摘要:

Drrk190719071910191 1191219131913191319141914191519161916191719171917191819181918191819191919192019201920192019211921192119211922192219231923192319231923192419241924192419241925192519261926192719271927GENERAL DISCUSSIONS OFTHE FARADAY SOCIETYSubjectOsmotic PressureHydrates in SolutionThe Constitution of WaterHigh Temperature WorkMagnetic Properties of AlloysColloids and their ViscosityThe Corrosion of Iron and SteelThe Passivity of MetalsOptical Rotary PowerThe Hardening of MetalsThe Transformation of Pure IronMethods and Appliances for the Attainment of High Temperatures in aRefractory MaterialsTraining and Work of the Chemical EngineerOsmotic PressurePyrometers and PyrometryThe Setting of Cements and PlastersElectrical FurnacesCo-ordination of Scientific PublicationThe Occlusion of Gases by MetalsThe Present Position of the Theory of IonizationThe Examination of Materials by X-RaysThe Microscope : Its Design, Construction and ApplicationsBasic Slags : Their Production and Utilization in AgriculturePhysics and Chemistry of ColloidsElectrodeposition and ElectroplatingCapiU ari tyThe Failure of Metals under Internal and Prolonged StressPhysico-Chemical Problems Relating to the SoilCatalysis with special reference to Newer Theories of Chemical ActionSome Properties of Powders with special reference to Grading byThe Generation and Utilization of ColdAlloys Resistant to CorrosionThe Physical Chemistry of the Photographic ProcessThe Electronic Theory of ValencyElectrode Reactions and EquilibriaAtmospheric Corrosion.First ReportInvestigation on Oppau Ammonium Sulphate-NitrateFluxes and Slags in Metal Melting and WorkingPhysical and Physico-Chemical Problems relating to Textile FibresThe Physical Chemistry of Igneous Rock FormationBase Exchange in SoilsThe Physical Chemistry of Steel-Making ProcessesPhotochemical Reactions in Liquids and GasesExplosive Reactions in Gaseous MediaPhysical Phenomena at Interfaces, with special reference to MolecularAtmospheric Corrosion. Second ReportThe Theory of Strong ElectrolytesCohesion and Related ProblemsLaboratoryElutriationOrientationVolumeTrans.33678999101011121213131314141414151516161616171717171818191919191920202020202121222223232GENERAL DISCUSSIONS OF THE FARADAY SOCIETYDateI3281929192919291930I9301931193219321933193319341934193519351936193619371937139819381939193919401941194119421943i 944i9351945I9461946194719471947194719481948194919491949195019501950195019511951195219521952195319531954I954SubjectHomogeneo-Js CatalysisCrystal Structure and Chemical ConstitutionAtmospheric Corrosion of Metals. Third ReportMolecular Spectra and Molecu!ar StructureOptical Rotatory PowerColloid Science Applied to BiologyPhotochemical ProcessesThe Adsorption of Gases by SolidsThe Colloid Aspects of Textile MaterialsLiquid Crystals and Anisotropic MeltsFree RadicalsDipole MontentsColloidal ElectrolytesThe Structure of Metallic Coatings, Films and SurfacesThe Phenomena of Polymerization and CondensationDisperse Systems in Gases : Dust, Smoke and FogStructure and Molecular Forces in (a) Pure Liquids, and (b) SolutionsThe Properties and Functions of Membranes, Natural and ArtificialRCX~~SU KineticsChemical Reactions Involving SolidsLuminescenceHydrocarbon ChemistryThe Electrical Double Layer (owing to the outbreak of war the meeting'The Hydrogen BondThe Oil-Water InterfaceThe Mechanism and Chemical Kinetics of Organic Reactions in LiquidThe Structure and Reactions of RubberModes of Drug ActionMolecular Weight and Molecular Weight Distribution in High Polymers.(Joint Meeting with the Plastics Group, Society of Chemical Industry)The Application of Infra-red Spectra to Chemical ProblemsOxidationDielectricsSwelling and ShrinkingElectrode ProcessesThe Labile Moleculewas abandoned, but the papers were printed in the Transactions)SystemsVolurrte2425252526262928292930303131323233333434353535363739383940414242 A42 BDisc.12Surface Chemistry. (Jointly with the Societe de Chimie Physique atColloidal Electrolytes and SolutionsThe lnteraction of Water and Porous MaterialsBordeaux.) Published by Butterworths Scientific Publications, Ltd.Trans.43Disc. 34Lipo-Proteins 6The Physical Chemistry of Process MetallurgyCrystal Growth 5Chromatographic Analysis 7Heterogeneous Cataiysis 8Physico-chemical Properties and Behaviour of Nuclear Acids Trans. 46Spectroscopy and Molecular Structure and Optical Methods of In-vestigating CeU Structure Disc. 9Electrical Double Layer Trans. 47Hydrocarbons Disc. 10The Size and Shape Factor in Coiioidai SystemsRadiation Chemistry 12The Reactivity of Free RadicalsThe Equilibrium Properties of Solutions of Pion-ElectrolytesThe Study of Fast Reactions11131415The Physical Chemistry of Dyeing and Tanning 1617Coagulatioa arid Flocculation 18The Physical Chemistry of ProteinGENBRAL DISCUSSIONS OF THE PARADAY SOCiElYDUk? Subject Volrcme1955I955195619561957195819571958195919591960196019611961196219621953196319641964196519651946196619671967Microwave and Radio-Frequency Spectroscopy 19Physical Cheniistry of Enzymes 20Membrane Phenomena 21Physical Chemistry of Processes at High Pressures 22Molecular Mechariism of Rate Processes in Solids 23Interactions in Ionic Solutions 24Configurations and Interactions of Macromolecules and Liquid Crystals 25Ions of the Transition Elements 26Energy Transfa v.ith special reference to Biological Systems 27Crystal Imperfections and the Chemical Reactivity of Solids 28Oxidation-Reduction Reactions in Ionizing Solvents 29The Physical Chemistry of Aerosols 30Radiation Effects in Inorganic Solids 31The Structure and Properties of Ionic 'Melts 32Inelastic Collisions of Atoms arid Simple Molecules 33High Resolution Nuclear Magnetic Resonance 34The Structure of Electronically-Excited Species in theilGas-Phase 35Fundamental Processes in Radiation Chemistry 36Chemical Reactions in the Atmosphere 37Dislocations in Solids 38The Kinetics of Proton Transfer Processes 39Intermolecular Forces 40The Role of the Adsorbed State in Heterogeneous Catalysis 41Colloid Stability in Aqueous and Non-Aqueous Media 42The Structure and Properties of Liquids 43Molecular Dynamics of the Chemical Reactions of Gases 44For current availability of Discussionvolumes, see back cover

ISSN:0366-9033    

DOI:10.1039/DF967440X001

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Analytical Mechanics and Almost Vibrationally-AdiabaticChemical ReactionsBY R. A. MARCUSNoyes Chemical Laboratory, University of Illinois, Urbana,Illinois 61801, USA.Receiued 14th June 1967Co-ordinates and " vibrationally-adiabatic " approximations are described for reactions in threedimensions. Some reactions may demonstrate a fairly strict adiabaticityfor certain degrees of freedomand a statistical adiabaticity for others. Some will simply be strongly vibrationally-nonadiabatic.Several topics in kinetics are considered from the viewpoint of vibrational-adiabaticity or mildnon-adiabaticity .1 .--IN T RODU c T I o NTo treat the mechanics of chenzical reactions it is desirable to have a set ofco-ordinates (natural collision co-ordinates) which pass smoothly from those suitedto reactants to those suited to products.Such a set was recently given for linearcollisions ar,d an approximate solution was obtained for nearly vibrationally-adiabatic reactions. A recent extemion to three dimensions is summarized below,and the probable nature of several nearly vibrationally-adiabatic solutions is outlined.In anticipation of the results of such a mode of solution, several problems areexamined : translation-vibration interaction inchemically-reactive collisions, mechanicsand a statistical-dynamical theory of reaction cross-sectionsy3 quantum correctionsto computer trajectories, and the quasi-equilibrium assumption in kinetics.2.-N A T U R A L C 0 L L I S I 0 N C 0 -OR D I N A T E SIn a three-centre rea~tion,~AB +C-+A+BCthere are six-co-ordinates in the centre-of-inass system.On introducing body-fixedco-ordinates via an Eulerian angle (0, 4, x) rotation matrix5 and then letting thebody-fixed y z plane be the instantaneous plane of the three atoms, there are obtainedseven co-ordinates of which one is redundant. This seventh is eliminated by definingthe orientation of the body-fured axes in the y z plane. A locally Cartesian methodwas used.6Curvilinear co-ordinates IZ and s were introduced to replace z and 2, as in fig. 1.(Here, z is a scaled 7z-component of the AB distance and 2 is a scaled z-componentof the distance between C and the centre of mass of AB.) A co-ordinate nz ofmagnitude (y2+ Y2)* was then defined.* It describes the nonlinearity of theconfigurations.The six independent co-ordinates are now s, n, m, 8, 4, x .Potential energyprofiles in the (my n, s) subspace have cross-sectional shapes indicated by the shadedregions in fig. 2. The reacting system typically moves through these regions. Forthis reason, co-ordinates (m, n) were transformed into polar co-ordinates (Y, r), wherem and n equal r sin y and r0-r cos y, ro being determined by the shape of the profilesand being a function of s, the reaction co-ordinate.Finally, to remove bothersome cross-terms at large 3-s between .i) and (8, 4, i )8 ANALYTICAL MECHANICSthe angles y and x can be transformed into angles describing orientation with respect tospace-fixed axes : a matrix (sin 0 cos @, sin 0 sin @, cos 0) was expressed in termsof (0, sin y, cos 7) by the Eulerian angle rotationmatrix.(However, for somepurposes,co-ordinates s, r, y, 0, 4, x may be more convenient than s, r, 0, a, 8, 4.)0FIG. 1.4rthogonal curvilinear co-ordinates (n, s) of any point P. n is the perpendicular distanceto any curve C and s is the distance from any fixed point on C to the foot of that perpendicular.LFIG. 2.-Cross-sections of a potential energy surface. The Zz plane, curve C, and skewed axes arethose of fig. 1. A reacting system moves thought the shaded regions, executing a rotation (free atfirst), a vibration and a translation in this (m. It, s) internal co-ordinate space. Initially, the rotationThe potential energy is a relatively simple function of (s, r, y).” The Hamiltonianis of the form given by (2*2), where T, denotes a number of cross-terms, of whichthose between the two sets of angle variables, (0, 4) and (0, @), are most prominent.These cross terms vanish when s~is about the 02 axis, the vibration is perpendicular to OZ and the translation is parallel to OZ.00.Here, p is [mAm,mc/(m, + mB +mc)*, y is essentially 1 + (ro-r)lc, ~ ( s ) is the curvature ofcurve C in fig.1 at any s, and I@) is essentially the largest moment of inertia at each sR . A . MARCUS 9The potential energy Y can be written as the value Vl(s) for any point on curveC plus the increment V2(s, r, y ) to go to any other point in the (m, n, s) space at thesame s. If the r-motion is assumed to be coupled mainly to the s-motion, (approxi-mation 3 of $3) thenV E Vl(s) + v2(s7 rO, 7) + YZ(s7 r7 O)- vZ(s7 r07 O).(2.3)TABLE l.-pHYSICAL NATURE OF THZ CO-ORDINATES FOR SEVERAL VALUESOF REACTION CO-ORDINATE Sco-ordinate nature at s = - 00 nature at s y s+ nature at s = +as radial translation of asym. stretching vibra- radial translation of Ation of ABC #r vibration of AB sym. stretching vibra- vibration of BCtion of ABCAB relative to C relative to BCorbital translation of rotation of ABC # orbital translation of AAB relative to C relative to BC8, #@,a rotation of AB doubly degenerate ben- rotations of BCding vibration of ABC #a This nature applies only the adiabatic case A in the text. For adiabatic case B these two itemsb 0-8 and @-+$ describe the bending.would be interchanged.The physical significance of the co-ordinates, in various regions of the six-dimensional configuration space, is given in table 1.At values of s intermediatebetween large +s and s*, the 0, <D motions are hindered rotations rather thanrotations or bending vibrations.3.-SOLUTION OF THE EQUATIONS OF MOTIONThe Hamilton-Jacobi and Schrodinger partial differential equations are obtainedfrom a Hamiltonian such as (2.2) in the standard way.l0* l1 Because of the closerelationship between the two partial differential equations, parallel treatments of theclassical and quantum mechanics can be made. As the extensive literature oncollisions and on molecular vibration-rotation interactions amply testifies, there is avariety of subsequent treatments of these equations which can be introduced.Anadiabatic-nonadiabatic approach is perhaps the most tractable. One adiabaticversion assumes2 : (1) all motions are adiabatic relative to the motion along s, ther-motion being the most adiabatic of all; (2) Case A the 0, @ motion is adiabaticrelative to the 8, $ one, or Case B the 8, # motion is adiabatic relative to the 0, @ one;(3) apparently minor r, y coupling terms can be neglected in the potential energy.Because of the assumed adiabaticity of the r-motion especially, approximation (3)could be fairly easily avoided, but it does lead to simpler results. Approximations(1) and (2) are made classically and quantum mechanically by writing (for case A,for example) +aw) +2(r; s> a; 0, 4, $1 +4(97 #; s) (3.1)(3.2) ws wl(s)+ w2(r; s)+ w,(o, m; e, 4, s)+ w4(e, 4; s),where W is Hamilton's characteristic function. The adiabatic approximation isactually made by neglecting the partial derivatives of each or W, with respect tothe variables to the right of the semicolon, which we shall call its weakly-dependentvariables.In case B, the 0, CD and the 8, $ in the above equations are interchanged.In the adiabatic approximation, variations in weakly-dependent variables ofa +, (or Wi) do not change the quantum numbers (or classical phase integrals)associated with it. Neverthless, in this adiabatic approximation large changes ma10 ANALYTICAL MECHANICSoccur in the physical nature of the strongly-dependent degrees of freedom (viz.,table l), in the shape of +i or Wi, and in the energy associated with the +i or W,.(One obtains, in fact, adiabatic correlation diagrams for the energy contributions.)In the next approximation the various approximations (1)-(3) can be examined.For example, (3) and others not mentioned can be examined by perturbation methods.The non-adiabatic corrections to (1) and (2) can be studied in several ways, oneapproximate one being of particular interest. It is relatively straightforward,maintains the parallelism of the classical and quantum calculations, and has intuitiveappeal : the weakly-dependent parameters in any $i or W, are replaced by the classicaltime-dependent solutions of the adiabatic equations, and the time-independentSchrodinger and Hamilton-Jacobi equations for a $i and a W, are replaced by thecorresponding time-dependent ones.In this way it is possible to calculate approximately the extent of excitation of thevibrational, rotational and orbital motions of the products.In this time-dependenttreatment of non-adiabatic calculations of cases A and B can be regarded as thechemical counterparts of the low mass and flywheel approximations, respectively.The latter were introduced by Cross and Herschbach l2 in their calculation of classicalrotational-translational energy exchange. Moreover, the adiabatic collision in thechemical reaction case can be regarded as the analogue of the elastic collision inphysical scattering : in both cases some event happens (e.g., reaction or scattering),but there is no change in quantum numbers or classical phase integrals of the periodicmotions.When a considerable change of reduced mass occurs, as, for example, inH + Cl,+HCl+ C1, (3.3)one limiting approximation might prevail before the activated complex region(s\<s#) and the other one after (s>s#). In (3.3) the flywheel analogue could prevailinitially and the low mass one finally.4.-VIBRATION-REACTION CO-ORDINATE INTERACTIONSIt appears that the present equations for the r, s motions can be made similar tothose derived for linear collisions elsewhere,l so that ailalogous deductions wouldthen follow.Several are summarized below.CURVILINEAR TUNNELLING.-The standard inetkod of computing tunnelling ratesin c.hemica1 reactions calculates the barrier along the " reaction path '' and assumesa one-dimensional Cartesian co-ordinate kinetic energy.It actually over-estimatesthe tunnelling rate for any given potential energy surface thereby.' The error issmall with energies just below the top of the barrier, but increases as the energydifference becomes increasingly negative. Two-dimensional calculations based onnatural collision co-ordinates r and s have been used to estimate the correct tunnellingrate, and may explain overestimates of tunnelling rates by the standard method.Detailed comparison with computer results would be of interest.(symmetric stretching) of ABC* are v and v#, the adiabatic change of vibrationalenergy is (v++) /z(v#-v)(quanturn), or J, (v# -v)(classical), where J, is prdr, thephase integral for the vibrational motion.' Typically, v# <Y, causing a decrease invibrational energy.According to the present r, s equations, the liberated energygoes into the energy of the s-motion, and so helps the system overcome the barrierby that amount. The agreement of this result with the results of numerical classicalmechanical integrations for linear collisions has been discussed previously.'CONTRIBUTION OF VIBRATION TO RATE.-If the vibration frequencies Of AB andR . A . MARCUS 11tional energy non-adiabatic changes can also occur, corresponding to changes invibrational quantumnumber or phase integral. A comparison of the calculated resultswith computer data will be interesting. The theory should break down when thenon-adiabatic effects become too large.VIBRATION EXCITATION OF PRODUCTS.-hl addition to adiabatic changes Of vibra-5.-RELATION TO A STATISTICAL-DYNAMICAL THEORY OF REACTIONCROSS-SECTIONSThe earlier analytical mechanical discussion indicates that for any given impactparameter and initial relative velocity of the colliding pair the barrier to reaction isseveral-fold : (1) the potential energy barrier ; (2) the centrifugal potential energybarrier, which is calculated in a simple way when approximate adiabaticity of theco-ordinate associated with the orbital angular momentum prevails ; (3) the change ofvibrational energy in forming the activated complex in the given vibrational state forthe r-motion ; and (4) the increment in energy when the transient bending modes areformed from the initial rotational ones (states for bending modes are more widelyspaced than those for the rotations).of these reactions, simple expressionswere obtained for the total reaction cross-section and for the reaction probabilityat any impact parameter, as a function of initial rotational, vibrational, and transla-tional state of the reactants.In the theory the above four contributions to the effectivebarrier were taken into account, the fourth in a statistically-adiabatic manner suchthat the quasi-equilibrium assumption in 97 was satisfied. Detailed classical computerdata have been given for the reaction cross-sections of the H + H2 reaction.14 In acomparison of the theory with these data the agreement between them was en-couraging, there being no adjustable parameter^.^ At high translational energies thecomparison indicated occurrence of some vibrational non-adiabaticity.With theaid of an approximate solution for the present equation for the s-motion it wouldbe possible to extend this simple model to estimate differential reaction cross-sections.In a recent statistical-dynamical theory6.+U AN TU M CORRECTIONS FOR CLASSIC A L TRAJECTORIESReaction cross-sections have been obtained from computer data only for classicalmechanical systems. Some quantum correction of the results for the H + H2 reactionwas made l4 by restricting the initial rotational-vibrational energy to values allowedby quantum mechanics. This point is now explored further.At any value of the reaction co-ordinate s before the system reacts, some quantumcorrections can be made via the WKB method. Namely, the initial vibrational, rota-tional, orbital, z-component of rotational and z-component of orbital phase integralsfpidq' are set equal to the values (u+ +)h, (j+Q)h, (I+ *)h, mjh, mlk, respecti~e1y.l~(These refer to the r, (0, (D), (8,4), (D and 4 motions, respectively.) If the adiabaticityoccurred at later values of s, i.e., if the equations were adiabatically separable into one-dimensional equations for all s, the above restrictions on the phase integral would beautomatically imposed by virtue of their having been imposed initially.The procedure of only restricting the initial rotational-vibrational energies tothose allowed by quantum mechanics amounts to imposing two of the above fivephase integral conditions.Because of the possible adiabatic correlation of theoriginal orbital-rotational motion of the reactants with the rotational-bending vibra-tional motion of the activated complex, it is desirable to impose the other phaseintegral conditions to achieve better results for the threshold region. Otherwise12 ANALYTICAL MECHANICSany zero-point, bending vibrational motion (if it indeed occurs) would be missed.However, outside of the threshold region the procedure originally used l4 shouldsuffice.As application of the WKB method to the adiabatically-separated equationsshows, there are additional quantum effects influencing the motion along the reactionco-ordinate (e.g., diffraction).However, they are less important for the reactionrate, except at low enough energies for tunnelling to occur.A further effort towards computing quantum effects might be made by comparingthe classical mechanical computer data of the original equations with those based onequations involving natural collision co-ordinates and into which various simplifyingapproximations have been introduced. If suitable agreement is achieved, computerstudies of the quantum analogues of the latter would be useful.7.-QUASI=EQUILIBRIUM, ADIABATICITY AND STATISTICAL ADIABATICITYIn the present section a quasi-equilibrium relation between reactants and activatedcomplexes of the same energy E is derived assuming adiabaticity.16 The resultingquasi-equilibrium equation has been tested by comparison with the computerdata.The results are summarized later.We consider a reacting pair whose total energy lies in (E, E+dE) and which is in aquantum state u for certain vibrational modes. The quantum numbers of the othervibrational and rotational modes are denoted by n. The remaining quantum numbersare I and ml, where Z describes the initial orbital angular momentum. If the initialmomentum along the reaction co-ordinate q isp and the initial energy of the rotational-vibrational modes of the reacting pairs in El,,, thenwhere El,, is independent of 1. The number of translational states of a pair in stateZun, in (E, E+dE) and in (4, q+dq) is dpdqlh, wherepdplp is dE. The reactive flux ofsuch states, computed by dividing by dq to obtain a density along q and by multiplyingby q and by the reaction probability wjvnp, is wIvnpdE/h, since dE is qdp.The reactiveflux from all states in (E, E+dE) is dEz(2Z+ l)wlvnp/h. The sum is over all n forwhich E,,, 5 E.A state Zvn adiabatically connects with one whose energy for all rotational-vibrational modes in the activated complexregion is E&. E now equals E$n +& /2p+,where the second term is the kinetic energy along the reaction co-ordinate. Of theoriginal states, those which lead to reaction are those for which EZn <E. The numberof states Zvn in dq# and in dE is dq#dp,/h, where dpz = p+dE/p,. The flux isQ#dp, / h and so is dE/h. An adiabatic noncrossing rule is now invoked, and therebythis flux expression is summed over all states for which E$,, <E and over all reactionpaths if there is more than one.Thus,E = E1V,+P212P, (7.1)nwhere r is a summation operator over all reaction paths. To avoid confusion wereplace mlZn on the right side by n*. (The designation m,Zn merely indicates originof a state in a correlation diagram.) Analysis by separation into states of differentJ, leads to eqn. (5) of ref. (17) and thence to (7.2).Eqn. (7.2) is a quantitative statement of a quasi-equilibrium between reactingpairs and activated complexes of the same E moving in the forward direction and wasoriginally derived l6 in that way. From it, an activated complex theory rate expressioncan also be derived if an equilibrium distribution of reactants’ states is assumedR .A. MARCUS 13In terms of reaction cross-sections uvnp, (7.2) becomesC ( k 2 / ~ ) ~ u n p = rc1,n n#msince cUnp equals" ( x / k 2 ) C (22+ l)wLvnp. In (7.3), k denotes p/h.1=0In a test of (7.3) using computer data and classical versions of the sums, the left-hand side was found to be 9.6, 24.5 and 55, while the right-side was 7.0, 22 and 50,when E was 15.5, 17.0 and 18.5 kcal/mole, respectively.l'* l9 The agreement isencouraging.20 The E are those of interest in thermal reaction studies.However, one might expect the rotational and orbital modes to be adiabatic in atbest a statistical sense rather than a rigorous one. This statistical adiabaticity wouldprevail when each relevant initial state yields at each s not necessarily a single statebut a group of states fairly symmetrically distributed in energy about the adiabaticallydetermined state at that s.It will be interesting to see if the computer data exhibit astatistical adiabaticity for those modes and to see when the quasi-equilibrium relation(7.3) can be derived from a suitable statistical adiabaticity.Acknowledgement is made to the donors of the Petroleum Research Fund,administered by the American Chemical Society, and to the National ScienceFoundation, for their support of this research.R. A. Marcus, J. Chem. Physics, 1966,45,4493,4500.R. A. Marcus, to be published.R. A. Marcus, J. Chem. Physics, 1966,45,2630; 1967,46,959.A reaction path which forms AC may also occur and would have its own set of natural collisionco-ordinates.E.B. Wilson, Jr., J. C. Decius and P. C. Cross, Molecular Vibrations (McGratv-Hill, New York1955), p. 286.In the subsequent notation, the orientation is altered until y sine = - Y cosc, where <(s) is chosento eliminate one of the vibrational angular momentum terms at each s. c = $ in fig. 1 whenS Z k f . ' The scaling factors in F. T. Smith, J. Chem. Physics, 1959,31, 1353, are used.* y = --m cos < (s), Y = m sin < (s).9y is related to 8, @ etc., via cosy = cos 8 cos @+sin 8 sin 0 cos ( c p - 4 ) .lo e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950), pp. 280 ff.l2 R. J. Cross, Jr., and D. R. Herschbach, J. Chem. Physics., 1965, 43, 3530.l3 R. A. Marcus, J . Chem.Physics., 1965, 43, 1598.l4 M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Physics, 1965,43,3259.l5 R. E. Langer, Physic. Rev., 1937,51,669 ; E. C. Kemble, The FundamentaIPrinciples of QuantumMechanics (Dover, New York, 1958) p. 155.l6 For simplicity of presentation, certain curvilinear effects are omitted, as is atom tunnelling.The former cause no difficulty and still lead to (7.3), the modified derivation being seen fromthe arguments leading to eqn. (3) of ref. (17). If tunnelling is also included, (7.3) is replacedby that eqn. (3).l7 R. A. Marcus, J. Chem. Physics, 1965,45,2138.l8 L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Addison-Wesley, Reading, Mass.,1958), p. 537.l9 A more accurate estimate of the right-side of (7.3) could have been made by taking into accountsome neglected rotation-vibration interactions.2o This agreement differs from the apparent disagreement in ref. (14) of a factor of 6 at 300°Kin trajectory-calculated and transition-state rate constants. It would be useful to replace theincomplete hybrid quantum-classical comparison in ref. (14) by an all-classical one and by themore complete hybrid proposed in 56. Above 1000"K, incidentally, quantum corrections in-fluenced mainly the stretching vibrations (judging from the partition functions of reactants andactivated complex), and the discrepancy in the k was only a factor of 1.25. Accurate tests of thequantum version of transition-state theory require quantum rather than classical computer-experiments.e.g. W. Pauli, Jr., in Handbuch der Physik., S. Hugge, ed. (Springer-Verlag, Berlin, 1958),vol. 5, p. 39

ISSN:0366-9033    

DOI:10.1039/DF9674400007

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Statistical Theory of Bimolecular Exchange Reactions*BY JOHN C. LIGHT?Dept. of Chemistry and The James Franck Institute,The University of Chicago, Chicago, Illinois 60637Received 26th June 1967A new concise derivation of the statistical theory of exchange reactions is given. The resultsof the theory are compared wih experiment for several reactions and for several types of information,and the reliability of such statistical calculations is assessed.The area of gas-phase collision phenomena, both energy transfer and chemicalkinetics, has become the subject of intensive experimental and theoretical investigationin recent years. Since the recent review articles 1-3 are both excellent and up to date,we shall concentrate, in this paper, only on the statistical (or phase space) theory ofsuch phenomena.It is, however, the detailed experimental information about theseprocesses, provided by various complementary experimental techniques, which hasprovided the impetus for further theoretical investigations.The experimental information available about some reactions such as the alkalimetal +halogen 4-6 and the hydrogen +halogen exchange reactions is rapidlyapproaching the ultimate of providing cross sections for reaction as a function ofvelocity from a spscified state (vibration-rotation) to a specified product state. Inany case, it is quantitative theoretical understanding on this level which now seemsto be the problem. If reasonable quantitative estimates of these cross sections couldbe made, all the information desired could be obtained by appropriate averages.The theoretical approaches to this problem fall into three rather well-definedcategories : quantum calculations,* * classical dynamical calcuIations, O - l andapproximate statistical calc~lations.~~-~ Although quantum mechanics and theformal apparatus of scattering theory provides the most rigorous approach to theproblem, they have not yet provided a large number of accurate results.There aretwo basic reasons for this: (a) the quantum dynamics of three-body (atomic size)systems are complex, often involving large numbers of states and strong interactions,and (b) the potential energy surfaces (presuming electronically adiabatic reactions)can only be estimated in a gross manner.The most successful approach to date has been the second; following classicaltrajectories on various potential energy surfaces to determine the most essentialfeatures affecting the dynamics of chemical reactions.Since the pioneering work ofWall, Hiller, and Mazur,l and Bunker,l O considerable information and confidencein the method have been produced. Although this approach has been successful,there are two limitations inherent in the method. First, specific quantum effectscannot be obtained. This is, almost certainly, a spurious criticism for most chemicalreactions. The fact that many quantum states are normally involved and availablealmost guarantees that classical mechanics is a very good approximation. Thesecond, more serious limitation is to electronically adiabatic potential energy surfaces.* This research was supported by grants from the National Science Foundation.-t Alfred P.Sloan Research Fellow.1JOHN C. LIGHT 15If more than one surface is present, a transition probability between surfaces must beassigned, normally zero or one. Although this has not been a problem in the reactionsconsidered to date, there is a large class of chemical reactions which are not adiabaticelectronically and which do produce electronically excited products.lgThe statistical approach to chemical kinetics is based on the hope that it may bepossible to bypass the complexities of quantum scattering theory and the uncertaintiesand labour of classical calculations (which consume non-trivial amounts of computertime) and yet to obtain quantitative results in good agreement with experiment.Thestatistical approach substitutes a hypothesis for much calculation, and the resultsmust be subjected to comparison with experiment in order to test the validity of thehypothesis. The comparison must be made in such a way that at worst the hypothesiswill be shown to be completely wrong, and at best it can be shown to be reasonablefor a well-defined class of reactions.There are several reasons for doing this. The most obvious is that it may providea relatively simple method of obtaining reasonable answers, particularly for complexsituations in which the other methods cannot be applied. In addition, the hypothesisis complementary to the other approaches. If, for a particular reaction, or class ofreactions, the statistical approach seems to yield valid results, it implies that thesystem can in no way be considered as weakly coupled.Thus the quantum approachand the classical trajectory approach probably will be even more difficult to carrythrough successfully for these reactions.There have now been enough calculations using the statistical theory and coin-parisons with experiment to provide a somewhat ambiguous answer as to its validity.In the following sections we shall try to provide a comprehensive outline of thefoundations, applications and results of the statistical approach, summarizing, atthis date, its validity and applicability.FOUNDATIONSAll statistical theories of dynamical events, from nuclear reactions to chemicalkinetics, involve two distinct types of approximations.The first is a definition,normally in phase space, of the limits of a strong coupling complex. The second is astatistical prescription, again usually involving phase space, for partitioning theproducts of the collision among the various possible products (or states).Statistical models differ from thermodynamic models of rate constants (such as" absolute " reaction rate theory 20) in several important respects. First, thestatistical theory is set up to describe the probabilities of possible outcomes of singlewell-defined collisions rather than the averages over equilibrium ensembles of theseprobabilities. Thus, in a statistical theory one may ask for the cross section at aparticular initial velocity for the transition from initial state i to final state j .Thisquestion has no meaning in a thermodynamic theory of kinetics. Secondly, in astatistical theory one is forced to make assumptions only on the limits of a strongcoupling complex but not on the lifetime or the configuration involved. By contrast,in a thermodynamic treatment, the complex or activated state must formally betreated as a well-defined " molecule ", having a particular unobservable equilibriumconfiguration with known (but unobservable) vibration-rotation states. Thus theabsolute reaction rate theory is parameterized to a much higher degree than isnecessary in a statistical theory. This makes a meaningful comparison of thepredictions of the statistical theory with experiment easier than that for the absolutereaction rate theory.A statistical theory of dynamical events was first proposed by Fermi 21 to describemultiple particle production in high energy nuclear collisions.This theory has bee16 STATISTICAL THEORYthe basis for much subsequent work in nuclear physics. It is, however, different indetail from the statistical theory of chemical kinetics described below.In 1958 Keck,14 while investigating the kinetics of recombination, gave a clear,correct description of the elements of a statistical theory of chemical reactions in termsof the fluxes in phase space. This theory is equivalent to the corrected version of thephase space (or statistical) theory presented by the author and co-workers in 1965,22-26although the language is different.The statistical theory of kinetics may be constructed in several ways : classically,from the S-matrix, etc.We shall give here a brief derivation of the quantummechanical version, emphasizing only the roles of symmetry and conservation require-ments about which there has been some More detailed derivations canbe found in the literature.23* 28We shall consider the problem of a bimolecular exchange reaction involving onlythree atoms. We denote by a, p, y, etc., the possible chemical species (channels)which can be formed in this reaction (a denotes AB+C, for instance). We denotethe quantum numbers of the internal states of the diatomic molecule in channel a bysa, the total rotational and orbital angular momenta by L, and L respectively, and therelative translational energy in this channel by Ea.To proceed to the essence of the statistical theory we now need to make onedefinition and one assumption. First, a “ strong coupling complex ” is defined by asingle property :(a) the mode of decomposition of a ‘‘ strong coupling complex ” is uncorrelatedwith the mode of formation except through conservation laws and detailed balancing.The conservation laws are conservation of energy, total angular momentum and itsprojection on one axis, and linear momentum.If so desired, one could also includeelectron spin angular momentum as being (approximately) separately conserved.Whether or not such complexes exist, even as an average over unobserved quantitiesis open to question (and experimental verification). The lack of correlation expressedin (a) is, however, a fundamental assumption of a statistical, as opposed to deter-ministic, theory.In the work of Keck and the author (although not in that of Euand Ross 7), one further assumption is made which, although not absolutely necessary,seems reasonable and greatly simplifies (indeed permits) calculations. It can bestated as follows :(b) the probability of formation of a “ strong coupling complex” from everychannel a is either unity or zero, depending upon the values of Ea, L, L a and Sa. ThusThe total cross section for formation of a “ strong coupling complex ” from thecollision specified by a, E,, L,, s,, is thenThe classical analogue of this equation is obtained by equating, for large I,,,,, theclassical and quantum mechanical expressions for orbital angular momentum(hL = p u b ) ,o=(classical)(a,Eu,La,s=) = nb&x(a,Ea,La,Sa), (2)where b is the impact parameterJOHN C.LIGHT 17Assumption (b) is certainly an approximation. Quantum mechanics, showsclearly that except for singular potentials, probabilities are " never " one or zero,i.e., there is always some penetration of a finite potential barrier into a non-classicalregion and a finite probability of reflection of a particle with enough energy to passover a barrier. However, for sufficiently massive particles, such as atoms, approxi-mations such as (b) are numerically very accurate except for L-LmaX, i.e., except fornearly orbiting collisions which constitute a minor fraction of the total number.Thus (b) is, in essence, a " classical " approximation.It implies that the density ofstates of the " complex " is so high that all collisions satisfying (b) can be accommo-dated.*We shall now indicate briefly how statements (a) and (b) are sufficient to determinethe statistical theory of chemical kinetics, specifically for bimolecular exchangereactions. The argument would be trivial in a two-dimensional world in which onlyone orientation of every angular momentum were allowed. In real space of threedimensions the coupling of L and La to form J J, make the algebra more complicatedbut do not change the basis of the derivation. We shall show that statements (a)and (b), together with the principle of detailed balancing, uniquely determine theprobabilities of decomposition of the complex into any final state.On this basis, detailed balancing requires thatwhere p a is the reduced mass of the atom-diatomic molecule in channel a, L a therotational angular momentum, and ( a ] stands for the entire set of quantum numbersEa, Sa, La.Because of the definition (a), we may write the cross sections as productsof a cross section for formation of the complex and the probabilities of decompositionof the complex. Thus, for a given total energy,where J is the total angular momentum of the complex.The cross section for formation of a complex of specified J is, from eqn. (l),nh2where the last factor is the fraction of collisions with L, L a which couple to yield atotal angular momentum J.The summation over L is restricted by the stronger of the two conditions (a) thatCombining eqn. (5), (4) and (3), we havei L-La I <J< I L+La 1, and (b) that LSL,,, (01, Ea, s,, L a ) .Since P(J+(B)) is, by assumption (a), independent of (a), we haveThe last sums are equal, by definition, to the numbers of states available in thespecified channel which can form, or be formed from, a complex of specified energyand angular momentum :* This is not true for electron-atom collisions in which well-defined resonances have been observed.However, no such resonances have been observed in chemical reactions18 STATISTICAL THEORYIt may now be proved by induction, starting with J = 0, that the only reasonablesolution of (7) is the obvious one :The requirement that the sum of the probabilities of decomposition of any complexbe unity provides the normalization and leads to the final result :where the summation in the denominator runs over all states accessible from thespecified complex.Thus Ntot is the total number of states available from a complexof given total energy and angular momentum. Thus the cross section for the overallprocess isIn the derivation the relative density of translational states enters only throughthe energy dependence of the cross sections and the microscopic reversibility relation.This is because we have dealt with probabilities rather than fluxes. However, thetwo results would be equivalent.Let P, be the radial relative momentum (theonly free variable since the angular momenta are specified). The flux into each stateper unit energy range will be the velocity times the probability times the density ofradial translational states per unit energy. Thus we haveThe above theory could be modified by changing one or both of the statements(a) and (b) to allow, for instance, for preferential coupling of certain initial and finalstates. This could most easily be done in the S matrix formulation of the theory.28However, unless such modifications were made with a firm theoretical basis theirvalue would be open to question since it would correspond to ad hoc parameterization.The purpose of such a theory is not to fit every experiment but to give a generalmodel for which a priori calculations can be made and used for comparison and(hopefully) reasonably accurate prediction.Large scale tampering with the modelserves neither purpose.THE “STRONG COUPLING COMPLEX”In the preceding section we used only the formal properties of the ‘‘ strong couplingcomplex ” in the general derivation of the statistical theory. In this section we shalldescribe briefly how the limits of the complex, which are needed before any computa-tion can be done, can be chosen. Since the choice of the limits to be used in state-ment (b) completely determines all the cross sections, this is the only point at whichthe physics of the process under consideration can be inserted. Again one wants tochoose the simplest limits which seem physically reasonable.For reactions without activation energy we have reasoned as follows, using theclassical impact parameter for simplicity.There should be a distance ro(ol) for eacJOHN C . LIGHT 19channel within which the atom and molecule are strongly coupled. Since there is noactivation energy, ro should be approximately independent of the internal and trans-lational energy and the rotational angular momentum. Thus the limit of the strongcoupling complex required in (b) is the maximum impact parameter, bmax(EJ forwhich the atom will approach within r,(a) of the molecule. For low energies withlong-range attractive forces between the atom and molecule and with ro about equalto the range of the repulsive (if any) forces, this leds towhere C, and 6 are determined by the form of the attractive potential (in eachchannel).23 This form seems appropriate for ion-molecule reactions and for someneutral reactions such as K+Br,, C1+Na2, and Hg*(3Po)+H2.At higher energies,however, b,,, given above may be less than ro. This should not be allowed to happenand the correct limits for unit probability of formation of the strong coupling areSeveral calculations using eqn, (13) have been carried out for ion-molecule and neutralreactiorrs. Some of the results will be given in the next section.For reactions with activation energy, the limits for formation ofthe strong couplingcomplex must depend on translational energy and probably internal energy asOnly a few calculations have been made for these reactions and the definitions usedwere :Here the effective activation energy Eeff(ct) was defined as the minimum translationalenergy necessary to have reaction for the molecule in a given internal state.Thespecific dependence on internal energy of the molecule was linear :26where E i is the activation energy from the ground state of the reactants, Evib andErot are the vibrational and rotational energies of the reactant molecule and theA are positive constants between 0 and 1. Although the parameterization in thiscase is higher than for reactions without activation energy, this form, which followsfrom simple collision theory of kinetics, seems as simple as is physically reasonable.Before going on to results, a physical description of what a “ strong couplingcomplex ” might be seems in order. The mathematical properties are clear : once asystem enters such a complex it ‘‘ forgets ” where it came from.In fact this does nothappen. In practice, however, there are certain quantities associated with a collisionwhich one cannot measure such as the z component of the orbital angular momentum,the magnitude of the orbital angular momentum, the exact translational energy, etc.If the coupling between all open channels is strong, the one might expect that crosssections, averaged over the unobserved quantities, to behave statistically. This wouldseem to be particularly likely if the complex lasts a reasonable length of time classically,i.e., for periods larger than the rotational period.Alternatively, if the potentialgradients are large and not smooth, rather random scattering could occur. Thusthe strong coupling approximation is essentially the oppsite of an adiabatic approxi-mation in which the system never changes state. We shall see in the next sectionthat the adequacy of the theory depends, to a large extent, on the type of informationdesired. For highly averaged quantities the statistical theory works well but fo20 STATISTICAL THEORYmore detailed information the deviations from the statistical predictions becomemore apparent.Finally the limits in statement (b) are not the actual limits of the strong couplingregion itself. They are the limits, in terms of the asymptotic states of the system,within which a strong coupling complex will, at some time and in some place, be formedwith unit probability.Thus they define asymptotic trajectories which, if followed,will lead through a strong coupling region. No information is required aboutthe time spent in this region-only that it has the property defined in (a).RESULTSThere has been a number of specific applications of the statistical theory tovarious reactive and inelastic proces~es.~~~ 30 We shall present here only fourexamples in any detail. These examples are chosen to illustrate the types of systemsto which the theory can be applied, and the types of information which can be obtainedand compared with experiment.The four cases are H +Cl,, K+ HBr, He + H;, and Hg* + CO. These illustratethe application to distinct types of reactions : exchange reactions with activationenergy, exchange reactions with negligible activation energy, ion-molecule reactions,10. I I I t I 1 \ I0 I 2 3 4 5 6nFIG.1.-Relative reaction rates kn/ka into vibrational states n for the reaction H+C12 -tHCl(n)+ C1. - - El - - calculated, Xo = Ar = 1.0 ; - - 0 - - d c . , hu = Ar = 0.97. (The experimental values forn<2 are lower than for n = 3 (ref. (34)).and excited atom-molecule reactions. These cases were all studied by the authorand co-workers 22-26 in an attempt to define the limits of applicability of the theoryby comparison with experiment.A. H+C1226The luminescent studies of this reaction by Polanyi 31-33 and co-workers haveprovided unparallelled information about the internal excitation of the products.The reaction has an activation energy (of about 2.0 kcal/mole) and thus the definitionof the complex in eqn.(15) was used in the statistical theory. The calculation wasdone with only the vibrational degree of freedom quantized, the angular momentabeing treated classically. Since the experimental information was for thermal systems,all quantities were averaged over the appropriate Maxwell-Boltzmann distribution.The only significant parameters to enter the calculation are 1, and &, which measurJOHN C. LIGHT 21the fractional contribution of internal energy in reducing the effective activationenergy, and the ro for each channel-the radii of the complex itself. The values ofro apparently have little effect in the statistical calculation except the gross crosssections.However, the values of ;I affect the internal energy distributions consider-ably. The internal calculated energy distributions of the product are plotted in fig.1 and 2.These results illustrate one apparently general result. With complexes definedonly by a single inequality on the impact parameter as a function of the energy variables,the statistical theory never predicts complete population inversions of products. ThereFIG. 2.-Rotational distributions of HCl. -a- calculated for HC1 (n = 2), hu = hr = 1.0,EOr = 0.1 eV; 0 , expt. (ref. (31)-(33)).are always more states available from a lower vibrational state than from a high oneand thus the populations of vibrational states of products fall monotonically withvibrational quantum number.This result has appeared in every statistical calculationto date. Since there is a large class of reactions proceeding with and without activa-tion energy for which the best indications are that vibrational population inversionsare produced, we find that the statistical theory is unreliable with respect to this typeof information.The rotational distribution given by the statistical theory cannot be directlycompared with experimental results since no firm experimental quantities are known. 34It seems, however, to be a reasonable distribution since high rotational excitation,after some relaxation, is still observed.33 The overall rate constants as a functionof temperature were computed.The result, from the statistical theory, isk(T) = 2.1 xThis seems in line with rates of similar reactions, but no good absolute values areavailable from experiment. One interesting point is that the value of 2.2 kcaljmoleappearing in the exponential factor of the rate coefficient is 10 larger than theactivation energy Eg assumed in the definition of the complex. This effect was alsofound in the studies of Karplus, Porter, and Sharrna.l3B. K+HBr 2 5 * 26This reaction, studied experimentally by molecular beam rnethod~,~~’~’ proceedswith an activation energy of less than 0.4 kcaljmole. Considerable information hasexp (-2200/RT) cm3 sec-’ molecule-l22 STATISTICAL THEORYbeen deduced from the experiments about the details of the reaction: probabilityof reaction against impact parameter, rotational, vibrational, and total energydistributions of the products, etc.In this reaction, the value of Ex = 0.15 kcal/mole(best experimental estimate) was used and A, = 0-8 and Lr = 0.4 were chosen to giveFIG.0.2 0.4 0.6 0-8R3.-Probability of reaction against reduced impact parameter for K+ HBr.experimentally deduced points, ref. (35). Ez = 0.08 eV.1 40 r-A0 = 0.8. 0,Et AeVjFIG. 4.Reaction cross section for KfHBr against translational energy, EG (eV). A, = 0.8,Ar = 0-4. -0- experimentally deduced, ref. (35). -@-, calc.the best agreement with the " experimental " probability of reaction against impactparameter. The value of ro = 3.8A for the product channel was taken fromS~plinskas.~~ The aim of the calculation was to see how much of the experimentaldata could be reproduced by the statistical theory with one set of 5 parameters(&, 4, G, e&,, r&))JOHN C .LIGHT 23In fig. 3-7 we plot the comparisons with experiment. It can be seen that for theprobability of reaction against impact parameter, the reaction cross section againsttranslational energy, and the rotational energy distribution, the agreement between--I---l-1-- rFIG. same04.maximum.nFIO. 6.-Vibrational distribution of KBr produced. -A- caIc., hv = 0-8, Ar = 0.4;-0- Cak. hv = 0.9, hr = 0.4; - " Typical " Monte Car10 result, ref. (10)24 STATISTICAL THEORYthe statistical calculation and the experimental estimates are good. We again find,however, that the vibrational distribution of the product molecule is probably notgiven correctly by the statistical theory, and this leads to a shift of the total internalenergy distribution of the product to lower energy.This is sensitive to the assumedvalue of Av and a value of A, = 0.9 improves the agreement considerably, but doesnot reproduce the curve of the probability of reaction against impact parameterquite as well. Since the only quantities varied were A, and A?, we can say that forthis reaction, with E' and the r0 given, the values = 0-8 or 0-9 and 3.r = 0.4reproduce well all the experimental information except, probably, the vibrationalenergy distribution.&ItCeVFIG. 7.-Total internal energy distribution of KBr produced. -D- --CI calc., Xu = 0.9,hr = 0.4; -0-0- calc., hu = 0-8, Ar == 0.04. Curves normalized to maximum = IEOr = 0.08 eV.Arrow indicates experimental maximum (ref. (36)).C. Hg*+CO 26The internal (vibrational) energy distribution for the CO produced in the quenchingreaction of Hg(3Po) has been studied by Karl, Kruse, and P ~ l a n y i . ~ ~ The statisticaltheory, with a value of A, = 0.98 and an activation energy of about zero (0.005 eV)gives moderately good agreement with the experimental results as shown in fig. 8.That this model is not unique in giving reasonable results has been shown by Karlef al., who used a model of impulsive energy release in a Hg*CO complex. Additionalcredibility is lent to the statistical calculation for this type of reaction by the recentwork of K.Yang and co-workers 30 on the Hg*+H, reaction. They used thestatistical theory to predict accurately the quantum yield of H atoms. Thus we mayhope that excited atom-molecule interactions approximate the strong coupling com-plex with some degree of accuracy.D. He+Hz24As one of the simplest ion-molecule reactions, this reaction, He + H:-+ HeHf $. H,has been studied in detail e~perirnentally,~~-~~ although no information concerningenergy distributions of products is available. For this reaction the limiting impactparameter for formation of the complex was taken from the Langevin 44 theory,i.e., it is determined by the requirement that the particles be able to pass over theangular momentum barrier. Since this is determined only by the polarizability oJOHN C.LIGHT 25the neutral species, there are no adjustable parameters. The initial vibrational dis-tribution of the H i was taken from the caIculations of W a c k ~ . ~ ~ It is critical sincethe reaction is endothermic from the ground state by 0.92 eV. Thus at low trans-lational energies, only vibrationally excited H i ions can react.FIG.W I 2 4 6 8 10 12 I4 16 18 2 0V8.-Vibrational energy distribution of CO from the reaction Hg(3Po)+ CO +Hg+ CO*. - calc., hu = 0-98. - - - - calc., hu = 1.0. - - -, expt., ref. (39).6l ' ' \ G + S iEcM(ev)FIG. 9.-Cross section for He+Hz -+HeH++H against Ea. Initial vibrational distribution ref. (45). - - - d c . , - ..-- total " strong coupling " cross section, -0-0 expt.ref. (41)26 STATISTICAL THEORY30-- 2.0-3230\" ir.0Xw* to-0'The cross sections for reactions from each vibrational state of H i were evaluated,and the overall cross section was obtained from the sum, weighted by the fractionalpopulation of each state. The results are compared with two sets of experimentaldata in fig. 9 and 10. In fig. 10, the calculated results were appropriately averaged1 I 1 I I 1-Q QQ --I I I I -0 1 2 3 4 5 6 71.0 2.0 3.0 4.0 5.0ECM(WFIG. 1 1.-Average isotope ratio against average translational energyHeH++D.He+HD++{ HeD++H7 calc., - .. 0 - - expt., ref. (42)JOHN C . LIGHT 27to conform to the experimental velocity distribution in a conventiona'l mass spectro-meter. Of interest is the fact that the calculated cross sections fall off at low energybecause of the purely statistical factor that more inelastic than reactive channelsare open there.Thus the correct result is given without assuming that a minimumtranslational energy (activation energy) is necessary.46 The isotope ratios for productsof the We-!-HD+ reaction were also calculated and compared with experiment.Fig. 11 shows that the qualitative dependence of this ratio on energy is reproducedwell but the absolute value of the ratio is in error by approximately, 20%.In general, the application of the stgtistical theory to low energy ion-mo!eculereactions has been productive. The most important effects seem to be predictedwith reasonable accuracy (such as ratios of possible products 29 and velocity depmdence of cross sections).It is likely that, because of the strong long range attractiveforces involved, the ion-molecule reactions at ZOW energies proceed through a complexwhich is strongly coupled. A good discussion of these processes is given by Futrelland Abrarnsox~.~~SUMMARY AND DISCUSSIONThe results presented in the previous section, while typical, represent only a smallfraction of the cases studied by the statistical method. As in the examples above,the results, in comparison with experiment, vary from being qualitatively andquantitatively in error to being quantitatively correct. It is now possible to setcertain limits on the applicability of the statistical theory. There are two factorswhich must be considered: first, the type of information desired, and second, thetype of reaction involved.In order to summarize the experience to date, we give, in table 1, the evaluationof the accuracy of the statistical theory for the specific types of reactions and infor-mation.The agreement is best for highly averaged quantities in ion-molecule reactionsTABLE 1 .-RELATIVE ACCURACY OF STATISTICAL THEORYatom-molecule atommolecule \ reaction ion-moIecule excited reactions reactionsreactions molecule no activation activation \ (Ex 10 eV) reactions energy energyinformationthermal rate constants good good faircross section against energy good goodisotope and product ratios fair to goodgoodtotal internal energy of products fair fair fairrotational energy distributions good fair( ?)vibrational energy distributions fair poor poorwhich are likely to have the strongest coupling, and worst for detailed informationabout neutral reactions with activation energy, even though one parameter Av wasvaried to some extent.There are three questions which it seems pertinent to attempt to answer (i) Arethere any types of reactive systems for which the basic assumptions of the statisticaltheory, (a) and (b), are approximately valid in detail, i.e., on a state for state basis?(ii) Are there any types of information about specific types of reactions for whichthe statistical theory gives reliable quantitative results ? (iii) When, if ever, should oneuse the statistical theory?The answer to the first question is easy, but ambiguous.No reactive system hasbeen shown to behave statistically in detail, but many have been shown to react in 28 STATISTICAL THEORYnon-statistical manner. The only types of reactions which might be truly statisticalare low energy ion-molecule reactions for which adequate experimental informationto give a definitive answer is not available. Thus we may conclude that it is possiblebut not likely that any real systems react statistically in detail.The answer to the second question is yes. Averaged information about ion-molecule reactions and excited atom-molecule reactions has usually been givenaccurately by the statistical theory.48 Quantities such as rate constants, crosssections against energy, and, perhaps most important, ratios of products can becomputed reliably by the statistical theory.For reactions of neutral, unexcitedspecies this holds true to a lesser degree.The third question is, perhaps, the most important. There appear to be tworeasonable uses of the statistical theory : as a model against which to compare experi-mental results, and as a predictive tool in the absence of much experimental informa-tion. The utility as a model was illustrated for the HefH; reaction. The fall-offin cross section for reaction as the translational energy is decreased is explained bythe statistical theory as the statistical behaviour for an endothermic reaction pro-ceeding from excited vibrational states. Although it cannot be asserted that thisis the correct explanation, its simplicity and accordance with known facts of thresholdbehaviour give it added weight vis-a-vis an alternative explanation in terms of akinetic energy threshold for ion-molecule reaction~.~~ In the opposite sense, thefailure of the statistical theory to account for the vibrational energy distribution inthe H-kCl, reaction means that a more deterministic model is both necessary andreasonable as the accurate approach.Finally, the author believes that the statistical theory can be used as a valuablepredictive tool in some cases.It is fairly simple, flexible, and not highly para-meterized so that it can be applied to systems for which little information is available.For example, the reactions H+HX (X = Cl, Br, I) yield some excited halogenatoms as product.The fraction of excited atoms produced is unknown. Thestatistical theory was applied to yield the following per cent excitation in thermal(300°K) reactions: Cl ("p,) 29 %; Br ("+) 17.5 %; I ("p,) 12 %.The statistical theory appears to be useful for those cases where alternative productsare available and one is interested in determining the ratios of possible products. Theinformation desired is highly averaged and so the results should be fairly reliable.In addition, little information about potential energy surfaces or their crossing pointsis needed so the computation can be carried out quickly and efficiently.We may thus consider the statistical theory as an '' interim " theory-a semi-quantitative model to be used in a comparative or predictive way until we have theability to make accurate a priori calculations of all kinetic quantities of interest.The author acknowledges the real contributions to the foundations, development,and applications of the theory made by Dr.Philip Pechukas and Dr. Jeong-long Lin.The author also gratefully acknowledges the general support of the Institute for theStudy of Metals (James Franck Institute) by the Advanced Research Projects Agency.D. L. Bunker, Theory of Elementary Gas Reaction Rates, (International Encyclopedia ofPhysical Chemistry and Chemical Physics, Topic, 19, Vol. 1) (Pergamon Press, New York,1966).K. J. Laidler and J. C. Polanyi, Prog. Reaction Kinetics 1965, 3, 1.J. Ross, ed., Ado. Chem. Physics, vol. X, (Interscience, New York, 1966).S.Datz and R. E. Minturn, J. Chem. Physics, 1964,41, 1153.K. R. Wilson, G. H. Kwei, J. A. Norris, R. R. Hem, J. H. Birely and D. R. Herschbach,J. Chem. Physics., 1964, 41, 1154.4T. T. Warnock, R. B. Bernstein, and A. E. Grosser, J. Chem. Physics, 1967, 46, 1685JOHN C . LIGHT 29J. C. Polanyi, Chem. in Britain, 1966, 2, 151.R. J. Suplinskas, Thesis, (Brown University, 1965).J. L. Magee, J. Chem. Physics, 1940,8, 687.lo N. C. Blais and D. L. Bunker, J. Chem. Physics, 1962, 37, 2713; ibid, 1963, 39, 315.l 1 J. C. Polanyi and S . D. Rosner, J. Chem. Physics, 1963, 38,1028.l2 M. Karplus and L. M. RaE, J. Chem. Physics, 1964,41, 1267.l 3 M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Physics, 1964, 40, 2033.l4 J. C. Keck, J.Chem. Physics, 1958,29,410.l5 0. B. Firsov, Zhur. Eksperim. i. Teor. Fiz. 1962, 42, 1307, (Eng. trans. ; Soviet Phys.-JEPTl 6 J. C. Light, J. Chem. Physics, 1964,40, 3221.l7 E. E. Nikitin, Theor. Expf. Chem., U.S.S.R., 1965, 1,428.l8 F. T. Wall, L. A. Hiller Jr., and J. Mazur, J. Chem. Physics, 1958, 29,255 ; ibid., 1961,35,1284.l9 K. E. Shuler, T. Carrington and J. C. Light, Appl. Optics. 1965, supp. 2, 81.2o S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes, (McGraw-Hill, New21 E. Fernii, Progr. Theor. Physics, Japan., 1950, 5, 570.22 J. C. Light, J. Chem. Physics., 1964, 40, 3221.23 P. Pechukas and J. C. Light, J. Chem. Physics, 1965, 42, 3281.24 J. C. Light and J. Lin, J. Chem. Physics, 1965, 43, 3209.2 5 P. Pechikas, J. C . Light and C . Rankin, J. Chem. Physics., 1966, 44,794.26 J. Lin and J. C. Light, J. Chem. Physics, 1966, 45,2545.27 B. C. Eu and J. Ross, J. Chem. Physics, 1966,44,2467.28 P. Pechukas, Thesis, (University of Chicago, 1966).29 F. Wolf, J, Chem. Physics., 1966,44, 1619.30 K. Yang, J. D. Pader and G. L. Hassell, private communication.31 J. C. Polanyi, J. Quant. Spectr. Radiative Transfer 1963, 3, 471.32 J. R. Airey, R. R. Getty, J. C. Polanyi and D. R. Snelling, J. Chern. Physics, 1964, 41,3255.33 J. R. Airey, F. D. Findley and J. C . Polanyi, Can. J. Chem. 1964,42,2193.34 The recent (unpublished) work of J. C . Polanyi et al., provides some information on the limits35 D. Beck, E. F. Gruen and J. Ross, J. Chem. Physics, 1962, 37, 2895.36 A. E. Grosser, A. R. Blythe and R. B. Bernstein, 9. Chem. Physics, 1965, 42, 1268 ; A. E.Grosser and R. B. Bernstein, ibid., 1965,43, 1140.37 S. Datz, D. R. Herschbach and E. H. Taylor, J. Chem. Physics, 1961,35, 1549 ; D. R. Hersch-bach, Disc. Faraday SOC., 1962,33, 149.38 R. J. Suplinskas, private communication.39 0. Karl, P. Kruus and J. C. Polanyi, J. Clzem. Physics, 1967, 46, 224.40 General reviews of ion-molecule reactions are available in Adv. Chem., vol. 58, (Amer. Chem.41 C. F. Giese and W. B. Maier 11, J. Chem. Physics, 1963, 39, 739.42 M. von Koch and L. Friedman, J. Chem. Physics, 1963,38, 1115.43 C. F. Giese, Adv. Chem. Physics., (Interscience, New York, 1966).44 P. Langevin, Ann. Chem. Physics., 1905,5,245 ; G. Gioumousis and D. P. Stevenson, J. Chem.45 M. E. Wacks, J. Res., Nat. Bur. Stand. 1964, 68A, 631.46 L. Friedman, Ado. Chem., vol. 58, (Amer. Chem. SOC., Washington, D.C. 1966), p. 87.47 J. H. Futrell and F. P. Abramson, Adv. Chem., vol. 58, (Amer. Chem. SOC., Washington, D.C.1966), p. 107.48 There are exceptions such as O++N2 (cf. C. Giese, ref. (40), p. 20) for which an activation energyappears to occur for an exothermic reaction. In these cases the normal definition of theion-molecule complex appears to be in error.1962, 15,906.York, 1941).of the initial distribution.SOC. Publ.) 1966, Washington, D.C.Physics, 1958, 29, 294

ISSN:0366-9033    

DOI:10.1039/DF9674400014

出版商:RSC    

年代:1967

数据来源: RSC

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Markovian Kassel TheoryBY M. R. HOARE* AND E. THIELE~Received 19th June 1967The possibility of extending Kassel’s theory to allow for Markovian fluctuations in the criticaloscillator energy is examined using a simple model based on pairwise interaction of normal modes.The results of this are numerically not very different from those of ordinary Kassel theory ; neverthe-less rate parameters can be calculated which include the effects of intramolecular relaxation andnon-random collisional excitation, quantities uninterpreted in the simple theory. The modelinvestigated shows that the condition of detaifed balance places such stringent requirements onallowable transition probabilities for the critical oscillator as to make it extremely difficult to extendthe notion of a “ collisional ” type intramolecular energy transfer much beyond the original Kasselterms.In the Kassell approach to unimolecular reaction theory 1* a necessary andsufficient condition for the reaction AB+A + B is the accumulation in some “ criticaloscillator ” (C.O.) in AB of an energy x greater than a threshold value Eo.Allmolecules with an energy E>Eo are held to be activated and, subject to an energy-dependent time-lag, will decompose if not first deactivated by collision. In thetraditional view reaction proceeds (i) via a Boltzmann distribution f ( E ) of energizedspecies with (ii) microcanonical distribution of critical oscillator energies bothimmediately after collision and throughout the lifetimes of a reacting ensemble, and(iii) complete deactivation to energies E<Eo should a collision take place before theattainment of the condition x>Eo.A further assumption (iv) that the probabilityof collision is uncorrelated to internal energy states is tacitly added to the above.In the simple Kassel theory, assumption (ii) leads by combinatorial considerationsalone to the intrinsic rate constant,which corresponds to the random lifetime distribution,g ( t ) = ka exp (-kat).Abandonment of the assumptions (i)-(iii) makes it necessary to treat the history ofthe individual molecules as a complex stochastic process with “ ladder-climbing ”replacing single-step activation and deactivation and an auto-correlated fluctuationof energy in the critical oscillator replacing Kassel’s simple randomization.Withthis it becomes possible to introduce the effect of intra-molecular relaxation timesand non-random initial distributions and it is the need to represent these quantitiesin a theory of lifetimes that justifies the considerable extra effort demanded by thestochastic approach. The simplest step, which retains much of Kasssl’s originalformulation, is to assume that the critical oscillator fluctations are Markovian incharacter and respond to the other degrees of freedom in the molecule as though to aheat-bath. The starting point of a lifetime theory will then be a transition kernelK(x, y [ E ) specifying the probability of a transition from x to dy about y in the criticaloscillator energy, given that the total energy in the excited molecule is E.* Bedford College, London, N.W. 1. t Johns Hopkins University, Baltimore, Md., U.S.A.3M. R . HOARE AND E. THIELE 31Our present contribution is concerned only with the removal of assumption (ii)within the established framework of assumptions (i) and (iii). We outline a Markoviantheory which starts from a simple assumption about the mode of energy transferbetween oscillators and leads, in effect, to the lifetime distribution g(E,Eo,t) forexcited molecules. This function, averaged over the collision-time distributionyields a specific rate contant,I@) = w y(E,t) exp (-wt)dt, (3) so which integrated in turn over the Boltzmann distributionf(E) gives a general first-orderrate constant kuni(m) with implicit " fall-off " behaviour (for details and generalbackground, see, e.g., Bunker 3 9However, first, the lifetime-distribution g has the full dependence g(E,Eo,s,c(x,O) ;t),i.e., in addition to the usual parameters it is a functional of c(x, 0) the distribution ofcritical oscillator energies immediately after collision (which could itself be a functionof E ) ; secondly it is the Laplace transform ;(a) appearing in (3) that we require, afunction that is more accessible mathematically than g(t) itself.and Slater.12)TRANSITION KERNELOur basic assumption is that the sequence of energies in the critical oscillator isnot the micrccanonically distributed random one postulated by Kassel but a Markovchain governed by a '' collisional " type of interaction between oscillators in pairs.We choose this partly for its simplicity, and partly because it would seem to maximizeany possible discrepancy with " ordinary " Kassel theory due to the Markovianproperty. In keeping with the spirit of the original theory we prescribe nothingabout the nature of the inter-oscillator " collisions " ; they are simply that effect whichon a certain time-scale effaces all meaory in the critical oscillator fluctuations, i.e.,makes them Markovian.We claim no special priority in using the Markovianzflproach which can be traced back to Zwolinski and Eyring via Wilson.' However,these authors used a '' quantum " formulation throughout and no treatment seemsto have been given in the original Kassel language with continuous energies. A pair-wise redistribution of energy between normal modes has also been treated by Bunkerbut on a purely deterministic basis different from the present model.alsoexcludes Markovian effects from his modified Kassel theory.In so far as normal mode energies are well-defined at any instant, a completeenergy-description of the excited molecule would involve knowledge of the successionof " microstates " : ( E ~ , e2 . . . E, I E), where the individual normal mode energiesE , , c2 . . . E, sum to E. To make any progress we must obtain a " macrostate "description ( E ~ 1 E ) with the energies E~ . . . E, unspecified. To achieve this there seemsno alternative but to assume that e2 . . . es, by virtue of their own interactions, presentto the critical oscillator an effectively microcanonical distribution at energy E- cl.There seems no objection to this, except perhaps for very large fluctuations and atthe smallest s values.solcWe first express the transition kernel K(x,y 1 E) for changes in in the form,K(x,F I Ejdy == F(xtEIQ(e,y)dcciy (4 r" (cf.Hoai-e *). P(x, E ) ~ E is the probability that the C.O. with energy x will collidewith another to give joint energy E, Q(E, y)dy is the probability that they will divideleaving energy y in the C.O.( 5 )By Kassel-type combinatorial arguments we findP(s, E) =; ( ~ - 2 2 ) ( E - : ; ) ' - - ~ / ( E - s ~ - ~ ; X<C <,E= o ; x>E32 MARKOVIAN KASSEL THEORYAt first sight there would seem to be a possible latitude in the choice of Q(8,y).However, if the final K(x, y I E) is to satisfy the detailed balance condition,K(x, Y I a@,(& E ) = K(Y, x I E)@o(Y, a.@O(X, E ) = (s- 1)(E-x)"-2/E"-',(6)(7)with @&, E) the equilibrium microcanonical distribution,then Q(E, y ) must be a uniform distribution :Thus the idea of pairwise interaction together with detailed balance must lead to thekernel,(A simple " time-reversal symmetry ", Q(&,y) = Q ( E , E - ~ ) is not sufficient to givedetailed balance.) The kernel (9) clearly satisfies the normalization condition,F Eand the equilibrium condition,The transport equation (Master equation) for the distribution function of criticaloscillator energy, c(x, t), at time t can now be written aswith K as above.Here we have scaled the time in units of an intra-molecular collisiontime 1 /v' (not necessarily equivalent to the Kassel factor 1 / v ) and have suppressed theE-dependence in the notation.K represents a coupling between oscillators which,although weaker than the ordinary Kassel randomization is still " strong " by anyother standard. Nevertheless, the only way in which it can be weakened withoutprejudice to the detailed balance condition is by regulating energy transfer through an" all or nothing " mechanism,KYx, Y I E ) = (1 -y)6(x-Y)+YK(x, Y 1 (1 3)with coupling parameter 0 <y <l. But the result of this on substitution into (12) isonly to extend the time scale by a factor l/y without significant effect on the overallform of the theory. There is possibility that y might itself be a function of E, butwe shall assume y = 1.RELAXATION OF AVERAGE ENERGYThe relatively strong coupling implicit in the model is best brought out in therelaxation equation for the ensemble-averaged energy in the critical oscillator,E<& 1 (t)> P( ~ ' ( t ) ) = 1 xc(x,t)dx.(14M. R . HOARE AND E. THIELE 33On multiplying eqn. (12) by x on both sides and integrating by parts, we obtaini.e., there is exponential relaxation of the mean energy with a characteristic time ofze = 2 (1 - 1 / s ) collision times, independently of total energy E and initial distributionc(x,O). The dependence on s is relatively weak : s-+ 00,2,+2 ; s+2 ; Te-+ 1.GENERAL RELAXATION PROBLEMThe solution of equation (12) to obtain the time-dependent distribution functionc(x,t) when no reaction occurs (Eo = E ) is of some intrinsic interest, but is alsoneeded in the more difficult problem of obtaining g (t).By separation of variables :c(x,t) = @(x)O(t) we findwhere Z is the integral operator acting on the right-hand side of (12). The eigenvalueproblem is most easily solved on converting to differential form. On twice differ-entiating (17) and rearranging, it is equivalent to(i/e)(aeiat) = A- I ; (16) XQ = a@, (17)(z = x/E). Eigenfunctions of the operators X and 9 are the modified Jacobipolynomials, with eigenval~es,~\Im(z) = F(-n, n+s-1, 1, z) =A, = (s - 2)/(n + 1)(n + s - 2).(20)(21)while (22)We have normalized the Qn such that COO is the microcanonical equilibrium distributionof eqn.(7). The genera1 solution for arbitrary initial conditions is then= [a !( 1 - ~)"-~]-l(d"/dz")[~"( 1 - ~)s+"-~),c(z,t) = cDO(z)+ ak@k(z) exp [-(l-&)t]k=lwith the constants ak to be determined by the orthogonality property,f lIn terms of the relaxation times zn = (1 -An)-' we have zo = co, z1 = 2(s- I)/$,z2 = 3~/2(s- l), z, = 1. Thus, apart from zo = GO, which represents equilibrium,for moderate s all relaxation times lie very nearly between two collision times and thefastest possible transient, one collision time. The ratio zl/z2 will govern the validityof an approximation which neglects all transients except exp (-t/zl) and for thepresent spectrum and moderate s such an approximation begins to hold to betterthan 10 % after some 10-20 collision times.LIFETIME PROBLEMThe lifetime problem consists in the solution of eqn.(12) in the perturbed formwhich results from replacing E by an activation energy Eo in the limit of the integral.The determination of a full time-dependent distribution function c(x, E, Eo; t ) in34 MARKOVIAN KASSEL THEORYclosed form is probably impossible, though numerical solutions could be obtainedrelatively easily by expanding the truncated kernel in the eigenfunctions of the ordinaryone.However, the main object of a lifetime theory, the determination of a generalrate-constant kUni(Eo,co) requires less than a complete knowledge of c(x,t). Weshall now consider a number of rate parameters arising from the " Jacobi '' kernel (9) :(a) the " equilibrium " rate constant keq, (b) the high-pressure rate constant k , ;(c) the specific rate constant k(E) ; (d) the mean first passage time for reaction i ; and(e) an upper-bound rate constant k,,,.EQUILIBRIUM RATE CONSTANTThe rate constant k,, is defined as the first-order constant which would governthe overall reaction with a given kernel, provided that an equilibrium distributionproportional to CDo(x,E) were maintained during the whole course of the reaction.Thus, if the total unreacted proportion of the ensemble at time t isC(t) = jEoc(x,t)dn, k,, is given by0k,,(E,,E) = - C-'dC/dt = />.(..E)[ Eo K(x,y 1 E)dydx// "CD0(~,E)dx.0 (25)This expression can equally well be a statement of the reacting flux across E, or afirst-order perturbation expression for the quantity (1 -Ao) which is shifted from zeroon truncation of the kernel at Eo.Evaluating the integrals by parts we find- EO .(26)Eok e q ( W 0 ) =The last integral has a simple asymptotic form for E,+E and using this it follows thatE0-EThis is identical with the Kassel intrinsic rate constant apart from the factor(s-~)/(s- l), which evidently reflects the peculiarity of the case s = 2 in our theoryand the fact that the critical oscillator is never affected as favourably as in Kassel'srandomization. Eqn. (26) is probably, though not necessarily, valid only in theregion of Eo/E where the simplification (27) would apply.HIGH-PRESSURE RATEIt follows from eqn. (3) that the high-pressure specific rate-constant k,(E) isEvidently, identical with g(E,O), the zero-time intercept of the lifetime distribution.EFor the special case where c(x,O) is a microcanonical distribution k , becomesidentical with keq.The general high-pressure constant follows by integration overthe Boltzmann distributionf(E)M. R. HOARE AND E . THIELE 35SPECIFIC RATE CONSTANT k(E,CI))We have not succeeded in solving the transformed Master eqn. (12) to obtaink(E,w) in finite form, but a series expansion can be found which illustrates thebehaviour of this function and would be suitable for computation. First, considerthe three Laplace transforms Z(x,co), ?(m) and c(co) corresponding to c(x,t), C(t)and g(t) (energy E suppressed). The following relations hold :c(m) = IEoF(x,m)dx, 0 (29);(#) = C(O)+W&).(30)Z(W) is also the moment generating function for the passage-times of molecules toreaction :As a first stage c"(x, m) will be found. Transforming eqn. (12) :(co + 1)xGm) - C(X,O) = K(x,y)?(y,m)dy. J:Differentiating twice and rearranging gives ( z = x/E),+(S-2)Z(z,co) O + l = S[c(z,O>]/(m+ l), (33)wher 53 is the same differential operator as appears in eqn. (18). The inhomogeneouseqn. (33) can now be solved by a series expansion using the already-determinedeigenfunctions of 9.' O Thus, puttingc(s,o) = zkak@,(z). (34)z(z,a) = xkbk(cD)@k(Z)and substituting into (33) we obtainc k h k f / k - p ) @ k ( Z ) = (1 + W)-'&Cfk/lk@k(Z)~ 1 . = - ( s - ~ ) / ( c o + 1 ) ; /.lk = -(k+ l ) J ( k . + ~ - 2 ) , (m#O)(37)(38)(39)withwhich, since the @k are linearly independent, implies thatThe expansion for;(x,co) is now known. Using a standard formula for the integra-tion of @k(Z), it follows from (36) thatbk = akclk/(w f 1)(pkC(U) = ( ~ * ~ ~ ) ~ b ~ ( ~ ) F ( 1 + k, - s -I- 2, 2, EO /E), (40)kwhich vanishes for Eo = E.Thus, finally, on writing out the b k :gUW,Eo, C(X,O), 4 =(EO/E) 2 a k ( I [ ( k + k ( k + s - + s-2)]}-1F(1 1 ) -I- k , k - s +2,2,E0/E). (40k = 36 MARKOVIAN KASSELL THEORYWe note the correct behaviour of the above series in as much as 5 +O as either E,+Eor cr)+co. By considering the limit of cog as a+co we can obtain a series expansionfor k, as an alternative to (28). Eqn. (41) thus contains all the information necessaryfor the computation of a fall-off curve.MEAN PASSAGE TIMEThe mean passage time f for reaction is a convenient measure of rate which,under random lifetime conditions, becomes equal to 1/keq.l1 It might be foundthrough the relation : t = (d?/dt),=, and the expansion (41), but is also obtainablein finite form.By a method of variation of parameters applied to the ci) = 0 formof eqn. (32) we found a closed form for c(x,O) and hence t by the property :8Eo0f = 1 Z((x,O)dx.The result iswhere ke, is as determined earlier and Pd is the total probability of making a down-transition from Eo :In fig. 1 values of l/f k,, are plotted for s = 9 and different ratios E/Eo. Thisquantity will be greater or less than unity according to whether reaction is on theaverage faster or slower than the random lifetime prediction.Curve B is for acompletely “ cold ” initial distribution c(x,O) = 6(x) ; curve A is for a micro-canonical distribution c(x,O) = (D,(x,E) ; x<Eo. Curve C is for the ‘‘ hottest ”possible initial conditions, a delta distribution c(x,O) = 6(x- E;) corresponding toall molecules initially at the threshold of reaction. For this case the integral in (43)vanishes and f = 1 +P,/k,,. Reaction from a microcanonical initial distribution ismarginally faster than from the “ cold ” distribution B (which must set a lower bound)and marginally slower than the random lifetime rate, but, as would be expected withsuch relatively strong coupling, the differences between all these cases are small evenat unrealistic values of E,/E.The condition C does give a noticeable increase inrate, but is physically unreal and of interest only as an upper bound.UPPER-BOUND RATEFor the special initial conditions C, we have also obtained a W t e expression foritself. This is again not of great physical importance, but nevertheless provides anupper bound to the general rate constant and illustrates the great complexity in thea-dependence of a non-random-lifetime rate expression. The result, which comesfrom a method of variation of parameters applied to eqn. (32) with the appropriatec(x, 0) is(45)1 - w(EolE)A(a,b,o)F(a,b,2,EolE),a+b = -(s-3); ab = -(s--)/(l+w),where a and b are the (interchangeable) solutions of the equations,(46M. R . HOARE AND E.THIELEand A is the function,37A(EO,E,CO) = (EoiE)pdbl ~ +cu)F(a,b,l,E,IE)-1 + 0(s-2)(EolE)F(1 - a , l - b,2,E0/E) JE (1 --:-3 dE}-l (47)EoClearly the first term in eqn. (45) represents the transient exp ( - t ) in g ( t ) due tosystems which react at the first collision while the second term is an explicit, thoughEOlEFIG. 1.-Mean passage time and initial distribution (s = 9) A, microcanonical distribution ; B, ‘‘ cold”distribution ; C, “ hot ” distribution.complicated, function of pressure. Nevertheless, on evaluating (d; /dw),= werecover the correct passage-time result of the previous section : t = 1 +Pd/keq.One further interest attaches to the result (47). If one draws a tenuous parallelbetween the present theory of critical oscillator fluctuations and the more detailedphase-space theory of the Slater New Approach l2 then the initial condition c(x, 0) =6(x- E;) would be the appropriate one for determining the “ gap-distribution ”h(t) and the expression obtained for g(co), weighted to remove the contribution ofthose systems which go unbound on the first collision, would be the transform of thisfunction.CONCLUSIONIn conclusion, the above is intended as a model theory illustrating that, withoutreference to any form of random lifetime assumption, a simple hypothesis about themode of intermolecular energy transfer can lead to an effective knowledge of lifetim38 MARKOVIAN KASSEL THEORYdistributions and consequent " fall-off" behaviour.At the same time-and thisis perhaps more important than the special results-we find that the restrictiveimplications of the detailed balance condition prevent any simple Markovian theoryfrom becoming very different from an ordinary Kassel theory.Whether, withinthis type of theory, the flow of energy between modes might be weakened without lossof detailed balance, for example, by the introduction of an energy-dependent couplingparameter, remains to be seen. Our general conclusion must be a further affirmationof the basic correctness of the random-lifetime assumption in systems with moderatelystrong coupling, and the belief that any significant advance in our understanding ofthe more subtle effects of intramolecular energy transfer must involve the hiddenvariables which underlie the simple energy-description of the reacting molecule usedhere and elsewhere.We are indebted to Mr. C. H. Kaplinsky for programming assistance and to theUniversity of London Institute of Computer Science for computing facilities.L. S. Kassel, J. Physic. Chem., 1928, 32, 225.1932).D. L. Bunker, Theory of Elementary Gas Reaction Rates, (Pergamon Press, 1966).B. J. Zwolinsky and H. Eyring, J. Amer. Chem. SOC., 1947, 69,2702.D. J. Wilson, J. Physic. Chem., 1960,64, 323.D. L. Bunker, J. Chem. Physics, 1964,40, 1946.M. solc, Mol. Physics., 1966, 11, 579.M. Hoare, Mol. Physics., 1961, 4,465.Handbook of Mathematical Functions, ed. M. Abramowitz, and I. E. Stegun, (N.B.S. Washing-ton, 1964,) chap. 22.lo Mathematical Methods of Physics, J. Matthews and R. L. Walker, (Benjamin, New York,1964), p. 255.11 B. Widom, J. Chem. Physics, 1959, 31, 1387.l2 N. B. Slater, The Theory of Unimolecidar Reactions (Methuen and Co., London, 1959), chap. 9.* L. S . Kassel, The Kinetics of Homogeneous Gas Reactions (Chemical Catalogue Co. New York

ISSN:0366-9033    

DOI:10.1039/DF9674400030

出版商:RSC    

年代:1967

数据来源: RSC

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Optical Potential for a Chemically Reactive System*BY B. C. Eut AND JOHN Ross$Receiued 12th h i e 1967The optical potential is investigated for a chemically reactive system, K-I- CHJ, with an assumedpotential of interaction among the three species, K, I, CH3. The complex optical potential satisfiesan integral equation with a kernel related to the solution of the three-body problem, as given byFaddeev. In order to obtain some information on the form of the optical potential without numericalanalysis we introduce a number of approximations. The imaginary part of the optical potential,which gives rise to absorption(reactive and inelastic collisions), is with these approximations essentiallya delta function of the K+CH3I distance. The real part of the optical potential is similar in formto the repulsive potential estimated from elastic scattering experiments on this system.1 .-INTRODUCTIONThe concept of the optical potential has played a useful role in the analysis ofthe scattering of many-body systems.The collision of the reactants of a chemicallyreactive system, such as K+CH31: which we consider here, may lead not only toreactive but also to elastic and inelastic scattering2 The optical potential is definedto be that effective two-body potential which predicts the same elastic scattering asthe solution of the exact many-body scattering equations. As such, the opticalpotential is complex; if the relative kinetic energy is above the threshold then theimaginary component provides for absorption from the elastic scattering channelleading to reaction and inelastic events.This concept serves as a phenomenologicalcorrelation of the observed elastic scattering in reactive systems with assumed analyticforms of real and imaginary components of the potential, each usually containingparameters of characteristic distances and energies of interaction for the reaction.Total reaction cross-sections and reaction probabilities as a function of energy andinitial orbital angular momentum may thereby be obtained.The optical potential is related to the potentials of interaction and is the solutionof an integral equation which contains the transition matrix of the many-bodyscattering prob1em.l In this paper we carry through an approximate evaluation ofthe optical potential for the system K+CH31, with the purpose of investigating theform of this potential for a chemically reactive system.The potentials of interactionbetween the units K, I and CH3 are not known but are taken to be those used inprevious classical calculation^.^ Thus, for the interaction of a K atom and an Iatom alone we take a Morse potential, and similarly for CH3 and I. The K-CH3interaction is assumed to be repulsive. In addition, we need switching functions torepresent the three-body interactions for the reaction. For the calculation of theT (transition) matrix we use Faddeev’s theory of three-particle ~cattering.~ Theintegral equation for the optical potential is then solved to lowest order by theFredholm rneth~d,~ which accounts for single, but not multiple, scattering contribu-tions.A number of further approximations are made in order to avoid numerical* This work was begun while the authors were at Brown University, Providence, Rhode Island,U.S.A., and was supported in part by the National Science Foundation and Project Squid, Office ofNaval Research. t Dept. of Chemistry, Harvard University, Cambridge, Massachusetts, U.S.A.$ Dept. of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.340 OPTICAL POTENTIALanalysis : first, we apply an adiabatic approximation to the Green's functionappearing in the T matrices of the Faddeev theory ; secondly, we take one of themasses of the reactants to be very large compared to the other two and expand allmatrix elements in power series in the mass ratio ; thirdly, we consider only acolinear configuration ; and finally, we approximate the wave function of the productKI by a square-well potential in the calculation of the imaginary component of thepotential.2.-FORMULATION OF OPTICAL POTENTIALIn the quantum theory of scattering, cross-sections are obtained from transitionmatrix elements (+f ] Vi I $:), or <t& I Vf I (6t), in which rPi and c # ~ are the initialand final wave functions, Vi and Vf the interaction potentials in the initial and finalconfiguration, and $: and $7 the solutions of the Lippman-Schwinger equationwith outgoing and incoming boundary conditions, respectively.Alternatively, thesematrix elements may be written as a transition between the initial and final wavefunctions with a transition operator Ti defined to obey the integral equationTi = K+T/,GFT = q+T,G'V;.,for which the Green's function operator isG' = ( E i - K - H i ) - l , (2.2)K being the kinetic energy operator for relative motion and Ift the Hamiltonian forbound particles in the initial configuration.Thus, the transition matrix element forelastic scattering iswhere ki is the wave vector for the initial relative motion and the letter i denotes theset of quantum numbers specifying the bound particles in the initial channel.The optical potential Yo is an effective two-body potential which must reproducethe elastic scattering in the many-body system. The effective two-body T matrix<ki7 i I T* 1 ki, 0 7 (2.3)Teff = < i i I i)serves to define the optical potential in the integral equationwith the Green's function operatorin which gi is the initial relative kinetic energy. If the complex optical potential isknown then we can evaluate the transition matrix elements for elastic scattering aswell as the absorption in a given initial channel.Here, however, we wish to solveeqn. (2.5) for the optical potentialThe perturbation parameter A, to be set equal to unity, is inserted for the convenienceof keeping track of the order of perturbation. We therefore consider Teff3$ as anintegra.1 kernel which in principle may be obtained from the solution of the many-bodyscattering equations. For this purpose we use Faddeev's theory4 and obtain anapproximate solution thereof.Since the postulated potentials of interaction (Morseand three-body switching functions) yield a transition matrix Ti and hence an integralkernal Tef&?i which are completely continuous and obey the Holder ~ondition,~the integral equation (2.7), although singular, has a solution.* The Fredholmvo = Te,-AT,,~,+Vo. (2.7B . C . EU AND JOHN ROSS 41solution to second order in A isK , = trace (K')h^ = Te@;;to lowest order we simply have Yo = T,,, and this is the quantity we shall calculatein 93.3.-CALCULATION OF OPTICAL POTENTIALThe calculation of the transition matrix T,, and from it T,,, = (i I Ti I i), requiresa solution of the many (3)-body problem. We use Faddeev's formalism for thistask which avoids certain convergence problems arising from an expansion of theT matrix in a Neumann series.the Tmatrix becomesand, following Faddeev, we expand this further in the formOn introducing the wave matrix lo GI+$: = GI+& (3.1)Ti = v,a+ (3.2)= C C TaR,+.a f ( 2 3 ) p#a(3.3)The symbols a, fl are running indices over all possible interactions of the three particleslabelled by arabic numbers; in terms of these numbers the reaction considered hereis 1 +(23)+(12) + 3.The T matrix for each interaction is given byCentre of mass co-ordinates are considered to have been separated in the following.The Faddeev equations can now be written for our initial conditionsT120TI 3T 1 2 3T120T2 3T123for which the lowest order solution reduces eqn.(3.3) simply to1 Ta.a # ( 2 3 )(3.71The neglect of multiple scattering events (lowest order solution) removes the couplingbetween T, and TB. We make this approximation here without further inquiry intothe rate of convergence of the convergent expansion of eqn. (3.6).The equation for T,, eqn. (3.4) can be rearranged slightlyTa = Va + VaGL Va,G,' = (E' - Ho- VJ-',(3.8)(3.9)so that the optical potential becomes (to lowest order Y0-Teff, see eqn. (2.8), (2.4))V , , = ( i ~ ~ ~ i ) = (i((Va+VaGZVa)Ii).a # ( 2 3 )(3.1042 OPTICAL POTENTIALNext we apply an adiabatic approximation to the Green's function G:GZ EZ (E' -Ho- I-',)-', (3.11)G; = ( E l -Ka-Va)-' Er-Kt +EZ - K,- V, (E:- K, - Va)2where Ea is the eigenvalue of the Hamiltonian Ka+ V, for the group a (a # (123))and Et is the eigenvalue of the relative kinetic energy operator Kr for the relativemotion of a to the remaining particle.For the three body forces used here a boundstate of the three particle system cannot be formed with first order solutions of theFaddeev equations. Hence we may write for the Green's function(3.12)in which j , denotes the set of internal quantum numbers of the interacting group a.The adiabatic approximation leads to local potentials whereas higher terms giverise to a non-local potential. With the use of this approximation in eqn. (3.10)we obtainOn separation of the principal part and pole contribution1E, - Ej==p-- nit@,- Eja),1E,* - Eja(3.13)(3.14)we havethe prime on the summation sign implies the condition Ea # Ejl, jEa means j , forEa = Ej,, and P(E,) is the number of quantum states in an interval of unit energyaround Ea.The first two terms of the optical potential, eqn. (3.15), are real whereas the lastterm, which leads to absorption, is imaginary and negative semi-definite.In theadiabatic approximation, absorption comes predominantly from those intermediatestate jEa with eigenvalues close to the internal eigenvalue of the initial channel.We turn now to an estimation of the optical potential, eqn. (3.15). First, wenote that in the second and third term contributions from intermediate states ofpotentials which are everywhere repulsive are smaller than those from states ofpotentids which can accomodate bound states.For repulsive potentials the inter-mediate wave function 1 jEa) is essentially sinusoidal and the matrix element formedwith such a wave function is therefore small compared to matrix elements formed withintermediate wave functions corresponding to bound states. Hence, for the systemconsidered here, the summation over a for these two terms is replaced in each caseby a single term, a = (12), because the assumed K-CH3 interaction and the switchingpotentials do not have bound states.For the imaginary term we need to estimate the matrix element <i J Vlz I j&,which for a linear confqpration of (123), ground vibrational (with fundamentaB . C. EU AND JOHN ROSS 43frequency wl and reduced mass pi) and rotational states of (23), and lowest rotationalstate of (12), becomesNl normalizes the wave function of (23), and we have changed variables in the secondequality by usingr12 = R-(m3/m23) r237 (3.17)see fig.1 . At this point we approximate the bound states potential of (12) by a square-FIG. l.--Co-ordinate system.well potential of depth V12, extent a’, and hard core radius b’. In this approximationthe wave function 4,E12(r12) within the square-well is4 . (r12) = N j sin kj(rI2 - b’)/r12, (3.18) J E i 2where k j = J 2 p x E j i ! ‘lo2 ’ is one of the roots of the transcendental equationkj(a‘ - b’) = -tan-l kj/Kj; K j = J(2pj I E j I /h), E,=E,,,. (3.19)m lm23 RWe also approximate 1/r23 w -2 - for large R. With the definitions of the followingquantitiesthe matrix element (i 1 Y12 I jE12) is(3.20)(3.2144 OPTICAL POTENTIALThe second term in this equation is partly an artifact of the simple square-wellpotential but mostly due to evaluation of the optical potential by a first-order pertruba-tion solution of the Faddeev equations.The first-order perturbation yields matrixelements to be evaluated over all space. A distorted wave treatment, or strippingapproximation, corrects for the main fault of the first-order perturbation, i.e., thepenetration of the undistorted wave functions into the interaction region. As thesecorrections are introduced, the second term, being proportional to the inner, hard1.0p0 58 0.513.CI UY 0 aT3W 3!20 5 10distance, AFIG. 2.Potential energy of interaction of K-CH31, in unit of DI2, the dissociation energy of KI.Solid line : the contribution to the real part of the optical potential given by eqn.(3-22). Dashedline : Repulsive potential as deduced from elastic scattering measurements on K-CH,IZb. (Ex-ponential-six potential, E = 0.51 kcal/mole. The values of rm equal to 5.0A and a = 14 arecore radius b’ of the square well, becomes vanishingly small. The dominant term ineqn. (3.21) is the first one. The range of the imaginary part of the optical potential,proportional to squares of matrix elements (i I V12 lj,,,}, eqn. (3.21, 3.15), isextremely small since the parameter p in the exponential of the first term of eqn.(3.21) is of the order of 1019 cm-2. Hence Im Yo is essentially a &function of thedistance R.When the first term is at its maximum, at R=a’+y//?=(a’ +0-8)A, thesecond term is negligible. This structure of Im Yo, and hence of the absorption, hassome similarity to that postulated in the spectator model of reactions.llTwo terms in eqn. (3.15) contribute to the real part of the optical potential. Weevaluate the matrix elements (i I Va I i> by again expanding quantities includingdistances such as r12 = R - ( r ~ ~ / r n ~ ~ ) r ~ ~ in a power series of the mass ratio m3/m23and retaining only the leading term; we obtaina 2 ( 2 3 ) c < i ~ v . l i ) ; z ~ , , ( ~ ) + ~ , , ( ~ ) + 4 ~ ~ ~ ~ e x p [ - ~ ] . jl-ttanh(aR+b))+assumed.)(3.22)The parameters a, b are introduced in the switching functions of the assumed potential ;OI2, 0 2 3 , a12 and a23 are Morse potential parameter^.^ The second term consistB .C. EU AND JOHN ROSS 45of matrix elements of the form already evaluated for the absorption term. We canstate, therefore, that the range of that term will be of the order of a &function; thesign or magnitude, however, is not obtained here.The contribution of the first term a&:23)<i 1 Kl i, to Re Yo is shown in fig. 2.For comparison we also plot the exp-six potential obtained for the system K-CHJfrom a study of the non-reactive scattering. Van der Waals attractive forces areneglected in the comparison. In view of the number of approximations made andthe arbitrary parameters introduced in the switching functions of the assumed potential,the similarity of the forms of the two potentials in fig.2 is considered to be satisfactory.K. M. Watson, Adv. Theor. Physics., ed. K. A. Brueckner, (Academic Press, New York, 1965),vol 1, p. 115 and references therein.(a) D. R. Herschbach, G. H. Kwei and J. A. Norris, J. Chem. Physics, 1961, 34, 1842,(b) J. R. Airey, E. F. Greene, J. P. Reck and J. Ross, J. Chem. Physics, 1967, 46, 3295.L. M. RaRand M. Karplus, J. Chem. Physics, 1966,44,1212 ; we are taking OR in their notationas the potential energy surface in our calculation.L. D. Faddeev,JETP-Soviet Physics, 1960,39,1459 ; Doklady 1961,138,565 ; ibid, 1962,145,301.P. M. Morse and H. Feshbach, Methods of Mathematical Physics, (McGaw-Hill Book Co.,Inc., New York, 1953), pp. 1018-1023.(a) B. Lippmann and J. Schwinger, Physic. Rev., 1950, 79,469.(b) M. Gell-Mann and M. L. Goldberger, Physic. Rev., 1953, 91, 398.(c) M. L. Goldberger and K. M. Watson, Collision Theory, (John Wiley and Sons, Inc., New York1964), or N. F. Mott and H. S . Massey, The Theory of Atomic Collisions, (Oxford UniversityPress, London, 3rd ed., 1965). ' N. I. Muskhelishvili, Singular Integral Equations. P. Noordhoff, (N. V.-Groningen, Holland,1953). * L. D. Faddeev, Mathematical Problems of the Quantum Theory of Scattering for a Three-particle System, trans. by J . B. Sykes, (H.M. Stationary Office, Harwell, England, 1964).S. Weinberg, Physic. Rev., 1964, 133, B232.R. E. Minturn, S. Datz and R. L. Becker, J. Chem. Physics, 1966, 44, 1149.lo C. Msller, Kgl. Danske Vid. Selsk. Mat.-fys. Medd, 1945, 23, 1

ISSN:0366-9033    

DOI:10.1039/DF9674400039

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Dynamics of Reaction of Monoenergetic Atomsin a Thermal Gas*BY &ON KUPPERMANN, JOHN STEVENSON AND PATRICIA O’mEFEGates and Crellin Laboratories of Chemistry t.California Institute of Technology, Pasadena, California 9 1 109Received 17th July, 1967When monoenergetic atoms are continuously introduced into a thermal gas, they can undergodeactivating, activating, and reactive collisions. The net result of such collisions is to establish asteady-state distribution of laboratory energies which, although not as sharp as the initial distribution,preserves some of its features, such as being centred at about the initial energy. The reactive collisionswhich occur under these conditions are characterized by the associated relative energy distributionfunction and the energy-dependent reaction cross section.As a result, as the initial laboratoryenergy of the atoms is experimentally varied, the relative energy distribution function can be madeto sample appropriately the reaction cross section curve. Therefore, from measurements of thecompetition between reaction and thermalization processes as a function of initial atom laboratoryenergies, and from a knowledge of the non-reactive differential scattering cross section, it is possibleto obtain information about the dependence on relative energy of the rotationally averaged reactioncross section. The appropriate Boltzmann steady-state equation needed to obtain this informationis derived in this paper and solved for an assumed set of reactive and non-reactive cross sections.Distribution functions of relative energies are thereby obtained and used to indicate the usefulnessof the suggested measurements.1.STEADY-STATE BOLTZMANN EQUATIONSWe consider atoms A of mass m, continuously introduced into a thermal gas ofmolecules B of mass nz2 which is at temperature T. Let Ro be the total rate of genera-tion of atonis A and let qo(vl) be the normalized distribution describing the laboratoryvelocity vectors v, with which A is introduced into the gas ; +o(vl) will be consideredspherically symmetrical but otherwise arbitrary. For the case of interest in thepresent paper it will be a rather narrow distribution function centred around aninitialspeed d?). Inparticular it can be a 6 function at d?). The rate of introductionRi(v,)dvl of atoms A in velocity range v1 to v1 +dv, is therefore Ro+o(vl)dvl. Understeady-state conditions, let g(vl)dv, be the concentration of D atoms in that samevelocity range.The rate R(-) (v,)dv, at which atoms are removed from that rangedue to reactive and non-reactive collisions is given bywhere* work supported in part by the U.S. Atomic Energy Commission. Report Code : CALT-532-13. t contribution no 3537.4A . KUPPERMAN, J . STEVENSON AND P. O’KEEFE 47In the expressions above, [B] is the number density of molecules B, v2 is theirlaboratory velocity, v is the relative velocity of A with respect to B, qT(v2) is thenormalized Maxwell distribution function of velocities v2 at temperature T, andS,(u), Snr(v), and Sr(u) are, respectively, the total, non-reactive and reactive crosssections of A and B at relative speed v.It is assumed that these cross sections havebeen averaged over the rotational motion of B at temperature T and that otherwiseonly one internal quantum state of A and B are involved in the collisions. Thiscondition is satisfied by the D + H2 system considered below. The present formalismcan easily be generalized so as to eliminate these restrictions.The rate R(+)(vl)dvl at which D atoms are inroduced into the range v, to v1 + dvl dueto non-reactive collisions of D atoms at other velocities with B molecules is given by13‘ + ’(v 1 )dv 1 = C B ~ J 9 (v; 1 Cp T(V; )u’onr(U’, x , q)d~dv;dvi 2 (1.6)where v; and vk are, respectively, the velocities of A and B before a non-reactivecollision which results in velocities v1 and v2 after collision, v’ is their relative speedbefore collision and Q = (x, q) is the direction of the relative velocity vector v aftercollision with respect to v’ and an arbitrarily chosen plane containing v’.Thereforex is the scattering angle and r j the angle between the v’, v plane and the reference plane.There are a total of four velocity vectors v;, v;, vl, v2 or twelve scalar componentswhich completely define the velocities of both particles before and after collision.Of these, however, only eight scalars are independent since conservation of centreof mass momentum as well as total energy establishes four relations among them.In eqn. (1.6) these eight scalars were taken as vi, vi, Q(ic, rj)? and X is a surface in thiseight-dimensional velocity space defined by the requirement that over itv, be a constant.The integral in this expression is therefore five-dimensional and includes the contri-butions of all collisions between A and B which form A in velocity range v1 tov1 + dv, after collision.These include both activating (u, > v i ) and thermalizing(u, <v;) collisions, although for u,. greater than the root mean square speed attemperature T, the effect of thermalizing collisions automatically predominates.Under steady-state conditions, the net rate of introduction of D atoms into rangev1 to v, +dvl due to all the processes described above is zero, by definition. Thesteady-state Boltzmann equation in laboratory velocity space is therefore2R,(v,)-R(-)(v,)+R(+)(v,) = 0, (1.7)or Ro(Po(v1) - k(v,)g(v,) +R(+)(vJ = 0.(1 -8)We can transform this equation into laboratory speed space by multiplying it byv; sin do, dq, and integratingit over the spherical polar angles 01, (pl of v, withrespect to an arbitrarily chosen laboratory system of reference. Making use of thespherically symmetric nature of the functions &, k and g we get :R,&(v,) - k’(v&’(v,) +R’(+)(ul) = 0, (1-9)where cpXul) = 47d~p0(~1), (1.10)g’(v1) = 4~u?g(vl), (1.11)k‘(u,) = k(vl). (1.12)An explicit expression for R’(+)(v,) can now be obtained by calculating its integralover v, : s., CBI J g(V;)~PT(Vi)u’a,,(u’,X,q)dv;dv~dR. (1.13)v; ,v>,f48 DYNAMICS OF MONOENERGETIC REACTIONSThe integral on the right-hand side of this expression now extends over the entireeight-dimensional velocity space v;, v;, a with no restriction on vl.We changeintegration variables from these to v;, v;, q;, vl, a. This is equivalent to changingfrom variable 0; to variable vl, the other seven scalar variables v;, v;, cp;, 0 remainingunchanged. Here (O;, 40;) = i2; represent the direction angles of v;. Let J be theJacobian of this transformation, for which(1.14)There results from this transformationsin O;dO; = I JI vfdu,.R‘(+)(vl) = [B][ g(v;)soT(v;)v’a,,(v’,x,r)l J lvtv;2dv;dv;dq;dS1. (1.15)The integrand of eqn. (1.15) is considered to be a function of the eight independentscalars v;, v;, q;, and ul. The integral is performed over the first seven of thesevariables, the last one remaining constant.Using eqn. (1.15) and (1.11) togetherwith eqn. (1.9) gives :(1.16)Vi,v;,&,nwhere(1.17)&(vi) = 4nv;2qT(v;). (1.18)To obtain eqn. (1.171, we chose as the axis for measuring O;, q; the direction of v;.Therefore, 0; became the angle between vi and v;, which we relabelled y’. Theintegrations over q;, O;, and (pi were performed explicitly and furnished factors of2n, 2, and 2n, respectively, which when combined with the 4n factors coming fromthe change from g to g’ and 4T to 4; gave the net factor of 1/2 in the right-handside of eqn. (1.17). The independent variables which appear in its integrand arethereby reduced to five, v;, ul, vh, x, q. The partial derivative of y’ with respect tou1 assumes therefore that ui, v;, x, and q are kept constant.We finally transform eqn.(1.16) into energy space by changing from variablesu, and v; toThere results the steady-state Boltzmann equation in energy spacewhere(1.19)(1.20)(1.21)(1.22)(1.23)(1.24)(1.25)In eqn. (1.22) through (1.25) the quantities vl and u; are considered functions ofEl and E ; , respectively, obtained from eqn. (1.19) and (1.20). The distributioA . KUPPERMAN, J . STEVENSON AND P. O'KEEFE 49functions q50 and cp+ satisfy the normalization conditions,(1.26)Jv2P~(v2)dv2 = 1. (1.27)The distribution function G(E,) as defined above is non-normalized. G(El)dElrepresents the steady-state concentration of A atoms in laboratory energy rangeEl to El +dEl.Once G(E,) is obtained from a solution of eqn. (1.21)' we can alsoobtain the normalized steady-state distribution function F(EJ of the A atom labora-tory energy :fIt is possible to give a simple physical interpretation to eqn. (1.21). The firstterm times dE, represents the initial rate of introduction of atoms A into laboratoryenergy range El to El +dEl. The second term times dEl represents the rate ofreinoval of these atoms from that range due to all types of collisions with molecules B,reactive and non-reactive. Finally, the third term times dEl represents the rate offormation of atoms A in that range due to all non-reactive collisions of such atomswhose laboratory energy before collision was E;, integrated over all Ei (larger thanEl for thermalizing collisions and smaller than it for activating ones).The following transformation permits one to obtain the distribution function inrelative energies f(E) from that in laboratory energies :(1.29)whereOn the right-hand side of eqn.(1.30), u1 and u are considered to be functions ofEl and E according to eqn. (1.19) and (l.31)y respectively.E = 3pv2, (1.31)where p is the reduced mass of the A, B particle pair. If all atoms A have the sameenergy E, i.e., are monoenergetic in the laboratory system, the distribution functionof their relative energy with respect to thermal molecules B is given by eqn. (1.30).This can be a very wide distribution, even if kT is small compared to El. Thereason is that if A and B have laboratory speeds u1 and u2, their relative speed canbe anywhere between vl +u2 and v 1 -u2:, depending on the directions of the corres-ponding velocity vectors. Therefore, their relative energies can be anywhere betweenEm,, == +p(vl +u2)2 and Emin = +p(ul -u2)2 corresponding to a spread in relativeenergies AE = Emax-Emin = 2pvlu2.Consequently, even though u2 may be rathersmall, AE can be quite large if vl is large. If u2 is substituted by (3kT/m2)*, theroot mean square speed of B at temperature T, the resulting AE is 2 p (6kTEl{mlm2)%This is a measure of the width of the distribution function given by eqn. (1.30).For El % kT this function has a peak at E approximately equal to El, and its widthis much larger than kT if m,{m2 and mJm, are not much smaller than unity.Forexample, for the D+H2 system, even at temperatures as low as 200"K, this widthis 0.45 eV for El = 2 eV (which corresponds to a most probable relative energ50 DYNAMICS OF MONOENERGETIC REACTIONSof 1 eV). In summary, a relatively small thermal spread in the B laboratory speedcan produce a large spread in the relative energy of A with respect to B, and if oneassumes that the B are stationary, very serious errors can result.It is possible from eqn. (1.21) to obtain a rigorous steady-state Boltzrnann equationdirectly in relative energies. Indeed, multiplication by qBT(EI,E)/K(E1) and integra-tion over El furnisheswhere(1.33)and G ( E ) is the non-normalized distribution function in relative energy. G'(E)dErepresents the steady-state concentration of atoms A whose energy relative to Bis in the range E to E+dE.G'(E) is related to G(E,) byThe inverse transform is(1.34)(1.35)whereIn the right-hand side of this last equation, vl and v are the functions of El and Egiven by eqn. (1.19) and (1.3 l), respectively. The following normalization propertiesare satisfied by (PBT and qbT :(1.37)Replacement of the dummy variable E by E' in eqn. (1.35) followed by substitutioninto the last term of the right-hand side of eqn. (1.32) and a change in the order ofthe integrations furnishes finallyRo&,(E) - G'(E) + f G'(E')H'(E',E)dE' = 0 (1.38)E'where(1.39)Eqn. (1.38) is the steady-state Boltzmann equation in relative energies sought.Since the functions &(E) and H'(E',E) are known, it can in principle be solved togive G'(E) directly.In terms of the latter, the normalized steady-state distributionfunctionf(E) is given byf ( ~ ) = GVW J GWW (1.40A. KUPPEMRAN, J . STEVENSON A N D P. 0' KEEFE 51An implied assumption in the derivation of eqn. (1.21) was that the amount ofenergy introduced into the A+B system via the kinetic energy of atoms A wassufficiently small so as not to disturb the thermal distribution of energies of moleculesB. It is always possible to satisfy this assumption by making [B] sdliciently large.Eqn. (1.21) can be easily modified to include the effect of one or more additionalreactants Ci. To this effect it suffices to add to the right-hand side of eqn. (1.2)and (1.17) similar terms for these substances.If only one additional reactant C isconsidered, and if it is much more effective than B in reacting with thermalizedA atoms, then a small [C]/[B] ratio can suffice to scavenge such thermalized atomswithout essentially contributing to the thermalization process. Under these condi-tions, the entire effect of introducing small amounts of C into the A+B system canbe approximated quite well by adding to the right-hand side of eqn. (1.2) a similarterm for C which, however, includes only the reactive part S:(u,) of the A+C crosssection, and leaving all other terms unchanged. Here u, is the relative speed of Awith respect to C . The calculations described in Section 2 correspond to this case.The generalization of the relative-energy Boltzmann eqn.(1.38) for a rnulti-reactant system is also easily obtained from the corresponding generalized form ofeqn. (1.21).2. SOLUTION OF THE STEADY-STATE BOLTZMANN EQUATION INLABORATORY ENERGYGiven the reactive cross-sections S,(v) and S:(u,) and the differential non-reactivecross section crnr(u, x, q), as well as the initial rate Ro of production of D atoms' initialdistribution function C$~(EJ, temperature T and B and C molecule number densities[B] and [C], it is possible to solve the corresponding generalized form of the steady-state Boltzmann eqn. (1.21). We call Ro& the source term, KG the sink term andJGHdE; the collision integral. This equation is a linear integral equation of the thirdkind, in which the known function H(E;, E l ) is called the kernel.It can be solvednumerically by several methods, one of which is the Liouville-Neumann series.2 Thisconsists essentially in writing the unknown function G(E,) asG,(E,) = li- l(E1)/K(E1) i = 1,2.. .We have obtained such solutions, but not for an exact kernel H as defined byeqn. (1.25), (1.17), and (1.18). Instead, we used an approximate kernel, as describedbelow. The reason for doing this is that the purpose of the calculations reported in thispaper was not to obtain accurate values off@) for any particular A+B system, butto develop instead information about the general shape off(E) and its variation withE(:), the initial laboratory energy with which the A atoms are injected into the system.This purpose is adequately and more easily accomplished by picking a simplified kernelwhich preserves, nevertheless, the main properties of kernels which describe realsystems.To do this we notice, from eqn. (1.17), that gnr(u', x, q) and p$(v;) appearas part of the integrand. We know that a,, is a rather sharp function of x, meanin52 DYNAMICS OF MONOENERGETIC REACTIONSthat the non-reactive differential cross sections for small angular deflections (corres-ponding to large impact parameters) tend to be quite large and fall off very rapidlyas the deflection angle increases. This means that the contributions to h’ from rela-tively small energy losses tend to play a major role. This suggests that H(E;, E l )be made a gaussian function centred at E; = E l . In addition, cp$(ui) may produce inE; a spread of the order ofAEi = ,/2rnlkT/m2.We therefore chose H(E;, E l ) as having the formwhereis a normalized gaussian function.In order to determine A , we used the followingrelationship between Knr and H :In this expression, K,, refers to the part of K obtained replacing &(u) by Snr(v) ineqn. (1.2), (1.12), and (1.23). Eqn. (2.6) is simple to prove and states essentially,when both sides of it are multiplied by G(E,)dE,, that the rate of removal of A atomsout of energy range El to El +dEl due to non-reactive collisions with molecules Bis equal to the rate at which atoms A are formed with energy E; due to such collisions,integrated over all possible values of that final energy. Substitution of eqn. (2.4) intoeqn.(2.6) givesWe conclude that for the kernel defined by eqn. (2.4), K,, must be a constant, inde-pendent of El. Returning to the definition of &,(El) given above, this implies thatwhere a is a constant independent of o. Indeed, substitution of eqn. (2.8) into thedefinition of Knr(E1) givesTherefore, eqn. (2.4) and (2.7) through (2.9) completely define the kernel and sinkterms for which we did our calculations. The main advantage in introducing thesesimplifications when making initial calculations is to avoid the time-consuming three-dimensional numerical integral indicated by eqn. (1.17) which would otherwisehave to be performed. In trying to obtain quantitative reaction cross sections fromexperimental data the use of that equation is required.In addition to the H and K,, just mentioned, we chose the following form forthe two cross sections describing the reaction of A with B and with the thermal atomscavenger C :Knr(E1) = A* (2.7)vSnr(u) = (2.8)A = a[B].(2.9)nb2(1 - E,/E) for E >Eo,for 0 <E <Eo,where E is related to v by eqn. (1.31), and(2.10A . KUPPERMAN, J . STEVENSON AND P. O'KEEFE 53where y is a constant independent of 21,. Eqn. (2.10) is the usual hard sphere line ofcentres reactive cross section with relative threshold energy Eo and eqn. (2.1 1) repre-sents a cross section which furnishes a thermal rate constant independent of tempera-ture, as is essentially the case for many scavengers of thermalized atoms.3. RESULTS AND DISCUSSIONCalculations off(E) were performed using the scheme described in 92 and pickingvalues of the several parameters of the correct order of magnitude to describe thesystem D+H2 with DI as a scavenger.The parameter a was chosen so as to makeSnr(v) = 46.5 Hi2 for v = 6-2 x lo5 cmlsec. [H,] was made 3.2 x 10l8 molecules/cm3corresponding to a partial pressure of 100 torr at the chosen temperature of 300°K.nb2 was chosen as 1 A2 and Eo was made equal to 0.3 eV. y was chosen so that atuc = 1.56 x lo5 cmlsec, S;(u,) = 0.1 A2. The value of Ro was taken as lofo Datoms/cm3 sec corresponding to the experimental conditions for photolyzing mixturesof DI+H2 used in a study of the D+Ht+DH+H rea~tion.~ The normalizedsource term +o(El) was chosen as a gaussian function centred at E ( f ) and withroot mean square width of 0.03 eV, also corresponding to the experimental conditionsjust menti~ned.~ The central D atom laboratory energy E(!' was given the values0.4, 0.7 and 1.2 eV, corresponding to most probable values of the relative energyof 0.20, 0.35, and 0.60 eV, respectively. Finally, the [DI]/[H,] ratio was variedbetween 0.1 and 2.5.It was found that the resulting distribution functions wereessentially independent of this ratio. This can be understood on the basis thatthe main constribution to K comes from Snr(u), the contributions due to S,(o) andS;(u,) being negligible in comparison. This means that the f ( E ) should be essentiallyindependent of the reaction cross sections, being determined mainly by the differentialnon-reactive cross section.This is an important point, since it enormously simpli-fies the procedure for extracting reaction cross sections from experimental data,as pointed out below.In fig. 1 are depicted several of the curves obtained. Curve S, represents thereaction cross section defined by eqn. (2.10) with values of the parameters as chosenabove. Curve fT/5 represents a normalized Maxwellian distribution of energiesat 300"K, with the ordinates divided by 5. For E>Eo = 0.3 eV, the values of itsordinates are essentially zero compared to its maximum value. Curves a, b, and ccorrespond to = 0.4, 0-7, and 1.2 eV. They have maxima close to E = 0.2,0.35, and 0.6 eV respectively, as expected, since the reduced mass of the D+H2system is 3 of the mass of a D atom.The most important point is that these curveshave large ordinates (in units of their maximum values) in regions where the crosssection S,(u) is appreciable. Therefore, scanning E(y) is an excellent way of samplingthis reaction cross section.In fig. 2, curve c is depicted again, together with curve d which is the correspondingsource distribution function 40(El) transformed into laboratory energies. Onecan see that the broadening in going from d to c, produced by the multiple thermalizingcollisions, is not large. This means that the results of experiments performed underthese conditions should not be too unlike those obtained when a D atom beam collideswith a thermal H2 gas target under conditions such that only a single collision occurs.In other words, the multiple collision nature of the photolysis experiments mentioneddoes not make them extremely different from single collision atomic beam experiments.Finally, we analyze the consequences of the distribution functions f(E) beingdetermined almost exclusively by the differential and total non-reactive cross sections,and essentially independent of the reactive ones. This means that once the non-reactive cross sections are known, thef(E) can be calculated by the methods describe54 DYNAMICS OF MONOENERGETIC REACTIONSabove. Then the reaction cross section S,(v) can be obtained simply by unfoldingit with respect to f(E) in such a way as to produce extents of hot reaction andthennalization which agree with experiment.In view of the nature of the dependenceE( kcal/mole)5.0 100 150 200 2 50E(eVFIG. 1 .-Variation of distribution functions and cross section with relative energy. f ~ , normalizedMaxwellian distribution at 300°K; ordinates have been divided by 5. a, b, and c, normalizedf(E) curves for E(P) = 0.4, 0.7, 1.2 eV, respectively ; see text for values of other paranieters. Sr,assumed reactive cross section curve for D+H2-+DH+H; see text for value of parameters.E(kcaljmo1e)E Oinitid distribution function ; c, steady-state distribution function.FIG. 2.-Initial and steady-state distribution functions of relative energies. E(P) = 1-2 eV; d,of f(E) on Ec!) described above, such unfolding should be devoid of difficulties andproduce accurate results. The procedures outlined above are presentfy being usedto determine SF(@) for the I) + H2 +DH + H reaction from experimental [DHI/[D,A . KUPPERMAN, J . STEVENSON AND P. O’KEEFE 55ratios obtained in the photolyses of DI +H, mixtures with monochromatic lightof variable wave-length.One of the author (A. K.) thanks Prof. R. N. Porter for stimulating discussionson this problem. We also express our appreciation to Prof. John M. White andMr. Joel Goldberg who participated in the early stages of this work.J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids, (JohnWiley and Sons, Inc., New York, 1964), p. 502.H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, (D. Van NostrandCompany, Inc., New York, 1956), p. 521.A. Kuppermann and J. M. White, J. Chem. Physics, 1966, 44, 4352

ISSN:0366-9033    

DOI:10.1039/DF9674400046

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Quantum-Mechanical Study of H + H, Reactive Scattering*BY M. KARpLust and K. T. TANGSDept. of Chemistry and Physics and IBM Watson Laboratory, Columbia University,New York, New York and Department of Chemistry, Harvard University,Cambridge, Mass.Received 22nd June 1967The results of an approximate quantum-mechanical treatment of the H+ H2 reaction are reported.Two limiting models in the distorted wave formulation (DWB) are compared ; in one, the moleculeis unperturbed by the incoming atom and in the other the molecule adiabatically follows the incomingatom. For thermal incident energies and the semi-empirical interaction potential employed previouslyfor a quasi-classical trajectory analysis of the reaction, the adiabatic model seems to be more appro-priate. To examine the total and differential cross section for a range of energies, a simplified(linear) version of the DWB method is used.Most of the results are similar to those obtainedquasi-classically. However, the quantum total cross section has a significantly higher thresholdthan does the classical cross section. In agreement with the quasi-classical treatment, the differentialcross section is strongly backward peaked at low energies and shifts in the forward direction as theenergy increases. The calculated variation of the total cross section with final rotational state raisesquestions concerning the standard procedure for determining para-H, to ortho-H, conversion rateconstants.The reactive or rearrangement scattering of hydrogen atoms by hydrogen mole-cules (H+H,-+H,+H) plays a fundamental role in the theory of chemical kinetics.lExact quasi-classical calculations of the cross sections and other reaction attributeshave been made with a realistic, though approximate, potential surface for the H3~ystem.~ Since it has been suggested that quantum effects are important for thisrea~tion,~ it is of interest to have direct estimates of their nature and magnitude.Inthis paper we report some results of an approximate quantum-mechanical treatmentwhich employs the same interaction potential as the classical calculation.Other quantum studies of H3 have been made by G ~ l d e n , ~ Yasumori and Sato,6Mortenson and Pitzer,' Nyeland and Bak,s M i ~ h a , ~ and Marcus.lo They alldiffer from the present work both in the choice of potential and in the simplifyingassumptions which reduce the problem to tractable form.METHODOLOGYWe consider the reaction A+ BC-+AB + C, where A, B, and C are structurelessparticles (except for nuclear spin) that interact through a known potential energysurface V.This surface may be either the exact or an approximate Born-Oppenheimer solution of the electronic Schroedinger equation for the real atomscorresponding to A, B, and C.The differential scattering cross section for rearrangement from the reactantchannel a to the product channel /? can be written in the centre-of-mass co-ordinate* supported in part by a contract with the U.S. Atomic Energy Commission. t present address : Department of Chemistry, Harvard University, Cambridge, Massachusetts.$ present address : Department of Physics, Pacific Lutheran University, Tacoma, Washington.This work is mainly from a Ph.D.thesis submitted by K. T. Tang to the Faculty of Pure ScienceColumbia University (1965).5M. KARPLUS A N D K . T . TANG 57system l2Here a corresponds to the quantum numbers for a particular rotation-vibration stateof molecule BC and to momentum ka of A relative to BC, the reduced mass (A, BC)being P a ; the index B has the same connotation for the final channel. The quantityTsa is the transition or scattering matrix (T matrix) defined by the two equivalentexpressionsTpa = <QB I YB I %"> = Vj-' I v a I @a>, (2)where $"'a, Vp are initial and final state interaction potentials ; i.e., Va is the part ofV that goes to zero as the initial atom-molecule relative co-ordinate goes to infinityand VB is correspondingly defined for the final channel. In terms of these, thetotal system Hamiltonian H i s writtenwhere K is the total (centre-of-mass system) kinetic energy operator, and Sa andZB are the non-interacting initial and final state Hamiltonians, which have solutions@a and 'DB (normalized to unit density) :Here R is the relative co-ordinate, r the molecular co-ordinate, and qBC(r), the mole-cular (rotation-vibration) wavefunction for the initial channel ; the quantities S, s,and vAB(~)B are defined correspondingly for the final channel.The functions Y$*),Yg* are, respectively, initial and final channel eigenfunctions of the entire Hamiltonian&' with energy E and outgoing (+) or incoming (-) spherical-wave boundaryconditions ; e.g., the Yi*) satisfy the Lippmann-Schwinger equations,12Z = K +V" = Z,+V", = Xp+ Vp, (3)@a = exp (ika * R)qBc(r)a, @p = exp (ikp S)vAB(s)p* (4)1with the positive infinitesimal E introducing the appropriate asymptotic behaviour.The functions ma, mB, Va, and Vfi being known, the difficulty in obtaining TBa, andfrom it the differential cross section Cafi(iB), occurs in the determination of Yp) orYb-).Since their exact evaluation, which is equivalent to solving the three-bodySchroedinger equation with appropriate boundary conditions, is not feasible atpresent, approximations must be introduced at this point. The simplest is to replaceYL') or Y$-' in TBa by d i a or d i p , respectively ; i.e.,which is the well-known Born approximation.12 Although it may be useful forsome atomic rearrangement problems (e.g., high energy, low activation barrier,)TBa(B) would yield incorrect results for the H+H2 reaction in the thermal region.To account in part for the strong repulsive interaction (high activation barrier) betweenthe approaching or receding atom and the molecule, it is appropriate to separateV", and Yp into two parts l2where rtrz(R) and V;(S) are distortion potentials that cannot produce rearrangement.They are chosen to account for the interaction as completely as possible, subject tothe condition that the Hamiltonians Se+Yz and x),+-y.,O have solutions xL*)TBa Tpa(B) = (@p I YB I @a) = <@p I v a I @a>, (6)Ya = v,O+ctr,:, VD = "Ir;+v-; (758 QUANTUM MECHANICS OF H 4- H2 REACTIONand xp)¶ respectively, which can be evaluated exactly or, at least, to a high degree ofapproximation.Here the xi*) satisfy the equationand a corresponding equation exists for xi*). Introduction of xi-) and use of eqn.(7) and (8) in the first (post) form of T’, [eqn. (2)] yields l2If now Yy) is approximated by x?), the so-called distorted wave Born approximation(DWB) is obtained; i.e.,T’, = (xj-1 I “yj I Yi”).TB, s T,,(DWB) = (xi-’ I V’i I xr)>.(9)(10)As compared with TBa(B) [eqn. (6)], eq. (10) should be considerably better forH+H2 because it includes distortion of the relative motion wavefunctions in boththe initial and final channels.However, the replacement of Yy) by x$+) in eqn. (9)to obtain (10) is generally a serious approximation, the accuracy of the resultingscattering matrix T’,(DWB) depending both on the nature of the problem and on thejudicious choice of Vz and Vi. The present work uses distorting potentials obtainedfrom two different approximations, which represent limiting cases as far as theresponse of the molecule to the incoming atom is concerned. In the first (I), thedistorting potentials, called VIO,(R) and V-&(s>, were calculated with the assumptionthat the molecule is unperturbed by the atom; i.e., the potential “t’, (and similarlyVfi) is expanded-tr,(R,r) =Cvan(R,r))P,(cos Y>, (11)(12)nwhere y is the angle (R, r ) and Yia(R) is writtenwith qBc(r), the vibrational wavefunction for the isolated molecule ; in the H3 system,the average over the ground vibrational state in eqn.(12) is very close to equilibriumvalue V,,(R, re), so that the latter was used for V;a(R). Thus, in case I (free moleculeapproximation), the wavefunctions xf:) and xfj-) can be writtenwith Frh+)(R) and Fb-)(S) determined by a standard partial wave expansion l2 for thespherically symmetric potentials VL(R) and V$(S), respectively. The secondapproach (II), which is in the spirit of the perturbed stationary state approximation,12bdetermines the distortion potentials Via@) and V&(S) with the assumption thatthe molecule adjusts adiabatically to the presence of the incoming atom ; e.g., V&(R)is the eigenvalue of the Schroedinger equationvia(R) = <VBdr)a I vaO(R?r> I VBdr)or),xf,“(R,r) = F‘,,+)(R)VBC(% X W ¶ 9 = F‘,i)(S)?*B(S)’, (13)where the eigenfunction qBc(R, r)= is the one which goes to qBc(r), as R+m.Tosimplify the solution of eqn. (14), only the first two harmonics in Y(R, r ) were includedand it was assumed that qBC(R, r), can be written for each value of the parameter Rin the separable form(16)V(RYr) = G ( R , r ) + ~2(R,r)Pz(cos y), (15)?!BC(R,r)a = (llr)zlBC(R?r)ar(R, cos y)aM. KARPLUS AND K . T. TANG 59Here (1 / r ) qBC (R, r), is a solution of the radial Schroedinger equation with the potentialV',,(R, r), and T(R, cos y) is a solution of the speroidal equation l3 obtained bysubstituting eqn.(15) and (16) into (14), multiplying by (l/r)qBc(R, r):, and inte-grating over r. Thus, for case I1 (adiabatic perturbation approximation), the initialand final state wavefuncticns arewhere F&:)(R) and Fhp)(S) are obtained from partial wave expansions with thepotentials "r/&(R) and Y&(S), respectively ; the unperturbed molecular wavefunctionqAB(~)B is used in xf,) to retain the form given in eqn. (10) for T,,(DWB).Although determination of the TP,(DWB) matrix elements by either of the pro-cedures (I or 11) outlined above is possible, the required integral evaluation is sotime-consuming that only a small number of such calculations were carried out.To permit a more general exploration of the nature of the reaction cross section, anadditional simplification was introduced.This was to assume, as justified in partby the DWB results, that reaction occurs only when the three atoms are in the neigh-bourhood of the linear geometry. The matrix element T',(DWB) is then approxi-mated by the expressionwhere y' is the angle (S, s) and A is the &function " strength " parameter. AlthoughA would be expected to vary as a function of the energy and of the initial and finalstates, a fixed value of A was chosen by comparison with a TB,(DWB) result. Becauseof the arbitrariness in A, all of the work with the DWBL model was based on thesimpler free molecule (case I) approximation for the distorted waves associated withthe relative motion.In the formulation outlined in this section, the particles (A, B, and C) are treatedas distinguishable and only the reaction A+BC+AB+C is considered.Since thereaction A + BC-+AC + B yields exactly equivalent results for the H + H2 system,the reported total cross sections include this factor of two. The symmetry conditionson the initial and final state wavefunctions due to the indistinguishability of theparticles are not explicitly introduced. This is permissible because the necessarysymmetrization can be applied to the T matrix elements obtained from the un-symmetrized calculation.14* l5 The details of the computations, which were pro-gramed for the IBM 7094 primarily in FORTRAN 11, will be given s~bsequent1y.l~dtk = F\tk(R)qBC(R,r)a, x\;b' = F$~)(~~AB(s)fi, (17)Tp,(DWBL) = A(x$-' I V;s(r' - n) I x:") (1 8)RESULTS AND DISCUSSIONIn this section we report some of the results obtained by the methods outlinedabove for the H+Hz reaction with the initial and final molecule in the groundvibrational state.We consider first the DWB approximation and then turn to themore extensive studies by the DWBL model.DISTORTED WAVE BORN APPROXIMATIONFor the free molecule (case I) and the adiabatically perturbed molecule (case 11)approximations, the effective potentials V&(R) and .rY;",,(R) are shown in fig. 1.Although the two potentials are similar, the difference in the region between 1-8and 3.5 a.u. is important for the reaction. That v & ( R ) is a " softer "potentialin this region results from both vibrational and rotational distortions of the moleculeby the atom; i.e., in the presence of the atom, the molecule has a smaller effectivezeropoint energy and its average orientation is shifted toward the linear geometry o60 QUANTUM MECHANICS OF H + H2 REACTIONminimum-energy .The elastic differential cross sections obtained with the twopotentials at an incident energy of 0.5 eV are shown in fig. 2. The difference betweenthem is relatively small. However, it is not negligible particularly for large1.0 2.0 3.0 4.0 5:O 6.0 7.0R (a.u.)potential Y&(R) corresponding to the adiabatically perturbed molecule.FIG. 1 .-Two-body potentials : -, potential .Yfa(R) corresponding to the free molecule ;0 30 60 90 I20 150 1808 (degrees)FIG. 2.-Differential elastic cross sections from two-body potentials as a function of the scatteringangle 8 for an incident energy of 0.5 eV : - from potential V&(R) corresponding to the freemolecule, - - - from potential V&(R) corresponding to the adiabatically perturbed molecule.scattering angles, which correspond to the small orbital angular momenta or impactparameters that contribute to reactive scattering.For both potentials the corres-ponding classical elastic cross sections are identical with the quantum results exceptin the small angle region (scattering angle O;55°).1M. KARPLUS AND K . T. TANG 61With the distorted wavefunctions from the effective two-body potentials, theDWB scattering matrix and cross sections were evaluated. Only the reaction withboth the initial and final molecule in the lowest rotational state (J = 0, J’ = 0) wasconsidered.The total cross sections Sap obtained by integrating ca(ia) over allangles and multiplying by two are listed in table 1 for a few incident energies. AtTABLE TO TOTAL CROSS SECTION (J = 0, J’ = 0) BY DWB APPROXIMATIONfree molecule perturbed moleculerelative energy,a approximation (I) approximation (11)eV a.u. a.u.0-5 0.009 0.200.33 - 0.0270-21 -a The barrier height is 0.396 eV.0-oO0103eV there is a profound difference between the two approximate models, theadiabatic perturbation of the molecule by the incoming atom yielding a 20-foldincrease in S,, over that corresponding to an unperturbed molecule. From thecollision time for this relatively low energy, it appears that the adiabatically perturbed0 30 60 90 120 150 180FIG.3.-Differential reaction cross section as a function of the scattering angle 8 for an incidentenergy of 0.5 eV : - from adiabatic model (11) for (J = 0, J’ = 0) ; - - - from exact quasi-classicalcalculation for (J= 0, all S).model should be the better approximation. Certainly, this is true for the vibrationaldistortion, although it is more questionable for the rotational reorientation.The differential cross section (in arbitrary units) obtained from the adiabaticmodel (11) at an energy of 0.5 eV is shown by the solid line in fig. 3. It correspondsto “ backward scattering ” in the centre-of-mass system ; i.e., the incoming atomstrikes the molecule, picks up an atom, and the newly formed molecule goes back0 (degrees62 QUANTUM MECHANICS OF H + H2 REACTIONdominantly in the direction from which the atom came. The differential crosssection for the free molecule model (I) is similar in shape to the adiabatic resultalthough the magnitude is much smaller.Also shown in fig. 3 by a dotted line isthe differential cross section (in arbitrary units) determined from the quasi-classicaltrajectory treatment at 0.5 eV ( J = O-+all J’). The form is again almost identicalto the model I1 result. Such strongly backward peaked cross sections are expectedwhen the quantum-mechanical wavefunction or classical path of the incoming atomis strongly distorted by a repulsive barrier. They contrast sharply with the Bornapproximation, which yields an oscillating cross section with its maximum in theforward direction.By an expansion of the total reaction cross section in terms of Contributions fromindividual partial waves 2, a comparison with the classical impact parameter (b)dependence can be made.For J = 0 to J’ = 0 and E = 0.5 eV the final state (exit)I values, which are essentially the same as those for the initial state, were found tomake contributions that decrease smoothly with increasing 2 and approach zero forZr lO(6 z2). This behaviour is very similar to that of the classical reaction proba-b i l i t ~ , ~ which goes to zero at b = 1-85 a.u.2 0 6 - &04 -0 2 -0 30 60 90 120 150 1.50I1.0 -0 8 -////0 2 - /04-I l l l t l l l l l l l lT (degrees)FIG. 4.-Fractional contribution of H3 configurations to the total reactive cross section (J = 0,J’ = 0) for model (I) at 0.5 eV.For each value of7, all configuration anglesy less than7 are included(see text).To obtain an idea of the configurations of the three nuclei which make the dominantcontributions to reaction, we used model I and considered a quantity Tpa(DWB, z)defined bywith H(x) the Heaviside function [H(x) = 0, x <O ; H(x) = I, x> 01 and OSz <n ;thus, Tpa(DWB, n) = Tpa(DWB). The corresponding total cross section Sap(T)provides a semi-classical measure of the contribution to reaction for atom, moleculeorientations with y in the range between 0 and 2. The quantity S’aa(z)/Sab. for theJ = 0 to J’ = 0 reaction at an energy of 0-5 eV is plotted as a function of z in fig.4.Only small angles contribute ; i.e., 80 % of the cross section is obtained with y < 40”.In the quasi-classical calculation,2 the average value of y is 24” at the same relativeenergy. It is these results, and the molecular reorientation in the adiabatic treatment,which supply some justification for the linear model discussed belowM. KARPLUS A N D K . T. TANG 63LINEAR APPROXIMATIONThe linear model [eqn. (IS)] has the constant A as a strength parameter for the&function. This was chosen so that T,,(DWBL) = TB,(DWB) at 0.5 eV for the(J = 0, J‘ = 0) reaction and was kept fixed at the same value for all of the calculations.For J = 0 to J’ = 0, the total cross section in the energy range between 0.25 and3.3 eV is shown in fig.5. Fig. 6a and 6b show the total cross sections for J = 0to J’ = 0, 1, and 2 in the threshold and the intermediate energy regions. In fig. 7are given the sum of the linear model cross sections (J = 0-d’ = 0, 1,2) and for COWL-parison the quasi-classical result (J = O-+all J’). The most important point is thatthe quantum calculations yield a significantly higher effective threshold than doesthe quasi-classical treatment. This is reasonable when one considers that the sameI0 I. 0 2.0 3.0incident energy (eV)FIG. 5.Total reaction cross section for (J = 0, J’ = 0) and incident energies between 0-25 and3.3 eV from the linear model.initial molecular zero-point energy is present in both approaches, but that the quantumconstraints in the saddle-point region may provide a limit on the vibrational energyavailable for crossiog the barrier which does not exist for the classical trajectories.The much higher values reached by the quantum cross sections at large energies mayresult from the breakdown of the adiabatic approximation, which invalidates anenergy-independent choice of the strength parameter.Also, since the DWB methodis a perturbation procedure, it becomes less valid as the magnitude of the total crosssection increases.16Of interest in fig. 6a and b are the relative values of the cross sections for differentfinal rotational states. While at threshold the (J = 0, J’ = 0) cross section is largest,at most energies the ratios are (J = 0, J’ = l ) > ( J = 0, J’ = 2)>(J = 0, J’ = 0).From the quasi-classical calculations at energies below 0.75 eV, the result (J = 0,J’ = 1)>(J = 0, J’ = 2)>(J = 0, J’ = 0) has been obtained by making a corres-pondence between the final state angular momentum (in units of fi) and the nearestinteger value of J.Since the nuclear spin and rotational states of the H2 moleculeme coupled by the Pauli principle, the variation in the cross sections for differen64 QUANTUM MECHANICS OF H -I- H2 REACTIONrotational states is significant for an appropriately symmetrized calculation of thepara ( t 5. ) to ortho ( t t ) hydrogen conversion in the H+H, exchange reaction.In particular, the present results raise questions concerning the essentially classicalassumption l7 of a simple (3 : 1) ratio for the (para-ortho) against (para+para)rate constants.1.20.20-3 0.4 0.5 0.6incident energy (eV)0.3 04 0.5 06 0.7 0.8 0.9 1-0 1.1incident energy (eV)FIG. 6.-Total reaction cross sections as a function of incident energy from the linear model:- corresponds to J = 0, J’ = 0, -.-.corresponds to J = 0, J’ = 1 ; - .. - corresponds to J = 0,J’ = 2. (a) threshold energy region ; (b) intermediate energy regionM. KARPLUS AND K. T. TANG 6586c? e,..4 8 + 0g 4El22 2v).Mc,00.2 0-4 0.6 0-8incident energy (eV)FIG. 7.-Total reaction cross sections as a function of incident energy : - from linear model forJ = 0 to J’ = 0, 1 and 2 ; - - - from quasi-classical calculation for J = 0 to all J’.0 30 60 90 120 150 1808 (degrees)FIG.8.-Differential reaction cross section as a function of the scattering angle 8 from the linearmodel at an incident energy of 0.5 eV : - corresponds to J = 0, J’ = 0 ; -.-. corresponds toJ = 0 S = 1 ; -..- corresponds to J = 0, J’ = 2.66 QUANTUM MECHANICS OF H -I- Hz REACTIONThe differential cross sections at an incident energy of 0.5 eV for J = 0 to J' =0, I, 2 are shown in fig. 8. All the curves are similar to each other and to the completeDWB result (fig. 3). More interesting is fig. 9 which presents a plot of the variationwith incident energy of the form of the (J = 0, J' = 0) differential cross section (inarbitrary units). As the energy increases from 0.4 to 1.5 eV, the primary peak in thecross section gradually shifts in the forward direction and a secondary peak appears." 0 30 60 90 120 150 1800 (degrees)FIG.9.-Differential reaction cross section as a function of the scattering angle 0 for J = 0, S = 0at a series of incident energies from the linear model.Thus, the incoming atom is strongly repelled by the barrier for low incident energies,but tends to '' remember " its initial direction of motion as the incident energy becomesgreater than the barrier height (0-396 eV). This trend can be obtained from a simplesemi-classical model which relates the elastic and reactive scattering. Although ithas not yet been observed in the molecular case, exactly corresponding results arefound for the (d, p) stripping reaction in a Coulomb field as the energy increases froma few MeV to a few hundred MeV.18CONCLUSIONSA distorted wave Born (DWB) approximation has been utilized to examinerearrangement scattering in the H + H2 system.The large difference in the total crosssection between the two limiting cases which have been considered (i.e., the moleculeis unperturbed or adiabatically perturbed by the incoming atom), show thatmolecular perturbations and their effect on the two-body potential for relativemotion can play a significant role in reactions with an activation barrier.The results for a variety of reaction attributes (i.e., energy dependence of differentialcross section, cross-section variation with final rotational state, geometry of atoms inreactive region) suggest by their similarity to the available quasi-classical calculationswith the same interaction potential that quantum effects are not very important.However, the energy dependence of the total reaction cross section is different inthe quantum and the classical treatments; in particular the effective threshold ishigher in the quantum calculation.To determine whether the difference is real oM. KARPLUS AND K. T. TANG 67is the consequence of the approximations in the quantum formulation will requirefurther study.The differential cross section is found to be strongly backward peaked at lowenergies in correspondence with expectations for a strongly repulsive barrier. Asthe energy increases and the effect of the barrier decreases, the peak in the differentialcross section moves forward and secondary maxima appear.This shows thatmeasurement of the differential cross section as a function of energy would yieldinformation concerning the potential surface.The dependence of the total reaction cross section on the final rotational state(J’ = 0, 1, 2) has been examined for molecules initially in the ground state (J = 0).The results question the usual procedure for obtaining para-H2 to ortho-€3, conversionrate constants. However, more extensive and reliable cross section calculations areneeded before an unequivocal conclusion is possible.We thank Dr. Keiji Morokuma for assistance with some of the calculations.We are grateful to the Columbia University Computing Centre for their co-operationin making available the machine time used in the project.(a) S.Glasstone, K. J. Laidler and H. Eyring, TIze Theory of Rate Processes (McGraw-HillBook Company, Inc., New York, 1941) ; (b) H. S . Johnston, Gas Phase Reaction Rate Theory(Ronald Press, New York, 1966) ; (c) K. J. Laidler and J. C. Polanyi, Prog. Reaction Kinetics,1965, 3, 3.M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Physics, 1964,40,2033 ; 1965,43,3259.R. N. Porter and M. Karplus, J. Chem. Physics, 1964, 40, 1105.see ref. (l), particularly (l(b)), for a discussion of this point.S. Golden, J. Chem. Physics, 1954,22, 1938.I. Yasumori and S. Sato, J. Chem. Physics, 1951,22, 1938.C. Nyeland and T. Bak, Trans. Faraday SOC., 1965,61, 1239.D. Micha, Ark. Fysik, 1965, 30,425, 437.’ E. Mortenson and K. S. Pitzer, J. Chem. Soc., Spec. Publ. 1962, 16, 57.lo R. A. Marcus, J . Chem. Physics, 1966, 45,4493.l1 The work of Micha is most closely related to certain parts of the present treatment. Inparticular, the general model used by Micha is similar, in principle, to one of the models(linear model) described by us ; however, the additional approximations introduced by Michain the actual calculation raise serious doubts concerning the results.l2 (a) M. L. Goldberger and K. M. Watson, ColZision Theory(John Wiley and Som,Inc., New York,1964); (b) T. Y. Wu and T. Ohmura, Quantum Theory of Scattering (Prentice-Hall, Inc.,Englewood Cliffs, New Jersey, 1962) ; (c) A. Messiah, Quantum Mechanics (John Wiley andSons, Inc., New York, 1962), chap. XIX.l3 J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little and F. J. Corbato, Spheroidal WaveFunctions (John Wiley and Sons, Inc., New York, 1956).l4 see ref, (12), particularly chap. 4 of 12(a).K. T. Tang and M. Karplus, to be published.l6 A. C. Allison and A. Dalgarno, Proc. Physic. SOC., 1967, 90, 609.l7 A. Farkas, Orthohydrogen, Parahydrogen, and Heavy Hydrogen (Cambridge University Press,1935).J. R. Erskine, W. W. Buechner, H. A. Enge, Physic. Rev., 1962, 128, 720; L. C. Biedenharn,K. Boyer and M. Goldstein, P/zysic. Rea., 1956, 104, 383

ISSN:0366-9033    

DOI:10.1039/DF9674400056

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

The H+H, Reaction in Two DimensionsBY M. S. CHILDPhysical Chemistry Laboratory, South Parks Road, OxfordReceived 31st May, 1967The correlation between reactant and product rotational energy, for levels which correlate witha true bending vibrational level of the transition state, is investigated for the H+H2 reaction in twodimensions. It is predicted that the product is formed in the same rotational state as the reactant.The energy variation between the reactants and the transition state provides an activation barrierfor each individual collision from which is calculated the range of impact parameters for whichreaction can occur, and the reaction rate constant k. The expression for k is identical with thatgiven by absolute reaction rate theory.All potential surface calculations for the H + H2 reaction indicate that the systempasses through a strongly preferred linear configuration or transition state. Thepurpose of this paper is to establish a connection between the rotational levels, j andj ' , of the reactant and product molecules respectively and the bending vibrationallevels of the transition state, and from this connection to calculate the reactionprobability, the reaction rate constant and the product rotational energy for a givenvalue of j .In order to bring out the essentials of the problem, certain assumptions andrestrictions are made.The system is constrained to lie in a fixed plane and it isassumed that the angular motion can be separated both from the purely vibrationalmotion of the atoms and from motion along the reaction co-ordinate.l# Distancealong the reaction co-ordinate is measured by the variable s used by Marcus ratherthan the corresponding quantity qr in ref.(1). Tunnelling contributions to thereaction rate constant are ignored.A satisfactory correlation is achieved only for reactant rotational levels whichcorrelate with a true bending vibrational level of the transition state, because, fora smooth transition from reactant to product angular co-ordinates, the free rotationwhich occurs at large distances must be effectively quenched at short range. Calcula-tions based on the potential surface of Porter and Karplus indicate a satisfactorycorrelation for the ljl = 0+3 rotational states of HZ. Since the reaction is sym-metrical and all memory of rotational motion is lost in passing through the transitionstate, the product is formed with the same angular momentum as the initial reactantmolecule.For a given value ofj, the effective activation energy for a particular collision, theenergy difference between the reactants and the transition state, is found to dependonly on the orbital angular momentum 1.The range of 1 values, or equivalently ofimpact parameters, Ab, for which reaction can occur at given translational energy E,can therefore be calculated. Ab takes the place in a two-dimensional reaction ofthe reaction cross-section in three dimensions. Its variation with E is shown in fig. 4 ;the smooth increase with increasing energy is interrupted by a cusp whenever thetotal rotational and translational energy of the reactants reaches the appropriatebending vibrational level of the transition state.Observation of such cusps wouldprovide important information about the structure of the transition state.6M. S . CHILD 69The reaction rate constant, calculated by averaging Ab over the Boltzmanndistributions of initial translational and rotational energy takes the form predictedby transition state theory,A& is the difference in zero-point energy (including the bending vibration) betweenthe reactants and the transition state. The origin of the probability factor (fv/'),is an increase in effective activation energy with increasing j . Only for reactants inlow rotational states does the system pass through a sufficiently low level of thetransition state to make a significant contribution to the rate constant.k = (f"/J;>Z exp (- AEo/RT).INTERNAL ROTATIONAL STATESWith these assumptions, we are concerned only with the angular motion of thesystem. The appropriate energy levels at a given point on the reaction path dependonly on the shape of the barrier to internal rotation, V(x,s) (fig.2). In the co-ordinatesystem shown in fig. 1 at a given value of s on the reactants side of the transitionstate (s < 0), the Schrodinger equation for angular motion iswhere m = m,(m, +m,)/(m, +m2 +m,), p = m2mJ(m2 +m3) and W(S) is theappropriate energy level.I 32(a (4)FIG. 1 .-The co-ordinate system : (a) reactants ; (6) products.We first consider free internal rotation, when the barrier V&,s) is negligibly small ;both orbital angular momentum I and rotational angular momentum j are goodquantum numbers, therefore $ can be written$Lj = exp (ile) exp (ij4), (2)where L = Z+j, and the energy becomesZ2fi2 j 2 h 2WLj = - +-- 2mr2 2pp2- ~~h~ (j-g)2h2+ 21" '--21(3)(4)where I = mr2+pp2, I* = mr2pp2/(mr2+pp2) and g = Lpp2/I.The subscript L is preferred to I because total angular momentum is always a goodquantum number.The first term in eqn. (4), the energy of overall rotation, isindependent of any barrier to internal rotation; it is the second termthe internal rotational energy, whose form is affected by the barrier V(x,s). In theabsence of this barrier, Wij shows a characteristic variation with s, which arises fromw;>~(s) = ( j - g)2h2/21*, ( 5 70 H+Hz REACTION IN TWO DIMENSIONSthe variation of g.rotational energy level of the reactant molecule,and all levels (except j = 0) are doubly degenerate.As the reaction proceeds how-ever, g increases and, according to eqn. (5), this degeneracy between + j levels isremoved, and degeneracies between other levels occur whenever g has an integral orhalf-integral value because the energy depends on l j - g I ; the Ievelsj, andj, related bycross each other at these points. j , and j , have the same or opposite parity accordingto whether g has an integral or half-odd integral value. This curve-crossing behaviouris characteristic of free internal rotation.Initially g is zero because I is infinite, Wij simply gives thejthwLj(- co) = j2h2/2pp2 (6)h+h = 29 (7)W’a b n+a ntb 2 n+aFIG. 2 .T h e barrier to internal rotation V(x,s).We turn now to the case when the barrier V(x,s) in eqn. (1) cannot be neglected.The trial solution+Lj = ~ X P (il@ ~ X P ( i j 4 ) t , j ( X > (8)leads to an equation for cLj,which is simplified by the subsitutionAn approximate solution to the equation forfLj,t L j(X) = ~ X P 1 - i(.i - g>XVi.j(x>*subject to the boundary condition that, because tLj(x+27r) = cLj(x),is then developed in terms of semi-classical wave-functions. By using standardconnection formulae to cross the non-semi-classical regions around a,b,n -t- a,7t + b in fig. 2, the two independent WKB wavefunctions in the region x < a,fL j ( x + 2 ~ ) = ~ X P [Wj - g ) i ] f L j W , (11)where p = ,/21*( Wf(.s) - V(x,s)), are continued into the region b < x < 27~ +a.boundary condition (1 1) is then applied to the arbitrary combinationThef = A+f++A-f- (13M.S . CHILD 71to determine an acceptable ratio A + / L and the corresponding energy W'. Twodistinct cases are considered.If W' is well below the maximum value of V, application of separate connectionformulae at each of the four classical turning points leads to the ftmctionsf;t in theregion b < x < 2rc i- a,f* = [sinh2y + cosh2y exp (T 2iz)I exp (k f S r ' i d x fi in/4)+ sinh y cosh y[1+ exp (& 2iz)l expwhereThe energy level condition appears as a condition on the integrals y and z,For energies W'Q V,,,, cosh y-+ GO and eqn.(15) reduces to cos z = 0 orcos z cash JJ = cos n( j - g ) = (- 1)' cos 7tg. (15)x+az = 's pdx = (~++)Tc, (16)h bfor all values of g. Eqn. (16) is the Bohr quantization condition for, in this case,a bending vibrational level of the transition state. According to eqn. (15), each levelis doubly degenerate, with one even and one odd value ofj, the degeneracy arising fromthe symmetry of the barrier V(x,s) (the incoming atom is equally attracted to eitherend of the reactant molecule). A set of equally spaced doubly-degenerate energylevels is therefore the characteristic of completely restricted internal rotation.The transition between the energy region W' Q V,,,, where the rotation is com-pletely quenched, and the region W' % V,,,, where the barrier can be ignored, is madewith the help of a different type of connection formula.The barrier V(x,s) in fig. 2is assumed quadratic in x around the maxima and a direct connection is establishedbetween the wave-functions in the regions x <a, b < x < n + a, n + b < 1 < 2n + a,without considering their forms in the non-classical regions a < x < b, n + a < x < n + b.In the region x < 2n + a the functions becomewhere E = (wl- Vmax)/fim, io being the imaginary frequency associated with thenegative curvature at the barrier maxima, and *Application of the boundary condition (1 1) yields the energy level condition4 ( ~ ) = ~ ( 1 -In E)+arg r(%+k) = -+(-E).cos (z - 4) J[ 1 + exp (- 27re)l = ( - 1)j cos ng.(18)(19)* The phase correction 9, given by Heading,4 was omitted from the corresponding analysis inref.(1) ; in eqn. (42), z should be replaced by z- 972 H+Hz REACTION I N TWO DIMENSIONSThere is a smooth connection between eqn. (1 5) and (19) in the energy region whereFor a quadratic both y and -2ne are sufficiently large that ey% e-y, e-2*t$= 1, 4 -h 0.barrier,--c = ; J j P / dX,hence eqn. (15) and (19) both reduce to the formcos (k J;+'pdx) exp (k J) p I dx) = (- l)j cos ng.The typical variation in internal rotational energy as s increases is shown on theleft-hand side of fig. 3. The calculation was based on the potential surface of Porterand Karplu~.~ The lower heavy line follows the zero-point level of the true vibration-.4 -.3 -9 -.I 0 -3 4s(amu+ A)FIG. 3.-The internal rotational energy levels for L = 15.of the system.The difference A V between upper and lower heavy lines is the maximumbarrier to internal rotation, the energy required to increase x from 0 to n/2 with rin fig. 1 fixed and p allowed to take its optimum (minimum energy) value. Con-tinuous and dashed refer to levels with even and odd parity respectively.This figure illustrates how the free internal rotational behaviour, typified by thecurve-crossing at s = -0.25, goes over for the lower levels to the characteristicbending vibrational behaviour with equally spaced even-odd degenerate levels ats = 0. The point s = -0.25 is an example of an integral crossing point (g = 2),where the levels which cross have the same parity ; the barrier causes an appreciableinteraction between them if W' < V,,, + A V.There are also half-odd crossing points,g = 2.5 and 3.5 at s = -0.1 and -0.01 respectively. Here the levels which crosshave opposite parity, and do not interact together ; this is a direct consequence of thetwo-fold symmetry of the barrier. The assignment o f j values is made by followingeach level outwards to the region where the barrier can be ignored and the energyis given by eqn. (5). If, in doing this, an integral crossing point is encountered, theupper or lower curve is followed if W'<V,,,+AV and a cross-over made if W'>V,,,+AV. By the same procedure it is seen in fig. 3 that the j values to be assigneM. S .CHILD 73to the bending vibrational levels at s = 0 are, in order of increasing energyyj = (+ 1 ,O),(+2,- l), (+3,+4), (-2,+5).Since the reaction is symmetrical the right-hand side of fig. 3, s>O, is the mirrorimage of the left. It is obtained by repeating the calculation using the co-ordinatesr’, p’, O’, 4‘ and x’ in fig. I, appropriate to the products side of the reaction. There isa smooth transition across the central line s = 0 for levels which are vibrationalin character, i.e., those with W’< V,,,, because a bending vibration can be describedequally well as an oscillation in x or an oscillation in 2’. No such connection isavailable, however, for levels with W’> V,,, because the motion then involves a fullrotation of either the reactant or the product molecule.It is clear from the sym-metry of fig. 3 that corresponding levels on the reactants and the products sides ofthe reaction must have the same j values. It follows that rotational angular momen-tum is conserved if the system passes through a bending vibrational level of thetransition state but no predictions can be made if the internal rotational energy istoo high to allow this.Fig. 3 refers to total angular monemtum L = 15. A similar analysis was madefor all total angular momenta between L = 0 and L = 20. It is found that theenergies of the two lowest levels at s = 0 are independent of L (6.43, 8.17 ltcal), thethird level varies between 9.72 and 9.78 kcal and the fourth level between 10433 and11.14 kcal, and also that because the frequency of crossing points increases with L(g is directly proportional to L) the correlation with initial rotational level changesslightly as L increases.The order o f j values at s = 0 isO<L<910 <L =a 1L = 1213 < L d 617<L<20(O+ 11, (+2, -I>, (-2, +3), (+4Y -3)(O+ 11, (+2Y - 11, (-2, +3), (+4, + 5 )(O+ 11, (+2, - 11, (+4, +3), ( - 2 7 +5)(0, +I)? (+2Y -I>, (+4, +3), (-2, +7)(07 +1)Y (+2, -11, (+4, +3)Y (+6, +7).A similar correlation is obtained for -2O<L<O by reversing the sign of j , becausethe energy levels given by eqn. (9, (15) and (19) depend only on Ij-91, and gccL.REACTION PROBABILITY AND REACTION RATE CONSTANTThe reaction probability for given L and j depends on the increase in internalenergy WLj(s) = Wij(s) +L2h2/21 compared with the available translational energy E.If tunnelling is ignored, the condition for chemical reaction is thatThe right-hand side of eqn.(23) is found to increase monotonically with ILI, so thata critical value IL* I exists above which no reaction will occur. From L* may becalculated a critical orbital angular momentum, I* = L*-j, and hence a criticalimpact parameterwhere mu is the initial reactant momentum. Fig. 4(a) shows the variation of b*with E forJ = 0, 1,2,3. The upper branch of each curve refers to Z>O and the lowerbranch to Z<O. The discontinuities for j # 0 arise because a given rotational levelcorrelates with one level of the transition state if j and L have the same sign and ahigher level if they have opposite signs.Fig.4(b) which is derived directly from 4(a), illustrates for each value of j therange of b values, Ab, (b2f <b<b*,) which lead to reaction, as a function of E. AbE> WLj(0)- WLj( - Go). (23)b* = l*ii/mvy (2474 H+Hd 7 REACTION IN TWO DIMENSIONStakes the place of the reaction cross-section for a reaction in three dimensions. Thecurves have been weighted according to a Boltzmann distribution of reactantrotational energy at 300°K ;where5 is the rotational partition function. From the sym-etry of the reaction,fig. 4(b) also gives the Ab for product formation in the prescribed rotational states.The threshold for each value of j is the increase above the initial rotational energyrequired to reach the lower vibrational level belonging to ljl ; the cusp occurs whenn, = exp (-j2fi2/210kT)lfr, (25)E(kca1)FIG.4.-(a) Critical impact parameters b* as a function of E ; (b) the range of impact parametersAb which lead to reaction, as a function of E.- j = o , - - - j = 1 , - . - . - . j = 2 , . . . j = 3.the energy reaches the upper level. The cusps are naturally less strongly marked inthe total Ab for all rotational states, but as the temperature is reduced below 80°Kthe level Jjl = 1 becomes most heavily populated and the cusp at E = 8 kcal beginsto stand out clearly. Qbservation of such a cusp would provide direct informationabout the structure of the transition state.Apart from tunnelling corrections, the rate constant for given j is derived fromAbj(E) by integrating over the Boltzmann distribution of relative velocity in twodimensionsAbj(E) exp (- mc2/2kT)c2dc, 4rnc;' = AEj,where AEJ is the activation barrier for the chosen value of j .Owing to the Boltzmannfactor, the main contribution to kj comes from low translational energies E ; foM. S . CHILD 75this low energy region, Ab,(E) may be derived directly from the range of total angularmomenta, 0 <L <L* for which reaction can occur, L* being given by the equationE = AEj + L*2222 121:It follows thatAbj(E) = ['m( 1 -9)T.(The use of the same AE, at all energies in eqn. (27) implies the same rotational-bending vibrational state correlation for all L values. At very high energies, L*becomes too high for this correlation to hold).With this expression for Ab,(E), k jis found by direct integration ;where 2 = J(2nlkTjm), the two-dimensional collision number.The mean rate constant i% is a Boltzmann average of k,,k, = 4 2 exp (- AE,/kT), (28)l * k? = - k j exp ( -j2h2/21kT).f r j=-coAccording to eqn. (28) and (29), the contribution from low j values (ijl = 0+3),which correlate with a true bending vibrational level of the transition state, is pro-portional to exp [-(AEj+j2fi2/21,)/kT)] = exp [-(AEo+Ev)/liT], where Ev is theappropriate bending vibrational energy. The contribution to k from a given rota-tional state therefore depends not on the rotational energy of the reactant, but on theenergy of the corresponding level of the transition state. It follows that only lowrotational states make a significant contribution to the sum in eqn. (29) and that ktakes the classical formwhere fy is the partition function for the bending vibrational level of the transitionk = Cfllfr)Z exp (-A&w), (30)the factor 3 counteracts the two-fold degeneracy of the vibrational levels. Theappearance of the probability factor c f v l f r ) in this collisional theory is due to an increasein effective activation barrier AEj as j increases so that the proportion of moleculeswith sufficient translational energy to cross the appropriate barrier decreases sharplywith j.l M. S . Child, Proc. Roy. Suc. A, 1966,292,272.R. A. Marcus, J . Chem. Physics, 1966, 45,4493.R. N. Porter and M. Karplus, J. Chem. Physics, 1964,40,1105.J. Heading, An Introduction to Phase-Integral Methods (Methuen, 1962)

ISSN:0366-9033    

DOI:10.1039/DF9674400068

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

GENERAL DISCUSSIONProf. M. Karplus (Harvard University) said: In the original formulation of thequasiclassical trajectory method for calculating cross-sections for exchange reactionsof the type A+BC+AB+C, the impact parameter b was selected from a randomdistribution between 0 and b,,,, the latter value being chosen so that the reactionprobability P, is zero for bzb,,,. On the other hand, the initial internal stateof the molecule BC was selected from the quantum-mechanically allowed valuesof the angular momentum and the rotation-vibration energy. We have now followedthe suggestion of Marcus and explored the effect of selecting b in accordance withthe quantum-mechanically allowed values of the orbital angular momentum L,given the initial relative velocity VR.When b is selected at random for a fixed V,, the calculated values of P, for thereaction H+H2+H2+H can be fitted to a good approximation by the simpletwo-parameter functionTreating b as a continuous variable and using eqn. (1) for P,(E), one may obtainthe reaction cross-section for the given value of VR by the simple integration(" ' ) a bi.bP,(b)db = 4 -Alternatively, restriction of impact parameter to the quantum-mechanical valuesof L [i.e., bl = (h/pVR){2(2+l))l/2, or b, = (h/pVR)(2+l/2), where 2 = 0, 1, 2, .. .]TABLE NUMERICAL COMPARISON OF EQN. (2) and (3).S4eqn. (311a VRa J 'rn 'ma, 'Aeqn. (2)1 L = [r(l+ l)l+h L = [i++)h1.2 0 0.39 1.85 11 1-940 a.u. 1.95 a.u. 1.945 a.u.0.95 0 0*13& 0.95 4 0.171 a.u. 0.178 a.u. 0.175 a.u.0.87 0 0.025 0.375 2 0.0051 a.u.0.0068 a.u. 0.0057 a.u.Q, velocity in units of 0.979 x lo6 cmlsec.b, this value is a correction to table 3 of ref. (1).and use of the quantum-mechanical degeneracy factor (22+ 1) together with eqn.(1) for P,(bi), gives S, as the sumNumerical comparisons of the reaction cross-section obtained from eqn. (2) and(3) at three values of V' are givenin table 1. Restriction of the impact parameterto quantum-mechanically allowed values does not produce a significant change inS, even for small b, values and velocities near threshold. Fig. 1 presents a graphicalrepresentation of eqn. (2) and (3) for values for the parameters a and b,, whichcorrespond to V' = 0.95 in units of 0.979 x lo6 cm sec-l. The initial molecularangular momentum is zero in each case.The area under the solid curve in theM. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Physics, 1965, 43, 3259.7GENERAL DISCUSSION 77figure is S, as given by eqn. (2); the area under the step function is S, as given byeqn. (3).The results reported here show that quantization of the impact parameter,particularly according to the semi-classical relationship L = pVRb = (1+1/2)h,0.3 c0 0.2 0.4 0.6 0-8 1.0b(a.u.)FIG. 1.-Probability of reaction Pr against impact parameter b. The curve corresponds to thecalculated probability and the bar graph to its approximation in eqn. (3).has a negligible effect on the reaction cross section for the H+H, system. Asto the quantization of the spatial orientation, which was also suggested by Marcus,we have made no quantitative tests ; however, qualitative considerations suggestthat the effect would be very small.Dr.D. A. Micha (University of Calgornia, Saiz Diego) said: One of the presentdifficulties in the quantum mechanical description of direct molecular reactionsis to aaccunt for the effect of inelastic processes (rotational and vibrational excitations)on the reaction cross-sections. A possible way of dealing with these effects wouldbe to use optical potentials. A formal exact theory of rearrangement collisionsin terms of optical potentials has been known for some time and we have thoughtit of interest to discuss its application to molecular reactions of the type A+BC+AB+C. The scattering wavefunction Yi+) for a collision where BC is in the stateva is given bywhere E(+) = E+k, E+O, CD, describe the system in the absence of the initial channelinteraction 0, and K , is given by 2 = IC,+O,.An optical (complex) potential u i + )for the initial channel is defined bywith = (y, I Y$+))q,. The real part of this potential describes the effect ofdistortion of the relative motion and of the internal motion of BC on the elasticscattering, and its imaginary part accounts for the effect of inelastic and reactivetransitions.Y:') = @,+(E'+'- tiJ-'o,Yi+),y:;) = c"D +(E'+'-li,)-'u~+'y~,+',It may be written explicitly asd+) = (7, I vnE+) I I?,>,M. H. Mittleman, Physic. Rev., 1961, 122, 1930; 1962, 126, 373.M. L. Goldberger and K. M. Watson, ColZision Tiwory (John Wley & Sons, Inc., New York,1964), chap.1178 GENERAL DISCUSSIONwhereSimilarly, a potential u r ) describes the elastic scattering of C colliding with ABin state yrs. Using these potentials the reaction scattering matrix takes the formwhere the symbol (-) indicates incoming waves and = U , - V $ + ) . This exactexpression is a useful starting point for the introduction of approximations. In areaction which proceeds through a direct mechanism it would be enough to keepthe first term in TB,, since the second term corresponds to reactions which proceedthrough multiple inelastic and rearrangement collisions. For direct reactionsIf, furthermore, inelastic and reactive collisions do not affect the initial and finalchannels, the optical potentials may be approximated by their lowest order in theinteractions,and similarly II~-)cz$, in which case Y:;) and Y($ reduce to the distorted wave-functions xi+) and xb-), and Tpa is given by the distorted wave Born approximationNevertheless, for reactions where the nature of the interaction or the range of relativekinetic energies studied lead to large probabilities of inelastic transitions it will bebetter to use T&), which takes into account those inelastic processes.Prof. M.Karplus (Harvard University) said: To provide a quantitative test oftransition state theory and its underlying assumptions, K. Morokuma and I havebeen examining the initial, transition, and final-state energy, and the phase-spacedistributions of appropriately selected classical trajectories for H, H2 reactivecollisions.Since this analysis is pertinent to the question of vibrational adiabaticity, we reportsome results for the simplest case of linear collisions.There are two degrees offreedom, which can be associated with the molecular vibration and relative atom-molecule co-ordinates in the initial state and in the final state and with the symmetricand asymmetric stretching vibration in the transition state. In fig. 1, (a) and (b),we plot the energy in the symmetric vibration at the saddle-point against the initiallypresent in molecular vibration. Each dot represents one reactive trajectory selectedfrom a set with fixed initial translational energy and a Boltzmann distribution ofvibrational energies (T = 900°K).The line corresponds to the energy partitioningpredicted by the adiabatic hypothesis. Significant deviations from adiabaticityoccur, and these are in the expected direction : if the initial translational energy isbelow the classical barrier (0.396 eV) the reactive trajectories have less than the adiabaticenergy in the symmetric mode; if the initial translational energy is higher than thebarrier, the symmetric mode energy is above the adiabatic value. An alternativecomparison can be made for a system with fixed initial vibrational energy (6.2 kcal,the zero-point energy of H2) and a Boltzmann distribution of translational energies(T = 900°K). Here the average symmetric mode energy of reactive trajectoriesis 2.3 kcal as compared with the adiabatic value of 3.1 kcal.Thus, more energyis available for crossing the barrier (i.e., in the asymmetric stretching mode) than issuggested by the adiabatic model. However, the final state molecular vibrationaGENERAL DISCUSSION 79energy in all cases is close to the initial state energy. Thus, the reaction is vibration-ally adiabatic in terms of initial and final-state variables. This implies that for theH, potential, which is symmetric about the saddle-point, the redistribution of energythat occurs on the incoming part of the trajectory is approximately counterbalancedon the outgoing portion.ER = 0.48 eVTvib = 900°Krc.04-rFIG. 1.-Relation between vibrational energy in the symmetric mode at the saddle-point and theinitial molecular vibrational energy (a) initial translational energy = 0.48 eV, (b) initial translationalenergy = 0.33 eV.The initial vibrational energy is selected from a Boltzmann distribution at 900°K.The solid line corresponds to the adiabatic result.Prof. R. A. Marcus (University of I h o i s ) said: I should like to indicate somespecific examples of the kinetic energy expression, based on natural collision co-ordinates, to illustrate the method of solution. For reaction in a plane one find80 GENERAL DISCUSSIONapart from several terms which are small at both small and large I s I. (Here,pR2 is I in my paper and f is cos(c- ~) there.) Evidently, p4 is a constant of themotion. Introduction of an adiabatic approximation for the r and y motions isrelatively straightforward.Non-adiabatic corrections can be calculated using amethod similar to that used in ref. (1) of the paper.For a reaction in space the “low mass” approximation is readily introduced asfollows. (For brevity of presentation the classical orbital plane, i.e., the 6, $-plane,which is constant in this zeroth order approximation, is taken to occur at 6 = rc/2).Apart from terms which are small at both small and large I s I one findsAgain, p4 is a constant and an adiabatic approximation is readily introduced foradiabatic-separation of the r- and the (y,X)-motions. The solution of the y,x-equation needs further analysis. It may involve methods similar to those used torelate, i.e., the asymmetric top to the symmetric top behaviour. A detailed dis-cussion of these equations and of approximate solutions will be given in a futurepublicat ion.Prof.R. Wolfgang (University o j Colorado) said: This comment regards theapplicability and usefulness of statistical or “ phase-space ” theory in treating directreactions, i.e., those occurring on the time scale of one molecular rotation or less.Perhaps the best single experimental criterion as to whether a reaction is direct, isthe angular distribution of products. If this is symmetric about the centre of mass,a long-lived intermediate is indicated. If it is asymmetric, the mechanism mustbe partly or wholly direct in that the products have retained a memory of where theycame from. This in turn means that the strong coupling assumption, which requiresthat decomposition of the intermediate complex be uncorrelated with its formation(except through the conservation laws) does not hold.Light acknowledges that statistical theory will hold best for reactions wherethere is a complex It is not clear to me that it can be useful at all for reactions,such as KfHBr, which have been shown to be direct by their angular distribution.It may be argued that even in these cases, fluctuation in product properties withsmall changes in reaction conditions (such as energy and impact parameter) willprovide some averaging over possible states in phase space.This may well be true.But it is also true that in almost any conceivable direct reaction certain large volumesof phase space are excluded or disproportionately populated.What these volumeswill be one cannot say a priori unless the mechanism of the reaction is alreadycompletely understood. Thus while it may be possible to fit the theory to a directreaction, it would seem that the parameters so fitted would have little physical signifi-cance. If the statistical theory is indeed applicable only to reactions in which along-lived complex decomposes independently of its mode of formation, it mightthen be appropriate to regard it as a theory of unimolecular decay.Prof. J . C. Light (University of Chicago) said: I agree with the basic premise ofWolfgang’s comment. If a particular reaction is known from experiment to yielda very asymmetric distribution of products in the centre of mass system, then nostrong coupling complex in the sense of the statistical theory is formed.In thiscase agreement between averaged quantities calculated from the statistical theoryand those found from experiment must be regarded as largely fortuitous. It caGENERAL DISCUSSION 81happen only if the numbers of states actually populated in each channel are pro-portional, on the average, to the total numbers of states (available) in each channelas calculated in the statistical theory. The reaction K+HBr may, however, beone in which a " direct " reaction yields a symmetric product distribution Inthis case the averaging over unobserved initial and final states may yield good agree-ment between quantities observed and calculated from the statistical theory (see alsocalculations for this reaction in ref.(25)).Finally there are two basic differences between the statistical theory and thenormal (RRKM) theory of unimolecular reactions. In the statistical theory aneffort is made to avoid parameterizing the theory in terms of unobservable molecularproperties of initial and final states. Secondly, the statistical theory includes theeffects of the total angular momentum of the complex. This often has a largeeffect on the mode of decomposition as well as the internal energy distributions ofthe products. Thus, although the statistical theory would be applicable to unimole-cular decomposition (particularly if alternative channels are available) it is differentfrom the RRKM theory.Prof. R. D. Levine (University of Wisconsin) said : The formalism of the statisticalapproximation is usually described in terms of a model where the colliding moleculesform a complex which then breaks down in a statistical fashion.It is important,however, to realize that this model is not necessary in order to derive the formalism,but is just a possible physical interpretation of the formal expressions obtained.that a possible parametrization of the average inelastic cross-section from channela to channel p , isUsing the concept of an average S matrix which we denote (S) one can showwherepa = 1- I ( S a a ) I 2*This proposed paramerization is independent of any specific model, and reducesto the Hauser-Feshbach formula for nuclear collisions and to the identical eqn.(1 1) of Light's paper when we adopt a collision complex model.In our derivationeqn. (1) does not hold for p = a, but<n-'k%a,a> = I 1 - (saor> I +PaPa/Cpp. (2)BTo evaluate Pa one needs to know (S). Bernstein, Dalgarno, Massey and Perciva14have shown that for strongly coupled channels it is reasonable to assume thatwhere R is an arbitrary orthogonal matrix. Taking R = -I it follows that(Saa> = 0 and Pa = 1. We shall define the statistical approximation by eqn. (3).With this definition eqn. (1) is equivalent to Light's theory. In this case = N,where N is the number of strongly coupled channels. (Only channels with thesame values of all good quantum numbers can be strongly coupled.)(S) = <RS) (3)Bsee C. Riley, K. T. Gillen and R.B. Bernstein, WIS-TCZ-256X, (24 July 1967).R. D. Levine, Quantum Mechanics of Molecular Rate Processes (Oxford University Press,Oxford, 1968), 9 3.51.W. Hauser and H. Feshbach, Physic. Rev., 1952, 87, 366.R. B. Bernstein, A. Dalgarno, H. S. W. Massey and I. C. Percival, Proc. Roy. Sue. A , 1963,274, 42782 GENERAL DISCUSSIONThere are two immediate experimental consequences of this statistical approxi-random-phase mation. (i) For N = 1, eqn. (2) reduces to the Massey-Mohrapproximationwhich predicts correctly the average elastic cross-section. (ii) Irrespective of thevalue of N, the total average cross-section obtained by summing eqn. (1) over p $: aand adding eqn. (2) satisfiesThus the total cross-section (for a given initial channel) has the random phase value,irrespective of the occurrence of inelastic collisions.2 The collision complex theoriesseem to omit the first contribution to eqn.(2) and thus obtain the erroneous valueof unity for eqn. (4) and (5).Eqn. (3) also implies( ~ i - kzc,,,) = 2, (4)(rc-'k:o,> = 2. ( 5 )< I sba i 2> = 1 / ~ * (6)Recent exact (close coupling) computations of I Spa J2 for strong coupling seemto confirm eqn. (6), in that the diagonal element can not be distinguished from otherelements of the same column.The point is that neither eqn. (1) nor eqn. (3) depend on the physical conceptof a collision complex, although they can be interpreted in terms of such a physicalmodel. Moreover, eqn. (I) is independent of eqn. (3). Eqn.(3) simply providesa computational prescription for the Pa. Further discussion will be publishedelse~here.~ It is a pleasure to acknowledge the benefit of many discussions withProf. R. B. Bernsteirz and Dr. W. A. Lester, Jr.Prof. J. C. Light (University of Chicago) said: Although the average S matrixapproach is pleasing because of its brevity and generality, one must still have aprescription for choosing the sub-set of strongly coupled channels from the entireset of channels allowed by the conservation laws. The value, if any, of a physicalprescription of a strong coupling complex lies in the ease with which physical intuitioncan be used to define the sub-set of strongly coupled channels. The alternativeof choosing the set of strongly coupled channels a priori and letting this choicedefine the " complex " seenis more difficult physically.Dr.J. Troe and Prof. H. Gg. Wagner (University of Giittingen) said: The statementof Hoare and Thiele, that the detailed balance condition prevents any simpleMarkovian theory from becoming very different from a simple Kassel theory forthe specific rate constants k(E), seems to be very important, because it is not easy tocheck this assumption experimentally. This is due to at least two reasons. (a) Theaccuracy of measured rate constants in the '' fall-off " region of unimolecular reactionsseems to be not yet sufficient to check deviations of the specific rate constants k(E)from those given by Kassel's expression, because different models of k(E) can resultin similar shapes of " fall-off " curves.However, from independent measurementsof k(E) it could be shown that Kassels description is essentially correct. (6) TheH. S . W. Massey and C . B. 0. Mohr, Proc. Roy. SOC. A , 1934,144, 183.R. B. Bernstein, A. Dalgarno, H. S . W. Massey and I. C. Percival, Proc. Roy. SOC. A , 1963,274, 427.W. A. Lester, Jr., and R. B. Bernstein, private communication, July 1967; see also Ckrern.Physics Letters, 1967, 1, 207.R. D. Levine, Quantum Mechanics of Molecular Rate Processes (Oxford University Press,Oxford, 1968), 3 3.51GENERAL DISCUSSION 83“ fall-off ” curves are effected by lifetime distributions (e.g., assumption (ii)).However, they are also influenced by intermolecular energy exchange processes(e.g., assumption (iii)).When interpreting experimental results on low pressurelimiting rates particularly of small molecules, it became evident,l that a much betterfitting of the data could be obtained, if assumption (iii) was replaced by a “ ladder-climbing ” model. Because of this additional complication the separation of theinfluences of lifetime distributions and intermolecular energy exchange processesin the “ fall-off ” region becomes very complicated. This would still hold, if onecould obtain very accurate experimental data in the transition region between low-and high-pressure limiting rates.Dr. E . R. Buckle (Imperial College, Londorz) said: I have been trying to devise akinetic theory of clustering in monatomic gases which is based on Kassel’s treatmentof unimolecular reactions.The ultimate aim is to provide an alternative to Volmer’squasi-thermodynamic treatment of condensation. The models I use for unimoleculardecay and bimolecular growth of clusters are necessarily crude, as they are intendedto apply at various stages of growth from the monomeric state. For example,cohesion is attributed to pairwise additive dispersion forces. The equilibrium distri-butions in argon are in rough qualitative agreement with the shape of Bentley’smass spectrometric curve for clusters up to the 22-mer in C02.2 The dimer molefraction at p = 0.44 atm and T = u/k deg is about 1 x in fair agreement withthe more exact calculations of Stogryn and Hir~chfelder,~ but while in both casesthe same value for u, the pair energy, is used their concentration falls less rapidlyat higher T.My model of the dimerization reaction closely resembles the three-atom radical-molecule mechanism described by P ~ r t e r . ~One of the simplifications of the cluster theory is to ignore, at all sizes, possibleeffects due to the distribution of angular momentum. In view of assumption (b)in Light’s paper it would be interesting to know in what way the results of ratecalculations using a theory which omits this assumption, but which otherwise agreeswith his definition (a) of a strong coupling theory, would be expected to differfrom those using a theory which includes assumption (b).With regard to the uniform distribution Q(EJ) found by Hoare and Thiele forthe energy of the critical oscillator after collision with another oscillator, are theyconfidient that detailed balancing can be applied without reservation to their hypo-thetical mechanism? They offer no physical description of a “collision ”, and Ifind it hard to think of reversibility in terms of normal vibrations.Dr.M . R. Hoare (Bedford College, London) said: In reply to Buckle, even if onehas doubts about the ultimate validity of the detailed balance condition, one cannotlegitimately raise them in the context of our theory. We are constructing a stochasticmodel which is supposed to mirror as closely as possible more detailed mechanicalprocesses for which microscopic reversibility is reasonably well established ; its useor not is hardly at our disposal.And in constructing valid transition kernels thecondition is virtually unavoidable. Although detailed balance (eqn. (6)) andnormalization (eqn. (10)) are sufficient rather than necessary conditions for passageto equilibrium (eqn. (11)) it is difficult to find kernels which lead to equilibrium,J. Troe and H. Gg. Wagner in Receirt Advances in Aerotherinochernistry (AGARD, Oslo1966) and Bc7r. Bunsenges. Physik. Chenz., 1967, in press.P. G. Bentley, Nature, 1961, 190, 432.D. E. Stogryn and J. 0. Hirschfklder, J. CJzeni. Physics, 1959, 31, 1531. ‘ G. Porter, Disc. Faraduy SOC., 1962, 33, 19884 GENERAL DISCUSSIONi.e., have an eigenvalue unity and the appropriate distribution as its eigenfunctionwithout this being due to the “ built-in ” property, (6).Dr.M. R. Hoare (Bedford College, London) said: The prospect of applying noisetheory to unimolecular reactions must have tempted many people and it is a pitythat the original work of Kramers has never been followed up, I admit to havingspent time searching for buried treasure in the electrical and mechanical engineeringliterature, without much success. Unfortunately mechanical engineers are interestedmainly in dissipative systems, usually with distributed mass, while electrical engineerslive in a linear universe admitting only the whitest of white noise.1 agree that the study of the driven harmonic oscillator will surely throw lighton the unirnolecular problem, but Kac would admit that the situation here is differentfrom the idealized one that he is treating.For example, in the molecule the drivingspectrum would not be white-there would be a high-frequency cut-off, peaks aroundthe normal-mode frequencies, plus side-bands, sub-harmonics etc. Even a Debyespectrum would be better than white noise. There is little evidence from computerexperiments that energy transfer occurs by a Brownian-motion-type mechanism.Thus, one loses the simplicity of the Kramers equation. Furthermore, one shouldreally be driving an anharmonic oscillator. Finally, there is the question of energyconservation, which would normally be violated in a driven system? To extractan energy-dependent rate constant the molecule would have to be large enoughfor heat-bath-like behaviour to occur. In this respect our model seems to be astep in the right direction. Because of the rather artificial energy-description weare able to “ drive ” a single oscillator with s- 1 other oscillators at constant totalenergy E.Actually, for s> -10, we can let the heat-bath become infinite, use acanonical ensemble formulation with D = s/E and obtain virtually indentical results.Altogether, in view of the heat-bath idea it is surprising how little attention is paid inconventional unimolecular theories to special results valid for large s.As to doubts about treating lifetimes in terms of a well-defined g(E,E, . . . etc.)we tend to agree. This is roughly what we had in mind in our concluding remarksabout the importance of “ hidden variables ” behind the energy-description.Prof.W. C. Gardiner (University of Texas) said : Kupperinan, Stevenson andO’Keefe find that their relative energy distribution function is determined by theelastic scattering cross-section and is essentially independent of the assumed reactivecross-section. Since the inelastic cross-sections corresponding to rotational andvibrational excitation of the H2 heat-bath molecules by the hot D atoms shouldbe comparable to the reactive cross-sections, it would seem that the lack of dependenceupon reactive cross-section implies a lack of dependence upon the inelastic non-reactive cross-section. Is this correct ?Prof. R. N. Porter (University of Arkansas) said: We have treated the generalproblem of reactions of hot atoms in thermal media by a method which makes useof the concept of integral reaction probabi1ity.l This is defined as the probabilityA,@) that an atom whose initial laboratory energy is E will be removed from thesystem by reaction i upon some future collision with component c.The integralequation satisfied by A&) has the simple forGENERAL DISCUSSION 85where Fc(E) is the collision fraction for the cth component, pi,(E) is the probabilityof reaction i per collision with component c, and the kernel K(E,E') is determinedby the total and differential cross-sections for the various possible collision processes.For a single reaction with a single reactive component, eqn. (1) takes the formA(E) = p(E) + [ 1 - p(E)]/ co P(E,E')A(E')dE', (2)0where P(E,E') is the normalized scattering kernel for a single non-reactive collision.It has been shown that eqn.(2) is mathematically equivalent to the integral equationfor the collision density n(E,E') viz.,n(E,E') = 6(E - E') + [l - p(E")~n(E,E'f)P(E'',Ef)dE", (3) sa combined with the Miller-Dodson equation,A(E) = /En(E,Ef)p(E')dE'. 0 (4)Recently, we have explored the compatibility of eqn. (3) and (4) with the steady-stateformalism which Kuppermann has employed and a similar formalism used by Kostinet al.3 Our formalism in terms of probabilities is equivalent to the steady-stateBoltzmann formalism, provided the probability for a given scattering process isdefined as the steady-state rate of the process divided by the total steady-state collisionrate.TABLE 1.A(E)eqn. (2) eqn. (3) and (4) E(eW0.6 0*01131 0.01 1290.9 0.05600 0.055961.0 0 07481 0.07477By way of numerical illustration, we compare in table 1 the results for A(E)calculated from eqn.(2) with those calculated from the steady-state Boltzmann for-malism [i.e., eqn. (3) and (4)]. The reactive and non-reactive cross-sections whichwere assumed for the calculations both have realistic energy dependence for thesystem D+H2. The small disagreement in the results is within the round-off andtruncation error for the trapezoidal integration routine. The computation timerequired for solving eqn. (2) to give A(E) at 0.001 eV energy intervals over the range0-6<E< 1.0 eV was slightly less than that required to obtain A(E) at one energyvalue by eqn. (3) and (4).Besides the ease of computation of A(E), an advantage offered by the formalismof eqn.(2) is that the equation is readily written in the formA(@-/ P(E,E')A(E')dE'1 - 1 E , P(E,E')A(E')dE'E' m =R. N. Porter, J. Chem. Physics, 1966, 45, 2284.J. M. Miller and R. W. Dodson, J . Chem. Physics, 1950, 18, 865.M. D. Kostin, J . Appl. Physics, 1965,36,850 ; R. M. Felder and M. D. Kostin, J. Chem. Physics,1965, 43, 3082 ; D. M. Chapin and M. D. Kostin, J . Chem. Physics, 1967, 46,250686 GENERAL DISCUSSIONThus, when experimental energy-dependent yields and differential non-react ivecross-sections are known, a simple integration gives the reaction probability percollision, a quantity closely related to the reaction cross-section.l A more completeand rigorous discussion will be published elsewhere.The stimulation of several discussions with Professor A.Kuppermann and thefinancial assistance of Public Health Service Research Grant no. GM 13253 aregratefully acknowledged.Prof. R. Wolfgang (University of Colorado) said : Theories of hot atom thermaliza-tion, such as that given by Kuppermann and that of Porter,2 and Kostin and Felder,3can potentially be useful in extracting information on the energy dependence ofreaction. However, although their formalism is complete, their present applicabilityseems limited by our ignorance of energy loss processes in non-reactive collisions.Such data are critical for any theory of hot atoms moderation. At low velocitiesand for simple systems (such as T+H,), energy loss appears to occur largely inelastic collision and can therefore be reasonably well estimated.But at higherenergies ( - 5 eV) hot-atom studies have shown that, with methane and more complexmolecules, energy degradation occurs largely in highly inelastic collision^.^ Suchcollisions may be expected to be velocity dependent, and are difficult to evaluatetheoretically. The consequent limitation on the usefulness of theories of hot-atommoderation emphasizes the need for detailed experimental information on inelasticinteractions in the range of a few electron volts and higher.Prof. I. Amdur (Massachusetts Iiwtitute of Technology) (communicated) : Thequestion has been raised as to the magnitude of non-reactive, inelastic scatteringin comparison with elastic scattering for systems such as those treated by Kuppermann.A.L. Smith and M. C. Fowler 5 * have recently completed experiments in ourlaboratory which bear directly on this question. They have measured total cross-sections for scattering of fast He atoms through laboratory angles greater than 0.1deg. by H2, D2, and HD at room temperature. Although the fast atoms had kineticenergies in the range 300-1900eV, it is the maximum energy lost by them inexperiencing their small-angle deflections, not their initial relative kinetic energy,which is available for possible transfer to internal energy of the H2 isotopes. Thisenergy loss which, for elastic scattering, is the potential energy of the system at thedistance of closest approach, is in the range 0.25-1-35 eV. This range is almost thesame as the range of initial laboratory energies of D atoms considered by Kuppermannfor the reaction D+H2 = DH+H.The results of our scattering experiments may be summarized as follows.(i) Theexperimental cross-sections for He + H2 and He + D, are identical over the entireenergy range, as are the average potential energies of interaction calculated on theassumption that the scattering is completely elastic. (ii). The measured He + H2(and He + D2) cross-sections are in excellent agreement with orientation-averagedcross-sections calculated from a theoretical angular dependent He + H2 potentialR. N. Porter, J. Chem. Physics,'1966, 45, 2284.R. N. Porter, J. Chem. Physics, 1966,45, 2284.M.D. Kostin, J . Appl. Physics, 1965,36,850 ; R. M. Felder and M. D. Kostin, J . Chem. Physics,1965,43, 3082 ; D. M. Chapin and M. D. Kostin, J. Chem. Physics, 1967,46,2506.J. W. Root and F. S. Rowland, J. Chem. Physics, 1963,38,2030; 1967,46,4299 ; R. Wolfgang,J. Chem. Physics, 1963, 39, 2983 ; A. H. Rosenberg and R. Wolfgang, J. Chem. Physics, 1964,41, 2166 ; D. Seewald and R. Wolfgang, J. Chem. Physics, 1967, 47, 151.A. L. Smith, Ph.D. Thesis, (M.I.T., 1965).M. C. Fqwler, Ph.D. Thesis, (M.I.T., 1967)GENERAL DISCUSSION 87obtained by Mies and Krauss. In calculating the cross-sections from the HefH,potential it was assumed that no inelastic scattering occurred. (iii) Several differentmethods for calculating cross-sections for rotational excitation for the He + M22nd He+D2 systems shown that these inelastic cross-sections are less than 0.5 %of the corresponding elastic cross-sections.(iv) The measured cross-sections forHe + HD are consistently higher than the measured cross-sections for He + H,or HefD,. The difference ranges from about 1 % at the lowest beam energy toabout 3 % at the highest. Because of the displacement of the centre of mass in HD,the He+HD has a larger anisotropic component than the He+H, or He+D2potentials. However, orientation-averaged cross-sections for He + HD calculatedon the assumption that all scattering is elastic are indistinguishable from the corres-ponding He+H2 or He+D2 cross-sections. On the other hand, inclusion in thecalculation of the effects of rotational excitation and de-excitation of HD by Heaccounts for most of the observed enhancement in cross-sections.Estimates of theeffects of vibrational excitation suggest that these are much smaller than the rotationaleffects.In summary, scattering of He by hydrogen isotopes indicates that for H2 or D2inelastic effects are negligible, and for HD, the detectable scattering resulting fromrotational excitation is of small magnitude, about 2 %, on the average. Kuppermann’sneglect of non-reactive inelastic scattering appears to be well justified.Prof. R. A. Marcus (University of Illinois) said : The vibrational energy resultsof Karplus (p. 78) are extremely interesting and can be used to test the near-adiabaticexpressions for the instantaneous vibrational energy of a reacting system, eqn.(33and 35, in ref. (1). The latter contain adiabatic, statistical adiabatic and non-adiabaticterms and, under certain conditions, display just the behaviour observed in fig. laas well as that in the “ annealing process” (an approximate return of the finalvibrational energy to the adiabatic value after undergoing statistical adiabaticand non-adiabatic effects).For example, at low initial vibrational energy E: the envelope of points in fig.la is seen to be a parabola about an axis parallel to but displaced upwards from adiagonally-drawn adiabatic line. This vertical shift, according to (35), is the non-adiabatic term and the scatter about the shifted-adiabatic line reflects the statistical-adiabatic term.When only the sine Fourier transform in (35) and (33) need beconsidered one finds that (i) the vibrational energy at the end of reaction shouldequal the adiabatic value ; (ii) the vibrational energy at s+ should equal (El+I)-2(E3* sin 6, where E% is the adiabatic value at s#, I is the square of the sine transform,and 6 is the initial vibrational phase angle. Both conclusions are modified somewhatwhen the cosine transform cannot be neglected.Item (i) agrees with the findings of Karplus. From item (ii) the shifted adiabaticline is seen to be E, = E,”+I and the envelope to be displaced 2(E;1)4 from it, inreasonable agreement with fig. la.3 A more detailed analysis of the computerR. A . Marcus, J. Clzem. Physics, 1966, 45, 4500. Altnough (35) was derived for the case ofi = O at s = s#, one can show that it remains applicable for # 0, provided w is replaced by w2.Also, as a first approximation Ireplaced x in the last term of (32) by xo.Note that the function in the transform is, like a sine term, small atE.g., the value of I at EI: = 0 is the intercept of the shifted-adiabatic line on the &-axis, andis seen from fig.l a to be about 0.04 eV. The value of 2(E3)* calculated from it at Et = 0,0.05, and 0-10 eV, is 0, 0.06 and 0.09 eV, respectively, while the observed vertical distancefrom the envelope to shifted adiabatic line in fig. la is 0. 0.055, and 0.085 eV, respectively.(Strictly speaking, the I varies with Ei and one should use the formula in item (ii), computingeach I.)= s#88 GENERAL DISCUSSIONresults, together with an Q priori calculation of the cosine and sine transforms usingthe method in ref.(l), will permit a more detailed testing of (33) and (35).At very low initial vibrational energies (<O.l25 eV) the data in fig. lb are belowthe adiabatic threshold. At these and at the higher Ei as well, the upper envelopeof points in fig. 16 corresponds to the maximum allowable by energy conservationunder these conditions.' Below this maximum, but at the higher Ei one couldagain test the equation in item (ii) above.Dr. D. A. Micha (University of California, San Diego) said: I would like tocomment briefly on the range of energies investigated in the contribution by Karplusand Tang. The distorted wave approximation for rearrangement collisions doesnot account for the coupling between inelastic and reactive scattering channels.For H and H2 colliding with relative kinetic energies above about 0.5 eV, vibrationalexcitations from ZI = 0 to ZI = 1, 2, .. . become energetically possible and bothrotational excitation and vibrational excitation cross-sections are comparablein magnitude to the reaction cross-section. This indicates that it would be important,to improve the quantum mechanical results at energies above 0.5 eV and to comparethem with classical calculations, to go beyond the distorted wave approximation byincluding the effects of vibrational and rotational excitations at those high energies.Prof. R. A. Marcus (University of ZZZinois) said: It is of interest to compare theresults of Karplus and Tang with those of a statistical-dynamical theory of reactioncross- section^.^" (The classical version of the latter and the classical mechanicalcomputer results agreed well in the energy region of interest, without introductionof adjustable parameter^.^') When the reaction co-ordinate is treated classically,but the vibrations of the activated complex are treated in a quantum manner, oneobtains a step-like cross-section against energy curve, instead of the smooth classicalone.The classical curve passes about mid-way through the ~tep-risers.~" Thestep-like nature of the new curve arises from the increasing accessibility of quantizedbending states of the activated complex with increasing energy and is closely relatedto the cusps mentioned by Child in his paper.When the reaction co-ordinateis treated quantum mechanically, there is a rounding-off at the foot of each stepbecause of tunnelling and a rounding-off at the top of each step because of theusual quantum mechanical reflection for motion just above the top of a barrier.4cAny non-adiabaticity of the bending modes also has a rounding-off effect.According to these arguments the quantum curve should be fairly close to theclassical one, except at threshold. Comparison with fig. 7 of Karplus and Tangshows that in both cases the threshold energy of the quantum case exceeds that ofthe classical but that their curve is appreciably below ours at the energies of interestin thermal reaction, a factor of 5 below, perhaps.At high energies, fig. 7 displaysthe usual " blow-up " of the distorted wave approximation.With the usual co-ordinates, the distorted wave approach at low energies givesa chemical reaction cross-section which is too low perhaps because of the difficultyE.g., the barrier in Karplus and Tang's paper is 0.396 eV, and so the initial energy defect is0.396-0-333 or 0.063 eV. The upper envelope in fig. lb is seen, in fact, to correspond ratherclosely to (E:-0*063) eV. The corresponding limit in fig. l a is (Ei+048-0*396) eV, whichis above the observed E,f at low Ei, but which may be a limiting factor at higher Et.A. C . Allison and A. Dalgarno, Proc. Physic. SOC., 1967, 90, 609.D. Rapp and T. E. Sharp, J . Chem. Physics, 1963, 38,2641(a) R.A. Marcus, J. Chem. Physics, 1966, 43, 2630 ; (b) J. Chem. Physics, 1967, 46, 959 ;(c) eqn. (19), (26), (29) and (31) in ref. (la) and eqn. ( 5 ) of ref. (lb) are used.M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Physics, 1965,43, 3259GENERAL DISCUSSION 89of choosing a wave function giving good overlap of the reactants and the productscontributions. Perhaps a calculation based on the natural collision co-ordinateswill offer an easier opportunity far getting the correct wavefunction.Dr. J. L. J. Rosenfeld (“ Shell ” Research Ltd. Chester) said: Hurle’s data onhydrogen atom recombination at temperatures between 2800 and 7000°K havebeen re-analyzed in detail, Fig. 1 shows the new results for the rate constant ofthe three-atom recombination reactionH + H + H -+HZ + H.The mean line through Hurle’s data and the extreme upper and lower envelopes(indicating approximately the scatter in the experimental results) are shown, togetherwith the results of Sutton,2 Patch,3 Rink,4 and Jacobs et aZ.j Patch and Rinkassumed a temperature-dependence of T-l in their analysis, and their results cantherefore only be used as confirmation of the magnitude of the rate constant atthese temperatures..\BENNETT and’-. -~B>ACKMORE15.0 .\. -. ‘.\ -0 1000 2000 3OOO 4000 5000 6000 7000temperature T, O KFIG. 1 .-Temperature-dependence of the rate constant of the three-atom recombination reaction,Both Sutton’s and Hurle’s results indicate a steep (P4 to P6) temperature-dependence at high temperatures, with a gradual reduction in the slope between4000 and 3000°K.There are indications of a maximum at about 3000°K. Thesefeatures gain significance in the light of the estimate at 300°K (also shown in fig. 1)by Bennett and Blackmore of an upper limit for the constant much lower thanthe shock-tube values at 3000°K. Other things being equal, one might expecthydrogen atoms to be about as efficient as Ar or H2 molecules in acting as energyH+ H+ H +H2 + H.’ I. R. Hurle, 11 th Int. Symp. Combustion (The Combustion Institute, Pittsburgh, 1964), p. 827.E. A. Sutton, J. Chern. Physics, 1962, 36, 2923.R. W. Patch, J. Chern. Physics., 1962, 36, 1919.J. P. Rink, J. Chern. Physics, 1962, 36, 262.T. A. Jacobs, R. R. Giedt and N. Cohen, 1.Chern. Physics, 1967,47, 54.J. E. Bennett and D. R. Blackmore, private communication (to be published)90 GENERAL DISCUSSIONsinks in the recombination. The rate constant kHZ for recombination with H2as third body is also shown in fig. 1 for comparison. These results taken togethersuggest the existence of a " resonance " in the three-atom rate constant due to somemechanism other than straightforward collisional stabilization. Stabilization byexchange is a likely possibility. If the vibrational mode is assumed to be adiabaticduring exchange then one may expect the activation energy E, for the exchangereaction to be independent of the degree of vibrational excitation. The detailedmechanism is then as follows.We imagine two hydrogen atoms, H(l) and H(2), to collide to form an orbitingpair.Nearly all the bond energy goes into the vibrational mode, the remainder,together with the relative kinetic energy Ety goes into the rotational mode. Whena third hydrogen atom, H(3), approaches, H(2) leaves with an excess kinetic energyequal to the activation energy, leaving the H2(l, 3) molecule with that much lessrotational energy. Thus,H(1) + H(2)+H;JH";(l,2)+H(3)-+H';J;(1,3)+H(2).The rotational energies areEj = E,,+D,; EjP = Ej-E,,where D, is the dissociation energy from the vth vibrational level.The products are stabilized only if EJ.cD,, and hence only collisions betweenH(l) and H(2) with E,,<E, must be counted. This leads to an effective three-bodyrecombination rate constant k,, wherecm6 sec-'.(I)Here [Hg*] and [HI are the concentrations of the orbiting pair and free atomsrespectively, and A exp (- E,/RT) equals the rate constant for the exchange reaction.When E, = 7-5 kcal mole-' (the experimental value when v = 0), the temperature-dependent part of k, goes through a maximum at 3000"K, in agreement with theobservations. Although the decrease of this function alone at higher temperaturesis much too slow, one might expect the ratio [H;*]/[Hl2 to diminish rapidly withincreasing temperature, as the effective potential gets shallower at higher energies.Thus it might still be possible to reconcile eqn. (1) with the experiments.To reproduce the observed magnitude of the rate constant at 3000"K, a valueof about lo3 c1n3 mole-' must be assumed for [H;*]/[Hl2.Under the experimentalconditions (typically 50 % Ar, 50 % H2, 10 dissociation at 3000°K and 1 atm)CH1-3 x mole ~ m - ~ , so that [H;*] N lo-* mole ~ m - ~ or one three-hundredthof the free atom concentration. This is physically acceptable in view of the approxi-mate nature of the calculations. It would seem therefore that exchange stabilizationwith vibrational adiabaticity may explain the observed " resonance " in three-atomrecombination rate constant.Prof. R. A. Marcus (University of Illinois) said : The results of Child are interesting,and some comparison with our related study is appropriate. The orbital-rotationalpart of the kinetic energy based on the natural collision co-ordinate system describedJ.C. Polanyi, Atomic and Molecular Processes, ed. D. R. Bates, (Academic Press Inc., NewYork, 1692), p. 807GENERAL DISCUSSION 91in my paper is similar to Child’s outside of the saddle-point region. Near thatregion, however, I was able to choose the co-ordinates so that the orbital-bendingkinetic energy cross-term became small, as it is for a typical molecule. (My co-ordinates pass snioothly from those in fig. l a of Child’s paper to those in hisfig. l b as s varies from - 03 to + m.) On this basis the details of Child’s calculationat the saddle-point are open to question, but this question does not effect theseat fairly negative or fairly positive s. Publication of a more detailed comparisonis planned.Child’s calculation of a rate constant which agrees with activated complex theoryis an interesting concrete example of a more general, formal result in the literature,viz., a vibrationally-adiabatic derivation of activated complex theory.Prof. M . Karplus (Harvard University) said : Child has presented an interestingtwo-dimensional calculation of the H + H2 exchange reaction which predicts thatthe product H2 molecule is formed in the same rotational state as the reactant molecule.This result is surprising, particularly in view of the rate of ortho-H, to para-H,conversion, which must involve a rotational state change if it proceeds via the exchangereaction. Moreover, molecular rotation, in contrast to vibration, is predicted tobe significantly non-adiabatic on the basis of an approximate quantum mechanicaltreatment and exact quasi-classical calculations on the potential surface used byChild. Would Child comment on this point, discuss the nature of the approximationswhich lead to rotational adiabaticity in his model, and indicate possible refinementswhich might alter the conclusions ?Dr. M. S . Child (Oxford University) said: Karplus has raised an importantquestion. The prediction he has questioned is based on the extreme adiabaticapproximation which rests on the neglect of certain coupling terms in deriving eqn.(1). The most important of these are determined by the rate of change of the internalrotational part of wavefunction and the velocity along the reaction co-ordinate asthe system enters the bending vibrational region; their effect also depends on theinternal rotational energy spacing in this region. The system H -t- H,, therefore,appears most favourable for the approximation because rotational energy spacingsare large, and the velocity along the reaction co-ordinate is reduced by the presenceof the activation barrier. The point can be settled however, only by further calcula-tion. Comparison with the exact quasi-classical calculations of Wyatt and Karpluswould then be of the greatest interest.In various degrees of generality (curvilinearity of reaction co-ordinate, etc.) this derivationmay be found in J. 0. Hirschfelder and E. Wigner, J. Chem. Physics, 1939, 7, 616; M. A.Eliason and J. 0. Hirschfelder, J. Chenz. Physics, 1959,30,1426 ; L. Hofacker, 2. Naturforsch. A,1963, 18, 607 acd R. A. Marcus, J. Chem. Physics, 1965,43, 1598 ; 1967, 46,959.M. Karplus and K. T. Tang, this Discussion ; see particularly fig. 6 and the related text.R. Wyatt and hf. Karplus, to be published

ISSN:0366-9033    

DOI:10.1039/DF9674400076

出版商:RSC    

年代:1967

数据来源: RSC

摘要:

Kinetics of the Four Basic Thermal Three-Centre ExchangeReactions of the Hydrogen IsotopesBY D. J. LE ROY, B. A. RIDLEY AND K. A. QUICKERTLash Miller Chemical Laboratories, University of Toronto,Toronto, CanadaReceived 5th June, 1967Rate constants have been determined for the reaction D+D2 = D2+D. These, together withour earlier results for the reactions H+Hz = H2+H, H+D2 = HD+D and D+H2 = DH+H,are considered in comparison with ab initio and semiempirical calculations. The experimentalresults for all four reactions can be treated in a satisfactory manner by a modified form of transitionstate theory, and suggest a barrier height of approximately 9.2 kcal mole-l and a reaction pathwhich, even for the heteronuclear complexes, is essentially coincident with the normal mode co-ordinate for the asymmetric vibration of the complex. On the basis of this treatment the forceconstants for the symmetric and asymmetric vibrations of the complex are approximately 2.6 x lo5and -0.27 x lo5 dyne cm-I, respectively.Basic to any theory of the chemical kinetics of elementary reactions is a completeunderstanding of the simplest atom-diatomic molecule exchange reactions,H+H2 = H2+HD+D2 = D2+DD+H2 = DH+HH+D2 = HD+DIn part A of this paper the results of an investigation of reaction (2) are reported ;those for reactions (l), (3) and (4) have already been r e ~ 0 r t e d .l ~ ~ In the previouspapers, Arrhenius plots * for reactions (1) and (3) showed definite non-linear inversetemperature dependence, and this curvature was interpreted in terms of a tunnellingcorrection to the semi-classical transition state theory.Although non-linear be-haviour was not observed for reactions (2) or (4), the temperature range covered ineach case was relatively narrow. Because of this, and the greater masses of thecomplexes involved, the absence of non-exponential " tunnelling " was not surprising.In part B an attempt is made to interpret the experimental results for the fourreactions. Although this is done within the framework of transition state theory,it is so done only for the lack of another model readily accessible to the experimentalist.Aside from the basic assumption of the theory, that there is thermal equilibriumbetween the reagents and the transition state complex, we attempt to clarify some ofthe misconceptions involved in its use.Specifically, we maintain, in accordance withBell,5 that if one uses transition state theory to the detail of quantum mechanicalzero-point energy, then one ought to allow for the quantum-mechanical effect resultingfrom curvature of the potential barrier. This is accomplished within the " imaginary "harmonic oscillator approximation by a method analogous to the usual use of real* Reactions (1) and (2) were observed through the p + o and o + p conversions, respectively.However, the rate constants kl and k2 reported in this paper are thc observed values based on theempirical relations k,[H] = -d In (p--p,)/df and k 2 w ] = -d In (o-o,)/dt. These are morereadily compared with most theoretical calculations (cf.ref. (2), (4)).9D. J. LE ROY, B. A. RIDLEY AND K. A. QUICKERT 93harmonic oscillator partition functions, without solving the Schroedinger equationfor barrier penetration in the usual manner.PART A. THE EXCHANGE REACTION D+D2 = DP+DEXPERIMENTALWith the exceptions of the reagent ortho-enriched deuterium supply and the use of acirculating system to conserve deuterium, the experimental arrangement was similar to thatused previously.2 The thermal conductivity analysis technique was ideally suited to thisstudy, in which only 0-D2 and p-D2 were present.The deuterium gas (Matheson, 99.5 %) was re-purified continuously on each cyclcby passage through two purifiers, each consisting of a white-hot tungsten filamentfollowed by a liquid-nitrogen trap.Between each such pair was placed a charcoal trap at% reagentFIG. 1 .-Analysis of o-D2 +n-Dz mixtures, showing proportionality between detector response andexcess o-Dz.- 196°C. The ortho-enriched deuterium was prepared using Burrell High Activity charcoalcooled with a pumped-down liquid-nitrogen bath. Deuterium gas from the high pressure(1 atm) side of the circulating system was passed through 2 mm int. diam. capillary tubinginto a series of three Pyrex traps 10 cm long and 1.8 cm int. diam. filled with the charcoaland then through another capillary tube to a Stokes High Vacuum needle valve and into thereaction vessel. Complete equilibration of the deuterium at the temperature of the pumped-down traps (approximately -218°C) corresponded to about 10 % o-D2, 90 % n-Dz.Separate experiments showed that the composition of the ortho-enriched deuteriumremained constant, without replenishing the refrigerant, for at least three times as long aswas required to perform the analyses.As shown in fig. 1, the detector response was linearin the concentration of ortho-enriched reagent, i.e., in the excess 0-D294 THREE-CENTRE EXCHANGE REACTIONSSince the reagent flow required was from 20 to 25 % of the total flow through the reactor,some back-diffusion of reagent into the dissociator chamber occurred. This had the effectof indicating too much reaction in the reaction vessel proper. The effect was eliminated asfollows. Instead of turning the dissociator filament completely off when measuring theresistance of the detector with no atoms present, it was left at a dull red heat. While thistemperature was nearly 100 % efficient in equilibrating 0- and p-D2, no atoms could bedetected on placing the detector as close as possible to the dissociator chamber.By followingthis procedure and using flow rates in the reactor in excess of 100 cm sec-', the back diffusioneffect was eliminated.RESULTS AND DISCUSSIONThe experimental results for k2 are presented in table 1 ; they are shown inArrhenius form in fig. 2, together with the results previously obtained for reactions (l),(3) and (4).1-3 Over the temperature range studied, 358-468"K, the plot for k2 islinear and is represented by the least-squares expressionThe standard deviation in log,, A is 0.06 that of Eact is 0.1 kcal mole-l.k2 = 1.22 x 1013 exp (-7.63 x 103/RT) cm3 mole-' sec-l.( 5 )TABLE 3.-RATE CONSTANTS FOR THE REACTION D+D2=D2+Drun37384041424344454647495052535455707172k2temp. total flow rate xo (o-o& B Z 9 WD cm3 mole-1pressure cmsec-1 cm (O-oe)t cm-1 cm-1 cm2sec-1 watt sec-1mm x 10-8359.0 2-74 1124 31.8 1.294 0.0114 0.251 469.2 0.248 2-65358.1 2.72 113.6 31-8 1.318 0.0140 0.255 471.6 0.250 279384-6 2.73 117.6 31.8 1.561 0*0085 0.232 525.5 0.251 5-194049 2.73 1247 31.8 1.949 0.0093 0.228 570.4 0241 9.08405.0 2-74 122-9 31.8 1.829 0.0095 0.226 568.1 0.219 8-78425.3 2-80 186.6 31-8 1.973 0.0359 0.317 600.1 0.318 15-39425.6 2.82 183.6 31.8 1.894 0.0059 0.313 596.9 0.318 1402425.3 2-81 185.1 27.8 1.759 0.0059 0.315 598.2 0.324 14.51425.3 2.81 185.1 23.8 1.677 0.0059 0.315 598.2 0.332 15.69425.3 3-81 185.1 19.8 1.524 0.0059 0.315 598.2 0-335 15.71425.0 3.69 192-6 28.8 1.643 0.0083 0.317 624-6 0-293 14.31425.6 2.76 183-0 27.8 1.887 0-0074 0.308 609.5 0.354 14.48447.4 2.80 200.5 27.8 1-814 0.0063 0.315 650.5 0.236 24.63467-4 2.80 210.0 27.8 2.257 0.0055 0.307 696.5 0.215 32.06467.5 2.81 210.3 27.8 2234 0.0051 0.308 693-7 0.275 31.87383.9 2.81 172.6 27.S 1.327 0.0064 0.345 509.4 0.339 5.94371.4 2.89 166.7 29.1 1.233 0.0102 0.364 470.7 0.303 4.13414.5 2.81 184.9 29.1 1.633 0.0103 0.336 574.6 0.322 11.58450.9 2-80 205.4 29.1 2.412 0.0078 0.320 657.8 0.338 24-80The quantities PI, and xo have been described previously 1, 3 ; 9 is the difl'usion coefficient for D atoms.To the authors' knowledge, the only other experimental determination of k2 wasaccomplished by Farkas and Farkas in 1935.6 They used a temperature range of903-98 1"K, but a considerable amount of scatter affected their results.Their averageresults at 903" and 981°K were 6-2 x loll and 7.4 x lo1' cm3 mole-l sec-l. Incomparison, the extrapolation of the present results using eqn. (5) gave 1-7 x 10I1and 2-5 x 10I1 cm3 mole-l sec-l. As their results are probably slightly too highY4*the agreement is reasonable but probably not significant.The reliability of the absolute values obtained for k2 depends on the efficiency ofthe detector in removing all of the D atoms.This can be tested most easily by changingthe value of .yo, the distance between the reagent inlet and the detector. As shown inruns 43-49 in table 1, the values of k2 are insensitive to the value of xo when it is20 crn or greater. If the detector had a constant efficiency less than 100 % this wouldaffect A but not Eact in eqn. (5)D . J . LE ROY, B . A . RIDLEY AND R. A . QUICKERT 95The successful elimination of any error arising from back-diffusion of reagentinto the dissociator chamber is illustrated by comparing the result for run 50 withthose for runs 43-49 in table 1. In run 50 the reagent gas accounted for only 14-5 %of the total flow, whereas for the other runs it amounted to approximately 22 %.I I I 1 I I I I II , , I I I I2.2 2 6 3 0 3.41000/T(T = 'I<)BFIG.2.-Rate constants for the reactions H+H2 = H2+H (I), D+D2 = D2+D (2), D+H2 =DH+H (3) and H+D, = HD+D (4). The curves for (1) and (3) are based on the rc.pf. method,using the parameters of table 2 ; those for (2) and (4) are given by eqn. (5) and (42), respectively.PART B. DISCUSSION OF THE FOUR EXCHANGEREACTIONSCRITIQUE OF TRANSITION STATE THEORYAccording to transition-state theory, in its exact but computationally impracticalform, the rate constant for a reaction of the typeis expressed asA+BC+[A-B-C]+AB+CAE;exp--, /<=------h QAQBC RTkT Qfwhere QA and Qsc are the complete partition functions for the reagents but omittingzero-point-energy contributions, Q * is the analogous partition function for thecomplex except for the contribution from motion along the reaction co-ordinate.AE; is the energy difference between the complex and the reagents measured fromthe zero-point-energy levels; it does not contain any contribution from the zero-point energy corresponding to motion along the reaction co-ordinate. In deriving (6)it is assumed that motion in the co-ordinate of decomposition is completely separable96 THREE-CENTRE EXCHANGE REACTIONSIn standard derivations, motion in the reaction co-ordinate is treated by anarbitrary mixture of classical and quantum mechanics.In one of the derivations ofthe quantity kT/h, motion in the reaction co-ordinate at the saddle point is treatedas a classical translation combined with the partition function for a particle in anon-existent one-dimensional box of length 6 with infinitely high potential walls.In another derivation the motion is treated as that of an harmonic oscillator for whichthe fundamental frequency v is appreciably less than kT/h, so that [l -exp (-hv/kT)]can be equated to hv/kT.At 300°K this is equivalent to requiring that v be appreciablyless than 210 cm-l, a requirement that is not satisfied by any of the recent calculations,either ab initio or semi-empirical. An additional complication, which is not normallytaken into account, is that the force constant corresponding to the partition functionkT/hv is negative, and hence the frequency v is imaginary. This latter problem willbe treated in a natural, although approximate, fashion in the succeeding section.If the harmonic oscillator-rigid rotator approximation is made, eqn.(6) may bewritten, conventionally, as the product of a temperature-independent factor and atemperature dependent factor :where a, m and c represent the reagent atom, the reagent molecule, and the complex,respectively ; o is a composite symmetry factor of magnitude 2 for reactions (1)-(4) ; r is the tunnelling factor, which can be considered to include the transmission co-efficient K ; i2 is the harmonic oscillator partition function ratio, defined byin which OM, 0, and Ob are the characteristic vibrational temperatures for the mole-cule and for the symmetric stretching and doubly-degenerate bending vibrations ofthe complex.In using eqn. (7) it is assumed that the motions of the complex on, say, a linear rl,rz potential surface can always be diagonalized in both the kinetic and potential energy,i.e., that there are no cross-terms in the vibrational degrees of freedom so that productsof one-dimensional vibrational partition functions can be used, one for each normalco-ordinate. While possibly valid for a classical treatment of very small displacementsfrom a saddle point, the assumption is of doubtful validity in treating reactions ofthe type (1)-(4).Nevertheless, it is customary to treat the asymmetric normalco-ordinate of the complex separately, as the reaction co-ordinate, and to associatethe corresponding imaginary frequency with the frequency with which complexesform products.If the asymmetricnormal co-ordinate is chosen as the reactionco-ordinate problemsarise in the calculation of the one-dimensional tunnelling corrections along thisco-ordinate, because the asymmetric normal co-ordinate depends upon the isotopiccomposition of the complex. The reagents in reactions (3) and (4) would faceunsymmetrical barriers with different side-wall repulsions in comparison with thosefor reactions (1) and (2).43 At low temperatures, one would expect the minimumenergy path to be the same for all four reactions since the potential energy surface,to an extremely good approximation, is independent of isotopic composition.Thus,one would expect the reaction co-ordinate, as distinct from the asymmetric normalco-ordinate to be the same for each reaction, viz.a line of slope -45" through thesaddle point. While such a co-ordinate could be defined for reactions (3) and (4) bytaking a linear combination of the symmetric and asymmetric normal co-ordinatesit is difficult to define true partition functions in terms of such a co-ordinateD. J . LE ROY, B . A . RIDLEY AND K . A. QUICKERT 97For uery small displacements of a linear complex A-B-C from the saddlepoint the potential energy may be expressed asand the kinetic energy asV = $(Fr: + 2F12rlr2 + Frt), (9)(10)12M T = - [mA(mB + mc>;4 + 2mAmcr,;2 + mc(mB + mA)ig],where M = mA+mB+rnc. By making the transformations,whereandand whereF(f - d ) F 2 ( f - d)2 (2fF12 - e”]’= eF-2dF12’[(eF-ZdF12)2- eF-2dFI2 ’1 2fF12-eFb eF-2dFI2’a = -d = rnA(rnB + rnJ; e = 2m,mc; f = mc(mB + mA), (1 5)V = $F,N: ++FUN:, (16)T = *m,N? + 3;m2N:, (17)then the potential and kinetic energies can be expressed in terms of the normal co-ordinates, N, for symmetric stretching, and N, for asymmetric stretching, as follows :where(dp2 + f - ep>.M(B2+1) ’(da2 + f - ea)m2 = M(a2 + 1) *ml =The sign of the second term in the expression for p in eqn.(13) must be chosen tomake negative.The symmetric and asymmetric force constants are given byFs =(F-z),andThe change in one bond length relative to the change in the other along N, and Nuare given bydr,/drl = - l/p = c,, say, along N,,= - l / a = c,, say, along Nu.(21)(22)498 THREE-CENTRE EXCHANGE REACTIONSThe frequencies corresponding to N, and AT, can be expressed in terms of the slopes c,and c, and the effective masses M, and M, for the two motions :whereSince the normal co-ordinate N, passes through a maximum at the saddle point,c, is negative, and FI2>F', so v, is imaginary.The normal co-ordinates N, and N,for reactions (1) and (2) lie along c, = + 1 and c, = - 1, i.e., at angles of +45"and -45" through the saddle point. The normal co-ordinates for reactions (3) and(4) will have different values of c, and c,.DHH HDDFIG. 3.-Comparison of the normal co-ordinates Na and N, with the reaction co-ordinates Ra and R,for reactions (3) and (4), assuming F = 0.94~ lo5 dyne an-' and Flz = 1-31 x lo5 dyne cm-'.Since it has been assumed that the reaction path for all four reactions will havea slope of -45" through the saddle point, it is convenient to define reaction pathco-ordinates, Rs and R, for which c, = 1 and c, = - 1.These co-ordinates willbe identical with N, and Nu for reactions (1) and (2) ; their relation to N, and Nu forreactions (3) and (4) is given by eqn. (26) and (27):(1 -PI (a- 1)-41 +P) (a+ 1)Rs = [2(/32+ l)]"S+ [2(a2+ 1)]-kN"9Ra = [2(a2+ l)yN"+ [2(a2+ l)]iN"The normal co-ordinates for reactions (3) and (4) shown in fig. 3 were calculatedby using the reasonable values lo P = 0.94 x lo5 dyne cm-l and F12 = 1.31 x lo5dyne cm-l. Any correlation of rate constants with theoretical potential energysurfaces must take into account the inherent problem of normal co-ordinates out-lined above.In calculating frequencies it is convenient to us2 a, b and c for isotopic factors ofthe atoms composing the complex A-B-C, and MH for the mass of the hydrogenatom.In this case the symmetric and asymmetric stretching frequencies equivalenD. J . LE ROY, B . A . RIDLEY AND K. A . QUICKERT 99I I I I 1 II 2 - -- -v! 1.0-2 -&-.+ 0(0943) 5 ( 0 866)“0 0.8-c --0( 0 707) ._ uc06- I- II I I I I I I I I I II 3 5 7 9 I 1to eqn. (23) and (24) are given by4n2v,2 = D+(D2-E)’/2MH,4n2~1,2 = D-(D2-E)li/2MH, (29)and the bending frequency by4n2$ = ((a + c)/ac+4/b)Fb/MH,where2 0 = [(a+c)/acl(F+ F,2)+[(a+c)/ac+4/b](F--12), (31)(32)andE = 4[(a + b + c)/abc](F+ FIJ(F- Fl 2).From eqn. (30), the ratio of the bending frequencies for any two complexes isalways given by the inverse of the square root of the effective mass.Also, from (28)and (29), the ratio of the symmetric, and of the asymmetic stretching frequencies canbe expressed as functions of (F+Fl,)/I F- F12 1, independent of the potential energysurface. The symmetric and asymmetric stretching frequencies for reactions (2), ( 3 )and (4), relative to the corresponding frequencies for reaction (l), are shown in fig. 4.The quantity (F+F12)/lF-F121 is equivalent to the quantity A,/lA,,I used byShavitt.lo It is seen from figs. 3 and 4 that for (F+Pl,)/( F-F12 I = 6, the ratiosof frequencies are effectively equal to the limiting values and the deviations of N,from R, for reactions (3) and (4) are only two degrees.For values greater than 6this deviation decreases. and Harris,Micha and Pohl l 2 predict a ratio of approximately 6 (6.1 and 6.7, respectively),while the semi-empirical calculations of Weston,’ Cashion and Her~chbach,’~and of Porter and Karplus l 5 gave appreciably smaller ratios (3.7, 2-3 and 2.7. re-spectively).The ab initio calculations of Boys and Shavit100 THREE-CENTRE EXCHANGE REACTIONSTUNNELLINGIn their treatment of transition-state theory, Hirshfelder, Eyring and Topleypointed out that if the potential energy barrier had an appreciable curvature at theconfiguration of the complex then the frequency factor kT/h would have to be multi-plied by the Wigner quantum mechanical correction factor (1 - (h~/kT)~/24). Sincethe de Broglie wave length at 300°K for an effective mass of MH/3 (see eqn.(25)) is1.7& quantum effects are inevitable for potentials such as those derived from abinitio calculations.ll* 12$ 17* l 8The justification for applying a one-dimensional quantum correction, implyingseparability, has been que~tioned,~ and there are divergent opinions as to whethersuch a treatment overestimates or underestimates the tunnelling.g* l9 However, inorder to test the transition state theory in the light of our experimental results, wepropose to use a one-dimensional approach, trusting in the results of the ab initiocalculations to the extent that (F+Fl2)/1F-Fl2 I is not less than 6.119 l2Aside from the shape of the potential energy barrier, there is a question as towhat effective mass should be used for tunnelling.Assuming that most systemsproceed along the path of minimum potential energy, diagonalization of both thekinetic and the potential energy requires that for reaction (1) the effective mass forrl = co (or r2 = co) be 2MH/3, while for rl = r2 it would be fkfH/3.Wigner 2o considered the problem of a Boltzmann distribution of wave packetsinteracting with a one-dimensional potential energy barrier, and showed that if thebarrier had a small curvature the tunnelling factor, which we call Tw, was given bythe expressionAt high temperatures or for very small curvature Tw approaches unity. Bell 21 hasdeveloped expressions for truncated parabolic barriers and Eckart 22 has obtaineda closed solution for the transmission coefficient through bell-shaped barriers of theform Y = E, sech2 (nxld) (Eckart barriers), Bawn and Ogden,23 and Johnston andco-w~rkers,~ have utilized Eckart barriers in problems of chemical kinetics.A more natural, although approximate, method of allowing for " tunnelling "in the sense of deviations from the strict kT/h frequency factor follows from transitionstate theory itself.In the development of transition state theory which involvestreating the motion leading to the formation of products as an harmonic vibrationof fundamental frequency v, the quantity kT/h appears as the product of a first-orderrate constant v* times the high temperature value of the harmonic oscillator partitionfunction [ 1 - exp (- hv,/kT)]-l from which the zero point energy term exp (- hva/2kThas been omitted.The frequency v* is most easily identified with the asymmetricstretching frequency along the reactionpath in the region of the saddle point ; like va,it will be imaginary.If, instead of taking the high-temperature limit of the partition function, we usethe " correct " harmonic oscillator partition function,Tw = 1 - (h~,/kT)~/24, (33)Q* = exp (- hVa/2kT)/[ 1 - exp ( - hv,/kT)], (34)thenv*Q* = v* exp (- hva/2kT)/[l - exp (- hv,/kT)],= (kT/h)I- c.p.f.where, if v* = v,,(35)(36) rc.p.f. = hlvJ - cosec ( h I V , I /2kT).2kD . J . LE ROY, B . A . RIDLEY AND K. A . QUICKERT 101For reactions (1) and (2), v* will be equal to ifa, since R, coincides with Na ; for re-actions (3) and (4) the ratio v * / ~ ~ may be derived using eqn.(24) and the appropriatevalues of c, (= - l/a) given by eqn. (13) and (14).Bell 5 * 21 obtained an expression identical with eqn. (36) as an approximation tothe penetration of a truncated parabolic barrier by a Boltzmann distribution of wavepackets. For T large and/or I v, I small,r c . p . f . = 1 +(~Iv,l/kT)2/24? (37)which is identical with Wigner's expression; rc.p.f. does not involve the barrierheight.If we treat the I' factor as representing the penetration of a potential energybarrier by a Boltzmann distribution of wave packets then, for a truncated parabolicbarrier, the form appropriate to the present problem is 21arparab =hA cosec(hIvaI/2kT)--- exp(a-pj,2kT p-awhere, in this instance, a = E,/kT ; /3 = 2EJh I v, I, and E, is the height of the potentialenergy barrier.Higher terms are present in the general expression for rparab, butthese are negligible in the present instance.The r factor for the penetration of an Eckart barrier is given by the expressionwhereand the permeability is given by= E/E, is the ratio of the energy of the incident particle to the barrier heightIf we tentatively assume that i2 is equal to unity then eqn. (7) can be written asIn k + 4 In T-ln r =In A-AEg/LT, (41 jBy using one of the three expressions for r (eqn. (36), (38) or (39)) together with theobserved values of k for reactions (1) or (3), it is possible, by iterating va and E, toobtain a best least-squares fit to the experimental data and to obtain the best valuesof In A and AEZ for each of the assumed forms of r, as well as of v, and E,.Thevalues of r, v, and E, so obtained are the result of curve fitting to an assumed func-tional form and contain no information about the separability of the reaction co-ordinate or of the effective mass ; instead, the derived quantities describe the equivalentone-dimensional barrier that is consistent with the observed data. The question ofeffective mass would only enter if one wished to calculate a force constant from v,.TREATMENT I N TERMS OF rc.p.f.The results obtained by applying eqn. (36) and (41) to all of the experimental datafor reactions (1) and (3) are shown in table 2. (Previously,2* in discussing reactions(1) and (3), we applied the rparab method but did not include the second term of eqn.(38); the treatment was therefore equivalent to the rc.e.f. method.) The valueobtained for the ratio vL3)/vi1) is 0.96, close to the limiting value (P+ F12)/l P- Pl I =0.94, given in fig.4.The Arrhenius plots for reactions (2) and (4) showed essentially no curvature overthe relatively narrow temperature range studied. The values of the parameters fo102 THREE-CENTRE EXCHANGE REACTIONSthese reactions given in table 2 were obtained from least-squares fits of the experi-mental data using the limiting values 0-745 and 0.707 for vi2)/v$') and v$~)/v!~) (cf.fig. 4). The fit for reaction (4) was slightly better than the equationused previ~usly.~ For reaction (2) the fit was no better than that given by eqn.(9,probably because of a greater scatter in the experimental data.k4 = 4-37 x 1OI2 exp (- 7 0 3 ~ x 103/RT) (42)P357°K 385°K reaction Va AE; logloA 313°K cm-1 kcal/mole(1) H+H2 117% 9.2 15.20 6.52 3-42 2.73(2) D+D2 8333' 8.9 14-86 2.03 1-7 1-55(3) D+H2 1136i 9.4 15.56 5.21 3.04 2.50(4) H+D2 87% 8.5 14.40 2.24 1.81 1.6The quantity A in tables 2,3 and 4 is the temperature-independent factor of eqn. (7).TREATMENT IN TERMS OF rparnbEqn. (38) is strictly valid only for a<P, a condition that is not fulfXled as Tapproaches 300°K for the values of v, and E, given in table 3. However, for vil) =1334i cm-I and E, = 9.9 kcal mole-' the difference between the right-hand sideof eqn. (38) and the numerically integrated general expression of Bell 21 was onlyTABLE 3.-REACTION PARAMETERS BASED ON THE rparab METHOD ASSUMING fi = 1r333°K 385°K 455°Kreaction vn EC A E ~cm-1 kcal/mole kcal/mole(1) H+H2 1334i 9.9 10.2 15-57 12.7 4.00 2-45(3) D+H2 1235 13-3 10.1 15.85 8-64 3.12 2.11(3) D+H2 1254i 9.9 * 9.9 15-70 8.96 3.30 2.19* arbitrarily set equal to value for reaction (1).slightly greater than 1 % even for T = 260°K.Since logarithmic values are used inthe fitting procedure the difference was neglected. To save computer time the inputvalues of kl and k3 were chosen from least-squares polynomial fits to the experimentalresults, rather than using the experimental data themselves.In the rparab method E, enters eqn. (38) only in the second term, which is smalland has the nature of a correction term.A statistical analysis of experimental datausing eqn. (38) will therefore lead to large probable errors in E,. This is illustratedby the results for reaction (3) given in table 3. The first value, 13.3 kcal mole-l,was obtained by a completely objective treatment of the data ; the second value wasarbitrarily set equal to that obtained for reaction (l), and resulted in only minorchanges in vi3) and rparab. As is seen from fig. 5, the fit obtained by arbitrarily settingE, = 9.9 kcal mole-1 is not as good as for the completely objective treatment, althoughthe value of v$~)/v!~) is identical with the limiting value, 0.94, shown by fig. 4D. J . I,E ROY, B. A . RIDLEY AND K. A. QUICKERT 1031000/T(T = O K )FIG. 5.-Treatment of reactions (1) and (3) by the rparab method.Solid curves for E p ) = E:') =9.9 kcal mole-', vi') = 1334i cm-', vL3) = 1254i cm-1 ; dashed curve for Er3) = 13-3 kcal mole-',vL3) = 12353,cm-'.TREATMENT I N TERMS OF rEckSeveral different procedures were used in applying eqn. (39) and (41) to obtainthe results shown in table 4. In every case, convergence was maintained to withinTABLE 4a-REACTION PARAMETERS BASED ON THE rEck METHOD ASSUMING a = 1reaction method vo Ec AE~" loglo A rcm-1 kcal/mole kcal/mole (temp. in parentheses)8995 407 46.6 (1) H+H2 (i) 2717i 15-0 16.2 17.31 (300) (364) (445)6516 303 26.6 (1) H+H2 (ii) 2564i 15.0 15.7 17-19 (294) (358) (445)46-6 9.0 4.4 (3) D+H2 (ii) 1673i 17.2 11-7 16-33 (308) (378) (445)4875 236 31.0 (3) D+H2 (iii) 26253.15.0 16.2 17.61 (302) (370) (455)115.0 20.2 8.2 (1) H+H2 (iv) 213% 11.0 12.0 15.93 (312) (385) (455)71-1 14.5 6.4 (3) D-tH-, (iv) 2005i 11.0 11.7 16.11 (312) (385) (455)1-2 % for both reactions (1) and (3). (i) Objective iteration for v, and E, using theexperimental values of k as input. (ii) Same as for (i) but using values of k taken fro104 THREE-CENTRE EXCHANGE REACTIONSa least-squares polynomial fit to the experimental data. (iii) Using experimentalvalues of k3 but arbitrarily setting E, = 15.0 and vi3) equal to 0.94 times the value ofdl) obtained by procedure (i). (iv) Using values of k taken from least-squaresI 1 I I I I I 1 1 11000/T(T= "K)FIG. 6.-Treatment of reactions (1) and (3) by the method using procedures (i), (ii), (iii), (iv).polynomial fits to the experimental data and iterating for vil) and vL3) using an arbi-trary value of 11.0 for E,.This particular choice for E, was based on a private com-munication from Dr. M. Krauss 24 giving us the results of an extension of the earliercalculations l7 which had been made in collaboration with Dr. C . Edminston. Theserecent calculations suggest that if allowance is made for asymptotic errors the barrierheight for the H3 system is about 11 kcal mole-I,THE r FACTOR AS A CRITERION OF THE PROPERTIES OF THE COMPLEXAlthough the general shape of the Eckart barrier is appealing, the extreme sensi-tivity of its derived parameters to minor fluctuations in the experimental data makesit difficult to use with confidence.Furthermore, the values of rEck derived are muchlarger than one would think, intuitively, to be realistic, and the values of E, wouldalso seem to be high, particularly in the light of the ab initio calculations of Conroyand Bruner.18However, if we now apply the three r approaches to calculate a rate constantat 1,00O0K, well beyond the experimental range, they all predict similar values. Ifwe apply the parameters for reaction (1) given in table 2, the rc.p.f. treatment predictsthat kl will be equal to at 1,OOO"K and that I' at that temperature will be 1.13.The rparab method yields the same results because the second term in eqn. (38) vanisheD. J . LE ROY, B . A . RIDLEY AND K . A. QUICKERT 105at this temperature. The rEck method using the parameters E, = 150 kcal mole-gand v, = 2717i cm-I (cf.table 4) gives the results k , = r = 2.1 ; using theparameters E, = 11.0 kcal mole-I, v, = 2138i cni-l, we find that k l = 1012.0, l? = 1.56.The only experimental results that can be compared with the above extrapolationsare those of F a r k a ~ , ~ ~ who found k l = 1012-4 at 1023°K. It has been suggestedthat a value of would allow for the fact that oxygen was undoubtedly present,but even with such a correction the high-temperature results remain ~ncertain.~Considered simply as a curve-fitting operation, the assumption of an Eckartbarrier with a height of 15 kcal mole-1 would seem to be unjustified on the basis ofthe extrapolation to 1,000"K. The assumption of an Eckart barrier of arbitraryheight 11 kcal mole-' would appear to be more in keeping with the likely value of klat 1,000"K, although it did not reproduce our own experimental data particularlywell. The most satisfactory way in which to treat the experimental data would seemto be in terms of either I'c.p,f.or rparab : the values of r are more realistic, and thevalues of v, are more in accord with those obtained in recent ab initio calculations.These (Boys and Shavitt l 1 (1361i), Harris, Micha and Pohl l2 (1410i), Edminstonand Krauss 26 (14009) are appreciably lower than those derived from semi-empiricalcalculations (Weston (1 9 1 89, Cashion and Herschbach l4 (2464i), Porter andKarplus (2295i), Johnston (1 6999). We will therefore utilize the rc.p.f, treatmentin our examination of transition state theory, recalling that we will be consideringequivalent parabolic barriers. The treatment will be confined to the temperaturerange of our experiments, inasmuch as rc.p.f.shows a discontinuity at T = li I v, I f2kz,(approximately 260°K). However, it will first be necessary to consider the possibleeffect of the bending frequency vb in causing curvature in Arrhenius plots.BENDING FREQUENCY OF THE COMPLEXSo far, in treating our data in terms of transition state theory, we have assumedthat the quantity R in eqn. (8) was equal to unity. The term most likely to causedeviations from unity is [l -exp (- Ob/T)]-2. This will be temperature dependentand it is necessary to consider how this might affect the values derived for AE; and v,,and whether it could account for the whole of the curvature observed in the Arrheniusplots for reactions (1) and (3).By neglecting that part of the temperature-dependent term in eqn.(7) having todo with the stretching vibrations of the reagent molecule and of the complex, anexperimental '' Arrhenius " activation energy can be defined as follows :in which Eb = 2RBb/[exp (&/i?)- 11 ; ET = I(272. Thevalues of Eexpt for reaction (1) derived from a least squares polynomial fit to theexperimental data were 4.37 kcal mole-l at 300°K and 8.64 kcal mole-' at 4543°K.Thus, in order to fit the experimental data (Eb-Er) would have to increase by 4-4 kcalmole-l between 300 and 45405°K. Now the maximum value of Eb (for 8,-+0 orT-+co) is 2RT, so the maximum increment in Eb between 300 and 45445OK is 0.6 kcalmole-'.A low bending frequency could therefore not contribute more than 14 %to the required change in (&I&).A low value of vb would affect the values derived for v, and A,??: using, say, the17c.p.f. method, but it would require a very low value to change these greatly. Webelieve that insofar as our results can be used as a test of transition state theory, itis justifiable to consider that Vb for reaction (1) is approximately 900 cm-', a roughmean of the ab initio l r n l2 and semi-empirical 13-15* calculations.Er = Rd In T/d(l IT) 106 THREE-CENTRE EXCHANGE REACTIONS8.0COMPARISON OF THE FOUR REACTIONS I N TERMS OF TRANSITIONSTATE THEORYWe believe that an adequate test of the transition state theory using our experi-mental results can be made by using the rCepmf.method and assuming that vb forreaction (1) is approximately 900 cm-l. This amounts to assuming that the potentialenergy surface of the complex is equivalent to a one-dimensional parabolic barrier.Since E, is the same for each of the four reactions and is related to AEG by the expres-sionone could, in principle, calculate E, as well as Bs for each reaction by using theassumed value of the bending force constant (corresponding to vil) = 900 cm-l andthe limitingratio for vii)/vS1)given in fig. 4. Such a calculation proved to be impracticalbecause of the severe demands it placed on the precision of the data. Nevertheless,it is possible to gain some knowledge of the properties of a potential energy surfacethat would be reasonably consistent with the data when interpreted in terms of theE, = A E ~ ( i ) + R ( B ~ ) / 2 - B ~ ) - B ~ i ) / 2 ) , i = f , 2,3,4, (44)form of the transition state theory.=;.I I I r I I 19.09.5 --8.5 -By equating !2 in eqn. (7) to [1 -exp (-S,/T)]-2 (i.e., neglecting the temperature-dependence of the contributions to S2 made by 8, and BJ, and incorporating thisas an extra term in eqn. (41), the values 9.0, 8.3, 9.2 and 8.2 were obtained for AE;for reactions (l), (2), (3) and (4), respectively ; this leaves Ec and 6s') as the onlyunknown quantities in eqn. (44). By summing the four equations of this type andutilizing the known values of 8, and the assumed values of Ob and v!')/v$') the follow-ing relation is obtained :This is plotted in fig.7 along with points representing various semi-empirical calcula-tions. Eqn. (45) is essentially a " best " relation between conjugate values of E,and vjl) consistent with our experimental data.It is evident from fig. 7 that the experimental data, so interpreted, are not consistentwith the " high " values of E, obtained by ab initio calculations 11* l7 However,E, = 11.76-1-229 x 10-3~!') (45D. J . LE ROY, B . A. RIDLEY AND K. A . QUICKERT 107since all of the theoretical calculations 11-14* are in agreement with a value of I):')in the vicinity of 2100 cm-', this would suggest that E, is about 9.2 kcal mole-'.A value of 2100 cm-l for v$l) corresponds to a value of approximately 9 for(F+ Fl 2)/1 P- Fl I (using our value of $I), which substantiates our previous con-clusion that R,=N, for reactions (3) and (4).CONCLUSIONSA modified formulation of transition state theory has been proposed whichincorporates a more consistent quantum mechanical interpretation of the asymmetricvibration of the complex.With the proposed model curvature in Arrhenius plots canbe accounted for without invoking the concept of tunnelling through the barrier.Despite logical difficulties associated with transition state theory, we have interpretedthe experimental results for reactions (l), (2), (3) and (4) in terms of the modifiedtheory. The normal co-ordinates for the asymmetric vibrations of the heteronuclearcomplexes of reactions (3) and (4) were found to be essentially coincident with thereaction co-ordinates.The results obtained by the present treatment suggest that the height of thepotential barrier is about 9.2 kcal mole-' and that the force constants for the sym-metric and asymmetric vibrations at the saddle point are approximately 2.6 x lo5 dynecm-' and -0.27 x lo5 dyne cm-l, in reasonable agreement with those calculated byBoys and Shavitt l1 and Harris, Micha and Pohl.12The authors acknowledge the financial support given by the National ResearchCouncil of Canada.Two of us (B. A. R. and K. A. Q.) express appreciation of theaward of National Research Council Bursaries and Studentships. We thank Dr.M. Krauss of the National Bureau of Standards for letting us have the results of thecalculations of himself and Dr. Edminston prior to their publication.W. R. Schulz and D. J. Le Roy, J. Chem. Physics, 1965,42, 3869.B. A. Ridley, W. R. Schulz and D. J. Le Roy, J. Chem. Physics, 1966,44, 3344.W. R. Schulz and D. J. Le Roy, Can. J. Chem., 1964,42,2480.H. S . Johnston, Gas Phase Reaction Rate Theory (Ronald Press, New York, 1966).R. P. Bell, The Proton in Chemistry (Cornell University Press, Ithaca, 1959).A. Farkas and L. Farkas, Proc. Roy. SOC. A , 1935, 152, 124.D. Rapp and R. E. Weston, Jr., J. Chem. Physics, 1962,36,2807;E. W. Schlag, J. Chem. Physics, 1963,38,2480.H. S . Johnston and D. Rapp, J. Amer. Chem. SOC., 1961, 83, 1;H. S. Johnston and J. Heicklen, J . Physic. Chem., 1962, 66, 532.' G. Boato, G. Careri, A. Cinimo, E. Molinari and G. G. Volpi, J. Chem. Physics, 1956, 24,783.lo I. Shavitt, J. Chem. Physics, 1959, 31, 1359.l1 S. F. Boys and I. Shavitt, Uniu. of Wisconsin Naval Research Lab. Report, 1959, WIS-AF-13.l2 F. E. Harris, D. A. Micha and H. A. Pohl, Arkiu Fysik, 1965, 30,259.l 3 R. E. Weston, Jr., J. Chem. Physics, 1959, 31, 892.l4 J. K. Cashion and D. R. Herschbach, J. Chem. Physics, 1964,40,2358.lS R. N. Porter and M. Karplus, J. Chem. Physics, 1964,40, 1105.l6 J. Hirschfelder, H. Eyring and B. Topley, J. Chem. Physics, 1936,4, 170.l7 C. Edminston and M. Krauss, J. Chem. Physics, 1965,42, 11 19.l8 H. Conroy and B. L. Bruner, J. Chem. Physics, 1965,42,4047.l9 E. M. Mortensen and K. S. Pitzer, Chem. SOC., Spec. Publ. no. 16, 1962, 57.2o E. P. Wigner, 2. physik. Chem., 1932, 19,203.21 R. P. Bell, Tram. Faraday SOC., 1959, 55, 1.*' C. Eckart, Physic. Reu., 1930,35, 1303.23 C. E. H. Bawn and G. Ogden, Trans. Faraday Soc., 1934,30,432.24 M. Krauss, private communication.2s A. Farkas, 2. physik. Chem. B, 1930,10,419 .26 calculated by us from data kindly offered by Dr. Krauss (ref. (24))

ISSN:0366-9033    

DOI:10.1039/DF9674400092

出版商:RSC    

年代:1967

数据来源: RSC