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Measurement of the rate of hydrogen atom abstraction from methane and ethane by muonium Snooks, Rodney J. 1993

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MEASUREMENT OF THE RATE OF HYDROGEN ATOMABSTRACTION FROM METHANE AND ETHANE BY MUONIUMByRodney James SnooksB.Sc. (hons) Saint Mary's University, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESCHEMISTRYWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIA1993© Rodney James Snooks, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of Ch The University of British ColumbiaVancouver, CanadaDate 477 / 30^/ 92,3DE-6 (2/88)AbstractReaction rates for the gas-phase reactionsMu+ CH4 —> MuH + CH 3andMu + C 2 H6 Mull + C 2 H 5have been measured using //SR over the temperature ranges 626-821 K and 511-729 Krespectively. The usual Arrhenius plots for each data set are linear. The measuredparameters A are 5.7+2. s x 10 -8 and 1.01 47 x 10 -9 cm3 molecule -1 s -1 , and the pa-rameters Ea , 24.661: 82 and 15.351:65 59 kcal/mol respectively. The Ea values are 11.5and 5.5 kcal/mol higher than for the corresponding H atom reactions. The very largeincreases in Ea seem to indicate drastic differences between the Mu and H variants ofthe title reactions, in location of the transition states on the potential energy surfaces.Also, for the Mu variants, the reaction rates seem to be reduced less for vibrationallyexcited states of CH 4 and C 2H 6 than for the ground states than is the case for thecorresponding H atom reactions, an effect which contributes to the large increase inactivation energy for Mu.iiTable of ContentsAbstract^ iiList of FiguresList of Tables^ viAcknowlegements^ vii1 Introduction^ 11.1 Background and Motivation  ^11.2 Organization of the Thesis  ^72 Bimolecular Reaction Kinetics^ 93 Electronic Structure^ 163.1 Introduction  163.2 Potential Energy Surfaces in Reaction Rate Theory ^ 183.3 Molecular Electronic Structure ^  203.3.1 SCF State Functions  233.3.2 Post-Hartree-Fock Procedures ^  273.3.3 Other Approaches to Electronic State Functions ^ 303.4 Constructing Potential Energy Surfaces ^  324 Reaction Rates from Theory^ 374.1 Introduction ^  37iii4.2 Transition State Theory ^  394.3 Some Forms of Collision Theory^  434.4 Theoretical Rate Parameters for the Title Reactions ^ 464.4.1 Mu + CH 4^Mull + CH3 ^  464.4.2 Mu + C 2 H6 —+ Mull + C 2 115 ^  495 Experimental Setup^ 515.1 Positive Muons and iLSR ^  515.1.1 ptSR and MSR  535.2 //SR Experiments at TRIUMF ^  645.3 Reaction Vessel for CH 4 and C 2H 6 Experiments ^  665.3.1 Thin Muon Entry Window ^  695.3.2 Heating of the Vessel ^  725.4 Measurement of the Title Reaction Rates ^  746 Results and Discussion^ 777 Conclusion^ 88Bibliography^ 89A Plots of Relaxation Rate Data^ 96ivList of Figures3.1 One "Sheet" of a Potential Energy Surface ^  194.2 Potential Energy "Cliff" and Comparison to Title Reaction ^ 404.3 Comparison of Several Sets of Theoretical Results and ExperimentalData for H CH 4 11 2 + CH3 ^  474.4 Geometry of Transition State for H CH 4 —p H2 + CH3 ^ 495.5 Muonium Hyperfine State Energies as a Function of Magnetic Field . . 585.6 MSR Signals for N2 and CH 4 Compared   615.7 Fit of Rate Coefficient of the Reaction Mu + CH 4 —> Mull + C11 3 at 821 K 625.8 M15 Secondary Beam Channel ^  675.9 Reaction Vessel Used in This Study  685.10 Design of the Muon Entry Window for Reaction Vessel Used in CH 4 andC 2H6 Experiments ^  706.11 Arrhenus Plot for Mu + CH 4 Mull + CH 3^  786.12 Arrhenus Plot for Mu + C2H 6 Mull + C 2H 5^  796.13 Correlation Diagram H CH 4 -4 CH 4 -H -4 H2 + CH3 ^ 82List of Tables1.1 Selected Rate Data for H + CH 4 11 2 + CH3  ^21.2 Selected Rate Data for H C 2 H6^112 C 2 H5  ^24.3 Comparison of Isotopic Reaction Rates for the POL-CI Potential EnergySurface ^  485.4 Tests of Inconel Muon Entry Windows ^  726.5 Measured Rate Coefficients for Mu + CH 4 MuH CH3 ^ 806.6 Measured Rate Coefficients for Mu + C 2 H6 MuH C 2 H5 ^ 806.7 Comparison of Mu and H Atom Data for Title Reactions ^ 806.8 Standard Enthalpies of Title Reactions and H Atom Variants ^ 836.9 Estimated Vibrational Wavenumbers of the CH 4—Mu Transition State ^ 856.10 Fit of CH 4 Data to Nonstandard Arrhenius-type Expression (Estimateof Excited State Rate Parameters) ^  86viAcknowlegementsI would like to thank my thesis advisor, Donald G. Fleming, for patient research di-rection and a "hands-off" attitude allowing me to pursue my many interests at U.B.C.Also owed thanks are the other members of our research group in my time at U.B.C.—Masa Senba, James Kempton, Susan Baer, Donald Arseneau, James Pan, and MeeShelley—all of whom were invaluable helpers with the present study, and offered muchhelp with technical details and theoretical interpretations.As well, this thesis would not have been possible without the help of many engi-neers and technical workers at TRIUMF. Particularly notable were the contributionsof George Clark, the designer of the reaction vessel used in the present study; KeithHoyle and Curtis Ballard, ,uSR technical workers, who went beyond the call of jobdescriptions on many occasions; and members of TRIUMF's machine shop, for qualitymetal work, often on very short notice.Lastly I would like to thank my favorite undergraduate professors, Jack Ginsburgand Robert Kruse, both of whom offered clear answers to my many questions, andwithout whom I may never have made it into graduate school.viiChapter 1Introduction1.1 Background and MotivationThe field of reaction kinetics calculations has become a particularly active area of chem-ical research in recent years (see [1-5] and references therein). Recent rapid advancesin computer technology have allowed theorists to contemplate the possibility of trulyaccurate computations of gas-phase reaction rates from first principles (ab initio).Experimental kinetics data are already available in abundance; see [6] and [7] forcompilation of rate data for the abstraction reactions:H + CH 4 —> H2 + CH3H +C 2 116^H2 C2H5^ (1.2)of which the title reactions^Mu + CH4 —4 MuH + CH3^(1.3)Mu + C 2 H6 —> MuH + C 2 H 5 (1.4)are isotopic variants. The data have been measured by a variety of methods. Anotherreport [8] on the reaction (1.1) has appeared since the beginning of the present study.Also studied have been the hot tritium (T*) variant of reaction (1.1) [9-12] and the H*and T* variants of reaction (1.2) [13,14]. The substitution reactionHa + CH4 CH3Ha H^ (1.5)1Chapter 1. Introduction^ 2Table 1.1: Selected rate data for H^CH4^11 2 + CH3 .T/K A/(10-1°cm' molecule -1 s -1 ) Ea /(kcal mole -1 ) Reference426-747 1.0 11.7 + 0.2 a673-763 5.3 15.1 b640-818 3.2 13.2 ± 0.8 c827-1729 1.8 12.8 + 0.2 d400-1800 1.3 11.9 + 0.2 e1100-1800 3.3 11.5 fa) Flow system and ESR detection of H atom [16].b) Inhibition of the first limit in 112 + 02 by C114; C113 + 02 and CH3 + H2 ignored [17].c) Flow discharge system; k obtained from a fitting of 14 reactions [18].d) Most recent study, measured by flash photolysis/shock tube technique [8].e) From global fit to several data sets given in reference [18]. Consistent as well with compilationof Shaw [7].f) Low pressure flames of CH4/H2/02/N20; OH + CH 4 and H2 + CH3 reactions neglected; krelative to k(H + N20) [19].Table 1.2: Selected rate data for H C2116^112 + C 2 1-1 5 .T/K A/(10-10cm3 molecule -1 s -1 ) Ea /(kcal mole -1 ) Reference281-347 0.83 9.1 a357-544 1.8 9.2 b503-753 3.1 9.8 c876-1016 8.7 11.0 + 0.2 da) H atoms produced by electrical discharge and monitored by spectroscopic measurement ofreaction product with HgO [20].b) Flow discharge system [21].c) Simultaneous determination with rates of H atom reactions with CH 3 and C 2 H 5 fragments [22].d) H atoms monitored by atomic resonance absorption [23].has recently [15] undergone a mechanistic study using deuterated methane (CD4),which shows the reaction to proceed by a standard inversion mechanism. Selectedexperimental thermal rate data for the reaction (1.1) are shown in Table 1.1, andfor reaction (1.2), in Table 1.2. The parameters listed are defined by the standardChapter 1. Introduction^ 3Arrhenius equation k = A exp(—Ea /RT), with Ea the activation energy and A the pre-exponential factor, reviewed in Chapter 2. The measured reaction rates are of coursegreater for C 2 H6 than for CH 4 , as expected from the simplest of chemical arguments.Thermal rate data for the T atom isotopic variant of reaction (1.1) is not avail-able, and two studies [24, 25] for the D variant, in which the rate parameters weredetermined indirectly, were complicated by calibration problems, resulting in absolutemeasured rates of uncertain accuracy. However, in the study of reference [25], an Ea of11.1 kcal/mol was found by comparison with H atom data of the same study, for whichan Ea of 11.7 kcal/mol was measured. This isotope effect is in reasonable agreementwith the theoretical calculations of Schatz et al. [26], discussed in Chapter 4.As calculated reaction rates approach measured rates in accuracy, a detailed com-parison between experimental data and the most accurate calculations should revealthe strengths and weaknesses of the theory underlining the calculations, allowing forfurther refinement of the theory and/or calculation methods. Well-characterized kineticisotope effects allow a particularly meaningful comparison between the experimentaland theoretical results. Isotopic substitution in reactants affects reaction rates in sev-eral ways. Of no real interest is the rate difference due to the difference in meanvelocities, known as the "trivial" isotope effect, owing to the mass difference of thereactants. However, isotopic substitution also affects reaction rates due to the shiftingof the energy levels of the rovibrational quantum states in reactants, products, andmost importantly, in transition states, resulting from the change in mass. The massdifference changes both the energy spacing (density of states) and the value of thezero point energies (the difference between the ground state energy and zero) of theselevels. In addition tunneling, the quantum phenomenon of particles passing throughclassically forbidden ("negative energy") regions, is more likely for lighter species thanfor heavier ones.Chapter 1. Introduction^ 4In general the zero point energy and/or tunneling effects on reaction kinetics aremanifest more strongly in lighter isotopes. Accounting for these effects is the challengefacing reaction rate theory. Because of the complicated results of the relevant iso-tope effects, theoretical calculations giving results comparable to experiment for morethan one isotopic combination for a given reaction are less likely to be in agreementfortuitously, and so examination of isotope effects gives a stringent test of theoreticalcalculations when compared to experiment.Aside from being the lightest atom, hydrogen also has the highest mass ratio be-tween its isotopes (T(3):D(2):H(1)) as compared to any other element; therefore it isthe element displaying the greatest degree of isotopic effects in its behaviour. A furtherextension to this series is made possible by ktSR, an experimental technique in use sincethe 1970's (see [27-30]) at the TRIUMF cyclotron in Vancouver. TRIUMF providesbeams of the particle known as the muon, and of its antiparticle //-, the latterbeing used in the present study.The positive muon at high (--MeV) energies can capture an electron to form asystem known as muonium (p +e- , abbreviated Mu) [27]. Since the muon mass is --1/9that of the proton, 206 times that of the electron, the reduced mass of the combinedsystem (defined as m i,me /(m i, + me ), with m's the particle masses of kt+ and e+) isvirtually unchanged from that of the electron, as is the case for the H atom. As well,the charge of the muon is the same as for a proton. From elementary physics, theequation of motion of the electron in Mu is then essentially the same as that of theelectron in H: that of an electron moving in a force field of a very heavy nucleus of unitpositive charge. Muonium is therefore chemically equivalent to, and can be consideredan isotope of hydrogen.The mass of Mu, being only 1/9 that of the mass 1 a.u. isotope (H), and 1/27 thatof tritium, leads to isotope effects of greater magnitude than those seen in conventionalChapter .1. Introduction^ 5chemical studies. For example, the standard heat of reaction AH° for the reaction (1.1)is [31] —2.6 kcal/mol; the zero point energy of MuH is [32] is —7.5 kcal/mol higher thanfor H2 resulting in a AH° of 4.9 kcal/mol for the Mu variant reaction (1.3). For thecorresponding reaction with C 2 H6 (1.2), Mu substitution raises AH° from —3.1 to4.3 kcal/mol l . In both cases an exothermic reaction becomes endothermic. A strikingexample of a kinetic isotope effect is the abstraction reactionMu -I- F 2 MuF F (1.6)in which pronounced tunneling is obvious from the kinetics data [34]. The kineticisotope effect is especially striking in comparison to data of corresponding H and Datom reactions, in which such tunneling is much less evident than for Mu.The ttSR technique allows observation of the chemical behaviour of Mu using equip-ment developed for nuclear and particle physics experiments. Together with conven-tional studies of H, D, and T, muonium studies can provide a wide range of kineticisotope effects.Being the simplest element, hydrogen is the most amenable to theoretical calculationof reaction rates. Hydrogen atom reactions have received more theoretical attentionthan those of any other species for this reason. The chemical reaction for which themost rate calculations exist is, predictably, the abstraction reactionHa + HbH, Hallb Hc (1.7)for which completely ab initio calculations have reached the point of chemical accuracy(see [35]). This includes the isotopic variantMu -I- H 2 MuH H^ (1.8)'Using AH° = 120 ± 3 kJ/mol for C2H 5 from reference [33].Chapter 1. Introduction^ 6for which there is remarkably close agreement between Schatz' reactive scattering cal-culations [36] and rates measured by 1iSR [37]. This agreement represents one of themost impressive demonstrations of the utility of ktSR.The same level of accuracy has not been reached for the abstraction reaction (1.1)or the substitution reaction (1.5) but these have received much attention, being thesimplest non-trivial reactions involving a hydrocarbon species, as well as key reactionsin the combustion of methane; see [38] for many references, and [39,40] for recent, abinitio, calculations. The calculations of Gonzalez-Lafont [39] et al. for the abstractionreaction (1.1) show fair general agreement with the shape of Arrhenius plots of the bestavailable rate data, but calculations of the accuracy of those performed by Schatz onMu + H2 are not available. Also, some calculations exist for the analogous H C 2 H6reactions, but at a much more approximate level (see [41] and references within).Clearly, theoretical study of the reaction dynamics of polyatomic molecules, thoughrapidly growing in feasibility, remains in its infancy. The complex nature of these dy-namics, as compared to those found in reactions of diatomics, may lead to featuresnot observed in the reactions of the smaller molecules, resulting in a need for approx-imations of a more general nature than those usually applied. Advances in laser anddetector technology have made possible state-selected and state-to-state reaction ratemeasurements (i.e. with preselected reactant and/or known product quantum statedistributions) on polyatomics (see [42] and references therein). These provide more de-tailed information on reaction dynamics than conventional, thermal rate measurements,and thus supplement, and can influence, theoretical treatments.Such measurements were recently reported [43] for the H CD 4 deuterium ab-straction reaction. This study shows the product distribution of HD states to have apositive correlation between vibrational and rotational energy, contrary to "all known"studies of reactions of atoms with diatomics. Initially termed "anomalous", the sameChapter 1. Introduction^ 7results were more recently observed [44], to a greater degree of magnitude, in a similarstudy of the H C2 116 abstraction reaction (1.2) and to a greater degree still in thecorresponding reaction with C 3H8 . These results, combined with the present resultsfor the Mu isotopic variants of reactions (1.1) and (1.2), and intrinsic interest in thereactions, should motivate theoretical treatments.1.2 Organization of the ThesisThis thesis consists of seven chapters including this Introduction, and an Appendix.The second chapter briefly reviews the standard empirical equations of bimolecularkinetics, with some interpretation of these equations in terms of molecular processes.The third chapter, which reviews electronic structure theory and recent progress in thisfield, is included for completeness and is optional. The fourth chapter comprises a briefoverview of the best-known methods for computation of chemical reaction rates, withdiscussion of recent progress and in particular, application to the title reactions. All ofthe popular theories of reaction rates share the need for preliminary computations ofelectronic state functions; progress in this area has continued since the 1920's [45] buthas been rapidly accelerated in recent years through advances in computer technologyand algorithm design; current methods are outlined.In the fifth chapter is outlined the theory of iiSR, with mention of some of thevarious applications of the technique, and the essentials of the setup of a //SR gaschemistry experiment, some of which go beyond experimental considerations. Alsodiscussed briefly are earlier results of Mu kinetics experiments. Included is a discussionof the design, fabrication, and testing of the reaction vessel used in the experimentshere reported.In the sixth chapter is discussed the results of the experiments, with a comparisonChapter 1. Introduction^ 8to older theoretical and experimental results for isotopic variants of the title reactions.The seventh chapter comprises concluding remarks. The Appendix consists of plots ofthe experimental data.Chapter 2Bimolecular Reaction KineticsIn this chapter is reviewed the basic equations of bimolecular reaction kinetics neededto interpret the experimental data reported herein'. The title reactions fall in thecategory of elementary bimolecular reactions, "elementary" meaning a "one-step" or"direct" mechanism. The equations describing the bulk kinetics of such a reaction aresimple in form and well-known. The general elementary bimolecular reaction is of theformC+D^ (2.9)and proceeds at the rate defined for a closed system asR^d[A]^d[B]dt^dt(2.10)where [A] represents the concentration of A and similarly for B.The reaction rate R for reaction (2.9) is proportional to the product of the concen-trations of A and B:R = k[A] [B]^ (2.11)where k is known as the rate constant or, more appropriately, the rate coefficient, since itdepends on temperature. Reaction conditions such as pressure can be accounted for bythe use of activities, rather than concentrations, of A and B. Equation (2.11) is said tobe a second order equation since the powers of the reactant concentrations appearing in'Standard constants used in thesis without explanation: k, Boltzmann constant, h, Planck constant,h = h/2ir, R, gas constant, c, speed of light.9Chapter 2. Bimolecular Reaction Kinetics^ 10the equation add to two. This equation, though of an empirical nature, has found wideapplicability in both gas- and liquid-phase bimolecular reactions and is generally obeyedfor elementary reactions. Compound (two or more step) reactions can be analyzedas combinations of elementary reactions obeying equation (2.11). Unimolecular andtrimolecular elementary reactions obey similar equations.Now, and for the remainder of this work, specializing reaction (2.9) to reactionswhere A is an atom, the parameter k can be considered an average of k2 , with i theindividual molecular quantum states of B:R i ki [A][B i  (2.12)defining k, as the rate coefficient between A and B with B in the ith internal state, andthe average weighted by the initial distribution of the B states. The fraction [B,]/[B],the weight of the ith state of B, is denoted w„ and is the probability for B to be in thestate i and so is called the distribution function for B. Then, the average k over the k 2takes the formk (k,) = E w,k, (2.13)B statesand similarly R represents an average of R2. These sums conventionally omit thetranslational energies of A and B, which are treated as separable from internal motion.The k, are measured by reactant state-selected reaction rate measurements, and arein general a function of temperature. This equation is correct for an ideal gas anddisplays explicitly that k is not a fundamental quantity but an average over distinctprocesses. The kinetic isotope effect will in general be different for each such process.As one example, recent [46] reactant state-selected measurements of the reaction02 + CH4 H CH 3 O2^ (2.14)at --ZOO K give a rate coefficient --17 times higher for vibrationally excited CH4 thanfor the ground state.Chapter 2. Bimolecular Reaction Kinetics^ 11The reaction cross-section a between A and B in state i can be defined usingk2(T) = Jdv w(T,v)a,(v)^ (2.15)where y is the reduced mass of A and 13 2 , v is the relative velocity (prior to collision"),which is assumed is separable from the other types of motion, and w is the distributionof v, a continuous function if translational motion is treated classically. The reactioncross-section can also be defined in terms of reaction probability for a given molecular"collision", which itself can be defined as an A-B encounter close enough to have anappreciably high probability of reaction, say, some arbitrary threshold. The integralserves to sum a jointly with the frequency of collision, resulting from relative transla-tional motion of A and B. For the reactant molecules far enough apart that they donot affect each others' collision processes appreciably, i.e. a gas, a can be classicallyinterpreted as a cross-sectional area, per reactant molecule, which needs to be collidedwith in order for reaction to occur.For an ideal gas at equilibrium, the internal states of B will be individually weightedby the Boltzmann distribution (see any statistical mechanics text), that is,[B, ]—tot[B]^qB^(2.16)where e t is the energy of the ith state of B, g, its degeneracy, co the energy of the groundstate, T the temperature of the system, and qB the molecular partition function of B.The partition function is computed by taking the sum of equation (2.16) over all statesand noting E[13 2 ]/[B] = 1. The translational energy of A or B obeys the same equationgiven our assumptions but the energy in this case is continuous.Using the distribution of equation (2.16) for translational energies of A and B. andapplying the result to equation (2.15), the resulting equation can be written as8kT) 1/2^7 Ek, = (— f — EJo^kT^kT) (E)--E IkT^(2.17)Chapter 2. Bimolecular Reaction Kinetics^ 12with the reduced mass of A and B 2 and E their relative translational energy. Theequation is in such a form as to explicitly display the dimensions, velocity times area,of lc,. This equation strongly displays the "trivial" kinetic isotope effect due to reactantmass; the factor multiplying the integral is inversely proportional to VT/.Another well-known empirical reaction rate equation is the Arrhenius equationk = Ae-Ea IRT (2.18)where k is as defined in equation (2.11) and T is the absolute temperature. Theparameter Ea is called the activation energy, and A, a constant, is the pre-exponentialfactor. The equation is frequently generalized to allow A to be a function of T. Whenfitted to experimentally measured thermal reaction rates, it holds very well for manyreactions. For bimolecular reactions for which equation (2.18) holds, Ea is of thesame order of magnitude (see Chapter 4) as the reaction's energy barrier per mole ofreactant, as suggested by its resemblance to the Boltzmann distribution of reactants inequation (2.16). Taking the natural logarithm of each side givesln k = ln A — RT(2.19)which, when plotted as ln k vs. 1/T, fits well to a straight line for many reactions,corresponding to constant A.The equation (2.18) is reproduced by use of simple models for the reactants. Forexample, suppose A and B are assumed to be hard spheres, ignoring any internalstructure, and all collisions with relative kinetic energy of below a threshold energy E 0do not result in reaction, while above E 0 ,a(E) = o-0 (1 — — 2E )^(2.20)with ao and E0 constants. This cross-section expression [47] is intended to include theeffect of the relative angular momentum of the reactants, and is commonly referred toChapter 2. Bimolecular Reaction Kinetics^ 13as the "line of centers" model. Using the equation (2.17), the equationk = ao (8kT777,) 1 / 2 6 -E0 /kT^(2.21)results [47], clearly a form of equation (2.18).The activation energy Ea can be interpreted [48] as the difference between theaverage energy (above ground state) of those reactant atoms A and molecules B thatactually react, commonly denoted (E* ), and the average for all A and B, ( E).Ea = (E*)—(E) (2.22)For relatively small molecules such as CH 4 , (E) is within an order of magnitude ofkT; with a large enough reaction energy barrier it can be ignored compared to ( E* ).If (E* ) is also only weakly temperature dependent, A is observed to be a constant.The observed A will in general have a temperature dependence where (E) is notnegligible compared to (E*), or where (E*) is reasonably temperature dependent,most notably in the case of a high degree of quantum tunneling. Where this is thecase, equation (2.18) is often used with A of the form of a constant times some powerof T, inspired mainly by the T" 2 dependence exhibited by the simple model withcross-section given by equation (2.21). Experimental data fitted to equation (2.18)with increased upward curvature at low T is often taken as an indication of reactionfacilitated by quantum tunneling since that corresponds to reaction with reactants ofenergy less than the barrier height; the extra curvature results since the presence oftunneling is obscured at higher temperatures by "normal" (classical) reaction paths.The equations (2.11), (2.12), and (2.16) are rigorously correct for an ideal gas inequilibrium; see any introductory text on reaction dynamics. For the experiments re-ported in this thesis, measurements were performed on CH 4 , C 2 H6 , and N2, whosecompressibility curves, found in many physical chemistry texts, show ideal gas be-haviour to a very good approximation up to the gas pressures used. A few sampleChapter 2. Bimolecular Reaction Kinetics^ 14calculations with functions fitted to these curves show negligible deviation from idealbehaviour for these gases compared to experimental error.Equation (2.11) therefore forms the starting point for the interpretation of the datafrom the //SR measurements. It does require a slight modification for this study. Forthe reactions (1.3) and (1.4):Mu + HR MuH R^ (2.23)equation (2.11) takes the formR = k[Mu][HR]^ (2.24)without any modifications. Now, the nature of the time-differential fiSR technique, inwhich muons enter the reaction vessel one at a time, dictates that the concentrationof Mu cannot be defined in the normal manner; this is dealt with in Chapter 5. Moreconsequentially, the number of muons , entering the reaction vessel in the course of anexperimental run dictate the maximum number of Mu atoms created in the run. Thenumber of muons per run is negligible compared to the number of reactant moleculesin a very small volume, therefore [HR] [Mu]. The result is that [HR] can be treatedas a constant in equation (2.24) to a very good approximation. It can therefore beabsorbed into the constant k giving the pseudo-first order equationR = V[Mu]^ (2.25)where k' = k[HR]. Recalling the definition of R in equation (2.10) and solving for [Mu]by integration,[Mu] = [mu] o e — k''^(2.26)where [Mu] o is the "concentration" of Mu at time zero. Note that, since H atoms arevery reactive and so tend to exist in very low concentrations, the same approximationapplies to kinetics studies done on the H, D, and T variants of the reaction (2.24).Chapter 2. Bimolecular Reaction Kinetics^ 15The parameters k' for the title reactions at various reaction temperatures are gottenfrom plots of the p,SR data, obtained in the experiments here reported. As detailedin Chapter 6, k' for each of the title reactions (there referred to as "k"), plotted as afunction of T, give very good fits to equation (2.18) with A constant.Chapter 3Electronic Structure3.1 IntroductionThe recent progress of the best-known chemical reaction rate theories provides themajor motivation for this thesis work. These theories depend on lower-level theoriesof molecular structure, which are reviewed in this chapter. This chapter thus servesas essentially optional background for the next, and is included for completeness sinceprogress in both has been highly correlated. Comprehensive reviews of virtually everytopic discussed in this chapter and the next will be found in the recently publishedNATO Advanced Studies Institute workshop proceedings, "Methods in ComputationalMolecular Physics," reference [1].For the problem of calculation of chemical reaction rates from theory it is possibleto define three main approaches. The rigorous statistical approach computes the rate ofreaction as a relaxation of the reacting system towards equilibrium, using the theoremsof non-equilibrium statistical mechanics. Other approaches consider the detailed inter-actions of reacting species, using the concept of the potential energy surface. Theseinclude transition state theory and generalizations of collision theory. For gas-phaserates, these have been more popular in the recent literature, and have undergone rapidtheoretical development. As well, transition state and collision theory results are oftenmore easily understood by experimental chemists. Only the latter two approaches willbe considered here.16Chapter 3. Electronic Structure^ 17The statistical approach may be more useful for condensed phase systems, in whichthe interactions of molecules are more complicated than for gas-phase systems. It is tobe hoped, however, that progress in gas-phase reaction rate theory can provide greaterunderstanding of the dynamics of condensed phase reactions, which are perhaps ofmore practical interest than gas-phase reactions. The relationship between gas-phaseand solution reaction dynamics is not well established but is under study [49-52].Following Johnson [48] a hierarchy for the computation of reaction rates can beestablished for theories which consider molecular level interactions:1. Calculate a potential energy surface for the reacting system.(a) Calculate the "potential energy" for a series of nuclear positions.(b) Join the points thus calculated by a function representing the molecularinteractions. The result is a multidimensional function of potential energy,a potential energy surface (PES).2. Calculate the reaction probability on the PES for all initial conditions likely tocontribute to the rate. These are the initial internal states of the reacting species.3. Average over an appropriate set of initial conditions. This is generally done usingthe Boltzmann distribution given by equation (2.16).Reaction rate theories, also applicable to atomic processes such as elastic scattering,are more difficult to apply for chemical reactions because chemical reactions simplyrepresent more complicated processes. Particularly for chemical reactions of moleculesof non-trivial size, each of the steps of the hierarchy is itself a rather complex problem.Consequently, approximations appear at each step. The interpretation of theoreticalrate data is then complicated by the need to consider the approximations made at eachlevel. The various approximations used to construct PES's are briefly reviewed in thisChapter 3. Electronic Structure^ 18chapter; those resulting from the assumptions of reaction rate theories using PES's arereviewed in the next.3.2 Potential Energy Surfaces in Reaction Rate TheoryFor TST and collision theory, it is conventional to first construct a so-called potentialenergy surface. A potential energy surface (PES) is actually a multidimensional plotof the total electronic (potential and kinetic) energy plus the electrostatic energy dueto the nuclei, as a function of the positions of the nuclei in all reacting molecules.Such a construction necessarily assumes the validity of the famous Born-Oppenheimerapproximation which holds that the electronic motion is separable from the motionof the nuclei. Thus nuclear motion is assumed electronically adiabatic; reactions areassumed to proceed without change in overall electronic state. Because of the approxi-mation, the surface is invariant to isotopic substitution, which allows isotopic variantsof chemical reactions to be easily compared.The validity of the approximation is dependent on the fact that the electrons aremuch lighter than the nuclei and so adjust their motion essentially instantaneouslyto that of the nuclei. The approximation has been found to hold very well for reac-tions involving molecules in their ground electronic states. In fact the calculations ofSchatz [36] for the rate of reaction (1.8), a chemical reaction for which the effects ofbreakdown of this approximation would be expected to be among the most serious,show virtually exact agreement with accurately measured experimental results [37].Within this approximation the potential energy surface is an effective force field fornuclear motion.The majority of chemists are familiar with the concept of a potential energy surface.For the reaction of an atom and a diatomic molecule such a surface allows for easyChapter 3. Electronic Structure^ 19visualization of the molecular interactions. This surface is a function of three relativenuclear positions. Most commonly, such a potential energy surface is represented as aseries of sheets, each with the same two nuclear positions as independent variables, theother being replaced by an angle variable 8. The dependent variable, energy, is thenplotted as a series of contour lines on each sheet. Figure 3.1 shows one sheet for thegeneral atom-diatom reactionAB+C--4 A+BC. (3.27)The part of the surface at the upper left, where A and B are close to each otherand far from C, represents the reactants in the reaction. Similarly the part at thelower right represents the products. (The opposite convention is also used.) ThisFigure 3.1: Potential energy surface for reaction (3.27) for a given angle ABC; takenfrom [53].Chapter 3. Electronic Structure^ 20easy visualization possible for an atom-diatom reaction is lost for more complicatedreactions, though insight can be gained by examining selected projections (i.e. cross-sections) of the surface.3.3 Molecular Electronic StructureThe problem of computing the electronic energy for a given point on the potentialenergy surface that is, a given set of relative nuclear positions, is itself a major compu-tational problem. Probably, it has represented the greatest bottleneck in the computa-tion of accurate reaction rates. The theory of electronic structure has undergone rapiddevelopment since the invention in the 1960's of the integrated circuit chip. Rapidimprovements in computer power and speed have greatly increased the feasibility oftruly accurate electronic structure calculations. Theoretical advances have also beenmade, most notably algorithmic developments designed to optimize the use of computerresources.The standard form of the complete nonrelativistic electronic Hamiltonian for amolecule or complex, with nuclear motion separated but including the electrostaticenergy due to the nuclei, isHel = E Za E 1 + Za,Z0(3.28)2 i a ,i nyi j,iO3 red co,a00 rcoin atomic units: h = 1, m e = 1, e = 1. The distance r ab is that between particles aand b. The Roman indices refer to the electrons, the Greek indices to the nuclei. Theenergies of the possible states are the constant eigenvalues E satisfying the equationHey = EV)^ (3.29)together with appropriate boundary conditions. The "potential energy" mapped by aPES is, at a given point, the lowest eigenvalue of this Hamiltonian.Chapter 3. Electronic Structure^ 21Equation (3.29) is the starting point of electronic structure theory. Note that, withinthe Born-Oppenheimer approximation, the last term of equation (3.28) has only theeffect of shifting the total energy by a constant, and so need not be considered in thecalculation, and can be simply added at the end. For systems of more than one electronmore levels of approximation are necessary to actually compute an electronic statefunction, since equation (3.29) can not be solved in general without them. This equationis a special case of the many-body problem of physics, whose many applications includeelementary particle interactions and the propagation of vibrations in solid matter.Conventionally, an electronic structure calculation is begun by separating the mo-tions of the individual electrons, giving a set of n one-electron Hamiltonians for ann-electron system. The one-electron state functions then must be combined to formthe total electronic state function in such a way as to take account of the factsthat electrons are indistinguishable and obey the Pauli exclusion principle. The totalelectronic state function for an atom or molecule can be most simply represented as alinear combination of Slater determinants of the form= 101(1) 02(2) • • On-i(n — 1) On(n)i (3.30)where the O's are the one-electron state functions of the n total electrons and all therows of the determinant are formed from the one shown by permuting the electronindices (parenthesized) in all possible combinations. The 0, are products of spatialfunctions with the two possible electron spin functions. The existence of electron spingives 2n possible Slater determinants for an n-electron configuration (e.g. a 1s1p atomicconfiguration). A linear combination of Slater determinants having the proper electronexchange symmetry is called a configuration state function (CSF).Basic to the theory of electronic state functions formed from CSF's is the variationtheorem, which holds [45] that the true ground state energy for any quantum systemChapter 3. Electronic Structure^ 22with a time-independent Hamiltonian lies below that of any calculated approximatestate function th approx whose energy is computed as the expectation value:( y''approx I k Oapprox ) ( iPapprox ikapprox )which is self-evident when it is realized that any approximate state function obeyingthe correct boundary conditions of the physical system is a linear combination of theground state plus states of higher E. This theorem leads to an approximate methodfor calculation of ground state functions: try a function with parameters which canbe varied, and minimize the function with respect to the expectation value (Eapprox )With the Born-Oppenheimer approximation, the theorem applies to equation (3.29).This provides a guide to creating algorithms for the calculation of electronic structure:systematically minimize ( EapprOX •When calculating equilibrium properties of molecules, the same theorem states thatthe equilibrium nuclear positions are those which give the lowest "potential energy"for nuclear motion, as defined above. To find this set of relative nuclear positions itis necessary to follow an optimization procedure which systematically varies the setpositions and does a series of electronic structure calculations, one at each set. Eachsuch calculation constitutes a point on a PES for the molecule. This PES can be usedto calculate the rovibrational state function(s) for the molecule by solving or estimat-ing the solutions of the nuclear motion Hamiltonian—the part of the Hamiltonian ofthe molecule omitted from Hei . The rovibrational state functions must be found forthe use of transition state theory, and are also of interest because they determine thespectroscopic constants for a molecule. Note that the rovibrational motion is not iso-topically invariant, which is of great importance to comparison of reaction dynamics.The theory of nuclear motion in molecules has lead to development of general compu-tational methods [54,55], including an adaptation of the SCF method [56] for electrons,(Eapprox) (3.31)Chapter 3. Electronic Structure^ 23described in the this section. Alternatively, or in combination, the use of analytic prop-erties of the electronic state function such as that expressed by the Hellmann-Feynmantheorem (see articles in reference [57]), can lead to these solutions. For electronic statefunctions from the various theories incorporating CSF's, this theory is particularly welldeveloped [58,59], but is not a "solved problem."3.3.1 SCF State FunctionsThere are many ways to form the 0, of the CSF depending on the accuracy desiredfor the calculation. If the CSF is to be used without modifications, it has been shown(references listed in [45]) that the best possible state function, in the sense of havingthe most accurate calculated energy, is given by the Hartree-Fock self-consistent field(SCF) theory. In this theory, the contribution to the one-electron Hamiltonian for anindividual electron due to the other electrons is replaced by a continuous charge dis-tribution. The continuous charge distribution due to electron j is just the probabilitydensity function, that is, p3 = ch . . This represents an average as opposed to instanta-neous interaction, and has the effect of raising the energy of the calculated electronicstate function as compared to the true value, because electron collisions, which allowelectronic energy transfer, are neglected. This "mean-field" approximation leads to aHamiltonian separable in the individual electron motions.Replacing the third term of the Hamiltonian in equation (3.28) with the contribu-tions due to the continuous charge distributions and substituting into equation (3.29),a series of n one-electron eigenvalue problems= Ei^ (3.32)Chapter 3. Electronic Structure^ 24is obtained, where=+^— ki]) of2^c, r i,^joi (3.33)with the Coulomb (J3 ) and exchange (f(j ) operators defined as^ij Oi = ck i j dr 1°^12^(3.34)3rteand= f dr (k2^(3.35)7-23with integration f dr over all space including the spin part of the 03 . The Fock operatorP, represents an effective one-electron Hamiltonian. It is dependent on its own eigen-functions and so must be calculated by a series of successive approximations, which aredone in such a way as to minimize the total energy, following the variational principle.The 0, are usually constrained to be orthogonal to each other. The resulting totalenergy is.E E Ei^E (Jj,— Ki,)^(3.36)where the Coulomb integral Jij is related to the corresponding operator ij byJij = f dr 40:^ (3.37)and similarly for the exchange integral Ku .The one-electron state functions furnished by SCF theory are easily interpretablein terms of electron interactions, and the theory gives an intuitively pleasing view ofthe total electronic state function. The spatial part of the one-electron state functionsobtained are the orbitals of atomic (AO) and molecular orbital (MO) theory. Togetherwith the spin-dependent parts, they are called spin-orbitals.The orbitals are frequently used in qualitative arguments to rationalize molecularstructure by synthetic chemists. The CSF obtained generally includes [601 —99% of theChapter 3. Electronic Structure^ 25total energy. However, as has been frequently noted (e.g. [45]), the error in the energyintroduced by the neglect of instantaneous electron interactions is comparable to theenergy change in chemical reactions.The difference between the energy of the SCF state function and the exact nonrel-ativistic energy is called the correlation energy after the term electron correlation forthe neglected instantaneous interactions. Therefore it is necessary to go beyond theSCF procedure to obtain an accurate potential energy surface. This is to be intuitivelyexpected since the electron interactions change drastically in the course of a chemicalreaction.For molecules, the spin-orbitals are almost universally built up as linear combina-tions of atomic orbitals centered on the various atoms in the molecule; this approach,abbreviated LCAO, provides molecular one-electron state functions with a very clearphysical interpretation in terms of the orbitals of the constituent atoms. As well thisapproach is relatively easy to implement as a computer algorithm. First proposed byRoothan in 1951 [45], this theory treats the atomic orbitals as basis functions of avector space. The molecular orbitals are then formed as projections onto this basis sat-isfying the one-electron eigenvalue problems of equation (3.32). The only disadvantage,greatly outweighed by the ease of computation, is that some accuracy is lost, becausethe basis set is incomplete. This can be alleviated by judicious choice of a basis set.Since the set of atom-centered functions used, called the basis set, is finite, thereforeincomplete, any functions which uniformly converge to zero at infinity, whether centeredon an atom or not, are legitimate basis functions, so it is advantageous to use thecomputationally most expedient functions. The functions actually used for the atomicorbitals are linear combinations of three-dimensional Gaussian functions (oc e -"2.., withc a positive constant) fitted by a least-squares procedure to Slater-type orbitals (STO's),which resemble the actual atomic orbitals. The resulting computation is comparablyChapter 3. Electronic Structure^ 26easy because "the product of two Gaussians is a Gaussian". Some calculations addfunctions not centered on atoms, but then the simple LCAO interpretation is lost.When the molecular orbitals are expanded in the basis functions x s0, = E cis x s^ (3.38)there results, inserting 0, into equation (3.32), the Roothan equationE ci sfrix s E E cisxs^ (3.39)or, premultiplying by Xr and taking expectation values, the matrix formFC ESC.^ (3.40)Without affecting the total energy computed [45] the Xs can be redefined to be orthog-onal to each other resulting inF'C'^EC'^ (3.41)the standard eigenvalue equation. The algorithm to find the Ei is well established andeven hardware-encoded in some computers.A basis set of as many AO's (with spin) as there are electrons is a minimal basisset. For especially large molecules, it may be desirable to constrain the lower energy(core) orbitals to be unperturbed from the initial AO's; this is the valence electronapproximation. In the other direction, when a larger-than-minimal basis set is used, anumber of orbitals beyond the "occupied" orbitals used in the CSF are obtained, calledvirtual orbitals. These are omitted from the potential energy terms of the Fock operatorbut are generally still calculated. Roughly speaking, the virtual orbitals represent one-electron "excited states"; the CSF with one or more 0, replaced by virtual orbitalsthen represents an excited electronic state of the molecule. This concept is the startingpoint of the configuration interaction (CI) theory of electron correlation.Chapter 3. Electronic Structure^ 273.3.2 Post -Hartree -Fock ProceduresAlthough Hartree-Fock CSF state functions are often useful in rationalizing molecularstructure and other static concepts, they are, as noted above, not generally of sufficientaccuracy for a potential energy surface of use in predicting chemical reaction rates. Forexample, for the reaction (1.1), calculation [61] with a DZ-SCF basis set predicted abarrier height of —34 kcal/mol compared to the experimental value [26] of —12 kcal/mol.This theoretical result was improved by post-Hartree-Fock theory (CI) to —17 kcal/mol.The three best-known procedures which start with a CSF state function and sys-tematically convert it to an improved state function including electron correlation arebriefly discussed here:1. Configuration Interaction or Mixing (CI or CM)2. Multi-Configuration SCF (MCSCF)3. Various perturbational approaches, in particular the Moller-Plesset (MP) Pertur-bation TheoryCombinations of these are possible, especially the first two [62].Much of the recent theoretical work in electronic structure (see [57]) has focused onthe use of the mathematical properties of CSF's to obtain unambiguous algorithms toobtain such properties as multipole moments and energy derivatives analytically. Theenergy derivatives are particularly useful in obtaining a potential energy surface withaccurate local curvature between fitted points. The multipole moments are required forthe accurate calculation of rovibrational molecular states. Analytic energy derivativesare now available in standard computational software for CI, MCSCF, and MP theory.The CI theory is based on the idea of forming new CSF's by "promotion" of one ormore electrons of the Hartree-Fock CSF from filled orbitals to virtual orbitals, calledChapter 3. Electronic Structure^ 28excitation. It has been shown that with a complete basis set, all possible CSF's forma complete vector space, whose properties can be used to algorithmic advantage. Thetrue state function b is a linear combination of the possible CSF's (I) /.Eci (D i. (3.42)including the Hartree-Fock calculated CSF and the C1 are found by substituting intoequation (3.28) and solving the resulting equations to minimize the total energy. Theexpansion is (in the nonrelativistic theory) exact for a complete basis set.To get reasonably accurate results, the initial basis set must be of reasonable sizesince this dictates the number of 1, available. Generally, for molecules the size of CH 4or C 2H 6 , only excitations of the highest energy (valence) orbitals are included in the(I) / ; this is called the frozen core approximation. The effect of the expansion on theHartree-Fock CSF, as succinctly explained in reference [63], is to place the "excited"electrons into a linear combination of so-called polarized orbital pairs which allows theelectrons to " 'avoid'" each other.In principle, in the CSF basis, the possible vectors C (= {C 1 }) are eigenvectorsof H. Since the forming of the matrix elements (Cf/((13.) is a time and computermemory-consuming process, a method has been developed, called direct CI, allowing thecomputation of the lowest energy eigenvector C without forming the full Hamiltonian.A detailed analysis is to be found in reference [64]. Generally, only a small subset of thepossible CSF's are included in the expansion. Much work has been done in determiningwhich excitations can be omitted with the least damage to the calculation's accuracy.Another advance has been development of alternate schemes to Hartree-Fock to getthe initial CSF, since there is no reason why the SCF orbitals will necessarily be thecomputationally most expedient in the CI calculation. Similar techniques are possiblefor other theories of electron correlation.Chapter 3. Electronic Structure^ 29The MCSCF theory, often used in combination with CI, is an expansion of the truestate function in a series of CSF's=EA.:Di (3.43)but in this case the CSF's 4. 1 are formed using different orbitals from each other, thatis, each CSF is separately optimized. So the resulting approximate state function isa combination of several electron configurations. As with CI, the state function isdetermined by iterations to minimize the total energy. In this case the chief challengesare in selecting the configurations making the greatest contribution to the true statefunction and in constraining the 1. 1 so that they do not collapse into one another; seereference [65]. As with CI, algorithms have been developed to obtain MCSCF statefunctions more easily (see, for example, [66]). MCSCF combined with CI is calledmulti-reference CI (MRCI).As described in reference [63], with variational theories such as CI and MCSCF,when an incomplete basis set is used, there is a problem of size inconsistency, meaningthat the average error for a given level of approximation will not produce the samefractional total energy error for different molecules. This is a problem if a potentialenergy surface describing a chemical reaction is to be constructed, since the parts of thesurface with the reactants near each other (the "transition state") will have a differentcorrelation energy error than the reactants, leading to extra error in the calculatedreaction rates.This problem does not occur for perturbational approaches such as the MP theory.MP theory treats the sum of the Fock operators E t F, as a reference Hamiltonian andHei — E, 1, as a perturbation on the reference Hamiltonian's eigenfunctions, leading bystandard methods (Rayleigh-Schriidinger perturbation theory [63]) to an infinite seriesof CSF's with, generally, diminishing contributions to the true state function as theChapter 3. Electronic Structure^ 30series is continued. Also a series in energy terms is obtained. The CSF series evaluatedup to nth order furnishes terms in the energy correction series up to order 2n -I- 1. Theseries are truncated after an arbitrary number of terms but within this approximationthe CSF's included in the expansion are non-arbitrary, as opposed to usual applicationof CI, in which informed guesses provide the list of CSF's to use. MP also gives aguide to which CSF's are most important in the expansion. For example, the secondorder correction to the single-CSF state function given by MP theory consists of doubleexcitations only. MP, up to fourth-order corrections in the energy, is the correlationtheory routinely used for molecules the size of CH 4 (see [39,67,68]).Recently, "benchmark" full CI calculations, in which every possible CSF arisingfrom a "moderate-sized" basis set, have set a standard for electron correlation calcula-tions. The full CI calculations, on H 2 O, CH2, N2 and F- , are critiqued in reference [60]in comparison with currently more widely applied MP calculations. It was found thatperturbational calculations, even through fourth order, failed to give an adequate de-scription of bond stretching in H 2 O, in which the error in the energy increased by anorder of magnitude on doubling the O-H bond length from equilibrium. The error isattributed to the use of a single reference function; the authors conclude that accuratedescriptions of dynamical processes will require the use of several configurations as thestarting point for electron correlation. Such calculations have not yet been performedon CH 4 to the knowledge of the present author.3.3.3 Other Approaches to Electronic State FunctionsAb initio molecular orbital theory coupled with the use of electron correlation theoriessuch as those described previously has been the subject of the majority of recent theo-retical developments in electronic structure theory, but other approaches have receivedsome attention.Chapter 3. Electronic Structure^ 31Semi-empirical state functions are normally calculated using the SCF-MO theorywith the valence-electron approximation, but with some integrals J23 and Kii neglectedand some others obtained by fitting state functions to experimental data. The fa-mous Hiickel 7 MO theory is the ultimate "semi-empirical" theory, using a total oftwo parameters to describe the state functions of every 7r-conjugated hydrocarbon sys-tem. Despite its deep approximations it is still used today in descriptions of electronstructure, often with useful qualitative accuracy. On the more accurate side, MNDO(modified neglect of differential overlap) and its successors such as AM1 and PI\43, fitthe integrals not neglected in such a way as to reproduce experimental heats of com-bustion. These theories are most useful for systems similar to the systems used toobtain the parameters, but for systems the size of CH 4 they have been superseded byab initio methods. Semi-empirical theory can be applied to much larger molecules thanis possible with ab initio calculations. Also, it can be expected that for many systemssemi-empirical calculations may perform better than ab initio calculations neglectingelectron correlation since the parameterized Ji, and K23 include electron correlation.Semi-empirical theories not using SCF have also been devised.In the valence bond VB theory, the approximate function consists not of MO's but oflinear combinations of atomic orbital product states. For example, a diatomic moleculeAB has its first-order VB state function given byOVB = 2(1 + ,S1B)1/2 [1SA(1)1SB(2) lsA (2)1sB (1)] [a(1)0(2) — a(2)0(1)] (3.44)where the electron numbers are parenthesized, and SAB is the overlap integral (1s A ilsB ).This approximate state function is a linear combination of two Slater determinants ofatomic orbitals 1sA (1)a(1) 1813(2)/3(2) and 1sA (1)0(1) 1sB(2)a(2). This type of statefunction is designed to describe the bonding in terms of electron exchange between theatomic orbitals of individual atoms. Each "exchange" which is allowed gives rise toChapter 3. Electronic Structure^ 32two Slater determinants of AO's. The linear combination is needed to account for theexchange symmetry of electrons.In the generalized VB theory, the atomic orbitals are replaced by variational func-tions and the total energy minimized. It is actually a form of MCSCF [63]. For apolyatomic molecule such as CH 4 the atomic orbitals would be replaced by "hybrid"orbitals, linear combinations of atomic orbitals on a single atom combined in such away as to conform to the geometry of the molecule. The hybridization allows the elec-trons to "avoid" one another and so the GVB theory has electron correlation built in.Because the AO's on different atoms cannot be be constrained to be orthogonal to eachother, GVB state functions are generally more difficult to obtain than MO CSF's.Another well-known theory of electronic structure is density-functional theory invarious forms, which are based on the fact that there is a unique functional relationshipbetween the electron density p and the state function 7/) [45]. The problem of density-functional theory is to find the electron density and also the form of the relationshipbetween p and the energy. The state function need not be calculated.For the theories and methods described in this section, casual examination of the lit-erature will reveal a huge number of acronyms and abbreviations used without explana-tion; in a modern quantum chemistry text such as I.N. Levine's "Quantum Chemistry",4th. ed. [45] will be found the majority of those in common use. An accessible reviewof recent progress is reference [69], and reference [63] gives an overview of physicalinterpretation of the theory.3.4 Constructing Potential Energy SurfacesOnce the "potential energy" has been calculated for a reasonably large series of points, apotential energy surface can be constructed. The electron motion has been "integratedChapter 3. Electronic Structure^ 33out" so the potential energy surface is a function of the relative nuclear positions. Ifthe Born-Oppenheimer approximation was not invoked, the topographical features ofthe potential energy function for the nuclei would be dictated mainly by the analyticalproperties of the electron interactions. For the practical problem of constructing apotential energy surface it is necessary to represent these electron interactions implicitlyby an essentially arbitrary function of the nuclear coordinates. The function is fittedto the points, normally using a standard least-squares procedure, which minimizes theaverage estimated error.The function used is dictated mainly by the intended use of the surface, sincepresently available potential energy surface calculations are of insufficient quality toavoid major trade-offs between accuracy in computing various quantities of interest.This arises partly due to the fact that only a limited number of points can be initiallycalculated due to limited computer resources. For example, the very accurate LSTHsurface for H 11 2 reaction has —300 points calculated. Also problematic are thepotentially rapid changes in energy at short distances, and the aforementioned "sizeinconsistency".For the best available potential energy surface for the H CH 4 abstraction reaction,published by Joseph et al. in 1987 [70], gives an excellent example of the trade-offs in-volved in the construction of a semi-empirical PES. For this surface, no attempt is madeto reproduce the the experimental rates for the substitution reaction (1.5). Used withthe transition state theory of reaction rates, it is in good agreement with experimentfor the abstraction reaction (1.1), over a reasonable temperature range. Rate coeffi-cients calculated with this surface, discussed later, are in very good agreement withexperiment. In contrast, both reactions are intended to be modelled by the surface ofRaff [71] et al., which reproduces the CH 4 and CH 3 experimental harmonic frequenciesquite well. However, it leads to transition state theory computed rate coefficients inChapter 3. Electronic Structure^ 34very poor agreement with experiment.The trade-offs are resolved by the choice of empirical parameters used to fit thesurface; modern potential energy surfaces must be fitted using some empirical param-eters and are therefore semi-empirical in nature even if the points are computed usingab initio theory. An example is the H CH 4 surface [26, 72, 73] of Walch et al. Forthis surface the experimental values of the reaction endoergicity of reaction (1.1) andvibrational frequencies of CH 4 were initially used, with the barrier height adjusted toreproduce experimental rate coefficients after other treatment of the data.It is notable that the first calculated potential energy surface of "chemical accuracy"(<-1 kcal/mol) was computed as early as 1973 [74]. An analytical function later fittedto this surface is the LSTH (Liu-Siegbahn-Truhlar-Horowitz) surface mentioned earlier.It is fully ab initio yet more accurate than any surface ever calculated for a largersystem, because the computational effort for constructing a PES increases very quicklywith the number of electrons. This surface was used in Schatz' accurate calculations [36]for the Mu + H2 reaction (1.8), and remains a benchmark for the quality of new PES's.Only on this surface can the accuracy of rate calculations be reliably attributed to therate theory used. The surface has undergone further development to the even moreaccurate DMBE (double many-body expansion) surface [74].A variety of functions and combinations thereof are in common use for the calcula-tion of potential energy surfaces. A few well-known ones, described in Johnston's 1965book [48] "Gas Phase Reaction Rate Theory", are still in routine use. The famousLennard-Jones 6-12 potential12 ^(  )61V(R) = 46[( 7R ) — (3.45)with a• obtained from experimental data or ab initio calculations, is found to reproducewell the interactions between two atoms at distance R in many cases. Also well knownChapter 3. Electronic Structure^ 35is the Morse functionV(R) = D e (e -"r — 2e -13r) (3.46)where D e is the dissociation energy from the energy minimum to infinity, /3 a parameter,and r is R—R e with Re the equilibrium (minimum energy) distance. Where this functionaccurately represents the potential between two atoms in an overall singlet electronicstate (total electron spin zero), the Sato-Morse potentialV(R) = D e (e -"r + 2e -13r) (3.47)may accurately represent the potential for the case of a triplet (total electron spin one).The LEPS potential energy theory uses for singlets and triplets respectively the energyexpressionsQES— ^a1 + AQ - ET=(3.48)(3.49)which are identical to the energy expressions obtained for VB state functions of diatomicmolecules from equation (3.44), where Q corresponds to the Coulomb integral 1r121JAB = (1s A (1)18 13 (2)a to the exchange integralKAB = (1S A (1)1SB (2)and A to the overlap integral1sA (1)1sB(2)),^(3.50)1sB(1)18A(2)),^(3.51)SAB = (13A 118B).^ (3.52)However these parameters are in general fit empirically rather than calculated.Calculated surfaces for polyatomic molecules may use modifications of these func-tions to represent the interactions, including generalizations to three- and more-bodyChapter 3. Electronic Structure^ 36interactions, which are needed [38] to properly represent the surface features for shortrelative nuclear distances. As well, it is common to use an average of singlet and tripletterms for pairwise interactions between nuclei since usually a spin-independent surfaceis desired, otherwise at least two surfaces would be needed to compute rates for a givenreaction. For example, the surface of Joseph et al. uses an average of functions (3.46)and (3.47) for pairwise interactions. This type of averaging has lead in the past [38] tospurious features on PES's arising from the difference in the properties for singlet andtriplet potentials, particularly false local minima near the point of closest approach ofreactants, and was not initially recognized as an artifact of the calculations. ModernPES's overcome this problem.Chapter 4Reaction Rates from Theory4.1 IntroductionIn this chapter are reviewed the best-known chemical reaction rate theories employingPES's. Many such theories have been proposed, and only a few are discussed here. Itis worth noting that many theories, such as transition state theory, are developed byintroducing rather restrictive assumptions, then trying to correct for the error producedby the assumptions. At the other extreme, development proceeds from formally exactequations to suitable approximations devised in such a way as to try to retain the rigourof the parent theory; an example is variational coupled channels theory.For all types of theories, many general types of approximations have been tried;most notably, restricting the degrees of freedom of the reactants leads to simpler equa-tions in every theory. For example, many atom-diatom reactions have been treated byrestricting the reactants to collinearity. Also, coordinate transformations [48] of varioustypes can simplify calculations.Transition state theory (TST), and its many variants, has long been the most popu-lar reaction rate theory. First proposed in 1935 by Eyring, Evans, and M. Polanyi [53].the theory combines equilibrium statistical mechanics with potential energy surfaces.A review, with definitions of the various versions of the theory, is to be found in ref-erence [75]. Conventional TST (CTST) focuses on the properties of the transitionstate (TS) in a given reaction. The TS can be defined as a "saddle point" or "col"37Chapter 4. Reaction Rates from Theory^ 38on the minimum (i.e. least extreme) energy path on a PES between the reactants andproducts.From the TS can be defined the vibrationally adiabatic (VA) reaction barrier E'asEvA = 17' +^—^ (4.53)with V b the (classical) PES barrier height and E, denoting the sum of energy in eachvibrational mode (with rotational states averaged), t. the TS, and i the reactants. Thisquantity denotes the actual energy barrier for reaction where the reactants evolve to theTS with no change of vibrational state. As normally seen in the literature, this quantityis usually quoted as the vibrationally adiabatic energy barrier with reactants and TSin their ground vibrational states. This quantity, denoted V G here, then correspondsto the sums E E1, counting zero-point energies only.The energy V G should give a better estimate of Ea than V b . It is possible for V Gto be less than V b . The two can in fact be significantly different, and the use of V G inany rate theory partially takes account of the quantization of internal states of reactingmolecules. It is certainly a better estimate of the true average energy barrier than isV b .For isotopic variants of a reaction, when the vibrational energy is included withthe electronic energy given by the PES, the energy profile of the reaction is no longerisotopically invariant. The V G differences for isotopic variants can be expected to givea reasonable estimate of Ea differences for given reaction. For example, comparingreactions of two isotopic variants A a and Ab of atom A:vG(A a.) — vG(A b )^—^ (4.54)v,a^vbso the difference in V G is just the difference in vibrational energy of the respectivetransition states, certainly an intuitively reasonable estimate of the Ea difference. For aChapter 4. Reaction Rates from Theory^ 39reaction with significant populations of excited reactants, such as the title reaction (1.3),the agreement of isotopic V G and Ea differences may not be as good since these reactiondynamics are more complicated than for reactions of only ground states.Comparing the D and H atom variants of reaction reaction (1.3), the VG value forthe D atom variant given by the ab initio surface of Walch et al., [26] is 0.93 kcal/mollower than that for H. This is reasonably close to their theoretical Ea difference (usingCTST at 500 K) of 0.8 kcal/mol, and consistent as well with an experimental value ofuncertain accuracy [25], (mentioned in Chapter 1), of 0.6 kcal/mol.Following a variational principle analogous to that for stable molecules, the TS canbe gotten using variational optimization of geometry to get a stationary value of (E),starting with a geometry near the expected TS. That is, the TS is located at the highestenergy point on a path requiring the lowest energy to transverse classically. This path,called the minimum energy path, is for the PES of Figure 3.1, the dotted path.4.2 Transition State TheoryEssentially, this theory assumes the following [53]:1. Once the path s has been followed to the col, the reactants do not turn back.This is called the no-recrossing rule.2. The reactant molecules obey the Boltzmann distribution of equation (2.16). Fur-thermore, equilibrium theory can be used to find the concentration of the TS inrelation to the reactants.3. The motion along s is separable from all others. Formally this results in animaginary frequency for this unbound motion.4. Motion along s is classical.k = K1  q:  -EDIRTh. 1-12 qi(4.55)Chapter 4. Reaction Rates from Theory^ 40Clearly, this theory will perform best for reactions with high reaction barriers withthe product side steeper than the reactant side, like the "cliff .' on the left side ofFigure 4.2. Compare the shape of the barriers Vb and VG from a PES for the H atomvariant of the title reaction (1.1). For a barrier of this shape the TS will be unlikely to goback to reactants. Also, when the TS is high in energy it will be formed rarely enoughthat it will be unlikely to exceed the calculated equilibrium value of concentration.These assumptions lead to a rate coefficientFigure 4.2: Left: potential energy "cliff"; ideal system for TST. Right: profile of clas-sical reaction barrier (called V b in text; here called VmEp , left scale) and vibrationallyadiabatic reaction barrier (V G , right scale) from a recent semi-empirical potential en-ergy surface for the title reaction (1.1); reproduced from [70].Chapter 4. Reaction Rates from Theory^ 41where q t is the partition function of the TS with motion in the s-direction omitted,.E0 the molar value of V G (not the reaction's threshold energy), and q2 the partitionfunctions of the reactant species i. This expression is, within a multiplicative constant,the formal equilibrium constant' of the TS species [53]; that is, the reaction rate isproportional to the concentration of the TS. Furthermore, the resemblance of thisequation to the Arrhenius equation (2.18) is rather striking.It is to be noted that the full PES is not at all needed to do this calculation, only theregions of the reactants and the TS, needed to calculate the partition functions q, andqt . These are weighted sums of the state functions for nuclear motion in the moleculei or t as discussed in Chapter 2. The nuclear state functions can be estimated withoutthe use of a PES for the reaction, by considering the motions of the reactants and TSseparately. Therefore a CTST rate coefficient can be gotten without a full PES.TST has the advantage over other rate theories of having a very simple form forkinetic isotope effects. The only difference in the TST k for isotopic variant reactantsis the difference in q t and the q 2 . It is to be expected that the isotopic ratios fromTST may be more accurate than the absolute rates since many factors cancel in thiscalculation, leaving [48]nr,= v t ,i2 ,—(E3—EDIRTk2for isotopic variants 1, 2 of any of the reactant species i. In the case of the presentstudy, some of the CH 4—Mu TS vibrational frequencies will be significantly higher thanfor CH4—H, and TST would be expected to predict correspondingly high ratios kH/km,,.In comparing Mu and H atom data, the initial partition functions are not changed andso there resultsMukmu^qt _(Et4 u^VRTkH =^e(4.57) (4.56)1 Following the use of the term "rate coefficient", the "equilibrium constant" might better be calledthe "equilibrium activity quotient".Chapter 4. Reaction Rates from Theory^ 42a simpler result still.More sophisticated versions of this theory partly or completely remove some ofthe assumptions of CTST. Generalized TST is TST with an arbitrary location of theTS. Use of an alternate TS can remove some of the inaccuracy introduced by the no-recrossing assumption, by moving the TS to a location where less recrossing occursfor the real system. Variational TST is based on the fact [75] that classically, thek of TST is an upper bound to the true rate coefficient. Therefore, VTST finds thegeneralized TS giving the lowest k. This is not true quantum-mechanically but VTST isstill generally more accurate than CTST. Of course, far more computation is required:an optimization procedure must be followed, with a TST calculation at each point.Many other procedures [53,75] have been tried to improve TST and cancel the errorsresulting from its assumptions. For example, even in the variational form, the no-recrossing assumption still leads to error. Also, the theory fails to account for quantumeffects such as tunneling. As well, various features of the PES such as that for H H2 ,can affect the behaviour of the TS. For example, the conventional TS on a completelysymmetric PES such as that of H + H2 is equally likely to go back to reactants as toproceed to products. It would then be appropriate to multiply expression (4.55) by one-half. On an essentially ad hoc basis, these effects are combined to form a transmissioncoefficient, defined such that the rate coefficient takes the formk =^akT .t  e-Eo /RTh fJ2 qi(4.58)The problem of the quantization of the reaction coordinate .s and its nonseparablilityfrom the other coordinate in the real TS have also been addressed. Reference [76] dis-cusses the requirements for "an exact quantum mechanical" TST. Practical applicationof many such methods is discussed in reference [77].Chapter 4. Reaction Rates from Theory^ 43The review [75] considers some of the ways used to seek an accurate value for n. , andreference [78] explores the requirements for "exact" tunneling corrections by comparingTST calculations to accurate scattering calculations. Frequently used is the Wignerfirst-order tunneling correction, whose accuracy is discussed in reference [79]. Thisgives the transmission coefficient2kw (T) = 1 + 1 (hcoI  )24 kT(4.59))which can improve some TST calculations (ibid.). It is to be noted though that thisexpression is most accurate at high T, where tunneling is less important in determiningthe reaction dynamics than at low T.4.3 Some Forms of Collision TheoryTheories which directly calculate the reaction cross-section of equation (2.15) are es-sentially variants of collision theory. For an elementary bimolecular reaction betweenspecies A and B the rate is given [53] by the expressionR a(v)vNANB^ (4.60)with NA and NB the number density of A and B, and other quantities defined inequation (2.15). Using the definition of k, that equation is obtained.Simple collision theory assumes that A and B are hard spheres, that the potentialenergy between A and B is zero beyond the sum of set radii for A and B, and infiniteat closer range. A "collision" then corresponds to the classical idea of solid objectshitting each other. The theory then assumes that all collisions result in reaction.These assumptions lead [53] to the expressionk (rA + n3)2 (87kT)1,2^(4.61)Chapter 4. Reaction Rates from Theory^ 44for the reaction, with it the reduced mass of the reactants. This agrees with theexpression obtained [48] by CTST for this potential. This is the same as the simple"line of centers" model discussed in Chapter 2, with zero energy threshold. Of course,even with appropriate values of r A and rB , the theory is often not even of qualitativeuse due to its deep approximations. Early work focused on correcting k by multiplyingby a "steric factor" P to correct for such things as the real geometry of molecules, butwas inadequate as a general theory.Trajectory theory calculates cr using a PES, essentially by placing the molecules inclose proximity, letting them go, and observing the result—a "black box" approach.A suitable set of initial conditions is used to devise the initial conditions for a setof trajectory calculations, whose results are then averaged. Frequently, the initialconditions are chosen at random assuming the Boltzmann distribution of initial states.The motion of the nuclei on the PES are generally found by numerical solution ofHamilton's classical equations of motion [80]:dq,(4.62)dt^Op,dp2 _an- (4.63)dt^aq,with H the classical Hamiltonian, q, the generalized coordinates of the system (not thepartition functions), and p, the momenta conjugate to the coordinates q,.The theoretical results are often competitive with those of TST in comparison withexperimental data. For one series of comparisons, of calculations on the isotopic variantsof reaction (1.7), see [81]. Normally, the initial states are quantized if the quantum-mechanically correct states are used to form them; the final states typically must be"binned" by an essentially arbitrary procedure to the correct quantized final states,since the reactants drift under Hamilton's equations to a continuum of states. Thisis quasi-classical trajectory theory. Extensions are possible to further increase theChapter 4. Reaction Rates from Theory^ 45quantum-mechanical correctness of the theory, the most obvious being the additionof zero-point molecular vibrational energies to the PES energies. Truhlar compares avariety of versions of this theory with TST in reference [82].Versions of reactive scattering theory have been widely used in the study of nuclearand particle interactions for many years, and the theory is well developed; a modernadvanced quantum mechanics text will contain much detail on this subject. Scatteringtheory calculates the cross-sections from rigorous quantum-mechanical principles, bycomputing the evolution from initial to final states by the solution of an appropriateversion of the SchrOdinger equation.Normally, the time dependence is factored out [83] by representing the total statefunction as a sum of ingoing and outgoing "waves", but a calculation has been done [84]for the rate of reaction (1.7) using the full time dependent wave packet formalism. The Smatrix, which when combined with appropriate boundary conditions, gives the relativeamplitudes of initial and final states, is obtained by substituting the ingoing/outgoingsum into the time-dependent SchrOdinger equation. The S matrix is essentially [85]an operator for transforming the initial state (reactants) at time —oo to the final state(products) at time oo.Many types of approximate methods exist for rigorous scattering theory, but forchemical reactions a key development has been the development of variational methods.The variational principle for scattering does not follow as simply as those for TST andelectronic energy, but as shown by Ramachandran and Wyatt [86], variational methodssimilar to those used in MO theory can give accurate answers with less computationaleffort than nonvariational methods. In the coupled channels theory, the state functionis expanded in a basis set as in SCF-MO theory, with the expansion coefficients foundby considering the boundary conditions matching initial to final states. This is a veryChapter 4. Reaction Rates from Theory^ 46complicated mathematical theory when applied to molecules, and a common approxi-mation is centrifugal sudden, which essentially [87] assumes rotational adiabaticity inthe reaction. This is the theory applied so successfully to reaction (1.8) by Schatz.4.4 Theoretical Rate Parameters for the Title Reactions4.4.1 Mu + CH 4 --+ MuH CH 3There are no available calculations for the Mu atom isotopic variant of the reac-tion (1.1). However a number of calculations exist for other isotopic variants. Forthe translationally excited ("hot") tritium (T*) atom variant trajectory calculationswere first done [88] using a semi-empirical PES that treated the CH 3 group in CH 4as a single mass point. Later calculations [89], treating all H atoms as equivalent andreactive, gave good agreement for T* above 400 kcal/mol. The semi-empirical surfaceof Joseph et al. [70] is a modification of this surface, and has been used to get themost accurate (albeit semi-empirical) available theoretical rates for reaction (1.7) us-ing VTST; see Figure 4.3, which compares these calculations to ab initio calculations ofSchatz et al. [73]. More recently [40,90], the surface has been used in studies of variousisotopic variants of this reaction, and its reverse, partly to establish benchmarks forapplication of VTST to polyatomic molecules.Ab initio calculations using a large basis set and CI (POL-CI), of Walch [72], ex-tended by Schatz et al. [26,73], on only the region of the reaction barrier (CH 4—H) andreactants (CH 4 ) were used (ibid.) to calculate conventional TST rate coefficients. In thesame reference, TST results for one other ab initio surface and several semi-empiricalones were calculated, and compared to the POL-CI results. These showed the latter tocompare the best with experiment. The Arrhenius fit is compared to the calculations ofJoseph et al. in Figure 4.3. Noting that the curve from the POL-CI calculations is tooChapter 4. Reaction Rates from Theory^ 47Figure 4.3: Arrhenius plot of experimental and theoretical rate coefficients for H CH 4abstraction reaction. Experimental data (solid line) from reference [18]. Dashed linedata from POL-CI CTST calculations of Schatz et al. [73]; dash-dot line from VTSTcalculations of Joseph et al. [70] on a semi-empirical surface. Reproduced from [31].steep, it is to be expected that the POL-CI calculated barrier of 13.5 kcal/mol is toohigh. Later work [26] shows that the (arbitrary) use of a barrier height of 12.5 kcal/molgives results in better agreement with experiment.The same POL-CI geometries were used [26] to calculate rates for a variety ofisotopic variants of reaction 1.1, shown in Table 4.3 for two forms of the (generalized)Arrhenius equation. As discussed in the previous chapter, comparing the H CH 4and D CH 4 results, the Ea difference is 0.8 kcal/mol compared to a VG difference of0.9 kcal/mol from the same reference; the D atom reaction is lower in each quantity,which would be expected since the TS involving D will have lower vibrational energyChapter 4. Reaction Rates from Theory^ 48Table 4.3:^Rate coefficients for H + CH4 and isotopic variants, usingVt = 13.5 kcal/mole. Parameters k, A' and A are in L/(mol s), B' in ln(K), andtemperatures in K. Activation energy at 500 K in kcal/mol. Table from reference  [26].In k = A' + B' ln T - C /71 = ln A(T) - Ea (T)/RTreaction A' B' C' k(500) Ea (500) A(500)H + CH4 10.91 1.974 5640 1.5x10 5 13.2 8.4x10"H + CD 4 13.10 1.692 6462 4.4x104 14.5 9.8x10"D + CH 4 9.21 2.169 5157 2.4x105 12.4 6.3x10"D + CD 4 11.31 1.898 5963 7.2x10 4 13.7 7.2x10"H + CH3 D^H2 10.81 1.951 5646 1.1x10 5 13.2 6.4x10"H + CH3 D --4 HD 11.0 1.781 6422 1.0x104 14.5 2.3x10"D + CH3 D^HD 9.103 2.146 5162 1.8x10 5 12.4 4.8x10"D + CH3 D^D2 9.217 1.986 5926 1.6x104 13.7 1.7x10"H + CH 2 D 2 -4 H2 10.59 1.928 5648 7.9x104 13.1 4.4x10"H + CH 2 D 2^HD 11.92 1.752 6434 2.1x10 4 14.5 4.6x10"D + CH 2 D 2 -+ DH 8.886 2.123 5167 1.3x10 5 12.4 3.2x10"D + CH 2 D 2 -f D2 10.14 1.958 5936 3.4x10 4 13.7 3.5x10 1°H + CHD 3^H2 10.10 1.903 5655 4.1 x 104 13.1 2.2 x10"H + CHD 3 -> HD 12.56 1.723 6446 3.2 x 10 4 14.5 7.1 x 10"D + CHD 3 --* HD 8.382 2.100 5169 6.6x104 12.4 1.7x10"D + CHD 3 -* D2 10.78 1.929 5948 5.3x10 4 13.7 5.3x10"due to the higher mass of the D atom compared to H. This is a good example of akinetic isotope effect resulting from the zero-point energy shift in the TS. Note thatthe D atom relative rate increase may be partially offset by its less ready tunneling,but this effect does not appear strong. On the basis of these results the Mu atomreaction TS might be expected to be raised by several kcal/mol, but it is important torecognize that this figure assumes that the TS is unchanged in location from that inthe H atom reaction. A significant change in the location of the TS for the Mu atomreaction could have other effects on the Ea depending on the detailed features of thePES and vibrational energy in the TS involving Mu.Chapter 4. Reaction Rates from Theory^ 49Figure 4.4: Diagram of TS for the H^CH4 reaction. Distances are in A, angles indegrees, and are for two different ab initio calculations. Reproduced from reference [39].Most recent ab initio CTST calculations [39] use only TS and reactant geometriesoptimized by SCF with MP theory to fourth order. The results are almost as good as thePOL-CI ones with a corrected barrier height, but k is systematically underestimated.This is attributed to the too-large barrier height of 13.0 kcal/mol. The geometry of theTS, shown in Figure 4.4, is similar to that calculated by the POL-CI calculations andthe geometry of a TS found [40] with VTST on the surface of Joseph et al.From the reaction profile in Figure 4.2, it is clear that this reaction is similar in itsreaction profile to the ideal "cliff" system; therefore TST should be expected to workwell, as the available calculations seem to indicate.4.4.2 Mu + C 2 1-16 MuH CJ1 5For the corresponding C 2 H6 reaction, calculations of the same quality as for CH 4 donot exist. As stated in reference [44], "nothing is known" about the PES for H C 2 H6 .The only recent calculation of any kind considering this reaction is a 1991 article [41],which gives the results of calculations using SCF/(second-order MP) for the reactants,TS, and products. The theory used is the "curve crossing model" which [91] essentiallycalculates the energy along the reaction coordinate from the overlap between the stateChapter 4. Reaction Rates from Theory^ 50functions of reactants, products, and TS along the reaction coordinate, giving a reactionbarrier. The vibrationally adiabatic barrier, which the authors called the "activationenergy", is computed to be 22.9 kcal/mol, but the relationship between this energy andthe experimentally measured Ea , though commented upon, is not given.This study also treated the corresponding methane reaction. Their barrier for thecorresponding reaction of CH4 is 25.6 kcal/mol, slightly lower than that given by thePES's of Walch et al. and Joseph et al. discussed above (both --26.5 kcal/mol). Thebarrier difference for these two reactions, 2.7 kcal/mol, is reasonably close to the Eadifference obtained from the experimental values of reference [6].Given the similarities in the electronic structures of CH4 and C 2H 6 , it is likely thatthe salient features of their respective PES's with respect to reaction with H (or Mu)are similar. Therefore, isotopic substitution H —+ Mu is likely to have similar effects onthe reaction dynamics, and therefore the bulk kinetics, of both reactions.Chapter 5Experimental Setup5.1 Positive Muons and //SRThe //SR technique is possible because of the parity-violating (chiral) nature of thedecay of the p+ parent, the positive pion 7r+:7r +^+ + (5.64)which arises because the neutrino v can have only negative helicity (the maximumspin projection opposite the direction of momentum). By conservation of angularmomentum the t.t+ (s = 1/2) is then also required to have negative helicity whenproduced from such a decay. This fact allows scientists to obtain nearly 100% polarizedbeams of and also /./- by the analogous 7r - decay, in modern particle accelerators.The muon decays with a mean lifetime of 2.2 /is:12+^e+^ve^(5.65)and the helicity of the neutrinos again leads to a chiral decay, resulting in a preferred di-rection of positron motion along the decaying muon's spin, with the number of positronsemitted in the angle patternN,(0) a (1 + A cos 0)^ (5.66)where 6 is the angle between a given direction and the maximum projection of the muonspin, and A is called the decay asymmetry. The average of A for emitted positrons of51Chapter 5. Experimental Setup^ 52all possible energies is 1/3. In experimental practice, the detected A is not 1/3, and isalways measured empirically.For an initially 100% polarized muon beam, the excess spin in the direction oppositethe initial motion with respect to the decaying pion constitutes left-handed polarization,when the muons are treated as an ensemble. The ergodic principle allows the muons inthe beam, an ensemble in time, to be treated the same as an ensemble in space. Theeffect is to treat the muons in a beam over a period of time as if they were a. single,macroscopic pulse of particles.When a beam of energetic (--MeV) ft+ enters an experimental target such as thegases under study here, these muons undergo [27] Bethe-Bloch ionization of the stoppingmedium, falling to an energy in the —100 keV [30] range for gases. This process does notaffect the muon polarization. The regime of slowing from --NO keV to thermal energydoes affect the polarization, through a repeated process of cyclic charge exchange:,a+ e- 4-4 Mu (5.67)with electrons from the molecules of the stopping medium. This process continuesdown to muon energy -.40 eV depending on the gas (ibid.). Assuming an initially 100%polarized muon beam (all ,u+ in state la„)) two spin states of Mu can be formed, with themuon and electron spins parallel lawa,), or antiparallel ja il3e ); in the antiparallel state,as discussed below, the muon spin is affected by the hyperfine interaction due to theelectron spin. In a dilute gas, residence times of a muon in a neutral (Mu) environmentmay be long compared to the time needed to depolarize the muon in Mu from the initialensemble polarization; in a denser medium collisions are more common and residencetimes will be correspondingly shorter. The characteristic time for depolarization is thetime 1/wo , where vo = wo /27r is the hyperfine frequency, 4463 MHz for Mu.Chapter 5. Experimental Setup^ 53Following the charge-exchange regime, muons slow down by elastic, inelastic, and re-active collisions in gases (ibid.) to thermal kinetic energies. After thermalization, somefraction of incident muons will be in the form of Mu and some in diamagnetic environ-ments, such as molecular ions (e.g. N 2 Mu+), or diamagnetic molecules such as MuH,which are indistinguishable from bare muons ji+ given current experimental resolution.In some gases muons may also thermalize in muonated free radicals such as C 2 H4 Mu.The fraction thermalizing in each environment is a strong function of the stopping gasdensity and composition. The phenomena contributing to the determination of the rel-ative fractions have recently received much experimental and theoretical study. Theyprovide a way, albeit model-dependent, to estimate hot atom (Mum) reaction rates [92],similar to that used in hot hydrogen (H*) and tritium (T*) studies. These rates havebeen measured for several reactions of Mu* including the title reactions; see ibid. for arecent summary.In gases, the muons will stop (i.e. thermalize) over a range of distances into the gastarget. The average stopping distance is inversely proportional to the electron densityin the medium since the energy loss processes of the muon principally involve collisionswith electrons. The entire thermalization process typically takes --50 ns [93] for gaspressures atm; less for higher pressures.5.1.1 ASR and MSRIf the muon target in which a polarized muon beam is stopped is in a uniform mag-netic field B, iLSR (muon spin rotation, or relaxation, or resonance) studies are possible.Conventionally, this field is either along the direction of initial muon polarization ("lon-gitudinal") or perpendicular to it ("transverse"). The acronym iLSR denotes severaltypes of studies employing muon polarization. In general each works by providing datawhich is fitted to the solution of appropriate equations of spin motion, with suitableChapter 5. Experimental Setup^ 54variable parameters.The asymmetry in the decay (equation (5.66)) has the same motion as the muonspin, and the motion is observed by the pattern of detection of the positrons from muondecay. In the limit of an infinite number of "events" (muon decays included in the fit),the ensemble will reproduce the equations of spin motion. A suitably high number ofevents for a reasonable approximation to this limit depends on the parameters of mostinterest in the particular experiment; for the experiments here reported approximatelytwo million events were collected for each experimental run.The detectors used in ,uSR are the same as those used in nuclear and particle physicsscattering experiments, which are also components of the experimental program atTRIUMF, as is detector development. Commonly called "counters" since they "count"decay events, they are made of a plastic which produces a "cascade" of light, called a"scintillation", when struck by a particle, such as e+, at high energy'. This is convertedto a current pulse by a photomultiplier and transmitted down a coaxial cable. For manyapplications it is desirable to reduce background signal from false counter triggeringby using counters in coincidence: requiring simultaneous triggering of several countersplaced parallel to each other to register a signal.The pulse from a counter is converted to a digital one and usually, treated by a "dis-criminator" which filters low height pulses, easily distinguishable from the true signalby visual observation. The signal is then treated by a logic circuit, which combines thesignals from the various counters to physically meaningful data. The data is collectedby a minicomputer program receiving input from the logic circuit.In time-differential fiSR experiments A is a function of time, called the "signal"S(t). For a particular muon, the time zero is set by the time at which it enters thetarget medium, through a thin scintillation counter, and t is defined by the time atcompared to thermal.Chapter 5. Experimental Setup^ 55which a decay positron is detected by another counter. With the radioactive decay ofthe muons due to the mean muon decay lifetime, denoted •, a normalization factorNo , and a background parameter B9 to account for noise in the positron counter andbackground radiation, the events are fitted to the functionN(t) = + S(t)] + B9 (5.68)for each positron counter (or counters in coincidence) in use. The time resolution isfinite, set in principle by the time for light to travel the spatial dimensions of thecounters, and the events are histogrammed into "bins", each of a given width, typicallyns. "Veto" mechanisms in the logic circuit ensure that the positron detected attime t indeed comes, with great certainty, from the muon detected at time zero. Thetotal length of the histogram is typically several muon mean lifetimes (-10 its), beyondwhich few muon decays are detected, and whose length is also determined to optimizethe veto mechanisms.In integral ,uSR S(t) is obtained by forming the quantityS(t) = N_ — aN+N_ aN+ (5.69)where the N's are numbers of detected decays over a reasonable period of time and +and - denote positron counters oppositely placed with respect to the target, with theempirical parameter a accounting for the difference in counting efficiency of the twocounters.The equation of motion for the muon spin in either the Mu or diamagnetic environ-ments have been solved many times (e.g. [27,30]) and will not be discussed in detailhere. Briefly, for a muon in a diamagnetic environment the muon spin evolves accordingto the Hamiltoniank p i, • B^ (5.70)Chapter 5. Experimental Setup^ 56where is the magnetic moment operator defined for any particle p aspg e p— 2mpc(5.71)with the quantities J the particle's angular momentum operator, m its mass, g itsquantum "g-factor" and e its charge. For a free muon, J is just its spin angularmomentum S, which has the same eigenvalues +h/2 as the electron. Then 1p,„1 takesthe value [27] 4.47 x 10 -23 erg/G, reduced from that of electron just by the ratio oftheir masses, 207.8. For later reference is definedep h= 2mpc(5.72)the "Bohr magneton".In a longitudinal field, either spin eigenstate of the bare muon is an eigenstate ofthe Hamiltonian and the signal detected by a positron detector is of the form (ibid.)S(t) = AD G(t)^ (5.73)where AD is a constant representing the amplitude of the diamagnetic signal and G(t)is a relaxation function accounting for loss of polarization due to interactions of themuon spin with the medium. While AD is intended to represent A of equation (5.66)integrated over the area of the positron detector, it is, as mentioned earlier, alwaysmeasured empirically by fitting S(t). For a tranverse field, the signal takes the form(ibid.):S(t) = AD COS(WDt + OD)^(5.74)for a detector in a direction perpendicular to the applied field. The fitted parameterOD accounts for rotation of the spin direction during beam delivery and thermalization.The parameter WD is the Larmor frequency 27rg,,IBI/h of the muon, corresponding toChapter 5. Experimental Setup^ 57classical magnetic moment "precession" in a magnetic field: the maximum muon spinprojection "sweeping" past the counter with the angular frequencyLop = 27r-y4 1B1^ (5.75)where the gyromagnetic ratio -y„, gotten from p p, and g„, is 0.0136 MHz/G.For free muonium (in the is electronic state) in a uniform magnetic field the Hamil-tonian is (ibid.)k 11,2 • B^B hwo S e • S i,^ (5.76)in which the first two terms represent the individual interactions of the muon andelectron spins with the external field, the third term the isotropic hyperfine interactionbetween their spins. The hyperfine frequency v o = w0 /277, as above, is 4463 MHz, givenby871110 = 3 gegw3ei3A101,(0)12, (5.77)proportional to the electron amplitude squared, P is (0)1 2 at the muon, and about threetimes that of the hydrogen atom (1420 MHz).The interaction results in coupling between the individual spin eigenstates. Theenergies of the four eigenstates of this Hamiltonian are shown, as a function of B, inFigure 5.5, which also details the asymptotic features of the eigenstates, labelled byusual fiSR convention. The most notable features of the figure are the approach toproduct eigenstates at high field as well as approach of the energies to linear functionsof B. It is also of note here that there is not any evidence for Mu surviving in excitedelectronic states after thermalization in gases, which would be obvious from a signal ofmuch different characteristic frequency, although some may be initially formed (ibid.)in excited states. As well it should be noted the frequencies v 34 and v14 are too highto be measured by itSR whose best time resolution is ns, leading to a Mu signalreduced in amplitude by just one-half.58Chapter 5. Experimental SetupFigure 5.5: Progressive construction of the Breit-Rabi diagram for eigenenergies ofstationary states of free muonium as a function of magnetic field. At low field, theeigenstates are close to the coupled representation states IFmF ) with F the total spinof muon and electron. At high fields the eigenstates approach the totally uncoupledindividual spin product eigenstates im,,e ms,,,). The magnetic-dipole allowed radiativetransitions are shown. The parameter "A" on the energy scales is hvo. Reproducedfrom [94].Chapter 5. Experimental Setup^ 59At very low fields (< 10 G), the frequencies v 12 and v23 are essentially degenerate,at a frequency just about half that of the electron precession frequency—the signalis dominated by the electron spin precession. This is the field regime of the presentstudy. Two frequencies are thus observed: the single coherent Mu frequency and the ,a+(diamagnetic) precession frequency. The frequencies v 12 and v23 lead to an observablesignal because Mu is not initially formed in its hyperfine eigenstates. This fiSR regime,weak field Mu precession, is sometimes called MSR (muonium spin rotation), and hasfound use as the method of choice for measurement of Mu reaction rates with gases.The signal resulting under these conditions isS(t) = AD COS(4.4) Dt - OD) + Amu Gmu (t) cos(wmu t — Omu) (5.78)with the A's separate asymmetry parameters for the diamagnetic and muonium frac-tions, and Gm„ the relaxation function of the muonium fraction. The Mu frequencywm„ has the opposite sense of w u and its value, given bywm. = 277mulB I (5.79)where rymu = v12/1B = v23/1131 = 1.39 MHz/G, is just half of the free electron preces-sion frequency, but 103 times the diamagnetic frequency (wmu 103wD = we /2). Withvery few exceptions [30], even for low ionization potential gases, in which 100% Mu for-mation might be expected by capture of the weakly bound electrons, the diamagneticfraction is always present for the pressure regimes (< 20 atm) of the present study. Forsuch low fields, no relaxation is observable for the diamagnetic fraction since it goesthrough less than a full cycle over several mean muon lifetimes.For chemical reactions which place Mu in diamagnetic environments, such as thetitle reactions (1.3) and (1.4), and under the pseudo-first order conditions mentionedin Chapter 2, the ratio Am u (t)/Amu (0) corresponds to [Mu](t)/[Mu] o . The "concentra-tion" [Mu] need not be known. Then, the relaxation function in the MSR regime, byChapter 5. Experimental Setup^ 60comparison of the result of chapter 2, is justGmu(t)^e-at^ (5.80)where the relaxation rate parameterA = k[HR] A o (5.81)and the "background" relaxation A o , is attributed to experimental artifacts, the mostlikely being an imperfectly homogenous magnetic field over the stopping range of themuons. Standard linear least-squares fitting procedures give the parameters A o andk and their estimated experimental uncertainties. This relaxation, due to a thermalchemical reaction is shown in Figure 5.6 in which MSR signals of N2 and CH4 gas at821 K are compared at equivalent muon stopping densities.The relaxation of the muonium ensemble for this type of chemical reaction comesabout because the process of chemical reaction, which places the muon in a differentmagnetic environment, removes the reacting Mu atom from the ensemble by changingits precession frequency to essentially random values during the course of reaction. Ingeneral, the molecular interaction times in the process of individual chemical reactionsare short (typically < 1 ns [30]) compared to the time resolution of the measurement,resulting in essentially instantaneous loss of coherence from the ensemble.The parameters k and A o can be obtained for a given gas temperature by measuringA at several concentrations (i.e. partial pressures) of the reactant gas HR. The reactionunder study has a relatively small rate coefficient and so high pressures (up to —17 atm)of reactant, and high temperatures (up to 821 K) were used. Under these conditions, inorder to maintain a constant magnetic environment of the reacting Mu, it is necessaryto add a inert "moderator" gas, N2 in this study, to maintain a constant average Mustopping distance at each concentration other than the highest. Also a run with justChapter 5. Experimental Setup^ 61Figure 5.6: MSR signals for N2 and CH4 at 6.1 and 8.6 atm pressure, respectively, bothat —821 K. The small relaxation in the N2 signal (top), 0.026 /is', is identified with the"background" relaxation A 0 . The relaxation 1.147 ps -1 in the CH 4 signal (bottom) isattributed to chemical reaction resulting in loss of Mu coherence. Signals are nonlinearleast-squares fits using the program MINUIT [95]. The A vs. concentration plot forCH4 at this temperature is shown in Figure 5.7.0.800.10 0.20 0.30 0.40 0.50 0.60 0.70[CH4 ]/(1020 molecule cm -3)Mu + CH4 -> MuH + CH 3 , T = 821±6 K1.351.201.050.900.750.600.450.300.150.000.00Chapter 5. Experimental Setup^ 62the moderator gas is performed to increase the accuracy in the determination of theintercept A o . Plots of the equation (5.81) at each experimental temperature, for eachof the two reactions studied, are to be found in the Appendix. Again for reference, thereactions of interest are:^Mu + CH4 Mull + CH3^(5.82)andMu + C 2 H6 —> Mull + C 2 H 5^(5.83)and a representative plot, for the Mu + CH 4 reaction at 821 K, is given in Figure 5.7.It should be noted that to attribute A to a chemical reaction, it is necessary to beFigure 5.7: Fit of the rate coefficient of the reaction Mu + CH 4 -4 Mull + CH 3 . Ratecoefficient k fitted on plot of measured relaxation rates A versus concentration of reac-tant, at 821 K, by linear least-squares. At each CH4 concentration, two measurementswere made, with that shown in the figure the average. All measurements included inthe fit of k.Chapter 5. Experimental Setup^ 63certain that no other processes occur to relax the Mu signal. In the present study,the only other processes competing with the reactions above to any degree are the Mubimolecular substitution reactions displacing H. In the case of CH 4 , experimental andtheoretical studies [31] show the bimolecular substitution reaction to be --20 kcal/molin Ea above that of abstraction, indicating that the signal relaxation due to substitutionis negligible compared to the abstraction reaction. For C 2 H 6 it is also to be expectedthat the substitution reaction would be far above the abstraction reaction in Ea .Earlier measurements of the same type have included the hydrogen abstraction fromH2, reaction (1.8) mentioned in Chapter 1, as well as the gas-phase reactions [34,96]Mu + X 2 MuX + X^ (5.84)andMu + HX MuH + X^ (5.85)with X = F, Cl, Br. Similar techniques have been used to measure Mu gas-phase addi-tion reactions such as [97]Mu + C2H4 C2H4Mu^ (5.86)as well as gas-phase spin exchange of Mu (spin flip of the Mu atom electron from colli-sions in the medium) with Cs atoms [98] and 0 2 [99]. Also measured have been manysolution-phase Mu reaction rates (e.g. [100]) and hyperfine parameters of gas-phasemuonated radicals [101] (some by integral pSR). As mentioned earlier with regards toit+ slowing down, epithermal it+ and Mu processes have been studied in gases, includingN2, CH4, C 2H6 , and C3 H8 [102,103]. Among the many muon experiments not employ-ing ,uSR, recent measurements of physical-chemical interest include observation [104]of chemiluminescence in ,u+-irradiated Ne gas, assigned to electronic transitions of theRydberg molecule NeMu.Chapter 5. Experimental Setup^ 645.2 /./SR Experiments at TRIUMFTRIUMF, a cyclotron particle accelerator, accelerates H - ions to a beam of high veloc-ities by repeatedly having them transverse a region of high voltage produced by a hugecapacitor-like device [106]. The ions are stripped of their electrons yielding protons ata kinetic energy of 500 MeV. The beam travels down a pipe of vacuum (a "beampipe")and then split several times, into a series of "beam channels." Some of these are avail-able for nuclear and particle physics experiments. Of more current interest at TRIUMFare experiments employing pion and muon beams, for which one part of the initial beamis directed into the experimental area called the Meson Hall. The proton beam hits aseries of pion production targets, of various substances such as carbon, beryllium. andwater, to produce pions, both positive and negative, of various kinetic energies. Someof the pions are directed into secondary beam channels ("beamlines") used for exper-iments or biomedical applications, or to produce high-energy muons from "in-flight -decay.For most fiSR gas chemistry experiments, the useful pions are those that, at rest.decay to muons on the surface of the production target, producing a muon beam withessentially all of the rest mass energy difference m ir c2 — m„c2 , 4.1 MeV, converted tomuon kinetic energy. Beams of muons in a small range about this average, called thenominal beam energy, are available for experiments, on TRIUMF beamlines M13, M15,and M20, and are called surface muon beams.All beamlines consist of a series of magnets to keep the beam focused, and in theright direction. Focusing quadrupole magnets direct the beam particles towards thecenter of a square array of four magnets, of alternating polarities. Bending dipolemagnets direct the particles in the beam, running between two pole faces of oppositepolarity, around a series of angles depending on the particle velocities; only thoseChapter 5. Experimental Setup^ 65particles in a narrow range of velocity stay in the beampipe. The form of the magneticforce law F ev x B/c dictates the effect of the magnets on a distribution of velocities.From this law and the momentum-velocity relationship p = my it should also be evidentthat the average momentum delivered by a beamline is proportional to the magneticfields, which are in turn proportional to the DC current delivered to the magnets. Itis also clear that across a cross-section of the beam the average momentum will varysmoothly after passing one or more bending magnets. Dipole magnets are momentumselectors only and so do not remove positron contamination (typically —100 times thenumber of muons). DC separators, found at the end of each muon beamline (M15 hastwo), are large, high voltage capacitors with a crossed magnetic field; these removethe positrons, and also narrow the momentum range of the beam reaching the target.They also "rotate" the spins of the beam muons by precession and can be used to rotatethe initial beam polarization to a preselected angle, typically 90° with respect to theinitial direction. Also, vertical slits can be narrowed and widened to narrow or widenthe momentum range sampled. Horizontal slits can be narrowed to further reducethe beam to optimum dictated by the logic circuit veto mechanisms. The magnets,separators and slits are collectively called beamline elements and are controlled by acomputer program called TICS.The M15 beamline is famous worldwide for its ability to deliver an intense beamof monoenergetic surface muons. In terms of "momentum bite", which is acceleratorjargon for narrowness of the momentum distribution about the average of the beam(nominally 29.8 MeV/c, corresponding to muon energy 4.1 MeV), M15 is the best inthe world. For gas chemistry experiments using pt+ this feature is attractive because anarrow momentum distribution leads to a small background relaxation A o , through amore homogenous B field in the muon stopping distribution, than would be obtainedfor a wider distribution, giving higher quality data. For very slow chemical reactionsChapter 5. Experimental Setup^ 66such as the title reactions, this feature is a necessity because the lowest measurablerate coefficients are of comparable magnitude to A o . Consequently the M15 beamlinewas requested and obtained for the present study. The beamline elements of M15 areshown in Figure 5.8.5.3 Reaction Vessel for CH 4 and C 2 H6 ExperimentsPreparation for the experiments reported in this thesis took far more work than thefinal collection of data, performed in two weeks in November 1992. The main taskwas the preparation of a reaction vessel appropriate to the conditions of the experi-ments, dictated by certain constraints. Specifically, the title reactions, as detailed inChapter 4, were expected to be rather slow, and therefore to give a very low relaxationrate parameter A. Consequently, moderately high reaction concentrations (partial pres-sures) and temperature range were considered necessary to achieve relaxation rates ofdetectable magnitude. The earlier experiments [37] of the same type for hydrogen gas(rate of reaction (1.8)) were used as a guide in deciding on the conditions to be used,estimated as temperatures 500-900 K and pressures up to 17 atm. The vessel had to bemade of a material which could withstand such conditions, while being nonmagnetic,so that there would be no interference with the weak applied field B. An earlier vesseldesigned for the hydrogen experiments was originally to be used, but developed a leakdue to a design flaw [105].Therefore, TRIUMF engineer G.S. Clark was asked to design a new, similar vessel,but this time following the world-recognized standards of the American Society of Me-chanical Engineers (ASME) pressure vessel code wherever possible, with the expectedbehavior of the vessel estimated from the observed behavior of the original hydrogenvessel. The new vessel, delivered in July 1992, is essentially as shown in Figure 5.9,015-017012-014EXPERIMENTALTARGETLOCATION(EL.291.5109-011 SEPARATOR 2SEPARATOR 1(7-ticntricocoCl)CD'LSIATI TARGETARRAY BEAMLINE M150I•asanna•A0•1s^-cael0Chapter 5. Experimental Setup 68Figure 5.9: Diagram of reaction vessel used in present study; reproduced from refer-ence [105]. Lengths are in inches.Chapter 5. Experimental Setup^ 69reproduced from Mr. Clark's design note, reference [105].The vessel is essentially a can with at one end, a thin muon entry window needed toallow the muon beam to pass into the reactant gas, and at the other, plumbing to allowthe introduction and removal of gas. The "vacuum vessel" in the figure is mentionedlater. The vessel, like the earlier counterpart was constructed of stainless steel becausethis material is essentially nonmagnetic. In the case of the present vessel, high carbon316 stainless steel was used, to conform to the ASME code. The finished vessel doesobey the code in every detail except for the window. Two details required much furtherdevelopment following receipt of the vessel: the window and the heating system.5.3.1 Thin Muon Entry WindowThe initial design of the window was essentially the same as that for the window usedfor the hydrogen vessel. It is shown in Figure 5.10. That window was made from0.051 mm thick foil of inconel X-750 alloy (purchased from Metal Men), chosen forits heat-resistant qualities. The foil was formed to a dome under 5.5 MPa hydrostaticpressure, and welded onto a specially shaped 316 stainless steel ring, which was thenitself welded to a hole at the head of the vessel. The dome forming gave a final productof approximately 0.043 mm thickness. No problems were ever experienced with thatwindow [31], but that may have been merely good fortune, because the first windowused on the present vessel study failed in high temperature/pressure safety testing. Asecond window broke in initial experimental runs. These occurred despite the fact thatthe windows were heat treated in an oven according to established ASME procedures toimprove their resistance to shear stress. The window failures were of a gradual nature,with cracks appearing around the outside edge near the weld to the vessel. The failureswere attributed to large forces developed in this region, which holds the entire windowto the vessel, although the alloy [107] "is known to be susceptible to heat treatment'71•-• •• Ot1Crqcrl• '—'co pCDAGre;1-4,0CDa0CD0QCD0C:)Chapter 5. Experimental Setup^ 71embrittlement."Following these failures in August 1992, a project was initiated to develop a win-dow which could withstand the experimental conditions long enough to complete theexperiments. The development essentially involved the use of different welding tech-niques (in TRIUMF's machine shop) to try to reduce the stress on the outside edgeof the window during welding and in use. The heat treatment was omitted becauseit was judged to be of no advantage. Windows were then tested at conditions of highpressure and temperature, given in Table 5.4, more severe than those to be used in theexperiments.The tests were carried out by welding the windows, in the same manner as withthe vessel, to specially-designed small, thick-walled vessels termed "jigs" with tubes toallow introduction of nitrogen gas for the tests. The jigs were placed in an oven withholes for the gas tubes, and heated to the temperatures given in the table. Nitrogenwas added to the given pressures, and the pressure monitored with a gauge added tothe plumbing. For some tests, after the pressure held for a while, the pressure wasincreased. For each test, the gas held at the given pressures for the times given. A fewtests, not shown, were also performed on two types of 0.076 mm thick inconel foil which,curiously, were found to be inferior to the 0.051 mm foil. The thicker foils developedcracks in their grain structure when formed to domes.Examination of the table clearly shows the progress made in improving the win-dows, though it is also clear that even the best of the windows can not be realisticallyexpected to indefinitely withstand the experimental conditions. Window leak test fail-ures immediately after welding, not shown on the table, also declined, and in the finalbatch of windows, none failed this test. In the beam time of November 1992, the win-dow on the vessel lasted for the entire two weeks, then failed as the setup was beingChapter 5. Experimental Setup^ 72Tab to o.4: selected results of tests of ( .051 mm thick inconel winetest window temperature (°F) pressure (psi) time held1 1250 255 3 min300 3 min340 4 min390 7 min450 5 min510 10 min2 1250 256 5 min313 7 min400 5 min3 1250 240 2.5 hr300 1 min4 1250 150 17 hr200 1.5 hr250 1 hr5 1100 150 4.2 hr250 3.5 days6 1100 250 3.5 days7 1250 150 4 days1150 300 2 daysprepared for dismantlement!5.3.2 Heating of the VesselThe metal used for the can, high carbon 316 stainless steel, while very nearly nonmag-netic, is a very poor heat conductor. To prevent heat loss, the vessel, like its predeces-sor, was mounted in a larger 316 stainless steel vessel which was continuously pumpedto a good (-10' torr at the pump) vacuum. Also, thin copper and aluminum heatshields, their thinness dictated by the necessity of detecting escaping decay positrons.ows.Chapter 5. Experimental Setup^ 73minimized radiative heat loss. Cylindrical shields covered the curved part of the (in-ner) vessel, and flat circular shields the flat window end. In the beam path, the heatshields had circular holes cut which were occupied by much thinner heat shields. Two,nearest the inconel window, were of gold foil 0.0013 mm thick; four shields beyond thiswere of 0.0051 mm thick aluminum foil. Beyond this, the vacuum vessel had a muonentry window, of 0.051 mm thick aluminized mylar. The vessel was heated by heatersmounted on the plumbing end. This was necessitated by the fact that heaters aroundthe curved surface would be likely to cause magnetic fields that would interfere withthe applied field B.The older hydrogen vessel, after considerable effort [37], had had a satisfactoryheating system, but it involved the use of tapped holes in the plumbing end. TRIUMFsafety officials required that, in keeping with the ASME code, no holes could be drilledinto the new vessel's plumbing end, and so a new way of heating the vessel neededto be found. An additional constraint was the fact that the plumbing end is rathersmall, 15 cm in diameter, with large tubes taking up some of this space. Initially,two flat electrical contact heaters, purchased from Omega Industries, Inc. were initially(August 1992) mounted to the plumbing end. These heaters had a series of tiny coilsof rather thin resistance wire, sheathed in a chromium alloy, and a maximum ratingof 1000 W each. When used at high temperature in vacuum, these tended to short-circuit, resulting in immediate destruction. When these failed in experimental use, itwas necessary to find another heating method, and to do some tests in advance of theNovember 1992 beam time, in order to be sure there would be no heater failure whentrying to collect data.While the technical details of heating the vessel are not of great scientific interest,the lack of data regarding the heating of a metal surface in a vacuum is remarkable.A large number of specialists were contacted, very few with any information. ContactChapter 5. Experimental Setup^ 74heaters specifically for vacuum were recently added to the Omega catalogue, but werejudged too expensive. The eventual conclusion was to use tubular contact heaters fromthe same manufacturer. These heaters, approximately 51 cm long, 1.2 cm in diameter,consisted of a single thick resistance wire, with a few cm at each end of low resistancewire, sheathed in an inconel alloy. They had rating 1000 W each; three were weldedto a thick stainless steel plate, mounted by bolts from another thick steel plate, tothe vessel. It was hoped that the welding would result in good heat transfer from theheaters, allowing them to heat the vessel without being damaged by excessive heat.Two of these heaters failed in initial testing; none during experiments.5.4 Measurement of the Title Reaction RatesFor the present study the vessel was placed, window facing the incoming ft+ beam,between a pair of Helmholz coils (parallel circular DC current loops) 1.5 m in diameter,which produce a field "homogenous to 0.1% over a volume of 10 L" [37]. A field ofG was applied; this field, fitted as a parameter of S(t), need not be initially knownwith great accuracy. Above and below the reaction vessel were two sets of two positroncounters, used to collect two separate double-coincidence MSR signals.All gas samples were purchased from Canadian Liquid Air. Nitrogen moderatorgas was ultra-high purity grade (> 99.999% pure), the ethane gas research grade(> 99.98%), and the methane, superpure grade (> 99.99%). Pressures over atmwere measured using a Borden test gauge (Matheson), and lower pressures with aBaratron capacitance pressure gauge (MKS Instruments, Inc.). Gas samples were in-troduced through a system of plumbing kept over 1 atm pressure at all times, exceptduring pumping, to prevent any possibility of inward leak of air. The density corre-sponding to a given pressure was different at different points in the vessel because theChapter 5. Experimental Setup^ 75temperature within was not uniform (see later discussion) but again invoking the verygood ideal gas behavior of the sample gases, it is to be expected that density would varysmoothly with temperature, resulting in uniform pressure throughout the gas system.Temperatures in the gas were measured by a series of type E thermocouples (Omega)inserted through tubes passing into the vessel. The thermocouples shown in the figureas being on the outside of the vessel were for monitoring for safety purposes as wellas to help identify temperature equilibrium. The temperature was controlled by anOmega temperature controller receiving its input from one of the thermocouples on theplumbing end of the vessel. At equilibrium the window was typically --200 K cooler thanthe plumbing end, with the gas somewhat hotter than the window. Interestingly, fora given temperature controller setting, equilibrium temperatures rose with increasingCH4 or C 2 H 6 concentration relative to N2.Some discussion is in order here regarding the temperature error bars used in theArrhenius plots of the next chapter. The addition of the heavy steel plates to mountthe heaters apparently greatly reduced the radiative heat loss at the plumbing end. Theresult was that fairly large temperature gradients developed in the gas, typically --50 Kfrom one end to the other of the vessel. It is therefore necessary to determine whichset of thermocouples gave the best indication of the actual temperature at the muonstopping region. Calculations using standard formulae (from reference [108]), standarddata (from reference [109]), and M15 calibration data, show the average muon range atthe lowest electron density in any of the experiments was less than 18.2 cm. The same/2+ momentum setting was used for all experiments, so for all others the average rangewas less.The stopping muons were closest, in every case, to a set of thermocouples in thegas located^cm from the window end. These thermocouples, one above and onebelow the level of the beam, generally differed from each other by^K. For a givenChapter 5. Experimental Setup^ 76temperature point on the Arrhenius plots, the temperature uncertainty reported is thedifference between the maximum and minimum readings for all of the runs at thattemperature. The resulting estimated uncertainties are still modest, ,,,5-10 K. Withthe muon range increasing with increasing T, because of decreasing electron densitywith the same initial muon momentum, T may be systematically underestimated as itincreases. However, this is expected to be a small effect, because the Arrhenius plotfits of the next chapter have reasonably small estimated uncertainties in the calculatedparameters.Chapter 6Results and DiscussionThe pseudo-first order rate coefficients (recall equation (2.25)) are calculated at eachexperimental temperature, for each of the two title reactions, according to the methodsdeveloped in the previous chapters. Two independent histograms for each run werecollected, and fitted as separate data points, but averaged in the final plot for clarity.The fits giving each of the experimental rate coefficients are shown in the Appendix,with the 821 K fit also shown in the previous chapter. For each reaction, the ratecoefficients are fitted by standard nonlinear least-squares procedures to the standardArrhenius form k = A exp(-Ea /RT), over the temperature ranges 626-821 K and511-729 K for Mu + CH 4 and Mu + C 2 H6 respectively. They are shown in Figures 6.11and 6.12. The data fit in the figures is shown in tabular form in Tables 6.5 and 6.6along with the calculated Arrhenius parameters, and compared with those of the Hatom variant, in Table 6.7, along with k values at selected temperatures. The errors inthe Mu rate coefficients are due to counting statistics only.Omitted from the C 2 H6 data are data taken above the temperature 729 K, whichdeviated very strongly from the straight line formed by the lower T points. Thisdeviation was attributed to "cracking" of C 2H 6 to form appreciable concentrationsof C 2H 4 , whose addition reaction with muonium is considerably faster than the Mureaction with C 2 H6 [97]. The C 2 H4 forms at an appreciable rate [23] from C 2 H6 ,compared to the reaction rate of Mu + C 2 H6 , the rate-limiting step of this processbeing, interestingly enough, the rate of the corresponding reaction of H C 2H 6 . Indeed,77Chapter 6. Results and Discussion^ 78Figure 6.11: Arrhenius Plot for Mu + CH 4 -4 MuH + CH3.Chapter 6. Results and Discussion^ 79Figure 6.12: Arrhenius Plot for Mu + C 2 116 -- MuH + C2H5.Chapter 6. Results and Discussion^ 80Table 6.5: Measured Rate Coefficients for Mu + CH4^MuH^CH3 .T/K k/(cm3 molecule -1 s -1 )626 + 9 1.890 + 0.243 x 10 -16634 ± 6 2.082 ± 0.272 x 10 -16662 + 6 3.710 ± 0.240 x 10 -16668 + 6 4.778 + 0.225 x 10 -16691 + 5 8.631 + 0.328 x 10 -16721 ± 6 1.748 + 0.044 x 10 -15732 + 6 2.250 ± 0.055 x 10 -15776 + 9 6.818 ± 0.117 x 10 -15821 ± 6 1.616 + 0.028 x 10 -14Ea = 24.6612 kcal/molA = 5.711 x 10 -8 cm3 molecule -1 s -1x 2 = 2.9Table 6.6: Measured Rate Coefficients for Mu + C 2H 6 MuH C2H5.T/K k/(cm3 molecule -1 s -1 )511 ± 4 3.066 + 0.202 x 10 -16599 + 5 2.229 + 0.049 x 10 -15630 + 4 4.793 ± 0.090 x 10 -15662 + 6 8.097 + 0.134 x 10 -15693 ± 8 1.500 ± 0.021 x 10 -14729 + 8 3.825 + 0.656 x 10 -14Ea = 15.35+g:2 kcal/molA = 1.01 74 x 10 -9 cm3 molecule-1 s-1X2 = 5.6Table 6.7: Comparison of Mu and H Atom Data for Title Reactions. H atom data fromreference [6].Reaction Ea(Mu) - Ea (H) Amu/Ax kmu/kH (620 K) kmu /kH (720 K)Mu(H) + CH 4 11.5 kcal/mol 180 1/62 1/17Mu(H)^C 2 H6 5.6 kcal/mol 3.2 1/28 1/15Chapter 6. Results and Discussion^ 81short runs taken above 729 K were observed to have increasing A with time from theintroduction of the gas. The highest T point retained for the fit, at 729 K, deviatesslightly above the line of the fit, indicating some C 2 H4 formation at this T.The most remarkable feature of both Arrhenius plots are the rather large valuesof Ea , particularly for CH 4 . The Ea values are also very large in comparison withthe values for the corresponding H atom reactions at comparable temperatures, takenfrom reference [6]. The Arrhenius parameters and k values at selected temperatures(from the plots) are compared, for both title reactions, in Table 6.7. The Ea value forthese reactions, 24.55 kcal/mol for CH 4 and 15.35 kcal/mol for C 2 H6 , are surely amongthe highest ever measured for abstraction reactions of H, as are the Ea differences,Ea (Mu) — Ea (H), of 11.5 and 5.6 kcal/mol respectively. For the C 2 H6 reaction, the ratioAmu /AH , 3.2, is near the 2.9 expected from the "trivial" isotope effect (see discussionof reaction cross-section in Chapter 2) due to isotopic mass ratios. The ratio for CH4 isless reasonable but within an order of magnitude; such a large error in A can be gottenfrom a very small error in slope (Ea ) on a logarithmic scale for k. However, both fitslook reasonable and linear, with no obvious sign of curvature. Quantum tunneling istherefore not likely to be an important factor in the dynamics these reactions, in thetemperature range studied.The Ea differences seem too high to be accounted for by the V G (vibrationallyadiabatic barrier) difference between the isotopic variants. To see this for CH 4 , it isnecessary to examine the vibrational states of CH 4 and the transition state CH 4 —H.(The corresponding information is not available for C 2H6—H.) These are shown in thecorrelation diagram of Figure 6.13, adapted from reference [26] with the wavenum-bers shown also from that reference. Three of the transition state frequencies arestrongly correlated with CH3 frequencies with almost no change in value and so shouldchange very little on substitution H Mu to give the transition state CH 4 —Mu. ThisChapter 6. Results and Discussion^ 82Figure 6.13: Correlation Diagram H + CH4 CH4-H H2 + CH3. At left are vibra-tional energy levels of CH 4 ; in centre, of CH4-H; at right, of CH 3 + H2. For each energylevel, symmetry species, degeneracies, and wavenumber in cm -1 are shown. The energylevels and degeneracies, which assume harmonicity, are from the POL - CI calculationsof Schatz et al., reference [24 The correlations, also given in that reference, are shownas dashed lines. For reference 349.75 cm- ' 1 kcal/mol.Chapter 6. Results and Discussion^ 83Table 6.8: Standard Enthalpies of Title Reactions and H Atom Variants. Also given inChapter 1.Reaction^ AH°H CH4 -> 112 + CH3 -2.6 kcal/molMu + CH4 MuH CH3 +4.9 kcal/molH + C2 H6 112 + C2H5 -3.1 kcal/molMu + C 2 H6 -> Mull + C 2 H 5 +4.4 kcal/molis borne out by the comparison of Schatz et al. [26] of the transition states CH 4 -H andCH 4-D. First noting that there is no zero-point energy difference in the reactants ingoing from H to Mu, and multiplying Schatz' et al. zero point energy difference by theratio ,In/D /rn i,, an estimated Ea difference of only 1.9 kcal/mol is obtained. Comparingthese TS's, the Ea for the reaction involving Mu is raised by much more than the zeropoint energy difference.However, this comparison assumes no change in the location of the transition stateCH4-Mu as compared to the TS CH4 -H. It is likely that the respective transition stateshave very different locations. To see this, compare the reaction enthalpies of the tworeactions, mentioned in Chapter 1, and shown here in Table 6.8. Also included in thetable are the reaction enthalpies for the corresponding reactions of C 2H 6 . Noting thatthe H atom reactions are exothermic, while the Mu reactions are endothermic, verydifferent reaction barrier locations can be expected in comparing CH 4-H to CH4-Muor C 2 H6-H to C 2 H6-Mu. As has been discussed many times, the geometrical characterof PES's causes exothermic reactions to have, in general, early barriers, in which the TSis relatively similar to the separated reactants. Similarly, for endothermic reactions, theTS will be relatively similar to the separated products. For the Mu + CH4 reaction,the TS CH 4-Mu can be expected to be shifted towards products, whose vibrationalChapter 6. Results and Discussion^ 84energy is much higher than that of the H + CH 4 transition state CH 4—H, due to themuch lighter mass of Mu compared to H, which results in much higher vibrationalfrequencies. This combined with the isotopic shifts in vibrational energy, chiefly due toenergy difference of MuH and H2 , could produce a drastic increase in the TS zero-pointenergy. A similar effect for H abstraction from C 2 H6 would be expected.Some comparison with the previous measurements [37] for the reactionMU + H2 MuH H (6.87)is in order here. In this case the H atom variant is thermoneutral whereas the Muvariant is endothermic by 7.5 kcal/mol, and the Ea is 4.8 kcal/mol higher for Mu thanfor H [74]. As with the reactions of the present study this likely represents differingTS location in the isotopic variants, given the very different thermochemistry of thetwo reactions. VTST calculations [110] on the Mu and H variants of this reaction showthe collinear TS to differ in geometry drastically, with the TS H 2 —Mu much closer toproducts than the H atom variant H 2—H, which is symmetric with respect to the twoH—H bond lengths.The .Ea difference, for CH 4 at least, is still too high to be explained by zero-pointenergy differences of the TS's. However, some of the difference might be explainablein terms of differences in dynamics for excited states of CH 4 with Mu compared to H.In the temperature range 626-821 K, the equilibrium concentration of vibrationallyexcited CH 4 is substantial. Even so, the extra vibrational energy is not comparable tothe Ea difference of Mu/H. Using experimental wavenumbers listed in reference [70],CH4 has an average of —0.4 kcal/mol excess vibrational energy at 626 K, and at 821 K,'-1.0 kcal/mol. For the transition state, assuming harmonicity of the vibrational motionand using the theoretical calculations of Schatz et al., the excess is kcal/mol at626 K,^kcal/mol at 821 K. Again, this is not comparable to the Ea difference.Chapter 6. Results and Discussion^ 85Table 6.9: Estimated vibrational wavenumbers of the CH 4-Mu transition state, esti-mated from the corresponding CH 4-H wavenumbers of Schatz et al., also given here,assuming harmonicity and using corresponding calculated CH 4-D wavenumbers.Mode (Degeneracy) Wavenumber in CH 4-H(cm -1 )Wavenumber in CH 4-Mu(cm-1)vl ( 1 ) 995 1017V2 ( 1 ) 1960 3480113 ( 1 ) 3228 3228V4 (2) 592 717v5 (2) 1146 1187v6 ( 2 ) 1534 1532(2) 3404 3406The CH 4-Mu transition state wavenumbers estimated from this reference are shown inTable 6.9.In the expected transition state structure CH 4-H, from the theoretical MP results(see Chapter 3), the leaving H atom is relatively far removed from the CH 3 group.Given this structure, it is certainly reasonable to expect that vibrationally excitedCH4 , in which the average C-H distance is higher than for the ground state, will havea higher reaction probability with H. For the Mu variant, the leaving H atom will likelybe even farther removed in the TS. As well, for Mu, with a higher average velocitythan H, it is also reasonable to assume that the reaction probability will increase fasterwith average C-H distance, and H-C-H angles, for Mu than for H; Mu is expected dueto higher average velocity to more easily approach the leaving H atom on noncollinearpaths than H, which should become more important for a longer C-H distance andgreater H-C-H angles.The result of this hypothesis is that the rate coefficient depression in the substitutionH -* Mu should be less for excited CH 4 states than for the ground state, leading toChapter 6. Results and Discussion^ 86Table 6.10: Fit of CH 4 rate data to equation (6.88).Parameter^ValueE 21.9 kcal/molA^3.6 x 10 -10 cm' molecule' s -1A4 16.6A2 109a high apparent Ea , since the excited states form an exponentially higher fractionof the total as temperature increases. The same hypothesis is also applicable to theabstraction reaction from C 2 H 5 , whose H/Mu Ea difference is also very high.To try to explore the source of the high value of Ea for the CH 4 reaction, the datawas fit to a function of the formAe -E/RT( i 3A4e-±4+AE4)/RT 2A2k^e-(v2-FAE2)/RT)1 3 e -v4/RT 2 e -v2/RT (6.88)with the wavenumbers v 4 and v2 converted to energy using 349.75 cm -1 1 kcal/mol.This expression is intended to show differing reaction rates for the first two excitedstates of CH4 , the only ones of significant population up to 821 K. In this equation,v4 (= 1405 cm', triply degenerate) and v2 (= 1573 cm', doubly degenerate) are thewavenumbers corresponding to these vibrational states. Thus the denominator is thepartition function. This expression is constructed to be of the form of equation (2.13),an average of k over excited states of the reactant CH4 . The parameters AE are thedifferences between the excited state energies above the ground state in CH 4 , and theenergies of the TS excited states to which they correlate. These energies are fromthe PES of Schatz et al. (The TS states to which v 4 correlate were averaged.) Theparameters A4 and A2 represent pre-exponential factors for v 4 and v2 relative to theground state of CH 4 . The results are shown in Table 6.10. The fit is not of good quality;Chapter 6. Results and Discussion^ 87the estimated uncertainties could not be established, and the fit certainly should notbe taken very seriously. However the results are at least in the expected direction—theground-state Ea is lowered, and the pre-exponential factors are higher for the excitedstates. Also, the overall pre-exponential factor is closer to that expected from theMu/H mass ratio than for the fit of the standard expression k = A exp(—Ea/RT).Chapter 7ConclusionIn this study, reaction rates for the gas-phase abstraction by Mu of H from each of CH 4and C 2 H6 have been measured using //SR over the temperature ranges 626-821 K and511-729 K respectively. The usual Arrhenius plots for each data set are linear, withreasonably small estimated uncertainties in the calculated parameters. Any large degreeof tunneling in these reactions for the temperature ranges studied is not evident. The Eafor the two reactions, 24.66 and 15.35 kcal/mol respectively, are 11.5 and 5.5 kcal/molhigher than for the corresponding H atom reactions.In the absence of theoretical calculations, the large Ea increases seem to indicatedrastic differences between the Mu and H variants of the title reactions, in location ofthe transition states on the potential energy surfaces. As well, the reaction rates ofthe excited vibrational states of CH 4 and C 2 H6 with Mu seem to be reduced less incomparison with the rates with H, than are their ground states.More cannot be said about the isotope effects on the reaction dynamics until the-oretical or state-selected data, or more Mu kinetic data with gas-phase polyatomicmolecules, are available. Since this is the first study of Mu kinetics in the gas-phasewith molecules of this size, it may be, like the product internal state correlations ofGermann et al. [43,44] on the same reactions, that these results are only "anomalous"in comparison with results from smaller molecules.88Bibliography[1] Wilson, S., Diercksen, G.H.F., eds., "Methods in Computational MolecularPhysics", Plenum Press, New York, 1991.[2] Truhlar, D.G., ed., "Potential Energy Surfaces and Dynamics Calculations",Plenum Press, New York, 1981.[3] Clary, D.C., ed., "The Theory of Chemical Reaction Dynamics", D. 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There are nine plots for the reactionMu + CH4 MuH + CH 3and six forMu + C 2 H6 MuH + C 2 H 5and the parameters k obtained from these are plotted in the Arrhenius plots of Chap-ter 6.In each plot, each point shown represents two independent measurements of A:one taken from a histogram collected by counters above the reaction vessel, and onefrom counters below the vessel. These are averaged in the plot for clarity but treatedas independent in fitting the straight line. In general these are within one standarddeviation of each other.Runs of zero reactant are pure N2 ("A0 ") runs. Others were done in every case suchthat muon stopping density was constant.Following each plot is a table showing the concentrations of reactant with the (av-eraged) measured relaxation rate A.961.0841.0841.4452.2120 0.006841 + 0.0039090.032131 ± 0.0042170.033157t 0.0047680.033575 ± 0.0031760.044450 + 0.003713Appendix A. Plots of Relaxation Rate Data^ 97[CH4 ]/(10 2° molecule cm -3 )^A/its-1Appendix A. Plots of Relaxation Rate Data^ 98[CH4 )/(102° molecule cm -3 )0^0.028330 + 0.006849^1.338 0.055432 + 0.0024411.964^0.069139 ± 0.0028080.37720.79211.0941.5091.8330 0.013785 + 0.0034160.031677 ± 0.0026560.044984 + 0.0042390.055439 ± 0.0041400.067621 + 0.0038620.087387 ± 0.004411Appendix A. Plots of Relaxation Rate Data^ 99[CH4]/(10 2° molecule cm -3 )^A/ps-1Appendix A. Plots of Relaxation Rate Data^ 100[CH4]/(102° molecule cm -3 )^Ahis'0^0.039469 + 0.0030380 0.039799 + 0.0074930.5710^0.063544 + 0.0061730.9328 0.088089 + 0.0033661.380^0.108702 + 0.0082521.832 0.123167 + 0.0042060.54200.90341.3371.7710 0.015220 + 0.0032890.059092 + 0.0051210.093238 + 0.0040160.133327 + 0.0052210.164787 + 0.006295k = 8.631±0.328x10 -1® C m 3molecule -1s -1Mu + CH, --* MuH + CH3 , T = 691±5 K0.180.160.140.120.10To4_ 0.080.060.040.020.000 0 0.2 0.4 0.6 0.8[CH 4 ]/(102°1.0^1.2^1.4^1.6molecule cm -3 )1.8 2 0[CH4 ]/(10 2° molecule cm -3 )^Ahis-1Appendix A. Plots of Relaxation Rate Data^ 101Appendix A. Plots of Relaxation Rate Data^ 102[C114 ]/(10 2° molecule cm -3 ) A/0 0.0392290 0.0476010.3160 0.1078150.6671 0.1432001.004 0.2306341.487 0.303956± 0.008633+ 0.006306± 0.005788± 0.004836+ 0.005752+ 0.0240550.32180.64810.98921.4670 0.010411 + 0.0036470.105085 ± 0.0045920.168750 + 0.0054120.265820 + 0.0078770.312369 + 0.009205Appendix A. Plots of Relaxation Rate Data^ 103[CH4 ]/(10 2° molecule cm -3 )1.80.2 0.4 0 6 0.8 1.0 1.2 1.4 1.6[CH 4 ]/(102° molecule cm 3)0.37500.79301.1751.5490 0.014257 + 0.0040890.284509 ± 0.0064730.554011 ± 0.0314050.794630 ± 0.0258301.025124 + 0.037914Appendix A. Plots of Relaxation Rate Data^ 104Mu + CH 4 -* MuH + CH 3 , T = 776±9 K1.201.050.900.75„, 0.60•-< 0.450.30k = 6.818±0.117x10 -t5 cm3molecule -is -10.150.000.0[CH4 ]/(10 2° molecule cm -3 )^Ahts-10.21340.40190.54740.75830 0.031125 + 0.0048470.367023 + 0.0113440.774750 + 0.0247080.891673 ± 0.0277941.164704 ± 0.040282Appendix A. Plots of Relaxation Rate Data^ 105[CH4 ]/(10 2° molecule cm -3 )^Ahts-1Appendix A. Plots of Relaxation Rate Data^ 106[C 2 H6 ]/(10 2° molecule cm -3 )^Ahis-10^0.004630 + 0.003799^1.426 0.054850 + 0.0042422.003^0.063688 + 0.0052972.442 0.077297 + 0.0057020.40850.81701.2171.2171.6340 0.011852 ± 0.0082640.097486 + 0.0047220.201049 ± 0.0077980.281719 ± 0.0098800.287138 ± 0.0110720.357131 ± 0.014504Appendix A. Plots of Relaxation Rate Data^ 107[C 2 116]/(102° molecule cm-3 )^A/ps-1Appendix A. Plots of Relaxation Rate Data^ 108[C 2 116]/(10 2° molecule cm -3 )^AhLs-10^0.006841 + 0.0039090.3976 0.215061 + 0.0076660.7952^0.394024 ± 0.0096621.105 0.504192 + 0.0139820.33140.71561.0471.6190 0.013774 + 0.0016810.300854 + 0.0069000.593513 + 0.0168790.828243 ± 0.0284401.260624 + 0.047012Appendix A. Plots of Relaxation Rate Data^ 109[C 2 H6]/(102° molecule cm -3 )^Ah.ts-10.25220.50430.75651.0091.2971.6570 0.015220 ± 0.0032890.428846 ± 0.0100480.830209 + 0.0223431.233618 + 0.0388231.380476 ± 0.0498291.687174 + 0.0764202.251196 ± 0.103084Appendix A. Plots of Relaxation Rate Data^ 110[C 2 H6 ]/(102° molecule cm -3 )^Ahis-10.20730.40390.61300.68500.78780 0.010411 ± 0.0036470.763854 + 0.0337791.986954 + 0.0733912.225998 + 0.1407492.818614 + 0.1496352.829827 + 0.201463Appendix A. Plots of Relaxation Rate Data^ 111[C 2 H6 ]/(10 2° molecule cm -3 )

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