain initial rotational states, for which Coriolis coupling is limited.' ' Thus, considering all data together, the evidence indicates that highly excited CH,O (z),above E z 5000-6000 cm-' and at times longer than a few ps after excitation, exhibits essentially sta- tistical vibrational dynamics.With these premises, we have initiasd a close investigation of the unimolecular dis- sociation of the highly excited CH,O (X) states. The unimolecular decay rates of single vibration-rotation quantum states, prepared by SEP with precisely defined values of E, J, r,etc., were studied over a wide range of energies (7000 < E/crn-' < lOO00). Prelimi-nary data for one excitation window were communicated in a recent p~blication.'~ The present paper now discusses the results obtained to date for the complete energy range. Data were collected for E < 8000 cm -from direct measurements with time-resolved detection of the decay of the highly excited states using a stimulated emission pumping laser induced fluorescence (SEP LIF) triple-resonance scheme and for E 2 8000 cm-' by analysing the homogeneously broadened line profiles in measured high-resolution SEP scans.The experimental approach is illustrated in Fig. 1. The experimental results which were obtained allow a first assessment regarding the variation of the quantum- state-specific dynamical properties of very highly vibrationally excited molecules as a function of excitation in a large energy range. This range comprises the regime below the classical threshold for the reaction (E < Eo), where one of the H atoms can tunnel through the small additional potential-energy barrier above the products H + H,CO, and extends into the regime above the classical threshold (E 3 Eo),where the reaction is a normal bond-fission process.State-speciJic Dissociation Rates qf CH30 (x2E) Experimental Section CH,O was generated by 248 nm photodissociation of CH30N0 in a pulsed Ne super- sonic free-jet expansion controlled by a solenoid valve (General Valve) with a 0.8 mm diameter pin-h~le.~ The excimer laser photolysis beam (Lambda Physik EMG-101) was focused ca. 1 mm below the pinhole using anf= 1 m lens. Efficient production of rota- tionally and vibrationally cold CH30 was observed with photolysis pulse energies of ca.50 mJ, Ne backing pressures of 10-15 bar, and with a CH30N0 reservoir temperature of -78°C. Under these conditions, judged from a simulation of the CH30 (A tz) LIF excitation spectrum, the rotational distribution of the molecules in the cold free-jet expansion could be characterized by a temperature of T,,, z 1.5 K. The molecular beam environment allows SEP and unimolecular decay measure-ments to be performed under collision-free conditions. SEP spectra were obtained using the fluorescence dip method employing two counter-propagating pulsed dye lasers (Lambda Physik FL-3002E) pumped by a frequency-tripled Nd : YAG laser (Spectra Physics GCR-3) and an XeCl excimer laser (Lambda Physik EMG-200), respectively.The pump laser crossed the cold region of the s_upersonic jet ca. 30 mm downstream from the valve orifice. Fluorescence from CH30(A) was monitored at right angles to the jet and excitation laser with a photomultiplier (Hamamatsu R-928) usingfll fused silica collection optics and suitable glass and dielectric bandpass filters (Schott) for reducing scattered light. The timing of the pulsed valve, the photolysis laser, the dye lasers and the detection gates was controlled by a digital delay generator (Stanford Research DG-535). To obtain SEP spectra, the dump laser was delayed relative to the pump by 400 ns. The fluorescence dip signals were accumulated for 100 laser shots using a gated integrator (Stanford Research SR-250) with a dual gate normalization scheme to com-pensate for shot-to-shot fluctuations of the pulsed valve opening time and photolysis and pump pulse energies.The normalized signals were digitized and transferred to a PC/AT microcomputer. Absolute wavenumbers were determined with accuracies of k0.5 to & 1 cm-' with the help of a calibrated pulsed UV wavemeter (Burleigh WA- 5500) or by recording simultaneously absorption spectra of I,' S or optogalvanic spectra of Ne and Fe.' High-resolution SEP scats were obtained for determining the line profiles of the resolved dissociative CH30 (X) states by operating the pump and dump dye lasers both with an intra-cavity etalon. Measurements with a high-finesse air-spaced monitor etalon (Lambda Physik) after careful laser alignment demonstrated linewidth reductions to 0.02-0.025 cm-in the visible region (ca.0.035 cm-' after frequency doubling). The experimental resolution w%s checked by measuring repeatedly SEP line profiles of stable or very long-lived CH30 (X) states at different energies. For direct time-resolved decay measurements, the pump and dump dye lasers were pumped using the second and third harmonic output of the Nd: YAG laser, and the XeCl excimer laser was used to pump independently a t_hird dye laser serving as probe. When necessary to ensure preparation of single CH30 (X) vibration-rotation states, the lasers were again operated with etalons. The pump and dump beams were combined using dielectric mirrors, while the probe beam was employed in a counter-propagating configuration. The data-acquisition system and timing of the lasers were contro_lled by the PC.The probe laser was used to detect the decay of the highl=y excited X states prepared by pump Cnd dump by exciting the molecules to a higher A _vibrational level. From this excited A level, one can selectively monitor LIF to the X 0, level using suitable bandpass filters (Laser Optik); this fluorescence is blue-shifted with respect to the pump, dump and probe lasers. The measurements were m2de in two steps. First, after locking the pump and dump lasers to a selected CH30 (X) state and setting the probe time delay to only a few ns, the probe wavelength was scanned and the induced fluorescence was recorded. The resulting single vibration-rotation level (SVRL) LIF Dertinyer et al.7440 7450 7460 7470 (b) I I I I Ill 7440 7450 7460 7470 111 I I 7440 7450 7460 7470 excitation energy/cm-' Fig. 2 High-resolution SEP spectra of CH,O (z)at E z 7460 cm-'. (a) Pump at 3:, pPl,(1.5,+ 1); (b) pump at 3;, 'R,1(0.5, 0); (c) pump at 3;, 'QZ1(0.5,0). excitatio? spectrum shows the possible rovibrational transitions from the selected highly excited X level to the higher vibrational levels. Then, the probe wavelength was fixed to a suitable transition. The SVRL LIF excitation signals were accumulated for a preset number of laser shots, incrementing the time delay between the probe and Cump in steps of a few ns. The SVRL fluorescence plotted us. probe delay time gives the X state decay profile.Background traces with the probe off resonance were subtracted to correct for scattered light. Between 10 and 30 decay profiles were averaged for each state. 8290 8295 8300 8305 8310 h 8885 8890 8895 8900 8905 (c) 9640 9645 9650 9655 9660 excitation energ y/cm -' Fig. 3 High-resolution SEP spectra ofCH,O (x).Pump at 3:, 1'PI1(1.5,+ 1). Table 1 Pump transitions and probed CH,O (x)Franck-Condon windows !2r+ m I energy window pump line pumped A level excited 2levels 5 0 E/cm -FC state branch notation u/cm -" K' J' J revr G '3 6000 31 642.65 0 0 0.5 0.5, 1.5 A2 0s 7000 32 305.01 0 0 0.5 0.5, 1.5 A2 0 " 7320 32 305.01 0 0 0.5 0.5, 1.5 A2 0 3. 7460 31 642.65 0 0 0.5 0.5, 1.5 A2 0 !3. 32 305.01 0 0 0.5 0.5, 1.5 A2 0s 32 317.17 2 1 1.5 0.5-2.5 E +1 32 3 12.66 1 1 0.5 0.5, 1.5 E +1 -5 rp32 306.47 1 0 1.5 0.5-2.5 A1 o"! 7800 31 642.65 0 0 0.5 0.5, 1.5 A2 0% 8300 33 576.92 0 0 0.5 0.5, 1.5 A2 00 8870 35 387.59 1 1 0.5, 1.5 0.5-2.5 --31 w 8900 34 260.25 0 0 0.5 0.5, 1.5 A2 0 0 9660 33 576.92 0 0 0.5 0.5, 1.5 A2 9780 33 576.92 0 0 0.5 0.5, 1.5 A2 Dertinyer et al.SEP Spectra SEP spectra of CH30 (2)were recorded over the complete energy range above the H-H,CO dissociation limit up to E(g)z 10000 cm-’, employing pump lines to several A state vibrational levels.” High-resolution scans over several FC windows selected for state-resolved decay-rate investigations are reproduced in Fig. 2 and 3. The observed spectra show the evolution of the density of states and the spectral widths of the resolved vibration-rotation levels as a function of 8 excitation energy. Dependins on the magnitude of the spectral widths, 6, of the observed lines in relation to the X nearest-neighbour level spacing (Le., the inverse density of states p-I) three different energy regimes can be identified: 6 c p-’, S z p-‘, or 6 > p-l.The prepared highly excited CH,O (8)states have well defined values of the overall excitation energy E, angular momentum J, rovibronic symmetry fevr, and the quantum number G, which is defined” as G = A + I -K modulo 3, with A, 1, and K representing the projections of the electronic orbital, vibrational and rotational angular momenta, respectively, on the molecular symmetry axis.The conditions used to reach the excited states are stmmarized in Table 1. With pump transitions to A levels with J‘ = 0.5 and 1.5, excited X levels can be reached by SEP with 0.5 < J < 2.5. For a large number of x levels, the ossible J values could be distinguished by cross-examining SEP spectra from adjacent R levels and by considering the rotational line structures in the SVRL LIF excitation spectra (in any case, the maximum uncertainty of J is & 1). The values of fevr =and G are determined by the selection rules fevrA, +,A, or E +,E and AG = 0.” It is important that by pumping AK’ = 0, one can prepare jf levels of a single rovibronic =symmetry class, since fevrA, ++A, and K‘ = 0 levels are A, or A, depending on whether N’ is even or odd.G remains a good quantum number (modulo 3) within the C3,(M) molecular symmetry group as long as hyperfine interaction can be ignored (Leo, nuclear spin of the H atoms is conserved) and isomerization of the highly excited CH30 (8)molecules to CH,OH can be neglected.” Considering, for example, the SEP spectra at E z 7460 cm-’ in Fig. 2, with three different pump lines employed, 36pP,1(J‘’ = 1.5, K” = + l), ‘R,,(J” = 0.5, K” = 0) and ‘QZ1(J”= 0.5, K” = 0), A rotational levels were isolated with K’ = 0, N’ = 0, J’ = 0.5 and K’ = 1, N’ = 2, J’ = 1.5 or N’ = 1, J’ = 0.5, respectively. Thus, SEP excitation with the first pump line prepares selectively =% vibration-rotation states with rev‘A, symmetry and J = 0.5 or 1.5.SEP with the =second and third pump transitions leads to % states with rev‘E symmetry and J = 0.5, 1.5, 2.5 and J = 0.5, 1.5, respectively. For the low J levels studied, it is not possible at this time to make a definite statement regarding the suitability of the K quantum number. Unimolecular Decay Measurements The observed CH,O (g)SEP spectra were practically fully quantum-state resolved up to an excitation energy of E z 9000 cm-’ (6 c p-I). At higher energies, neighbouring reso- nances start to overlap (6 z p-’), until finally a resolution of single quantum states becomes difficult (6 > p -’). Experimental state-specific dissociation rates k,, were deter- mined for different 2 energy regions from either time-resolved direct measurements or SEP line-profile measurements.7000 < E@)/cm-< 8000 Direct decay rate inJestigations can be performed with ns time resolution for excited metastable CH,O (X) quantum states with excitation energies not too far above the H-H,CO bond-dissociation limit of the molecule. For these measurements, SVRL LIF excitation spectra were recorded to identify suitable probe transitions for a large number State-spec@ Dissociation Rates of CH,O (x2E) of excited states with energies ranging from E M 3000 cm-' to E x 8000 cm-'. A portion of a typical SVRL excitation spectrum is reproduced in Fig. 4. For the selec- ted % level with E = 7458.4 cm-', J = 1.5, revr= A, [reached with the 3:, PSI1(J'= 1.5, K' + 1) pump], o?e can see probe rotational transitions to several excited vibra-tional states.These A vibrational levels were identified as 36, 21325' and 3451. The respective rotational lines were assigned based on previously observed LIF excitation spectra from the Oo ground state.17 The rovibrational line structure in the SVRL excitation spectra thus encodes quantitatively information on the exact vibrational com- position of the highly excited CH,O (X) states. The observation of intense transitions to excited A levels with excitation of u5 in Fig. 4,for instance, is a direct consequence of the strong vibrational mixing of the individual X levels; the probed X level must have sig- nificant u5 character for the above probe transitions to be observable. Investigations of the temporal decay profiles of the excited states were initiated first for a number of bound levels in the E = 6000-7000 cm-' x energy region, below the H-H,CO bond-dissociation energy, to check loss processes other than unimolecular dissociation.Collisional relaxation is essentially ruled out in the molecular beam expan- sion. On the timescale of interest, radiative transitions of the 2 levels in the IR also do not play a role. This leaves the supersonic free-jet expansion which removes the excited molecules from the detection volume. For all excited states up to E M 7000 cm-' that were probed, no sign of unimolecular dissociation could be found. All time profiles up to this excitation energy could be described with a common effective exponential decay constant of k' = 5 x lo5 s-'.Considering the chances for deconvolution, this sets a lower limit for the smallest experimentally detectable unimolecular decay constant of k,,(E = 7000 cm-'; J = 0.5) z 3 x lo5 s-'. With these premises, time-resolved direct decay rate measurements were performed for 27 excited single CH30 (X) vibration-rotation states in a selected FC window at E M 7460 cm- '.High-resolution SEP scans over this FC window are depicted in Fig. 2. The probed vibration-rotation states in this range are marked. In addition, six states were studied at slightly lower or higher energies (E M 7320 and 7800 cm-'). Experimentally measured decay profiles for a number of dissociative single CH,O (%) vibration-rotation states at E x 7460 cm-' are reproduced in Fig.5. The different panels are for different individual quantum states. The measured time profiles show the unimolecular dissociation of the prepared CH30 (X) levels. The profiles of all probed Fig. 4 Example of an SVRL LIF excitation spectrum for a selected highly ezcited CH,O (x)vibration-rotation level. The labels identify the final Lovibrational levels in the A electronic state. Pump at pQ,1(1.5,+ 1); E(X) = 7458.4 cm-". Dertinger et al. 0 E = 7471.8 crn-'imKlmk,= 8.8 x 106 6' ks= 1.3 x 106 s-1 CT) , . 1.: < ,.., .. 01230123-1012 ti me/v s Fig. 5 Measured unimolecular decay profiles for a series of highly excited CH,O (2)vibration-rotation levels levels but one were found to be single exponential.For the one exception, a bi-exponential decay was observed. The dissociation reaction was found to proceed on a micro- to submicro-second timescale. Respective state-specific decay constants k,, were determined by fitting to the observed curves calculated exponential decay curves convo- luted with a Gaussian describing the temporal laser pulse. The true unimolecular disso- ciation constants follow by subtracting the small effective experimental loss constant (k' = 5 x 10' s-I). The individual decay rates of the states probed in Fig. 5 are given in the respective panels. For one state, it was possible only to determine a lower limit to the decay rate. For the level which showed a bi-exponential decay, the two time con- stants were found to be kssl = 1.3 x 106s-' and kss2 = 6.3 x 106s-'.The experimental data demonstrate a striking degree of quantum-state specificity of the unimolecular reaction: For 27 single states in the narrow energy window around E z 7460 cm-', the decay constants were found to exhibit strong, seemingly erratic, state-specific variations. Differences were observed in the range from k,, = 8 x lo5 s-' to 24 x lo7 s-', i.e. the decay constants of neighbouring quantum states with virtually the same overall excitation energy (fca. 20 cm-. ') and the same value of J (sometimes with AJ = f1) vary by up to a factor of at least 50. For states only 0.3 cm-' apart, decay rate differences were measured up to a factor of 8. Averaging the decay constants of the individual quantum states in the E z 7460 cm -window, energy-, angular momentum- and symmetry-specific unimolecular reac- tion rate coefficients were obtained k(E z 7460 cm-'; J = 0.5 and 1.5; fevrA2) = 9.2 x lo6 s-'= k(E z 7460 cm-'; 0.5 < J < 2.5; fevr= E) = 4.6 x lo6 s-' Thus, the average rate coefficients for rovibronic states of fevr= E symmetry were found to be higher than those for rovibronic states of fevrA, symmetry, by a factor of 2.= These values will later be compared with predictions of statistical unimolecular rate theory. The averaged results for the several states probed at slightly lower and higher energies were k(E z 7320 cm-'; J = 0.5 and 1.5; fevrA2) = 6.9 x lo6 s-l= k(E z 7800 cm-'; J = 0.5 and 1.5; fevrA2) = 2.1 x lo7 s-'= 40 State-specijic Dissociation Rates cf CH30 (8'E) Since only few states were probed at these energies, these data may not be statistically representative, however.In fact, presumably because the decay was too fast, it was impossible to detect SVRL LIF excitation signals for most of the excited states in the E = 7800 cm-' region. Thus, the above k value for this energy is likely to be biased (to smaller values). 8000 < E(Z)/cm-' < WOO Above excitation energies of E w 8000 cm-', the CH30 (8)unimolecular dissociation is too fast to be monitored with time resolution. However, the decay of the highly excited quantum states of the molecule is reflected by corresponding line broadening in the SEP spectra. The observed SEP spectra are, in principle, Doppler-free.Under the condition that individual CH30 (g)resonances are isolated and in the absence of other contribu- tions to the SEP linewidths (e.g., radiative transitions of the A and 2 levels, collision- induced broadening, po_wer broadening), the homogeneous (Lorentzian) widths, 6Jcm-',of the excited X levels are proportional to their state-specific unimolecular decay rates, k,, = 2nc6,. The experimental resolution of the SEP spectra which was achieved is illustrated in Fig. 6(a). This figure shows line profiles of three long-lived CH30 (2)states at E w 7800 cm-'. At this energy, the homogeneous linewidths are small compared with the spectral widths of the pump and dump lasers. The observed line profiles can be seen to be fully resolved.Stronger SEP lines (fluorescence dip signals 230%))were observed to be affected by power broadening. To avoid power broadening, lineshape measurements were therefore performed using low dump laser fluences until the observed fluorescence dip signals were below ca. 10-15%. Under these conditions, the measured profiles of all probed bound levels could be approximated by a Gaussian with a width (FWHM) of 6, = 0.04-0.045 cm -Using deconvolution techniques, homogeneous contributions to the linewidths could be determined down to a limit of 6, z 0.01 cm-'. Detailed line-profile measurements were carried out for 42 excited single CH30 (2)quantum states at energies of E = 8300 and 8900 cm-'. The SEP spectra for these regions are shown in Fig.3(a) and (b). The individual resonances which were investi- gated were usually well resolved. A number of line profiles which were analysed are reproduced in Fig. 6(b) and (c). As shown, the observed lineshapes could be described by Voigt profiles. The Voigt shapes were obtained by convoluting Lorentzians with widths 6, with a Gaussian of 6, = 0.044 cm-' width, accounting for the experimental resolution. The homogeneous widths 6, for the individual resonances were determined using a non-linear least-squares fitting routine based on the Marquardt algorithm.20 To accommodate cases of overlapping transitions, the fitting routine was written to handle up to five incoherently added Voigt profiles at once. A worst-case example with four overlapping lines is shown in Fig.6(c). Nevertheless, the homogeneous widths could be determined with 20 standard deviations on the fits of <5-10(%. The observed homogeneous resonance widths in the two energy regions studied were found to vary between 0.005 < 6,/cm-' < 0.18 (E = 8300 cm -') and 0.06 < 6,/cm-' < 0.26 (E w 8900 cm-'). Thus, as found from the time-resolved measurements above, the linewidth results reveal strongly quantum-state-specific dynamical properties. The average unimolecular dissociation rate coefficients in the two energy windows studied were found to be k,,(E = 8300 cm-'; J = 0.5 and 1.5; fevrA2) = 7.2 x lo9 s-l= k,,(E z 8900 cm-'; J = 0.5 and 1.5; fevrA2) = 2.5 x 10" s-'= Dertinyer et al.7982.5 7983.0 8292.0 .' 8293.0 -095 022 Fig. 6 Measured SEP line profiles for several highly excited CH,O (z)resonances. (a) S, ycm- l); (b) 6, (/cm.--');(c)6,-(/cm-I). E(%) 3 9000 cm-' At excitation energies above E = 9000 cm ',the linewidths in the observed SEP spectra reach the same order of magnitude as the level spacing [see Fig. 3(c)]. Thus, it becomes difficult to resolve single CH,O (8)quantum states. However, as long as the linewidths and the spacing between lines are compar_able (6 z y -'), and provided that the strongest SEP peaks belong essentially to single X levels, the line profiles can still be taken to determine approximately state-specific unimolecular decay rates. Lineshapes were investigated for 50 reasonably well resolved peaks in the SEP spectra for two energy windows with E x9660 or 9780 cm- '.As seen immediately from Fig.3(c), the observed profiles are much broader compared with those at the lower energies. State-specific linewidths for the selected peaks were determined as far as pos- sible by least-squares fitting to the observed spectra either single Lorentzian functions or, in case of overlapping peaks, a corresponding number of incoherently added Lor- entzians. An example of a well resolved peak for which a single Lorentzian with a width of 6, = 0.41 cm-' was fitted is shown in Fig. 6(d). The inferred Lorentzian widths of the 50 measured levels were found to lie in the range 0.21 < S,/cm-' < 1.54. The average over all levels corresponds to a specific unimolecular rate coefficient of k,,(E = 9700 cm-'; J = 0.5 and 1.5; revrA2) = 1.4 x 10'' s-'= Discussion The last few years have witnessed a tremendous expansion of our abilities to probe the dynamics of highly vibrationally excited molecules in single quantum states.However, connected experimental data on the unimolecular dissociation of molecules in specific vibration-rotation states and the related intramolecular vibrational dynamics, expecially IVR processes after excitation of a specific molecular vibration, for molecules which can State-specific Dissociution Rates of CH30 (x2E) be taken as model systems have been The present state-resolved e_xperi- ments on the unimolecular dissociation of highly vibrationally excited CH30 (X) are important here because they have been performed for a typical, reasonably large (five- atom) molecule and cover a wide range of excitation energies.Highly excited dissociative molecular states were studied from the energy region close to the asymptotic disso- ciation limit, where neighbouring states are fully isolated, into the overlapping resonance regime far above the reaction threshold. The results are thus also of interest considering unimolecular reaction mechanisms of larger molecules. As opposed to the situation for some other systems,21 *22*24which have also been investigated over a range of excitation energies,-spectroscopic evidence shows that, essentially, the vibrational dynamics of CH30 (X) in the energy region and on the timescale of interest is statistical, ergodic, chaotic, etc.'' Measured State-specific Unimolecular Dissociation Rates The experimentally measured decay profiles of the specific highly excited states of CH30 (2)which were probed, and the observed strong line broadening in the SEP spectra at higher energies, have to be attributed to the unimolecular dissociation of the molecules, CH30 (%; E, J, r)+ H + H2C0 (1) The SEP spectra were essentially fully state resolved up to excitation energies of E z 9000 cm-'.Examples of a few accidentally unresolvable states cannot significantly change the picture which is found. Doppler or power-broadening effects were ruled out. Linewidth broadening due to IVR is excluded in fully vibration-rotation eigenstate resolved spectra; it plays a role in large molecules where eigenstates cannot be resolved.Unimolecular isomerization, CH30 +CH20H, could be a competing reaction. In the large-molecule limit of radiationless transition theory,27 isomerization would result in corresponding spectral line broadening; the reaction would be irreversible on the time- scale of the experiment. However, at the level of single eigenstates, isomerization would be reflected by line splittings rather than line broadening, similar to the tunnelling split- tings observed for molecules with double-minima potentials. The experimentally observed density of states is inconsistent with the assumption that the prepared highly excited molecules can access the CH20H part of phase space.Also, as noted'?' the isomerization threshold is considerably higher than that for the dissociation process. The observed decay profiles of the individual vibration-rotation states which were probed were single exponential. Conversely, the SEP line profiles which were analysed could be approximated using simple Lorentzian lineshape functions. This situation is the expected one for isolated molecular resonance states. The unimolecular decay rate and the Lorentzian homogeneous width of an isolated resonance state are related by k,, = 2nc&. The experimental data demonstrate without ambiguity that drastic quantum-state- specific differenses exist between the unimolecular decay rates of highly vibrationally excited CH30 (X) in isolated molecular vibration-rotation states with practically equal excitation energy and identical angular momentum and rovibronic symmetry.Seemingly erratic fluctuations of the state-specific rates were observed of up to at least two orders of magnitude. The quantum-state-specific variations were observed over the complete range of excitation energies which was investigated. The state-specific experimental results thus contrast strongly with the traditional picture of statistical kinetics assumed for highly vibrationally excited molecules. Unimolecular rate theories, as commonly used, predict the specific reaction rate coefficients k(E, J) of the molecules to increase smoothly with increasing E2v3 Dertinger et al. The measured individual decay constants are plotted us.the overall excitation energy of the molecules in Fig. 7. In addition to the detailed data described above, a few points have been included in the plot for levels with values of J higher than from our previous work.'*'' The average unimolecular dissociation rates in narrow energy windows can be seen to increase with increasing excitation energy, as one would expect. However, the state-specific variations persist over the complete energy range, from the H-H2C0 bond dissociation limit, on through the tunnelling regime, up to the range well above the classical reaction threshold energy. This offers a nearly unique opportunity for inves- tigating the magnitude of the state-specific fluctuations of the dissociating molecules as a function of energy quantitatively employing statistical measures, e.g. the first and second moments of the decay rate distributions (see below).It is of central importance that in view of the established extensive vibration-rotation mixing mechanisms,' the state- specific decay rate fluctuations cannot be attributed to regular, assignable, vibrationally mode-specific dynamic properties. RRKM Model The H-H2C0 bond fission reaction proceeds via a tight transition state that is local- ized at the maximum of the potential barrier to the products. Hence, the specific rate coeficients can be evaluated using the conventional RRKM expression with modifi- cation to allow for tunnelling of the H atom through the barrier, The T*(E-E,, J)are adiabatic channel transmission coefficients, h is Planck's constant and p(E, J)is the symmetry-specific density of rovibronic states of the molecules, 1°12 6000 7000 8000 9000 10000 11000 excitation energy/cm-' Fig.7 CH,O (2)unimolecular decay rates us. molecular excitation energy State-specijic Dissociation Rates of CH30 (z'E) Table 2 RRKM model paramete2 for the unimolecular dissociation of CH,O (X) (in cm..') CH,O [H--H2COIS H2C0 + H Ar H: and AE, 0 8500 6950 A 5.210 3.925 9.405 B 0.932 1.007 1.295 C 0.932 0.962 1.134 tT 3 1 2 ui s(CHH) G(CHH) J(OCH2) S(C0)6(OCH') a(CHH) 6(CH H H ') 2840 1359 1437 1045 lo00 2809 1437 2798 1593 1173 1414 58 1 2892 740 281 1 1500 1170 1756 286 1 -- y( CH H H ') lo00 1232 1251 EZ s(CH') 2809 7868 830i 6212 5675 - gL gs 2 2 1 2 1 2 The molecular parameters needed for calculating the statistical specific rate coeffi- cients are compiled in Table 2.The density of rovibronic states of the molecule was determined by exact state counting taking the molecular vibrational frequencies given in our earlier work." Corrections for anharmonicity which have been suggested,' were below 20%. For CH,O, the Pauli principle dictates that excited levels of A, or A, compared with E rovibronic symmetry correspond to different nuclear spin configu- rations (I, = 3/2 us. I, = 1/2). Hence, symmetry and nuclear spin conservation was taken into account within the C3,(M) molecular symmetry group." In the high-energy limit, one obtains the expected ratio29 p(E, J, A,) :p(E, J, A2) : p(E, J, E) = 1 :1 :4, where the two-fold degeneracy of the E levels has been formally resolved.However, since G is conserved in addition to fevr in the CH30 system in C3,(M) symmetry, only half of the E symmetry levels contribute to the accessible volume of phase space for a given E and J. The rotational constants and vibrational frequencies for the transition state were adopted from the ab initio work by Walch.9 The H-H,CO bond energy was adjusted as described below. Adiabatic channel potential curves a were constructed with individ- ual channel threshold energies E,, taking into account the correlation rules of the C3,(M) group. The transmission coefficients T*(E-E,, J) for tunnelling through the channel barriers were evaluated by approximating the potential barrier with an Eckart potential for which analytical solutions for T*(E-E,, J) are known.30 At energies exceeding E,, the sum in the nominator in eqn.(3) approaches the usual step function for the number of open reaction channels N*(E, J) in the absence of tunnelling. The imaginary frequency describing the motion along the reaction coordinate has been cal- culated by Page' and Walch.' However, at the SCF level of ab initio theory which they used, the height of the potential barrier is overestimated. Hence, the imaginary frequency was adjusted in this work by scaling it to the barrier height based on the values of Ar HZ and E, assuming a constant barrie_r height over force constant ratio.Only the A' com-ponent of the doubly degenerate (X 'E) electronic potential surface of CH,O correlates through to ground-state H + H,CO. Hence, one obtains electronic degeneracy factors for the transition state of gL = 1 (compared with 2 for the molecule in its equilibrium conformation) and g, = 2. Dertinger et al. The calculated k(E, J) curve for J = 0.5 (f= A2) is shown in Fig. 7 for comparison with the measured data. It can be seen that on average the increase of the experimental decay constants with increasing energy is very well described by the model. It is noted that the calculated curve for J = 1.5 which has not been plotted for clarity is almost identical to that for J = 0.5; the difference would in fact be hardly visible on the scale of Fig. 7.Hence, considering the magnitude of the state-specific variations of the experi- mental data, the fact that J = 0.5 and 1.5 levels were excited by SEP in the above experiments is unimportant here. However, it is interesting that, because of the corre- spondingly higher density of states (see above), the calculated k(E, J) curve for rovi- bronic symmetry f = E was found to be lower than that for f = A, by a factor of 2. This reduction was confirmed precisely by the experimental difference between the aver- aged decay constants for A, and E levels in the E = 7460 cm-' region. Thus, the RRKM model appears very satisfactory. State-specific deviations are not predicted. Tunnelling of the H atom through the potential barrier can be seen to be of great importance considering the range of excitation energies below the classical threshold energy E,.E, is determined by the H-H,CO bond energy A, Hg and the height of the additional potential-energy barrier above the H + H,CO product level with zero-point energy corrections. A zero-point-energy corrected threshold energy for addition of an H atom to H,CO of 1560 _+ 150 cm-' was evaluated from our measurement of the activa- tion energy for H-D isotope exchange in the reaction D + H,C0.7 In the calculated k(E, J) curve in Fig. 7, the classical reaction threshold le_ads only to a slight bend. The H-H,CO bond energy (A, HE) of the CH30 (X) molecule and the height of the classical dissociation threshold energy E, had not been known precisely before the present work.Values of A, HE have been quoted in the range 6100 d Ar H$cm-' d 8400.31 A strict upper limt of 7300 cm-' is determined by the excitation energy of the first level which was found to undergo decay. A value of Ar HE(H-H,CO) = (6950 150) cm-' leads to a best fit of the calculated specific rate coefficients to the experimental data (see Fig. 7). This result is identical to the value of Ar HE = (6930 & 200) cm-' determined very recently by Neumark and co-workers3' from translational-energy release spectra-of the photodissociation products of CH,O (X). For the heat of formation of CH30 (X), one obtains a value of A, H:98 = +21 kJ mol- '. This value is ca. 4 kJ mol-' higher than assumed previously.Taking the data as given above, the threshold dissociation energy is E,= (8500 150) cm-'. For comparison, from a measurement of the thermal CH,O (X)disso-ciation rate one would derive a value of E, z 8360 cm-', assuming that a suitable extrapolation can be made from the experimental activation energy, which refers to the low-pressure limit of thermal unimolecular reactions. Both the values of Ar HE and E, are close to those calculated by Walch.' State-specific Decay Rate Fluctuations in the Isolated Resonance Regime and Random Matrix Transition-state Theory Models In order to rationalize the quantum-state-spectific experimental data, it is important to realize that the specific rate coefficients k(E, J) calculated by traditional unimolecular rate theory are obtained for a microcanonical ensemble of molecules and not for single molecular quantum states.As shown, neighbourinj quantum states can have vastly dif- ferent decay rates. The present data for CH,O (X) point out, in particular, that state- specific dynamical properties can be observed over wide ranges of excitation energies. On a linear scale, the magnitude of the fluctuations increases with increasing energy, since the average decay rates increase. On the other hand, the extent of vibration-rotation level mixing was found to be close to the statistical limit, in contrast to our Stute-specific Dissociation Rates of CH30 (82E) notion of vibrationally mode-specific dynamics. Thus, the observed decay fluctuations are understood as a sort of statistical-state ~pecificity.~~ In a quantum-state-resolved context, the interpretation of statistical chemical properties of highly vibrationally excited molecules is revised.For further explanations, one has to consider the projections of the molecular wave- functions, which are usually given in terms of the normal mode coordinates Qi,on the reaction coordinate q. Taking into account the ensuing strong level mixings, the pseudo molecular eigenstates I 'Yj) embedded in the dissociation quasicontinuum are, in the isolated resonance state regime (6 4 [I-'), described as stochastic mixtures of normal mode states I qi), I Considering the effective vibrational coupling selection rules,' the allowed mechanisms which govern the interaction of the normal mode states work highly specifically, even at high excitation energies.The resulting coupling matrices are relatively sparse. Under these conditions, the level mixing can be statistically modelled within the framework of random matrix theory. In particular, the Gaussian orthogonal ensemble (GOE)ansatz' has been used successfully. The unimolecular decay of the highly excited states results as a consequence of the coupling between the molecular pseudo eigenstates and the disso- ciation continuum. Thus, the observed fluctuations of the state-specific decay constants in the isolated resonance state regime simply reflect the variations in the overlap between the stochastic molecular wavefunctions defined by eqn.(4) and the continuum wavefunctions. This interpretation is very similar to that for the observed intensity fluc- tuations in eigenstate-resolved SEP spectra of highly excited molecules.35 The intensity fluctuations simply demonstrate the statistical state-specific properties of the highly excited molecules manifest in the overlap of the molecular wavefunctions to those in other electronic states. The statistical distribution of the state-specific decay constants can be described by a random matrix-transition-state theory (RM-TST) model as developed by Miller, Moore and co-worker~.~~ In the isolated resonance state regime, the decay of the highly excited molecular vibration-rotation states is. represented by a complex Hamiltonian, The real part of H:!',,. describes the zero-order energies and anharmonic couplings of the vibrational levels using some meaningful basis set In), while the imaginary part describes the coupling to the dissociation continuum.The complex eigenvalues, E, -ifj/2, contain the energies Ej and decay constants fjfor the individual statesj. Using perturbation theory with the random matrix ansalz for the level coupling, the distribu- tion of the state-specific unimolecular decay rates can be evaluated based on transition- state parameters obtained from ah initio quantum chemistry.36 The variance (second moment) of the distribution of the experimental decay constants is connected with the number of open reaction channels and thus provides information on the transition-state region of the potential-energy surface which governs the reaction dynamics.The present experimental data have so far been analysed in a somewhat simplified picture. Assuming that the reaction proceeds uia v independent channels, the distribution of the reduced decay constants x = k,Jk can be approximated by a x2 distribution func- tion with v degrees of freed~m,~~?~~ with the gamma function G(v/2).Histograms which illustrate the statistical distributions of the experimental data are depicted in Fig. 8. Except for the lower-energy window Dertinger et al. 41 E = 7460cm-" E = 8300 cm-' vcalc = 4 vexpt = 3 1- 1- vcalc = 4 -*.-.,.: '.,-. A =.* -0 0 (E z 7460 cm-I), it can be seen that these histograms can be approximated by x2 dis-tribution functions with different v.The numbers for v which could be derived from the experimental data are given in the different panels. These values can be compared with the numbers of open reaction channels N*(E,J) from the RRKM model. The experimental histograms show convincingly that the magnitude of the fluctua- tions of the reduced decay constants k,,/k decreases with increasing excitation energy of the molecules approximately in accordance with the increasing number of NS(E,J). With increasing E, the distribution narrows and peaks at the value predicted by the RRKM model (even though the fluctuations of the absolute decay constants k,, had been found to increase in magnitude). Thus, statistical unimolecular theory is seen to work increas- ingly better at higher energies above the reaction threshold.This conclusion is under- standable because the state-specific decay constants are summed over a larger number of channels, which leads to averaging and smoothing over the existing stronger single- channel variations. Unimolecular Decay Rates in the Overlapping Resonance Regime The notion of an isolated resonance state encompasses not only that it has an energy separation to neighbouring states which exceeds the widths of the states but also that there is negligible overlap between the respective wavefunctions in the continuum region of the potential surface. The open reaction channels must be independent. The spectrum of resonance states with no wavefunction overlap in the continuum corresponds to an incoherent sum of Lorentzians, whose individual widths reflect the true state-specific rates.At higher excitation energies, the density of states and the unimolecular decay rates both increase and neighbouring states start to overlap within their widths. In the limit, where statistical rate theory applies, the overlapping resonance regime is reached when NS(E,J) 2 2n. This limit is reached especially rapidly for reactions with loose transition State-specific Dissociation Rates of CH,O (22E) states. Hence, it may be difficult to observe quantum state-specific effects for reaction systems without clearly defined transition-state structures, e.8. for most bond-fission reactions.Interferences between the wavefunctions in the continuum lead to asymmetric, Fano-type line shape^.^^,^^ The temporal decay profiles of overlapping resonance states may deviate from single exponential, and the decay rates for the individual states cannot be determined as easily. In principle, asymmetric lineshapes arise because wavefunction amplitudes have to be coherently added rather than adding probabilities. Related experimental data have been reported for D2C04' and FN0.41 In the present work, one example of a bi-exponential temporal decay profile which may be due to interfering channels was found experimentally. The observed SEP line- shapes could usually be described with single Lorentzians or incoherent sums of single Lorentzians within the signal-to-noise limitations of the fluorescence dip SEP method.However, it cannot be excluded that asymmetries in profiles, which were observed occasionally and ascribed, for simplicity, to one or several additional non-interfering states at nearly coincident energy, should really be attributed to interferences between decay channels and thus described with Fano-type lineshapes. Further time-resolved measurements and, in particular, higher-resolution SEP studies with Fourier-limited laser bandwidths (dO.003cm-') would be of considerable interest to elucidate com- petition and interferences between different decay channels. Strong linewidth variations were observed in the SEP spectra in the energy range where neighbouring molecular resonance states do start to overlap (E = 9600 cm-I).At least for the most part, these variations were attributed to corresponding decay rate fluctuations. The question of a survival of non-monotonic variations in the dynamics of highly excited states in the overlapping resonance regime is controversial. The fluctuations of the individual resonance decay rates may persist, but the relation between spectral line- widths and single-resonance decay rates is blurred. An extension of the RM-TST model .~discussed above to overlapping states has been put forward by Peskin et ~1 Further ~ suggestions may be obtained from models developed for nuclear reactions.43 Non- monotonic variations of the cross-sections of overlapping resonances are known in nuclear physics as Ericson fluctuation^.^^ The observed scattering amplitudes f (E) can be described by sums of terms over individual resonances, with the complex amplitudes A, (which include a phase factor) and the energies Er,,and widths rpfor the single resonances.Possible Deviations from Statistical Behaviour Apart from the quantum-state-specific deviations, a difference between a factor of 2-3 can be seen between the average decay constants inferred from the SEP line-profile measurements and the calculated RRKM rate coefficients in Fig. 7, considering the range of excitation energies above E z 9000 cm- '. The inferred experimental decay rates grow more rapidly with increasing E than the calculated ones. It is understood that the discrepancy has to be considered with due caution because of possible contami- nation by overlapping levels.However, it is of interest that at E z9600 cm- ', one can also find deviations in the statistics of the decay rate fluctuations (see Fig. 8). The widths of tne distribution appears wider than it may be expected. On the other hand, the life- times of the highly excited molecules at these energies are only of the order of a few ps. The characteristic times for IVR processes had been found to be in the same region. Hence, one may approach the limits for the application of statistical theory here. At the energies of interest, the highly energized molecules may not sample the complete volume Dertinyer et al. of phase space before dissociation. Hence, the density of active states is smaller than calculated from state counts.The comparison between the experimentally observed and calculated densities of states which is shown in Fig. 9 seems to support such a picture. Computations of the specific unimolecular rate coefficients for the isomerization reac- tion CH,O (X) +CH,OH with the ab initio value for the reaction threshold show that this channel can be ruled out as an explanation for the deviation. The K quantum number ceases to be a suitable quantum number at the high excita- tion energies, although it is noted that certain restrictions on K in combination with A and 1 are imposed by nuclear-spin conservation and the strict selection rule for G. However, it is not possible at this time to make a definite statement regarding the extent of K state mixing.The experimentally observed total density of rovibronic states reaches a value of p z 3.2 cm-’ in the SEP spectrum of the E z 7460 cm-’ region with the pPll(J”= 1.5, K” = + 1) pump. As shown in Fig. 9, this value is close to the calculated symmetry-specific total rovibronic state density with the assumption of complete K mixing. On the other hand, it is unlikely that the Coriolis interaction is strong enough for complete K mixing of the low J states considered here.” Information on the values of K is encoded in the recorded SVRL LIF excitation spectra. These spectra seem to indicate that for many highly excited states, K remains defined. Note that only low J states were considered here (J < 2.5).Decay rates inferred for some levels with higher J (Fig. 7) are slightly above the calculated RRKM values, which might also be due to incomplete K mixing. However, it is difficult to rule out inhomogeneous contributions in view of the higher density of rovibronic states. Future time-resolved experiments have to be awaited, which are underway. Comparison to Other Molecules Fully quantum-state-resolved investigations of the unimolecular kinetics of highly vibra- tionally excited molecules have only recently become possible. Thus, extensive experi- mental data are rare. In one extreme, vibrational predissociation measurements have been reported for a number of van der Waals molecule^.^^ In these weakly bound complexes, coupling of the molecular vibrations to the intermolecular bond is extremely weak.Considering chemically bound molecules, vibrationally mode-specific decay dynamics have been established for HC0.2 ’ For comparison, the analogous dissociation 100.0 0.1 2000 4000 6000 8000 10000 excitation energy/cm-’ Fig. 9 Comparison of the measured and calculated densities of rovibronic states as a function of excitation energy State-specific Dissociation Rates of’ CH30 (i?l ’E) reaction of DCO, which we have been studying,22 is different because a near 1 : 1 :2 resonance between the vibrational frequencies in this isotopomer leads to strong vibra- tional level mixing. Fluctuations of the decay constants for D2C0 similar to those for CH,O (8)had first been reported by Moore and co-~orkers.~~ They investigated Stark anti-crossing spectra of D,CO in the tunnelling region below the classical reaction threshold.The large number of resonances detected in their work stimulated the devel- opment of the RM-TST concept.36 However, the experimental analysis is complicated by the interaction between the S, and So electronic states and the presence of an electric field which ultimately leads to a breakdown of the J quantum number. On the other hand, HFCO, which was also studied by Moore and co-w~rkers,~~ was found to exhibit assignable mode-specific dynamics up to energies of ca. 8000 cm-above the threshold for dissociation to HF + CO. Stephenson and King investigated the dissociation of HN, to NH(,C) + N, by overtone excitation of NH stretching levels with u = 4 to 6.25 This reaction is a spin-forbidden process that is governed by the coupling between the respective electronic potential surfaces. Abel et al.found evidence for fluctuations in the linewidths in the photofragment yield spectrum of NO2.26 Of interest, furthermore, are fluctuations in the product-state distributions observed for the dissociation of NO, and ~~~0.~~9~’ The present study of the unimolecular dissociation of CH,O (i?l) to H + H,CO, a system with nine vibrational degrees of freedom which are strongly coupled, develops an especially detailed picture for unimolecular reactions of moderate-sized molecules from several points of interest. The unimolecular dynamics were probed over a wide range of excitation energies. The results have a far-reaching bearing which extends to larger mol- ecules. Viewed in conjunction with the data on the ensuing IVR kinetics, the CH30 system constitutes a model system for elucidating the mechanisms of unimolecular reac- tions.Conclusions As evidenced by a large body of papers over the last years, the understanding of sta- tistical chemical properties of molecules at a quantum-state-resolved level is not triviaL6 The experimentally observed strongly quantum-state-specific decay kinetics contrast with a traditional picture. However, the state-specific variations should not be taken as a sign of non-statistical molecular properties. At least under certain conditions, the notion of statistical state specificity appears to be appropriate. In this context, statistical unimolecular rate theory generally works at high energies on average because of the averaging over many open reaction channels and, less so because of the large degree of level mixing in the molecules. The state-specific fluctuations in narrow energy windows may be rationalized to an extent in the RM-TST picture.36 The statistical distribution of the decay constants pro- vides information on the transition-state region of the potential-energy hypersurface which governs the reaction dynamics. Eventually, however, one would like to make comparisons with results of exact quantum dynamics calculations on accurate ah initio potential-energy surfaces. Realistic quantum calculations are feasible today for triatomic molecules, as shown by the excellent agreement between experimental and theoretical results for the unimolecular decay of HCO and DC0.48*4y Whether the picture obtained here for CH30 (R) is generally valid is a question that needs further consideration.Note that using the current technique, it is impossible to measure the decay of all states in a given energy window. A computer-controlled pro- cedure which allows the dump and probe laser to be scanned simultaneously with differ- ent time delays will allow a more complete coverage. The highly excited quantum states prepared by SEP had significant v3 character. Whether or not this introduces bias remains to be checked. Also, very rapidly dissociating states are difficult to detect using Dertinyer et al. 51 the SVRL LIF scheme.Experiments with direct detection of the H,CO and H products would be complementary. Furthermore, it is of interest to study states with large amounts of initial excitation in the dissociating bond. Here, experiments are in progress to prepare levels containing a large amount of energy in the v4 mode, which is closely related to the reaction coordinate. Information on the vibrational composition of each individual quantum state may be extracted from the SVRL LIF excitation spectra of the highly excited states. Last but not least, the question of symmetry and nuclear-spin conservation is addressed. Whether statistical state specificity can be observed in much larger molecules and whether or not there are consequences for modelling unimolecular reactions in thermal systems remains to be seen.It is necessary to investigate the importance of a k(E, J) distribution for thermal reactions. State-specific variations may have an effect in the fall-off region and for the limiting high-pressure rate coefficients. Very rapidly reacting states are accessed at high pressures through fast collision-induced energy randomiza- tion. This could mean that the high-pressure thermal rate constant could be pushed towards higher and higher values. On the other hand, at very high pressures, collisions may also lead to a collisional decoupling effect similar to the well known collisional linewidth narrowing effect.We thank Professor H. Gg. Wagner for his continuous interest and support and for many stimulating discussions. Support of this work by Deutsche Forschungsgemein- schaft (SFB 357 ‘Molekulare Mechanismen unimolekularer Reaktionen’) is gratefully acknowledged. We also thank Lambda Physik for the loan of a high-finesse air-spaced monitor etalon. References 1 J. Troe and H. Gg. Wagner, Ber. Bunsen-Ges. Phys. Chem., 1976,71,937. 2 J. Troe, J.Phys. Chem., 1979,83, 114. 3 R. G. Gilbert and S. C. Smith, Theory c?fUnimolecular Reactions, Blackwell, Oxford, 1990. 4 J. Troe, J. Chem. Soc., Faraday Trans., 1994,90,2303. 5 F. Temps, in Molecular Spectroscopy and Dynumics by Stimulated Emission Pumping, ed. H-L. Dai and R. W. Field, World Scientific, Singapore, 1995, p.375. 4 Ref. 5, whole volume. 7 S. Dobe, C. Oehlers, F. Temps, H. Gg. Wagner and H. Ziemer, Ber. Bunsen-Ges. Phys. Chem., 1994, 98, 754; S. DO&, C. Oehlers, F. Temps, H. Gg. Wagner and H. Ziemer, to be published. 8 M. Page, M. C. Lin, Y. He and T. K. Choudhury, J. Phys. Chem., 1989,93,4404. 9 S. P. Walch, J. Chem. Phys., 1993,98, 3076. 10 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101,3618. 11 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994,101,3634. 12 E. P. Wigner, SIAM, 1967,9, I. 13 M. Carmeli, Statistical Theory and Random Matrices, Marcel Dekker, New York, 1983. 14 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1993,99,2271. 15 S. Gerstenkorn, J.Verges and J. Chevillard, Atlas du Spectre de la Molecule de L’lode, Laboratoire Aime Cotton, CNRS, Orsay, France. 16 F. M. Phelps 111, M.I.T. Wavelength Tables, Band. 2, MIT Press, Cambridge, 1982. 17 A. Geers, PhD Thesis, Report 16/93, Max-Planck-Institut fur Stromungsforschung, Gottingen, 1993. 18 J. T. Hougen, J. Chem. Phys., 1962,37, 1433. 19 P. R. Bunker, Molecular Symmetry and Spectroscopy, Academic Press, New York, 1979. 20 P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, 1969. 21 G. W. Adamson, X.Zhao and R. W. Field, J. Mol. Specrrosc., 1993, 160, 1 1 ;D. W. Neyer, X. Luo and P. L. Houston, J. Chem. Phys., 1993, 98, 5095; J. D. Tobiason, J. R. Dunlop and E. A. Rohlfing, J.Chem. Phys., 1995, 103, 1448. 22 H-M. Keller, X. Li, R. Schinke, C. Stock, R. Schinke and F. Temps, to be published. 23 W. F. Polik, C. B. Moore and W. H. Miller, J. Chem. Phys., 1988, 89, 3584; W. F. Polik, D. R. Guyer and C. B. Moore, J. Chem. Phys., 1990,92, 3453. 24 Y. S. Choi and C. B. Moore, in ref. 5, p. 433. 25 B. R. Foy, M. P. Cassassa, J. C. Stephenson and D. S. King, J. Chem. Phys., 1989,90,7037. State-spec@ Dissociation Rates of CH,O (% 2E) 26 B. Abel, personal communication. 27 M. Bixon and J. Jortner, J. Chem. Phys., 1968, 48, 715; M. Bixon and J. Jortner, J. Chem. Phys., 1969, 50,3284. 28 J. Troe, J. Chem. Phys., 1977,66,4758. 29 A. Sinha and J. L. Kinsey, J. Chem. Phys., 1984,80,2029. 30 C. Eckart, Phys. Rev., 1930,35, 1303; L.D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon, Oxford, 1965; B. C. Garrett and D. G. Truhlar, J. Phys. Chem., 1979,83,2921. 31 F. Temps, Report 1 1/95, Max-Planck-Institut fur Stromungsforschung, Gottingen, 1995. 32 D. L. Osborn, D. J. Leahy, E. M. Ross and D. M. Neumark, Chem. Phys. Lett., 1995,235,484. 33 P. J. Wantuck, R. C. Oldenborg, S. L. Baughcum and K. R. Winn, 22nd Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, p. 973. 34 W. L. Hase, S-W. Cho, D-H. Lu and K. N. Swamy, Chem. Phys., 1989,139, 1. 35 R. D. Levine, Adv. Chem. Phys., 1987,70,53. 36 W. H. Miller, R. Hernandez, C. B. Moore and W. F. Polik, J. Chem. Phys., 1990, 93, 5657; R. Hernan- dez, W. H. Miller, C. B. Moore and W. F.Polik, J. Chem. Phys., 1993,99,950. 37 R. D. Levine, Ber. Bunsen-Ges. Phys. Chem., 1988,92,222. 38 U. Fano, Phys. Rev., 1961,124, 1866. 39 F. H. Mies and M. Krauss, J. Chem. Phys., 1967,45,410; F. H. Mies, Phys. Rev., 1968, 175, 164. 40 W. F. Polik, C. B. Moore and W. H. Miller, J. Chem. Phys., 1988,89, 3584. 41 J. T. Brandon, S. A. Reid, D. C. Robie and H. Reisler, J. Chem. Phys., 1993,97, 1992. 42 U. Peskin, H. Reisler and W. H. Miller, J. Chem. Phys., 1994, 101, 9672; U. Peskin, H. Reisler and W. H. Miller, J. Chem. Phys., 1995, 102, 8874. 43 C. Mahaux and H. A. Weidenmuller, Annu. Rev. Nucl. Part. Sci., 1979, 29, 1 ;H. A. Weidenmiiller, Ann. Phys., 1984, 158, 158; J. J. Verbaarshot, H. A. Weidenmuller and M. R. Zirnbauer, Phys. Rep., 1985, 129, 367. 44 T. Ericson, Ann. Phys., 1963,23, 390; I. Rotter, Rep. Proyr. Phys., 1991,54, 635. 45 D. N. Nesbitt, Faraday Discuss.,1994,97, 1, and following papers. 46 M. Hunter, S. A. Reid, D. C. Robie and H. Reisler, J. Chem. Phys., 1993,99, 1093. 47 R. D. van Zee, C. D. Pibel, T. J. Butenhoff and C. B. Moore, J. Chern. Phys., 1992,97,3235. 48 H-J. Werner, C. Bauer, P. Rosmus, H-M. Keller, R. Schinke and M. Stumpf, J. Chem. Phys., 1995, 102, 3593. 49 H-M. Keller, X. Li, R. Schinke, C. Stock, M. Stumpf, F. Temps, C. Bauer, P. Rosmus and H-J. Werner, to be published. Paper 5f06356F; Received 26th September, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950200031

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss., 1995,102, 53-63 Quantum Mechanical Theory of Collisional Recombination Rates Part 2.-Beyond the Strong Collision Approximation William H. Miller Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley, California 94720, USA A quantum mechanical theory of collisional recombination (within the Lin- demann mechanism, A + B f-) AB*, AB* + M -+ AB + M) is presented which provides a proper quantum description of the A + B collision dynamics and treats the M + AB* inelastic scattering within the impact approximation (the quantum analogue of a classical master equation treatment). The most rigorous version of the theory is similar in structure to the impact theory of spectral line broadening and involves generalized (four- index) rate constants for describing M + AB* collisions.A simplified version is also presented which involves only the normal (two-index) inelastic rate constants for M + AB* scattering but which also retains a proper quantum description of the A + B dynamics. 1. Introduction Collisional recombination reactions, i.e. A + B + M -+ AB + M have recently been receiving considerable attention,' one reason being their importance in combustion processes (e.g. H + 0, -+ HO,, H + CO -+ HCO, H + CO, HOCO). The standard Lindemann mechanism for collisional recombination,T A + Bt,AB* (l.la) AB* + M -+ AB + M (1.16) leads [via the steady-state approximation and with the strong collision approximation (SCA) for the deactivation step, eqn.(l.lb)]to the following well known expression for the recombination rate constant ( =cm3 molecule- sec-') kr(T, 0) = Qr(T 1exp(-BEJmk,/(m + kd (1 -2)I where {El) and {k,) are the energies and unimolecular decay rates of the metastable states of AB*, B = (kT)-',Q, is the reactant (A + B) partition function per unit volume and o = [M]kdeac,is the frequency of 'strong collisions' (which is proportional to the pressure of the bath gas M), where kdeactis the bimolecular rate constant for the M + AB* collisional deactivation step in eqn. (1.lb).Most applications of this theory are t See Ref. 2: Note that these references, and others cited later, more often discuss collisional dissociation, the reverse of the reaction in eqn.(l.l), but the two rates are related by a detailed balance. 53 Quantum Mechanical Theory of Collisional Recombination Rates within the RRKM framework,2 whereby one assumes a classical continuum of meta- stable AB* states, i.e. E, .+ E and (1.3~) where p(E) is the density of AB* states, and uses microcanonical transition-state theory for the unimolecular decay rates, (1.3b) At tempts ‘‘9 d* at a more rigorous, quantum mechanical description of the A + B collision dynamics in eqn. (l.la) [still with the phenomenological SCA for the relaxation step in eqn. (l.lb)] have focused on identifying the energies (E,) and unimolecular decay rates {k,) in eqn. (1.2) as the energies and widths (r,= AkJ of scattering resonances of the A + B system.In some cases the lifetime (or time delay) matrix introduced by Smith4 has been used to describe the quantum dynamics of the A + B system. Though these approaches are appropriate when the A + B collision dynamics are dominated by long- lived resonances, if the separation of the resonant and non-resonant A + B scattering is ambiguous and/or if non-resonant scattering makes a significant contribution, then these approaches are ill-defined and can give unphysical results that are clearly not correct. A recent paper’ has presented a more rigorous quantum mechanical description of the A + B collision dynamics which is physically correct whether or not resonances dominate. It is based on flux correlation functions6 (uide infra) and does not require that one separately identifies resonant and non-resonant contributions to the A + B scat- tering.The initial version of this theory utilized the strong collision approximation (SCA) for the relaxation step, eqn. (l.lb), and it is the purpose of the present paper to go beyond the SCA, i.e. to show how a more general treatment of the M + AB* inelastic scattering can be combined with this rigorous quantum treatment of the A + B collision dynamics. Section 2 first summarizes the quantum theory of Ref. 5; this provides a rigorous quantum description of the A + B collision dynamics in eqn. (1.la) but utilizes the SCA for the relaxation step in eqn. (1.lb). Section 3 next briefly reviews the classical master equation description? for going beyond the SCA; this provides a more general treatment of the M + AB* energy-transfer process in eqn.(l.lb) but relies on a classical description of the A + B dynamics in eqn. (1.la). Section 4 then presents the new development of the paper, a synthesis of the two previous sections, combining a rigorous quantum descrip- tion of the A + B collision dynamics with the appropriate quantum generalization of the master equation treatment of the M + AB* energy transfer. Finally, an approximation to the general result of Section 4 is identified which leads to a much simpler result that also provides a synthesis of Sections 2 and 3, i.e. in the SCA it reverts to the quantum theory of Section 2, and in the classical limit it becomes the master equation treatment of Section 3.2. Review of Quantum Theory of Recombination within the SCA Referring to Fig. 1, the quantum theory of Ref. 5 is the quantum analogue of a classical trajectory simulation in which one would begin trajectories inward (i‘ < 0) at time t = 0 from the ‘dividing surface’ Y = a (sampled from a Boltzmann distribution at temperature T)and follow them until the time z at which they return to the dividing surface, weight- ing each trajectory by [l -exp(-mcoz)], the probability of a ‘strong’ (i.e. stabilizing) -f See, for example, Ref. 2(c),p. 268 et seq. W. H. Miller Fig. 1 One-dimensional schematic diagram of the interaction potential for the A + B system; r is the centre-of-mass separation of A and B.r = a is the dividing surface with respect to which the flux operator of eqn. (2.2)is defined, and ~(r)is the absorbing potential in eqn. (2.3). collision sometime within the time interval (0, z). The quantum mechanical expression for the recombination rate constant corresponding to this classical picture was shown to be k, Q, = rdt exp( -cot)C,(t) (2.14 where Cf is the flux autocorrelation function6 c,(t)= tr[E exp(iii*t,*/h)f exp( -iAt,/h)l (2.1b) where t, = t -ih/3/2. Here E is the usual flux operator,t related to the dividing surface in Fig. 1, -1P = -h [A,L] = 6(i -a) + 6(i -u) -i (2.2)m where h = h(a -Y) is the Heaviside function (1 for r < a, 0 for r > a), and H is the Hamiltonian for the A-B system, augmented by the absorbing potential E(q)(see Fig.l), A-H = H,, -k(q) which enforces outgoing wave boundary conditions7 for the time evolution operators (and Green’s functions). The correlation function C,(t) is a property solely of the A + B collision system; the only effect of the ‘stabilizing’ collisions is the factor exp(-cot) in eqn. (2.1~). Since the A + B collision system is a non-reactive system [i.e. everything that goes in through the dividing surface must come out (if there are no collisions)] one has t For future reference, we note that matrix elements of the flux operator are skew-symmetric,(x2 I fi I xl) = -(x,I k‘I x2),. either for the real wavefunctions x, and xz or if one uses (as we do below) the complex-symmetric convention of not complex-conjugating functions in the bra state.Quantum Mechanical Theory of Collisional Recombination Rates so that eqn. (2.1)gives k, = 0 if co = 0, an obvious physical requirement. Eqn. (2.4) also allows one to write eqn. (2.1~)as k, Q, = rdt[exp( -cot) -l]C,(t) which explicitly enforces the limit k, -,0 if co -+0 even for an approximate correlation function. The quantum rate expression can be written more explicitly if one diagonalizes a matrix representation of H (a complex-symmetric matrix), yielding the complex eigen- values (El -iTJ2} and eigenfunctions, ( I $,)}. The time evolution operator can then be expressed as follows where we note that, unless explicitly indicated, the wavefunction is not complex-conjugated in the bra state ($1I in this complex-symmetric algebra.' Eqn.(2.lb)for the flux correlation function thus becomes Cf@)= 1exPC-P(E1 + &)/21($l I F I $3 1. 1' the time integral of which gives the following expression for the rate From eqn. (2.7),it is easy to see what approximations are necessary to degrade it to the classical result in eqn. (1.2):one neglects the off-diagonal interference terms, 1 # l', in eqn. (2.7~)for t > 0 and makes the identification = -($1($1" I F I $1) I FI$3= rl/h= kl (2.8) whereby eqn. (2.7~)becomes Cf(t)= -exp( -BE,)k? exp( -k, t)1 Since this approximation is valid only for t > 0, eqn. (2.5)is used? to obtain the rate, Pm k, Qr = J --dt[ 1 -exp( -cot)] C exp(-PEl)k? exp( -kl t) (2.10~) 0 I (2.10b) (2.10c) which is the classical result, eqn.(1.2). 3. Summary of Classical Description of Energy Transfer Even within the classical RRKM description of the A + B collision dynamics, i.e. eqn. (1.2)and (1.3), it is common nowadays to go beyond the strong collision approximation t One may equivalently make the classical approximation directly to eqn. (2.7b)provided one subtracts from the result so obtained its o = 0 limit ; i.e. k, Q, = -XI exp( -flE,)(kt/w+ k,)-[-1,exp(-PE,)k,] = c,exp(-PE,)ok,/(o + k,). This is also equivalent to adding to eqn. (2.9) the very short time transition-state theory contribution, d(t) c,exp(-PE,)k,;see Ref. 5 for more discussion of this point. W. H. Miller in treating the collisional relaxation in eqn. (1.1b). This is typically done? via a classical master equation: if (c1(t))denotes the populations of the energy levels (E,) of AB* at time t, then the master equation is il(t) = -k,c,(t) -C caP,*+,c,(t)+ UP,+,*cJt) (3.1)I’ I’ (3.2~) it being noted that the state-to-state transition probabilities depend on temperature.[Note that the diagonal terms, 1’ = 1, in eqn. (3.1) cancel, but for convenience I leave them in.] The terms on the right-hand side of eqn. (3.1)are, respectively, the rate of loss from state 1 due to unimolecular decay (to A + B), the rate of loss from state 1 due to collisions that transfer population to other states 1’, and the rate of gain of population in state 1 due to transitions from other states. The state-to-state transition probabilities are normalized as and they satisfy the detailed balance relation (The normalization condition is essentially a definition of the diagonal element PI,,= 1 -PlI+,.) It is also useful to introduce the symmetric matrix PI#,I, 6 , I = expCB(E1 + E1*)/214t+IexP(-BE,) = exp(PElt/2)P,+, exp(-PEl/2) (3.34 or in matrix notation P = exp(~~,/2)~exp( -PH,/~) (3.36) where H,is the diagonal matrix of energy levels I(Ho)rfl, = 4,I Ef Because of the normalization, eqn.(3.2b),the master equation reads W)= -(4,I’ kI + us,, I’ -wp,,Ih4) I’ and its solution is conveniently written in matrix form c(t)= exp{ -[k + 41 -P)]t)c(O) where k is the diagonal matrix of unimolecular decay rates (k)r,I = h,I kl (3.7) Therefore, if I‘ is the initial state at t = 0, i.e.cI(0)= &,,, then the populations at time t are cr(t)= qtlt(t) = (expi -[k + a(1 -p)]t}),, (3.8)I‘ The classical flux correlation function which corresponds to the generalization of the classical SCA result in eqn. (2.9)[including the collisional factor exp(-at)], is thus given t See Ref. 2(c),p. 268 et seq. Quantum Mechanical Theory of Collisional Recombination Rates by (3.9) (3.10) 1 I, I’ -P)]-1 ‘(1 -(3.11)1, I‘ Finally, this can be written in a more symmetrical form by noting that the similarity transformation relating P and p in eqn. (3.3b) is also true for any power of these matrices, i.e. P”= exp(-PH,/~)Pexp( + PH,/~) (3.12) and thus also for any function of the matrices, so that in terms of the symmetrized transition probability matrix p,eqn.(3.1 1) becomes -41-(1 -F)co}l,exp( -PE,)/2)k, Q, = exp( -I. I’ = -[k + ~(lP)]-’ ‘(1 -F) bo (3.134 where b is the ‘Boltzmann vector’ (41 = exp( -PEl/2) (3.13b) Eqn. (3.13) [or eqn. (3.1 l)J is thus the desired generalization of the classical SCA result, eqn. (1.2); the SCA result is obtained when P or p-+0. (Plr+l-+ 0 in the SCA because an infinite number of final states I’ are populated.) Note also that the rate vanishes if the collisions are purely elastic, i.e. p-+ 1, since inelastic (i.e. energy loss) collisions are clearly necessary for recombination to take place. 4. Quantum Theory beyond the SCA The recombination rate given by eqn.(2.7b) of Section 2 contains a rigorous quantum description of the A + B collision dynamics but with the (rather crude) strong collision approximation for energy transfer in M + A-B collisions. Eqn. (3.11) or (3.13) of Section 3, on the other hand, gives the rate with a more general and realistic (classical master equation) treatment of M + A-B inelastic scattering but neglects quantum effects in the A + B dynamics. Here we wish to combine these two approaches, i.e. to have a rigorous quantum description of the A + B dynamics together with a master equation-like treat- ment of the M + A-B inelastic scattering. We expect this more general theory to reduce to eqn. (2.7h)if the SCA is made and to eqn. (3.11) if one neglects quantum effects in the A + B dynamics.The general treatment presented here is based on the impact approximation for M + A-B scattering and follows very closely the collisional line broadening theory of t See footnote on p. 53. W. H. Miller Baranger* (see also fan^,^ Ben Reuven" and Gordonll). The recombination rate is still given by the time integral of the flux autocorrelation function? k, Q, = rdt tr[exp( -bfi)BO(t)tEO(t)] where the evolution operator O(t) includes the effect of collisions by the bath gas M. Within the impact approximation (which assumes that the collisions of A-B with M are isolated, independent and random events) if there are n collisions, at times t, < t, < -< t, ,then this propagator is exp[ -iA(tn-t, -l)/~~~(n-W(t)= exp[ -iA(t -~,)/A]S(~) l) --s(,) x expC-iii(t, -tl)/fi]s(')expC-iA(tl -o)/~I (4.2) where = A,, -ii: and S(")is the Smatrix for an M + A-B collision at time t,.If the classical path approximation1 is used to treat the M + A-B collisions, then S(")is a matrix between states of A-B and is a function of impact parameter and relative velocity for the M + A-B collision, Sl,l,(b,, u,); eqn. (4.1) is to be averaged over impact param- eters in the usual way and also over a Boltzmann distribution of relative M + A-B velocities. (The superscript index in S(,) indicates that the impact parameter b, and velocity u, are independent variables for each collision.) Eqn. (4.1)must also be averaged', over the various collision times, and finally averaged', over the number of collisions n, weighted by the probability of having n collisions if cu is the collision frequency, (4.4) (4.54 t This follows from the analysis in Ref.5. The recombination rate is k, Q, = -tr[exp( -P&')Epr], i.e. the Boltzmann flux incident through the dividing surface, where P, is the recombination probability, which is given by P, = lim,+m O'(t)hO(t), i.e. the probability that the system is inside the dividing surface as t + a; this can also be expressed as J; dt(d/dt)O+(t)hi?(r)= -J; dto '(t)PO(t).1It is no problem to relax this approximation and utilize a fully quantum S matrix description of M + A-B scattering; see Ref. 11. 60 Quantum Mechanical Theory of Collisional Recombination Rates where Wl',1 = (4,-El)/h (4.56) with w<'t',1k -] ' 'lk, lk -1) CMlvPlmdE, 6"d62n6Sz',1k-]'(',exp(-pE,)(PE,) ')',k, ,,-I(', ') (4'6) where E, = &v2 is the relative translational energy for the M + A-B collision and V = J(87rT/np) the average thermal velocity.The average over impact parameter 6 produces a (generalized) cross-section, and the average over translational energy a (generalized) rate constant. The integral over t in eqn. (4.5~1)is recognized12 to be the Laplace trans-form of an n-fold convolution, so the result is the product of the n individual Laplace transforms; eqn. (4.5~)thus becomes Further progress in simplifying the general result is made by using a Liouville (or tetradic) vector space:8-" each pair of indices (li, Ik) is considered a composite vector index.The following vectors are thus defined 1.'. In - <$;JI,, I F I $1,) (4.8~) Flo%)(P)= exPC-P(Elo~ + El0)/2I<&$I F I $lo> (4.8b) and the matrices 91k'lk, lk-I'lk-1 = <'t',lk-1' 'lk, lk-1) (4.9a) (4.96) (4.94 the latter two of which are diagonal. In this notation eqn. (4.7) reads k, Q, = -1 9 [w(o + K + in)-91" (o + K + in)-F(/3) (4.10) n=O so that the geometric series over n can be summed to give krQ, = -P [w(l-9)+ K+ iR]-' *9(/3) (4.1 la) which in component reads I F I $i)Ca(l-9)+ K + iQI17/10gi0k, Q, = 1 I<$? lo, lo' 1, 1' x ($lo I F I $;",,>exPC-B(E10 + Elom (4.116) Eqn. (4.1 1) can also be written in the following time-dependent form, dtF exp{ -[~(l9)+ K + inlt)-F(P) (4.1 lc) W.H. Miller Eqn. (4.11) is the desired result that generalizes eqn. (2.7b) and (3.11); it treats the A + B dynamics fully quantum mechanically and treats energy transfer (from collisions with M) more generally (via the impact approximation). In the SCA (i.e.9--+0),it is easy to see that eqn. (2.7b) is recovered, and if one keeps only the 'semi-diagonal' ele- ments Ek = lkfor all k in eqn. (4.7)-(4.10), then eqn. (3.1 1) is obtained (noting that plklk, lk- Ilk- 1 = ( I 'lk, 1k-1 12> 'Ik, lk-1 (4.12) is the transition probability matrix of Section 3). This more general result, eqn. (4.1 l), has been achieved at a heavy price, however, for it involves the generalized 'four-index' transition probability matrix of eqn.(4.9a), a 'supermatrix' in Liouville space. This involves the phases of the Smatrix, as well as their magnitudes, and thus requires much more information about the inelastic M + A-B scattering than simply the transition probability matrix of Section 3. It would thus be very desirable to have a theory, necessarily more approximate, that involves only the transition probability matrix but nevertheless contains both eqn. (2.7b)and (3.1 1) in the appropriate limits. One way to achieve such a result is to make a random phase approximation for the phases of the S matrix elements in eqn. (4.6); i.e. with all the averaging (over impact parameter and relative velocity) that is involved in eqn. (4.6),one assumes that the most oscillatory terms, those involving the phases of the S matrix elements, average to zero.This corresponds to the following approximation for the generalized transition probabil- ities matrix of eqn. (4.94 -p~k'~k~~k-l'lk-l= ('$',~k-1''lk,~k-l) 'lk',Ikd1k-1',lk-1 'lk,lk-l (4.13) which is now to be used in eqn. (4.7). Separating off the n = 0 term of eqn. (4.7) [since it does not involve any Smatrix factors] eqn. (4.7)thus becomes The first term above is recognized as the quantum SCA result of Section 2, i.e. eqn. (2.'7b),and with the identifications of eqn. (2.8) and recognition of the sums over 1, ,...,I,, in the second term above as a matrix product, eqn. (4.14)becomes The sum over n can be evaluated, C[(o+k)-'*Pw]"=-1 +[k+co(l -P)]-'-(w+k) (4.16) n=l sothat the eqn.(4.15)becomes (4.17) The second term-on the right-hand side above is recognized (once the o = 0 limit is subtracted off?) as the classical SCA result, eqn. (2.10), and the third term as the classical *f See footnote on p. 53. 62 Quantum Mechanical Theory of Collisional Recombination Rates result of Section 3, eqn. (3.1 l), which includes the description of energy transfer via the classical master equation. The final result of this random phase approximation may thus be written as kr = kQM SCA -kc, SCA + kc, (4.18) where the first term on the right-hand side is the rigorous quantum result within the SCA, eqn. (2.7h),thc second term is classical result within the SCA, eqn. (2.10c), and the third term the classical result which describes energy transfer via the classical master equation, eqn.(3.1 1) or (3.13). If the SCA is valid, then the latter two terms in eqn. (4.18) cancel each other, and one obtains the quantum SCA result. If quantum effects are negligible, then the first two terms cancel, and one obtains the classical master equation result. Eqn. (4.18) thus does successfully combine the results of Sections 2 and 3 in the appropriate limits. In various applications one may wish to write eqn. (4.18) as kr = kcL + AkQ (4.19~) where AkQ = k,, SCA -k,, SCA is a ‘quantum correction’ to the classical master equation result, the correction being made within the SCA, or as kr = kQMSCA + Akcoi, (4.19b) where Akco,, = kc, -kcLscA is a ‘collisional correction’ to the SCA, the correction being made classically.In either case one notes the important practical feature that eqn. (4.18) requires one to carry out the quantum calculation only within the SCA, so that the more complicated aspects of the energy-transfer step are described via the classical master equation. This latter feature is not true, of course, for the more rigorous result given by eqn. (4.1 1). 5. Summarizing Remarks Thus it has been possible to combine a proper quantum mechanical treatment of the A + B collision dynamics, eqn. (1.1a), with a more general description of the energy- transfer step of the Lindemann mechanism, eqn. (1.lb). The general result, eqn. (4.11), is essentially an adapted version of Baranger’s impact theory of spectral line broadening. Unfortunately, however, this result involves the generalized rate constants of eqn.(4.6) for M + A-B collisions, considerably more detailed quantities than the inelastic rates UP,,,,that are commonly used? to model the classical master equation description of M + A-B energy transfer. A random phase approximation for the S matrix elements in eqn. (4.6), though, leads to a very simple result, eqn. (4.18), which incorporates both a quantum description of the A + B dynamics and the master equation description of M + A-B energy transfer. It will be interesting to see in applications the extent to which these more general theories can be applied and the nature of the corrections to the simpler treatments. I am grateful to Prof.David Chandler for some helpful discussions regarding the sta- tistical averagings in the impact approximation. This work has been supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the US Department of Energy under contract no. DE-AC03-76SF00098 and also by the National Science Foundation under grant no. CHE-9422559. References 1 (u) W. L. Hase, S. L. Mondro, R. J. Duchovic and D. M. Hirst, J. Am. Chem. SOC., 1987, 109, 2916; (h) C. R. Gallucci and G. C. Schatz, J. Phys. Chem., 1982, 86, 2352; (c) S-W. Cho, A. F. Wagner, B. t See Ref. 2(c),p. 268. W.H. Miller 63 Gazdy and J. M. Bowman, J. Phys. Chem., 1991,95, 9897; (d)B. Kendrick and R. T. Pack, Chem. Phys.Lett., 1995, 235, 291. 2 (a) P. J. Robinson and K. A. Holbrook, Unimolecular Reactions, Wiley, New York, 1972; (h) W. Forst, Theory of Unimolecular Reactions, Academic Press, New York, 1973;(c) R. G. Gilbert and S. C. Smith, Theory of Unimolecular and Recombinution Reuctions, Blackwell, Oxford, 1990. 3 R. E. Roberts, R. B. Bernstein and C. F. Curtiss, J. Chem. Yhys., 1969, 50, 5163. 4 F. T. Smith, in Kinetic Processes in Gases and Plasmas, ed. A. R. Hochstim, Academic Press, New York, 1969, ch. 9. 5 W. H. Miller, J. Phys. Chem., 1995,99, 12387. 6 W. H. Miller, S. D. Schwartz and J. W. Tromp, J. Chem. Phys., 1983,79,4889. 7 T. Seideman and W. H. Miller, J. Chem. Phys., 1992, %, 4412; 1992,97,2499. 8 M. Baranger, Phys Rev., 1958,111,481,494; 112,855. 9 U. Fano, Phys. Rev., 1963, 131, 259. 10 A. Ben Reuven, Phys. Rev., 1966, 145, 7. 11 R. G. Gordon, Ad;. Magn. Reson., 1968, 3, 1; R. Shafer and R. G. Gordon, J. Chem. Phys., 1973, 58, 5422. 12 D. Chandler, J. Chem. Phys., 1974,60, 3500 (Appendix A). Puper 5/05047B; Received 28th July, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950200053

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss., 1995,102,65-83 Canonical Flexible Transition-state Theory for Generalized Reaction Paths Struan H. Robertson? School of Chemistry, University of Leeds, Leeds, UK, LS2 9JT Albert F. Wagner$ Chemistry Division, Argonne Nutional Laboratory, Aryonne, I L 60439, USA David M. Wardlaw$ Department of Chemistry, Queen’s University, Kingston, ON K 7L3N6, Canada ~~~~ In previous work (J. Chem. Phys., 995, 103, 2917), simple yet exact formulae for the canonical flexible transition-state theory expression for the thermal reaction-rate constant were derived for all pairings of atomic, linear rigid top, and non-linear rigid top fragments when the distance between the centres of mass of the fragments serves as the reaction coordinate. In this paper, we derive the fundamental modifications required to generalize the reaction coordinate for all fragment-type pairings.That is, the hinge point about which each fragment rotates is no longer constrained to be the centre of mass of the fragment, being in general arbitrarily displaced from the frag- ment centre of mass. The generalized reaction coordinate is the line connect- ing the displaced hinge points. It is shown that only the kinetic energy associated with the internal relative motion of the two fragments is affected by a generalized reaction coordinate. The correction to this kinetic energy has a simple functional form whose evaluation for a given fragment-type pairing is straightforward. The ensuing formulae for the canonical rate con- stant will not be as simple as for the centres-of-mass reaction coordinate case, but are not more dificult to employ computationally.The new theory is applied to the simplest possible fragment-type pairing, atom-diatom, which serves to illustrate the essential features of the method and some of the implications of a generalized reaction coordinate. 1. Introduction Reactions, either dissociative or associative, whose transistion states or ‘bottlenecks’ are characterized by large amplitude motion, constitute a large and important class of chemical processes. This situation most commonly arises when there is no pronounced potential barrier for the formation of a parent molecular from a pair of constituent molecular fragments. Such ‘barrierless’ reactions are commonplace in chemistry, partic- ularly in combustion, atmospheric, and interstellar processes.Large amplitude transition-state motion can also occur in cases where there is a barrier, for example, in t Work supported by the UK Engineering and Physical Sciences Research Council. 1Work supported by the US Department of Energy, Division of Chemical Sciences under Contract No. W-3 1-109-Eng-38. 9 Work supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada. 65 Canonical Flexible Transition-state Theory heavy-light-heavy reactions such as C1’ + HCl, where the bending degree of freedom at the transition state might be thought of as a hindered rotation of HCl (e.g., ‘figure eight’ reaction trajectories).Despite ongoing advances in computer technology, classical tra- jectories and quantum dynamics, the most widespread theories for modelling reaction- rate coefficients are statistical theories. Such theories are generally one of two interrelated but distinct types : variational transition-state theory (VTST) and the quantum adiabatic channel model (QACM). The incorporation of large amplitude motion in QACM has been the subject of a recent study.2 Here we present further developments for accurately incorporating such large amplitude motion in VTST. The first version of VTST to account satisfactorily for large amplitude motion was flexible transition-state theory (FTST), which was introduced by Wardlaw and Marcus circa 1984.3 Like all variational theories, FTST allows the transition state to be mobile and not associated with any particular potential-energy surface (PES) feature; it typi- cally occurs at larger interfragment separations for low energies and angular momenta and at smaller separations for high energies and angular momenta. However, FTST also treats the transition state as floppy and not associated with any particular geometry; there are large amplitude motions (transitional modes) which are coupled to each other and to overall rotation of the molecular system.Typically, transitional modes corre- spond to free rotations of the separated fragments that evolve into vibrational motions of the parent molecule. The strength of the original FTST is the avoidance of approximations concerning the transitional modes by inclusion of the full transitional mode potential (or, the entire PES, if available) and by an exact enumeration of the transitional mode contribution to the transition-state sum of states. This enumeration can be feasibly accomplished only by a classical treatment of the transitional modes through phase-space integrals, which neglects quantum effects (such as zero-point energy) and which makes an individual variational location of the reaction bottleneck for each quantum state impossible. In contrast, QACM treats the transitional-mode potential in a highly approximate manner to achieve a quantum mechanical accounting of all levels in the transition state.A thor-ough discussion of the advantages, limitations, and approximations of original FTST is provided in recent review article^.^ Subsequent fundamental developments in FTST have been made by Aubanel and Wardlaw,’ Klippenstein and Marcus,6 Klip- ~enstein,~--’Srnith,l2-’ Aubanel et a1.,I6 and the present authors.17 This paper will blend two different improvements to FTST put forward by ourselves17 and Klip- penstein.* The price to be paid for inclusion of the potential-energy surface in FTST is complex- ity of implementation, intensive numerical calculations and loss of physical insight into the effect of PES features on the rate coefficient.These impediments prevent its routine use, e.y. in complex kinetic schemes consisting of many elementary steps involving rad- icals, and its casual use to interpret trends in the dependence of rate coefficients on PES features, temperature, size and type of radicals, etc.It is therefore desirable to develop versions of the original FTST which are easier to implement and are physically more transparent, yet which retain the essential features. Significant improvements along these lines have been made by Klippensteing and by Srnithl3 at the microcanonical level, and by ourselves’ ’at the canonical level. An explicit connection between Smith’s approach and our approach is established in Ref. 17. A common feature of all three approaches is that the momentum integrals in the classical phase-space expression for the sum of states along the reaction path can be done analytically.The remaining coordinate space integrals must, in general, be done numerically. The advantage of Smith’s13 and our17 reformulations of FTST is twofold: (1) The dimensionality of the integral expression for the sum of states is reduced by a factor of two compared with the corresponding original FTST expression; (2) the variables in the coordinate space integral are Jacobi internal angles which describe the transitional modes by specifying the orientation of fragments S. H. Robertson et al. relative to the vector R,, connecting the centre of mass of the two fragments. We point out that use of R,, as the reaction coordinate is implicit in the treatments of Ref. 13 and 17. The clear relationship between these angles and bond internal coordinates consider- ably reduces the effort devoted to the transformation of coordinates for the purpose of evaluating the PES which is most often expressed in bond coordinates.Indeed, the PES may be expressed in terms of Jacobi coordinates in the first place, thus eliminating the need for a coordinate transformation. In original FTST, on the other hand, the sum of states is expressed as an integral over angular momenta and their conjugate angles; a complicated transformation is required to go from these variables to bond internal coor- dinates. In a recent paperI7 (hereafter denoted part I) we derived a new version of canonical FTST (CFTST) which is easier to implement, involves less computation, and provides more physical insight than the original FTST.Application of this theory provides an expression for the thermal rate coefficient, k( T).The canonical approach requires only one flux minimization along the reaction coordinate for each temperature, as opposed to the (correct) microcanonical approach in which N,, is first minimized for each energy and angular momentum, and then summed over J and Boltzmann-averaged over E to obtain a transition-state partition function at each temperature. A canonical estimation of the rate constant must always be higher than the true microcanonical one. For radical-radical association reactions for which this error has been studied, the over- estimation of k(T) by a canonical treatment is typically 5-20?4, and is weakly tem- perature dependent.8 For ion-molecule reactions, the overestimation is expected to be larger, and for a particular case, is known to exceed 50% at some temperatures.’ Although our treatment in part I is formally equivalent to a canonical extension of Smith’s microcanonical treatment, it is distinctive in that no reference is made to a microcanonical formulation and some of the physically important features of flexible transition states are presented in a conceptually appealing manner.As it stands, our new version of CFTST is restricted to the reaction coordinate being the centres-of-mass separation R,, . Extensive work by Klippenstein7t8 within the micro- canonical FTST framework has revealed the limitations of using R,, as the reaction coordinate for a handful of representative systems.Klippenstein has shown that use of a traditional bond reaction coordinate (i.e. atom-atom separation distance) can dramat- ically lower the corresponding sum of states at intermediate and small interfragment separations. Moreover, he has introduced the concept of an optimized reaction coordin- ate and implemented this’idea for a restricted set of reaction coordinate variations. The motivation for the present paper is to incorporate a completely general reaction coordinate at the canonical level of FTST, the precise choice of reaction coordinate being optimized, at a given temperature and for a given system, by the usual variational minimization of the sum of states orthogonal to this coordinate. We do so by extending our new version of FTST, thus retaining the essential features and advantages of that treatment.In particular, the present treatment of generalized reaction paths within the framework of CFTST, and indeed within the broader context of canonical VTST, makes no reference to a microcanonical formulation and appears to be the first at the canconi- cal level. Furthermore, the extension is formulated so that the resulting formulae and their numerical implementations involve relatively minor modifications of the existing centres-of-mass reaction coordinate treatment described in part 1. The conceptual and computational advantages of the present treatment become especially important when employing non-traditional reaction coordinates whose optimization requires computa- tional effort absent in a sum of states calculation using a traditional (non-optimized) reaction coordinate, such as a bond coordinate or the centre-of-mass coordinate.The rest of this paper is divided as follows. Our canonical reformulation of part I is reviewed in Section 2. The extension to arbitrary reaction coordinates is described in Section 3, which is the focal point of the paper. Case-specific formulae for a simple Canonical Flexible Transition-state Theory system (atom-diatom) are derived in Section 4. The complete range of case-specific for- mulae and selected numerical examples are deferred to a subsequent publication. The results of Section 4 are discussed in Section 5 and the paper concludes with a summary in Section 6.2. Review of CFTST for the Centre-of-mass Reaction Path 2.1 Degrees of Freedom Before giving explicit results from part I, it is useful to review the allocation of degrees of freedom for the various combinations of fragment types. A summary is provided in Table 1. All fragments are modelled as one of the following types of rigid tops: atomic, linear or non-linear. The total number of atoms in both fragments is N. The number of internal angles, N4,n,,describing the relative orientation of the two (rigid) fragments, is equal to the number of transitional modes. One coordinate, the reaction coordinate, specifies the separation of the fragments. There are three external rotation coordinates and N, = N4inr+ 3 transitional/external rotational coordinates (q = ql, .. .,qNq)in terms of which one constructs a kinetic-energy operator in matrix form. In our canonical formulation of the thermal rate constant in CFTST, the determinant of this matrix carries all relevant information about the kinetic energy for the transitional/external rotational modes. The number of conserved modes, N,,,, = 3N -3 -(N, + l), is equal to the number of vibrational degrees of freedom in the two fragments. Table 1 also reveals that the problem under consideration is a finite one. That is, there are only five different cases to consider and, once case-specific formulae for a general reaction coordinate are obtained for each, the overall problem is solved. 2.2. Summary of Results The canonical VTST expression for the rate coefficient is (with /3 = l/kT) The quantity Q;, = QTs(R:,, T)is the pseudo-partition function of the system at the transition state, the transition-state location RA, being that value of the reaction coordinate R,, which, at a given T, minimizes QTS(Rcm,T).This partition function has its zero of energy on the reaction path where the potential is Vt = V(Rlm).The form of QTS will be described in detail below. (IreaCtis the partition function of the reactants; c/ot is the ratio of reactant and transition-state symmetry factors and ge is the ratio of elec- tronic degeneracy factors for the reactants and transition state. The incorporation of large amplitude transition-state motion is through QTS . In FTST, the overall rotation of the system and the large amplitude transitional modes (as a group) are separable from the small amplitude conserved modes (as a group), i.e.Table 1 Summary of degrees of freedom for various fragment types case description N*,", N, Ncom 1 atom + lin. -top 1 4 3N-8 2 atom + non-lin. -top 2 5 3N-9 3 lin. -top + lin. -top 3 6 3N-10 4 lin. -top + non-lin. -top 4 7 3N-11 5 non-lin. -top + non-lin. -top 5 8 3N-12 S. H. Robertson et al. and where Qc and Qtr are the conserved mode and transitional/external rotational mode partition functions, respectively. Qc is evaluated quantum mechanically (almost always as independent harmonic oscillators from normal modes, which are assumed to be orthogonal to the reaction coordinate).However, Qtris treated in the classical limit, i.e. where H is the classical Hamiltonian for the N, transitional/external rotational modes (excluding, of course, the reaction coordinate). The integration variables are the gener- alized coordinates q and their conjugate momenta p. H can be written as the sum of a transitional-mode kinetic energy T,and a potential energy V,, . In describing ?;, and V,, , the fragments will be assumed to be rigid bodies whose shapes are those at the given value of Rcmalong the reaction path. That is to say, at any point on the reaction path, each fragment is to some degree adjusting its optimized internal geometry in response to the presence of the other fragment. When either fragment departs from the reaction path in transitional-mode directions, its internal geometry will be presumed fixed and only its orientation relative to the other fragment will change.This assumption means that each fragment, no matter how complicated, can be treated as either an atom or a linear, spherical, symmetric, or asymmetric top for the purposes of transitional motion. Prin- cipal moment of inertia analysis of each fragment at position R,, fixes the locations of the atoms of the fragment with respect to the principal axes. The relative orientation of the fragments to each other, i.e. the transitional modes, are then determined only by the angles that govern the relative orientation of the principal axes of each top. When decomposed from the full PES, K,can ultimately be formulated as a function of only those orientation angles and R,, .With the above assumptions, large amplitude motion is contained exclusively in Q,, , which can be drastically simplified, as described in detail in part I. The first step in the simplification is to analytically do all the momentum integrals. Since V,, cannot depend on p, eqn. (4) can be rewritten as When the kinetic energy is written as a sum of quadratic terms in the generalized veloc- ities, q = (dl ,42,. . .), N, 2Tr = c Aijqidj= 27;, = QT.A,Qi, j=1 the integral over the conjugate momenta in eqn. (5) can be performed without having to express p explicitly,20 with the result, Qt, =&I...I($y21Ao/1i2 exp(-pV,,)dq j... 1= (G)Nq'2I A, exp( -BV,,)dq (7)Ph 70 Canonical Flexible Transition-stute Theory where I A, I is the determinant of the matrix whose coeficients are Aij.The expression for 1 A, I depends on the definition of the reaction coordinate, since A, describes transitional motion referred to that coordinate. In part I, specific expres- sions were derived for lAol for when R,, is the reaction coordinate. These expressions have a relatively simple form and are given for cases 1-5 in Table 2. To provide an example of the generalized coordinates, the associated matrix A,, and its determinant, we use case 5 as considered in detail in part I, i.e. two dissimilar sym- metric top fragments with principal moments of inertia (Ila, Ila, Ilb) and (I2a,12a,12b). In general, these moments (and hence A,) are functions of Rcm ,owing to the dependence of a fragment’s equilibrium geometry on the reaction coordinate, but this dependence is suppressed in what follows.Determinants for cases 1-4 can be independently derived but can also be inferred by systematically reducing the symmetric tops of case 5 to linear tops and atoms. The extension of the symmetric-top version of case 5 to dissimilar asymmetric or spherical tops is outlined in part I. It is useful to divide the kinetic energy for angular motion, Tr,into three terms:21 2Tr = 2(Tr,int + Tr, int-ext + Tr, ext) --qT . A,*q (8) The first term in eqn. (8) is the internal kinetic energy, Tr,int, the second term represents coupling between internal and external angular momenta (Coriolis coupling), Tr,int-ext, and the last term is the energy associated with the motion of the body-fixed axes, Tr,ext.For the case of two non-linear tops, q = (Y, 6,6, Ocml, 4cm2,Ocm2, xcm)where (Y,0and O)are external angles describing the orientation of the entire system relative to space-fixed axes, and (4cml,4cm2,Ocml , Ocm2, and xcm) are internal (Jacobi) angles specifying the relative orientation of the two rigid symmetric tops. The elements A, of A, are given in Table 1 of part I. After lengthy algebra (performed via the MAPLE algebraic manipulator software package), the determinant I A, I required in eqn. (7), is I A, I = (~R~~)~z~~sin2 Ocm2 sin2 O (9)Ilb~;, 12, sin2 Ocml Note that the R,, dependence of the moments of inertia have been suppressed. Substitution of eqn.(9) into eqn. (7) followed by analytical integration over O(0 to n), “(0 to 271) and O(0 to 2n) yields, Qtr(Rcm 9 7‘)= VRcm 9 T)QpdRcm9 T)Qfr,l(Rcm 9 T)Qfr,2(Rcm 7 7’) (10) where QPd(Rcm,T)= 2pR&,//?h2 is the partition function for the pseudo-diatomic formed from the centres of mass of the two fragments, Qjr,i(Rcm, T)= (2nZib/Bh2))”2(2ZiJBhZ)is the partition for the free rotation of symmetric top i, and T(Rc,,T)is a hindering function in the form of the configuration integral: r(Rcm, T) (257131-1 [d~ctnl sin Ocml decrn2 sin 0cm2 lffd4crnl Lffd4cm2 lndxcm s: x exP(-PVr) (1 1) eqn. (10) has been derived for two symmetric tops. However, because the modifications to I A, I are so simple for other tops or atomic fragments, it turns out that the equation is generally valid if Qfr,i(R,m, T) is correctly evaluated.For example, in the case of two dissimilar asymmetric top fragments (Zi, # Ii, # Zic, i = 1, 2), eqn. (10)should be evalu- ated with Qfr,i(Rcm,T)of the form: Table 2 Determinants for centres-of-mass reaction coordinate moments of inertia internal case angle fragment 1 fragment 2 Canonical Flexible Transition-state Theory This expression of Q,, is exact within the framework of canonical FTST with the centres- of-mass separation as the reaction coordinate. The expression has a particularly appeal- ing form: it is a product of a kinematics factor and a hindering function. The kinematic factor is itself a product of standard rotational partition functions for non-interacting fragments.These partition functions are completely independent of K, . They depend only on the masses and on the shape changes of the fragments as they progress down the reaction path. The hindering function, eqn. (1 l), is independent of the masses and is exclusively dependent on Kr. The hindering function is just exp( -p Kr) averaged over the (internal) angles required to specify the relative orientation of the two fragments. Note that when v, = 0 (i.e.no inter-fragment interaction), T(R,,, T)= 1. By definition, Kr is zero at the reaction-path geometry. If the reaction-path geometry has the most attractive geometry at a given value of R,, (as is generally the case), then at that value of R,,, l/;, will by definition never be less than zero and consequently the hindering func- tion will always fall between 0 and 1.Thus Qtr is generally a potential-dependent frac- tion times a shape- and mass-dependent kinematic factor. Minimization of the partition function product Q,(R,,, T)Q,,(R,,, T)[eqn. (2)] with respect to R,, yields Qs( T)Q:,(T)= Q:(T)Tt(T)QA,(T)Q!r,l(T)Q!,,2( T). Substitution of the latter expression into the rate-coefficient expression [eqn. (3)] gives; For an association reaction, Qreacc = Qtrans Qvib, Qfr, Qfr, and k(T)becomesQvjb, Of the last three factors in this expression, the first contains all the vibrational informa- tion, the second all the shape information, and the last all the transitional-mode poten- tial information.3. Extension to Arbitrary Reaction Paths 3.1. Discussion of Approach In part I, we choose the CM separation as the reaction coordinate and locate a particu- lar point on the reaction path by fixing the value of R,,. The relative orientation of the fragments is specified by rotating the fragments about hinge points located at either end of the line connecting the centres of mass of the two fragments. However, a more general approach is to allow for arbitrary hinge points about which to rotate the fragments. The associated reaction coordinate is the line connecting the displaced hinge points; a point on the reaction path is now located by fixing the value of R. Such a generalization is depicted in Fig. 1 for the atom-diatom case (case 1).As a consequence, the distance R,, for case 1 in Fig. 1) for a given value of R: R,, = F(interna1 angles; R) (1 5) where F is some completely flexible function capable of representing all ways that R,, could depend on angles while holding a separation distance R fixed. Explicit expressions for this general constraint of R,, will be given in the next section for a simple case. This situation implies that when time derivatives of Cartesian coordinates are taken, as is required in the construction of the kinetic-energy matrix, R,, can in general have a non-zero time derivative since it is now a function of the internal angles which do have a non-zero time derivative. It is only through such derivatives that new terms arise in the kinetic-energy matrix.These new terms will affect the determinant. Note that because 8,, depends on the internal angles (just S. H. Robertson et al. diatom centre of mass Fig. 1 Schematic representation of the geometric constraints on R,, for a fixed separation R and a completely arbitrary reaction coordinate parameterized by d and y~ the new terms enter only through the time derivatives of internal angles, they only appear in that part of the matrix connected to the coriolis and internal kinetic-energy terms. These new terms become kinetically interesting when they lower the value of the determinant (at least some values of internal angles), for then there is at least a chance that the rate constant can be further minimized. It will be shown in the next section that it is possible to select a coordinate system such that these additional terms arise only in the internal kinetic-energy part of the matrix and never in the coriolis part of the matrix. The objectives of this section are: (a) to express a completely general F function; (b) to select an appropriate coordinate system in which these additional terms only appear in the internal kinetic energy; (c) to establish the form of the modified A matrix whose determinant we seek.3.2. General Formulae The basic formula for the kinetic energy is N N = c rniii.ij + 2o. 1rniri x ij+ oT-Z*o i= 1 i= 1 = (dcxt 7 (iintIT * A (Ocxt 9 dint) = (0, * A’ . (0,Ojn,) (16) where ois the angular velocity vector in the internal coordinate system, ri is the carte- sian vector for each particle of mass miand 3, is the time derivative of ri .A is the matrix whose determinant we seek but it is simpler to work with A’ which is related to A by a similarity transformation involving the time derivatives of qext= (Y, 0, @). This transformation matrix connecting oand qexIis derived in Ref. 22. Since deter- minants of similarity transforms are products of the determinants of the original matrix, the transformation matrix, and its inverse, we have I A I = sin2 0I A’ I. The formula for 2Tr is based on the notion that rigid body kinematics can be rep- resented by motion of a fixed point attached to the body plus rotation about the fixed point. The most convenient selection of an internal coordinate system is one that has its origin at this fixed point.The actual fixed point selected will change A’ and the determinant. It can be shown that the best fixed point is the collision system’s centre of mass. Any other fixed point will increase [A’[for all angles and thus is not kinetically interesting. 74 Cunonicul Flexible Trunsition-state Theory With the fixed point selected, the orientation of the axes at the fixed point is com- pletely optional. A similarity transform that reorients the axes via orthogonal transform- ations leaves the determinant of the transformed A’ unchanged from that of the original A’. Thus, as in part I, we choose Rc, as the zaxis of the coordinate system. This has a number of desirable qualities which are quite relevant to the question of including the F function constrain on R,, .We write A’ in a very general form. To begin, we define the following quantities: N N N N N N Here H,, ,H,, . . . are related to the usual moment of inertia expressions (e.g. I,, = Hyy+ H,, ;I,, = -H,,), whereas Hxxj, Hxjxk, etc. are derivatives of the Hs with respect to internal coordinates. Here A,,, is that part of A’ that is multiplied by products of cu components. It comes from the third term for rigid rotation in the basic formula [eqn. (16)] and for all cases is a 3 x 3 matrix. A,,, is that part of A’ that involves cross products of o components and qintcomponents. It comes from the second term for Coriolis coupling in the basic formula and is a rectangular Nqinlx 3 matrix, where N,,t is the number of internal angles which varies from 1 to 5, as indicated in Table 1.Aintis that part of A’ that involves products of qintcomponents. It comes from the first term in the basic formula and is an Nqin,x Nqinlmatrix. With the above definitions, N . . .]C miii * ii = [dint, 1 dint, 2 i= 1 ...Hxlxl + Hylyl + Hzlzl Hxlx2 + H,ly2 + Hzlz2 ‘lint, 1 ...HAx1 + HY2Yl + Hz2zl Hx2x2 + Hy2y2+ Hz2z2 dint, 2 X ...1 ... .. Based on eqn. (16), (21), (22), and (23), the following identifications are readily made: Arot = I (24) S. H. Robertsonet al. ... Hxlxl +Hylyl +Hzlzl Hxlx.2 +Hyly2 + Hzlz2 ... A. = Hx2xl + Hy2yl + Hz2zl Hx2x2 + Hy2y2 + HZ222 ... (25)In1 ...1 These formulae have already been implicitly used in part I for all five cases in order to get the determinants for a centres-of-mass reaction coordinate.Now we want to take explicit account of eqn. (15) and the details of the coordinate system to determine how Aro1,Ainland A,,, change. To begin with, every place where R,, appears in A,,[, Aintand A,,, requires a substitution of the F function from eqn. (15). However, for our purposes, we do not have to do that before deriving the determinant. One can make this substitution in the determinant directly. Thus everywhere R,, appears in I A'[, we can substitute an expression, as yet unspecified, of angles and the separation R. Before constructing the determinant, it is necessary to develop the new terms that come from the fact that R,, is now an explicit function of the internal angles.These new terms only arise through the derivative expressions involving Hriyj,etc. This can be illustrated by analysing Hyxl.This has the form: N N Hyx1 = C mi yi(dxi/dqint, 1) = C mi ~1 C('xi/aqint, 1) +(axi/aRcmXaRcm/aqint, 111 (27)i= 1 i= 1 where (8xJ8qinl,1) indicates the explicit dependence of xion qinl, (with R,, and all other 1)qin,s held constant) and (~xi/~R,,)(~R,m/~qin,,indicates the implicit dependence through the dependence of R,, on qinl, uia eqn. (15) (with all qintsheld constant in the first factor and with R and all other qintsheld constant in the second). This second term does not exist in part I because the F in eqn. (15) has always been treated as a constant, making aR,,,Jdqinl, zero.Now we consider it not to be zero. Even though eqn. (15) is a very general constraint, making (dR,,,Jaqin,,1) of as yet undetermined complexity, the specific selection of the coordinate system has implica- tions for the terms (aui/aR,,) where u = x, y, or z. In particular, since R,, is the z axis of the coordinate system, (ax,/M,,) = (ay,/aR,,) = 0 (28) because the common x and y axes for both reactants are perpendicular to the z axis. Furthermore, changing R,, only causes a common displacement for every atom on one reactant and a different but common displacement for every atom on the second react- ant. Consequently, if u stands for either x or y: N Huzj = C mi uiC(azi/aqint, j) +(azi/aRcm)(aRcm/aqint, j)Ii= 1 = 2 mi .i((aZi/aqint, j) i= 1 +Cmi, uiz(aZiJaRcm) (29)+(aRcm/aqint, j) C mi,~i~(aZi~/aRcnJK: Nz I}i2 where i, and i2 label atoms in fragments 1 and 2, respectively, and N, +N, = N.Since (dz,,/c?R,,) and (c7ziz/dR,,) are constant within each fragment, they can be set equal to Canonical Flexible Trunsition-state Theory Azl and Az2,respectively. Now eqn. (29) becomes where the fact that Rc, connects centres of mass for each reactant guarantees that 1miui = 0 when the sum is restricted to either reactant and u = x or y. Eqn. (30) is the expression we implicitly use in part I and thus the constraint of eqn. (15) in this coordin- ate system has no effect on any HuUj,where u and u are x, y and z. The above arguments imply that in the selected coordinate system, no new terms arising from eqn.(15) develop in A,,, and A,,,. Given the nature of the terms in the remaining component of A‘, namely Aint,the only new terms that occur are, via Hzjzk : N Hzjzk = 1mi[(azi/aqint, j) i= 1 + (azi/dRcmXJRcm/ciqin,, j)lC(azj/aqint,k) + (Jzi/dRcmXaRcm/aqint, dl N = 1mi(azi/aqint, j)(azi/aqint, k)i= 1 N2 + (aRcIn/aqinl. k)[Azl NI mil(aZil/dqinl, k) + Az2 1miz(azi,/aqint, k) 1ii i2 N + (aRcrn/dqint, j)(dRcrn/aqint,k) C mi(azi/JRaJ2 (32) I where (dzil/dRcm)and (dzi2/dR,,) have been replaced by Azl and Az2 in going from eqn. (31) to eqn. (32). To proceed further, we note that in our coordinate system: Zil = -fRc, + Zi1 ziz = (1 -f)Rc, + Zi2 (33) where z’ is measured in the local fragment centres-of-mass coordinate system such that S.H.Robertson et al. This equation makes the middle two terms in eqn. (32) become zero. The last term of eqm. (32) assumes the form: = ~(aRcm/aqint.jXaRcJaqint, A (35) where p = rn, m,/M is the reduced mass of fragments 1 and 2. If we define A; as the form of A’ when the F function is just a constant and if we define the vector VF by: then the A’ matrix is related to the A; matrix by 1 (37)A’ = A’,(R,, = F) + [07 pVFVFT IO where 1 is the 3 x 3 identity matrix, 0 is the 3 x N, null matrix, and A’,(R,, = F) means the old matrix where F is substituted for Rcm.Thus, if we can write down a general form for the constraint F, we need only evaluate F, add it to A’,, form the determinant, and then substitute F for Rcm.Since we have already the form of A’, in the coordinate system used here, all we need to evaluate a completely general determinant is an expression for F of eqn.(I 5). 4. Case-specific Formulae The constraint F varies from case to case, but a specific expression for each of cases 1 through 5 can be formulated without undue difficulty. Since both the dimension of the original A’, matrix and the complexity of the modification embodied in pVFVFT increases from case 1 to case 5, determination of a compact analytical form for the determinant of the new A’ matrix [eqn. (37)J by hand is generally not feasible. As was done in part I for ]&I, it is expedient to employ an algebraic manipulator to obtain A’ I.Even so, the resulting analytical expressions for I A‘ I are not as compact or physi- cally transparent as for I A; I (see Table 2 for the latter). This is not, however, a matter of great concern: The IA’ I expressions (i) are nevertheless analytical, which is a significant improvement over any existing variable reaction coordinate treatments in the broad context of VTST; and (ii) will ultimately appear in a configuration integral [analogous to eqn. (7)J which will be evaluated numerically. It is nevertheless of pedagogical impor- tance to illustrate the implementation of the method derived in Section 3. We choose the simplest possible case, case 1 for an atom-diatom system, since there is only one transi- tional degree of freedom and since all algebraic manipulations are readily performed by hand and the ensuing expressions are not too cumbersome.The various distances and angles involved in specifying an arbitrary reaction coordinate for the atom-diatom case are indicated in Fig. 1. (R,,, OCm) denote the centres-of-mass reaction coordinate and the (Jacobi) internal angle associated with the single transitional degree of freedom, respectively. The generalized reaction coordinate and its associated internal angle are designated (R, 0). The reaction coordinate is general, in that the hinge point at the diatomic end of R is at the tip of the vector d whose magnitude d and orientation q with respect to the diatomic axis are arbitrary. A particular reaction coordinate is selected by fixing d and q.The associated transitional Canonical Flexible Transition-state Theory mode now consists of all possible rotations of d about the displaced hinge point; a particular point on the reaction coordinate is now specified by fixing R. Here we confine d to the triatomic plane but, at least in principle, d can lie out of this plane; one needs to specify only an azimuthal angle in addition to the polar angle q. The out-of-plane hinge point is not further considered because its physical significance for an atom-diatom case is not apparent and because it unnecessarily complicates our example without providing additional insight into the essential features. The out-of-plane case has obvious physical relevance for the non-linear topatom case (case 2), whose treat- ment we defer to a subsequent paper.From Fig. 1 it can be seen that R,, depends on 0 or, equivalently, on i= q -O,, for fixed R. Choosing the latter as the independent internal angle variable is consistent with the approach described in Section 3, The relationship between R, R,, and O,, is simply expressed using the law of cosines: R’ = R:, + d2 -2dR,, cos 3, (38) or R,, = d cos 3, + [R’ -d2 sin2 A]’/’ = F(O,,; R) (39) We are now in a position to derive the determinant expression for an arbitrary reaction coordinate for the atom-diatom case by modifying Adderived in part I by using the above F function in eqn. (37). In this case Ad is a 4 x 4 matrix since there are three external rotational degrees of freedom and one internal (transitional) degree of freedom.With the z axis of the coordinate system pointing along R,,, and with the y axis in the triatomic plane, it has the following components : I + pR,2, 0 0 Ah = 0 0 pR:, + 1 cos’ O,, -I cos O,, sin O,, -I cos O,, sin O,, I sin2 O,, I 0 0 The kinetic energy, Tr,is given by eqn. (8) with q = (w,qinJ= (cox, coy, w,, O,,) and A; by eqn. (40). The determinant, I A; I, for a centres-of-mass reaction coordinate for case 1 is given in Table 2. Next one needs VF, which in this case has the single element of ~R,,/~O,,. From eqn. (39),this is VF = aR,jao,, = -aR,jan = d sin 1, + d2 sin 1cos 3,/[R,, -d cos i] = R,,d sin i/[R,, -d cos A] To evaluate eqn.(37), we will need the square of this expression: (VF)’ = RZmd2sin’ i/[R,, -d cos 112 Thus from eqn. (37), the final general expression is I + pR:m 0 0 I 0 pR;, + I COS’ Ocm -I cos O,, sin O,, A‘ = 0 -1 cos O,, sin O,, I sin’ O,, 0I 0 1 pRZmd sin2 1,0 0 I+P [R,, -d cos S. H. Robertson et al. Here each R,, could be replaced by eqn. (39) but it would be simpler to form the determinant and then to make the replacement. Accordingly, where F(OC,;R) is given by eqn. (39). Implications of this result are investigated in the next section. 5. Discussion In order to assess the conceptual and computational implications of an arbitrary reac- tion coordinate in the context of CFTST, it is useful to compare with the centres-of- mass reaction coordinate results derived in part I and summarized in Section 2.To make a specific comparison, we use the atom-diatom system of Section 4 and consider the transitional/external-rotational partition function, Qtr. For a centres-of-mass reac- tion coordinate with N, = 4 and A, = sin2 @A; with A; from case 1 in Table 2. = Qfr(Rcrn , T)Qpd(Rcm 3 T)r(Rcm 9 T) (48) In eqn. (48) a kinematic factor consisting of the free rotor and pseudo-diatomic rotor partition functions Qfr= (21/Ph2)and Qpd = (2pRZJPh2) can be identified, and a hin- dering function r in the form of a configurational integral has been defined: r(Rcm T)= + poem sin ocm exp[-Pl/;r(Ocm Rcm11 (49) The kinematic factor is a product of standard rotational partition functions for non- interacting fragments, depends only on the masses and the geometry, and is completely independent of Kr.The hindering function, on the other hand, depends exclusively on qr,exp(-/3Vr) being averaged over the internal angle d,, weighted by sin 8,, . Cunonicul Flexible Transition-state Theory The corresponding partition function expression for an arbitrary reaction coordinate for the atom-diatom system is obtained by substituting A' [eqn. (45)]for Ah in eqn. (46): = ---do,,; I"Q,,(R, d, q, T) sin 8,, (;;2)(;:2) + #,,) -cos(qd eqn. (50) reduces to eqn. (47) when d -+ 0 (and hence R + R,,). The complete factor- ization of Qtrobserved for the centres-of-mass reaction coordinate and expressed by eqn. (48) is not achieved for an arbitrary reaction coordinate.The hindering function now depends on masses and geometry via the ratio pd2/Z. The kinematic factor remains just that: it still depends only on masses and geometry but no longer contains the pseudo- diatomic partition function, Qpd. Nevertheless, the numerical evaluation of the configu- ration integral in eqn. (50) is not more difficult or time consuming than the original configuration integral in eqn. (49) and the cost of evaluating the latter is trivial no matter what the nature of the potential v,. The same quadrature method and the same grid size can be used. The same form of the potential, namely V,, = v,(O,,, R,,), appears in both configuration integrals. Thus any coordinate transformation (e.g. from centres- of-mass coordinates to bond coordinates) required to evaluate the potential must be applied to obtain both integrands.The difference between these two configuration integ- rals is the relative weighting of different regions of the O,, angular space due to the factor FZ(O,, ; R)'( R2 + F2(t),, ; R)(pd2/l)sin2(q-Oo) R2 -d2 sin2(q -o,,) in the integrand of eqn. (50).This circumstance makes it possible that with F(B,, ; R) [R2 -d2 sin2(q 0cm)]1'2.It is easy to verify that = -QTS(R4 V, T)= QAR, d,'13T)Qtr(R, V, T) < QTs(Rcm, T) = Qc(Rcm7 T)Qtr(Rcm, T) (52) for a range of separations S of fragments (i.e. S = R = R,,) and for a range of (d, q) values. If this range includes Rim, the value of R,, that minimizes Q,,, then a better thermal rate constant prediction emerges.We note that evaluation of Q, relies on know- ledge of the frequencies of the conserved modes orthogonal to the reaction coordinate. How to identify these modes for arbitrary reaction coordinates and how to obtain reli- able estimates of their frequencies is a matter for further study and is beyond the scope of this paper. Improved canonical rate constants can be obtained by applying the varia- tional criterion to the transition-state partition function, QTS,as the position along the reaction coordinate and the definition of the reaction coordinate are varied. The resulting reaction path can be said to be optimized in that the definition of the reaction coordinate (i.e. the value of d) is continuously varied as the fragments approach or separate.There is a hierarchy of such paths based on the constraints placed on the magnitude d and orientation q of the vector d, whose tip specifies the hinge point for relative orientation of the diatomic fragment: (i) both d and q vary; (ii) one is fixed while the other varies; (iii) both are fixed. We consider the three scenarios in turn. (i) Fully Optimized Reaction Palh Definition This treatment involves evaluating Q,,(R, d, q, T) [eqn. (50)] on a 3D (R, d, q) grid, finding the minimum value of QTS [eqn. (52)] on the grid, identifying this minimum as S. H. Robertson et al. the transition state value Q&, and using the latter in the CFTST rate constant expres- sion [eqn. (l)]. Instead of the transition state being characterized only by a separation Rt, it is now characterized by a location (Rt, d', qt) in the space of all possible reaction coordinates and displacements along these coordinates.Implementation of a fully optimized reaction path treatment has, to the best of our knowledge, not yet been undertaken at the microcanconical or canonical levels, even for the atom-diatom system. The method proposed herein makes such treatment feasible. Being at the canon- ical level, only one minimization is required at each tempcraturc. For a particular choice of reaction coordinate (this includes the centres-of-mass reaction coordinate as a special case), one evaluates eqn. (50) on a ID grid of nK valucs of R. For a fully optimized treatment, the workload goes up by a factor of nd x n, where nd and n, are the sizes of the d and q grids.This factor would probably be of the order of 10, to lo3, which is substantial but not a cause for pessimism because each evaluation of the configuration integral in eqn. (50) is computationally trivial. (ii) Partly Optimized Reaction Path Definition Here Q,,(R, d, q, T) is evaluated, and the minimum value of QTS determined, on a 2D grid of (R, d) with q fixed or (R, v) with d fixed. Connection with the work of Klip-penstein is made if we choose the former with q fixed at a value of 0 or n. That is, the diatomic hinge point is displaced a distance d from the diatomic centre-of-mass along the diatomic's symmetry axis. Klippenstein has applied this level of reaction path optim- ization to Li + HF --+ LiF + H 'la and to He + H2+--+ HeH' + H 'Ib using his imple- mentation of microcanonical FTST.He finds that, on significant portions of the ranges of energy and total angular momentum considered, the transition-state 'location' (Rt, dt) is not on the centres-of-mass reaction coordinate (i.e. dt # 0), and sometimes lies outside the diatom (i.e. dt > m2r/(ml + m,) or m,r/(rn, + m2),where r is the diatomic bond length). Klippenstein has applied the same level of reaction path optimization to NCNO --+ CN + NO'" with similar results. In this case, the hinge point in diatoms 1 and 2 is allowed to be displaced by distances d, and d, along the corresponding diatom's symmetry axis. This situation corresponds to our case 3 (see Table 1) with q, and 77, fixed at 0 or n.(iii) Fixed Reaction Path Definition Fixing d( #O) and q specifies a particular reaction coordinate and one that is different from the centres-of-mass reaction coordinate. There is an infinite number of such choices. A particularly interesting choice is a bond reaction coordinate, obtained by specifying (d = m2r/(m, + rn,), q = n) or (d = rn, r/(rnl + m,),q = 0). Minimizing QTS with respect to R, which is now an atom-atom separation. Klippenstein has used a bond reaction coordinate at the microcanonical level of FTST for NCNO + CN + NO" and CH,CO + CH, + CO." 6. Summary This paper represents a major step toward our overall objective of an improved and rigorous version of canonical flexible transition-state theory, CFTST.By rigorous, we mean that no new approximations or assumptions are introduced beyond those con- taiaed in the original FTST of Wardlaw and Marcus. By improved, we mean conceptual and computational advantages over the original FTST. Both advantages stem from the fact that, at the canonical level of FTST, the multidimensional integral over the phase Canonicul Flexible Transit ion-state Theory space of the transitional/external rotational modes analytically simplifies to an expres- sion containing a configuration integral only over positional coordinates involved in large amplitude (transitional) motion. These coordinates are internal (Jacobi) angles in terms of which the transitional-mode potential energy, Fr,may be directly expressed or which are easily related to the traditional internal bond angle coordinates.In original FTST, on the other hand, the phase-space integral is expressed in terms of angular momenta and their conjugate angles, the momentum integrals cannot be done analyti- cally, and the angle variables have no intuitively obvious connection to the system’s geometry. The dimension of the FTST phase-space integral is more than twice that of the configuration integral in our new version of CFTST; specifically, the former has dimension 2(NqinI+ 1) and the latter has dimension N4,”,,where Nqinlis the number of transitional modes. It is thus generally necessary to evaluate the phase-space integral in original FTST with a Monte Carlo method.Since the maximum value of NYln,is 5 (for two non-linear top fragments), the configuration integral in our new CFTST can be economically evaluated using an appropriate quadrature scheme. This has the further advantage that Vr needs to be known only on a suficient number of grid points to converge the integral. The randomly selected geometries arising in a Monte Carlo evalu- ation necessitate knowing crglobally and this may involve fitting to a functional form, a task which can be quite difficult in itself. Our program for improving CFTST has its origins in Ref. 16, where a method for handling the kinetic energy of the transitional/external rotational modes was proposed. This method is rigorous [all internal-external (Coriolis) couplings included] yet leads to simplification (the integrals over the momenta associated with this kinetic energy can be done analytically).The treatment in Ref. 16 was limited to a particular pair of fragments; at atom plus a asymmetric top (a disc). In Ref. 17 (part I), this limitation was removed by treating two dissimilar symmetric tops explicitly and showing how to infer the results for all fragment combinations (any two of atom, linear, top, spherical top, symmetric top, asymmetric top). Since the transitional/external rotational kinetic energy depends only on the types of rigid tops and on their moments of inertia, there is a finite set of phase- space integrals, each of whose momentum integrations can be done analytically. Since all systems fall into one of the fragment combination categories, the new CFTST can be applied to any system without additional formal development.A central feature of Ref. 17 is the restriction to a reaction coordinate connecting the centres of mass of the two fragments. This has the advantage of leading to particularly appealing forms for the transitional/external rotational partition function : it is a compact expression that remarkably factorizes into a purely kinematic factor and a purely configurational factor determined by the potential. The disadvantage is that a centres-of-mass reaction coordinate can be a poor choice in the sense that other choices may lead to a smaller reactive flux. To address this shortcoming, we have extended, in rigorous fashion, the theory of part I to include all possible reaction coordinates for all fragment combinations.The formal theoretical development accomplishing this task is given in Section 3. The modi- fication to the matrix A, of the coefficicnts of the momenta associated with the transitional/external-rotationalkinetic energy for a centres-of-mass reaction coordinate has a simple form [eqn. (37)]. Only the internal kinetic energy quadrant of this matrix is affected, owing to a judicious choice of coordinate systems in Section 3. The modifi- cation involves the gradient of F, where F specifies how R,, depends on the internal angles and the value of the chosen reaction coordinate R. The general results of Section 3 are applied only to the simplest possible system atom-diatom, to illustrate the nature of the resulting formula for the transitional/external rotational partition function [eqn.(50)]. This formula is more complicated than its centres-of-mass reaction coordinate counterpart [eqn. (47)] and the complete factorization of kinematic and potential contri- butions achieved for the latter is not achieved for the former. Nevertheless, the com- S. H. Robertson et al. puter time and programming required to evaluate eqn. (50) will be only marginally greater than that to evaluate eqn. (47), so that the computational advantage of our new version of CFTST survives the generalization to arbitrary reaction coordinates. Of course, the computer time required to optimize the reaction coordinate will scale lin- early with the volume of the parameter space defining the generalized coordinate.Case- specific results for all fragment combinations and applications to selected systems will be the subject of future publications. It is worth noting that microcanonical sums of states N(E) for optimized reaction coordinates could be readily obtained by inverse Laplace transformation of the transition-state partition function, QTS,discussed in this paper. References 1 B. Amaee, J. N. L. Connor, J. C. Whitehead, W. Jakubetz and G. C. Schatz, Faraday Discuss. Chem. Soc., 1987,84, 387. 2 J. Troe, J. Chem. Soc., Faraday Trans., 1991,87,2299. 3 D.M. Wardlaw and R. A. Marcus, Chem. Phys. Lett., 1984, 110,230; J. Chem. Phys., 1985,83,3462.4 W. L. Hase and D. M. Wardlaw, in Bimolecufar Collisions, ed. M. N. R. Ashfold and J. E. Baggott Chem. Soc., London, 1989;D. M. Wardlaw and R. A. Marcus, Adv. Chem. Phys., 1988,70,231,part 1. 5 E.E.Aubanel and D. M. Wardlaw, J. Phys. Chem., 1989,93,3117. 6 (a)S.J. Klippenstein and R. A. Marcus, J. Chem. Phys., 1987, 87, 3410; (h)Phys. Chem., 1988,92, 3105. 7 S.J. Klippenstein, (a) Chem. Phys. Lett., 1990, 170, 71 ;(h)J. Chem. Phys., 1991,94, 6469. 8 S.J. Klippenstein, (a)J. Chem. Phys., 1992,%, 367; (b)J. Phys. Chem., 1994,98, 11459. 9 S.J. Klippenstein, Chem. Phys. Lett., 1993, 214,418. LO S.J. Klippenstein, J. Phys. Chem., 1991, 95, 9882. I1 S.J. Klippenstein, (a)J. Chem. Phys., 1994, 100,4917; (h)J.Chem. Phys., 1992,%, 8164. 12 S.C.Smith, J. Chem. Phys., 1991,953404. 13 S.C. Smith, J. Chem. Phys.,'1992,97,2406. 14 S.C. Smith, J. Phys. Chem., 1993,97,7034. 15 S. C. Smith, J. Phys. Chem., 1994,98,6496. 16. E.E.Aubanel, S. H. Robertson and D. M. Wardlaw, J. Chem. Soc., Furaday Trans., 1991,87,2291. 17 S. H.Robertson, A. F. Wagner and D. M. Wardlaw, J. Chem. Phys., 1995,103,2917. 18 D.M.Wardlaw and R. A. Marcus, J. Phys. Chrm., 1986,90,5383. 19 E.E.Aubanel and D. M. Wardlaw, Chem. Phys. Lett., 1990,167, 145. 20 M.L.Eidinoff and J. G. Aston, J. Chem. Phys., 1935,3, 379. 21 Maple, Waterloo Maple Software, University of Waterloo, Waterloo, 1991. 22 H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1950. Paper 5107063E;Received 26th October, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950200065

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss., 1995,102,85-115 GENERAL DISCUSSION Prof. Troe opened the discussion: The curves of k(E) = W(E)/hp(E)seem to show a step-like structure which is often attributed to the energy pattern, W(E),of the activated complex states or open channels. I wonder whether these structures should not instead mostly be attributed to fluctuations in the density, p(E), of the states of the dissociating molecule which still may be distributed in clumps of levels. Prof. Moore responded: For ketene there are no spectroscopic data mapping out the level densities near 28000 cm-'. However, it would require a rather improbable saw- tooth pattern in p(E), doubling through the first step, with each successive tooth becom- ing smaller in amplitude, to yield a stepped k(E)from a smoothly increasing W(E).Prof. Reisler commented : Regarding the colder than statistical rotational distribu- tions in the singlet methylene fragment, what do you think are the relative contributions of translational-to-rotational energy transfer and non-adiabatic curve crossings in popu- lating high J states. These states have low translational energies and therefore are likely to follow the adiabatic curves. ]Prof. Moore responded: I can offer only speculation in reply. The source of non- adiabatic coupling which populates product rotational states of higher energy than pre- dicted for strictly adiabatic dynamics on non-crossing curves is not clear. The rotational and translational energies involved range between lo2 and lo3 cm-', where the observed distributions are much colder than statistical.Certainly some translation-to- rotation energy transfer must occur from what we know of rotational relaxation in collisions. However, I suspect that curve crossing is also important and perhaps domi- nant. Although the translational velocity of the highest J states is small, asymptotically the motion along the reaction coordinate probably involves 100 cm-' or more where the adiabatic curves cross. Since non-adiabatic hopping through avoided curve crossings tends to preserve the rotational wavefunction, it may be that a Franck-Condon model in which one simply projects the wavefunction at the transition state onto the free rotor states of the fragments will have some value. Prof.Stoke said: Your exciting review about very detailed observations of the product state distribution obtained in your laboratory in a study of the post threshold photochemistry of ketene raises the question whether information can be acquired in terms of detailed reaction dynamics. I am wondering if it is possible to extract from the rather statistical (phase space) rotational state distribution of the products, 'CH, and CO, quantitative information about the torques (anisotropic strengths of the relevant potentials) exerted onto the separating products carrying themselves through the tran- sition state? Prof. Moore replied: I think we can expect that ub initio potentials and dynamical calculations will ultimately allow us to understand the mechanisms of energy flow and provide a quantitative fit to the data.I do not think that we can ever invert data to derive potentials, especially where distributions are close to the phase space theory (PS'T)predictions. Prof. Rizzo asked: My question concerns the issue of the goodness of the K quantum number. You indicated that the dissociation rate would be lower if K is not conserved. 85 General Discussion It seems to me that there should be two effects caused by the non-conservation of the K quantum number. One, of course, is the 25 + 1 increase in the density of states that could couple with a given level. The other, however, is the increase in rotational energy that would be available to flow into the reaction coordinate owing to the looseness of the transition state: (B -B.f.)J(J+ 1).Would you expect these two factors, which are in different directions, to cause a change in the slope of the rate as a function of energy? I would expect this to certainly be the case if you were able to keep K fixed and increase the energy by increasing J.'12 1 L. Brouwer, C. J. Cobos, J. Troe, H-R. Dubal and F. F. Crim, J. Chem. Phys., 1987,86,6171. 2 X. Luo and T. R. Rizzo, J. Chem. Phys., 1991,94,889. Prof. Moore answered: The key point here is that the A rotational constant is so much larger than the B and C rotational constants. Thus at 4 and 30 K there is a negligible population of excited K, levels but J ranges up to 4 and 10, respectively. Now if K, is strongly mixed for the highly excited molecule away from the transition state, 2J + 1 appears with the density of states in the denominator. However, AJ.J2$-(B -Bj-)J(J + 1) and thus for the higher K, (K in the symmetric top limit) channels the rotational energy at the transition state is much greater than the thermal energy of rotation.For K conserved, these channels are also closed but there is no 25 + 1 multiplying the density of states. Thus the calculated rate changes little with increasing molecular beam temperature in the absence of K mixing and actually decreases for strong K mixing. Prof. Troe said: The question of K conservation may be complicated in the case of the spin-forbidden dissociation of ketene into CO and triplet methylene. If the singlet- triplet crossing hypersurface has a complicated shape, the specific rate constants k(E, J) of the crossing may have a J dependence which looks like partial K conservation.However, experimental thermal rate constants of unimolecular bond fissions in the low pressure limit have rotational factors F $-1 (see the analysis in ref. 1, 2) which clearly reflect the non-conservation of K during the lifetime of the dissociating exciting mol- ecule. 1 J. Troe, J. Chem. Phys., 1977,66,4758. 2 J. Troe, J. Phys. Chem., 1979,83,114. Prof. Moore replied : The selection rules for first-order spin-orbit coupling give AK = 0 and for spin-orbit/vibronic and spin-orbit/rotation AK = 0, f1, f2. Also, the optical excitation selection rules are AK = 0, +_ 1. So this does lead to some increase in K above the thermal population at the temperature of the molecular beam.Our experi- ments show that K is a reasonably good quantum number for values of J up to at least 6. Coriolis mixing is very much stronger for the J states important in thermal unimolecular reactions; our results do not bear on that regime. Prof. Simons asked: How is K conservation/non-conservation to be understood in a situation where the structural changes attending a unimolecular decomposition lead to rotation of the inertial axes? Prof. Moore answered: This certainly can be a mechanism for non-conservation of K. However, for low values of K it takes a large rotation of the inertial axes as the molecule moves to the transition state to give significant amplitudes for a change in K of more than one unit if one simply projects the original rotational wavefunction onto the rotated basis functions.General Discussion Prof. Quack contributed: In relation to the question concerning K, (or K)conserva-tion in unimolecular dissociation, if the principal axes of inertia change during the reac- tion process, I note that some approximately good quantum numbers such as K,K, or others, may be defined and empirically observed as conserved, quite independently of whether they look dynamically conserved when we model the reaction process, A sufi-cient condition for such a conservation would be the empirical observation that the S-matrix for the scattering process proceeding through the intermediate collision complex is approximately blockdiagonal in the presumably conserved quantum number, and similarly the resonances related to long lived states connect only to certain product channels.Such observations would be conclusive quite independently of what happens dynamically during the process of separating the fragments. It is true, however, that such conclusive experiments were not carried out by Moore and co-workers. By noting this, I do not want to question their conclusions, for which they have provided substantia evidence. Note that the point of view presented in my remark has been at the origin of intro ducing nuclear spin symmetry as an approximately good quantum number.lq2 I. M. Quack, Mol. Phys., 1977,34,477. 2 M. Quack, Stud.Phys. Theoretical Chem., 1983,23,355. 3 Y. He, J. Pochert, M. Quack, R. Ranz and G. Seyfang, Faraday Discuss., 1995, 102,275. Prof. Moore responded: I certainly concur. While we plan to look further for corre- lations between the rotational states of ketene and the states of the methylene and CO fragments, I strongly suspect that the only observable quantum number to be conserved from beginning to end will continue to be that for the nuclear spin state of the two H atoms. Prof. Herman communicated : Another interesting framework would be the situation of a linear molecule reacting through a non-linear transition state. There is genuine and predictable Coriolis-type coupling in linear species, called the rotational /-doubling, which mixes different K values at higher J values.How do such levels correlate in the non-linear TST geometry? How does their mixing transfer into Coriolis coupling in the non-linear TST, and influence K? Prof. Moore communicated in reply: The case in which vibrational angular momen- tum in a linear molecule correlates to A-axis rotation at the transition state would certainly be interesting to study. Prof. Clary opened the discussion on Dr. Smith’s paper: The approach you describe is appropriate for the ‘hindered rotor’ region of the interaction but is more difficult to apply for the shorter-range torsional or bending regions. It could be that a different Hamiltonian is needed in these regions appropriate for a more rigid interaction. Dr. Smith replied: Your comment is correct.Our calculations indicate that the coupled Wigner basis allows reasonably facile calculation of eigenfunctions etc. at centre-of-mass separations of 4.0 8, or greater, but as the anisotropy rises steeply at smaller separations, the efficiency of the calculations is degraded. The key problem with altering the representation is how to do this without necessitating storage of the Hamil- tonian matrix. The pseudospectral methods of Leforestier and co-workers’ attempt to achieve this by a (short) sequence of sequential one-dimensional diagonalization/ truncation procedures and show some promise for adaptation to the present case. 1 For example, J. Antikainen, R. Friesner and C. Leforestier, J. Chem. Phys., 1995, 102, 1270. 88 General Discussion Prof.Troe contributed: As your method requires a large amount of computer time, I would like to recommend our alternative technique.' We calculate adiabatic channel potential curves for the transitional modes by perturbation methods in the perturbed rotor limit at large distances and in the anharmonic oscillator limit at short distances. A suitable switching is assured by one matrix diagonalisation or semi-classical calculation at one intermediate distance. The calculation for selected model anisotropies is then scaled to represent the realistic situation. 1 A. I. Maergoiz. J. Troe and Ch. Weiss, J. Chem. Phys., 1994, 101, 1885. Dr. Smith replied : Approximate methods for computing adiabatic channel potentials will clearly play an important role in interpreting a wide range of reactions.Our focus here is two-fold: (1) to generate an algorithm capable of computing exact results for arbitrary anisotropies, thus eliminating the uncertainties associated with switching per- turbed rotor channels to anharmonic oscillator channels and the uncertainties associ- ated with scaling the results for simple model anisotropies to approximate those for more complex anisotropies and (2) to build an algorithmic kernel which can then be readily augmented by inclusion of a basis for the radial coordinate to enable the disso- ciation dynamics to be directly simulated. This investment of human and computer time clearly cannot be made for all reactions of interest. However, we feel that benchmark systems such as ketene, for which extremely detailed and precise experimental data are available, warrant such detailed theoretical investigation.The experience obtained from the approximate methods you have mentioned will certainly be of great value in enhanc- ing the efficiency of the exact calculations, as it indicates what sort of basis set represen- tations we should incorporate into the algorithm. Prof. Quack said: Professor Smith has presented impressive methods for calculating adiabatic channels in polyatomic systems. In complementing the very nice summary he presented, I might note that Quantum Monte Carlo techniques can be, and have been, used not only for the low-lying bound states of (HF), ,(HF), and other polymers (HF), and their isotopomers, but also for low-lying quasiadiabatic channels by means of the quasiadiabatic channel Quantum Monte Carlo technique.' ,2 This method is particularly useful for large polyatomic systems and benchmark calculations, as it provides accurate results with upper and lower bounds.1 M. Quack and M. Suhm, .I. Chem. Phys., 1991,95,28. 2 M. Quack, J. Stohner and M. Suhm, J. Mol. Struct., 1993,294,33; and work to be published. Dr. Smith answered: It would indeed be very interesting to compare the results of the Quantum Monte Carlo (QMC) method of computing low-lying channels with those of the Lanczos method. The Lanczos method has the advantage that it can be precondi- tioned to enhance performance using techniques such as those we have discussed in response to Prof.Clary's question, whereas QMC has the advantage that it can be pushed to higher-dimensional problems for which even iterative methods such as Lanczos become unfeasible owing to core memory limitations. Investigating channel potentials and dissociation dynamics at higher energies, where the density of states is very large, is another matter. QMC has not yet shown promise for enabling studies in the higher energy regime. Dr. Baht-Kurti stated: Dr. Smith has discussed several interesting methods for finding the exact vibrational energies and wavefunctions of polyatomic systems. I would like to mention some related methods which seem to show great promise. In a recent paper' we have presented test calculations for the vibrational states of formaldehyde using a grid based Davidson method approach.This is an iterative approach which General Discussion 89 requires an initial guess for the wavefunction of interest. We have recently developed a coupled cluster vibrational approach which together with the Davidson method permits us to compute exact eigenstates up to 5500 cm-’ above the ground state. 1 G. G. Balint-Kurti and P. Pulay, Theochern., 1995, I, 3431. Dr. Smith replied: The work to which you refer represents a major advance in the applicability of the Davidson method, which has traditionally been successful only for low-lying states, for the calculation of high-lying vibrational eigenfunctions. The David- son method requires substantial effort to be invested in computing a good starting approximation for the seed vector, such as the coupled cluster approach you have devel- oped.The methods which I have discussed in this paper contrast in the sense that they are extremely robust and can start with a random seed vector if necessary. There are definite similarities between the pseudo-spectral sequential diagonalization/truncation methods which we are moving towards (discussed above in response to Prof. Clary’s question) and the self-consistent-field methods that you have developed. The difference is that, for the Davidson method, one applies them to improve the seed vector whereas in Lanczos-based methods it is better to use them to improve the quality of the representa- tion of the Hamiltonian itself.Prof. W. H. Miller opened the discussion on Dr. Temps’ paper: I have two com- ments regarding Dr. Temps paper. (1) In Fig. 8 the theoretical distribution of decay rates at the highest energy seems to be significantly more sharply peaked than the experimental distribution; i.e., the inferred ‘number of decay channels’ vCalcis about twice vexPC.One explanation for this might be that the K states (the body-fixed projection of total angular momentum) are not strongly mixed (cf. the second part of ref. 1 dealing with the effects of K mixing on the decay rate distribution). In other words, the theoretical value vcalc presumably assumes that K is statistically mixed, but if in fhct it is not, then this theoretical value would be too large.Is this possibly an explanation for the lack of agreement in Fig. 8? (2) Regarding the discussion on p. 48 on Fano-type (asymmetric) line shapes, in ref. 2 it was shown that the distribution of phase angles describing the asymmetric lineshapes is directly related to the distribution of decay rates. Specifically, the width of the phase angle distribution, AO, is given by A0 zv1I4/4,where v is the effective number of decay channels. 1 W. H. Miller, C. B. Moore and W. F. Polik, J. Chem. Phys., 1993,99,95. 2 W. F. Polik et al., J. Chem Phys., 1990,92, 3471. Dr. Temps replied: Regarding the first question, the data plotted in the different panels in Fig. 8 of the paper have been obtained from SEP spectra with the same pump line (cf.Table 1 of our paper) so that the spectra should exhibit the same type of rota- tional levels. Otherwise, a comparison of the histograms would not be possible. It is important that the only differences between the panels are the energy ranges. Hence, the observable differences between the histograms do indeed show that there is an energy dependence in the decay rate distributions. This, we think, is a major result of our work. Considering the selection rules, the observed SEP spectra can exhibit states with J = 0.5 and J = 1.5. From comparisons with spectra obtained with different pump lines we conclude that the states taken into account for Fig. 8 are, for the main part, J = 0.5 states. Contributions from J = 1.5 states cannot be ruled out, however.Nevertheless, this does not change our conclusion that there is an energy dependence of the decay rate distributions. The theoretical decay rate distributions were calculated for J = 0.5, assuming com- plete scrambling of the quantum number P, which is the analogue of K for a molecule General Discussion with spin 3 in Hund’s case a (P = K 3). This assumption can indeed be questioned. For instance, in our accompanying investigation of the rate and extent of intramolecular vibrational redistribution in CH,O (ref. 1 and 2), we found evidence for incomplete IVR for specific initial states with P = J up to J = 7.5. We are pursuing further investiga- tions. In principle, decay rate measurements can be performed using the pumpdump probe scheme for different J values, at least up to J = 12.5.Regarding the second question, the line profiles observed in the SEP spectra could be described assuming simple Lorentzian lineshape functions as expected for the case of isolated resonance states which exhibit no overlap with other resonance wavefunctions in the continuum region of the potential-energy surface. With the exception of one, the measured decay time profiles were single exponential. Hence, we do not have data regarding the phase angles. 1 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101,3618. 2 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101,3634. Prof. Quack commented: The terminology and some aspects of the treatment of symmetry in the paper presented by Dr.Temps deserve comment. The interpretation of the commonly used acronym IVR as intramolecular vibrational energy randomization in the paper by Temps seems undesirable to me. I strongly prefer intramolecular vibra- tional redistribution,‘ because this language allows for redistribution phenomena that are not necessarily ‘random’ (whatever is meant by this) and also it can be used for both the time independent and the time dependent descriptions of the underlying physical phenomenon (in contrast to relaxation, which is also sometimes used and stresses time dependence and irreversibility). A review of this matter including some terminology can be found in ref. 1. A second question, also of terminology, concerns the use of the four letter symbol RRKM as essentially synonymous with ‘statistical theories of unimolecular reactions’ (the paper of Dertinger et a1.2 attributes results to RRKM, which were originally intro- duced by other statistical theories, not by RRKM).I do not want to discuss whether it should be ‘LH-RRKM’3 or ‘QET-RRKM’, or whether one of the Rs should be really given to Rabinovitch. But I strongly suggest the use, as general notation, of the rather neutral terminology ‘statistical theories (of unimolecular reactions)’, which comprises many different statistical models or theories which were developed over the years, such as, in particular, RRKM, RRKM-AM,4 SACM (statistical adiabatic channel m~del),~ PST (phase space theory6) and others.By clearly distinguishing such models in the terminology, one can, for instance, clearly discuss novel features, differences in assump- tions and results from one model to the other, which is not the case if one lumps together everything under RRKM etc. For instance, it may well be true that some sta- tistical RRKM models require a smooth increase of k(E) with energy, as suggested by Dertinger et al. in their paper under ‘commonly used’. However, in SACM it was always clearly recognized and stated that the calculated k(E, J) should be interpreted as an average rate constant, averaged over an energy interval AE >> p(E, J) -comprising many isolated resonances (if any), with p(E, J) being similarly an average density of states.Around these averages, fluctuations may occur (see, for instance, the discussion in section I1 of ref. 7 and the discussion in a review,8 which anticipates much of what has been treated in other papers on fluctuations and averaging later with greater length, but not necessarily greater insight). My third comment refers to the way in which symmetry, in particular nuclear spin symmetry conservation in the theory of the unimolecular reaction of CH,O, is dealt with in the paper by Dertinger et ul. The consequences of the principle of nuclear spin General Discussion Table 1 Symmetry correlations for CH,X(C,,) mol-ecules dissociating in CH3(D3,,)+ X and H + CH,X(C,,) (the forbidden species are given in par- entheses; for CD,X the correlation would be modified correspondingly) CH,X CH,X CH, + X C3V rrn t ST (ST) CH3X CH,X + H SIS' s2 c,, ST symmetry and parity conservation for chemical reactions including statistical theories of unirnolecular reactions were worked out originally about two decades ago and have been publishedg-" (see also ref.8). An aspect which is missing in the paper of Dertinger et al., is the possibility of assigning a well defined parity to certain levels in CH,X (CF,X etc.) molecules of C,, point group symmetry, when one uses the group Sg instead of the ordinary point group C3, to classify the rovibronic levels. The situation is shown in Table 1 [from Tables 4 and 5 of ref. 10, see also ref. 91, which is obtained by calculating the induced representation T,(C,,) t S: , where rovibronic parity is explicitly indicated by + and -and permutation symmetry species by Al, A, and E following the conven- tion of ref.9. The symmetry species in S: which are forbidden by the Pauli principle for the three protons have been put in parentheses (they would occur for deuterons). Some care is needed when reading the shorthand notations for the correlations, which we cannot explain here for brevity (see ref. 9 and 10). However, one central result, which is obvious, when displaying explicitly parity and permutation symmetry, is that there are significant restrictions for reactant (CH,X) and product (CH, + X or CH,X + H) states for the CH,X isotopomers, because some of the C,, symmetry labels have well defined parity (and permutation symmetry).This would be different for the CD,X isotopomers. For the example of Dertinger et al., this implies that the A, and A, levels of CH,O correlate with B' levels of CH,O, whereas the E levels correlate with both A' and B'. Rather than dwelling futher on these established results, I would like to point out some of my more recent results concerning possible consequences of violations of both nuclear spin symmetry and parity. The violation of nuclear spin symmetry is expected by analysis of nurovibronic couplings on timescales of ns to ms. An experiment where one could search for such an effect would consist of exciting first a molecule to a selected rovibronic symmetry (e.g.,in our examples, CH,D or CD,H) by overtone excitation, per- haps with double resonance preselection following the work of Rizzo and co-workers,12 and subsequent dissociation by UV one photon13 or IR multiphoton excitation (similar to IRLAPS14) detecting CH, products, for instance, or CH,D with spectroscopic state Generul Discussion selected techniques.If there is a long delay between initial selection of nuclear spin symmetry in the highly excited CH3D and the final dissociation, one may probe for potential nuclear spin symmetry violation, provided that the density of states is high enough, at least locally, e.g. 103-10'0 states per cm-'. Similarly, one could look at reactions of CF,H (to give CF, + HF or CF, + H). Much more fundamental would be a test of parity violation, which can be carried out by the general scheme of ref.15, which can be further extended to a test of CPT symmetry.' In connection to 'control' in dynamics I would like to take here a general point of view in terms of symmetries (see Scheme 1). We could start with control of some sym- metries in an initial state and follow their time dependence. This can be used as a test of fundamental symmetries, such as parity, P, time reversal symmetry, T, CP and CPT, or else we can use the procedure to discover and analyse certain approximate symmetries of the molecular dynamics such as nuclear spin symmetry species,' or certain structural vibrational, rotational symmetries.' I would like to draw attention here to some work on chiral molecules, which allows fundamental tests of symmetries in physics and chemistry.The experiment outlined in Scheme 215 allows us to generate, by laser control, states of well defined parity in mol- ecules, which are ordinarily chiral [left handed (L) or right handed (R)] in their ground states. By watching the time evolution of parity one can test for parity violation and I have discussed in detai1'5*'6~'s~'9 how parity violating potentials A& might be mea- sured, even to as small as J mol-'. Timescales corresponding to such symmetry violation are of the order of hours to days. On the other hand, one can also generate a very short lived time dependent chiral excited state, even if the potential has a minimum at an achiral geometry. If the excited state potential is harmonic, for example, one may derive interesting results on chirality in relation to harmonic oscillator dynamics.' 'This includes femtosecond evolution of chirality and control of stereomutation in chiral mol- ecules.O On the most fundamental level, we have shown how this experimental scheme might be used for a fundamental test of CPT symmetry violation." While still somewhat hypothetical at present, this would constitute the most sensitive currently proposed test on CPT symmetry. The sensitivity expressed as a baryon mass difference Am between particles and antiparticles (with mass m)would be of the order of Am/m x .l6 The best currently proposed other experiment is on antihydrogen spectroscopy at CERN (not yet carried out) with Am/m z and the best existing result for the proton- antiproton pair is Am/m < 10-9.21 Complementary to a selection of parity one can also consider chirality (quantum number L or R) as being a selected symmetry in Scheme 2.In the ground state potential it would be a very good, long lived quantum number, whereas in the excited state poten- tial it would be a very short lived symmetry (violation on the fs timescale). control of symmetry of initial state 1 time dependence of the symmetry properties? -1 test of fundamental symmetries in nature (such as P, T, CP, CPT) or test of approximate symmetries of the dynamics Scheme 1 Control of symmetries in dynamics General Discussion I lparity '1 1' parity/ ->q Scheme 2 Control of parity and chirality in the scheme of ref.15 1 M. Quack and W. Kutzelnigg, Ber. Bunsen-Ges. Phys. Chem., 1995,99,231. 2 S. Dertinger, A. Geers, J. Kappert, J. Wiebrecht and F. Temps, Faraday Discuss., 1995, 102, 31. 3 M. Frey, personal communication. 4 S. A. Safron, N. D. Weinstein, D. Herschbach and J. C. Tully, Chem. Phys. Lett., 1972, 12, 564. 5 M. Quack and J. Troe, Ber. Bunsen-Ges. Phys. Chem., 1974,78,240. 6 P. Pechukas and J. C. Light, J. Chem. Phys., 1969,42, 3281; E. E. Nikitin, Theor. Exp. Chem., 1965, 1, 144. 7 M. Quack and J. Troe, Ber. Bunsen-Ges. Phys. Chem., 1975,79, 170. 8 M. Quack and J. Troe, Theor. Chem. Adu., 1981,6B, 199. 9 M. Quack, Mol. Phys., 1977,34, 477. 10 M. Quack, Stud. Phys. Theor. Chem., 1983,23,355.11 M. Quack, J. Chem. Phys., 1985,82,3277. 12 L. Lubich, 0.V. Boyarkin, R. D. F. Settle, D. S. Perry and T. R. Rizzo, Faraday Discuss., 1995, 102, 167. 13 M. Hippler and M. Quack, Chem. Phys. Lett., 1994,231,75; J. Chem. Phys., 1996,104,7426. 14 0.V. Boyarkin, R. D. F. Settle and T. R. Rizzo, Ber. Bunsen-Ges. Phys. Chem., 19'95,99, 504. 15 M. Quack, Chem. Phys. Lett., 1986, 132, 147. 16 M. Quack, Verh. DPG VI, 1993,28,244;Chem. Phys. Lett., 1994,231,421. 17 A. Beil, D. Luckhaus, R. Marquardt and M. Quack, Faraday Discuss., 1994,99, 504. 18 M. Quack, Angew. Chem., 1989, 101,588; Angew. Chem., Int. Ed. Enyl., 1989,28, 571. 19 M. Quack, in Femtosecond chemistry, ed. J. Manz and L. Woeste, Verlag Chernie, Weinheim, 1994, ch. 27, pp. 781-818. 20 R.Marquardt and M. Quack, J. Phys. Chem., 1989,90, 6320; J. Phys. Chem., 1994,98, 3486; Z. Physik D, 1996,36,229. 21 J. Groebner, H. Kalinowsky, D. Phillips, W. Quint and G. Gabrielse, Phys. Rel;. Lett., 1993, 28, 315; G. Gabrielse, D. Phillips, W. Quint, H. Kalinowski, G. Rouleau and W. Jhe, Phys. Reo. Lett., 1995, 74, 3544. General Discussion Dr. Temps responded: We thank Prof. Quack for his remarks on the historical devel- opment of statistical unimolecular rate theory and his comments on the terminology, which stress our own view. Regarding the phenomenon of intramolecular vibrational redistribution (IVR), we would like to use this opportunity to draw attention to our first two papers on the unimolecular dynamics of CH,O (ref.1 and 2) which focus on this issue and give further explanations. The present paper is to be viewed in connection with these two papers. It is indeed important that fluctuations of the state specific decay constants per se are not in contrast with the concepts of statistical theories but are actually part of it. This point is still not yet fully realized everywhere. Attributing state specific fluctuations to non-RRKM behaviour without having other measures for statistical dynamics, which have been established, e.y. nearest-neighbour level spacing distributions etc., cannot be justified and should be avoided (cf: ref. 1 and 2). This is a point that we make in our paper. The possibility for state specific decay rate fluctuations has been mentioned pre- viously in discussions of statistical theories as pointed out by Prof.Quack. However, what is important is that, for several molecules (e.g. D,CO by Moore and co-workers and CH,O by us), decay rate fluctuations have now actually been established experi- mentally. In particular, we have presented our experimental data for a reasonably large molecule, which indicate that the strength of the fluctuations changes with excitation energy. It is an advantage of SACM theory which Prof. Quack refers to that it concep- tually allows for state specific dynamics by considering individual adiabatic channels. For systems for which exact adiabatic channel potential curves may be calculated, one may ultimately be able in the future to make state specific dynamical predictions using SACM theory.At present, this is not yet possible. On the other hand, quantum dynamics calculations for unimolecular reactions on high quality a6 initio potential-energy surfaces are becoming available, (see, e.g., the paper by Schinke et al. at this discussion) which throw new light on the subject. In our laboratory, we are investigating the unimolecular decay reaction of DCO radicals. This system is also being investigated theoretically (e.g. Schinke and coworkers). We observe very good agreement of the quantum dynamics results of Schinke et af. with our experi- mental data. The issue of symmetry conservation in the unimolecular decay reaction of CH,O has not been fully exploited in the present paper for several reasons, but the correlation diagrams presented by Prof.Quack have been worked out and used by us previ~usly.~ These correlation diagrams relate to the transition between initial states of the CH,O molecule and the final states of the H,CO molecule. They are important when one considers the H2C0 product state distribution. Experiments to probe the H2C0 product state distributions are in preparation; these are, however, difficult. Considering the specific unimolecular rate constants k(E, J), one has to consider the correlation between the initial states and the states of the activated complex (transition state) located at the saddle point of the potential-energy surface and then consider how these activated complex states correlate to product states.For the purpose of calculating the specific unimolecular rate constants k(E, J), we have considered the correlation between the initial states and the states of the activated complex within the induced representation, i.e. Sf.Sz is isomorphous to the D3h(M) molecular symmetry group. We have concluded that no further restrictions arise for the calculated value of the specific unimolecular rate constant (but not for the product state distribution) other than those originating from the division of the volume of phase space because of the correspon- dence of the A, and A, states in C,, with A$ in Ss and the correspondence of E states with E', provided that all levels of the transition state configuration can be adia- batically correlated with product states and that these exit channels are energetically accessible. This assumption can be justified because of the potential barrier of the poten- tial surface.Since the potential-energy surface has a clear maximum at the transition General Discussion 95 state configuration, it is reasonable, at least for energies not too far above threshold, that the adiabatic channel potential curves leading from the transition state to product states are energetically open. The situation may be different for reactions with loose activated complexes, i.e. systems without a potential barrier, where specific exit channels which are dictated by the symmetry correlations may become energetically inaccessible. Of course, for the product state distribution, one will have to follow the correlations explicitly through to product states, but this was not the task of the present paper.The main complications of the statistical rate calculations arise from the question of K conservation. Note that, in our spectra, J = 0.5 and 1.5. In a Hund’s case b represen- tation it would suffice to calculate the decay rate for angular momentum without elec- tron spin N = 0 or 1; the electron spin would introduce a degeneracy factor of 2 in the density of states and the number of open channels. For a Hund’s case a molecule such as CH30 one can argue, as we did, that the statistical unimolecular rate calculations should be done with P = K instead of K. It is questionable, however, whether states of different P for a given value of J are fully mixed for J as low as 0.5 and 1.5.If P or K were conserved, the calculated unimolecular rate constants would change correspond- ingly. Furthermore, we assumed that of the two components of the E symmetry ground electronic state potential surface, only one correlates adiabatically to products H + H,CO. On the other hand, we assumed that both components contribute to the density of states. This assumption is justified by the Jahn-Teller interaction, but may be questionable. These problems have to be addressed in detail in the future. 1 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101,3618. 2 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101, 3634. 3 J. Kappert, PhD Thesis, University of Gottingen, 1992.Mr. Sanov said: I would like to make yet another comment on extracting rates from the linewidths in the regime of overlapping resonances. When only isolated spectral features are analysed, there is a bias towards narrower resonances, simply because these resonances have more chance to appear isolated. As a result, the extracted average rate is a lower bound of the actual rate. Because of this, the discrepancy between theory and experiment in Fig. 7 of the paper at E > 9000 cm-l may in fact be even larger than shown. Dr. Temps responded: The extraction of unimolecular decay rates from linewidths in SEP spectra indeed has to be done with caution. We have carefully examined our results to check for inhomogeneous contributions to the widths and to rule out saturation effects.We are confident, considering the range of the state-to-state decay rate fluctua- tions, in the spectra at E z9000 cm-’. The results for the highest range of excitation energies have to be considered more carefully. Very broad resonances can indeed be lost in the background. Narrow line profiles can be contaminated by overlapping features. Hence, there is some bias towards an ‘average’ decay rate. However, the discrepancy between experiment and statistical theory at the highest energies remains. I would like to emphasize that the results from the direct, time-resolved decay rate measurements should not be affected by inhomogeneous contributions to the spectra line profiles.It would be of great interest to obtain direct time-resolved data at higher energies. However, this requires a shorter laser pulse than is currently available in our laboratory. Prof. Moore asked: Your experimental density of states is equal to that calculated by direct count if K, is fully mixed. If K, is a good quantum number, how much larger is the observed density of states than the calculatcd one? 96 General Discussion Dr Temps replied: For the SEP spectra with low J considered in the present paper, it is difficult to make a judgement regarding the ‘goodness’ of the K quantum number. At excitation energies of ca. 6000 cm-’,the experimental density of states is slightly lower (50%)than th5calculated one if we assume that the SEP spectra from J = 0.5 in the A state show all X rotational levels with J = 0.5 and 1.5, which could be reached according to the J selection rule (cf: Fig.9 of the paper). For CH30, one also has to consider the projection quantum nucber C = &+ for the electron spin. In Hund’s case a notation which is appropriate for X CH30 in the vibrational ground state and K combine to give the quantum number P = K & C. If we assume P to be conserved, the experimental density of states would be higher that the calculated one by a factor of 3 if the SEP spectra really show all J = 0.5 and J = 1.5 levels. The difference is clearer for higher J. In ref. 1 and 2, we consider spectra for J = 7.5 to 9.5. In that case, the experimental density of states is higher than the calculated pure vibrational density of states by a factor of 10-20.Anharmonicity corrections are thought to be small (30%). 1 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101,3618. 2 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101,3634. Prof. Rizzo said: In your talk you quoted an IVR time of ca. 2 ps but in Fig. 2 of your paper you show lines spread out over a range of 30-40 cm-’. If you truly have pure sequence spectra, this means that the timescale for IVR must be much shorter that 2 ps in this energy range. Are you sure that the spectrum of Fig. 2 represents a single zeroth-order rovibrational state? Dr. Temps communicated in response: The SEP spectrum in Fig. 2 of our paper does not arise from a single zeroth-order bright state.Thus, one cannot derive an IVR rate from these spectra. The IVR timescale of 2 ps was found in ref. 1 and 2. There, we did investigate pure sequence spectra so that conclusions on the IVR rates could be drawn. The results of ref. 1 and 2 indicate that IVR occurs via a two-tier mechanism with timescales of 0.25 and 2 ps, respectively. These results are for pure CO stretching overtone states, J = 7.5-9.5. It is highly unlikely that IVR is faster for the case of J = 0.5-1.5 considered in the present paper. Thus, the value of 2 ps can be used as a reference with which the unimolecular decay rates may be compared. 1 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101, 3618.2 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101, 3634. Prof. R. E. Miller said: The large fluctuations you report in this paper are observed routinely in the vibrational predissociation of weakly bound complexes. Although not always the case, these fluctuations can often be understood in terms of the nature of the vibrational state that is initially excited. In the weakly bound complexes that have been studied in detail, the density of states is often low enough to permit such assignments to be made. In your case the vibrational density of states is much higher and may even be so high as to prevent such assignments entirely. Nevertheless, I am not sure why you say that ‘the state-specific decay rate fluctuations cannot be attributed to regular, assignable, vibrationally mode specific dynamical properties’.The existence of such fluctuations does not preclude the assignment of the spectra. Dr. Temps answered: The vibrational dynamics of a molecule such as CH,O with a deep potential well is fundamentally different from the dynamics of loosely bound van der Waals molecules. At the vibrational excitation energies of interest for the General Discussion unimolecular decay reaction of CH30, the SEP spectra of the molecule are unassignable in the traditional sense. Note that the densities of states are in the range 1-10 cm-’. Thus, the average energy spacing between neighbouring states is of the order of 1-0.1 cm-’. This has to be compared with the size of anharmonicity coupling matrix elements, which may be of the order of 50 cm-’.Furthermore, in CH30 one has a highly efficient level coupling mechanism due to the Jahn-Teller effect. Under these conditions, a full vibrational assignment is not only impossible in practice, it is indeed physically meaning- less considering the number of parameters which need to be determined. In ref. 1 and 2, we applied different statistical tests to the SEP spectra of CH30 which indicate that the vibrational dynamics of the molecule is close to the ‘statistical’ or ‘chaotic’ limit. In contrast, the vibrational dynamics of van der Waals molecules are ‘regular’. Spectra of van der Waals molecules usually do not meet the criteria for statistical dynamics.1 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101, 3618. 2 A. Geers, J. Kappert, F. Temps and J. W. Wiebrecht, J. Chem. Phys., 1994, 101, 3634. Prof. W. H. Miller commented: If the coupling between the modes in the molecules is weak, as, for example, in van der Waals complexes, then the distribution of unimolecular decay rates will indeed show fluctuations; in fact very large fluctuations. In this situation vibration quantum number assignments can often be made, and progres- sions in various modes identified, thus explaining the fluctuations. The point of the random matrix/transition state theory (RM/TST) of Polik, Moore, and myself,’-3 however, is that there are still fluctuations in the state-specification decay rates even when the quantum states are as strongly mixed (and therefore as statistical) as they can possibly be.To be sure, the fluctuations are less when the states are strongly mixed, but they nevertheless exist. When one observes fluctuations in state-specification decay rates, therefore, it is not possible to conclude immediately whether the states are strongly mixed or not; it is necessary to quantify the magnitude of the fluctuations to see if they can be described by the strong coupling (RM/TST) model or are larger than that and thus presumably due to weak coupling between relevant modes of the molecule. 1 W. F. Polik, C. B. Moore and W. H. Miller, J. Chsm. Phys., 1988, 89, 3584; W. F.Polik, D. R. Guyer and C. B. Moore, J. Chem.Phys., 1990,92, 3453. 2 W. H. Miller, R. Hernandez, C. B. Moore and W. F. Polik, J. Chem. Phys., 1990,93, 5657; R. Hernandez, W. H. Miller, C. B. Moore and W. F. Polik, J. Chem. Phys., 1993,99,950. 3 W.F. Polik, C. B. Moore and W. H. Miller, J. Chem. Phys., 1988, 89, 3584. Dr. Demianenko said: As we have shown in our work presented in the poster session, a molecule as a dynamic system possesses a family of conserving quantities in addition to the ordinary first integrals of motion (energy, momentum and angular momentum). The presence of these quantities gives rise to the following: ‘energy levels in molecules exist, related to the zero-order vibrational Hamiltonian, that after excitation retain their energy on the timescale very much longer than the inverse value of the slightest non- linearity ’.The results of our study can be related to the results of the discussed work in the following way. After excitation of the level of the above type, the energy flows only between vibrational modes related to this level, while other modes remain unexcited. Thus, depending on the level excited, the bond under study may possess, on average, energy substantially more or substantially less than the statistical limit E, = E,/N, where E, is total excitation energy and N is the total number of vibrational modes. Hence, the decay rate of the bond under study may differ significantly from that predict- ed by RRKM theory. Dr. Temps responded: Your argument may be true. However, the point of our paper (and the papers by Polik, Miller and Moore, ref.I, 2, 3) is that decay rate fluctuations General Discussion exist even if there is strong coupling between vibrational modes. The existence of fluc- tuations of the decay rates must not be taken as a sign for incomplete level coupling. Indeed, in the spirit of random matrix concepts, they must not be taken as a sign for non-statistical dynamics. As it is well established, one does need separate tests to eluci- date the assumption of statistical dynamics. 1 W. F. Polik, C. B. Moore and W. H. Miller, J. Chem. Phys., 1988, 89, 3584; W. F. Polik, D. R. Guyer and C. B. Moore, J. Chem. Phys., 1990,92, 3453. 2 W. H. Miller, R. Hernandez, C. B. Moore and W. F. Polik, J. Chem. Phys., 1990,93, 5657; R.Hernandez, W. H. Miller, C. B. Moore and W. F. Polik, J. Chem. Phys., 1993,99,950. 3 W. F. Polik, C. B. Moore and W. H. Miller, J. Chem. Phys., 1988,89, 3584. Prof. Wardlaw asked: I have two questions based on figures appearing in the paper. (i) For Fig. 5, in the upper middle panel (E = 7443.5 cm-’) the points are widely scat- tered leading me to question the significance of a fit of any function to the data. What is the error associated with the extracted value of k,, and how does this error compare with the fluctuations of k,, in this energy range? Can we be sure that the decay is actually exponential? How many plots with such widely scattered points are employed in the statistical analysis? (ii) For Fig. 8, the histograms appear not to be converged.How many decay rates contribute to each (this should be stated in the paper.) What are the error bars on each histogram bin? Any appraisal of the significance of the degree of freedom parameters extracted from the fits of x2 distributions relies on the extent to which these histograms are converged. Dr. Temps answered: Regarding Fig. 5 of our paper, we derived (as stated) a lower limit to the decay rate for the level at E = 7443.4 cm-. (ks, > 4.0 x lo7 s-’), The observation of the single vibrational rotational level LIF signal with the probe laser was firmly verified by repeated scans of a spectrum such as in Fig. 4. The error limits of the k,, values were generally between one and a few per cent. Thus, the error limits are negligible compared with the magnitude of the state-to-state fluctuations.As seen from the examples stated and shown in Fig. 5 in the paper, the decay profiles were observed to be single exponential with one exception out of 27 states probed. Regarding Fig. 8 of our paper, the numbers of states within each of the energy intervals considered are given in the text. These numbers are the ones used to derive the histograms. Even if the histograms are not fully converged, the trend for the widths of the fluctuations to decrease with increasing excitation energy is real. Only a limited number of states of the molecule can be probed. Dr. Smith opened the discussion on Prof. W. H. Miller’s paper: In the solution of the classical master equation for recombination reactions at atmospheric temperatures, one finds that there is typically a dramatic separation of timescales for the relaxation of the molecular population distribution over energies and angular momenta. This is manifest- ed in a large separation between the smallest eigenvalue of the stochastic propagation matrix, which represents the ‘pseudo-steady-state’ rate coefficient, and its higher eigen- values which represent timescales for relaxation to the steady state.One expects that this may indeed also be the case for the fully quantum mechanical expressions for the recom- bination rate coefficient which you have derived. Is it possible that one could use this eigenvalue separation to advantage when designing methods to compute the quantum mechanical trace? Prof.W. H. Miller replied: This is a very good idea. The relevant eigenvalue problem in this case would of course be in the Liouville vector space described in Section 4 of the paper. By looking at eqn. (4.11~) (the time-dependent version of the equations) one can see that the smallest eigenvalues of the operator in the exponent will make the greatest General Discussion contribution to the integral over time, so that if there is a large separation of a few low eigenvalues from the others, then they will make the dominant contribution. Prof. Hynes said: Could you elaborate on how it is that what you call a random phase approximation produces a rate constant beyond that of a master equation (ME) approach? In particular, the approximation appears to restrict the required information about energy transitions to only two values (and not four as in your general case) and thus would seem to involve only ME level of information.Prof. W. H. Miller responded: This is a very good question! If one were to make this random phase approximation (i.e. neglect of quantum off-diagonal elements of the energy transfer matrix) fully to eqn. (4.1la), then you are correct that the classical master equation (CLME) result would be obtained. Thus what has been done is to expand the inverse matrix in eqn. (4.1la) in powers of 9,to give a geometric series (02-09+ K + ill)--' = (coZ+ K + iQ)+' + term in -9+ term in 2P2 + and to make the random phase approximation in all the above terms except the first one which does not involve the energy transfer matrix 9.Since 2P = 0 corresponds to the strong collision approximation (SCA), this means that the first term above gives the quantum SCA, and the sum of all the remaining terms gives the classical ME result minus the classical SCA, thus yielding eqn.(4.18). Prof. Quack said: Prof, W. H. Miller has addressed the question of whether one uses relaxation eigenvalues of a master equation or the reactive flux through a hypersurface separating reactants from products in configuration space for the definition of rate coef- ficients (he used the latter in his paper). I looked at this question some time ago in relation to unimolecular isomerization reactions,' but the results and conclusions can largely be transferred to the recombination-dissociation reaction systems discussed by Prof.W. H. Miller. I summarize some of the conclusions' briefly. (1) The definition of rate coefficients one uses in the comparison of experiment and theory depends upon the type of the experiment considered. (2) Most kinetic experiments actually measure the chemical relaxation process, not the reactive flux. Indeed the detailed balance relation is in fact widely used to define the time independent steady-state forward and reverse rate coefficients from the measured chemical relaxation rate coefficient and the equilibrium constant, although this fact is frequently not realized explicitly. (3) In particular unimolecular fall-off of the steady-state relaxation rate coefficient results frequently from the reverse reaction, being important aiready in the early stages of the reaction due to overpopulation of reactive product states, and not only because of a lack of population of reactant states (which would be the common mechanistic picture for fall-off).(4) Experiments measuring reactive flux result in time-dependent rate coefficients. (5) Spectroscopic experiments can be conceived, which measure the reactive flux at equilibrium. In such experiments one would measure the high pressure limiting rate coefficient k, at all pressures, including low pressures, thus circumventing the Linde- mann falI-off.2 While we have tried to implement this experimentally, this work has been delayed due to unfortunate difficulties (but we have not given up).This work is part of our general approach to derive intramolecular dynamics from time-independent high resolution spectr~scopy.~'~ One might think that by adding a sink to the product side of the reaction system, the relaxation rate coefficient and the reactive flux rate wefficient can be forced to be the same. This is true, however, only in the trivial case of placing the sink at the dividing surface between reactants and products, which would provide a quite distorted picture General Discussion of the real reaction process. I think that in general one must use the reversible mechanis- tic description of unimolecular reactions to obtain a good picture.’ I might mention here also a little extension derived some time ago’ for the irrevers- ible Lindemann mechanism steady-state rate coefficient of a unimolecular reaction ki A+M---+A*+M; activation kl = CMIk k2 A*+M -+A+M; deactivation k2 = CMlkd k3 A* ------, products; reaction (3) The usual ‘text-book’ expression for the quasistationary state rate coefficient in the fall-off regime is given for this simple mechanism by; kl k3 kerf = -(4)k2 + k3 The correct steady-state (‘chemical relaxation’) rate coefficient for this mechanism is Eqn. (4) and (5) are only the same if kl < k2 + k3, which is often, but not always, the case.While it is trivial’ to derive the correct eqn. (9,it is strikingly enough not general knowledge. I might note that some of our ideas were further pursued by Green et a1.6 and Aguda and Pritchard7 recently, who confirmed the results of ref.1. 1 M. Quack, Ber. Bunsen-Ges. Phys. Chrm., 1984,88, 94. 2 F. A. Lindemann, Trans. Faruday SOC.,1922,17,598. 3 M. Quack, Mode Selective Chemistry, ed. J. Jortner, R. D. Levine and B. Pullman, Reidel Publishers, Dordrecht, 1991, vol. 24, p. 47. 4 M. Quack, in Femtosecond Chemistry, ed. J. Manz and L. Woste, Verlag Chemie, Weinheim, 1994, pp. 78 1-8 18. 5 M. Quack and S. Jans-Burli, Molekulure Themodynumik und Kinetik: Teil I, Chemische Reaktionskine- tik, Zurich, 1986. 6 N. B. Green, P. J. Marchant, M. J. Perona, M. J. Pilling and S. H. Robertson, J. Chem. Phys., 1992, %, 5896. 7 B. D. Aguda and H. 0.Pritchard, J.Chem. Phys., 1992,96, 5908. Prof. W. H. Miller replied: The objective of my paper was to try to combine a proper quantum mechanical description of the A + B t-)AB* collision dynamics (step 1 of the Lindemann mechanism) with a classical master equation description of the relaxation, AB* + M -+ AB + M (step 2 of the Lindemann mechanism). The net result of the paper is that one cannot combine them without going to a quantum version of the master equation-like description : the quantum impact approximation. This development was carried out, producing the desired quantum generalization of the master equation-like treatment of ‘step 2’. In the course of the development I summarized (in Section 3) the classical master equation treatment of ‘step 2’ (see also my communication on the following page).There is nothing new here, simply a summary of the standard classical master equation [to which the general quantum mechanical result, eqn. (4.1l), reduces with appropriate approximations]. Thus Prof. Quack’s comments about the classical master equation are really not germane to the subject of my paper. Nevertheless, I do feel it necessary to respond to some of the points he raises with which I disagree: General Discussion (1) Recombination is a bimolecular reaction (A + B +products), and like all bimolecular reactions its rate constant (or cross-section) can be rigorously defined as a ratio of fluxes. In this sense recombination is less ambiguous than unimolecular decom- position.(2) This definition does not lead to time-dependent rate constants. All the rate con- stant expressions in my paper are manifestly time independent. Dr. Smith commented: For a recombination process with a single arrangement channel, the master equation is describing a closed system. Hence, ultimately (typically on a timescale that is long compared with that which is of interest when defining the non-equilibrium recombination rate coefficient) the system will approach equilibrium with zero net flux. One can extract the correct non-equilibrium recombination rate coef- ficient either by imposing suitable boundary conditions which enforce steady state (whilst still allowing for the effect of collisional reactivation of molecules from energies in the vicinity of the threshold) as shown, for example, in the work of Green et al.' or by examining the eigenvalue spectrum of the superoperator that Prof.Miller has derived which gives the recombination rate coefficient. 1 M. A. Hanning-Lee, N. J. B. Green, M. J. Pilling and S. H. Robertson, J. Phys Chem., 1993,97,860. Prof. W. H. Miller communicated in reply: The description in Section 3 of the col-lisional stabilization of AB* [eqn. (l.lb)] by a classical master equation needs further discussion. To simplify notation, the matrix 52 is defined by Q = m(l-P) (la) with matrix elements 521, I' = 44,1' -&lo I' and one notes the property c 521,I' = 0 1 Eqn. (3.11)for the recombination rate thus reads and with the matrix identity? k * (k + Q)-l * Q = 52 -(k +a)-'* k (4) it can also be written as k, Qr = 1 [Q (k + a)--kIl, exp(-BElt) 1, I' = 1 Q1,lt(k+ Q);,irfklflexp( -PE,,,) I, l', I" t For square matrices A and B, one has A (A + B)-' .B= A * [(l + BA-')AJ-' . B = (1 + BA-')-' B = [B(B-' + A-')]-' * B = (B-1 + A-y which is manifestly symmetric on interchange of A and B, so that A .(A + B)-. B = B .(A + B)-. A General Discussion But in light of eqn. (2), one sees that eqn. (3c) vanishes, i.e., the recombination rate is zero ! This (initially unsettling) result is actually physically correct because the above description has included no sink to remove the stable species AB from the kinetic system. Thus collisions which deactivate (i.e. stabilize) AB* to AB, also reexcite AB to AB*, which redissociates to A + B; all trajectories incident through the dividing surface in Fig.1 in the paper will therefore ultimately recross it on the way out, so that j2 dt Cf(t)= 0, and there is no net recombination, i.e. no trajectories remain trapped. (This situation does not arise in the strong collision approximation of Section 2 since the stabilization step is irreversible within the SCA.) To have recombination it is thus necessary to introduce a sink to remove the stable, recombined states of AB. (See, for example, the recent work by Pilling and co-workers.') Let {r,) be the removal rates, so that the diagonal matrix Y(r,. = 8,. rl)is added to k in the master equation eqn. (3.1) and its solution eqn. (3.8). It is useful to write the relevant matrices in partitioned form: the stable states of AB, those with essentially real eigen- values of I? -ii., whose energy eigenvalues {E,) lie below the dissociation limit, are denoted as group 1 (with index E < lo), and the metastable states (AB*), those with non- zero imaginary parts of their energy eigenvalues, and with the real parts of their energy eigenvalues above the dissociation limit, are denoted as group 2 (with index 1 > lo).The diagonal matrices r and k thus have the structure .=(; x>.=(x :) C2 is a full matrix, The recrossing flux in eqn. (3.10)thus becomes kYcroSSQr= -1 kl(k + r + a),,! klt exp( -PElt) 1, I' >lo where the sums are restricted because {k,) is zero for 1 -= I,. Thus only the 2-2 block of the matrix inverse in eqn.(6) is required, and by using standard partitioning algebra it is given by (k + r + a),: = (k + @lf2)-' (74 where aeff 2,2 = a2,2-fi2, 1 * (r + 521, 'I-' -a,,2 Using this result in eqn. (6),and adding in the incident flux, kPQr = 1kl exp( -PE,*) ,=-lo gives the following general result for the recombination rate which replaces eqn. (3.11). General Discussion The removal rates (rl) may have physical significances, e.g. the rate of removal by wall collisions, or the reaction rate with another species in the kinetic system. A particu-lar simple case to consider is the limit rI-+ co,i.e. the assumption that the stable states are removed instantaneously once they are formed.In this case sZ:y2 of eqn. (7b)becomes simple Q,,, , and the only difference of eqn. (9) from the original result of Section 3, eqn. (3.1 l), is that one interprets the matrix product in eqn. (3.1 1) to include only the metasta- ble states I, E' > lo. It is useful to check the low pressure limit of eqn. (9), lim k, Q,= 1 sZ1, exp(-BElc) [MI+0 1, I' >lo which, in light of eqn. (2), can also be written as showing explicitly that it is proportional to the rate of deactivation from incident meta- stable states (I' > lo) to stable states (I < lo). Finally, the analogous interpretation of the quantum result in eqn. (4.11) should also be made. 1 S. H. Robertson, M. J. Pilling, D. L. Baulch and N. J. B. Green, J. Phys. Chem., 1995,99, 13452.Dr. Green commented: I would like to try to clarify the problems of separating forward and reverse rate constants in a reversible reaction discussed by Prof. Quack and Prof. W. H. Miller. Prof. Quack has pointed out for an isomerisation reaction (originally in 1977) that the actual fluxes from reactants to products and back are transient and relax towards their equilibrium values (the respective high pressure limits) with the same time constant as the relaxation to equilibrium. He therefore calls into question the valid- ity of the classical separation of a reversible reaction into forward and reverse com- ponent reactions. Prof. W. H. Miller has analysed a recombination reaction using just such a separated scheme, in which the reaction rate is the steady-state flux into (deactivated) product.This method has been criticised because it does not include the possibility of reactivation and dissociation once the product has been deactivated. Both Prof. Quack and Prof. W. H. Miller are correct, and their methods are not incompatible as long as the dividing surface, below which products are considered to be deactivated, is drawn in a sensible place. Operationally this means that the population distribution below the dividing surface should be essentially Boltzmann, and therefore the dividing surface should be well below the reaction threshold (e.g. 10kT below threshold). If this condition can be met then the reversible scheme can be simply separat- ed into irreversible forward and reverse reactions : for the recombination reaction the deactivated product is a sink, whereas for the dissociation reaction it provides a steady- state source by activation from the Boltzmann 'bath'.The dissociation flux of Prof. Quack is then a weighted average of the flux from molecules activated from the Bolt- zmann 'bath' and the flux from molecules redissociating before deactivation; in other words the population distribution above the dividing surface is a weighted sum of the steady-state population distributions from the forward and reverse reactions, which can be obtained separately in the manner suggested by Prof, W. H. Miller. The overall population distribution does not attain a steady shape because it is a weighted sum of two different steady-state distributions and relaxes towards the Boltzmann distribution and chemical equilibrium.This analysis breaks down in two places. (i) At very short times the Boltzmann 'bath' General Discussion has not yet been set up. But at these times the kinetics is described by many time constants, whose effects normally die out rapidly leaving the classical relaxation time constant. (ii) At very high temperatures there may not be a single dominant time con- stant because several eigenvalues of the master equation may be of similar magnitude. Under these conditions it is also impossible to draw an adequate dividing surface below which everything is Boltzmann-like because the effects of reaction will extend too far below the threshold.However, under both sets of conditions the reaction will not follow a simple rate law and more detailed analysis is necessary. Under all ‘normal’ conditions the decomposition of a reaction into steady-state forward and reverse reactions is essen- tially exact. Prof. W. H. Miller replied: I agree with your analysis and also refer to my communi- cated reply which points out the necessity of having a sink to remove the recombined molecule. Prof. Quack communicated: I am sorry if the impression has been generated that my comments implied that there is something wrong with Prof. Miller’s paper: This was far from my intention! Indeed, my comments were stimulated by a remark made by Prof. W. H. Miller (on relaxation eigenvalues and reactive flux) and were not meant to be any criticism of his paper. Rather, in my remark I tried to constructively point out that a completely fresh (and different) look at the mechanism of unimolecular reactions is pos- sible.I do not question the possibility of defining time independent rate coefficients, but I point out that one might look also at a different observable (the time dependent flux). I stress here the significant, new idea of deriving in this way ‘high pressure limiting rate constants at all, including low pressures ’ (from observables at equilibrium). It may be good to illustrate this with a real molecular example from some of our work.‘ Hexafluoroisopropanol exists in three conformers, two of which are chiral (C,) and one achiral (C,)(with a symmetrical position of the CF, groups).H The two C1 conformers give the same IR spectrum (except for parity violation and VCD) and thus may be lumped together into one ‘C1’in a spectroscopic investigation of conformational kinetics. Fig. 1 shows a graphical representation of the torsional poten- i -7r 0 a __3 +n Fig. 1 Potential scheme for the conformational isomerization in hexafluoroisopropanol General Discussion 1 .u (4 Fig. 2 (a) CH stretching fundamentals in (CF,), CHOH (high frequency band for C, conformer) (b)OH stretching fundamentals in (CF,), CHOH (low frequency band for C, conformer) tial interconverting these conformers, Fig. 2, 3 and 4 show the IR fundamental and overtone spectra of the CH and the OH chromophores.In the fundamentals, an analysis of the spectra as a function of temperature leads to the assignment of the weaker band in each case to the higher energy C, conformer (the relative band strength increases with temperature because of the higher statistical weight), with an energy difference of the zero point levels of AE: =ca. (4 f1) kJ mol-'. More interestingly, a somewhat oversimplified analysis of the temperature dependent bandshapes with an extrapolated coalescence at temperatures between 550 and 600 K leads to a rate constant k 2 2 x 1OI2 s-' and an energy barrier E, of somewhat less than 10 kJ mol-I. This rate constant is evaluated from a flux exchange process at thermal equilibrium, i.e. as a Maxwell-Boltzmann equilibrium average and therefore identical to the high pressure limit k,, although the spectra are taken at low pressures (mbar range).To achieve the high pressure limit in an ordinary kinetic relaxation experi- ment (say T-jump), one would have to use pressures in the 100 bar range, and then still have to extrapolate, if one wished to get k,. Although the proper analysis of k, from spectra is much more complex (it relates to obtaining ultimately molecular quantum motion from spectra according to an approach developed since 1984),2 the present simple discussion may be sufficient to illustrate the new point of view, even though the final numbers for this particular example may change in a more detailed analysis (but not the orders of magnitude). The spectra shown serve also to illustrate another point.The OH overtone spectra show nicely the same isolated chromophore doublet from the two conformers, with the General Discussion n E 10400 10500 fi/cm-' Fig. 3 Overtone absorption of the OH chromophore in (CF,), CHOH splitting increasing about linearly (ca.40, 80, 120 cm-' for N = 1, 2, 3) as expected. This could be used to study even higher interconversion rates (up to ca. 6 x 10l2 s-'). The simple structures show the approximate adiabatic separation of the high frequency OH chromophore from the low frequency molecular frame modes, which is quite commonly observed also in other alkanols (the separation is not quite as good as in the better studied acetylenes, tho~gh).~,~ On the other hand, the CH chromophore overtone spectra show a very complex fractured structure, which is due to the interaction between very fast vibrational redistribution due to the CH stretching-bending Fermi re~onapce~*~and the conformational redistribution. At the time the experiments' were originally done, we were not in a position to unravel these structures due to the limi- tations in experimental technique and in the analysis.In the meantime we have learned to analyse quantitatively such structures both for C, and C, (chiral) molecule^^*^ and thus we hope to make progress in the near future on this very interesting interplay of vibrational redistribution rates and conformational, unimolecular interconversion rates, which is very pertinent, indeed, to the topic of this discussion. 1 H.Hollenstein, M. Quack and N. Spirig, 1984, unpublished results; N. Spirig, Diploma Thesis, ETH Zurich, 1984. 2 M. Quack, Femtosecond Chemistry, ed. J. Manz and L. Woeste, Berlin, Verlag Chemie, Weinheim, 1994, ch. 27, pp. 781-818. 3 K. von Puttkamer, H. R. Dubal and M. Quack, Faraday Discuss. Chem. SOC., 1983,75,197. 4 M. Quack, Jerusalem Symp., 1991,24,47. 5 A. Beil, D. Luckhaus, R. Marquardt and M. Quack, Faraday Discuss., 1994,99,49. 6 D. Luckhaus and M. Quack, Chem. Phys. Lett., 1992,190,581. General Discussion ncy 0.20 0.15 0.10 0.05 7800 8300 8800 l7/cm-l Fig. 4 Overtone absorption of the CH chromophore in (CF,), CHOH Prof. Oref said: During previous comments on the present paper, it was pointed out that the solution of the quantum master equations presented is somewhat involved and it yields the whole set of eigenvalues.It was indicated that there is an efficient solution of the master equation in the steady-state regime which yields the reaction rate coefficient which is simply the lowest absolute value of the eigenvalue. I would like to point out that at high temperatures steady state is not obtained and even in the case of the clas- sical master equation, an inversion of the whole matrix is required and the rate coeffi- cient is composed sometimes of as many as ten or twenty eigenvalues.' 1 V. Bernshtein and I. Oref, J. Phys. Chem., 1993,97,6830. Prof. W. H. Miller responded: I do not disagree with any of what you say.However, if I wished to evaluate the classical master equation rate expression, eqn. (3.13a) (and also see communicated reply), then I would definitely not proceed by finding all the eigenvalues of the matrix k + co(Z -P),for all that is needed is the inverse of this matrix multiplying a single vector, and this latter calculation is much easier than finding all the eigenvalues of the matrix. Prof. Wardlaw asked : Within your exact quantum master equation treatment, it appears possible to incorporate S-matrix models based on scattering calculations or General Discussion based on physical approximations such as chaotic dynamics or a particular direct mechanism. Some such approach could simplify eqn. (4.7) and (4.11), or at least the implementation of them.For example, a statistical S-matrix model based on random matrix theory would presumably obviate the need for such detailed knowledge of the scattering. What are your thoughts on this? Prof. W. H. Miller replied: The result of the exact quantum master equation (actually the quantum impact approximation) does indeed require the S-matrix for M + AB* inelastic scattering, a formidable requirement, so that I agree with you that in practical applications one will often use simple models to obtain it. A statistical approx- imation would of course be appropriate if one thought that the M + AB* collision itself formed a long-lived collision complex, but I imagine that a more common assumption would be that the M + AB* scattering is 'direct' or impulsive, so that sudden-like approximations (e.g.10s = infinite order sudden) would be most appropriate. Prof. Smith said : As well as collisionally stabilised association, there is considerable interest in radiative association, not least because of its importance in the chemistry of interstellar clouds. It would seem that the approach described in Prof. W. H. Miller's paper would be well suited to calculating rates of radiative association. Would he care to comment? Prof. Smith also communicated: I should also like to ask if Prof. W. H. Miller would be willing to speculate on how large quantum mechanical effects are likely to be in typical association reactions and whether there are likely to be particular kinds of reac- tions where such effects are likely to be largest. Prof.W. H. Miller communicated in reply: You are correct that this description of collisional recombination can be applied to radiative recombination. This was in fact pointed out in ref. 1. The second question is extremely relevant, i.e. when does one expect this more rigor- ous (and thus difficult) quantum treatment to be necessary? Several applications are in progress, H + 0, -+ HO, (with H. Taylor and V. Mandelshtam,) and H+ CO -+HCO (with J. Bowman), and they will help to answer this. My intuitive feeling is that this more rigorous quantum theory will be most significant for small molecular systems (where quantum effects will not be so heavily averaged), and also for cases where the AB* collision complex is not so long lived, i.e.where the contribution from direct A + B scattering is not negligible. 1 W. H. Miller, J. Phys. Chem., 1995,99, 12387. 2 H. Taylor and V. Mandelshtam, J. Chem. Phys., in press. Prof. Schatz asked: An important question raised by your treatment of collisional effects is whether there are circumstances when memory (non-Markovian) effects are important in collisional relaxation. In earlier work,' Bruehl and Schatz examined suc- cessive collisions of hot CS2 with bath atoms, comparing (AE) values obtained with and without vibrational phase randomization between collisions. There were important differences whenever IVR was slow compared with the mean time between collisions. Is this memory effect recovered in your theory if the random phase approximation is not applied ? 1 M.Bruehl and G. C. Schatz, J. Chem. Phys., 1988,89,442. Prof. W. H. Miller answered: The quantum impact approximation assumes that suc- cessive collisions of the bath gas M with AB* are random and statistically independent, General Discussion but it does keep proper account of the phase information between collisions; cJ: the effective quantum propagator in eqn. (4.2). Thus I believe that the classical memory effects you describe are contained in the quantum impact approximation (if the random phase approximation is not made). If one wrote out the semiclassical expression for the propagator in eqn. (4.2), i.e. using the classical S-matrix for each S-matrix factor and performing all sums over intermediate states by the stationary phase approximation, than I think that this ‘memory’ version of your classical treatment would result.Prof. Clary opened the discussion on Prof. Wardlaw’s paper : Conventional tran- sition state theory breaks down in certain cases. A good example is the heavy-light- heavy reaction involving transfer of protons or hydrogen atoms. This type of reaction requires special reaction paths. Does your more generalised approach offer any advan- tages to a reaction type such as this? Prof. Wardlaw responded : Our generalized approach to the reaction path has been formulated within the framework of variational transition state theory as it applies to unimolecular dissociation/bimolecular association in which the reactant/product is a stable molecule, i.e.there is a deep potential well. The conventional transition state theory (TST) to which you refer applies to direct reactions where there is an identifiable potential feature (usually a saddle point) with which to associate the transition state and no potential well. We have not considered the extent, if any, to which our approach is applicable to the conventional case but it warrants consideration. Truhlar and Garrett’ have developed numerous modifications to deal with tunnelling and ‘corner cutting’ corrections to conventional TST. How our treatment and its associated reaction paths are related to those of Truhlar et al. is currently unknown. 1 D. G. Truhlar and B. C. Garrett, Annu.Rev. Phys. Chem., 1984,35, 159. Prof. Troe asked: Does your new approach provide easier access to the activated complex partition function or does it also lead to different results? Prof. Wardlaw replied: Our new approach provides much better access to the tran- sition state partition function. The computational savings relative to the original Wardlaw-Marcus method are enormous and there is improved conceptual access to the system properties controlling the temperature dependence and magnitude of this parti- tion function. Formally, the original and the new methods are equivalent. For CH,+ I3 -P CH, we have verified this equivalence numerically: thermal rate constants cal- culated by each method agree to within the Monte Carlo uncertainty inherent in all implementations of the original method.This comparison is reported in ref. 1. 1 S. H. Robertson, A. F. Wagner and D. M. Wardlaw, J. Chem.Phys., 1995,103,7917. Dr. Smith said: In relation to the issue of defining a reaction coordinate as the separation between two generalized points, which may or may not lie on the molecular frame of the recombining (or dissociating) fragments, are there any topological con- straints on the generality of this definition imposed by the fact that one may obtain non-physical results in some cases? Prof. Wardlaw answered: We do not yet know if non-physical results can be obtained for some choices of reaction coordinate. It is hoped that numerical implemen- tation of the method (which has not yet been undertaken) for one or two typical systems might shed some light on this.There is nothing in the existing mathematical framework which hints at topological constraints on the general definition of the reaction coordi- General Discussion nate. I agree that some reaction coordinates seem to be intuitively unphysical, such as one connecting the atom and a point out of the triatomic plane in an A + BC associ- ation reaction. It is remotely possible that such strange reaction coordinates might arise when excited electronic states are involved but I cannot cite any particular examples. It is worth noting that for a particular diatom + diatom reaction Klippenstein' has shown that the optimized reaction path has endpoints lying on the internuclear axis of each diatom but outside the diatom.1 S. J. Klippenstein, J. Chem. Phys., 1992,96, 367. Prof. Quack stated: I have a few comments on the treatment of non-rigid or 'flexible transition states',' which are common in unimolecular simple bond fission reactions. In the adiabatic channel model2 an explicit quantum treatment of such transition states with large amplitude motions was put forward, originally using interpolation procedures for channels, as the accurate numerical calculations of channels was out of reach at the time when the original theory was proposed. From there, two lines of development have been pursued. The one was an extremely simple canonical variational transition state theory3i4 with a quantum treatment being based on interpolation of partition functions. The other line of work was to implement numerical techniques to calculate accurately adiabatic channels.An early example which was used to check the adiabatic channel interpolation procedures was published a long time More recently we have devel- oped the very powerful quasiadiabatic channel quantum Monte Carlo meth~d.~'~ While the method is described in detail in the original papers cited, I might mention here some highlights and results, as these developments seem to have not been noted by Robertson, Wagner and Wardlaw. (1) The quasiadiabatic channel quantum Monte Carlo method can be used to solve accurately the Schrodinger equation for low-energy adiabatic channels in very high dimensional systems (we have done calculations for systems up to six atoms, explicitly treating 14 rovibrational degrees of freedom, but larger systems could be dealt with, if potential hypersurfaces are known). (2) In particular one may obtain a rigorous, fully numerical, non-variational result for the lowest adiabatic channel (of a given symmetry), including upper and lower sta- tistical error bounds for the computational result, which makes the method superior to variational upper-bound-only results.(3) One thus obtains accurate vibrational quantum states for polyatomic molecules at high angular momenta J, which can be compared with spectroscopic results at a similar level of theoretical accuracy as is usually only performed for diatomic molecules. The method has been used for some fundamental tests of structural properties and dipole moments of (HF), and CH, isotop~mers.~~~ One also obtains adiabatic channel maxima as a function of J, which indicate the importance of quantum effects even for the rotational degrees of freedom due to the intimate coupling with zero-point m~tion.~ This is in agreement with qualitative considerations presented in ref.9 and contradicts the proposal made by Robertson et al." that a classical mechanical treatment of the transitional modes is adequate. Indeed classical mechanical centrifugal barriers are terri- bly misleading (see Table XI and the discussion in chapter V.F of ref. 7). (4) One may furthermore check the original channel interpolation procedures used in the adiabatic channel mode12.11 with accurate solutions from the quasiadiabatic channel quantum Monte Carlo technique.The comparison is found to be generally quite excel- lent for (HF), (see Fig. lb of ref. 7 and also the earlier test for a potential function of methane12). We would suggest that the use of approximate quantum adiabatic channels with interpolation techniques (adjusted to the potential) is much more accurate than using rigorous classical treatments of the transitional modes. Both the adiabatic channel model itself and the interpolation have also been tested spectroscopically by looking at General Discussion the coupling of one transitional mode (v,) with the hydrogen bond fission reaction coor- dinate in the bound state region of the spectrum.Again the approximations of the adia- batic channel model were found to give excellent predictions (see section V. C. of ref. 7).(5) Finally, the quasiadiabatic channel quantum Monte Carlo technique can also be used for unimolecular isomerisations. In particular one can solve the multidimensional tunnelling problem for isomerisation reactions, which implies an exact solution for the dynamics of isomerisations in the tunnelling regime. For instance, the very first results in full 6 (9) dimensional space were carried out for the tunnelling isomerisation of (HF), in 1990/91637 and recently confirmed by our~elves'~ and others14 at higher accuracy. I might mention that Clary and co-workers have also recently applied our quasiadiabatic channel quantum Monte Carlo methods to a number of further problems.In summary, I would suggest that the classical mechanical treatment of transitional niodes in canonical variational transition state theory may perhaps not be a very good strategy for obtaining accurate results, except perhaps at high temperatures. I would suggest the following better strategy (in direction of going from difficult to simple met hods) quasiadiabatic channel quantum Monte Carlo7 (and related techniques) 1 adiabatic channel interpolation schemes2*' ' 1 canonical variational quantum transition state theory with interpolation of partition functions3 In contrast to the early work,2 the simple interpolation schemes could nowadays be adjusted to potential hypersurfaces, where available, and to benchmark calculations by the quasiadiabatic channel quantum Monte Carlo method.They are then expected to give accurate results in spite of their exceptional simplicity. I have two further small comments. One concerns the use of symmetry numbers (ot in the paper by Robertson et al.) for non-rigid or 'flexible' transition states. The problem is slightly less trivial than seems to be assumed by some" and I have given a full discussion elsewhere, to which I may refer here.15*16 'The last comment refers to the use of Laplace transforms to recover W(E)[N(E) in the nomenclature of Robertson et al.]. As we have already discussed long ago,3 this method cannot be used in practice to recover to important J dependence in W(E,J) and thus the highly non-trivial J dependence of the statistical rate coefficient k(E, J), orig-inally introduced by a proper consideration of angular momentum conservation in ref.2 and now widely accepted : Thus, although the Laplace transform method looks good formally, it is much less useful, in practice, to obtain specific rate constants K(E,.J). 1 M. Quack, Faraday Discuss. Chem. SOC.,1981,71,309. 2 M. Quack and J. Troe, Ber. Bunsen-Ges. Phys. Chern., 1974,78,240. 3 M. Quack and J. Troe, Ber. Bunsen-Ges. Phys. Chem., 1977,81 329. 4 M. Quack and J. Troe, in Gas Kinetics and Energy Transfer, The Chemical Society, London, 1977, vol. 2, p. 175. 5 M: Quack, J. Phys. Chem., 1979,83,150. 6 M. Quack and M. A. Suhm, Mol. Phys., 1990,69,791. 7 M.Quack and M. A. Suhm, J. Chem. Phys., 1991,9S, 28. 8 H. Hollenstein, R. Marquardt, M. Quack and M. Suhm, J. Chem. Phys., 1994,101,3588. 9 M. Quack, Philos. Trans. R. SOC.London A, 1990,332,203. 10 S. F. Robertson, A. F. Wagner and D. M. Wardlaw, Faraday Discuss., 1995,102,65. General Discussion 11 M. Quack and J. Troe, in Theoretical Chemistry: Advances and Perspectives, ed. D. Henderson, Aca- demic Press, New York, 1981, voi. 6 B, pp. 199-276. 12 M. Lewerenz and M. Quack, J. Chem. Phys., 1988,88,5408. 13 M. Quack and M. A. Suhm, Chem. Phys. Lett., 1995,234,71. 14 D. H. Zhang, Q. Wu, J. Z. H. Zhang, M. von Dirke and Z. Bacic, J. Chem. Phys., 1995,102,2315. 15 M. Quack, Mol. Phys., 1977,34,477. 16 M. Quack, J.Chem. Phys., 1985,82,3277. Prof. Wardlaw replied: There are comments on a variety of topics. Some are simply statements concerning methods and applications undertaken by Quack et aE. and require no further discussion. There are several issues which do require clarification and/or rebuttal. Rather than respond on a point by point basis to Prof. Quack's num- bered comments, it is more compact to respond to the major issues emerging from this comment taken as a whole. Classical treatment of transitional modes in J'lexible transition state theory (FTST).It is asserted that such a treatment is inadequate based on several features of adiabatic channel models. There is, however, a large body of evidence which suggests otherwise and which has been overlooked in the above comment. It is worth indicating at the outset that all versions of FTST treat the conserved modes quantum mechanically and thus account for their zero-point energy.It is standard practice in FTST to treat the transitional modes classically. The physical basis for this approach is that the transition- al modes are generally low-frequency, large amplitude motions whose character evolves considerably along the reaction coordinate. A precise quantum treatment of such modes over a significant range of energies E, total angular momenta J, and reaction coordinate values R, is still well beyond current computational capabilities. Given the relatively large density of states (not too far above threshold and beyond) and the small zero-point energy (relative to the conserved modes) of the transitional modes, a classical treatment cannot be dismissed as unreasonable (but should certainly be checked; see below).The strength of the classical treatment is the ability to enumerate the number of transitional modes states at a given E and J 'exactly', i.e. without making any approximations to the associated potential energy, other than separating it from the conserved mode potential energy. One simply evaluates a phase space integral. This advantage comes at the expense of a quantum treatment of these modes and reveals an important aspect of the philosophy of FTST: namely, it is better to base the (classical) state count of the transi- tional modes on the best available potential surface than to approximate the potential so as to facilitate a quantum state count.A variety of studies at the microcanonical and canonical levels have addressed directly or indirectly the adequacy of a classical treat- ment of the transitional modes in FTST. These are briefly summarized below, in chro- nological order : (1) In the inaugural work on FTST by Wardlaw and Marcus in 1985,' a detailed comparison between FTST and SCAM was undertaken at the microcanonical (E, J) level for NO, +NO + 0. Sums of states, N;TJST and Wgt?", were compared from J!$hreshold to 40 kcal mol -for J = 0-40. The SACM results were obtained following the procedure of Quack and Tree,, i.e. reactant and product eigenvalues were interpolated with a single exponential function to create the adiabatic channel curves. The FTST sum of states arises from a classical treatment of the transitional modes on a model potential- energy surface based on the corresponding interpolation of force constants.In other words, the potential surface was chosen so that it would give rise to actual adiabatic channel curves which would be very close to those obtained from the SACM inter-polation scheme in order to make the comparison as definitive as possible. The results are plotted in Fig. 5 of ref. 1. The agreement is generally quite good, and is poorest just above threshold where Wtf?" is ca. 10% greater than NET;=. (2) Klippenstein and Marcus3 undertook a study of a quantum correction factor for the transitional modes for a radical recombination reaction (CH, + CH, +C,H,) at GePzeral Discussion the canonical level of FTST.This was calculated via Monte Carlo path integral evalu- ation of the appropriate classical/quantum partition function ratio. The results are sum- marized in Table I of ref. 3. The quantum correction to the recombination rate constant k(T) ranges from 1 to 2% in the 300-2000 K temperature range, shows no systematic temperature dependence, and is less than the Monte Carlo statistical error. The authors rationalize this outcome in terms of the energy dependence of the transition state loca- tion. (3) At the canonical level of FTST, the effect of transitional mode zero-point energy, cZp,on NiT;T has been studied by Aubanel and Wardlaw4 for the recombination CH3 + H -+ CH4.Using a semiclassical ansatz proposed by Tr~e,~ phase space for which the transitional mode energy E is less than E,~is inaccessible and NET;' + 1 as E-+ E,~.Both classical and semiclassical rate constants k(E, J) were then summed over J and Bolt- zmann averaged over E to obtain thermal rate constants k(T).The classical and semi- classical k(T)s agree within the Monte Carlo error bars arising from their numerical calculation on the temperature range 500-2500 K. (The relative errors are 1-2% at all temperatures.) At 300 and 400 K, the semiclassical results exceed the classical results by ca. 4%. Interestingly, this semiclassical correction would act to reduce the low tem- perature discrepancy between FTST and SACM noted in ref.1. (4) At the microcanonical level Klippenstein et have compared accurate three- ~1.~9~ dimensional quantum scattering calculations for the cumulative reaction probability (on a range of energies and with J = 0) for Li + HF -+ LiF + H6 and He + H, +--+HeH'+ H7 with variational transition state theory calculations in which the transitional modes are treated classically. Generally good agreement is obtained, particularly at lower energies where one expects complex-forming collision mechanisms to predominate (when there are potential wells as is the case here) and statistical models to be most appropriate. Some differences between the quantum scattering results and transition state theory are attributable to the explicit neglect of resonance effects in the statistical calculations.In summary, there is no compelling evidence that classical treatment of transitional modes results in significant error at the microcanonical or canonical levels, even at lower energies and temperatures, or for larger systems. In fact, the evidence indicates that a classical treatment is, perhaps surprisingly, rather robust and that its relatively small errors are well worth incurring in exchange for computational tractability. Centrifugal barriers. I agree that classical mechanical centrifugal barriers are terribly misleading when it comes to flexible transition states or adiabatic channels. Assuming the transition state to be atop the centrifugal barrier is essentially phase space theory but is not part of FTST.In the latter treatment, the transition state is determined by minimizing the sum of states orthogonal to the reaction coordinate. The sum of states is determined by a Hamiltonian which includes all the kinetic- and potential-energy contri- butions, including the L2/(2pR2)term for mutual rotation of the fragments about the centre-of-mass. The variational transition state coincides with the centrifugal barrier only in certain limiting energy and angular momentum regimes. Symmetry numbers. That our paper' appears to render this issue trivial is regrettable and will be corrected by appropriate referencing, including the relevant papers of Prof. Quack. That we have long recognized this as a difficult and potentially important issue is revealed by discussion and implementation of a modified symmetry correction factor in ref.4 and 9. Use of Laplace transforms to obtain N(E).I agree that this looks good formally but is much less useful in practice because (a) numerical implementation is generally dificult, and (b) one gets N(E) and not N(E, J). There are nevertheless situations where N(E)may suffice. One is in the evaluation of pressure-dependent rates via a 1D master equation in energy. This is valid if a strong collision approximation is valid for rotational energy transfer or angular momentum transfer. Such an approach is not expected to be gener- General Discussion ally valid but may work for, or provide the desired level of treatment for, certain systems or classes of system.Another situation where N(E) suffices arises when the flexible tran- sition state location depends strongly on energy and only weakly on angular momen- tum. Then one may sum N(E, J) over J and subsequently minimize N(E)with respect to the reaction coordinate en route to k(E) or k(T).The reaction CH, + CH, 4C,H, pro-vides an excellent example.' Comment on exponential interpolation model. In Fig. 5(b) and 16 of a paper on (HF), by Quack and Suhm,1° one finds logarithmic plots of hindered rotor barrier heights, the out-of-plane bend, and several of the lowest energy quasiadiabatic channel curves as a function of distance between the HF centres-of-mass (Rab). None of these plots is linear, indicating that a simple exponential interpolation is a rather crude approximation.A feature common to all plots is a steeper (negative) slope at small R,, and a shallower (less negative) slope at larger Rab. As is discussed in ref. 10, the steeper slope corresponds to an exponential parameter of a M 0.8-1.0 A-' and the shallower slope to a M 0.3 A-'. In a study of properties of the minimum energy path for CH, + CH, -+C,H, by Robertson et aE.,ll the same type of behaviour as a function of separation was found for most of the vibrational frequencies and several geometrical parameters of the methyl radicals. Other properties were found to obey the single exponential model, including the torsional barrier height. Assuming a single exponential for all properties (regardless of how good or bad the fit) results in a wide range of a values, all larger than the standard value of 1.0 A -'.1 D. M. Wardlaw and R. A. Marcus, J. Chem. Phys., 1985,83,3462. 2 M. Quack and J. Troe, Ber. Bunsen-Ges. Phys. Chem., 1974,78,240. 3 S. J. Klippenstein and R. A. Marcus, J. Chem. Phys., 1987,87,3410. 4 E. E. Aubanel and D. M. Wardlaw, J. Phys. Chem., 1989,93,3117. 5 J. Troe, J. Phys. Chem., 1979,83, 114. 6 C-Y. Yang, S. J. Klippenstein, J. D. Kress, R. T. Pack and A. Lagana, J. Chem. Phys., 1994,100,4917. 7 S. J. Klippenstein and J. D. Kress, J. Chem. Phys., 1992,96, 8164. 8 S. H. Robertson, A. F. Wagner and D. M. Wardlaw, Faraday Discuss., 1995,102,65. 9 D. M. Wardlaw and R. A. Marcus, J. Phys. Chem., 1986,90,5383. 10 M. Quack and M. A. Suhm, J.Chem. Phys., 1991,95,28. 11 S. H. Robertson, D. M. Wardlaw and D. M. Hirst, J. Chem. Phys., 1994,99,7748. Prof. Troe said: I would like to point out that canonical variational transition state theory can lead to substantial errors if one compares the results with those from clas- sical trajectory calculations on the same potential-energy surface. In a series of model comparisons for potentials of the charge-dipole interaction type, I have recently shown' that the errors can be up to a factor of three; truncation methods' help only to some extent. Thermal averaging over E-and J-microcanonical variational TST improves the results considerably. However, only the statistical adiabatic channel model brings exact agreement with the trajectory calculations.1 J. Troe, Adu. Chem. Phys., 1996, in the press. 2. S. N. Rai and D. G. Truhler, J. Chem. Phys., 1983,79, 6046. Prof. Wardlaw responded : The extent to which canonical transition state theory is in error depends strongly on the system under study, in particular on the strength of the long-range interactions. When the latter is very strong, as it is when ions are involved, the variational transition state location varies considerably with energy E and angular momentum J. Canonical rate constants are obtained by summing over J and integrating over E first, then minimizing the resulting sum of states with respect to R. The correct microcanonical rate constant is obtained by minimizing an E-J sum of states on a grid of (E, J) values and then summing over J, integrating over E.Thus in the canonical approach there is a single transition state location Rt (T)whereas there are numerous General Discussion locations Rt (E, J) in the microcanonical approach. Consequently, in such systems, the canonical rate constant substantially exceeds the corresponding microcanonical result. The ion-dipole system represents a most severe case and errors up to a factor of three do not surprise me. To assess fully the conclusions and implications of the ion-dipole study by Troe requires the cited article which was not available to me at the time of writing. We have applied microcanonical and canonical FTST to two ion-molecule reactions1 and found that the canonical rate constant exceeds the microcanonical one by factors up to 50%, depending on temperature.The smaller difference here us. the ion- dipole system is attributable to weaker long-range forces in the ion-molecule systems. For a neutral radical-neutral radical system (NO + 0 -+ NO,), good agreement between SACM and FTST results has been found at the microcanonical level., 1 E. E. Aubanel and D. M. Wardlaw, Chem. Phys. Lett., 1990,167, 145. 2 D. M. Wardlaw and R. A. Marcus, J. Chem. Phys., 1985,83,3462. Dr. Jordan said: It would seem sensible to define a generalised reaction coordinate in terms of the physical features of the particular molecular system. For example, you point out that in an atom-diatomic system the high point should be in the plane, and indeed in this case the generalised coordinate is defined in terms of the atom. My question is, then, how would you extend the notion of a generalised reaction coordinate to more cornplicated systems in which neither fragment is an atom? Prof. Wardlaw answered: This is an open question. The most complicated system yet studied for which variations in the reaction coordinate definition were explored is two diatomics,' In this case variations in the hinge points were confined to displacements along an axis coinciding with the bond axis of each diatom and the optimum hinge points were found to lie outside the diatoms. We are in the process of extending our variable reaction coordinate treatment to polyatomic partners, The theory will allow for hinge points anywhere inside (or outside) each fragment. There are, however, no theo- rems (that we know of) or clearly delineated physical criteria which can be used to restrict the variations in the hinge points. I suspect that useful indications will emerge from numerical calculations but this is always somewhat unsatisfactory. Hopefully as people begin to think about this kind of extension of variational transition state theory, some progress will be made towards answering this question. 1 S. J. Klippenstein, J. Chem. Phys., 1992,96, 367.

ISSN:1359-6640    

DOI:10.1039/FD9950200085

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss., 1995,102,117-128 Fourier-transform Analysis of the NOz Spectrum Antoine Delon, Robert Georges, Bernd Kirmse and Remy Jost High Magnetic Field Laboratory, CNRS-MPI, BP 166,38042 Grenoble Cedex 9, France An almost complete knowledge of the NO, energy spectrum is now avail- able, from the ground state up to the first limit of dissociation at 25 130 cm-l. This allows the determination, by Fourier transformation, of the dynamic behaviour of this molecule as a function of energy. We have used the windowed Fourier-transformation technique introduced by Johson and Kinsey (and named ‘vibrogram’ by Gaspard). Four ranges can be distin- guished in the NO, energy-level ladder: Below 10000 cm-l, the 192 vibra- tional levels of the ground state are assignable and ‘regular’ even if some specific resonances can be observed. Above 10000 cm- ’, the vibrational levels of the first electronic excited state, whose origin lies at 9732 cm-’, interact with the highly excited vibrational levels of the ground state.Above about 17000 cm-’, these vibronic interactions are so strong, that vibronic quantum chaos exists, while the rotational degrees of freedom are only weakly coupled. In contrast, rovibronic chaos seems to exist just below the limit of dissociation. The low-resolution absorption spectrum of NOz has also been analysed; it shows a periodic structure which corresponds to the bending frequency in the A2B, state. In summary, we discuss the interplay of the frequency and time domains in various energy ranges of the NO, molecule.1. Introduction Time-frequency duality allows the dynamics of polyatomic molecules to be studied by two experimental methods : fs laser pumpprobe experiments and high-resolution resolved spectroscopy, as presented here. Understanding the notorious complexity of the NO, spectrum has been a challenge for decades.lP4 We have now, thanks to advances in both theoretical and experimental approaches, a semiquantitative description of the main characteristics of this molecule. The high density of lines in the ‘visible’ NO, spectrum (from 10000 cm-l up to the limit of dissociation at 25 130 cm-’) and of resonances lying above this limit is due both to the usual effects, like rotational congestion, spin and hyperfine splittings and hot bands, and to two more specific effects.These two effects are the conical between the potential-energy surfaces (PESs) of the ground and first excited electronic states and the rovibronic interactions which can mix all levels of the same symmetry.6y7 In this paper we deal only with the vibronic properties of NO,. The conical intersection induces a strong non-Born-Oppenheimer (NBO) interaction which, in turn, mixes the high vibrational levels of the ground state (X 2A,) with those of the electronically excited state (X2B2).*-l0 The resulting density of ‘bright’ vibronic levels is of the order of one level every 10 cm-’ around 16000 cm-’ and one level every 2 cm-’ around 25000 cm-l 10-12 These average vibronic energy spacings (or spacings between nearby band origins) are much smaller than the typical spectral width (ca.50-100 cm-l) of the rota- tional structure of each band at room temperature, leading to a very congested absorp- tion spectrum, not resolved at room temperature even at Doppler So, the 117 Fourier-transform Analysis of the NO, Spectrum NO, room-temperature spectrum is treated as a continuous cross-section (as for a large molecule) even at energies as low as 1.5 or 2 eV. 20 years ago, Smalley et al.I3 made an experimental breakthrough in terms of the understanding of the NO, spectrum, by using a supersonic jet which allows the rotational degrees of freedom of the NO, mol-ecule to be cooled to just a few K. Since then, several groups have improved, extended and completed the jet-cooled NO, spectrum, from the ground state up to the limit of dissociation and even above where resonances are resolved.7~9~10~14-18 At low tem- perature in a supersonic jet and at high resolution, the NO, spectrum is well resolved and the lines (levels) are assignable rotationally. However, there are numerous ro-vibronic interactions which perturb the usual regular rotational pattern of each vibronic band (level) located above 10000 cm-’.The rotational structures are more and more irregular with increasing energy and may be ‘chaotic’ near the limit of dissociation; this last point is important for the quantitative analysis of the photodissociation rate near the threshold, as discussed by Reid and Reisler and by Lange et al.(this Discussion); see also Ref. 7. Here we will focus our attention mainly on the electronic and vibrational degrees of freedom and we will mention rotational degrees of freedom only when neces- sary. Our goal is to link the properties of the energy levels and of the corresponding intensities (for transitions from the ground state) with the dynamical properties of the molecule. We will see that Fourier transformation allows the behaviour of an energy spectrum to be characterized into chaotic, regular or intermediate between these two. The Fourier transform (in fact, the square of the modulus of the Fourier transform) allows the survival 1,) FT(I of an energy spectrum (here the vibronic energies of NO,) 1, I c $(t)IP(t)probability, $(t = 0) > to be calculated where $(t = 0) represents an = initial wavefunction.The choice of this initial wavefun~tion’~ is (more or less) arbitrary and the observed dynamics may be dramatically different according to this choice, as discussed in Section 2-5. So, it is necessary to define carefully this initial wavefunction (wavepacket) either from experimental or from theoretical considerations. Note that in most of the experiments the molecule is excited from its ground state (vibrationless if the temperature is low enough) and the initial wavefunction (at t = 0) is a linear com- bination of the states observed in the absorption spectrum. For NO, this absorption is very broad (it spans continuously from 12000 to 35 000 cm-’) and a laser pulse centred around 400 nm and as short as 1 fs duration is necessary to cover the whole absorption range to the A2B, state.Obviously, only a ‘numerical’ experiment is possible in that case, namely by using the Fourier transform of the absorption spectrum. The behaviour of the survival probability is intimately linked with the distribution of energy spacings through the interplay of interference effects: the ‘nayve’ idea that there are destructive interferences when the energy levels are randomly distributed leading to ‘speckle’ noise and constructive interferences (recurrences) when the energy levels are regularly spaced is correct but misses a critical point: when a quantum system has ‘chaotic properties ’ (see below) there are very important correlation properties between its energy levels which, in turn, with the interplay of destructive interferences effects, produce a loss of memory, much stronger than the loss expected for a random distribu- tion.These abovementioned correlation properties between energy levels are a brand of quantum chaos and are intimately linked to the loss of memory. The prototype of com- plete chaos is a random matrix model named the Gaussian orthogonal ensemble, GOE. O This random matrix model has elegant mathematical properties but is not fully rele- vant for real physical systems. The GOE model has well defined correlation properties of its eigenvalues (and eigenvectors) which can be observed in other kinds of random matrices,, and also in model Hamilt~nians.~~~~~ So, the correlation properties of the GOE model should be considered as an ultimate limit, the real systems having ‘weaker’ correlation properties.For example, the GOE model assumes an instantaneous loss of A. Delon et al. memory, while a real system is expected to lose its memory only after a finite time (often not measurable), There are several measures of these correlation properties. The most popular and robust measure is the nearest neighbour spacing distribution (NNSD), but it is of limited relevance for dynamics. The long-range correlations play a more impor- tant role in short-time dynamics.,’ These correlations can be described by the two-point correlation function, Y,(L),,’ but this function is difficult to handle and it is easier to deal with its Fourier transform, b2(t),where L is a reduced energy variable (the average energy spacing is taken as unity) and t is the corresponding reduced time.Two other standard functions, C2(L)and A3(L), can be used but they are not convenient for describing the time evolution of a system. Moreover, there are exact analytical relation- ships between Y,(L), C2(L),A3(L) and b2(t).20721When we consider the Y,(L) function for a stick spectrum, or its Fourier transform, b,(t), it is a very noisy function. In con- trast, X2(L)and A3(L), which are two smoothed versions of Y2(IJ9 seem less noisy, but contain less information. An important distinction should be made: when we want to study only the corre- lation properties (and not the molecular dynamics) we have to consider all the levels (of a given symmetry and quantum number) at the same time. This means that when we want to characterize the correlation properties we have to consider a stick energy spec- trum with equal intensities.In contrast, in molecular dynamics and in real experiments (or for the corresponding simulations) the intensities play a crucial role as discussed below. Here, we present first the Fourier transformation of a large set of vibronic energy levels in order to characterize the correlation properties. We then present the I FT l2 of Ithe absorption cross-section of NO,. We will interpret these FT1,in terms of molecu- lar dynamics on different timescales and relate the dynamical properties (which take intensities into account) with the correlation properties of energy levels (which do not tak:e intensities into account).The windowed Fourier transform ‘vibrogram’, which has recently been introduced by Gaspard et al.,19 has been used to analyse the properties of NO, spectra: we have compared the vibrogram of the raw, room-temperature absorp- tion spectrum with the vibrogram of our stick spectrum (obtained with ICLAS and LIF experiments, see below) in which the experimental intensities are taken into account. 2. Correlation Properties of the NO, Vibronic Levels around 18 000 cm-I using Fourier Transformation Th.e visible NO2 spectrum (observed either by or by laser-induced fluorescence) is not resolved at room temperature.Owing to the rotational cooling of a supersonic jet, it is possible to resolve the NO, spectrum and to analyse the observed transitions in terms of rotational lines grouped into vibronic bands. The observed density of vibronic cold bands (i.e. vibronic levels) is much higher than expected if we take into account only the A2B, state (the first excited electronic state) and is inter- preted as the sum of the vibronic density of the A2B2 and the x2A, electronic states. Note that highly excited vibrational levels of the electronic ground state g2A, are expected to be dark levels. The density of observed vibronic levels is due to the conical intersection between potential-energy surfaces of the 2A, and A 2B, electronic states and to the corresponding NBO interactions.Furthermore, quantum chaos is expected if the interaction between these two states is strong enough. The standard way to evidence quantum chaos is to analyse the correlation properties within a sequence of energy levels of a given symmetry. However, this statistical analysis of correlations requires completeness and purity (no spurious level^)^^'^ of the sequence of energy levels. These two requirements are very difficult to fulfil from an experimental point of view, mainly because the intensities vary by several orders of magnitude, even in the case of the Porter Thomas distribution which corresponds to the narrowest distribution. This Fourier-transform Analysis of the NO, Spectrum means that a very high signal-to-noise ratio should be achieved to detect (almost) all the lines (levels) in a given energy (spectral) range.Owing to the high sensitivity of the LIF technique (a signal-to-noise ratio of 5000 has been achieved in our experiment) combined with efficient rotational cooling of the supersonic jet, the vibronic energy levels of NO, are now properly known in the visible range. 10.13 -1 8 As a result, more than 500 cold bands (i.e. vibronic levels) have been observed between 11 200 and 19 360 cm-', either by intracavity laser absorption spec- troscopy (ICLAS)" or by LIF.9*10913-17 However, the completeness of this set of vib- ronic energy levels is satisfactory (but not perfect!) only between 16580 and 19360 cm-l; in this range a set of 315 vibronic levels has been observed and studied in detail." New LIF measurements using a slit jet, a germanium detector and a Ti : sapphire laser are under progress in the red and near-IR ranges in order to improve the completeness of the NO, vibronic spectrum in these regions.Two kinds of Fourier- transform analysis have been carried out on the observed set of vibronic energy levels: (i) Fourier transformation of the stick spectrum with observed intensities, discussed in Section 3 ; (ii) Fourier transformation of the stick spectrum (unfolded see below) with equal intensities, discussed in this section. This second type of FT analysis is well adapted to reveal correlation properties between energy levels. The proper method for analysing these correlations necessitates the initial energy spectrum to be deconvoluted in order to obtain a new, 'unfolded' spectrum with uniform level density normalised to unity.The technical details of the unfolding procedure are described el~ewhere.~*~~*' FT I2 of the unfolded NO2 vib-The I ronic spectra of 315 levels, shown in Fig. 1, displays a strong (deep) correlation hole, which demonstrates the existence of strong correlations within this set of 315 vibronic levels, in global agreement with the predictions of the standard GOE model. Several remarks should be made in order to interpret this figure correctly and quantitatively. First, the 1 FT1,of any stick spectrum is a noisy function with 100% fluctuations around its average value.This phenomenon is similar to speckle noise. Consequently, some 300 200 1 FTI 0 2 4 6 time/ps Fig. 1 I FT 1' for the stick spectrum (all the sticks have unit amplitude) between 16 580 and 19 360 cm-'. I FT 1' has been smoothed with a Gaussian function (see text) in order to observe the global shape related to the correlation properties of the system. The experimental I FT 1' presents a deep correlation hole close to that observed using the model, the GOE but some regularities still exist: the two recurrent peaks at 0.05 and 0.2 ps correspond to periodic motions. The peak at 0.05 ps can easily be interpreted as the bending period of the A 2B2 electronic state, while the peak at 0.2 ps remains unassigned. A. DeEon et al. smoothing should be performed in order to obtain a clear comparison with the standard cases of Poisson and GOE statistics for which only the averaged curves are meaningful.These two curves are displayed in Fig. 1. Since the smoothing procedure is not defined 1, I FTby some standard recipe, we have chosen to smooth our with a Gaussian function having a full width at half maximum (FWHM) proportional to its centre abscissa (which is interpreted as a time t, see below). In this way, the smoothing gives rise to constant 1, I FTSecondly, the value of At/t. for the normalized stick spectrum is (315), at t = 0, and drops very rapidly (as a function of time) to 315 for Poisson statistics and almost to zero for GOE statistics. This corresponds to the ‘loss of memory’ expected for a chaotic system (for both classical and quantum systems).The following increase of I FT l2 us. time is a pure quantum effect which occurs for a typical time oft = l/AE, the recovery time, where AE is the average energy spacing (of the order of 8.8 cm-l in our set of 315 levels of NO,). At the classical limit (h-O),1, A crucial characteristic of I FTinfinity. AE-0 and the ‘recovery’ time goes to is the existence of a ‘correlation hole’, the upper limit of which is defined by the Poisson curve (horizontal line) and the lower one by the GOE curve. However, the statistical fluctuations, owing to the limiting size of our set (315 levels), are not completely smoothed out and allow these two limits to be exceeded locally.Note that, strictly speaking, the horizontal axis is the conjugate variable of the ‘reduced’ energy, defined in the unfolding procedure (see above), which is a dimensionless quan- tity. However, it is possible to interpret the horizontal axis physically as a ‘time’ by using the inverse of the average energy spacing of 8.8 cm-’ for our set of 315 vibronic levels. One unit corresponds to 3.8 ps and allows the horizontal axis to be graduated in time units. This is correct as long as the density of states does not change too much along the considered sequence of levels; between 16580 and 19 360 cm-’, the density of states increases by a factor 1.6, i.e. it varies by, at most, 30% compared with the average value. So, this allows us to interpret the horizontal axis approximately as a time.1, The important question is to determine which characteristics of I FT (e.g. peaks), are significant, i.e. are not due to fluctuations. The first very important feature occurs when I FTI2 drops close to zero for a very short time. The ultimate short-time limit is the reverse of the energy window considered. Note that the abovementioned ‘unfolding’ procedure may significantly affect the short-time behaviour of I FTI,, i.e. introduce a spurious correlation hole, or equivalently, spurious long-range correlations. This point is discussed in detail in Appendix C of Ref. 9. We have carefully checked that our unfolding procedureg-10 does not produce the observed correlation hole. As a first important result, the observed correlation hole in I FT l2 means that the vibronic levels of NO, located around 18 500 cm- are correlated at least over several hundred cm- ’.The second very important characteristic is the existence of two peaks around 0.05 and 0.2 ps which are significant (i.e. not due to speckle noise, see above) and are inter- preted in terms of residual order, i.e. periodic motions in the classical description associ- ated with the sequence of energy levels of NO,. The peak at 0.05 ps (or ca. 700 cm-’) can easily be interpreted as the bending period in the x2B, electronic state, while the peak at 0.2 ps remains unassigned. Quantum dynamics calculations may be useful to confirm and/or to interpret these periodic notions. The results presented in Section 3 and 4 reinforce the interpretation of the peak at 0.05 ps as a bending vibration. 3.Vibrogram of the NO, Vibronic Spectrum A vibrogram (introduced first by Johson and Kin~ey,~ and more recently by Gaspardlg) is a windowed Fourier transform ( I FT 12), of a spectrum. The basic idea is to follow the characteristic frequencies of a spectrum (related to classical periodic orbits) as a function of energy, i.e. to study the dynamics of a system as a function of energy. 122 Fourier-transform Analysis of the NO, Spectrum The finite width of the energy windows results in a compromise between the energy resolution (high-energy resolution requires narrow energy windows) and ‘frequency’ range (high ‘frequencies’, i.e. short periods, requires large energy windows).We have used a Gaussian window in order to avoid spurious oscillations. In contrast with the correlation analysis of Section I1 (which discards the intensities), the intensities play an important role in the analysis of the vibrogram: as a consequence, the missing levels (see Section 2), as long as they have a weak intensity, do not play an important role. A second important difference between the vibrogram analysis and the correlation analysis used in Section 2, is the fact that it is not necessary to ‘unfold’ the energy axis. Conse- quently, the conjugated variable is a time. In contrast to the correlation analysis present- ed in Section 2, the vibrogram analysis is well adapted to study the ‘regularity’ of a spectrum, i.e.to detect (if any !) the vibrational progression(s) and resonances between vibrational modes which are related with classical periodic orbit(s). We will consider here a set of 516 vibronic levels of NO, observed between 11 200 and 19 360 cm- either by ICLAS (175 levels between 11 200 and 16 125 cm-l) or by LIF (350 levels between 16000 and 19 360 cm-I). Note that nine levels have been observed twice (by LIF and by ICLAS) between 16000 and 16 125 cm-’ The stick energy spectrum is displayed in Fig. 2 with experimental relative inten- sities. The normalisation of ICLAS and LIF intensities has been carried out by using a third spectrum, the absorption spectrum analysed in Section 4 : experimental ICLAS and LIF stick spectra have been convoluted with a Gaussian of 100 cm-’ width (in order to obtain roughly the apparent resolution of the absorption spectrum) and have been cali- brated to the absorption spectrum in their respective ranges.This allows the ICLAS and LIF spectra to be concatenated into a single stick spectrum in the 11 200-19 360 cm-l energy range as shown on Fig. 2. Obviously, the LIF and ICLAS intensities are not strictly equivalent (mainly because the lifetimes and spectral ranges of fluorescence depend on the level) but they do not differ too much on average. The vibrogram of this spectrum has been constructed as follows: We first calculated each windowed spectrum by multiplying the stick spectrum by a Gaussian window centred at Ei and having an FWHM of 3770 cm-’, and the 1 FT l2 of each windowed spectrum was analysed.The vibrogram is a plot (as a function of the centre, Ei, of the energy window) of the position (in time units) of the I FT l2 peaks. ” 11000 12000 13000 14000 15000 16000 17000 18000 19000 energy/ cm-’ Fig. 2 Concatenated ICLAS and LIF stick spectra. Comparison with the absorption spectrum allowed the relative intensities to be calibrated between the ICLAS and LIF parts: first, each part has been convoluted with a Gaussian function (FWHM = 100 cm-’) in order to obtain about the same resolution as the absorption spectrum; secondly, we calibrated each part separately with the absorption spectrum and concatenated both. A. Delon et al. I L I0.0 ' 12000 13000 14000 15000 16000 17000 18000 19000 energy/ cm-' Fig.3 Vibrogram of concatenated ICLAS and LIF spectra. Each windowed spectrum used to construct the vibrogram is the product of the initial stick spectrum (see Fig. 2) and a series of Gaussian windows (FWHM = 3770 cm-l) with various centres, Ei, separated by 500 cm-l. Then the positions of the I FT I2 peaks of the windowed spectra are plotted vs. the centres, E,, of the windows. The amplitudes of the peaks are proportional to the width of the continuous lines. The main feature at t = 0.047 ps (ca. 720 cm-l) is due to the bending motion in the A2B2 electronic state. For each window centre, Ei, the peak of I FT l2 at the origin (t = 0) gives a normal- isation factor which allows the fact that the stick spectrum is stronger and denser at high energies than at low energy to be taken into account.The position and amplitude of the 1, other peaks of each 1 FT have been fitted for each windowed spectrum. We have used steps of 500 cm-' for the window centres (much less than the FWHM of each window at 3770 cm-l) in order to correlate the peaks of consecutive energy windows easily. These peaks are linked on the vibrogram and appear as a continuous line, the width 1, I FTof which is proprotional to the amplitude of the peak. When two peaks overlap, the corresponding lines on the vibrogram may not be resolved. The vibrogram presented in Fig. 3 can be interpreted as follows: Each peak on the FT l2 (or each line on the vibrogram) should not necessarily be interpreted as a periodic 1, motion in NO,.It is well known that the I FT of any spectrum contains peaks, and even a purely random spectrum (with no regularities) will give peaks (these are often termed 'speckle noise'). The question is how to discriminate between the peaks due to speckle noise and the those that are physically significant. Two rough criteria can be used: (a) the peaks due to speckle noise are expected to exist only in an energy range (horizontal axis) of the order of the width of the window. Conversely, the peaks due to a periodic motion may exist over a much larger energy range. The broad horizontal line which goes from 12000 to 19000 cm-' at t = 0.047 ps can hardly be attributed to speckle noise and is due to the bending motion in the A 2B2 electronic state.This strong peak in the I FT l2 is stable and corresponds to the intensity modulation of the NO, s ectrum envelope (see Fig. 2) which is attributed to the bending progression of the f213, state. The other lines of the vibrogram are weaker and/or shorter, and have not been interpreted in terms of periodic motion. Trajectory calculations or wavepacket propagations in the PES of NO, are highly desirable in order to interpret this vibrogram. However, periodic orbits do not necessar- ily exist and, if quantum chaos is well developed in NO,, no periodic orbits are expected. We have shown in Section 2 that NO2 around 18000 cm-'is significantly but not fully chaotic (because there is some residual order). The vibrogram confirms the existence of the periodic motion associated with the bending, but does not give additional information.124 Fourier-transform Analysis of the NO, Spectrum 4. Vibrogram of the Room-temperature NO2 Absorption Spectrum At low resolution (1-10 nm) the NO, absorption cross-section,26 which is shown in Fig. 4, has a bell shape which is approximately a Gaussian multiplied by the energy when the cross-section is plotted as a function of wavenumber. The centre and width of this Gaussian envelope can be explained by the reflection method2' which takes into account the vibrationless wavefunction of the ele_ctronic ground state and the shape of the PES of the first electronically excited state, A 2B2. Here we will focus our interest only on the spectral oscillations which are superimposed on the envelope and which may hold some meaningful dynamical information.The main difference between the stick spectrum analysed in Section 3 and this room-temperature absorption spectrum is the contribution of the numerous rotational lines associated with each vibronic band. This rotational contribution can be modelled by a convolution with a Gaussian function having an FWHM of about 100 cm-'. This means that the small spacings between vibronic levels are blurred by the rotational structures, but that the information linked to larger energy spacing (2100 cm-l) may be extracted from the room-temperature absorption spectrum. We have employed the same Gaussian windows (FWHM = 3770 cm-') as those used in Section 3.A typical example of Fourier transformation is shown in Fig. 5 for a Gaussian window centred at 22 500 ern-'. The peak at the origin has been used as a normalisation factor and 14 peaks have been fitted in the displayed time range (0-300 fs) among which the peaks A, B, C and D are the four strongest ones. The vibrogram shown in Fig. 6 has been constructed with 36 such I FT l2 for which the window centre goes from 12000 to 29500 cm-' in steps of 500 cm-'. The peaks A, B, C and D of the I FT 1' of the windowed spectrum centred at 22 500 cm-' are reported in the vibrogram for the sake of clarity, but there are, in fact, 14 I FT l2 peaks at this energy. On the left-hand side of Fig. 6 we have reported the time (i.e.the period) and on the opposite side the corresponding wavenumber (in cm-') in order to interpret easily (if possible!) the peaks in terms of vibrational frequencies. Peak A, which is very strong at 0 ,/ , , , , , , , , , 1 2 3 4 energy/l o4 cm-' Fig.4 Experimental absorption cross-section (divided by the energy) of NO, at low resolution (between 1 and 10 nm) as a function of energy. The global shape can be approximated well using a Gaussian function according to the reflection principle.27 One can clearly observe the spectral oscillations (of period close to 720 cm-') that are superimposed on the envelope. These oscil- lations are attributed to the bending period in A ,B2. A. Delon et al. 2000 1500 500 0 0.1 0.2 0.3 tim e/ps Fig.5 Fourier transform of the windowed absorption spectrum centred at 22 500 cm-(FWHM = 3770 cm-I). 14 peaks have been fitted between 0 and 0.3 ps. The peaks A, B, C and D are the strongest ones. low energy, decreases in intensity with increasing energy, but is stable in position (over a period of 47 fs or 720 cm-') up to about 25000 crn-'. Peak B, with a period of about 115 fs, is weaker but seems to be significant, i.e. it is not due to speckle noise. These two lines on the vibrogram (or peaks in I FT 12) coincide with first two peaks of the 1 FT l2 of the stick spectrum analys_ed in Section 2. Once again, peak A can be assigned to the bending frequency of the A 2B2 electronic state, but peak B remains unassigned, as do all the other lines of the vibrogram.v)Q \ Q, .-E c) 12000 15000 18000 21000 24000 27000 energy/ cm-' Fig. 6 This vibrogram has been constructed from 36 individual power spectra (like that shown in Fig. 5). The centres of windows run from 12000 to 29 500 cm-' in steps of 500 cm-'. All the power spectra are normalised with respect to their peaks at the origin. The peaks labelled A, B, C and D on Fig. 5 are reported on the vibrogram. Peak A, which is stable in position us. energy (t x 0.047 ps), corresponds to the main spectral oscillation superimposed on the absorption spec- trum envelope (see Fig. 4). Fourier-transform Analysis of the NO, Spectrum To conclude, this vibrogram confirms our previous analysis but gives no additional information.5. Fourier Transform of the Photoelectron Spectrum of NO2 A very interesting spectrum of the highly excited vibronic levels of NO, was obtained by photoelectron detachment spectroscopy of NO, by Weaver et The spectrum ~1.~~9~~ contains information on three electronic states (X ,A1, A 2B, and "Ad, but we will consider only the energy range, displayed in Fig. 7, corresponding to A2B2 between 9500 and 16000 cm-l. The I FT of this range displayed in Fig. 8 shows one strong progression with a period, T,, of 46 fs (726 & 5 crn-l) associated with the bending I1.o 2678 (040) 13396 (050)0.8 >r 0.6 c.-VJ Ca, c .& 0.4 0.2 I I I0.0 10000 11000 12000 13000 14000 15000 energy/ cm-' Fig. 7 Photoelectron detachment spectrum from Weaver et ~21.~~1~'The equilibrium angle of NO2-is 124" (cJ 134"for 8 2A, and 102" for A 2B,); the Franck-Condon envelope with A ,B2 is a maximum around n, = 4.t I\ 0.015 0.05 0.10 0.15 0.20 time/ps Fig. 8 I FT l2 in the 9500-16000 cm-' sub-range of the photoelectron detachment spectrum, cor- responding to the A2B, electronic state of NO, (see Fig. 7). In addition to the strong rogression of period T2 = 0.046 ps (ca. 726 cm-I), corresponding to the bending period in % 2B2, one observes a much weaker progression of period TI = 0.023 ps (ca. 1470 cm-I), strongly resonant with the bending motion. A. Delon et al. motion of A ,B, and a superimposed and much weaker progression with a period, T,, of 23 fs (1470 f.20 cm-'), which is strongly resonant with the bending motion (LO' z 2 w,). The bending frequency, cu, = 726 5 cm-', is an average over bending energy spac- ings located between 9500 and 16000 cm-l, i.e. 0 < n, < 8. In contrast, the previous values of co2 are an average over the bending energy spacings located at higher energies (see Sections II-IV). The peaks observed from 9500 to 20000 cm-' are described well by a pseudo-Dunham expansion, En = E, + weff(n2+ %)+ xeff(n+ $)2, n being the polyad numier, meff and xeffthe pseudo-Dunham coefficients and E, the electronic energy of the A2B2 state. We have avoided using n,, LO,and x22because there are some strong indications that the combination bands (1, n2 -2,O); (2,n, -4, 0);... play an important role, in addition to the pure bending progression (0, n2,0).Note that: (a) the photoelectron spectrum from the ground state of NO,-(which is bent at an angle of 124") has a maximum around 12700 cm--' [which corresponds to a maximum Franck-Condon factor with the (0, 4, 0) level of A ,B,], while the absorption from the ground state of NO, (bending angle of 134") peaks around 25 000 cm-'. (b) The peaks of the photoelectron spectrum are significantly narrower than those of the absorption spectrum. Taking into account the low experimental resolution of the photoelectron technique it is difficult to know if there is more than one vibronic level, (0, n2,0), under each peak. (c) The peaks around 14 800 cm-(0, 7, 0) and 15500 cm-' (0, 8, 0) are significantly structured: This means tkat above ca. 15000 cm-', either the combination bands are important or the z2A,-A2B2 vibronic interaction is strong enough to scramble the bending vibrational progression of A ,B2.Further ab initio calculations (including Franck-Condon calculations) are necessary to interpret the observed spectra in more detail. 6. Conclusions Owing to Fourier-transform analysis, we now have a semiquantitative interpretation of the main properties of the visible spectrum of NOz. (a) The strong correlations between the vibronic levels above ca. 17000 cm-' demonstrate the existence of quantum chaos. (b) There are some residual regularities within the energy level (observed in long- range energy spacings) which indicate that the chaos is not instantaneously established but that there is a residual regular vibrational motion for a few tens of fs.(c) The chaotic behaviour seems to be stronger at higher energies and there is prob- ably a smooth evolution from regularity to chaos with increasing energy, but additional investigations need to be carried out in the red and near-IR ranges to confirm this statement. (d) A real, dynamic (pumpprobe) experiment on NO, is (to date) not possible, because most of the characteristic times are too short. (e) The implications of the chaotic behaviour of the vibronic degrees of freedom of NO, near the limit of dissociation are not well understood because the rotational (and electronic spin) degrees of freedom need to be taken into account.Some preliminary experiments (not presented here) have indicated the existence of rovibronic chaos near the limit of dissociation. This kind of formation is crucial in order to interpret the NO, photodissociation processes just above the limit of dissociation, a subject that is con- sidered twice in this Faraday Discussion. References 1 D. K. Hsu, D. L. Monts and R. N. Zare, in Spectral Atlas of Nitrogen Dioxide, Academic Press, New York, 1978. 128 Fourier-transform Analysis of the NO, Spectrum 2 K. Uehara and H. Sasada, in High-resolution Spectral Atlas of Nitrogen Dioxide, 559-597 nm, Springer Series in Chemical Physics 41, Springer Verlag, Berlin, 1985. 3 C. F. Jackels and E. R. Davidson, J. Chem. Phys., 1976,64,2908. 4 G.D. Gillispie, A. V. Khan, A. C. Wahl, R. P. Hosteny and M. Krauss, J. Chem. Phys., 1975,63, 3425. 5 (a) H. Koppel, W. Domcke and L. S. Cederbaum, Adu. Chem. Phys., 1984, 57, 59; (b) E. Haller, H. Koppel and L. S. Cederbaum, J. Mol. Spectrosc., 1985, 111, 377. 6 A. Delon, P. Dupre and R. Jost, J. Chem. Phys., 1993,99,9482. 7 A. Delon, R. Georges and R. Jost, J. Chem. Phys., 1995, in the press. 8 A. Delon and R. Jost, J. Chem. Phys., 1991,95, 5686. 9 A. Delon, R. Jost and M. Lombardi, J. Chem. Phys., 1991,95, 5701. 10 R. Georges, A. Delon and R. Jost, J. Chem. Phys., 1995, 103, 1732. 11 S. I. Ionov, G. A. Brucker, C. Jaques, Y. Chen and Wittig, J. Chem. Phys., 1993,99, 3420. 12 J. Miyawaki, K. Yamanouchi and S. Tsuchiya, J. Chem. Phys., 1993,99,254. 13 R.E. Smalley, L. Wharton and D. H. Levy, J. Chem. Phys., 1975,63,4977. 14 G. Persch, E. Mehdizadeh, W. Demtroder, Th. Zimmermann, H. Koppel and L. S. Cederbaum, Ber Bunsen-Ges. Phys. Chem., 1988,92,312. 15 (a) S. Hiraoka, K. Shibuya and K. Obi, J. Mol. Spectrosc, 1987, 126, 427, (b) K. Aoki, M. Nagai, K. Hoshina and K. Shibuya, J. Phys. Chem., 1993,97, 8889. 16 H. J. Foth, H. J. Vedder and W. Demtroder, J. Mol. Spectrosc., 1981,88, 109. 17 H. J. Vedder, M. Schwarz, H. J. Foth and W. Demtroder, J. Mol. Spectrosc., 1987,123, 356. 18 R. Georges, A. Delon, F. Bylicki, R. Jost, A. Campargue, A. Charvat, M. Chenevier and F. Stoeckel, Chem. Phys., 1995, 190,207. 19 P. Gaspard, D. Alonso and I. Burghardt, Adu. Chem. Phys., submitted; Phys.Reu. A, 1993,47, 3468; J. Chem. Phys., 1994, 100,9. 20 M. L. Mehta, in Random Matrix, Academic Press, New York, 1967. 21 T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pendey and S. S. M. Wong., Rev. Mod. Phys., 1981, 53, 385. 22 R. Jost, in Stochasticity and lntramolecular Redistribution of Energy, ed. R. Lefebvre and S. Muckamel, NATO AS1 Series C, Reidel, Dordrecht, 1986, vol. 200, p. 31. 23 0.Bohigas, M. J. Giannoni and C. Schmidt, Phys. Rev. Lett., 1984,52, 1. 24 E. Haller, K. Koppel and L. S. Cederbaum, Phys. Reu. Lett., 1984,52, 1665. 25 B. R. Johson and J. L. Kinsey, J. Chem. Phys., 1989,91,7638. 26 B. Kirmse, Diplomarbeit Universitat Fridericiana zu Karlsruhe, 1994. 27 R. Schinke, in Photodissociation Dynamics, ed. A. Dalgarno, P. L. Knight, F. H. Read and R. N. Zare, Cambridge University Press, Cambridge, 1993, p. 109. 28 A. Weaver, R. B. Metz, S. E. Bradforth and D. M. Neumark, J. Chem. Phys., 1989,90,2070. 29 A. Weaver, Thesis, University of California, Berkeley, 1991. Paper 51061925; Receiued 19th September, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950200117

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss.,1995, 102, 129-146 Resonances and Fluctuations in the Unimolecular Reaction of NO2 Scott A. Reid, Andrei Sanov and Hanna Reisler Department of Chemistry, University of Southern California, Los Angeles, CA 90089-0482, USA Fluctuations and oscillations in the unimolecular reaction of NO,, and their manifestations in photofragment yield (PHOFRY) spectra, NO rotational state distributions and decomposition rates are examined. Comparisons between experimental and simulated PHOFRY spectra show that extraction of rates from linewidths in state-selected spectra is unjustified in the regime of overlapping resonances. Measurements of the alignment parameter of the NO product in the excess energy range EE= 200-500 cm-' evidence the existence of fluctuations in the decay rate.Changes in the patterns of fluc-tuations and oscillations in the NO rotational state distributions reveal the progressive tightening of the transition state (TS) as the excess energy increases and the importance of exit-channel interactions beyond the TS. Distributions well fit on the average by phase-space calculations can be obtained even when the transition state is tight. 1. Introduction The unimolecular decomposition of NO, has long served as a prototypical system for addressing experimentally at the state-to-state level many issues central to statistical theories. Experimental studies include : (i) spectroscopic interrogation of vibronic coup- ling between the ground 1 ,A, and excited 1,B, electronic states and the exploration of chaos (vibronic and rovibronic) at high excitation energies,'-', (ii) measurements of unimolecular reactions rates, k(E) and k(E, J),l 3-1 and (iii) determination of detailed product state distributions including vector properties and correlated distribution^.'^-^^ Although for NO, full dynamical calculations on a realistic PES are still beyond reach, significant progress in theoretical descriptions has been made on two fronts : (i) restricted calculations on PESs limited to the region of the TS have yielded realistic descriptions of the unimolecular reaction rates, the vibrational energy distributions, and the correlated spin-orbit distribution^,^^-^' and (ii) extension of the random matrix model to the case of overlapping resonances and its application to tight and loose TSs enabled simulations of state-selected photofragment yield spectra, and investigation of the issue of rates us.widths in the case of overlapping resonance^.^'^^^ Following much work and some controversy, several aspects of the dissociation of NO2 are now well established. The dissociation takes place from mixed 12A,-1 2B2 states which possess predominantly ground-state character. The near threshold rates at excess energies EE= 0-1500 cm-' (Do = 25 130 cm-')34~36~37are well described by sta- tistical theories [Rice-Ramspeiger-Kassel-Marcus (RRKM), statistical adiabatic channel model (SACM)].43-48 The rotational distributions can be fit on the average by phase space theory (PST),48and the vibrational distributions and their dependence on EEare adequately described by variational RRKM theory.43 The NO and 0 spin-orbit distributions are on the average colder than statistical.,' Deviations from sta- 928,29,33 tistical predictions appear to be more pronounced at high excess energies (e.g.EE> 7000 cm-l ).26,28 Both experimental evidence and theoretical calculations suggest that the TS tightens progressively as EEincreases. 129 Resonances and Fluctuations Despite the good agreement (on the average) with statistical models, measurements carried out from well defined initial states and with complete final-state characterization reveal pronounced fluctuations and ~scillations.~~~~~~~~-~~The oscillatory patterns change qualitatively as the excess energy increases, thus providing insights into the decomposition mechanism and the nature of the TS.In this paper we shall emphasize areas of unimolecular dynamics of small molecules that are best revealed through oscil- lations and fluctuations. In particular, we concentrate on three issues requiring further scrutiny: (i) fluctuations in the state-specific rates and the connection between rates, resonance widths and the observed widths of spectral features; (ii) the effects of averag- ing and the transition from oscillating/fluctuating state-to-state behaviour to smooth averages; and (iii) exit channel interactions during product evolution ;in particular, what statistical theories should internal state distributions be tested against? These general issues will be addressed both by experimental work on NO, and by illustrative calcu- lations that qualitatively capture salient features of specific observations.2. Experimental A 0.5% or 4% NO,-He mixture at 1200 Torr was expanded into a vacuum chamber [evacuated to ca. (2-3) x low6Torr] via a piezoelectrically actuated pulsed nozzle (0.5 mm diameter orifice, 180 ps pulse d~ration).~’-~~ The molecular beam was intersected ca. 15 mm downstream from the orifice by collinear and counter-propagating photolysis and probe laser beams, each produced from an excimer laser pumped dye laser system. For the alignment measurements, tunable radiation at 1= 398-370 nm (QUI/dioxane) was used for photolysis, and typical pulse energies were ca.1-5 mJ in a 2-3 mm diam- eter beam. The double-resonance excitation scheme has been described in detail else- where.32 In this method, specific rotational states in the (101 +000) band of NO, were excited with tunable IR radiation from an LiNbO, optical parametric oscillator (OPO), and the vibrationally excited molecules were further excited with a tunable laser to energies above Do. Tunable radiation at 3, = 224-227 nm, generated by frequency doub- ling the fundamental dye output (C450/methanol) in a BBO crystal, was used to excite NO fragments on the diagonal bands of the y-system. Fluorescence collected by a three- lens telescope32 was imaged through a ‘solar blind’ filter onto the surface of a photo- multiplier tube (PMT, Hamamatsu RU166H).To avoid partial saturation of the strong NO transitions, which is important for alignment and product distribution measure- ments, typical probe pulse energies were kept at 0.1-0.5 pJ in a 5 mm diameter beam. For other experiments, probe energies 540 pJ were used. Three types of experiments were carried out. First, state-specific one-photon or TR- visible PHOFRY spectra were obtained by scanning the photolysis laser frequency while probing a specific NO quantum Secondly, NO product state distributions were determined by fixing the photolysis laser and scanning the probe la~er.~’’~’ Finally, alignment measurements of the NO fragment at specific photolysis wavelengths were obtained. Here the polarization of the nominally vertically polarized photolysis beam was further purified by a Rochon prism polarizer before passing through a photoelastic modulator (PEM) configured to rotate the plane of polarization by 90” on alternate laser shots.This allowed direct shot-to-shot collection of signal for parallel and perpen- dicular photolysis/probe polarizations. Typical polarization extinction ratios of > 100 were obtained as measured by a pyroelectric detector (Molectron 53-05) before the chamber entrace window. At each photolysis wavelength, typically (1-2) x lo5 laser shots were averaged at each laser polarization. 3. Widths of Spectral Features in the Case of Overlapping Resonances Recently, it has become possible to observe the manifestations of overlapping resonances in state-to-state unimolecular reactions.When resonances overlap and interact via a S. A. Reid et al. common continuum, they interfere and such interference will modify the lineshapes and positions, giving rise to asymmetric line shape^?'-^^ Above Do, each resonance has a characteristic width, and in unimolecular reactions several resonances often overlap.54 By monitoring specific final states of the products, partial absorption spectra [i.e. the so called PHOFRY, photofragment excitation (PHOFEX), or action spectra] can be obtained. If the final states are largely uncorrelated, each may derive from a slightly different combination of resonance amplitudes and phases (for a fixed photolysis energy), giving rise to a somewhat different interference pattern and thus different lineshapes.This is precisely what is observed in the state-specific PHOFRY spectra of NOz .30-35 Examples of representative PHOFRY spectra obtained at excess energies EE= 2000-2500 cm-' by monitoring different NO product states are shown in Fig. 1. The PHOFRY spectra can be qualitatively simulated using a model based on assumptions inherent in statistical theories, i.e. the parent eigenstates are ergodic and formation and decay of the activated complex have uncorrelated phase^.^^^^' 942 Conse-quently, the parent nearest neighbour level spacings are assumed to obey a Wigner-like distribution.1*6i10 Since the parent wavefunctions can be described with a basis set of normal modes with random coefficients, we also assume that, in analogy with the case of isolated resonances, the coupling matrix elements to the continuum of fragment chan- nels are random, and that this in turn leads to randomly fluctuating resonance widths obeying a chi-square-like distribution with n degrees of freedom.Here, n is the number of states independently coupled to the zeroth-order for a loose TS n equals 2000 21.00 2200 2300 2400 2500 excess energy/cm-' Fig. 1 Representative NO, TR-visible PHOFRY spectra at EE= 2000-2525 cm-' obtained by monitoring (a) NO(v = 0) Ql,(33.5) and (b) R,,(23.5) transitions, and (c) NO(v = 1) R,,(7.5) tran- sition Resonances and Fluctuations the number of open fragment channels, while for a tight TS n is the number of energeti-cally accessible states of the TS.The amplitude of each resonance, R,(E), is given by *42,49 where Emis the resonance energy, a, is the resonance excitation amplitude and r, is the sum of the partial widths of the resonance m into all final channelsf: The coefficient ,/(r,,J27c) ensures normalization, i.e., SFrnI R,(E) l2 dE = a:. In the case of isolated res- onances, r, can be measured directly from the linewidth and is related to the decay rate k(E) by the uncertainty principle, Tm= hk(EJ271. In contrast, when the resonances overlap, the decay width is an unobservable quantity, and the relationship between (T(E)) (the average decay width) and k(E) is not The goals of the present simulations are: (i) to demonstrate the correlation between the partial and total spectra under conditions similar to those used in recent state-to- state experiments on the decomposition of NO, ,30-32 and (ii) to explore the relationship between the resonance width and the widths of individual spectral features.This is important since extraction of decay rates from line widths is often attempted. In the ~'~~~calculations described by Peskin et ~1. as well as here, a, are chosen as uniform random deviates and the nearest-neighbour level spacings of the resonances obey a Wigner-like distribution. We use a chi-square distribution of decay widths in our simula- tion~,~~despite the fact that Peskin et al. demonstrated a progressive deviation from this form as the degree of resonance overlap increases.42 We find that for a fixed (T(E)),the simulations are not sensitive to the exact form of the width distrib~tion.~~,~~ It has been shown that for a given set of resonances (i.e.for pre-selected arrays of Em, a, and r,) the probability of producing a final statefat energy E, P,-(E), is given by the general form:32v42 where the amplitudes C,, and phases 4fmdefining the projection of the resonance rn on the final statesfare real numbers taken as uniform random deviates in the intervals 0-1 and 0-271, respectively.This expression has been recently obtained from a random matrix version of Feshbach's optical model, where the coupling of the TS to the molecu- lar complex is modelled via a universal random matrix effective Hamilt~nian.~, This Hamiltonian matrix, whose diagonalization provides the resonance eigenstates and widths and describes correctly the average unimolecular decay rate,41*42 can be used in both loose and tight TS cases.42 The simplified description of eqn.(2) yields spectra qualitatively similar to ones obtained by the more rigorous treatment of Ref. 41 and 42. The simulated spectra are compared to experimental PHOFRY spectra obtained in two excess energy regimes: EE= 2000-2525 and 475-650 cm-1.30932 The PHOFRY spectra still contain some averaging despite full N0(211,; v, J, A) quantum state speci- ficity; i.e. we average over the three spin-orbit states of the correlated atomic oxygen fragment (j= 2, 1, 0), as well as over several parent rotational states. (Even with double- resonance excitation it was not possible to excite a single rotational state of For EE= 2000-2525 cm-', we have recorded PHOFRY spectra for 15 different NO final and some of these spectra are shown in Fig.1. Although not all populated NO product channels have been monitored, qualitative conclusions about the total absorption spectrum can be drawn by summing these 15 spectra (Fig. 2). Clearly, the main effects of summation are: (i) an increase in background, (ii) broadening of the peaks, and (iii) a decrease in the range of widths in the summed spectrum. We note that based on the reactive density of states there should be ca. 250 resonances in this 133S. A. Reid et al. I I I I I I 2000 2100 2200 2300 2400 2500 excess energy/cm-' Fig.2 NO, IR-visible yield spectrum obtained by summing 15 individual PHOFRY spectra of NO(v = 0,l; J) obtained at EE= 2000-2525 cm-regi~n;'*-''*~~*~~yet fewer than ca. 20 peaks are observed in either partial or total spectra, It is intriguing that residual structure is apparent in the summed spectrum despite the extensive averaging of seemingly rather different spectra, suggesting the spectra are correlated. In order to simulate PHOFRY spectra in a given energy range, we use eqn. (2) with a pre-selected random set of resonances, whose number reflects the vibronic density of states (p). Resonance widths (with average (r))are assigned from a chi-square-like dis- tribution with n degrees of freedom, where n is the number of energetically accessible TS levels or independent decay channels.Since in the regime of overlapping resonances the decay rate k(E) is not related to (r) simply by the uncertainty prin~iple,~'~~~ the RRKM equation is not used here, and (r),p and n are treated as independent param- eters taken either from calculations, 38960 or chosen to fit experimental data.' 1*29-35 To simulate the spectra shown in Fig. 1, we use n = 20, (r)= 25 cm-' and p = 0.5 per cm--'. The simulations are less sensitive to n and p and more sensitive to (r).Given a set of resonances, each array of random coefficients C,, and q5,m in eqn. (2) generates a synthetic PHOFRY spectrum for a single fully quantum-state-resolved decay channel. To account properly for experimental averaging over oxygen spin-orbit states and parent rotational states, nine single-channel spectra are used to simulate PHOFRY spectra for individual NO quantum states.First, spectra weighted to simulate the contributions from the three spin-orbit states of oxygen are summed. The average spin-orbit ratios are 3P2: 3P1: 3P0= 1 : 0.2 : 0.04 at EEc 2000 ~rn-'.~',~~However, since these ratios are known to fluctuate about this average,33 we assign them at random from the interval cj/g-cjg were cj is the average spin-orbit ratio and g is the fluctuation parameter, which is assumed to be 3. Secondly, three spectra thus summed over the oxygen spin-orbit contributions are averaged to account for incoherent averag- ing over parent rotational states in the double-resonance experiment^.^^ Several repre- sentative synthetic spectra are shown in Fig.3, and the sum of 15 such spectra is shown as the solid line in Fig. 4. Note the qualitative similarities between the simulated and experimental spectra in terms of the number of peaks, range of widths and the differ- ences among individual spectra. Since the number of summed spectra represented in Fig. 4 is large, it is instructive to see the effect of incoherent superposition of the underlying 250 resonances in this spec- tral region, i.e. Resonances and Fluctuations I 1'1' I I I 2000 2100 2200 2300 2400 2500 excess energy/cm-' (b> 11'1' I I 2000 I0 2200 2300 2400 2500 excess energy/cm-' Fig. 3 Four representative simulated spectra corresponding qualitatively to IR-visible PHOFRY spectra of NO, at EE = 2000-2525 cm-'.The parameters used in the simulations, n = 20,(r)= 25 cm-', and p = 0.5 per cm-', have been chosen to simulate best the experimental PHOFRY spectra. The level of incoherent averaging included in the calculated spectra corre- sponds roughly to that involved in the experimental IR-visible spectra when monitoring a single NO level. 2000 2100 2200 2300 2400 2500 excess energy/cm-' Fig. 4 Solid line: sum of 15 individual simulated spectra such as those shown in Fig. 3. Dotted line : spectrum obtained by incoherent superposition of the resonances used in calculating the simulated spectra [eqn. (3)]. S. A. Reid et al. The incoherent superposition is generated by the single sum given by eqn.(3), and is shown as the dotted line in Fig. 4. The two spectra in Fig. 4 are quite similar, indicating tha.t the summation of 15 individual spectra is sufficient to largely average out the effects of phase and interference. Thus, although some differences between the coherent and incoherent superpositions can still be discerned, the random-phase approximation is justified at this level of averaging. Using eqn. (2), we have successfully simulated the experimental PHOFRY spectra in other regions of excess en erg^.^^.^^ In most cases, when a large number of spectra are summed the result reproduces quite well the spectra obtained by incoherent super- position of the underlying resonances, in a similar manner to the spectra shown in Fig.4. In spectra such as those shown in Fig, 3 and 4, as well as in others with (r)p > 5, it is clear by inspection that the resonances are indeed overlapped. The situation is more ambiguous for smaller overlap. Take for example the synthetic spectra simulating the region close to threshold. Fig. 5(a) displays two simulated spectra that correspond qual- itatively to PHOFRY spectra in the region EE = 100-300 cm-'. The parameters in these simulations are n = 8, p = 1 per cm-' and (r)= 3 cm-l. The individual spectra exhibit some differences, yet the sum of 15 such spectra still displays many sharp spectral -. .-5 11'1 11 I I I I I 100 120 140 160 180 200 220 240 260 280 300 excess energy/cm-' IIIIIIIIII 100 120 140 160 180 200 220 240 260 280 300 excess energy/cm-' Fig.5 (a) Two representative simulated spectra corresponding qualitatively to IR-visible PHOFRY spectra of NO, at EE = 100-300 cm-'. The parameters used in the simulations are n = 8, (r)= 3 cm-',and p = 1 per cm- '. (b) Solid line: sum of 15 individual simulated spectra such as those in panel (a). Dotted line: spectrum obtained by incoherent superposition of the resonances used in calculating the simulated spectra [eqn. (3)]. Resonances and Fluctuations features, as shown in Fig. 5(b). Note that for this lower degree of overlap, both the summation of the individual spectra and the incoherent superposition of the 225 under-lying resonances produce similar results; i.e. ca. 30 peaks, many of which appear rather narrow.It is instructive therefore to extract ‘widths‘ for these spectral features by treat- ing them as single Lorentzians. The average width of the individual spectral features is 1.8 & 0.1 cm-’ in the state-selected spectra and 2.6 cm-l in the summed spectrum, a substantial broadening. The average widths obtained from features in the state-specific spectra are thus much narrower than the input average width (3 cm-I), reflecting both the effects of interference and a bias in favour of narrow peaks when selecting peaks that appear ‘isolated’. Similar behaviour is observed in the experimental results. For example, PHOFRY spectra obtained with double-resonance excitation exhibit a greater number of narrow features than those obtained with one-photon e~citation.~~*’~ Also, spectral features in PHOFRY spectra obtained by monitoring the 0 fragment are on average broader than those obtained by monitoring single NO ~tate~.~~*~~v~~,~~This reflects the fact that the former arise from superposition of spectra correlating with many NO states.54 We also point out that Peskin et aE.have recently shown that when resonances overlap, the average width cannot be linearly related to the average rate, but in fact the calculated rate is always smaller than that inferred from the average ~idth.~’ Thus, in the regime of overlapping resonances (i.e. beyond the tunnelling regime) it is unjustified to extract average unimolecular decay rates from the widths of spectral features, even when many features in the state-specific spectra appear sharp and ‘isolated’.In summary, both the experimental PHOFRY spectra and the simulated spectra show clearly the effects of interference due to overlapping resonances. These effects manifest themselves as marked differences among spectra obtained for different NO final states. The widths of the spectral features in general do not correspond to the widths of the underlying resonances. Summation of a sufficiently large number of PHOFRY spectra yields a spectrum similar to that obtained by incoherent superposition of all the underlying resonances ;thus the random-phase approximation is usually justified when dealing with averaged spectra. The extraction of average decay rates from linewidths in the regime of overlapping resonances is unjustified both on experimental and theoretical grounds.4. Fluctuations in the Alignment Parameter, Af) Having established that the linewidths in state-selected spectra are not a good measure of the dissociation rates, we need another way to explore the existence of state-to-state fluctuations in the rates. Direct time-resolved measurements require short photolysis and probe pulses, and thus typically yield rates averaged over a range of photolysis energies and summed over many product channels. However, time-dependent informa- tion can also be obtained indirectly from time-independent measurements via corre-lations in vector properties of fast dissociating m01ecules.~~-~~ In order to relate the decomposition lifetimes to the rotational period of NO,, we measured the state-specific alignment parameter Ah2) of the NO fragment.Fluctuations in the value of Ah2) as a function of EEcan signify fluctuations in the state-to-state rates. Fluctuations in the rates of unimolecular reactions in the regime of isolated reso- nances have been inferred from linewidth fluctuations in the unimolecular reactions of formaldehyde and the methoxyl radical. Moore and co-workers have used Stark level crossing spectroscopy to examine the decomposition rates of single eigenstates in formal- dehyde.66 The observed rates varied by over one order of magnitude for states differing in energy by only fractions of a wavenumber; however, average values of the state- specific rate constants were reproduced by RRKM calculations with no adjustable parameters.More recently, Temps and co-workers have examined dissociation lifetimes S. A. Reid et al. of individual rovibrational eigenstates of CH,O in the vicinity of the classical barrier to H +-H,CO fragmentation, using a triple resonance stimulated emission pumping (SEP) scheme combined with LIF.67 Strong fluctuations in the state-specific linewidths with small variations in energy were observed. When the linewidths were converted to rates, the average rate were well reproduced by RRKM calculations. A mathematical treatment of rotational alignment has been developed in several paper^.^^-^'*^^ The alignment parameter Ab2)characterizes the correlation between the vibronic transition moment p in the parent and the fragment angular momentum J.Ab2)= 0.8 and -0.4 represent, respectively, the limiting cases of fragment J vector paral-lel and perpendicular to the parent vibronic transition In NO,, the 12B, t 1,A, transition moment lies in the molecular plane; thus, we expect A‘,,) to approach the limiting value of -0.4 for dissociation on a timescale fast compared with parent r0tation.~~3~’Fig. 6 shows the reduction in Ah2) expected for longer decomposi-tion lifetimes at several NO, rotational temperatures. This calculation follows the treat-ment of and assumes a Boltzmann distribution of ground-state level populations. For each decomposition rate, as the temperature increases (i.e.the average period of rotation decreases) the A‘,,) value deviates further from its limiting value. Our measurements were carried out mostly at a parent rotational temperature of 4 K. ‘To obtain Ah2) from our experimental LIF intensities, we follow the treatment of Di~on,~’in which the integrated intensity I(G, Qi) of a rotational line in a photofrag-ment fluorescence excitation spectrum is expressed as : where G refers to the specific photoysis-detection laser geometry, Qi specifies the quantum numbers of the initial state which has population Ni, C is a constant, S is the rotational line strength factor, bo and b, are constants which depend on J, the rotational branch, and the specific laser-detection geometry, and Bb2)(02)is related to the rotational alignment by 8b2)(02)= 5/4Ab2).To determine Bb2)(02)(and thus Ah2))for a given product NO quantum state, we measure the NO(A ’Z+ +-X 2111,2)LIF intensity for different geometries (I and I1 as denoted by Dixon)6s on a fixed transition.This was accom-plished by rotating the photolysis laser polarization, as described in Section 2. As seen in Fig. 6, at Tot= 4 K a sharp dependence of Ab2) on decay lifetime z is obtained for z = 3-15 ps. According to the k(E) measurements of Ionov et uE.,’~~ these lifetimes correspond to EE < 350 cm-’. However, at these excess energies only low J -O.‘O 1 15 K -0.15 -1 -0.20 --0.25 --0.30 --0.35 --0.40 0.1 1 10 100 Iifetime/ps Fig. 6 Dependence of the alignment parameter Ab2)on the dissociation lifetime of NO, at several rotational temperatures Resonances and Fluctuations levels of NO can be populated for which hyperfine depolarization is a prominent effect that markedly reduces the measured alignment.63 The choice of EEin our measurements represents a compromise between these two conflicting requirements [i.e.low EE and high NO(J) states]; we therefore selected EE= 200-500 cm-l where the lifetimes are 2.5-1.4 ps, re~pectively,’~~ and J = 10.5 is the highest J level populated at the lowest EE. Also, at T,,, = 4 K there is a larger change of Af) with z in this range than at lower temperatures (Fig. 6). Fig. 7 displays both a series of alignment measurements and a state-specific PHOFRY spectrum obtained by monitoring the Q11( 10.5) transition of N0(211,,,, v = 0).To correct for hyperfine depolarization we have measured Ah2) for both Qll(10.5) and Q11(20.5) at higher EE. In this normalization we assume that depo- larization is not significant for J,, > 20.5. This conclusion is confirmed by a study of NO, dissociation at EE= 3038 cm-1,22 where maximum alignment (i.e. Ah2)z -0.4) is obtained for NO(v = 1,J > 17.5), while a clear reduction in Ah2)is observed at lower J. The Ah2) values for selected peaks in the PHOFRY spectrum, corrected and uncor- rected for hyperfine depolarization, are summarized in Table 1. The measured Ah2) values fluctuate with energy in a reproducible manner. The corrected values range from ca. -0.3 at EE= 240-280 cm-l to ca.-0.35 at EE= 400-450 cm-l, in agreement with the values reported by Miyawaki et al. in this energy region.35 Similar average values and pattern of fluctuations were also obtained when monitoring NO(J = 6.5) and cor- recting for depolarization. The error bars, k lo,represent the range of values obtained after many repetitions and during a period of several weeks. They refer to the precision of the absolute value for each Ah2).Although the absolute values will sometimes vary from day to day, as a result perhaps of incomplete polarization or slight saturation effects, the relative values, which represent the fluctuations, are very reproducible and thus more robust than the absolute values. There are several points worth discussing regarding the absolute values of Ah2) and the nature and magnitude of their fluctuations.The limiting values of the averaged Ah2) (-0.38, Fig. 6) is not reached even after correcting for depolarization. One reason for the deviation from the limiting value may be the simultaneous excitation of the perpen- dicular 1 2Bl t1,Al transition, e.g. a 5% contribution from this transition can explain our reduced alignment. However, this is not considered very likely; all the experimental -0.15 -0.2 -0.25 -SO T -0.3 -0.35 -0.4I I I -7-7 1 200 250 300 350 400 450 5 1 excess energy/cm-’ Fig. 7 One-photon PHOFRY spectrum (solid line) and measured Ab2)values (filled circles) in the range EE= 200-500 cm-’ obtained by monitoring the Q,,(10.5) transition of NO(u = 0).The Ab2) values are corrected for hyperfine depolarization as described in the text; error bars correspond to *la. S. A. Reid et al. Table 1 Alignment parameters of N0(211,,2;u = 0; J = 10.5), and relative translational energies and velocities of fragments A(2) cm-'EEI cm-Ed uncorrected correctedn %I/A ps-' 242 41 -0.208 f0.022 -0.313 f0.035 3.1 258 57 -0.208 & 0.010 -0.313 f0.015 3.6 282 81 -0.192 f0.020 -0.289 0.030 4.3 326 125 -0.260 k0.020 -0.392 f0.030 5.3 364 163 -0.196 f0.017 -0.295 & 0.026 6.1 379 196 -0.231 0.005 -0.348 & 0.008 6.7 408 207 -0.214 & 0.019 -0.323 f0.028 6.9 452 25 1 -0.220 f0.020 -0.332 & 0.029 7.6 a Corrected for hyperfine depolarization. evidence points to the 2B, state as the only bright state efficiently coupled via the 1,A, state to the dissociation continuum.Moreover, the band positions of the 1,B, t12A1 systems are well known,70 and dispersed fluorescence studies show that highly localized ,B, vibronic states exist even at excitation energies as high as 22000 cm-I (i.e. ca. 3000 cm-I below Thus, any possible influence of the 2B, state would be localized to distinct known wavelengths characterizing absorption to this state. Another probable cause for the reduced alignment is rotation of NO, prior to disso- ciation, which can change the plane of product NO rotation relative to the initial NO, molecular plane. Based on Fig. 6 and considering the average decomposition lifetime of NO, (i.e.2.5-1.4 ps), we do not expect a significant reduction in Ah2)at EE= 200-500 ern-.' and To,NN 4 K. However, we must also consider the timescale for the final NO(,n,, J) + O(3Pj) product separation as a function of internal energy, ENo(J)+ Eo(j). For the reasons specified above, we preferentially monitor the highest NO states allowed energetically, which are states with low relative recoil velocity. The centre of mass (CM) translational energy available to the recoiling products, E,,, is given by energy conservation : JAt EE> 359 crn-', ND(2111,2; = 10.5) can be correlated with oxygen atoms in both the ground (3P2)and excited (3P, or 3P0)spin-orbit states. However, the average popu- lations of O(3P,) and O(3P0)are small (even though state-to-state fluctuations in the spin-orbit populations are significant both as a function of EE and the monitored NO quantum ~tate),,~.~~ and thus their contributions, which can further reduce E,, , are ignored here.The values of E,, calculated for NQ(211,,2;J = 10.5) + O(3P2)are included in Table 1. We also calculate the NO-0 recoil velocity, vrel= [~EJP]'/~(where p is the reduced mass of the NO-0 system), and these values are listed in Table 1as well. The recoil velocities, assuming dissociation solely to N0(2111,2,J) + O(3P2),are in the range 3.1-7.6 A ps-'. If long-range interactions are important for NO,, as our NO rotational energy distributions and PST calculations indi~ate,~' then some correlated orbital motion involving the NO-0 complex may persist at large separations (i.e.for a longer timescales than indicated by the measured decay time z), thereby altering the plane of rotation of NO relative to that of NO, at the time of excitation. This effect may reduce the alignment from the expected value based solely on k(E). A dependence of vector properties on the internal energy of the NO fragment has also been seen in measurements of the recoil anisotropy parameters, brec,in NO, photo-dissociation. These measurements, which yield in the limit of fast dissociation the direc- tion of u relative to p, show a reduction of Prec with increasing ENo(J) both near Resonances and Fluctuations threshold,36 and at higher EE.72Thus, it appears that both Ah2)and Preccan be reduced when the relative separation velocity of the fragments is small.It will be interesting to verify this effect in molecules which dissociate from isolated resonances. For example, in the photodissociation of HCO, isolated resonances of varying widths are ~bserved;~~,~~ would the alignment and recoil anisotropy parameters for each resonance vary as a function of the monitored state of the CO product? The reduction in the values of Ah2) could also derive from preferential dissociation from high parent rotational levels. The rotational temperature in our experiments (4 K) was estimated from analysis of the structure in PHOFRY spectra right at Do,32 which yields values consistent with both direct LIF measurements of NO,, and the rotational temperature of NO contaminant in the beam.Nevertheless, we also checked whether the reduction in alignment may arise from preferential contributions from high rotational states of NO, at the selected values of the photolysis energy for which the Ab2) values were determined. To this end, we compared PHOFRY spectra obtained with 4% and 0.5% NO, in He (T,,, = 8-10 and 4 K, respectively). The peaks in the spectra did not change in shapes or relative magnitude; only some low-intensity ‘background’ peaks diminished in intensity in the more dilute samples. Our Ah2)measurements were taken at prominent peaks which did not display a marked dependence of their structure on the dilution. We thus discount an effect due to preferential bias from high parent rotational levels on our results. In the experiments reported here, the 0.5% NO,-He mixture was used because it gave colder T,,, .Finally, we comment on the magnitude of the fluctuations in Ah2). As mentioned above, the range of the average NO, decay lifetimes at EE= 240-450 cm-’ is (7) = 1.4-2.5 ps. However, even an order of magnitude fluctuation about these lifetimes will correspond to only a modest change in Ab2). Let (7) = 2.0 ps; the corresponding limiting value of Ah2) is -0.38 (Fig. 6). An increase in z by a factor of five will yield Ah2)= -0.27 for z = 10 ps, whereas a decrease by a factor of five (to 0.4 ps) corresponds to the asymptotic value Ab2)= -0.4. Thus, under our experimental conditions, an order of magnitude fluctuation about the average lifetime will only lead to a change of Ah2) between -0.4 and -0.27; moreover, our measurements are sensitive mainly to lifetimes longer than the average.In addition, with one-photon photolysis fluctuations tend to be further diminished due to incoherent averaging over several excited NO, rotational level^.^^^^^ Thus, the magnitude of the fluctuations in the Ah2) values reported here seems reasonable, and we believe that they reflect fluctuations in the unimolecular decay rates. Measurements using IR-visible double resonance excitations were also attempted, since they define the initial NO, state better; however, they resulted in poor signal to noise ratios. 5. Adiabaticity beyond the Transition State: Fluctuations in the NO Product State Dis- tributions In this section we comment on the relation between fluctuations and oscillations in PSDs and the statistical descriptions of unimolecular reactions.To test the validity of statistical theories, both microcanonical unimolecular reaction rates and PSDs are needed. In the absence of an activation barrier, as is common when the products are free radicals, such comparisons are more complex. The rate is usually calculated by finding variationally the dividing surface of minimum flux perpendicular to the reaction coordi- nate (the TS) for each EE.47It is now well established that the TS moves inward from atop the centrifugal barrier as EEin~reases.~~*~~?~~Thus, in a triatomic molecule the TS levels will resemble those of a free rotor at low EE,and be best described as hindered rotors or low-frequency bends at higher EE.38955*75776 While rate calculations are rather straightforward even for a barrierless dissociation, the calculated PSDs must be examined more carefully.Near Do, where the TS is very loose and its levels resemble those of the products, PST becomes the natural choice. This S. A. Reid et al. 141 theory apportions product populations as per the degeneracies of product quantum states, subject only to energy and angular momentum constraint^.^^ As the TS tightens, but a barrier is absent, the situation is less clear and the issue of adiabaticity beyond the TS arises; i.e. do product states still exchange energy beyond the TS? In other words, how important are exit channel interactions beyond the TS? The current view is that product vibrations, especially those of a diatomic fragment, evolve rather adia-bati~ally.~~?~~Rotational and spin-orbit states usually have small energy separations; do these evolve adiabatically, and if so where along the reaction coordinate does this happen? What are valid tests for statistical behaviour of the PSDs? We shall show that fluctuations and oscillation patterns in the PSDs provide insights into these important issues. Although the PHOFRY spectra and NO product state distributions are complemen- tary ways of viewing the same information, the latter sometimes better reveal patterns and regularities.We have obtained complete (ie.rotational, vibrational and spin-orbit) state distributions of the nascent NO fragment for dissociation at specific excess energies in the range EE= 0-3038 cm-1.29933 In this section we examine the oscillations and fluctuations in the NO PSDs, compare the PSDs with statistical predictions, and examine the implications of the observations to the dissociation mechanism and to properties of the TS. We focus here on NO rotational state distributions, and emphasize the excess energy region where the TS is expected to tighten signficantly. It appears that for each NO vibration, the TS is loose near its appearance threshold and tightens progressively as EE in~rea~e~.~~*~~~~~*~~Thus, we examine NO(v = 1) rotational distributions from near threshold (at EE= 1876 cm-l) to 1160 cm-l above this threshold.Displayed in Fig. 8 are N0(2111,2,v = 1) rotational distributions for the two (A’, A”) A-doublet states of N0(211,,2) obtained using one-photon excitation. Fig. 9 presents semilog plots of the distributions summed over A-doublet levels, compared with PST calculation^.^^^^^ Inspection of each A-doublet level distribution reveals fluctuations in the rotational populations; however, the behaviour at low and high excess energies is different. Near the v = 1 threshold, rotational distributions for each A-doublet component fluctuate bl 2 0.5 2.5 4.5 6.5 8.5 0.5 2.5 4.5 6.5 8.5 10.5 12.5 14.5 0CL I I 0.5 5.5 10.5 15.5 20.5 2: 5 0.5 5.5 10.5 15.5 20.5 25.5 JNO Fig.8 N0(2rI,,,; u = 1) rotational distributions obtained in one-photon dissociation of NO, at EE = (a) 2061, (b) 2200, (c) 2700 and (d) 3038 crn-’. The vibrational energy of NO(v = 1) is 1876 cm-‘. Distributions for the two A-doublet states (A’, A and A”, 0)are shown. Resonances and Fluctuations 6 5 6-(a> (b) -F+o 5 0 100 200 300 100 200 300 cu v2 v56 3 S -4 4-2 2-0 I 0.1. 0 200 400 600 800 lo001200 I I1 I NO rotational energy/cm-' Fig. 9 Semilog plots of the rotational distributions presented in Fig. 8, summed over the A-doublet states of NO and compared with PST calculations (solid lines) considerably, but the fluctuations are significantly diminished when summing over the A-doublet levels. These observations are also characteristic of NO(v = 0) rotational state distributions at EE= 0-400 cm-',29y32 and are consistent with near threshold PHOFRY ~pectra.~~?~~ 1 rotational distributions obtained at higher excess The v = energies exhibit different behaviour.Here, the rotational distributions show pronounced oscillatory structures which are well reproduced for the two A-doublet states [as well as for spectra obtained when monitoring NO(2113,2)],2g-32 and therefore are not dimin- ished upon summation of the fine-structure states. This behaviour is particularly notice- able in Fig. 8(d) and 9(d) for NO(v = 1) at 1162 cm-' above its threshold, and also in NO(u = 0) rotational distributions obtained at EE> 1800 cm-' (e.g. Fig. 3 and 4 of Ref.31). The oscillatory structures depend sensitively on EE.They are more easily identified in distributions obtained via one-photon excitation than in the double-resonance experi- ments, since fast fluctuating structures are averaged out by incoherent superpositions of distributions correlating with several parent rotational states. It is instructive to compare the product state distributions with predictions of sta- tistical models, and discuss the implications of our findings to the TS. Studies of the decompositions of NCN076-78 and CH2C055,79t80 have shown good agreement with PST for diatom rotational distributions, indicating that the basic assumptions of the statistical theories are justified even in the fast dissociation of these small molecules.However, each quantum state of the diatomic fragment is typically correlated with many states of the other fragment, so averaging is inherent and no fluctuations or oscillations are observed. The distributions shown in Fig. 9 all agree fairly well, on the average, with the predictions of PST. We have shown in previous work that the prominent oscillatory structures observed at high EE can be modelled qualitatively by mapping bending-like wavefunctions associated with a tight TS into free rotor states of the NO fragment.32y81 Why, then do the oscillations consistently cluster about the statistical predictions of PST, which assumes a very loose TS? This can be reconciled by recognizing two aspects of product evolution. First, the range of rotational excitations allowed in the Franck- 143S. A.Reid et al. Condon mapping is a sensitive function of the TS bending angle and The TS parameters used in the mapping calculations for NO, are based on ab initio calcu-lation~,~~?~'and they happen to give rise to rotational excitations that span the full range allowed by energy conservation at all excess energies st~died.~~,~' The qualitative agreement observed with PST even at high excess energies thus reflects in part the spe- cific geometry of the tight TS, and does not constitute a valid test for the looseness of the TS. This is an important recurrent theme; distributions which can be fit by a sta- tistical model do not prove that such a model correctly describes the dissociation mechanism.A second reason why PST-like distributions are observed even when the TS has tightened considerably is that exit-channel interactions beyond the TS can wash out oscillations and lead to population of all levels allowed by energy and angular momen- tum conservation. This effect is probably more important at lower energies, where the relative velocities of the recoiling products are small. Thus, the PSDs can be further modified beyond the TS via exit-channel interactions governed by a subtle interplay between the size of the energy quantum to be transferred and the relative recoil velocity. In barrierless reactions at low excess energies, the relative recoil velocities are small (see Table 1) and only low J levels (whose energy separation is small) are populated; thus, rotational energy transfer in the exit channel should be quite efficient.As the excess energy increases, the recoil velocity of the fragments increases, and the probability of rotational energy transfer in the exit channel decreases. Products in high rotational levels (whose energy separation is large) will undergo less efficient energy transfer, while products in low rotational levels will have higher recoil velocities and their evolutions may be well described by the sudden approximation. Previous studies have shown that the long-range attractive forces in NO, may extend to rather large 0-NO internuclear separation^,^^ thereby facilitating energy transfer. Thus, we expect that for each product vibration, at relatively low excess energies rotational energy transfer beyond the TS will be efficient and will smooth over osciliations arising from mappings of TS wavefunc-tions, resulting in a more PST-like behaviour.The oscillatory structures will remain prominent at higher excess energies where the sudden approximation is more justified. The NO(v = 1) distributions suggest that the transition between the two regimes occurs between 400 and 1000 cm-'. However, more work is clearly needed to explore the importance of long-range interactions beyond the TS. In summary, the progressive tightening of the TS is revealed by the change in fluc- tuation patterns in the NO rotational state distributions from random to oscillatory as EE increases. When the TS is loose (i.e. low EE), it is not surprising that the rotational distributions are well described by PST.When the TS tightens and its levels become bending-like, agreement with PST may signify either eficient exit-channel interactions beyond the TS and/or mappings of wavefunctions of a TS whose geometry happens to produce PST-like distributions. In the case of NO,, it is possible that both factors contribute to the average PST-like appearance of the rotational distributions, while the appearance of prominent oscillations whose shapes depend sensitively on EE are the prime indicators of the tightening of the TS. The similarity between oscillatory patterns obtained in rotational distributions of the two spin-orbit states of NO has led us to suggest that the spin-orbit distributions are fixed last, at long range, where the potential curves correlating with different spin-orbit pairs are c10se.~' Recent ab initio calculations confirm this interpretati~n,~~ while explaining also why the lowest energy channel, N0(211,!2) + O(3P,), is fav~ured.~~~~~ Thus, the PSDs in NO, decomposition indicate that a hierarchy of adiabaticity exists.NO vibrations are fixed first, at the shortest 0-NO separations, followed by the rota- tional distributions with their typical oscillatory patterns at higher EE. The different electronic channels correlated with the different spin-orbit states of NO and 0 are determined only at large internuclear separation^.^^ Resonances and Fluctuations 6. Concluding Remarks The unimolecular reaction of NO, exhibits fluctuations and oscillations evidenced in the PHOFRY spectra, the state-specific rates, and the PSDs.However, only those manifest- ed in the state-selected PHOFRY spectra as fluctuations in the line positions and widths of spectral features obtained when monitoring different final states can be ascribed con- clusively to overlapping resonances. Fluctuations in rates and PSDs have also been observed in dissociation from isolated resonances. For NO, , comparisons between experimental and simulated PHOFRY spectra show that the effects of overlapping reso- nances are best revealed at moderate excess energies. Our simulations show that effects due to interference are most prominent when the overlap parameter (r>p is large (i.e. at high EE)but the number of independent decay channels is modest (e.g.n = 10-30). In NO,, this corresponds to the excess energy range EE= 200-3000 cm-l, where most of our experiments were carried out. It is intriguing that similar conditions (ie. high EE and modest n) are also required for the observation of prominent oscillations in the rotational distributions that depend sensitively on excess energy. These oscillatory structures derive from Franck-Condon mappings of bending-like TS wavefunction into rotational distributions, and have been interpreted as a manifestation of the tightening of the TS. Their experimental observ- ation depends on the extent of exit-channel interactions, which are presently not taken into account in the simulations.32942 Energy transfer beyond the TS may wash out fluc- tuations and oscillations, particularly at low excess energies.Thus, agreement between the measured rotational distributions and PST predictions does not necessary imply a loose TS. Energy transfer beyond the TS and specific geometries of a tight TS may both lead to the appearance of PST-like rotational distributions. More work is clearly needed in order to understand long-range forces and exit channel interactions in simple bond- fission reactions. We wish to thank our collaborators in these experiments, Martin Hunter, Dan Robie, Craig Bieler, Jenny Bates-Merlic, Uri Peskin and William H. Miller for their enthusiastic and valuable contributions to the understanding of the complex behaviour of NO2 dis-sociation.We are indebted to Martin Hunter for providing the data used in Fig. 8 and 9. We benefited greatly from discussions with Curt Wittig, Howard Taylor, Jurgen Troe, Vladimir Mandelshtam and Reinhard Schinke. This research is supported by the US National Science Foundation, the Army Research Office and the Department of Energy, Basic Energy Sciences. References 1 J. Miyawaki, K. Yamanouchi and S. Tsuchiya, J. Chem. Phys., 1994,101,4505. 2 (a)A. E. Douglas and K. P. Huber, Can. J. Phys., 1965, 43, 74; (b) A. E. Douglas, J. Chem. Phys., 1966, 45, 1007. 3 (a)D. Hsu, D. L. Monts and R. N. 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ISSN:1359-6640    

DOI:10.1039/FD9950200129

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Fu~u~u~ 1995,102, 147-166 D~SCUSS., State-resolved Dynamics in Highly Excited States of NO2: Collisional Relaxation and Unimolecular Dissociation Bernd Abel,* Hilmar H. Hamann and Norbert Lange lnstitut fur Physikalische Chemie der Universitat Gottingen, Tammannstr. 6, 37077 Gottingen, Germany Recent state-resolved investigations of unimolecular dissociation and col- lisional relaxation of NO, at chemically significant internal energies are out- lined. Two powerful double-resonance techniques are described which permit the investigation of these processes on a quantum-state-resolved level of detail. A sequential optical double-resonance technique with sensitive laser-induced fluorescence detection has been employed for assignments of the molecular eigenstates of NO, in the energy range at 17 700 cm- '.Subse-quently, we were able to measure state-to-state rotational and vibrational energy transfer in NO,-NO, self-collisions using a time-resolved double- resonance technique. From these data, direct information about propensity rules and intermolecular interactions for rotational and vibrational energy transfer in NO, self-collisions at high vibrational excitation could be obtained. In addition, we have used a folded high-resolution V-type double- resonance technique in a free jet to access and to assign rovibronic states of NO, above and below the dissociation threshold, E,. From the double- resonance spectra, linewidths at around 25 130 cm-' as a function of inter- nal energy, E, and total angular momentum, J, could be extracted.Specific rate constants, k(E, J), calculated from the homogeneous linewidths, have been compared with results from SACM calculations, predictions from a statistical random matrix model, and ps time-domain measurements. 1. Introduction The mechanisms of collisional energy transfer and unimolecular dissociation of small molecules at high levels of vibrational excitation are of fundamental interest for the understanding of reaction dynamics and chemical reactivity, in particular, in situations where collisional energy transfer competes with unimolecular reaction.' In this article, we will discuss powerful double-resonance techniques in the frequency and time domain, which can be employed for the investigation of these processes at chemically significant energies at a state-resolved level of detail.A particularly appealing aspect of these tech- niques is their ability to provide information about state mixing, 'good' quantum numbers and a detailed insight into relaxation and dissociation mechanism^.^-^ Several experimental approaches today permit the investigation of unimolecular reaction from single reactant rovibrational states.2 Most of these use double-resonance techniques2-l3 or molecular beam methods'4915 for the preselection of molecular quantum states. Double-resonance techniques, in general, have the advantage over jet techniques that they can prepare molecules in a wide range of rotational states, limited only by the room-temperature Boltzmann distribution. This allows detailed investiga- tions of rotational effects on unimolecular dynamics.147 Highly Excited States of NO2 In order to calculate and predict unimolecular reaction rates of polyatomic mol- ecules at high internal excitation by exact quantum mechanical,16 trajectory17 or sta- tistical calculation^,^ 871 g the complete molecular potential-energy surface (PES), or its main features have to be known or calculated. Recently, a few important theoretical concepts and developments, including quantum reactive scattering,l6P2' and wave-packet propagation calculations,21 as well as calculations of the global potential-energy surface22 have enabled a detailed quantum-mechanical treatment of some small systems.However, in many cases the assumption of a statistical single-exponential decay of all quantum states is a fair approximation (and the only feasible way of calculating rate constants), such that decomposition rates of polyatomic molecules can be calculated from the molecular density of states, p(E), and the number of open reaction channels, W(E)(both depend sensitively on details of the PES), via In this case individual (channel) specific rate constants k(E, J) as well as product-state distributions can be calculated using a state-resolved statistical theory like the statistical adiabatic channel model (SACM).". l9 The assumption that the dissociation proceeds statistically can be subjected to stringent tests, even without knowing the exact potential-energy surface, when state densities at the reaction threshold and state-resolved lifetime data from time and frequency domains are available close to E, ,where only the first few open channels or transition-state levels are involved.In addition to reaction rates, the mechanisms of collisional energy transfer of small molecules at high levels of vibrational excitation are of considerable importance for reaction dynamics and kinetic^."^'^^ The basic experimental approaches which have been used to study collisions at chemically significant energies tend to fall into two general categories. In many of the studies in the past, chemical activation or internal conversion was used initially to prepare highly excited large molecules of essentially arbitrary complexity.They monitored the energized species or their collision partners and extracted quantities such as the average energy transferred in a collision, (E), and the second moment of energy transfer, (E2), over a wide energy range as a function of the complexity of the parent molecule and the nature of the collision partner using time- and energy-resolved pumpprobe techniques. Hippler and Troe, and Flynn and Weston reviewed the state-of-the-art experiments in this field in Ref. 3 and 23. In the second approach relatively small molecules, whose quantum states can be resolved, were pro- duced and probed in highly excited states by various double-resonance tech-nique~.~*~~-~*In spite of the fundamental interest, only a very limited number of direct, state-resolved experimental data on the relaxation of polyatomic molecules at high vibrational excitation exist.This resulted mostly from experimental dificulties in the state-selective preparation and the detection of the energized molecules. However, recent laser-based, double-resonance-type experiments on C2H2,25 HCN24 and H2C0,26 which directly monitored the energy and population redistribution (after collisions) within the highly excited polyatomic molecules, provided a first picture of the collisional energy transfer at energies between 6000 and 11000 cm-'. In contrast, at low internal energies detailed studies are much more common, leading to a consistent physical picture of inelastic processes in this regime.28931 In general, state-to-state cross-sections for inelastic collisions can be derived from formal scattering theory or classical trajectory calculation^.^^^ While straightforward, 7923 such calculations can be quite tedious, especially, for heavy systems with many degrees of freedom and large quantum numbers.In any case, an accurate intermolecular poten- tial function is required if the calculations are to yield meaningful results. Such poten- tials are by no means well particularly if one of the collision partners is a B. AbeE et al. 149 radical or in an electronically or vibrationally excited state. For these reasons, a number of attempts have been made during the past several years to find semiempirical system- atizations or condensations of state-resolved data, in which a small number of measured data would suffice to determine an entire state-to-state array.Empirical scaling relations based solely on the magnitude of the energy defect in a rovibrational energy exchange have been successful in a variety of systems. The simplest approach to scale the inelastic collision rates connecting initial and final states is one in which the rate decreases expo- nentially with the amount of energy transferred in a ~ollision.~~~~~ A power law in AE is an empirical alternative to the EGL for fitting the experimental data.33 In contrast to the preceding scaling laws, which are statistical in nature and based on the amount of energy transferred, are the angular momentum scaling laws, which are derived from expressions based on the sudden approximation in inelastic scattering the~ry.~~-~~ The ultimate goal of such a data condensation or a data inversion is always to obtain direct information about propensities in vibrational and rotational relaxation, intermolecular interactions and information about the intermolecular potential surface itself at high internal energies.We have chosen the NO, molecule as the subject of our studies because it is an excellent (realistic) model system for quantum-state-resolved investigations of unimolecular dissociation and collisional relaxation of small molecules (radicals) at chemically significant internal energies. The spectroscopy and photodissociation dynamics of this molecule have been extensively studied in the past, but not necessarily completely understood. The spectroscopy of NO,, which challenged spectroscopists for many decades, has been studied more than that of any other small mole~ule.~~-~~ In addition to spectroscopic studies, a large number of photodissociation experiments in the frequen~y’~.~~-~’ and time domain48 have been carried out in order to understand the nature of the unimolecular dissociation process, product-state distributions’ 2*43946 and related quantities such as average densities of The NO, molecule has also been the subject of numerous non-state-resolved energy-transfer measurements.The competition between unimolecular reaction and energy transfer was investigated in a series of experiments by Troe and co-~orkers~~ some time ago.In recent experimental studies, single-photon excitation was employed and the time evolution of the system was followed by time-resolved IR emission,50a fluorescencesob and photoacoustic detec- tion.” These experiments provided quite detailed information about vibrational relax- ation processes, but they were unable to provide sufficient resolution to distinguish individual rotational states. 2. Experimental 2.1. Double-resonance-detected Collisional Energy Transfer and Assignments of Eigenstates Fig. 1B shows the sequential optical double-resonance technique employed for the assignment of mixed eigenstates. This technique was originally developed by Shibuya, Obi and c~-workers.~~*~~ Two laser pulses for the preparation and probing of the popu- lated states were employed.29 Double-resonance signals were monitored by observing the UV fluorescence from the final state [2,B,(O, 0, O)] to the ground state (X2A,).A complication of the double-resonance scheme (Fig. 1, box) is, that the first excitation step., in general, mostly prepares intermediate eigenstates with small Franck-Condon factors to any level of the excited 2 ,B, electronic manifold. Therefore, strong, detectable double-resonance signals arise from perturbations and state mixing, where optically bright zero-order states are coupled to other (optically dark) zero-order states (e.g. C ’A,, X ,A,) with more favourable Franck-Condon factors to the 2 ,B2 state. Optically dark states can, in principle, be discovered with this technique from isoenergetic scans of both lasers (vl + v2 = constant).This technique is similar to the approach of Crim and Highly Excited States of NO2 2 *B~states strongly perturbed predissociating J,N ’, K’ A \\ / J“, N”= 2,K“ = 0 J”, N” = 0, K” = 0 x 2A, Kla = K”,4-1 N”+ 1 N‘‘ N”-1 LL N::+ 1 N bright state dark state ”’-1 X 2A, Fig. 1 A, V-Type (fluorescence depletion pumping) double-resonance scheme. B, Sequential double-resonance scheme (for details see text). co-workers2’ and Orr and co-worker~.~’ In the experiments, two kinds of double- resonance spectra could be obtained for a single unknown intermediate rovibronic level.40 The A,-scanned spectrum was obtained by fixing Al and scanning A2. Alterna-tively, the A,-scanned spectrum could be measured by fixing A2 and scanning Al.The conclusive assignment of the intermediate state could be obtained independently from the two spectra. With a variable time delay between the two laser pulses, the growth and decay of population in initially prepared or neighbouring states coupled to the first one by collisions could be monitored. The fluorescence lifetime was determined for the eigen- states in this energy region and found to be long compared with the timescale of the e~perirnents.~~The experiments have been performed in a (quasi)-static stainless-steel cell (with a slow flow) at 300 K. The excitation and the probe laser pulses were obtained from two commercial dye lasers (Lambda Physik, bandwidth: ca.0.03 cm-’ in the high- and 0.2 cm-’ in the low-resolution mode) pumped by an excimer and an Nd : YAG (second harmonic) laser. The two parallel polarized laser beams at A, and 1, counter-propagated collinearly through the cell. The UV-fluorescence emitted from the central part of the cell was collected by two quartz lenses (fl1.2) and imaged onto a variable B. Abel et al. 151 & -scanned double resonance spectrum A* ~l.l'l'I'l' 445.7 445.8 445.9 446.0 446.1 hlnm A,-scanned double resonance spectrum * I I I I I I I I 1 r--a 563.2 563.4 563.6 563.8 564.0 Vnm Fig. 2 ill (lower trace) and A2 (upper trace) scanned double-resonance spectra. The spectra were obtained by fixing one wavelength on a transition marked with an asterisk (*).The assignments for the intermediate state in this case are N(&) = 3, N(R,) = 4,K = 0, J = 3.5. For details see text. pinhole and a photomultiplier (EM1 9345). The signal was pre-amplified, integrated in a boxcar integrator (EG&G), and a microcomputer collected and processed the signals. 2.2. Double-resonance Spectra and Linewidth in a Free Jet For the state-resolved linewidth measurements we have used the double-resonance scheme depicted in Fig. lA.47 In this 'V-type' double-resonance scheme, a (weak probe) laser is tuned to a transition of the 2 2B2+-X 2A, band.35 The excitation laser is scanned and induces transitions in the heavily perturbed 2A1/2B2tX 2Al electronic system. If the two transitions share a common level (e.g. ground-state level) the pump laser creates a population hole, and the fluorescence down to the ground state (22B2 -+X2A1) is decreased.In our scheme the lower- and upper-state quantum numbers could be unam- biguously assigned, even in the case of strong state mixing. In the experiments, we observed double-resonance spectra which are labelled by the quantum number N", but which still contained an unresolved spin splitting, because the spin splitting of the ground state was too small to be resolved with our laser resolution. NO2 molecules at rotational temperatures between 2 and 10 K have been prepared in a supersonic jet by expanding 2-5% NO, in Ar at a backing pressure of 2-4 bar through the 0.4 mm diameter orifice of a solenoid-actuated pulsed nozzle (IOTA 1, Highly Excited States of NOz General Valve).The two counterpropagating beams (parallel polarization) at R, and 1, crossed the jet expansion ca. 10-12 mm downstream from the orifice in the vacuum chamber. The excitation and probe laser pulses were obtained from high-resolution pulsed dye lasers (Lambda Physik), which were pumped by the third harmonics of a pulsed Nd : YAG laser and an excimer laser. In this experiment we used the same UV- fluorescence imaging optics and signal processing devices as in the previous experiment. 3. Results 3.1. Characterizationof Highly Excited Mixed States of NO, Mixed molecular eigenstates in the energy region near 17 700 cm-' had to be assigned prior to the collisional energy-transfer studies.Typical double-resonance spectra obtained in our experiments are shown in Fig. 2, where the two spectra (lower and upper trace) correspond to the A, -and A,-scanned double-resonance spectra, respec- tively. The A,-scanned spectrum was measured by tuning A1 to the line in the lower spectrum which was marked with an asterisk. This spectrum corresponds to the tran- sitions from a single 'unknown' intermediate level to final levels of the 22B, electronic state. With the spectroscopic data of the 2,B2 the three lines could be easily assigned to the A, transitions of perpendicular type terminating on the 31,2, 4', and 51,4 rotational levels of 2,B2(O, 0, 0). Thus, the rotational quantum numbers of the intermediate level could be assigned to be NKa,K,(R2) = 40,4. The lower trace of Fig.2 shows the 1,-scanned spectrum tuning 1, to the transition in the upper spectrum marked with an asterisk. Using the rotational constants for the ground the two transitions of parallel type originating from the 40, and 20, states of the ground state could be easily assigned. The rotational quantum numbers of the intermediate state = 30, ,which does not accord with from this spectrum were determined to be NKa,Kc(ll) NK?,&,) = 40,4. We have found this for most of the states in this energy region. Table 1 lists all the observed states and their assignments detected by our double-resonance technique in the region around 17700 cm-'. All intermediate levels have the rotational Table 1 Term energies and assignments of eigenstates in the range 17 800-17 700 cm-', populated and probed in collisional energy-transfer experiments" assignments eigenstateb term energy /cm-' "4) K, K, J(4,&) FA, A,) ref.a 17 717.80 3 0 3 3.5 b 17 716.40 3 0 3 3.5 C 17 724.30 5 0 5 5.5 d 17 726.41 5 0 5 5.5 e 17 736.06 7 0 7 7.5 f 17 737.80 7 0 7 7.5 gh 17 750.07 17 780.30 9 7 0 0 9 7 9.5 7.5 1 17 794.80 9 0 9 9.5 jk 17 771.01 17 750.59 9 8 1 1 8 8 9.5 8.5 I 17 757.45 9 1 8 9.5 m 17 777.20 11 1 10 11.5 a Rovibrational states that belong to the same vibrational state can be found from a plot of the term energy us. N(N + l).29 Index. In vacuum.This work. B. Abel et al. 153 I I I I 6000 0 0.0 0.2 0.4 0.6 0.8 ffP Fig. 3 Total depopulation for the N = 3, K, = 0 state at 17716.40 cm-' (state b in Table 1) at 500 pbar; t, delay between laser pulses; (0)experiment, (-) fit of the total depopulation rate; for details see text. quantum numbers "(A,) = odd, N'(A2)= even and Ku(Al)= &(A2) = 0 and 1. The dif- ference in the N quantum numbers was AN = "(A,) -"(A2) = & 1, while all of the KA quantum numbers were identi~al.~' This unambiguously shows that the quantum number N is spoiled in this energy region. Although, we have determined different N quantum numbers for the intermediate level, the total angular momentum, J, must be conserved and well defined in an isolated molecule (e.g.J = 3.5 in Fig. 2). 3.2. State-resolved Vibrational and Rotational Energy Transfer In Fig. 3 the N = 3, K = 0 level at 17 716.40 cm-' is populated and probed by optical double resonance. From these traces total depopulation rates of 28 6 ps-l Torr-' corresponding to k,, = 9 Ifr 2 x 10-l' cm-3 s-' (where the subscript td refers to total depopulation) could be determined. No systematic variation of ktd upon the rotational quantum numbers could be found." Our measurements of k,, displayed in Fig. 3 are a direct measure of collision numbers, 2, in highly excited states. This rate can be com- pared with the Lennard-Jones collision number k,, = 4.2 x cm-3 s-' (13.4 ps-l Torr -'). In order to obtain quantitative estimates of individual state-to-state energy- transfer rate constants and cross-sections we used a master equation approach which enabled us to model population transfer among a specified set of levels.If N is the vector bath of rot levels ,,I I I nvib. levels t N,K=O N, K= 1 N,K=2 Fig. 4 Schematic differential equation energy-transfer model (for details see text) Highly Excited States of NO, 600, , I -(d 0.0 0.2 0.4 0.0 0.2 0:4 600 400 200 0 J, 0.0 0.2 0.4 0:o 0:2 0.4 ffP ffP Fig. 5 Experimental rotational energy-transfer traces (open circles) and master equation model calculations (dashed lines). The signal amplitudes are normalized. For details see text. (a) g -k, 150 pbar; (b) g+e, 150 pbar; (c) g-c, 150 pbar; (d) g +b, 150 pbar (for assignments of states a-m, see also Table 1).of level populations and K is the matrix of rates connecting these levels, the time evolu- tion of N is given by N=KxN (2) Although the initial and final states are completely resolved in our experiments, the complete state-to-state rate matrix, K, can be obtained only by modelling the population changes in the detected levels. This is an inherent problem of this type of experiment, in which one is always summing over a number of possible pathways connecting the initial and final states. In order to reduce the number of energy levels and differential equa- tions, we used a simplified model which is depicted in Fig. 4. In this case, level families which have not been probed were grouped together in global baths of energy levels (e.g.the K = 3 stack, N 15 in the K = 1 and 2 stacks, rotational states of vibrations which could not be probed). In this particular case, two vibrational levels are very close to each other in the considered energy range. The fundamental problem in rotational energy- transfer studies on polyatomic molecules is that the number of levels, and the number of collisional rates can be very large. Not only would it be difficult to measure all of these rates, but would also be especially unrealistic to vary every constant individually in the simulations. A common practical approach is to scale the state-to-state rates by a scaling or fitting law. For scaling the J and K changing rate coefficients in our model, we used a simple and widely used exponential gap law (EGL)32333 ki,, = (2Jf + l)k, x exp(-C Ai,, Elk, T) (3) for downward transitions.The upward transitions were calculated using the principle of detailed balan~e.~,.~~ The program used the EGL values for the rate constants as a first choice, then the rates could be adjusted individually, if necessary, to account for specific deviations from the EGL. The master equation was integrated numerically with a B. Abel et al. 155 Y -8-64-202468 v)c9 A J(AK=O)6 -40 -20 0 20 40 AEIC~ Fig. 6 (a) Individual rotational state-to-state energy transfer rates (R-T, R) for the initial level N = 9, K, = 0, J = 9.5 at 17 750.07 cm-' (level g from Table 1) as a function of exchanged angular momentum, AJ.(0)Experimental data; (-) EGL approximation. For more details see text. (b) Individual rotational state-to-state energy-transfer rates (R-T, R) for the initial level N = 9, K, = 0, J = 9.5 at 17 750.07 cm-(level g from Table 1)as a function of energy separation. (0)Experimental data and (-) EGL approximation. Transitions (N,K, t--,N, Ka):(1) 9, 0 o3, 0, AJ = 6, AK = 0; (2) 9, 0-4, 1, AJ = 5, AK = 1;(3) 9, 0-5, 0, AJ = 4, AK = 0;(4)9, 0-5, 1, AJ = 4, AK = 1; (5) 9, 007, 0, AJ = 2, AK = 0; (6) 9, 0-6, 1, AJ = 3, AK = 1; (7) 9, 0-7, 1, AJ = 2, AK = 1; (8) 9, 0-8, 1, AJ = 1, AK = 1;(9) 9, 0-9, 1, AJ = 0, AK = 1; (10)9, 0-10, 1, AJ = 1, AK = 1; (11) 9, 0-11, 0, AJ = 2, AK = 0; (12) 9, 0-11, 1, AJ = 2, AK = 1; (13) 9, 00 12, 1, AJ = 3, AK = 1; (14) 9, 00 13, 0, AJ = 4, AK = 0.The rates for transitions (2), (4),(6), (7), (lo), (1l), (13) and (14) are predicted by the model. For more details see text. FORTRAN program using a standard Runge-Kutta algorithm with adaptive step size. The results of this procedure are displayed in Fig. 5. The individual state-to-state rate constants for an initial N = 9, K = 0 rovibrational level at 17750.07 cm-l are given in Fig. 6(a) and (b). The solid line of Fig. 6(a) and (b) is an EGL fit (ko = 0.50 x 10-l' cm3 s-~,C = 17 _+ 2). In the case of rotational and vibrational energy transfer we modelled the state- resolved traces by assuming a ladder of vibrational levels each with fast rotational energy transfer within their rotational level manifold.The energy positions of known vibrational levels in this range have been taken from Ref. 37 and 39. Finally, we included the (dark) vibrational levels we have found in this region by our double-resonance tech- nique. With this stepladder model we modelled the experimental traces (Fig. 7), and we determined the rotationally resolved rate constants for vibrational (and rotational) Highly Excited States of NO, 6000 6000-r b) 3000 3000: h v) OO O c).-c co 3 00 no 0 -0q)od T 00 0.4 0.0 0.4 0.8 f/PS ffw Fig. 7 Experimental traces monitoring rotational and vibrational energy transfer (0)and step- ladder model calculations (-). For details see text. (a) h +f, 300 pbar; (b) a +by 150 pbar; (c) i +e, 300 pbar ;(d) c +d, 150 pbar (for assignments of states a-my see also Table 1).energy transfer shown in Fig. 8. The straight line is an energy gap law fit to the experi- mental data (k, = 1.7 x lo-'' cm3 s-' and C = 13.5 f2). Because the number of levels in this range is still somewhat uncertain (missing dark levels), the energy-transfer data derived solely from the simple model would have quite large error bars as well. To improve this situation we recorded double-resonance spectra in the single-collision 0 10 20 30 40 50 60 A Edownlcm-' Fig. 8 Individual rotationally resolved vibrational energy-transfer rates (V-T, V, R-T, R). (1) Transition c -+ d (Table 1);(2) transition i +j (Table 1);(3) transition h +f (Table 1);(4)transition i +f (Table 1).For details see text.B. Abel et al. 157 regime (collision number 2 = 0.1-0.2). The derived rate state-to-state constants are con- sistent with those derived from the time-resolved data and the model within the quoted error bars.29 3.3. Double-resonance Spectra, Linewidths and Specific Rate Constants of Jet-cooled NO, close to the Dissociation Threshold The Fig. 9 (lower trace) shows a high-resolution double-resonance (DR) spectrum of ~~NO, with N" = 0, [v~ =,40~126.78 cm-', 2,B,(O, 0, 0), lo, +-X2Al(0, 0, 0), Oo,o]where the laser frequency, v2 ,is scanned across the dissociation threshold. Owing to the unresolved spin splitting in the ground state, the DR spectrum consists of two sequences of transitions with J' = 0.5 and 1.5.For N" = 0, the DR spectra can be compared with laser-induced fluorescence (LIF) (below ,To), photofragment excitation (PI-IOFEX) spectra of ultracold NO, molecules,45 and laser-induced grating spectra (LIGS).ll In order to exclude effects on the line-shapes and -widths from power and saturation broadening, we measured the spectra at different laser intensities and found no influence on the lineshapes as long as the depletion depth of the lines remained below 40%. Tuning to the 2 2B2(0,0, 0), 30, +-X 2A,(0, 0, 0), 20,,probe transition at 40 128.10 cm-l we were able to record high-resolution, DR spectra with N" = 2. The DR spectra simulation J I I I I 25128 25130 25132 25134 25136 I spectrum I'I'I'I'T 25128 25130 25132 25134 25136 excitation energy/cm-' Fig.9 Lower trace: Experimental double-resonance spectrum just below and above the disso- ciation threshold. The lower state is N" = 0 and the upper state is J' = 0.5 and 1.5, owing to the unresolved spin splitting in the gound state. The horizontal axis represents the vacuum photon energy. Upper trace: Simulation of the double-resonance spectrum (J' = 0.5, 1.5) with a multi-line fit (Lorentz-profiles). The baseline has been derived from the region below the dissociation thresh- old, E,. E, has been determined to be 25128.5 & 0.1 cm-'. For details see text. 158 Highly Excited States of NO, in this case contained transitions from N” = 2 to J’ = 0.5-3.5. Owing to the fact that the spectrum of NO2 at large excess energies is very congested and the lines heavily overlap we limited ourselves to the spectral range very close to the dissociation threshold.However, even in this range the exclusive fit of single relatively isolated lines to a single Lorentzian is not appropriate, because of the superposition of the wings of adjacent lines and the biased selection of these ‘special’ lines. Instead, we decided to fit all lines glob- ally in a (narrow) wavelength interval to a series of Lorentzians, with the sacrifice of somewhat larger error bars compared with the fit of single isolated lines. The procedure for the extraction of linewidths from the DR spectra (Fig. 9) has been described in detail in Ref. 47. Comparing the N” = 0 and N” = 2 DR spectra, we could sort out transitions which terminate on J’ = 0.5 and 1.5, and J = 2.5 and 3.5, respectively, in the N” = 2 double-resonance spectra.This technique can also be applied to DR spectra with N” > 2. For a detailed investigation of the rotational dependence of DR spectra and linewidths we refer to ref. 47. Fig. 9 shows a comparison of the observed DR spectrum (lower trace) for N’ = 0 and a simulation (upper trace, multiple-line Lorentzian fit). In this approach it is still possible that several lines form one peak, especially in the case of broader peaks, therefore we regard the derived linewidths from all spectra as upper bounds. From the line densities in the N” = 0 and N” = 2 DR spectra we found some evidence for K-mixing; however, these studies are currently still in progress.Fig. lqa) and (b) display the observed linewidths for J = 0.5, 1.5 and 2.5, 3.5 close to the disso- ciation threshold. The average linewidths above threshold for J = 0.5, 1.5 and 2.5, 3.5 25128 25130 25132 25134 25136f 0.01 25128 25130 25132 25134 25136 term energy/cm-’ Fig. 10 (a) Linewidths of NO, states with J’ = 0.5 and 1.5 close to the dissociation threshold, E,, as a function of internal energy, E (term energy = vacuum photon energy). (b) Linewidths of NO, states with J’ = 2.5 and 3.5 close to the dissociation threshold, E,, as a function of internal energy, E [term energy = Ephoton(vac.)+ EJ. B. Abel et al. 159 I I I -OICOO I 0 I 0 50 100 150 200 250 300 -I I 0 10 20 30 40 E-Edcrn-’ Fig.11 Experimental rate constants (0,this work) close to the dissociation threshold for J = 0.5, 1.5, experimental rate constants, k(E), from time-resolved ps experiments (0,Wittig and co- workers4*), and SACM model calculations for J = (0.5 + 1.5)/2 = 1 (marked A and B). B, SACM model calculations of specific rate constants, k(E, J), for J = (0.5 + 1.5)/2 = 1 with a calculated density of states, p(E, J) (and a calculated energy dependence), as a function of excess energy E -E,. A, SACM model calculation of specific rate constants, k(E, J), for J = (0.5 + 1.5)/2 with an experimental threshold density of states, p(E, J), from Ref. 45, 47, 49 (with a calculated energy dependence as in B), as a function of the excess energy E -E, .For details see text. The box in the picture shows rate constants from the frequency (this work) and time domain (Ref. 48) and the calculations A and €3 on a larger energy scale (similar axis units). For more details see text. 1 0 10 20 30 40 E-Edcrn-’ Fig. 12 Experimental rate constants (0)close to the dissociation threshold for J = 2.5 and 3.5 and a SACM model calculation of specific rate constants k(E, J) (-) for J = (2.5 + 3.5)/2 = 3 as a function of the excess energy, E -E, . For this calculation a calculated density of states, p(E,J), and its energy dependence was used (full K-mixing). The average threshold rate constant is indi- cated by an arrow. For details see text. Highly Excited States of NO, are 0.11 and 0.09 cm-', respectively.The observed linewidths (FWHM) can be con- verted into state-specific rate constants by using the relationship k(E, J) = r/h, where ti is Planck's constant over 27~.Fig. 11 and 12 show experimental specific rate constants, k(E, J), of NO, above E, as a function of internal energy, E,. From the average line- widths of 0.11 and 0.09 cm-' above E, = 25 128.5 cm-' we calculate average threshold rates of 2.0 x 10" s-' and 1.65 x 10" s-l. Increasing J obviously decreases the average specific rates slightly, owing to the increased rovibronic density of states (for K-mixing), a result which has been predi~ted.'~'~~ 4. Discussion 4.1. Eigenstates in the 17 700 cm-' Region As has been shown in the Experimental section, the 'good' quantum numbers of heavily mixed eigenstates can be assigned from the DR spectra.However, the rotational quantum numbers, N, determined independently from two spectra turned out to differ by 1. Our results indicate that some state mixing connects two rotational levels of two zero-order states with N = 3 and 4 (Fig. 2). Our results, as well as the results from Shibuya and co-~orkers,~~~~~ lead to the conclusion that the intermediate rovibronic states detected by the DR method are mixtures of the bright state and a dark state, which can be expressed schematically as I molecular eigenstate) = c1 I bright state) + c, I dark state) (4) where c1 and c, are mixing coefficients. The coefficients, c1 and c,, of the zero-order states are probed by L1 and A,, respectively (to obtain a strong DR signal, c1 and c, have to be larger than 0 and of the same order of magnitude!).As discussed in Ref. 39 and 40 the symmetries of the interacting states can be obtained from the DR spectra as well (evB2and eVA2).In the considered energy region (17700 cm-l), where collisional energy transfer has been measured the detected intermediate states are verified to be mixed states of dark C 2A, and bright A ,B,, coupled by spin-orbit interaction with the selection rule AN = 1 and AK, = 0. Therefore, the 17 700 cm-' region turned out to be a spectral region where at least two, or most likely three, electronic states (X2Al, C ,A, and A ,B,) contribute to the spectrum and the dynamics of NO2. 4.2. Rotational and Vibrational Energy Transfer in NO,-NO, Self-collisions The experimental results on vibrational and rotational energy transfer show that it is possible to obtain state-resolved energy-transfer data at internal energies of ca. 70% of the dissociation energy, even in the case of strong interstate mixing. The data on NO, self-relaxation presented in Section 3.2 are far from being complete, such that many more relaxation channels, vibrational states and collision partners will be investigated in the future.Nevertheless, the data provide a first picture of state-to-state relaxation in highly excited NO,, Collisional energy transfer at high levels of vibrational excitation is very important for chemical reactivity, especially for thermal unimolecular reactions. In order to model or predict these processes it is necessary to know something about col- lisional energy transfer rates and mechanisms.Beyond global deactivation rates and energy-transfer parameters like the first and second moment of the relaxing population distribution, one would like to know the magnitude of single state-to-state rates and collision numbers, 2,as a function of the good quantum numbers. This information can be used for two-dimensional master equation models (in E and J). Furthermore, the question of whether state-to-state propensity rules for vibrational and rotational energy transfer are still valid in the high-energy (intermediate state density) region is still not answered yet. B. Abel et al. 161 From the modelling of rotational energy transfer at about 17700 cm-' we find that the total relaxation is quite fast (as expected for a polar molecule) and occurs at about 2-2$ times the Lennard-Jones collision rate (Fig.3). The individual rates for rotational- energy transfer (R-R, T) are fast as well, as can be inspected in Fig. 6. The state-to-state rate constants show a weak but interesting propensity pattern, which slightly favours transitions with AJ = AK, = 0, AJ = 2 (AKa = 0), AJ = AKa = 1, AJ = 0 (AKa = l), although multi-quanta transitions with AJ = 3-6 occur as well. The individual rates seem to follow (on average) a simple energy-gap law. Deviations and a more detailed analysis will be given elsewhere.29 These preliminary results are consistent with a leading dipole-dipole term in the interaction potential which governs the collision dynamics, and much smaller higher-order terms.In this picture, the AJ = AKa = 1 tran- sitions correspond to 'dipole-type' transitions and the multi-quanta transitions to high-order dipole-dipole interaction or higher-order multipole interactions. A detailed evaluation of the potential-energy surface parameters of the two collision partners is, in principle, possible, but not trivial at all for polyatomic collision partners. The magnitude of the cross-sections for rotational relaxation indicates that many of the primary relax- ation channels of single eigenstates are rotational in nature. However, owing to the relatively high density of vibrational levels (0.2 levels per cm-', Ref.47) relaxation to nearby vibrational levels cannot be neglected. The quite large constants for population transfer with AJ = AK, = 0, which corresponds to vibrational energy transfer between rovibronic level that are 2 cm-' apart, demonstrate this. 'The dependence of rovibrational energy transfer on the energy separation, AE, is depicted in Fig. 8. The relatively high rate constants (V-T process) for small energy gaps (e.g. 2 cm-') can be understood in terms of time-dependent perturbation the~ry.~.~~ Such a transfer can be brought about easily by the long-range attractive part of the intermolecular potential. As can be seen in Fig. 8, the rates connecting two rovibrational states in two different vibrations do not show a pronounced rotational dependence (as long as the energy separation is comparable) and the overall magnitude of the relaxation rates seems to be dependent mostly on the energy gap between the states.The quoted error bars correspond to the uncertainties in the ET-model and the time-dependent DR spectra. 4.3. Comparison of the Measured Specific Rate Constants with Statistical Adiabatic Channel Model Calculations and Time-domain Measurements The linewidths for J = 0.5 and 1.5, as well as their average value of 0.11 cm-', agree quite well with the experimental results from Tsuchiya and co-~orkers~~ (obtained with a different technique), although the sudden increase in linewidths or average linewidths when a new product channel opens is not as pronounced as in their PHOFEX study.Nevertheless, their conclusion that the dynamics are governed by a loose transition state and that the dynamics can be described by phase-space theory (PST) seems to be in accord with our data and our measured average product-state distribution^.^^ If we compare our results with the time-resolved data from Wittig's group in Ref. 48 we think that the results are consistent with ours, although the reported threshold rate constants are quite different. These differences can be explained by the fact that the ps experiments most probably probe the rapidly dissociating molecules in an ensemble of molecules with different internal energies (bandwidth of the ps laser) and even more different life- times (lifetime distribution), whereas we sample the state-resolved distribution of life- times (biased toward long lifetimes and small linewidths).In order to calculate state-resolved unimolecular reaction rates as a function of inter- nal energy and total angular momentum, J, which can be compared with our experi- mental data, we have used eqn. (1) within the framework of the statistical adiabatic channel model (SACM).'8*'9 Here p(E, J) and W(E, J) represent the rovibrational Highly Excited States of NO, density of states and the number of open channels, respectively. The key to a successful modelling of the energy dependence of k(E, J) (from all available frequency- and time- domain measurement^^^) with eqn. (1) is to use W(E,J) and p(E, J) instead of W(E, J = 0) and p(E, J = 0).We calculated the specific rate constants, k(E, J) (without spin), of NO2 for J = 1 and 3, in order to compare them with our measured average rate constants close to the dissociation threshold and time-resolved data at higher energies.48 While the number of open channels was calculated straightforwardly within the SACM- PST frame~ork,'~*'~ the calculation of the density of states, p(E, J), turned out to be a non-trivial task.'8c In the model calculations we used two densities of states, the calcu- lated density of states from an anharmonic stretch-bend-coupling model'8c and its energy dependence, and the experimental density of states close to the dissociation thresh~ld?~*~~*~~with the energy dependence predicted from Troe's model.The results are shown for an average J = 1 [J = (0.5 + 1.5)/2] and J = 3 [J = (2.5 + 3.5)/2] in Fig. 11 and 12. The influence of spin on the results is negligibly small and can easily be estimated from Ref. 18(b).Details of the calculations can be found in Ref. 18(b),(c)and 47. The calculated density of states, p(E, J), matches the experimentally determined density of states close to E, within a factor of 1.5. The latter (experimental density of states derived from line densities) may be regarded to be more accurate close to the threshold, but unfortunately the energy dependence of the density of states, p(E, J), can hardly be determined from the spectra in this case. Because both calculated k(E, J) curves in Fig. 11 are already very close to the experimental data, we take both curves as the upper and lower limit for the calculated specific rate constants [representing the error bars in p(E, J)].Actually, a finer adjustment of k(E, J) to the experimental data would give additional information about the energy dependence of p(E, J) and/or the PES itself above the dissociation threshold. The reason for the observed decrease in the rate constants for increasing J is the increase in the density of states, p(E, J). This effect is indirect evidence for K-mixing and the breakdown of the K quantum number. While the overall trend of decreasing experimental average rate constant (J = 2.5, 3.5) is pre- dicted correctly, the deviation between the calculation for J = 3 and the experiment (average threshold rate constant) is slightly larger than in the case of J = 0.5, 1.5.However, the calculated threshold rate constant matches the average experimental rate constant (which is difficult to determine owing to linewidth fluctuations) within a factor of two. Unfortunately, time-resolved data at higher energies for the higher angular momentum states are not available yet, such that further tests are difficult. The results of this section show that our data and the energy dependence of the specific rate constants, k(E, J), of low-J states in the dissociation of NO, can be under- stood by a straightforward modelling by SACM-PST. It is clear from the discussion above that the total angular momentum, J, has to be taken into account carefully.4.4. Linewidth Distributions From the linewidths above the dissociation threshold we determined lifetime distribu- tions P(r/rav)as a function of T/T,,. The results are displayed in Fig. 13(a) and (b). In order to interpret these distributions one must keep in mind that there is a lower limit of the observable T/T,,,owing to the limited resolution of our laser system (0.04 cm-l). Lines with linewidths smaller than this limit would show up in the pile close to the resolution limit, T/T,,. On the other hand, there are some difficulties in the determi- nation of large linewidths, owing to the 'background' problem. This, and the fact that we also sample linewidths in a range where an additional channel opens (JNo= 1.5 in addition to J,, = 0.5), should be considered when comparing these distributions with calculated ;C2-distributions with n degrees of freedom, However, n should be closely related to the number of open channels, W(E,J), from Section 4.3.It should also be kept in mind that our distributions are those of two sequences of transitions to the upper B. Abel et al. 163 2012345678 cO.30 I I, I,, , I , I, I, 0.25 3 3b, 012345678 Ura" Fig. 13 (a) Reduced lifetime distributions P(r/raV)for J' = 0.5 and 1.5 between 25 128.5 and 25 136.5 cm-'. The average linewidth, Tav,is 0.11 cm-'. (b) Reduced linewidth distributions, P(T), for J' = 2.5 and 3.5 between 25 128.5 and 25 134.5 cm-l. The average linewidth, Tav,is 0.09 cm-l. states, J = 0.5, 1.5 and J = 2.5, 3.5, respectively.Although we found deviations from a theoretically predicted X2-distribution, the observed lifetimes seem to be in reasonable agreement with Miller's theoretical model' derived from random matrix the~ry.~~-~ The results presented here are consistent with a complete intramolecular energy flow (IVR) prior to reaction, and the interpretation of the observed average dissociation rates in terms of statistical unirnolecular reaction theory. The fluctuations in the linewidths (and product-state distributions1 2747) reflect only the characters of the individual eigen- states and wavefunctions excited, and their projection onto the transition state. 4.5. Rate Constants from Linewidths in the Case of Overlapping Lines Mies and Kra~s,~~ Peskin et and Someda et al.58 have shown that unimolecular decay processes of densely distributed quasi-bound states can be studied numerically by randomly generating the Hamiltonian matrices.The decay rates can be obtained from the Feshbach theory of resonance scattering in the case of non-overlapping and overlap- ping resonance^.^^^^^ In unimalecular dissociation studies in the frequency domain the standard prescription for the conversion of linewidths, I?, into specific rate constants, k(E, J), is k(E, J) = T/h, where h is Planck's constant over 271. In the case of non-overlapping resonances this relation holds rigorously, but in the regime of overlapping resonances, deviations due to interferences may be expected. The requirement that the resonances be non-overlapping is that the average width, (r),is less than the average 164 Highly Excited States of NO, energy spacing, AE, (T)/AE < 1 E (T)p(E, J) < 1 (5) In our observed spectra, close to the dissociation threshold we noted about 10 lines per wavenumber.It is clear that they consist of two sequences of transitions to different upper states with different J' quantum (0.5, 1.5) numbers, which belong to J-states that do not interfere. If we observe an average linewidth of about 0.1 cm-' and a vibronic density of states of 1.66 cm, we assume that the condition for strongly overlapping resonances is not yet fulfilled. 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Phys., 1988,89,3584. Paper 5/06181D; Received 19th September, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950200147

出版商:RSC    

年代:1995

数据来源: RSC