摘要:

FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 75 1983 Intramolecular Kinetics THE FAR AD AY DIVISION THE ROYAL SOCIETY OF CHEMISTRY LONDONFARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 75 1983 Intramolecular Kinetics -WE FARADAY DIVISION 2OYAL SOCIETY OF CHEMISTRY JONA GENERAL DISCUSSION ON Intramolecular Kinetics 18th, 19th and 20th April, 1983 A GENERAL DISCUSSION on Intramolecular Kinetics was held at the University of Warwick on 18th, 19th and 20th April, 1983. The President of the Faraday Division, Professor D. H. Whiffen, FRS, was in the chair: about 150 Fellows of the Faraday Division and visitors from overseas attended the meeting. Among the overseas visitors were : Mr. A. Aspiala, Finland Prof. T. Baer, U.S.A. Dr. J. E. Baggott, U.S.A. Dr. G . D. Billing, Denmark Dr.Biron, France Prof. M. T. Bowers, U.S.A. Dr. W. A. Brand, West Germany Prof. J. Brickmann, West Germany Prof. P. Brumer, Canada Dr. L. Carlsen, Denmark Dr. E. Castellucci, ItaZy Mr. J. H. Catanzarite, U.S.A. Prof. F. F. Crim, U.S.A. Prof. X. De Hemptinne, Belgium Mr. H-R. Dubal, West Germany Dr. G. Dujardin, France Prof. G. E. Ewing, U.S.A. Dr. J. M. Figuera, Spain Prof. A. Foffani, Italy Dr. C. Fotakis, Greece Prof. Y. Haas, Israel Dr. P. Hackett, Canada Dr. E. J. Heller, U.S.A. Prof. J. Hepburn, Canada Prof. R. M. Hochstrasser, U.S.A. Dr. L. Holmlid, Sweden Dr. J. M. Hutson, Canada Dr. J. M. Jasinski, U.S.A. Prof. S . Koda, Japan Dr. P. J. Kuntz, West Germany Dr. S . Leach, France Prof. E. K. C. Lee, U.S.A. Mr. M. Lewerenz, West Germany Prof.C. Lifshitz, Israel Dr. P. Longin, France Prof. J. P. Maier, SwitzerZand Prof. R. A. Marcus, U.S.A. Dr. R. D. McAlpine, Canada Prof. C. B. Moore, U.S.A. Prof. J. T. Muckerman, U.S.A. Dr. R. Naaman, Israel Dr. G. S . Ondrey, West Germany Prof. I. Oref, Israel Prof. D. W. Oxtoby, U.S.A. Prof. C . S . Parmenter, U.S.A. Dr. E. Pollak, Israel Prof. M. Quack, West Germany Prof. R. P. H. Rettschnick, The Netherlands Prof. S . A. Rice, U.S.A. Mr. T. R. Rizzo, U.S.A. Dr. J. Rostas, France Prof. F. S . Rowland, U.S.A. Mr. S . Ruhman, Israel Dr. K. Rynefors, Sweden Prof. J. Santamaria, Spain Dr. J. L. G. Suijker, The Netherlands Prof. J. Troe, West Germany Prof. S . Tsuchiya, Japan Dr. C. A. G. 0. Varma, The Netherlands Dr. G. Vazquez, Mexico Miss K. von Puttkamer, West Germany Mrs. E.Weger, West Germany Dr. B. Whitaker, France Prof. C . Wittig, U.S.A. Prof. I. Yamazaki, Japan Mrs. T. Yamazaki, Japan Prof. R. N. Zare, U.S.A. Prof. A. H. Zewail, U.S.A.Organising Committee Prof. J. P. Simons (Chairman) Dr. M. S. Child Prof. R. J. Donovan Dr. G. Hancock Dr. D. M. Hirst Prof. K. R. Jennings Dr. R. Walsh ISBN: 0-85186-658-1 ISSN : 030 1-7249 Printed in Great Britain by Fletcher & Son Ltd., Norwich.CONTENTS Page 7 23 45 57 77 89 103 117 131 141 155 173 183 197 21 1 223 The Spiers Memorial Lecture; Vibrational Redistribution within Excited Electronic States of Polyatomic Molecules by C. S . Parmenter Intramolecular Relaxation of Excited States of C6F6 + by G. Dujardin and S. Leach Isomerization of Internal-energy-selected Ions by T.Baer, W. A. Brand, T. L. Bunn and J. J. Butler Kinetics of Ion-Molecule Collision Complexes in the Gas Phase. Experiment and Theory by M. T. Bowers, M. F. Jarrold, W. Wagner-Redeker, P. R. Kemper and L. M. Bass Intramolecular Decay of Some Open-shell Polyatomic Cations by J. P. Maier, M. Ochsner and F. Thommen GENERAL DISCUSSION On the Theory of Intramolecular Energy Transfer by R. A. Marcus Pulsed Laser Preparation and Quantum Superposition State Evolution in Regular and Irregular Systems by R. D. Taylor and P. Brumer A Quantum-mechanical Internal-collision Model for State-selected Uni- molecular Decomposition by B. K. Holmer and M. S. Child The Correspondence Principle and Intramolecular Dynamics by E. J. Heller GENERAL DISCUSSION Intramolecular Dephasing.Picosecond Evolution of Wavepacket States in a Molecule with Intermediate-case level Structure by D. D. Smith, S. A. Rice and W. Struve Energy Conversion in Van der Waals Complexes of s-Tetrazine and Argon by J. J. F. Ramaekers, H. K. van Dijk, J. Langelaar and R. P. H. Rettschnick Time-dependent Processes in Polyatomic Molecules During and After Intense Infrared Irradiation by K. von Puttkamer, H-R. Dubal and M. Quack Energy Distributions in the CN(X 'C') Fragment from the Infrared Multiple- photon Dissociation of CF,CN. A Comparison between Experimental Results and the Predictions of Statistical Theories by J. R. Beresford, G. Hancock, A. J. MacRobert, J. Catanzarite, G. Radhakrishnan, H. Reisler and C. Wittig Product Energy Partitioning in the Decomposition of State-selectively Excited HOON and HOOD by T.R. Rizzo, C. C. Hayden and F. F. Crim239 251 289 30 1 315 33 1 341 365 377 387 395 407 424 428 430 Low-power Infrared Laser Photolysis of Tetramethyldioxetan by S. Ruhman, 0. Anner and Y. Haas GENERAL DISCUSSION Unirnolecular Reactions Induced by Vibrational Overtone Excitation by J. M. Jasinski, J. K. Frisoli and C. B. Moore Unimolecular Decomposition of t-Butylhydroperoxide by Direct Excitation of the 6-0 0-H Stretching Overtone- by M-C. Chuang, J. E. Baggott, D. W. Chandler, W. E. Farneth and R. N. Zare Picosecond-jet Spectroscopy and Photochemistry. Energy Redistribution and its Impact on Coherence, Isomerization, Dissociation and Solvation by A. H. Zewail Energy Redistribution in Large Molecules. Relaxation in the Gas Phase with Picosecond Gating by R. Moore, F. E. Doany, E. J. Heilweil and R. M. Hochstrasser GENERAL DISCUSSION Direct Study of Intramolecular Rotation-dependent Intramolecular Processes of S02(A ' A z ) in a Supersonic Jet by H. Watanabe, S. Tsuchiya and S. Koda Role of Rotation- Vibration Interaction in Vibrational Relaxation. Energy Redistribution in Excited Singlet Formaldehyde by N. L. Garland and E. K. C. Lee Sub- Doppler Spectroscopy of Benzene in the " Channel-three " Region by E. Riedle, H. J. Neusser and E. W. Schlag Intramolecular Electronic Relaxation and Photoisornerization Processes in the Isolated Azabenzene Molecules Pyridine, Pyrazine and Pyrimidine by I. Yamazaki, T. Murao, T. Yamanaka and K. Yoshihara GENERAL DISCUSSION ADDITIONAL REMARKS LIST OF POSTERS INDEX OF NAMES

ISSN:0301-7249    

DOI:10.1039/DC9837500001

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1983, 75, 7-22 Introductory Lecture Vibrational Redistribution within Excited Electronic States of Polyatomic Molecules BY CHARLES S. PARMENTER Department of Chemistry, Indiana University, Bloomington, Indiana 47405, U.S.A. Received 18th April, 1983 The diversity of intramolecular vibrational redistribution (IVR) studies displayed in this Discussion testifies to the high interest in this fundamental aspect of polyatomic reaction dynamics. Moreover, the fact that the Discussion has attracted so many responses on a problem that has been under active study for at least 30 years demon- strates both the importance and the elusive nature of IVR. IVR is truly a child of theory, first arising within the competing R.R.K.M. and Slater theories of thermal unimolecular dissociation.' R.R.K.M.theory included IVR as a necessary prerequisite to dissociation, whereas IVR was explicitly excluded in the Slater picture. The dichotomy sparked lively discussions and much experi- mental interest, leading to the demonstration in 1960 by Butler and Kistiakowsky of IVR as in experimental reality in molecules with high (reactive) vibrational excitation. While the R.R.K.M.-Slater debates are now quiet, in part because of such demon- strations, theoretical discussions retain their intensity. Fundamental descriptions of quasiperiodic and chaotic behaviour are being sought,' and better descriptions of dense field coupling and 1VR dynamics are being developed. The papers in this Discussion and a recent issue of the Journal of Physical Chemistry devoted almost entirely to the topic of chaos illustrate the point. The early experimental approach to IVR was dominated by the productive efforts of Rabinovitch and coworkers that still continue^.^ Their work provided several methods to populate energy-selected regions high in the vibrational manifold where excitation is sufficient for chemical reaction.Coupled with tests for subsequent IVR that used the chemical clocks of reactivity, their studies revealed the ubiquity and fast time scales (picoseconds) of IVR in these high regions. They also probed the issue of ergodicity, demonstrating how difficult it is to hold IVR to a limited region of vibrational phase space. In contrast, much of the recent experimental progress on IVR has been down and out, so to speak. It is down in the sense that attention has increasingly focused on ground-state systems with lower vibrational energies.It is out in the sense that new technologies have allowed experiments to branch out with a diversity of techniques, many of them spectroscopic. Again, this Discussion reflects well these ingenious adventures, as do several summaries.'*6 Some of the newer experiments, such as high vibrational overtone spectroscopy and photodissociation, probe with relatively precise preparation, IVR in the high reactive regions of the vibrational manif~ld.~ Interest in infrared multiphoton8 VIBRATIONAL REDISTRIBUTION WITHIN POLYATOMIC MOLECULES dissociation has generated a variety of IVR studies both in beams and bulk gases t h d probe IVR at much lower energies.' Vibrational product distributions as seen in infrared fluorescence also search lower regions,6 as do infrared fluorescence experi- ments after pumping the first C-H overtone in variety of polyatomic~.~ Most recently, infrared fluorescence after excitation of C-H fundamentals has brought the IVR probe to the 3000 cm-' region of ground-state rnanifolds.'O As an ensemble, these and other experiments on ground-state systems are com- pelling on several issues. Most importantly, they show that IVR is a general property of ground-state manifolds, found in regions with low state densities (< 100 per cm-l) and occurring all the way up to high regions of the quasi-continuum responsible for chemical reactivity.Exceptions exist, but they are few.6 In spite of these triumphs, the experimental characterization of IVR remains largely incomplete in the sense that no comprehensive study of IVR from various vibrational regions within a molecule has yet been possible.The systematic explora- tion of the molecular parameters that control IVR dynamics has remained largely out of reach. This in turn has contributed to the most pressing deficit in discussions of IVR, namely the absence of a predictive theory or even empirical correlations. The dilemma stems in large part from the experimental difficulty of monitoring ground-electronic-st ate vibrational populations with the required sensi tivi ty . The problem is particularly compounded by the need to impress fast time resolution (nanosecond, picosecond) on these measurements.Substantial experimental advances will certainly be brought to these problems, and perhaps much of the progress will derive from fast non-linear spectroscopies. In the meanwhile, one can retreat (or advance) to IVR in excited electronic states where some of these experi- mental problems are greatly eased. INTRAMOLECULAR VIBRATIONAL RELAXATION WITHIN EXCITED ELECTRONIC STATES Bright ultraviolet laser sources tunable over wide regions with controlled band widths in either pulsed or C.W. operation offer close control over initial vibrational excitation. By selective pumping of S1 t So absorption features, opportunity occurs in many molecules for exploration of IVR characteristics after preparation of zero- order levels varying systematically in mode structure and energy.Resolved S1 -+ So fluorescence spectroscopy provides a monitor of Sl vibrational identities which, when combined with bright pump lasers, generates detection sensitivities that exceed those in the infrared by orders of magnitude. Furthermore, the short S, radiative lifetime (commonly nanoseconds) provides a built-in clock for excited-state processes that is often made even more useful by faster non-radiative electronic-state decays. Thus nanosecond or sub-nanosecond time limits are imposed on detection even without deliberate efforts. The short radiative lifetime has an even more important con- sequence, however, that is now being exploited in several laboratories. As several papers of this Discussion illustrate, sufficient optical sensitivity exists to monitor vibrational flow with imposed picosecond fluorescence timing.These experiments are the forerunners of direct explorations of IVR dynamics. Excited-state studies also encounter substantial experimental liabilities. Thermal congestion or inhomogeneous broadening in electronic absorption spectra is far more extensive than encountered with ground-state infrared pumping and is a severe problem for room-temperature experiments. Even with narrow-band sources, the ability to restrict pumping to single zero-order S1 vibronic levels or even narrow regions is severely compromised by sequence- and hot-band structure. Thus room-CHARLES S. PARMENTER 9 temperature studies are restricted to low resolution with respect to single-level preparation or restricted to molecules with special characteristics (such as high symmetry) that endow them with reduced congestion.For this problem, the supersonic nozzle expansion is the great redeemer. The reduction of inhomogeneous broadening by rotational cooling to near 1 K and vibrational cooling to tens of K enormously expands the opportunities. As an example, it has opened remarkably large molecules such as the macro-ring system free-base phthalocyanine l1 or ten-ring aromatics l2 to spectroscopic study of IVR. Supersonic beams will be the dominant technology for excited-state IVR studies. A second limitation inherent in spectroscopic probes of excited-state IVR stems from the restricted span of the Franck-Condon absorption envelope. The severe Franck-Condon attenuation of S1 t So transitions that terminate on higher S1 levels restricts most studies to the first 3000 or 4000 cm-I of the S1 vibrational manifold. In fact the bulk of present S1 IVR data concerns levels below 2000 cm-' excess energy.This is quite a different vibrational world from that historically considered in the context of IVR. In this low region, level densities can be calculated by direct counting, anharmonic normal modes seem to provide acceptable vibrational descriptions and one can discuss the zero-order vibrational identity of the levels initially pumped. Thus the language of IVR in these studies is more detailed than in ground- state probes of higher regions. The interconnections between the low-region IVR results of these studies and the picture that emerges from the high-energy ground-state work will surely become a topic of lively discussion.As recent papers ill~strate,'~*'~ information about IVR within excited electronic states can come from the photophysics of excited-state decay rather than from explicit spectroscopy. IVR is manifested in these decay measurements by the control that redistribution has over the details of radiationless transitions, and inference about IVR characteristics from photophysical measurements follows appropriate modelling. A most interesting suggestion from early discussions of IVR and photo- physics concerned the wide variation in IVR lifetimes, extending in cases to as long as hundreds of nanosecond^.^^ The more recent IVR characterizations from spectro- scopic measurements do not yet confirm these suggestions.Much more work is required before enough is known about IVR from independent sources to secure its role in non-radiative decay. In contrast, we might term the spectroscopic probes of S, IVR as " direct '' studies, and in spirit this is correct although some of these also rely on rather subtle modelling. Three complementary spectroscopies have been used both with supersonic beams and in room-temperature measurements. Homogeneous linewidths as inferred from the rotational contours of S, t So absorption bands may reveal rovibronic interactions within the S, manifold that, given proper initial preparation, are associated with IVR. The prime challenge is (a) to get an accurate measure of this width from complicated spectra and (b) to understand the cause of that width.With respect to the latter, IVR becomes a compelling cause when the linewidths correspond to relaxation times much faster than the known S, electronic state lifetime. Naphthalene is such a case.16 IVR lifetimes on order of lo-', s are inferred from the absorption linewidths in supersonic-beam experiments. A striking example has been provided recently by Riedle et al. for IVR in S, benzene,17 and in fact the work is described in one of the papers of this Discussion.18 Two-photon sub-Doppler spectroscopy has exposed many individual rotational lines in vibronic bands of the benzene S, +- So transition. The spectra show most bluntly a rotationally selective broadening of rotational lines in a band associated with ca. 3400 cm-' of excess vibrational energy.K = 0 structure remains sharp at the present10 VIBRATIONAL REDISTRIBUTION WITHIN POLYATOMIC MOLECULES resolution whereas K > 0 structure is severely broadened. The authors note that linewidths in many bands reaching thse vibrational regions l9 correspond to relaxation times far shorter than the relaxation time of the S, population itself.20 Hence the broadening must be associated with intrastate coupling, namely IVR. These remark- able data show that rovibronic coupling rather than purely vibronic coupling is basic to IVR in this domain. The other spectroscopic approaches to excited-state IVR use S1 -+ So fluorescence spectroscopy in two variants. Collsion-free S, + So fluorescence spectra can reveal IVR within the S1 electronic state by extra structure in fluorescence that usually appears in the form of unresolved congested emission after pumping a single zero- order vibronic state.The natural clock of the s, lifetime puts a long limit on the time scales of IVR that produces such congestion. The more direct approach is to impose timing on these collision-free spectra. Picosecond fluorescence spectroscopy is just beginning to display its role in discovering S, IVR characteristics. There is little doubt that as improved techniques are worked out, these experiments will be among the more important consequences of picosecond technology. Reviews have described the operation of these 1VR spectroscopies.21p22 Our present interest concerns the results that are coming from such experiments. There now exists a sufficient corpus of S, IVR data to reveal some general aspects of 1VR within excited states.These and related issues are discussed in the sections to follow. LOW IVR THRESHOLDS The ability to excite a wide variety of known initial S, vibrational distributions in a given molecule is among the principal virtues of excited-state IVR studies. Such excitation allows a systematic study of IVR as functions of excess vibrational energy and initial vibrational identity. The first questions asked in such explorations have been extremely simple. As one climbs the S, vibrational manifold, at what point does one first observe indications of IVR? Answers are now available for many molecules, and there is uniform agreement. The IVR thresholds are always low, in fact amazingly low in the historic context of ground state IVR studies.The thresholds in a variety of aromatics and heteroaromatics occur within the first few thousand cm-I of excess vibrational energy where vibrational level densities range from a few states (or less) per cm-I to hundreds of states per cm-’. Before commenting on these findings it is useful to discuss the meaning of “ IVR thresholds ” as derived from these fluorescence experiments. The collision-free fluorescence experiment is understood within the context of the level-mixing model shown in fig. 1. Consider a field of zero-order (e.g., harmonic) levels, not necessarily overlapping by their widths, that are mixed with average coupling matrix elements VLl FZ Vsl. The Is) level has large oscillator strength (on account of a favourable Franck-Condon factor) to a thermally populated level in the ground electronic state whereas the { \ I ) } levels do not.In the absence of coupling, the Is) level would give rise to a bright and discrete feature in the S, t So absorption spectrum even though that level is surrounded by a thick field of S, levels. In real life, the coupling disperses the Is) character in a quasi-Lorentzian distribution over the field, so that the absorption now has the structure of the dispersion. If the level structure is sufficiently sparse, the dispersed multi-line structure can be directly observed, but this has been a rare situation for cases other than few-level Fermi resonances.* More commonly, the coupling width is so small relative to the experimental resolution (often limited by * An impressive recent example is the observation of absorption structure as a result of mixing between an SI rovibronic 1s) state and rovibronic states { 11)) in the triplet manifold of pyra~ine.~’CHARLES S .PARMENTER 11 thermal inhomogeneous broadening) that the Is} state dispersion is not readily detected. In these cases the absorption spectrum may still appear " sharp " under conditions of only moderate resolution. The fluorescence spectrum, however, is extremely sensitive to such mixing. It can reveal mixing that is quite undetectable in absorption. If, for example, the excitation resolution was sufficient to pump only one of the absorption components (a single mixed molecular eigenstate), the ensuing collision-free fluorescence spectrum will be rich in structure since it is, in effect, a superposition of zeroth-order (e.g.harmonic) I s> Fig. 1. Schematic diagram of the mixing of zero-order vibrational levels (or rovibrational levels) within a single electronic state. The average coupling matrix elements Vsl and Vzz are expected to be equal. Within the S1 state, the state 1s) has a large Franck-Condon factor so that it is optically accessible from the So zero-point level. The bath levels { Il)} are " dark " on account of small Franck-Condon factors. single vibronic level fluorescence spectra, one from each zero-order vibrational identity contained in the mixed state. In the more common situation, the excitation bandwidth spans much or all of the coupling width, and the ensuing fluorescence is a superposition of so many SVL spectra that the fluorescence congestion becomes a quasi-continuum. By this means, " sharp " absorption can yield unstructured collision-free fluorescence. The threshold measurements generally rely on the appearance of unstructured emission as the indicator of VIR when an excitation laser is tuned to successively higher S1 regions.Fluorescence congestion is a sensitive marker, and the appearance of congestion beyond that expected from thermal inhomogeneous broadening is a sure indication of the extensive level mixing that is a necessary prerequisite of IVR. Drawing further conclusions about IVR from such collision-free spectra can be a bit tricky. The IVR dynamics are particularly difficult to elucidate since, as many have pointed out, the dynamics of IVR are sensitive to the state p r e p a r a t i ~ n .~ ~ - ~ ~ In the limit of single molecular eigenstate pumping no time evolution occurs, but the state mixing will produce congested fluorescence. At the opposite extreme, coherent pumping of the entire eigenstate package will result in evolution from pure Is} at the time of excitation to the full mixed-state vibrational identity at later times. The collision-free fluorescence will reflect this evolution with an abundance of unstructured emission if the evolution time-scale is much shorter than the S, fluorescence lifetime.12 VIBRATIONAL REDISTRIBUTION WITHIN POLYATOMIC MOLECULES In contrast structured fluorescence from Is) will occur if the evolution is slow compared with the S, lifetime.Structured emission may also occur if the excitation bandwidth limits pumping to only a small subset of levels over which the Is) character is distri- buted, and this structure will occur even if the vibrational time evolution is much faster than the S1 fluorescence lifetime. Alternatively, structure can persist in the presence of extensive mixing and fast IVR if the set {IZ)} is small. From these choices it is clear that relative contributions of Is) and unstructured (IZ}} emission to collision-free fluorescence cannot by themselves be particularly informative about the time-dependence of the S1 vibrational identity of IVR dynamics. Congested fluorescence only reveals the necessary state mixing of IVR. It is in this sense and only this sense that we talk about IVR thresholds as derived from collision- free fluorescence.Table 1. Upper limit of IVR thresholds within S1 electronic states as determined from collision-free S1 -+ So fluorescence molecule threshold vibrational p v l b vibrational a source /cm-l degrees of per cm-' temperature freedom /K p-difluorobenzene g-fluorotoluene coumarone 1 -azaindolizine indole ethyl benzene naphthalene azulene (S,) t-butyl benzene anthracene stilbene tetracene pentacene ovalene p-O-CH3 P-NHZ-V)-C~H~ v-CEFEC-C~H~~ 1760 1590 720 1110 1440 760 930 2570 670 530 1380 1200 730 1900 500 1060 1280 30 39 39 39 39 42 48 48 48 66 66 72 72 84 87 102 132 10 300 10 300 <1 300 -1 300 3 300 cold cold 400 cold <1 300 cold cold cold cold cold cold cold cold ref. (29) b C C C d ref.(28) ref. (33) ref. (28) ref. (39) ref. (48) e g C f .f f ' 300 K are room-temperature bulk experiments. " Cold " refers to measurements from super- K. W. Holtzclaw, A. E. W. J. B. Hopkins, D. E. Powers and D. E. Powers, J. B. Hopkins and R. E. Smalley, A. Amirav, U. Even and J. Jortner, J. Chem. Phys., 1981,75,3370; D. E. Powers, J. B. Hopkins and sonic beams. b C. s. Parmenter and B. M. Stone, unpublished results. Knight, C . S. Parmenter and B. M. Stone, to be published. R. E. Smalley, J. Chem. Phys., 1981, 74, 6986. J. Chem. Phys., 1980,72,5721. Opt. Comrnun., 1980,32, 266; Chem. Phys. Lett., 1980,69, 14. R. E. Smalley, J. Chem. Phys., 1981, 74, 5971. Table 1 summarizes many of the IVR thresholds that have been observed by the criteria of fluorescence congestion in collision-free experiments.These thresholds are in fact conservative limits, because other sources of congestion (such as thermal inhomogeneous broadening) tend to obscure the first IVR onsets. The quoted thresholds merely cite the lowest energies where it can be determined with certainty that the congestion exceeds that possible from non-IVR sources. True IVR onsets will generally be lower. The remarkable aspect of these data concerns the ubiquity of low thresholds amongCHARLES S. PARMENTER 13 these aromatic and heterocyclic systems. Low thresholds appear in every molecule, and IVR at low energies is clearly the rule. IVR turns on at first chance as one climbs the S1 vibrational ladder. The low energies are further emphasized by the threshold vibrational level densities estimated for a few cases in table 1.IVR occurs in truly sparse regions of the vibrational manifold, where densities are built primarily on the lowest ring frequencies. Stewart and McDonald lo have concluded that a rovibronic level density of 5-30 per cm-' characterizes the IVR thresholds in ground-state systems. S, thresholds are too indistinct to set such figures, but the data in table 1, based on vibronic level densities, could well be in accord. Zewail 27 in this Discussion has offered an intriguing correlation of thresholds with low-frequency modes of large polyatomics. FROZEN MODES AND PROMOTING MODES A singular aspect of collision-free fluorescence concerns the persistence of IVR congestion from all levels that have been probed above the threshold.The experi- ments have yet to reveal " frozen modes " above the threshold, i.e. zero-order modes for which level mixing is so restricted that the IVR indicators are absent. Whereas frozen modes have so far failed to appear, it must be noted that the searches are forced to sample only a few types of modes in any molecule. The vibrational excitation depends on choosing those S, levels that have unusually large Franck-Condon factors with the So zero-point level. Thus the IVR probes see mainly the progression forming modes (often appearing in combination levels) as excitation climbs the S, ladder. Smalley's group have reported a particularly interesting series of probes designed to look for frozen modes or barriers to IVR in S, substituted benzenes.2s As reported in table 1, early thresholds (530 and 930 cm-') are observed in cold-beam fluorescence as excitation pumps a benzene-ring mode coupled to an alkyl side chain that provides level density.If that chain is isolated from the ring by a -C=C- linkage or a -0- linkage, the early thresholds persist and inhibitions to the IVR coupling fail to appear. An even more pervasive coupling was discovered in S, p-alkylanilines, where pumping of an -NH2 inversion mode overtone (730 cm-') displayed IVR coupling through the ring to ap-alkyl side chain. These studies emphasize the message of table 1 : IVR couplings are indeed ubiquitous in low-S, regions of ring systems. Refinements of these probes, especially with time-resolved experiments, will surely reveal evidence of the variable coupling strengths and fields that must exist in these systems.In fact, collision-free experiments presently give indications of sub- stantial sensitivity of IVR to excitation of a low-frequency out-of-plane ring mode ( ~ 3 0 ) in S, p-diflu~robenzene.'~ The sensitivity is seen as a marked boost in fluores- cence congestion as excitation pumps combination bands carrying increasing quanta of ~ 3 0 . As described below, time-resolved studies confirm the special activity of v~~ in IVR. Further suggestions of this activity appear also in the high-resolution S, f- So absorption of p-difluor~benzene.~~ One observes that the apparent sharpness of a combination band 5',31, (&b = 2069 cm-l) is markedly degraded by the addition of a vi0 component to give the band 51,3;30,1 (&b = 2189 cm-').While quantitative estimates of the linewidths are yet to be reported, it is known that any additional linewidth must come from intrastate IVR couplings since the S, electronic state lifetimes are themselves about the same for each S , level.31 We find further comments about this elusive aspect of IVR in several papers of this Discussion. The experimental work of Chuang et al.32u describes an intriguing14 VIBRATIONAL REDISTRIBUTION WITHIN POLYATOMIC MOLECULES exploration of variable IVR couplings in t-butylhydroperoxide (Bu'OOH) as energy is pumped into the 6th 0-H overtone (ground electronic state) and -0-0- bond dissociation provides a molecular clock. Marcus 32b considers theoretically the possibility of damping coupling among a system of oscillators -C-C-C-M- C-C-C- by a massive central atom M.ERGODIC MIXING The IVR mixing model presented earlier allows us to view congested fluorescence as a superposition of SVL fluorescence spectra. As such, the fluorescence provides opportunity for experimental comment on whether IVR coupling is ergodic. Two probes have been completed, both using the same approach. Smalley and co- workers 33 have modelled by computer simulation the congestion appearing in the cold-beam fluorescence spectrum of naphthalene pumped to S, &vib = 3068 cm-', where the level density ranges from 400 to 2000 per cm-' depending upon the sym- metries allowed for inclusion in the calculation. They observe that the lesser density is adequate to simulate the observed congestion, but that a subset of these states would be insufficient.By this test, IVR appears ergodic, at least within a limited symmetry class. It is not possible to learn about the further participation of other field states. Dolson 34 has completed analogous simulations for fluorescence from S, p- difluorobenzene with dlb = 2189 cm-l, where the ungerade level density is ca. 40 per cm-l (an ungerade S1 level is initially pumped via a sequence band). These results are markedly different. The density falls far short of being sufficient to provide the required congestion. Ergodicity becomes a moot issue, for it is apparent that an additional source of level density is being used in IVR by S, p-difluorobenzene. That density is, by all odds, provided by rotational levels through Coriolis vibration- rotation coupling.ROTATIONAL LEVEL CONTRIBUTIONS TO IVR There is extensive documentation of vi bration-rotation coupling among dense fields of levels in a closely related problem, namely that of S1-T radiationless transi- tions between electronic states. The intersystem crossing (ISC) is modelled with the same phenomenology as the IVR problem of fig. 1, with the exception that Is) and ( ] I ) ) now are zero-order levels of different spin states. Early indications of rotational- level participation in the { \ I ) ) state density came from studies of ISC dynamics in intermediate-case molecules such as methylglyo~a1,3~9~~ biacetyl 35 and ~yrimidine.~~ In each molecule, analysis of kinetic data showed that the density of levels in the coupled field far exceeded that available from vibrational levels alone.There is no clear source of additional level density other than rotational levels brought in by Coriolis rotation-vibration coupling. Separate studies of quantum beats due to S,-T coupling in methylgly~xal,~~ pyrazine 39y40 and other molecules further emphasize the importance of rovibronic states in the intrastate coupling. An antilevel-crossing study of S,-T-level mixing in glyoxal 41 shows coupling with rovibronic mixing. Additionally, an increasing body of data illustrates the rotational dependence of decay dynamics in S, states where intermediate case S,-T coupling has important control over the S, r e l a ~ a t i o n . ~ ~ * ~ ~ * ~ ~ ~ ~ ~ Another demonstration is provided by super- sonic-beam spectra of the s, +- So pyrazine where the individual molecular eigenstates of fig.1 are explicitly resolved (in pyrazine Is) is a pure spin singlet and (IZ)} are triplet rovibronic levels). These rovibronic couplings are entirely consistent with the pyrazine quantum beats found earlier by Zewail's g r o ~ p . ~ ~ * ~ OCHARLES S. PARMENTER 15 With such a body of evidence concerning the importance of rotation-vibration mixing in electronic-state combination, it would be quite remarkable if such mixing did not contribute also to the IVR characteristics in both ground and electronic states. Several lines of evidence now show S, rotational-vibronic mixing in IVR coupling. The beautiful sub-Doppler experiments of Riedle et al.discussed earlier are per- haps the most explicit.18 The selective disappearance (broadening) of K > 0 rotational structure in absorption bands reaching to higher S1 demonstrates extensive participa- tion of rotation-vibration coupling within the S, state. An equally compelling, but indirect, indication of rotation-vibration coupling in IVR is provided by the threshold data in table 1. As described above, the threshold vibrational level densities in room-temperature studies are too low to support the extent of congestion observed in the fluorescence, and by default it is difficult to find the required level density from a source other than rotation-vibration coupling. including one of this In symmetric and near-symmetric tops, for example, the rotation-vibration coupling precludes K from being a good rotational quantum number.Thus while overall angular momentum and hence J must be conserved among the coupled levels, the AK = 0 requirement is relaxed. Since each rovibronic level is (2J+ 1) degenerate, the accessible level density is increased by order J. In an alternativeview, if the AK = 0 rule amongst coupled levels is relaxed, then an initial rovibronic level can mix with nearly resonant rovibronic levels built on vibronic levels that span a large energy range around the initially pumped level. Consider, for example, the near-prolate top p-difluorobenzene with S, rotational constants ( A - B) = 0.13 cm-'. An initial rovibronic level with, say, J = 50, will be nearly resonant with rovibronic members of vibronic levels whose energies are some- where within the K stack energy for K = 50.The boost in density by such coupling has been discussed in several From E(rot) z J(J + l ) B + ( A - B)K2 the K stack energy for a given J is ( A - B)J2. This energy is 325 cm-l for p-difluoro- benzene with J = 50. Thus some rovibronic level with J = 50 from every vibronic floor within a span 325 cm-l will be accessible for near-resonant coupling (in the absence of AK and symmetry restrictions). Thus in the approximation that the vibronic level density is constant over this span, the rotation-vibration coupling boosts the effective density by the factor ( A - B)J2. In either treatment, mixing scales a s f ( J ) . If selective excitation or cooling can limit the initial J to low values, a marked inhibition on IVR coupling should be apparent when contrast is made with high-J excitation. Thus a general test for rota- tional participation in LVR is derived from the J dependence of IVR.Fig. 2 shows two comparisons using collision-free fluorescence from S, p-difluoro- benzene. Temperature has been used to bias initial J values, using the extremes of room-temperature fluorescence, where J z 30-50 excitation is common, and super- sonic cold-beam fluorescence, where the rotational temperature is (presumably) a few K. After pumping a lower level (&vib = 2069 cm-'; P v i b z 30 per cm-l), the reduc- tion in unstructured background emission in the cold-beam spectrum is dramatic and substantially exceeds the change that occurs from merely reducing thermal inhomo- geneous broadening. There is little doubt about the contributions of rotations to IVR after this initial excitation.The second pair of spectra contrast fluorescence after exciting a higher level In this case the difference in fluorescence is small and possibly nil. Thus there is sufficient vibronic coupling at this level density = 2888 cm-l; P v i b z 300 per cm-').16 VIBRATIONAL REDISTRIBUTION WITHIN POLYATOMIC MOLECULES F to sustain nearly complete fluorescence congestion even at low-Jexcitation. Whether rotations provide an additional coupled level density cannot be discerned from these spectra alone. For this case it would be necessary to see the appropriate absorption band at sub-Doppler resolution as per the benzene example,lS or alternatively, to monitor the IVR dynamics directly.IVR DYNAMICS The interplay between experiment and theory shows a nice inversion symmetry in working out the related processes of IVR and electronic radiationless transition. Radiationless transitions in isolated molecules were first exposed by experiment, and subsequently many lively discussions concerning the theoretical interpretation of these transitions occurred before the theory was in place. The theory then provided new expectations for the dynamics of radiationless transitions that were soon confirmed by experiments. These experiments are, in fact, still continuing today with increasing The history of IVR is in many aspectsjust the opposite. Theory in the form of the R.R.K.M. discussions of unimolecular reactions emphasized the central importance of IVR to reaction dynamics before any explicit experimental evidence was available. The theory inspired an abundance of ingneious experimental approaches to the problem, with the work of Rabinovitch's group being particularly important in early (and continuing) experimental demonstrations of IVR.Radiationless-transition theory later provided a detailed quantum-mechanical description of IVR with specific predictions that IVR dynamics should display the full spectrum of characteristics sublety. 17 9 23,40.42.45 - 47CHARLES S. PARMENTER 17 found in radiationless transitions. Experiments that probed these IVR dynamics in any detail have come forward only in the past few years. To date, these experiments are only from studies of IVR within excited electronic states.Whereas experimental work on radiationless transitions gave early suggestions that excited-state IVR might operate with sub-microsecond time-scales in low vibra- tional regions of a number of large polyatomics,15 no IVRs have proved so leisurely in subsequent experiments. The time-scales seem to be rather those of nanoseconds and picoseconds. The fast IVR times have proved difficult to handle, and the first direct probes of IVR dynamics have only now begun to emerge. Those first results show that IVR is quite obedient to radiationless-transition theory, as expected. Zewail’s group have observed S, LVR dynamics at &vib% 1400 cm-’ in a cold anthracene beam.27*39 In this threshold region the collision-free spectrum (without time resolution) shows sharp structure attributable to Is) emission and a congested background of { Il}} emission.With picosecond pulsed excitation the time dependence of a structured component shows quantum beats of about nanosecond frequency. These dynamics are much different than the decay of the electronic state itself, and hence the behaviour must be that of IVR. In this aspect the IVR is consistent with the sparse-intermediate case of radiationless transitions with coupling of the Is) state to a small-field {IZ)) and coherent excitation of a small number of these mixed states. Stilbene has provided another example of IVR dynamics.48 In experiments with time resolution of 100-300 ps, IVR can be observed at the S, trans-cis isomerization threshold of ca.1200 cm-’. A third example of direct observation is reported in this Discussion 49 by Hoch- strasser’s group. In this case, time resolution of several picoseconds has been imposed on spectrally dispersed fluorescence. In S , p-difluorobenzene they have been able to observe the time evolution of structured 1s) emission as well as background (Il)) emission. The room-temperature experiments reveal IVR with a time-scale of < 10 ps operating at &,ib % 1616 ern-', where the vibrational-state density is less than ten levels per cm-l.- Very likely this IVR is operating on rotation-vibration mixing, as is observed so commonly in the low-threshold regions. This fast IVR should be contrasted with a much slower time evolution (500 ns) for an S, p-difluorobenzene level at &vib FZ 2190 cm-’ reported by Halberstat and Tramer,” and other indications of slower IVR from levels with &,ib > 2000 cm-l in this molecule.Perhaps the fast IVR at &kib% 1616 cm-’ is associated with pumping of an initial level 3,303, which contains high excitation in via, a probable IVR promoting mode. These time-resolved experiments are the first direct view of IVR dynamics, and they foreshadow much work to come, particularly as picosecond methods become more generally accessible. It is certain that a much fuller understanding of IVR will be an important product of picosecond technology. In the meantime, an alternative approach to picosecond fluorescence spectroscopy has been developed at Indiana University. It is easily operated with conventional fluorescence technology using either C.W.or pulsed laser excitation. While the method is indirect, it appears to give a view of fluorescence spectra with time resolution reaching to times shorter than 10 ps. Whereas nothing yet reveals that the method gives time-resolved spectra that differ from direct picosecond spectra, an explicit comparison has yet to be made. The comparisons await complete direct picosecond spectra. The indirect method uses molecular or chemical timing imposed by gas collisions in a fluorescence cell. A gas with the property of quenching the S, electronic state with a large cross-section is added to the fluorescence cell. In the presence of this gas, emission occurs only from molecules that have not suffered a deactivating18 VIBRATIONAL REDISTRIBUTION WITHIN POLYATOMIC MOLECULES collision between the absorption act and spontaneous emission.Thus by adjusting the collision interval with added gas pressure, the fluorescence window can be con- trolled from the full s, lifetime (often nanoseconds) to < 10 ps. Molecular oxygen, with an S, quenching cross-section of about half gas-kinetic, has been the most convenient timing gas, but others (NO, CS,) also appear effective. Examples of the time-resolved spectra have been provided with benzene,22 p-difluoro- benzene 2 2 v 5 1 and p-fluor~toluene.~~*~~~~~ Typically when high added gas pressures establish short timing windows, IVR is revealed by a boost in Is) emission intensity relative to unstructured {IZ)) emission. In cases where IVR is so extensive that no discrete structure occurs in untimed emission, (for example the &vib = 2888 cm-I p-difluorobenzene emission of fig.2) the time-resolved emission reveals structure consistent with that expected from initially pumped ls).22944 Since short timing requires high added gas pressures (typically ca. 30 kTorr or 4 MPa of O2 for 10 ps), this transformation is pressure line narrowing as opposed the more conventional broadening. Other examples where Is) structure appears even in untimed spectra are shown in fig. 3. PUMP 3'5' ~ ~ I I I I I I I ~ I 30 000 38 000 Elcm - PUMP 3' 5'30' I [ l " l l " ~ l F 5000 Y- L l l l l l l l l l l 30 000 38 000 E/cm-' Fig. 3. Time-resolved S1-So fluorescence from p-difluorobenzene excited to S, levels &,ib x 2070 cm-' (3'5l) or &,ib % 2190 Cm-l (3l5'30') The average emission time after excitation is shown to the left, with 5000 ps being the collision-free S, lifetime.Shorter times are achieved by quenching with added O2 at pressures ranging from ca. 2.6 kTorr (90 ps) to 30 kTorr (7 ps). Reabsorption due to increased p-difluorobenzene pressures has removed the higher-energy fluorescence structure from the timed 3l5' spectra. The spectra have been provided by K. W. Holtzclaw.CHARLES S. PARMENTER 19 It is interesting to contemplate the extent to which chemical timing introduces collisional vibrational redistribution or other collisional perturbations into the observed spectra. If the quenching cross-section was much larger than that for other collisional processes, the method would appear safe.Molecules would disappear from experimental view well short of other state-changing perturbations. It is known, however, that cross-sections for vibrational relaxation into fields of S, levels are generally as large or larger than hard-sphere even for small gases such as CO, N2 or Ar.47 Thus collisional redistribution would seem a serious complication in chemical timing. The observed pressure narrowing of unstructured emission immediately reveals that collisional redistribution is not dominant. In addition, examination of spectra such as those of fig. 3 at, say 10 cm-I fluorescence resolution, shows abundant detail in [ s) emission that remains remarkably insensitive to added O2 pressures exceeding 30 kTorr, where the congested background is severely Thus the collisional redistribution fails to appear in these specific searches.Why is collisional redistribution absent ? A rationale is found in measurements of collisional vibrational relaxation within S, states of molecules decaying under intermediate case radiationless transition^.^^ The vibrational relaxation cross- sections (for, say, the partner N2) are consistently orders-of-magnitude less than that in statistical-case S, states. We understand this reduction on the basis of mixed electronic states in the intermediate-case molecules. Those states typically have dominant triplet components 25 (whereas statistical-case states do not). Hence there is small probability of vibrational-state change to produce other vibrational levels in fluorescing singZets (there is correspondingly high probability of producing another vibrational state in the " dark " triplet manifold).In the O2 problem, the electronic quenching interaction necessarily involves mixed electronic states. Although the intermediate or even final states may not be well understood, the important aspect is the development of restricted S, character in the molecule under attack with ensuing immunity to collisional vibrational redistribution. It has been commented 54 that even a spectrologist can recognize the opportunity to follow Is) decay by IVR in such spectra as those of fig. 3. It is apparent that a qualitative difference separates the two cases. Emission from the level 3,5' (cVib = 2069 cm-l) in p-difluorobenzene shows a marked change in the ratio of structured to unstructured intensity in the early stages of timing (5000-500 ps) without further evolution as the window becomes shorter.The opposite behaviour exists for emission from the higher level 3'5,301 where the spectrum is largely insensitive to timing until the window becomes shorter than ca. 500 ps. Clearly the IVR times are widely separated for the two cases. Further, since the structure persists at 5000 ps even when the time evolution is much faster, the IVR must be non-exponential. Such decay is not surprising since both levels are in the threshold region. Both sets of spectra have been analysed 55 within kinetic models based on radiation- less transition (i) In neither case does Is) decay, as seen in the time-dependent ratios of structured and unstructured fluorescence intensities, fit the statistical limit.(ii) Both decays are entirely consistent with inter- mediate case IVR, implying nonexponential decays and recursion of Is) character. (iii) The initial Is) decay times or dephasing times are, as anticipated, widely separated. That from the lower level is ca. 1000 ps. That from the higher level is ca. 50 ps. Both IVR times are short relative to the collision-free S, lifetime. The persistence of Is) structure at the full S1 lifetime demonstrates the intermediate case recursions or non-exponential decay. Details aside, these spectra provide additional evidence that radiationless transition Several findings emerge.20 VIBRATIONAL REDISTRIBUTION WITHIN POLYATOMIC MOLECULES theory is an appropriate description of IVR.With details admitted, however, some interesting points arise. The variation in initial dephasing times (50 vs 1000 ps) is remarkable when it is placed in context of initial-level energies and vibrational densities. The levels are separated by only ca. 120 cm-l and the densities (30 vs 40 per cm-l) are almost the same. Thus attention must be placed on the initially excited mode. The two levels Is) differ only by the presence of a quantum of vi0 in the higher level. The acceleration of IVR dephasing from this higher level reinforces the proposition 29 that vi0 is an IVR promoting mode. A second detail concerns the size of the rovibronic-coupling matrix element (VsI w VIl of fig. 1). Analysis of the data from the higher level by the intermediate case model gives V,, z value far below the anharmonic matrix elements of 1-10 cm-l commonly found in analyses of Fermi resonances. One suspects that cm-I is far more representative of typical interactions among members of the field rather than those few special interactions so large that they can be observed by strong level perturbations in spectra.It must be recalled that the room-temperature- cold-beam comparisons of fig. 2 suggest that the derived V,, w cm-l is largely dominated by rovibronic couplings. Many of those couplings must be of high order so as to relax effectively the AK restrictions on rotation-vibrational level mixings. The chemical-timing IVR studies in p-difluorobenzene provide an additional demonstration of rotational participation in IVR.The demonstrations occur in a study of the effect of initial level preparation on the IVR dynamics from the 3'5'30' level at 2189 cm-l. The vibrational level density is pvib z 40 per crn-l, so that the levels are >0.02 cm-I apart on average. Since this distance exceeds the intrinsic level width, it should be possible to limit coherent pumping to a single (mixed) level. In this limit, no time evolution should occur (although the fluorescence may well show congestion since the levels are mixed). Such an experiment is in progress 56 using a single mode of a C.W. argon-ion laser. While a full analysis is not yet available, the timed spectra show distinct evolution of structure, not markedly different from the experiments in fig. 3 that used the same laser on all longitudinal modes.The single- mode time evolution makes it clear that IVR operates on a density of states far larger than that provided by vibrational levels alone, i.e. on the rovibronic density as dis- cussed above. The single-mode experiments emphasize a point described in several papers of this Discussion 57*58 and elsewhere.26 Although it is yet to be demonstrated, the IVR dynamics must be sensitive to the initial conditions of level preparation. For the S1 experiments, such sensitivity centres on the coherence bandwidth of the exciting laser and the size of the coherence relative to the intrinsic IVR coupling width in the molecule. If the laser coherence width is substantially less than the IVR width, the dynamics may reveal more about the laser than the intrinsic IVR character.This is a difficult experimental issue in IVR dynamics that will require substantial future attention. cm-l, IVR: EXCITED VS GROUND ELECTRONIC STATES A pressing question concerns the correspondence between IVR characteristics as seen in the ground and upper electronic states. There will be of course differences due to vibrational frequency changes. Differences may also occur in specific cases where geometries change or where vibronic coupling becomes dominant in the upper electronic state. In addition, fast non-radiative processes in upper electronic states may broaden levels to overlap in regions where discrete molecular eigenstates wouldCHARLES S. PARMENTER 21 otherwise exist. All of these effects will produce differences in detail, but none introduce true uniqueness that is general to any aspect of S1 IVR behaviour. Several resourceful studies now specifically illustrate the similarities between IVR within different electronic states.Both concern thresholds. Smalley and co- workers 59 have been able to probe IVR state mixing in the ground electronic state of the alkylbenzenes for which S, thresholds are established. The result is “ n o difference”. Low thresholds persist also in the So states, and the data in the two electronic states look remarkably alike. Stewart and McDonald lo have looked for the IVR state mixing in the ground electronic state of 23 molecules of widely varying types by pumping a C-H fundamental and searching for subsequent infrared fluorescence from states other than that initially pumped.Since these are collision- free cold-beam experiments, such fluorescence is certain revelation of IVR mixing. Again, low thresholds are the rule in these systems just as in the excited-state studies. These data are so extensive that they will probably become the benchmark for IVR thresholds. This work is supported by the National Science Foundation (U.S.A.). I am grateful to the present and former members of my research group and to colleagues elsewhere for enlightening discussions and instruction. D. W. Noid, M. L. Koszykowski and R. A. Marcus, Annu. Rev. Phys. Chem., 1981,32, 267. J. N. Butler and G. B. Kistakowsky, J. Am. Chem. Suc., 1960,82, 759. J . Phys. Chem., 1982,86, issue 12. I. Oref and B. S. Rabinovitch, Acc.Chem. Res., 1979, 12, 1966. S. P. Wrigley and B. S. Rabinovitch, Chem. Pnys. Lett., 1983, 95, 363. K. V. Reddy, D. F. Heller and M. J. Berry, J. Chem. Phys., 1982, 86, 2814. P. A. Schulz, Aa. S. Sudbla, D. J. Krajnovich, H. S. Kwok, Y . R. Shen and T. Y . Lee, Annu. Rev. Phys. Chem., 1979, 30, 379. D. J. Nesbitt and S. R. Leone, Chem. Phys. Lett., 1982, 87, 123. P. S. H. Fitch, L. Wharton and D. H. Levy, J . Chem. Phys., 1979,70, 2018. ti D. J. McDonald, Annu. Rev. Phys. Chem., 1979, 30, 29. lo G. M. Stewart and D. J. McDonald, J. Chem. Phys., 1981,75, 5949; 1983,78, 3907. l2 A. Amirav, U. Even and J. Jortner, Opt. Cummun,. 1980,32, 266. l3 G. Dujardin and S. Leach, Faraday Discuss. Chem. SOC., 1983, 75, 23. l4 M. D. Morse, A. C. Puiu and R. E. Smalley, J.Chem. Phys., 1983, 78, 3735. l 5 C. Tric, Chem. Phys., 1976, 14, 189. l6 S. M. Beck, D. L. Monts, M. G. Liverman and R. E. Smalley, J. Chem. Phys., 1979, 70, 1062. l7 E. Riedle, H. J. Neusser and E. W. Schlag, J. Phys. Chem., 1982, 86,4847. l8 E. Riedle, H. J. Neusser and E. W. Schlag, Faraday Discuss. Chem. Soc., 1983, 75, 387. l9 J. H. Callomon, J. E. Parkin and R. Lopez-Delgado, Chem. Phys. Lett., 1972,13, 125. 2o L. Wunsch, H. J. Neusser and E. W. Schlag, 2. Naturfursch., Teil A , 1981,36, 1340. 21 R. E. Smalley, J . Phys. Chem., 1982, 86, 3504. 22 C. S. Parmenter, J. Phys. Chem., 1982, 86, 1735. 23 B. J. van der Meer, H.Th. Jonkman, J. Kommandeur, W. L. Meertsand and W. A. Majewskie, 24 K. F. Freed and A. Nitzan, J. Chem. Phys., 1980, 73, 4765. 25 An example from the analogous problem of intraelectronic state mixing occurs in this Dis- cussion: D.D. Smith, S. A. Rice and W. Struve, Faraday Discuss. Chem. SOC., 1983, 75, 173. z6 W. Rhodes, J. Phys. Chem., 1983,87, 30. 27 A. Zewail, Faraday Discuss. Chem. SOC., 1983, 75, paper 19. 28 J. B. Hopkins, D. E. Powers, S. Mukamel and R. E. Smalley, J . Chem. Phys., 1980, 72, 5049 29 S. H. Kable, W. D. Lawrance and A. E. W. Knight, J. Phys. Chem., 1982, 86, 1244. 30 T. M. Dunn, personal communication and as reported in ref. 22. 31 C. Guttman and S. A. Rice, J. Chem. Phys., 1974, 61, 661; L. J. Volk and E. K. C . Lee, J. Chem. Phys. Lett., 1982, 92, 565. and references therein. Chem. Phys., 1977, 76, 236.22 VIBRATIONAL REDISTRIBUTION WITHIN POLYATOMIC MOLECULES 32 (a) M.C. Chuang, J. E. Baggott, D. W. Chandler, W. E. Farneth and R. N. Zare, Faraday (6) R. A. Marcus, Faraday Discuss, Chem. Soc., 1983, Discuss. Chem. Soc., 1983, 75, 301. 75, 103. 33 S. M. Beck, J. B. Hopkins, D. E. Powers and R. E. Smalley, J. Chem. Phys., 1981, 74, 43. 34 D. A. Dolson, Ph.D. Thesis (Indiana University, 1980). 35 R. van der Werf and J. Kommandeur, Chem. Phys., 1976, 16, 125; R. Van Der Werf, E. Schutten and J. Kommandeur, Chem. Phys., 1976,16, 151. 36 R. A. Coveleskie and J. T. Yardley, Chem. Phys., 1976, 13, 441, and earlier references therein. 37 K. G. Spears and M. El-Manugch, Chem. Phys., 1977,24,65. 38 J. Chaiken, M. Gurnick and J. D. McDonald, J. Chem. Phys., 1981, 74, 106. 39 W. R. Lambert, P. M. Felker and A. H. Zewail, J. Chem. Phys., 1981,75, 5958; A. H. Zewail, 4n P. M. Felker, W. R. Lambert and A. H. Zewail, Chem. Phys. Lett., 1982, 89, 309. 41 M. Lombardi, R. Jost, C Michel and A. Tramer, Chem. Phys., 1981, 57, 355. 42 See for example: S. Okajima, H. Saigusa and E. C. Lim, J. Chem. Phys., 1982, 76, 2096; G. ter Hoort, D. W. Pratt and J. Kommandeur, J. Chem. Phys., 1981, 74, 3616; H. Baba, M. Fujita and K. Uchida, Chem. Phys. Lett., 1980,73,425; G. ter Hoorst and J. Kommandeur, J. Chem. Phys., 1982,76, 137. W. R. Lambert, P. M. Felker, J. Perry and W. Warren, J. Phys. Chem., 1982, 86, 1184. 43 D. B. McDonald, G. R. Fleming and S. A. Rice, Chem. Phys., 1981,60, 335. 44 D. A. Dolson, C. S. Parmenter and B. M. Stone, Chem. Phys. Lett., 1981, 81, 360. 45 Y. Matsumoto, L. H. SpangIer and D. W. Pratt, Chem. Phys. Lett., 1983, 95, 343. 46 I. Yamazaki, T. Murao, T. Yamanaka and K. Yoshihara, Faraday Discuss. Chem. SOC., 1983, 47 A. E. W. Knight, J. T. Jones and C. S . Parmenter, J. Phys. Chem., 1983, 87, 973. 48 J. A. Syage, W. R. Lambert, P. M. Felker, A. H. Zewail and R. M. Hochstrasser, Chem. Phys. 49 R. Moore, F. E. Doany, E J. Heilweil and R. M. Hochstrasser, Faraday Discuss. Chem. Soc., 75,395. Lett., 1982, 88, 226. 1983, 75, 331. N. Halberstadt and A. Tramer, J. Chem. Phys., 1980, 73, 6343. 51 R. A. Coveleskie, D. A. Dolson and C . S. Parmenter, J. Chem. Phys., 1980,72, 5774. 52 D. A. Dolson, C. S. Parmenter and B. M. Stone, Proc. NATO Advanced Study Institute: Fast Reactions in Energetic Systems (Reidel, Amsterdam, 1981), p. 433. 53 A. E. W. Knight and C. S . Parmenter, J. Phys. Chem., 1983, 87, 417. 54 J. H. Callomon and R. W. Field, personal communication. 55 R A Coveleskie, D. A. Dolson, K. W. Holtzclaw and C. S. Parmenter, to be published. 56 K. W. Holtzclaw (Indiana University), personal communication. 57 R. D. Taylor and P. Brumer, Faraday Discuss. Chem. Soc., 1983, 75, 117. 58 K. von Puttkamer, H. R. Diibal and M. Quack, Faraday Discuss. Chem. Soc., 1983,75,197. 59 J. B. Hopkins, P. P. R. Langridge-Smith and R. E. Smalley, J. Chem. Phys., 1983, 78, 3410.

ISSN:0301-7249    

DOI:10.1039/DC9837500007

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1983, 75, 23-43 Intramolecular Relaxation of Excited States of BY GBRALD DUJARDIN AND SYDNEY LEACH * Laboratoire de Photophysique Moleculaire du CNRS,t Bitiment 21 3, UniversitC de Paris-Sud, 91405 Orsay Cedex, France Received 27th January, 1983 Fluorescence quantum yields, qF(u’), and lifetimes, ~ ( v ’ ) , of selected vibrational levels of electronic excited states of C6F6’ have been measured by counting coincidences between threshold photoelectrons and ion-fluocescence photons (T-PEFCO technique). The derived non-radiative rates k,,(u’) for the B 2A2u state correspond to intramolecular electronic non-radiative transitions (ENRT) via coupling to high vibrational levels of the X *Elg ground state. Radiative rates kr and their variation with internal energy reflect ezectronic properties whereas the variations of knr also reveal vibrational dynamics of the ion.The knr(u’) values exhibit (i) a monotonic quasi-exponential increase for each of the two vibrational progres- sions 1” and 1”2l with increasing vibrational energy E, and (ii) mode-selective behaviour, knr(u’) being enhanced when mode 2 is excited. Model calculations of the vibrational part of knr(u’) were carried out on a non-communicating, harmonic-oscillator basis ; the results reproduce qualitatively the experimental findings. Good quantitative agreement is found between experimental and calculated values of knr(U’) for low values of E,. Progressive deviations which occur as E, incre_ases are interpreted as indicating the gradual onset of vibrational redistribution in the B state.This constitutes a new method for studying vibrational non-radiative transitions (VNRT). With increasing E,, the VNRT processes are considered to be analogous to the sparse-level (I; E, = 0-2500 cm-’), intermediate (11; E, = 2500-4500 cm-’) and statistical-limit (111; E, > 4500 cm-’) cases, respectively, of ENRT. Emission in coincidence with ex_cit_ation of the c 2B2u state is discussed and shown to involve coupling between and mixed B,X vibronic levels. 1. INTRODUCTION This study concerns two types of intramolecular relaxation processes of excited electronic states. These are (i) electronic non-radiative transitions (ENRT) in which coupling occurs between two or more electronic states (fig. 1) and (ii) vibrational non- radiative transitions (VNRT), involving intramolecular vibrational redistribution due to mode coupling within a single electronic state.ENRT can give rise to fluorescence quantum yields, qF, of the initially excited state, which are less than unity, especially in the case of large polyatomic species. VNRT can occur both in the optically excited state Is), which couples to the final states { I } , and in the latter. We will dis- tinguish these as VNRT(s) and VNRT(Z) processes. The species studied is a molecular ion, C6F6+ ; we have reviewed elsewhere the specific advantages of molecular ions for the study of intramolecular radiationless transitions.’,* We report experimental data on fluorescence quantum yields qF(j,u’) and life- * Also at Departement d’Astrophysique Fondamentale, Observatoire de Paris-Meudon, 921 90 t Laboratoire associe a I’UniversitC de Paris-Sud.Meudon, France.24 RELAXATION OF C6F,+ EXCITED STATES times z(j,u') of energy-selected levels of hexafluorobenzene cations, where j is the electronic and 0' the vibrational state initially excited. Radiative and non-radiative relaxation rates, k,(j,v') and k,,,(j,v'), respectively, are derived and are used to explore both ENRT and VNRT processes in C6F6'. Variations in the k, rate as a function of internal energy are shown to reflect electronic properties of the ion, whereas variations in k,, chiefly reveal its vibrational dynamics. A preliminary account of this work has been given el~ewhere.~ Two methods have been used to measure qF and z of energy-selected levels of 2 , v" X " \\\\ Fig.1. Vibronic coupling scheme of electronic non-radiative transitions in C6F6' as applied in model calculations (see text). C6F6+. Maier and Thommen4 have photoionized C6F6 with He(1) radiation and measured coincidences between energy-analysed photoelectrons (resolution E 100 meV) and fluorescence photons (PEFCO technique). The present work uses a different technique; level selection is by monochromatized V.U.V. radiation from a synchrotron-radiation source coupled with detection of photoelectrons having zero kinetic energy, and coincidence measurements are made between the threshold photoelectrons and ion-fluorescence photons (T-PEFC0).5 The energy resolution in our T-PEFCO study ((45 meV) makes it possible to study specific vibronic levels.Our time resolution enabled us to measure shorter lifetimes (> 1 ns) than in the PEFCO experiments, where the lower limit is of the order of 15 ns. This made it possible to determine directly decay rates of a greater number of levels in the T-PEFCO experi- ments. A further characteristic of the T-PEFCO method is that ion states can be formed not only by direct ionization, as in the PEFCO technique, but also via auto- ionization processes. As will be seen from our results on hexafluorobenzene, this makes accessible some vibronic states of the ion that are difficult or impossible to form by direct ionization alone, e.g. in Franck-Condon gap 2. EXPERIMENTAL The experimental technique for obtaining threshold photoelectron spectra (TPES) and for counting coincidences between threshold photoelectrons and the fluorescence photons emitted by the C6F6' ion excited into selected vibronic levels has been described in detaiL5 We recall here only the salient features of the experiment.G.DUJARDIN AND S. LEACH 25 The photon source is synchrotron radiation from the Orsay storage ring (ACO), dispersed by a 1 m normal-incidence grating monochromator (McPherson, 2400 lines/mm). This continuously tunable light source was operated with a band pass of 2 A. Photoelectrons produced by photoionization of ajet of C6F6 ( p zi 5 x Torr) * are accelerated towards a time-of-flight electron spectrometer by a 1.4 V cm-’ electric field and are detected by a microchannel-plate electron multiplier. The kinetic-energy spectrum of the photoelectrons is determined by measurement of the time interval between a synchrotron radiation pulse (1.2 ns f.w.h.m., 13.6 MHz) and detection of the photoelectrons.Threshold photoelectrons (TPE) can therefore be detected by gating the electron detector after each synchrotron radiation pulse with a time window centred on the specific arrival time of the zero-kinetic-energy photoelectrons. We recall that our electron energy resolution is better than 45 meV. The time-of-flight analysis requires a number of calibrations and corrections which are detailed elsewhere.’ Threshold photoelectron spectra are obtained by counting the TPE as a func- tion of the photon excitation energy. At certain excitaJion energies the C6F6’ ion emits fluorescence photons mainly correspond- ing to the B 2A2,-X 2Elg transition in the 400-600 nm spectral region.6 T-PEFCO experi- ments are done as follows.Coincidence detection and counting are carried out using a time-to-amplitude converter whose start input and stop input are, respectively, the TPE signal and the fluorescence photon signal. The latter was detected by an R943 Hamamatsu photo- multiplier with appropriate electronics. The coincidence signals are stored in a multichannel analyser. The time scale is calibrated by the method used by Dujardin et al.7 The lifetime z of the energy-selected state is determined by fitting the accumulated coincidence curve to an exponential-decay curve using a least-squares method. The total number of true coin- cidences, Nc, during the time of the coincidence count is obtained by time integration of the T-PEFCO curve after subtraction of the false coincidence background.During the same time interval we count N,, the total number of threshold photoelectron_s. The TPE can result from direct ionization producing the emitting level of CbF6+, e.g. B 2A2u(u), and also from autoionization processes which can form not only the fluorescing state of the ion but also nzn-radiative isoenergetic vibrational levels belonging to lower electronic states, notably the X 2E1g ground state.’ Methods for obtaining the branching ratios for these processes in CsF6+ are described in detail elsewhere.’ These methods enable us to determine N,(j,u’), the number of TPE corresponding to formation of the j,u’ state of the ion. With the T-PEFCO technique, autoionization processes can increase the relative number of ions formed at certain j,u’ levels with respect to the PEFCO technique, in which only direct ionization processes are measured.This is particularly true in the so-called Franck-Condon gaps ; the T-PEFCO technique therefore permits us to measure the fluorescence decay characteristics of j,u’ levels whose access is difficult with the PEFCO technique. The fluorescence quantum yield &,d) is then determined from the relation: VF(j,V’) = N c / ~ e ( j , u ’ l h v (1) where f h v is the fluorescence photon detection efficiency, whose method of determination is described elsewhere. l9’ 3. RESULTS The TPES of C6F6 has been measured over the 9-29 eV excitation energy range.’ In fig. 2 is reproduced only the 11.8-13.8 eV region of the TPES.The B” 2A2u state origin of the C6F6’ ion is at 12.595 eV; two vibrational progressions in this state are indicated; these involve the two totally symmetric vibrational modes vl(alg), which is a C-F stretching mode of frequency 1515 cm-l, and v2(alg), a ring vibration of frequency 525 cm-’. Peaks assigned to autoionizing levels of a Rydberg series converging to the 2; 2B2u state at 13.88 eV are also indicated. They have the same energies as known Rydberg bands in the V.U.V. absorption spectrum of C6F6 ;’ analysis * 1 Torr = 101 325/760 Pa.26 RELAXATION OF C6F6' EXCITED STATES r 1 - 0.5 - 112.738 13 I 14 1 12 1.R (19) 1.E (C) photon excitation energy/eV Fig. 2. Threshold photoelectron spectrum of C6F6 in the region of the state of the ion.Two vibrational progressions (1" and l"2l) are indicated, involving the two totally symmetric vibrational modes v1 and v2. Also shown is a Rydberg series (n is the principal quantum numier; quantum defect 6 = -0.02) with its vibrational components v and converging to the C 2Bzu state at 13.88 eV. The circled numbers correspond to excitation energies at which pF and r measurements were made. The zero signal at 12.39 eV corresponds to a check of detector signal background in the absence of photon excitation. of these TPES and V.U.V. absorption Rydberg features is discussed el~ewhere.~ These Rydberg peaks are also quasi-identical in energy with &state vibronic levels of C6F6+ ; threshold autoionization to form the ion levels is thus fa~oured.~ A remark on the energy resolution is in order.The symmetry of C6F6 is high (&) and is retained in the electronic states of concern to us in the ion. Furthermore, mode distortion or displacement is relatively small for all known vibrational modes in going from C,F6 to the Consequently, apart from sequence bands (whose maximum relative intensity we estimate as no greater than 30% of the B(0Qband in the PES or TPES), only the two totally symmetric modes are excited in the B 2A2u state. The energy resolution in our TPES and TPEFCO experiments, although not high in comparison with normal optical spectroscopy techniques, is quite sufficient to resolve the bands corresponding to these vibronic excitations. The 10 circled numbers in fig. 2 correspond to ten of the photon excitation energies Eexc at which T-PEFCO measurements were made.A measurement was also carried out at 13.88 eV, the band energy of the 2; 2B2u state origin. These excitation energies are given in column 1 of table 1. Columns 2 and 3 give the measured fluorescence quantum yields qF and lifetimes z. Columns 4 and 5 give, respectively, the radiative (k,) and non-radiative (k",) rates for the 8 and state levels, derived from relations (2) and (3): 2A2u state of the k r = PF/T (2) knr = (1 - M Z . (3)G. DUJARDIN AND S. LEACH 27 Table 1. Fluorescence quantum yields, qF, lifetimes, z, and radiative (k,) and non-radiative (knr) rates of selected vibronic levels of C6F6+. The band assignments are discussed in ref. (5). photon band energ y/eV VF zjns kr/106 s-' kn,/106 s-l Ev/cm-' assignment 12.59 12.65 12.77 12.84 12.92 12.98 13.04 13.22 13.40 13.68 13.88 1.07 f 0.10 0.81 f 0.12 0.97 & 0.10 0.75 f 0.15 0.81 f 0.12 0.81 f 0.12 0.54 f 0.08 0.45 f 0.07 0.25 f 0.04 0.07 f 0.01 0.008 f 0.001 49 f 2 41 f4 46 f 4 42 f 4 41 h4 41 & 4 43 f 4 29 f 2 17 f 2 8 f 1 1.7 f 1 20.4 f 3.0 19.8 f 5.3 21.1 f 4.3 17.9 f 5.7 19.8 & 5.3 0 to 0.6 3.7 4.6 & 3.1 2.4 0.6 & o,6 4.6 6.0 f 3.8 3.7 4.6 & 3.1 19.8 f 5.3 12.6 & 3.2 4.6 f 3.7 3.1 3.1 10.7 f 2,6 4.0 19.0 f 3.5 8.5 44.1 & 6.8 18 116 f 14 15.5 & 3.7 14.7 f 4.6 8.8 f 2.7 835 584 & 216 4.7 f 2.0 0 500 1500 2000 2650 3150 3650 5100 6550 8800 10 400 B: oo 2' 1' 1'2l valley; (1 122 ?) l2 1 221 1321 1421 1522 ? c' oo Column 6 gives the selected internal energy, E,, of the B and c states with respect to the Oo level of the B" state at 12.595 eV: the corresponding vibrational assignments are given in column 7.4. RADIATIVE RATE kr ( j , ~ ' ) In fig. 3 we compare our T-PEFCO k, results with those reported by Maier and Thommen from their PEFCO results;4 for clarity in the figure we have not reproduced the error limits of the Maier and Thommen values, which are of the order of *2 x lo6 s-l. There is excellent agreement between the two sets of values for common excit- ation energies. From table 1 and fig. 3 we see that the radiative decay rate is approxi- mately constant, with a value k, = (18 &- 3) x lo6 s-l for the nine &v' levels up to and including the photon excitation energy EeXc = 13.40 eV.It decreases to ca. 9 x lo6 s-l at Eex, = 13.68 eV in the vicinity of the c state and further drops to ca. 5 x lo6 s-l at the c-state origin. We note that the k, rate for the C(Oo) level is ca. 25% of that of the &OD) state. This result shows that corrections must be applied to the non-radiative rates for the c state region of C6F6+ (Eexc = 13.88-14.28 eV) determined by Maier and Thommen from the relation k,, = k,(l - v)F)/v)F with the assumption that k, is the same for the B and ~ t a t e s . ~ 5 . NON-RADIATIVE RATE knr(B,v') 5.1. PRINCIPAL OBSERVATIONS The experimental values of the non-radiative rate k,, in the h a t e region (table 1) are plotted in fig. 4 as a function of internal vibrational energy E,. This figure also28 l o 8 3 I \ 2 m 10'- lo6 RELAXATION OF CsF,j+ EXCITED STATES - 2' 1' 2' 1*2' 13z1 142' (15 22) t oo I 1 I I 1 @f wp .P p I! f u OP 1' l 2 I I 1 lo9 l o 8 - - ~ ~ 2 ~ 1 a 1 V 1* 2' l 3 2'r-b...1' 2' 1 o6 10' - 10000 E,/cm-' Fig. 4. Non-radiative decay rate k,, of the &' levels of C6F6'. Rectangles indicate the uncertainty limits of experimental measurements. T-PEFCO values are taken from table 1 and PEFCO values are taken from Maier and Th~mrnen.~ Data on levels involving mode 2 excitation are given by cross-hatched rectangles.G. DUJARDIN AND S. LEACH 29 l"2l progression as well as 0' and l", with a resolution better than 45 meV. The agreement is good between the PEFCO and T-PEFCO results for the common values of E,. Our two principal observations (table 1 and fig. 4) are as follows: (1) We note first of all that the 1" (n = 0-4) and l"2' (n = 0-4) progressions, taken separately, each exhibit a monotonic, quasi-exponential, increase in k,, as E, increases.(2) We see from table 1 and fig. 4 that the k,, rate depends not only on the vibrational energy of the optically excitedjp' state but also on the specific optical vibrational mode excited. This type of behaviour has previously been observed in neutral species, e.g. benzene.11*12 In the case of C6F6' it is clear from fig. 4 that the k,, rate is enhanced, with respect to the 1" progression, when the v2 mode is excited simultaneously with 1" in the 1"2l combinations. Most of the rest of this paper is devoted to interpretation of these two results and requires first of all the presentation of an adequate theoretical model of ekxctronic non- radiative transitions.We will use the simplified notation knr(u') = k,,(B,u'). 5.2. MODEL CALCULATIONS OF k,,(u') RATES : THEORY Dujardin et al.,7 using the photoion-fluorescence-photon coincidence technique,lp7 have previously shown that ENRT in Auorobenzene cations corresponds to the so- called large-molecule case of radiationless transitions.12 ENRT from vibronic &' levels was shown to be due to internal conversion to isoenergetic 2,u" vibronic levels of the electronic ground state. In a model appropriate to this case (see e.g. fig. l), each initial Born-Oppenheimer Is) state (i.e. each 8,u' level) is coupled by the intra- molecular non-adiabatic potential uSz to the set of I states (2,~" levels).We remark that for C6F6' other non-radiative transitions such as intersystem crossing to quartet states or predissociation are not expected to exist in the energy region studied here. In particular, competitive dissociation processes do not set in until ca. 15.3 eV.13 Furthermore the low pressure (ca. 5 x lov4 Torr) in our experiments makes the effect of collisions negligible in comparison with the electronic intramolecular pro- cesses, as has been verified by pressure-variation experiment^.^ The irreversible transition of a vibronic Is) state of energy E, to quasi-isoenergetic E states (energy El) occurs with a rate given by We note that in this formulation the non-radiative rate kSl is equivalent to the experi- mentally determined k,, rate, which is extracted from the fluorescence quantum yield q&) and lifetime z(s) of the corresponding Is) vibronic level by relation (3).We stress the fact that for open-shell species such as C6F6+, coupling with high vibrational levels of the ground state is the only non-radiative relaxation pathway from the first excited state of the same multiplicity as the ground state, whereas in closed-shell species (e.g. C6H6) one or more triplet levels lies below the s1 level and can give rise to several other relaxation channels competitive with S1-So coupling. We therefore consider that interpretation of non-radiative transitions in open-shell species such as C6F6+ can be less equivocal than for analogous studies in closed-shell species.l Theoretical studies predicting the monotonic increase of knr(u') as E, increases have previously been carried out by a number of authors.12 Such studies have shown that this behaviour is expected for moderate and large values of the electronic energy gap30 RELAXATION OF C,F, EXCITED STATES Eo (we recall that E, = Eo + &,).I4 In our case, the electronic energy gap is large (Eo = 21 601 cm-'> and our experimental findings on the trend of k,, as a function of E, for each of the two progressions 1" and 1*2' are thus qualitatively consistent with the general theoretical results.Our own theoretical work reported here concerns more specifically model cal- culations of relative k,, rates for each of the two progressions. Our aim is to confirm the monotonic trend for each progression and also, more importantly, to see whether we can reproduce the experimentalIy observed vibrational mode selective dependence of knr.The starting point of model calculations of the relative k,,(u') values is eqn (4). The key is in the assumption that the overall interaction energies usl can be factorized into single products of electronic and vibrational factors.12a*b The electronic factor, whose calculation would be compIicated, is taken to be constant for different vib- rational levels of the B state [the Condon approximation following from the Born- Oppenheimer (B.O.) approximation; problems arising in its application to radiation- less transitions are discussed by Freed and Lin "]. The electronic factor can there- fore be factorized out in the relative rate caIculations. The latter then reduces to a calculation of the appropriate Franck-Condon factors taking into account the energy conservation condition represented by the &function in eqn (4). The electronic-vibrational factorization approximation requires that the Born- Oppenheimer separation of the electronic and nuclear motions be valid.This is reasonably the case in describing the initial vibronic &' states in which onIy totally symmetric modes are excited. However, the orbitally degenerate ground state of C,F,+ is affected by dynamic Jahn-Teller distortion via four non-totally symmetric degenerate eZs modes,6*16 so that the €3.0. separation may no longer be valid for the final 2,u" states. Nevertheless, we consider, and assume in the folIowing, that it is reasonable to caIculate the k,, rates wchin the B.O.approximation, in view of the fact that selection rules restrict accepting X,d' levels to those that are totally symmetric. PROMOTING MODES AND ACCEPTING MODES In order to carry out the model calculations we need to know which are the promoting and accepting modes (as defined by Lin)." Fur this we shall make use of group-theoretical selection rules. A promoting mode K is considered to be any vibrational mode which induces extensive non-adiabatic coupling between initial S and final L electronic states via a the electronic matrix element (ySl -1 yL) l2 occurring in usi, where QK is the nuclear 8 Q K normal coordinate associated with mode K. From the form of this matrix element the selection rules given by Nitzan and Jortner l4 imply that the direct symmetry products Ts x rL x rK and Ti x rp x r, must contain the totally symmetric represent- ation of the C6F6+ (&) point group, where Ti and rp are the vibrational symmetries of the initial.and final states. Since rs = A2u, and r, = El,, it follows that r, = elu. Three vibrational modes of C6F, have el,, symmetry.6 it has been shown that the promoting mode K acts to reduce the effective energy gap ED, the reduc- tion being equal to the vibrational energy wK. It has been shown by many authors l2 that k,, can be expressed as an exponentially decreasing function of the electronic energy gap E,,. From this energy-gap law we would expect that the greater is the reduction of the effective energy gap (i.e. the smaller is Eo - mK>, the higher is the non- radiative rate.In the absence of information as to how the electronic factor From an explicit calculationG. DUJARDIN AND S. LEACH 31 a a Q K (ty&--~tyL) varies with different el, modes, we thus consider the most effective promo& mode to be the el, vibrational mode with the highest frequency. In the following we shall assume that only this promoting mode is effective [mode v20 in ref. ( 6 ) ] ; its frequency is taken to be that of the same mode in neutral C6F6, (0.1~~ = 1530 cm-I), since its value for the C6F,+ cation is unknown. The propensity rule derived by Freed and Jortner l8 predicts that the transition probability is highest when one quantum of the promoting mode is found in the final vibronic state (cf. also Nielsen and Berry,19 who first propounded a similar vibrational hexis for the radiationless process of autoionization).We recall that only the totally symmetric v1 and v2 modes are optically excited, so that Ti = alg. The vibrational selection rule then gives us that in addition to the promoting mode, the final vibrational state contains only v, and v2 totally symmetric modes as accepting modes (or totally symmetric combinations involving non-totally symmetric modes, the latter giving rise to weaker higher-order couplings). In going from the initial S state to the final L state, vibrational modes could be distorted, i.e. change in frequency, and/or be displaced, i.e. change in equilibrium position. Since the C6F6+ cation retains its D6,, symmetry group in relaxing from the S electronic state to the final L state (as can be assumed from optical spectroscopy results 6*10p16) symmetry conditions restrict the possibility of displaced modes to the two totally symmetric modes.Furthermore, apart from the special case of the eZg Jahn-Teller modes in the ground state of C6F6+, mode-frequency changes appear to be relatively minor between the B and 8 states.6*10 We therefore considered all modes other than the totally symmetric modes 1 and 2 to be effectively constrained to being undistorted as well as undisplaced. In this case they would have the same quantum number in both S and L, and so we factorize them out of our relative k,, expression. THE NON-INTEGRAL OCCUPATION NUMBER (NION) METHOD The specific method we used for calculating relative k,,(v’) rates is derived from the non-integral occupation number (NION) method developed by Kiihn et a1.20 (see also Prais et a1.).21 It enables us to confront in practical fashion the delicate problem of ensuring that energy is conserved in the radiationless transition, i.e.to deal adequately with the &function in eqn (4). The NION method has been shown to be consistent with other methods used for relative k,, rate calculations, such as the “ saddle-point approximation ” 12b*20 but is of simpler application. It is well known that the evaluation of rate expressions can be achieved by the use of generating functions.18 The &function in eqn (4) can be represented by a Fourier integral. Evaluation of this integral by the saddle-point approximation requires the analytical evaluation of infinite sums.This is avoided in the NION version of the generating function method, in which the generalized density- of-states functions do not have to be factorized into products of matrix elements and the vibronic density of states. Furthermore, conservation of energy and evaluation of a sum in vibrational quantum numbers in the final state are facilitated by introduc- tion of a non-integral vibrational quantum number y. The NION method is presented in detail in ref. (20), where it is shown how one can obtained closed form expressions for the Franck-Condon factors involving y ; see also the discussion in ref. (12b). To calculate the k,,(u’) values we have to determine the vibrational dependence of ks+l from eqn (4). From our previous analysis it results that the accepting modes are the two optical modes vl and v2 of C6F6+.For an initial vibronic B,v’ state in-32 RELAXATION OF C6F6' EXCITED STATES cluding n, and 12, quanta of modes v, and v,, the vibrational part of the non-radiative rate is given by a sum of products of Franck-Condon factors: The sum is over all the integer values of Il from 0 to (Eo + n,co; + n20; - wK)/o; where the primed and double-primed co refer, respectively, to the initial B,u' and final 2,u" states. The non-integral quantum number y is defined by where AE(I,) = Eo + n,o; + n,oi - coK - I,co," (7) is the vibrational energy to be redistributed into mode 2 in the final state. We derived relation ( 5 ) for the case of two optical modes by extension 22 of eqn (2.8) of ref.(20) in which the NION method of calculation of non-radiative rates is expounded. We remark also that we hEve used the normal-mode description in characterizing the vibrational levels of the B (and 2) states. A local-mode description might be more appropriate, especially for the higher members of the 1" and 1"2l progressions. However, from the results of Kiihn et al.,O there is not expected to be any significant consequence in our NION calculations, which are carried out on a harmonic- oscillator basis, in the use of a normal- rather than a local-mode description. 5.3. MODEL CALCULATIONS OF k,, (U'): APPLICATION TO C6F6+ In calculating the non-radiative rate knr(nl,nz) by the NION technique, mode number 2 in relation (5) is conventionally assigned to the best accepting mode.20 In the well studied case of non-radiative transitions from the 2 1B2y state of neutral C6H6, the v, = 3073 cm'l C-H mode has generally been considered to be a far better acceptor than the other al, vibration, the C-C mode v2 = 992 crn-l;,O [however, recent calculations of Hornburger and Brand 23 for S,+So internal conversion show that the two modes are of comparable accepting quality and that other accepting modes can be important on a fully (vibrational) communicating model].From the symmetry selection rules we deduced that for non-radiative transitions from the B state of C6F6' the totally symmetric modes v, and v2 are the principal accepting modes, and we shall now discuss which is the best accepting one. The accepting character of a given mode is related to its degree of excitation in the final vibrational states (I).Many different isoenergetic ( I ) states are non- radiatively coupled to the initial Is) state, and consequently the relative non- radiative rate k,, is expressed in relation (5) by a sum of products of Franck- Condon factors. We denote k;,"" as the maximum term value of this sum and p a x as the final state associated with k;,"". We shall assume in the following that the accepting character of v1 and v2 is related to the vibrational composition of the Pax state. Table 2 shows in column 3 and 4 the numbers L1 and L2 of v1 and v2 vibrational quanta included in the I""" state for different J,u' optically excited levels. The corresponding vibrational energies El = Llcoz and E2 = L2co," are listed in columns 5 and 6.v1 will be the best accepting mode when El > E2 and v2 when E, > El. Table 2 shows that for the 1" progression, v2 is the best accepting mode for n = 0,1,2 but that v1 is a better accepting mode for n = 3 and 4; for the 1"2' progression v2 is a better acceptor than v1 for n = 0,1,2,3 and v1 better than v2 when n = 4.G. DUJARDIN AND S. LEACH 33 Table 2. Numbers L1 and L2 of v1 and v2 vibrational quanta included in the Zmnx state (_see text) and corresponding vibrational energies El and E2 for different optically excited B,v’ levels of C,5F,5+ optically excited B,v’ level EJcm - L1 L1 = El = L~o’; E2 = L20i v‘ = 1”12”2 /em- /cm- O0 2l 1’ 1 l2l l2 1 221 13 1321 14 1421 0 525 1515 2040 3030 3555 4545 5070 6060 6585 4 4 6 5 7 7 9 8 10 10 24.6 25.6 21.9 25.6 21.9 22.9 19.3 22.9 19.3 20.2 6 060 6 060 9 090 7 575 10 605 10 605 13 635 12 120 15 150 15 150 13 874 13 440 11 497 14 438 11 497 12 916 10 132 12 916 10 132 11 393 Whether vL or v2 is a better accepting mode therefore depends on the initial B,u’ level, so that no intrinsic relative accepting character can be given overall for vl and v2.We remark that for the 2 lBz,-8 ‘Al, internal conversion process in neutral benzene it has also been shown that the relative accepting qualities of particular modes can vary with Ev.23 Although the relative importance of v1 and v2 as accepting mode changes with excess vibrational energy E, in the state, we have uniformly taken the C-C mode v2 as the best accepting mode in the calculations reported in table 3.This does not affect the relative values of kn,(v’) since the latter are diminished by a constant factor of 10 when mode numbered 2 in relation (5) is switched from the C-C mode to the C-F mode. This result is interesting and requires further study. Table 3. Experimental and calculated relative non-radiative rates k,,(c’) of selected 1 state vibronic levels of CsF6+. Data used to determine relative experimental values of kn,(V’) are taken from table 1. The calculated knr(v’) values are normalized to the knr(v2) experimental value. optically relative k,,(o’) excited level experimental calculated 0 = oo v2 = 2‘ V 1 = 1‘ v1 + v2 = 1121 2v1= l2 2Vl + v2 = 1221 3Vl ---r 13 3V1+ v2 = 1321 4Vl= 14 4Vl+ v2 = 1421 <0.13 0.03 1 1 0.13 0.17 1.3 5 1 0.67 2.3 16.7 1.7 4.1 45 5 9.6 105 - - NION calculation parameter values (see text) : electronic energy gap, Eo = 21 495 cm-I ; accep- = 525 cm-’; 0: = 564 cm-’; promotor tor mode, w: = o: = 1515 cm-’; best acceptor mode, mode, W K = 1530 cm-l; reduced displacements, x1 = 0.0027; x2 = 0.284 (see Appendix 2).34 RELAXATION OF CsFs -i- EXCITED STATES The NION calculation k,,(u') values presented in coIumn 3 of table 3 were obtained from relation ( 5 ) by calculating Franck-Condon factors for dispIaced and distorted harmonic oscilIators.The harmonic F.C.-cakulation method of Katriel 24 was generalized to the non-integral quantum-number case (Appendix I). The molecular parameters used are shown in table 3. The energy gap Eo and vibrational frequencies of the [Note that the Eo value used here was taken from our TPES data.It differs by ca. 100 cm-l from the optically determined valuem6 This difference is insignificant in the present context of the NION calculations of k,,(v') rates.] Since the w; value is uncertain we took w; = w;. The reduced dispIacements state are taken from data presented in ref. (5). miw; 2fi xi (i = 1,2) = - (Q; - Q ; ) 2 of the normal coordinate Qi were determined 22 from the photoelectron spectrum of Duke et aLZ5 using the Franck-Condon-analysis method given in Appendix 2. We remark that mode 1 corresponds to a displaced oscillator, whereas the mode-2 oscillator is both displaced and distorted. Some comments on the sensitivity of the non-radiative rate calculations to mole- cular parameters are in order.Calculated k,, values have been shown 20-21*26 to 0 5000 EJcm - ' 10000 Fig. 5. Compari2on of experimental and calculated non-radiative relaxation rate k,, of selected levels B,d of C6F6+. VaIues from tables 1 and 3 and ref. (4). Calculated values were normalised to the experimental value knr(21). 0, Experimental ; 0, calculated. depend strongly on the vibrational frequencies in initial and final states and on vib- rational anharmonicities when the electronic energy gap is large. Further calculations in the case of C,F6+ require a better knowledge of the vibrational frequency of the v1 (C-F) mode and of the anharmmicity parameters involving the v1 and v2 modes, which are unknown at present. The present results are very satisfying in that they reproduce the two principal experimental observations on k,, behaviour.The calculated non-radiative rates given in table 3 and in fig. 5 have been normalized to the k,,,(2') experimental value.G. DUJARDIN AND S. LEACH 35 The results show a monotonic increase of k,, with E, for each vibrational progression 1" and 1"2' taken separately, in accordance with experiment. The calculations re- produce the mode-selective behaviour of k,, in predicting that for " equal excitation energy," the k,, (l"2') values should be greater than the k,,(l") values. For the 1" progression the quantitative agreement between theory and experiment is good, but for the l"2' progression we observe an increasing deviation of the experi- mental values from the calculated ones. This important result will be further dis- cussed.We also note that with k,,(2') normalized to the experimental value our calculations lead to a value of k,,(Oo) = 1.4 x lo5 s-' for the B 2A2u origin level, consistent with our experimental results and with those of Maier and Th~mmen.~ Before ending this section we briefly discuss, and establish as non-operative in our case, another possible source of apparent mode-selective behaviour of k,,. This relates to the case where a promoting mode is excited via sequences that overlap with the band b corresponding to excitation of one quantum of the optical mode. A theoretical treatment by Nitzan and Jortner l4 has shown that the k,,(b) rate could then be enhanced by a factor of 3. This cannot be the origin of the marked increase of the k,,(v2) rate in our case, because the frequencies of the three (possible promoting) el, modes in C6F6, and their expected changes in going to the B state of CsF6+, are unlikely to create a sequence band in the ~ ( v J region under our experimental con- ditions.We note also that the experimental " enhancement factor " is rather greater than 3 (see table 3). 5.4 VIBRATIONAL REDISTRIBUTION We have seen (fig. 5) that the model calculations reproduce qualitatively the two principal results, i.e. the observed increase of k,, with E, for each vibrational pro- gression, and the mode-selective behaviour of k,,. However, quantitative agreement falls off as E, increases, especially for the l"2' progression. In fig. 5 the dashed and full lines join, respectively, the successive calculated and experimental points up to the last calculated level, 14F.The dotted line links the experimental values for E, >, E(14Y). The experimental k,, values oscillate with smaller and smaller amplitude as E, increases and show little variation from an expo- nential increase above E, = 4500 cm-'. We note in particular that there is no marked discontinuity at the onset of the state. The damping of the k,, value oscillations is in marked contrast to the calculated values. The calculations on relative k,,(&u') rates were carried out using the model scheme of fig. 1, on a harmonic-oscillator basis. This implies the absence of vibra- tional redistribution in the optically excited state. The calculated values thus provide a non-communicating oscillator limit, from which the progressively increasing devi- ations of experimental results can be used to argue that vibrational redistribution does indeed occur with increasing importance as E, grows.The model represented by fig. 1 is therefore insufficient. A more appropriate model scheme is illustrated in fig. 6. Optically excited vibrational states {s} in the excited electronic state S ( d ) can undergo coupling via matrix element D , , ~ to other quasi-isoenergetic vibrational levels in the {s'} manifold of the S state. These can be considered to form nuclear molecular eigenstates {.?I in the excited electronic state, which can couple in turn to isoenergetic highly excited vibrational levels {I} of the L(=8) ground state. However, it is more useful to continue our discussion in the zero-order basis of ($1, {s'] and ( I ) ; coupling will be discussed in terms of the matrix elements vSl and36 RELAXATION OF CsFs' EXCITED STATES In the present scheme we have no need to consider vibrational redistribution in the {I} states.The effect of VNRT ( I ) would be to add to the level width of the {I) states. The density of the { I } states isoenergetic with a particular s level is very high (ca. 10' per cm-l according to a Haarhoff-type calculation'), so that the effective quasi- continuum of ( I } states would be little affected by the introduction of VNRT(I). We remark that information on the VNRT(I) process can be obtained in studies of the so- called intermediate case of ENRT. An example we have studied is the chloro- acetylene cation.27 The operators in the various matrix elements will have the following physical significance : (i) vss': vibrational anharmonicity and Coriolis (rotational-electronic) S Fig. 6.Vibronic coupling scheme including electronic non-radiative transitions (ENRT) (coupling matrix elements uSz and uStZ) and the vibrational non-radiative transition [VNRT(s)] (coupling matrix element t],,*). coupling; we may use the term Fermi resonance as a generic name for any interaction which takes place between close-lying zero-order vibrational levels and which is caused by terms in the vibrational potential-energy expression cubic or higher in the vibrational coordinates. We remark that our concern is with anharmonicity in the optically excited S state and not, as in previous studies, 12*20*21 with the effects of an- harmonicity in the L state acceptor modes.(ii) vsl and vsnl : nuclear kinetic-energy operators resulting from the incomplete Hamiltonian which is implied in deriving the zero-order states. No spin-orbit operator contributions are necessary in our case of open-shell molecular ions, where the lowest accessible quartet state is above the first, and possibly the second, excited state. vsl and v s r l correspond to coupling with the quasi-continuum of ( t } states and thus lead to irreversible transitions whose rates are, respectively, ksl and ksfl. Let us consider successively I, I1 and 111, three different regions of E, corresponding to increasing level density p(E,v'). The boundaries between these regions are not rigid. Our experimental results (e.g.fig. 5) lead us to propose as approximate ranges in the B state of C6F6' : E,(I) = o to ca. 2500 cm-', E,(II) = ca. 2500 to ca. 4500 cm-', E,(III) > 4500 cm-l. From the viewpoint of VNRT(s), these regions corres-G. DUJARDIN AND S. LEACH 37 pond, respectively, to the sparse level (I), intermediate case (11) and statistical limit (111) analogues of ENRT.28-31. At low-level densities p(8,v') (region I), and in the absence of Fermi resonance and neglecting Coriolis coupling [both of which can lead to mixing of zero-order (de- perturbed) levels within the B" vibrational manifold], we expect the optically excited &I' levels to be isolated in the B" state and to couple individually to g,;v". Thus mode; selective behaviour of k,,, reflecting mode specific coupling matrix elements between B and 2, is reasonable to observe at small Eu.At higher internal energies we enter region IT. As E, increases, anharmonicity factors will become more important and p(B,u') will also increase; more Fermi reson- ance and Coriolis coupling possibilities will result. Greater zero-order level mixing and dynamic vibrational redistribution within the 8 vibrational manifold will gradually set in. This can also be envisaged in terms of a gradual increase, with Eo, of the phase- space occupancy.32 The marked decrease of observed k,,(1"21) with increasing n as compared with the calculated values is in contrast to the behaviour of the 1" levels, and indicates that the presence of one quantum of mode 2 facilitates vibrational redistribution towards levels that have poorer Franck-Condon factors in the ENRT process.At very high E, (region 111) we approach the situation of a quasi-continuous set of 8,v' levels which would correspond to the fully communicating " statistical limit " case of complete vibrational redistribution. The existence of such a continuum can be predicted for energies E, such that the mean energy interval p-l between levels in the (s} vibrational manifold is smaller than the average width Ts of these levels. The lower limit of I?, (in cm-l) is (27~c.r)"~, where z is the lifetime observed in fluorescence and reflects only those processes which depopulate the fluorescing electronic state S. Vibrational redistribution certainly adds a further contribution to r,.We can thus estimate an-upper limit EI,, beyond which B,v' levels form a quasi-continuum. Cal- culating p(B,v') by the Haarhoff approximation l and using the lifetimes @,d) from table 1, we find that p-' = (27~.cz)-~ at energy E,,, = 6900 cm-l. This calculated upper limit, neglecting vibrational redistribution, is compatible with the 4500 cm-l value estimated for the onset energy of region 111, as evidenced by the quasi-collinearity of the k,,(l") and k,,(1n21) values as a function of E, above 4500 cm-I (fig. 5). Relaxation processes mentioned above have their analogues in models of uni- molecular dissociation (UMD) reactions.33 In our case the ground-state vibrational quasi-continuum (I> plays the role of the dissociation continuum in unimolecular dissociation.The operator responsible for the s-+Z and s'-Z transitions will have the same nature in ENRT and in electronic predissociation (neglecting spin changes), whereas it will be of the VNRT(s) type in vibrational predissociation. In another area where similar problems arise, that of multiphoton dissociation (MPD),34 the initial optical excited vibrational level has access to a denser set of vibrational levels (analogous to the isoenergetic s-Z and d-+Z processes) by the non-isoenergetic mechanism of photon absorption. In both UMD and MPD, vibrational redistri- bution is essential in order to achieve the adequate nuclear configurations for tran- sition to the dissociation continuum; for example, in UMD the R.R.K.M. model supposes the existence of very rapid VNRT prior to dissociation, so that the whole phase space corresponding to all vibrational (and rotational) substates is equally accessible, subject only to conservation of energy and of angular-momentum res- traints.Recent approaches to these problems equate the onset of rapid intramolecular vibrational redistribution with the onset of dynamical but no quantitative correlations have been reported for specific examples.38 RELAXATION OF C6F,5+ EXCITED STATES Rett~chnik,~~ Tramer and Voltz,12c Parmenter 36 and Smalley 37 have reviewed results on VNRT in excited electronic states of large neutral molecules. Three different techniques which have given information on VNRT in neutral species 36 are (i) a comparison of level widths determined by high-resolution absorption spectroscopy and from the fluorescence decay constant; (ii) a study of the fluores- cence spectrum as a function of E,, in single vibrational-level excitation experiments, where the development of a background and the broadening of spectral features are considered as indicators of the onset of VNRT processes; (iii) studies based on the time-dependent development of background emission from the excited species.For single-ring aromatics without side chains, VNRT sets in at E,, z 2000 cm-1.36 For example, in the case of the S1 state of p-difluorobenzene, Covaleskie et al.38 have inferred that no significant VNRT occurs below E,, = 1600 cm-l, but that a high VNRT rate, ca. 10l1 s-', exists at E,, > 2400 cm-'; Halberstadt and Tramer 39 also showed that at E, = 2190 cm-I the VNRT rate is greater than 3 x lo9 s-l.In our work, evidence for vibrational redistribution is obtained by a new method, (inherently also based on time-dependent phenomena since radiative and non-radiative rates act as internal clocks and probes), in which the criterion of VNRT is deviation of experimentally determined radiationless rates from values calculated with a model where vibrational redistribution is excluded. This has led us to evaluate the onset of significant VNRT in the B 'Azu state of C6F6+ to occur at E, z 2500 cm-', which is of the same order of magnitude as the vibrational energy for onset of VNRT in the S1 The model illustrated in fig. 6 suggests that VNRT in C6F6+ could also be followed by time- and energy-dependent spectral analysis of the fluorescence.Emission from unrelaxed (insignificant VNRT) s levels should have a different spectral distribution than the emission from relaxed levels. Such observations require a different experi- mental approach than the T-PEFCO method, where the number of fluorescence photons produced is at present too small for studies of dispersed spectra. state Of C6H6 40 and P - C ~ H ~ F ~ . ~ ~ ' ~ ~ 6. EMISSION IN COINCIDENCE WITH Zf-STATE EXCITATION Emission resulting from excitation in the c-state region of C6F6+ was first observed by Ames et aL41 in PEFCO experiments and was later confirmed by Maier and Thom- men, who showed that the emission in coincidence with excitation of the c,O* level was in the 400-650 nm spectral region.4 We will now discuss what is the nature of the emission connected with the c state of C6F6+.Three possible origins of this emission _have previously been proposed :' (i) the Zf+8 radiative transition, (ii) rapid c-+B cascade process followed by emission &8, (iii) c,B vibronic mixing and subsequent emission to the 8 state. Our T-PEFCO results throw new light on this problem and we will therefore examine the three possible mechanisms in more detail than in our earlier discu~sion.~ (i) The c 2B2,,-2 'El, transition is electric dipole forbidden, but could be vibronically induced. For a Herzberg-Teller-induced transition, the c+8 vibronically induced origin band should occur in the 320 nm_region, but because of possible relative dis- placement and distortion of the c and X potential hypersurfaces, the c-8 Franck- Condon emission might occur to high vibrational levels of the 8 state in the same 400- 600 nm region as that of the normal Franck-Condon 'Azu-+Z '3, emission. The closest allowed state to the c 'BZu state is indeed the lower-lying B 'Azu state whose electronic origin is some 10000 cm-l away.However, the D,, species C6F6' is vibrationally deficient in that it possesses no vibration of appropriate symmetry capable of inducing Herzberg-Teller coupling between the c 2B2u and 2A2uG. DUJARDIN AND S. LEACH 39 The next-closest-allowed state is the P2E1, state, which lies ca. 20 000 cm-' above the Herzberg-Teller coupling to this state would involve excitation of odd quacta of e29 vibrations. The situation is similar to that of the well known A 'BZu- 2 lA,, forbidden transition in neutral where eZg vibrations are Herzkerg- Teller active and the perturbing allowed state is ca.16 500 cm-l above the A 'Bzu state. If we assume that a Herzberg-Teller effect of similar magnitude occurred for the c 2B2u-f 2Elg transition in C6Fs+, the oscillator strength would be fz The corresponding radiative decay rate would then be of the order k , ( C ) z 7 x lo5 s-l (calculated from the relation k, = QF/z = v2fl.5, using the appropriate unitsP6 where v is the mean transition frequency). Our experimentally determined value for c,Oo (table 1) is several times greater than this value. We therefore do not consider direct &f transition, albeit Herzberg-Teller-induced, as having a dominant contri- bution to the emission in coincidence with c state excitation. 2Bzu+B 2Azu optical transition would have an origin band in the 970 nm region, beyond the range of our photomultiplier, Furthermore, since this is a_forbidden dipole transition, whose frequency (ca.10 000 crn-l) is relatively low, a c-+B radiative cascade would consequently be a very slow process (analysis of Herzberg-Teller possibilities leads us to expect an even smaller k, rate than that derived for c-2 above), and we will therefore neglect this relaxation pathway. (iii) Let us now examine the possibility of a B-state contribution to the emission. Vibronic coupling between the c and 8 states should be a weak process since the promoting mode would have to have bl, symmetry (see the theoretical discussion in section 5.2), and this is not a proper vibrational mode for a DCh species.42 We note that non-radiative relaxation of the state can also occur by direct coupling, via e,, vibrations as promoting modes, to high vibrational levels of the 2 state.For the state to be involved in the radiative relaxation process occurring subsequent to c- state excitation the molecular (ion) states must have three components, namely c,R,v' and 2 , v " zero-order states. In an imperfect sense, but consistent with symmetry properties, Cne can consider that c-state intramolecular relaxation occurs by inter- action with B-state levels already contaminated by mixing with high vibrational levels of the 8 2Els ground state. This discussion leads us to propose that the radiative transitions in the 400-600 nrn spectral region resulting from (?-state excitation arise from molecular states derived from vibronic coupling between c, B and 3 states.(This refines our original con- cl~sion,~ later adopted by Maier and Th~mmen,~ that the emission in coincidence with the (? state involves a very rapid c+8 irreversible non-radiative transition, and amounts to a &z emission from high vibrational levels of the state to high vib- rational levels of the lower 2 state.) The kr and k,, results reported in table 1 add further support for our interpretation of the process of emission in coincidence with the c state. We discuss first the radiative rate k,. It is approximately constant for most vibrational levels of the state, but decreases for energies close to and at the origin of f-state excitation.The k, values in the c-state region cannot therefore be the radiative rates of pure high vibrational levels of the 8 state. We have found that the same situation holds for the analogous case af 1,3,5-C6F,H,+, where the relevant experimental evidence is even more extensive in that not only the 0' level can be measured for this ion in the state but also several higher vibronic le~eXs.~'-~ The lessened k, rate for the c-state region confirms that in this energy region the (ion) molecular states are mixed. The situ- ation is similar to that found in small molecules exhibiting the " lifetime-lengthening " Douglas effect, except that in the mixed states, here, in contrast to the reported small- molecule cases,44 it is the (low) oscillator strength of the optically excited state (c) 2B2u (ii) The40 RELAXATION OF C6F6 -i- EXCITED STATES that is apparently reinforced by dilution with a perturbing state (B).Consequently, again in contrast to the original Douglas-effect cases (NO2, SO2, CS2),44 it is a final " acceptor "-state component whose radiative rate kr is " reduced " by this dilution. Fig. 4 and 5 show that there is no significant discontinuity in the k,, value on passage to the c,Oo level, where the excess vibrational energy in the fi state is 10.400 cm-l. This can be taken as evidence that very efficient intramolecular vibrational redistri- bution takes place in the vibronic levels to which the c,OO state is coupled, similar to the explanation of the continuity in k,, found for P-naphthol and P-naphthylamine on crossing the threshold to the S3 To sum up, the spectral range of the emission in coincidence with the c state and the radiative and non-radiative rates associated with this emission decay indicate that high vibrational levels of the B state are involved along with the 2; state.The reduced apparent radiative rate k, implies that the molecular state, resulting from excitation at the 2;-state energy, is compounded of zero-order, emitting, B vibronic states plus some other non- (or weaker) emitting states that " dilute " the oscillator strength. From our previous discussion we propose that these contributing states include both the state and high vibrational levels of the 8 state, and that efficient vibrational redistribution occurs in the molecular ion states to which c,O0 is coupled.If a quartet state exists that is both isoenergetic and accessible to the 2B2u it might also provide a component for the molecular state. We thank the LURE synchrotron radiation facility for technical support concern- ing the experimental work reported here. APPENDIX 1 : CALCULATION OF FRANCK-CONDON FACTORS WITH NON-INTEGRAL QUANTUM NUMBERS We calculated the Franck-Condon factors with non-integral quantum numbers by generalising the Katriel method 24 available for the usual case of integral quantum numbers. Let us consider a harmonic vibrational mode which is both displaced and distorted. The frequencies in the initial and final electronic states are, respectively, o' and o" and x is the reduced displacement of the normal coordinate between the two electronic states (see Appendix 2).The Franck-Condon factor for this vibrational mode with n and I quanta, respectively, in the initial and final states is l(n\Z)12. For non-integral values of I, the overlap integral (nll) was calculated from the following expression: with B = o'/c"'. ( I - 2iz -- i3) are positive or zero. function of I for both integral and non-integral values of 1. The SU-M is over all positive integral values of il, iz and 4 such that (n - 2il - 4) and We remark that from this expression (nll) is a continuous APPENDIX 2 : DETERMINATION OF THE REDUCED DISPLACEMENT x OF A NORMAL COORDINATE Q BETWEEN TWO ELECTRONIC STATES OF A POLYATOMIC CATION Heilbronner et al.46 proposed a semi-quantitative method for estimating the displacement of a normal coordinate Q between the electronic state v' of a cation M+ and the groundG.DUJARDIN AND S . LEACH 41 state X of the corresponding neutral molecule M (fig. 7). We used the following similar but simpler method to estimate this displacement, and to obtain the reduced displacement x of the normal coordinate Q between the electronic states v/’ and v/” of the ion. The relative intensity Zzo/Iyo of two vibrational components z and y of the t,v’ electronic Fig. 7. Schematic potential-energy curves showing the displacement of normal coordinate Q of cation M+ in state v/’ relative to the corresponding neutral molecule M in its ground state. band in the photoelectron spectrum is given in the Born-Oppenheimer approximation by the ratio of Franck-Condon factors: 46 ]q’(u’)) and lp(O)), respectively, represent the M+(t,v’) vibrational state d and the M(X) ground state II = 0.For harmonic displaced and distorted vibrational modes, the expression (Al) with z = 1 and y = 0 gives: !!!! = ( -) 4o’ (g (Q’ - Q o ) 2 ) zoo (1 + PI2 Q’ and Qo are the Q values in the M+( v/’) and M(X) states at equilibrium. rn is the reduced mass of the vibrational mode. co’ and oo are the angular frequencies of this mode in the M+(y/’) state and in the neutral ground-state, respectively (/3 = o’/wo). The reduc_ed displacements x1 and x2 of the two totally symmetric modes of C6F6+ between the X and B states of the cation were derived as follows. For each vibrational mode we first determined 2/(rn/2h)JQJ - QO), using expression (A2), from the corresponding vibrational progression in the B band of the C6F6 He(1) photoelectron ~pectrum.~’ The sign of (Q’ - Qo) was obtained according to the following rule: (Q’ - Qo) is positive when o’ < oo and negative when or > coo.The same method was used20 deteLmine -v‘(rn/2h) (Q” - Qo) (the prime and double prime refer, respectively, to the B and X states of the cation). From these two quantities we then calculated the reduced displacement x = (rno’/2h) (Q’ - Q”)’. The values x1 and x2 (corresponding to the two totally symmetric vibrational modes) obtained by this method are given in table 3. As an illustration of the method we give in table 4 the results of calculations of x and AQ = Q” - Qo for ionization of H2 and 02. The experimental PES band intensities Zlo/loo were taken from Gardner and Samson 47 and the vibrational frequencies from Huber and H e r ~ b e r g .~ ~ The values of AQpes derived from photoelectron spectra are in good agreement with the change of internuclear distance AQop, determined by optical spectroscopy.42 RELAXATION OF C6F6' EXCITED STATES Table 4. Ionization of H2 and 02. Reduced displacement x and change in internuclear distance derived from photoelectron (AQpes) and optical (AQopt) spectral data. ionization process ~ 1 0 / ~ 0 0 B = woo X A QpeslA A Qop t I A ~~ ~ Hz(X 'Z;) + Hz (X'Z,') 2.06 0.53 1.21 0.26 0.31 0,+(X211g)+ 02(X 'Xi) 2.32 1.21 2.80 -0.1 1 -0.09 S. Leach, G. Dujardin and G. Taieb, J. Chim. Phys., 1980,77, 705. Findley and R. Huebner (D. Reidel, Dordrecht, 1983).G. Dujardin and S. Leach, J. Chem. Phys., in press. J. P. Maier and F. Thommen, Chem. Phys., 1981,57, 319. G. Dujardin, S. Leach, 0. Dutuit, T. Govers and P. M. Guyon, J. Chem. Phys., in press. C. 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Mass Spectrom., 1977, 12,297. l4 A. Nitzan and J. Jortner, J. Chem. Phys., 1971,55, 1355. l5 K. F. Freed and S . H. Lin, Chem. Phys., 1975, 11, 409. l6 V. E. Bondybey and T. A. Miller, J . Chem. Phys., 1979,70, 138. l7 S. H. Lin, J. Chem. Phys., 1966, 44, 3759. l9 S. E. Nielsen and R. S. Berry, Phys. Rev., 1969, 180, 139. 2o I. H. Kuhn, D. F. Heller and W. M. Gelbart, Chem. Phys., 1977,22,435. 21 M. G. Prais, D. F. Heller and K. F. Freed, Chem. Phys., 1974,6, 331. 22 G. Dujardin, ThPse d'Etat (UniversitC Paris-Sud, Orsay, 1982). 23 H. Hornburger and J. Brand, Chem. Phys. Lett., 1982, 88, 153. 24 J. Katriel, J. Phys. B, 1970, 3, 1315. 25 C. B. Duke, K. L. Yip and G. P. Ceasar, J. Chem. Phys., 1977, 66,256. 26 D. F. Heller, K. F. Freed and W. M. Gelbart, J . Chem. Phys., 1972, 56, 2309. 27 G. Dujardin, S. Leach, G. Taieb, J. P. Maier and W. M. Gelbart, J. Chem. Phys., 1980,73,4987. 28 A. Amirav, U. Even and J. Jortner, Chem. Phys. Lett., 1980,71, 12. 29 P. R. Stannard and W. M. Gelbart, J. Phys. Chem., 1981, 75, 3592. 30 C. Tric, Chem. Phys., 1976, 14, 189. 31 K. F. Freed and A. Nitzan, J. Chem. Phys., 1980,73,4765. 32 See e.g. E. J. Heller and M. J. Davis, J. Phys. Chem., 1982,86,2118; S. A. Rice and R. Kosloff, J. Phys. Chem., 1982,86,2153; W. P. Reinhardt, J. Phys. Chem., 1982,86,2158; P. Pechukas, J . Phys. Chern., 1982, 86, 2239. ' S. Leach, in Photophysics and Photochemistry in the Vacuum Ultraviolet, ed. S . McGlynn, G. * K. E. Ockenga, P. Gurtler, S. S. Hasnain, V. Saile and E. E. Koch, VI Int. Conf. VUV Radiation K. F. Freed and J. Jortner, J. Chem. Phys., 1970, 52, 6272. 33 P. J. Robinson and K. A. Holbrook, Unimolecular Reactions (Wiley, New York, 1972). 34 R. V. Ambartzumian and V. S . Letokhov, in Chemical and Biochemical Application of Lasers, 35 R. P. H. Rettschnick, in Radiationless Transitions, ed. S. H. Lin (Academic Press, New York, 36 C . S . Parmenter, J . Phys. Chem., 1982, 86, 1735. 37 R. E. Smalley, J. Phys. Chem., 1982, 86, 3504. 38 R. A. Covaleskie, D. A. Dolson and C. S . Parmenter, J. Chem. Phys., 1980,72, 5774. 39 N. Halberstadt and A. Tramer, J. Chem. Phys., 1980, 73, 6343. 40 A. E. W. Knight, C. S . Parmenter and M. W. Schuyler, J . Am. Chem. SOC., 1975,97, 1993. ed. C. B. Moore (Academic Press, New York, 1977), vol. 111, p. 167. 1980), p. 185.G. DUJARDIN AND S. LEACH 43 41 D. L. Ames, M. Bloch, H. Q. Porter and D. W. Turner, in Proceedings of the Sixfh Conference on Molecular Specrroscopy, ed. A. R. West (Institute of Petroleum, Heyden, 19771, p. 399. 42 G. Herzberg, Electronic Spectra and Electronic Structure of Polyaromic Molecules (Van Nostrand, New York, 1966). 43 G. Bieri, L, Asbrink and W. von Niessen, J. EZectron Spectrosc. Related Phenomena, 1981, 23, 281. 44 A. E. Douglas, J. Chem. Phys., 1966, 45, 1007. *’ 3. C. Hsieh, C. S. Huang and E. C. Lim, J. Chem. Phys., 1974, 60,4345. 46 E. Heilbronner, K. A. Muszkat and J. Schaiublin, Helv. Chim. Acta, 1971,54,58. ‘’ J. L. Gardner and J. A. R. Samson, J. Electron. Spectrosc. Related Phenomena, 1976,8,123. 48 K. P. Huber and G. Herzberg, Constanrs of Diaromic Molecules (Van Nostrand, New York, 1979).

ISSN:0301-7249    

DOI:10.1039/DC9837500023

出版商:RSC    

年代:1983

数据来源: RSC

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Faraday Discuss. Chem. Soc., 1983, 75,45-55 Isomerizati on of Internal -energy- selected Ions BY TOMAS BAER, WILLI A. BRAND,* THOMAS L. Bum AND JAMES J. BUTLER? Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27514, U.S.A. Receiued 23rd December, 1982 Solutions of coupled differential equations for the decay rates of energy-selected ions which competitively dissociate and isornerize show that under appropriate circumstances two-component decay rates can be expected. Although the requirements for the observation of such non-exponential decay is rather restrictive, it has now been observed in a number of cases, including the dissociation of butene and pentene ions, as well as l,Coxathian, 1,4- dithian and dioxan. Photoelectron-photoion coincidence spectroscopy has been used to energy-select ions and to measure the rate of dissociation.Examples of such non-exponential decay rates are presented for the above-mentioned ions. INTRODUCTION Non-exponential or two-component decay resulting from competitive decay and internal conversion has been known for some time now for the case of ion fluorescence.l This effect can be observed in the case where the fluorescence rate and radiationless transition rate are comparable, and the ion is sufficiently small so that the probability for reconstitution of the excited electronic state and subsequent fluorescence is appreciable. A similar situation exists in the case of metastableion dissociation reactions. Metastable ions are those ions which dissociate during the time of mass analysis in a mass spectrometer.This generally means dissociating ions with life times of the order of microseconds. Two-, or perhaps multi-, component exponential decay has been observed in a few dissociations of metastable and in the case of propargyl chloride,2 CH3- CCCl+ , this was attributed to the competition between isomerization and dissociation. The dissociation of a series of energy-selected butene ions gave further evidence for the connection between isomerization and dissociation, and two-component decay ratesms We present here the mathematical analysis of the expected decay rates and some additional examples which we have recently discovered. Photoelectron-photoion coincidence (PEPICU) is a technique fur energy-selecting ions and measuring their dissociation rate, the branching ratio to the various products, and the translational energy released in the dissociation reaction.6 The principle of the experiment is based on the conservation of energy and momentum, which ensures that for a given photon energy the ion internal energy, E,, is given by EI = hv - E, + Eth in which E, is the energy of the departing photoelectron and Eth is the residual thermal energy of the ion as a resuIt of the initial thermal energy of the precursor neutral * Present address : Institut fiir Physikalische Chemie, Universitat Bonn, 5300 Bonn, West Germany.7 Present address: National Bureau of Standards, Washington D.C. 20234, U.S.A.46 ISOMERIZATION OF ENERGY-SELECTED IONS molecule. In large molecules such as brornobenzene Eth can be as much as U.1 eV.Two types of PEPICO experiments are being carried out in various laboratories. One employs a fixed-energy light source and the ion energy is selected by varying the electron energy with which the ion is ass~ciated.~-~ The other type is a threshold PEPICO experiment in which a variable-energy light source is used and only threshold photoelectrons are co11ected.2-6* 'Om The latter approach is utilized fur this study. EXPERIMENTAL In the threshold PEPICO experiment, electrons of nominally zero energy are measured in coincidence with ions. The experimental set-up is illustrated in fig. 1. The electron signal ELECTRONS t START To STOP DISCRIM- - - DELAY >PULSE h€IGHT 4 - INATOR DISCRI M - INATOR CONVERTER MULTICHANNEL I I I ANAWSER PDP 11/03 Fig.1. Experimental set-up fur the photoelectron-photoion coincidence spectrometer. provides the start for measuring the ion time-of-flight distribution, which in addition to its role as a mass-analyser contains a considerable amount of dynamical information. For instance, if a dissociation is accompanied by release of translational energy, the ion time-of- flight distribution will be symmetrically broadened. Metastable ions are identified by their asymmetric time-of-flight distributions, which arise from the dissociation of the ion during acceleration in the ionization region. ISOMERIZATION AND TWO-COMPONENT DECAY RATES Consider the potential-energy curve in fig. 2, which relates isomers A+ and B+ (the + signs are omitted in the fig.), each of which can dissociate to products PI and P2, respectively.These products could be either a single or several ion-neutral pairs. Furthermore PI and P2 could be the same or different products. Initially, ionization will take place to either A+ or B+ [eqn (l)], depending on the starting neutral isomer: A + k v + A + ( 1 4 €3 + hY --f B+. (1b)T. BAER, W. A. BRAND, T. L. BUNN AND J. J. BUTLER 47 Fig. 2. Model potential for a competitive dissociation/isomerization reaction. The reaction then proceeds via k2 A+ 7 B+ B+ A P2+ + N2. k Now the rate of production of P1+ and P2+ is (3) (4) In order to determine the rate, we need to know the time dependences of A+ and These are given by the coupled, linear, homogeneous differential equations (6). B+. (The plus sign indicating ions will not be included for the remainder of this paper.) (6) dA - = (-kl - k2) A + k3 B = alA + blB dt @ = k2A + (-k3 - k4) B = a2A + b2B.dt Solutions are of the form: A = ae-lf , B =pe-A'. Substituting these solutions into eqn (6) yields (7) This matrix equation only has solutions when the determinant of the 2 x 2 matrix is zero. This condition yields the following values of A: ' 2 = -+(a, + b2) & +d[(a, + b2I2 + 4a2b1 - 4a&]. (9) Now the general solutions for A and B are given by h e a r combinations of the + and - solutions of eqn (7) and (9), which result in:48 ISOMERIZATION OF ENERGY-SELECTED IONS The a and j3 coefficients are related by the conditions set by eqn (8). However, only the ratio of a and p is determined, so that we can arbitrarily set, say, the p coefficients equal to 1.Suppose that A is the initially formed ion, at an energy above the dissociation limit of both P, and P2. As a result B(t = 0) = 0 and A ( t ) and B(t) reduce to To go beyond eqn (10) we need to consider the initial conditions. Eqn (11) has been written with the slower rate term, A-, first. rates are not simple exponential functions. exponentials. limiting cases. The rate of product formation is given by eqn (5) and (1 1). It is evident that these Rather they are the sums or differences of Because A, are rather complicated, it is helpful to approximate them for various Case (A) then k2 x k3 9 kl x k4. This results in Suppose that isomerization is very fast compared with dissociation. We have Solving eqn (8) for the a in terms of the j3 leads to dP2 -- dt - kk44B = (e-IE-t - e-A+t).Because k3 x k2 and A+ 9 A-, the second terms, involving the fast rates, are neg- ligible at times other than t z 0, so that the two products are formed with the single rate of A-. This is expected for the conditions imposed, which imply a complete equilibration of isomers A and B prior to dissociation. A variation of this case is k2 > k3 so that L is just k4. These latter conditions are apparently the ones most frequently encountered, such as in the dissociation of isomers of C4H6+,12 C6H6+ ,13 C8H8+ l4 and C4H5N+.15 Case (B) We can imagine a situation in which isomerization of A is in competition with direct dissociation. That is, k, x k2 $ k3 = k4. The two rates derived from eqn (9) are This again yields a fast rate, A+, and a slow rate, A- x k3 z k4.The resulting decay rates are - dP2 dt = k4B = k4(e-a-t - e-A+t).T. BAER, W. A. BRAND, T. L. BUNN AND J. J. BUTLER 49 In this case ion A will produce P, with a two-component rate because the co- efficient for each rate is of the order of magnitude of the rate itself. On the other hand, the product P2 will be formed essentially at the slow rate. We will show with the accompanying examples, that conditions such as this are reasonable and can account for the observed two-component behaviour. Case ( C ) Suppose that photoionization initially produces ion B, and that the rate constants are as they were in case (B), namely k, z k2 9 k, = k4 so that ,l* are as in case (B). However, now A(t = 0) = 0 so that the rates of P1 and P2 production are given by It is evident that the product P, will be formed to any significant extent only with the slow rate because the coefficient of the fast rate is small.Similarly product P, can be formed only with the slow rate. Cases (B) and (C) therefore offer a convenient means of testing a two-component mechanism because if the most stable isomer is one which can be formed directly from a stable molecule, its decay rate can be shown to be one- component [case (C)]. Furthermore this rate should be equal to the slow component of the higher-energy isomer [case (B)]. EXPERIMENTAL RESULTS DISSOCIATION OF C5H10' The most clear-cut case of two-component decay to date has been found for the dissociation of energy-selected C,H,,+ isomers.16 Both 13C and deuterium labelling 1 1 Fig.3. Potential energies of three C5H10+ isomers and their dissociation to C3H6+ and C4H7+. The energies of the isomers and the dissociation limits are known. The isomerization barriers are only estimated.50 ISOMERIZATION OF ENERGY-SELECTED IONS studies have shown that extensive rearrangement of the ions prior to dissociation results in scrambling of the carbon and hydrogen atoms.17 In addition recent photo- dissociation studies of several C5H10+ isomers, initially produced by electron impact,18 as well as ion-molecule reaction and collisional-activation investigation l9 suggested that some of the ions rearrange to more stable isomers. However, none of the data lend themselves to a simple interpretation. Two reactions are energetically possible within 0.5 eV of the dissociation threshold. These involve the loss of CH3 and C2H4 to produce C4H7+ and C3H6+, respectively. The dissociation threshold to both of these ions lies essentially at the thermochemical dissociation limit, so that there is no reverse activation barrier. Fig.3 shows the energies relevant for the dissociation of three of the isomers. 11 12 13 I4 15 ion time of flight Fig. 4. Time-of-flight distributions C3Hd + and C4H7 + fragment ions originating from the indicated parent C5HI0+ ions. The solid lines are calculated time-of-flight distributions assuming the indicated rates, A- and d+ . The same values of d are used for both fragments. The energies are the 0 K absolute energies based on the common heat-of-formation scale.The time-of-flight distributions of fragments from the three isomers obtained at several ion internal energies are shown in fig. 4. These data are examples of the remarkable variety observed for all six isomers investigated over a range of internal energies. Not only do the ratios of the C3H6+ to C4H,+ signals vary from one isomerT. BAER, W. A. BRAND, T. L. BUNN AND J. J. BUTLER 51 to the next, but even for one isomer at a single energy it is evident that the two products are both produced with two-component rates. The thermochemistry of the C5H10+ isomers is well known.16*20*21 The cyclo- pentane ion, although quite stable as a neutral molecule, is less stable than the simple branched and linear pentenes. The most stable ion is 2-methylbut-2-ene.At all energies it decays via a one-component decay rate, as predicted for the case (C) type of dissociation. All other, higher-energy isomers are consistent with case (B) dissociation. That is, they decay via a two-component rate. The slow rate is the same for all ions at a given ion internal energy and it increases with ion energy. This slow rate is associated with the rate of dissociation from the lowest isomeric structure, that of the 2-met hylbut-2-ene ion. Because of the numerous isomeric structures which are involved in the dissociation, it is not very useful to analyse the data in more than a qualitative manner. There are too many unknown reaction steps, so that the actual mechanism is far more compli- cated than the models of fig. 2 or 3. DISSOCIATION OF C4HsOS' AND C4HsS2' The total rate of ion decay to several product ions can be measured by monitoring the time-of-flight distribution of any one of the fragments.For a number of ions, such as isomers of C6H6+,13 C4H5N+ l5 and dioxan (C4H802+),22 the measured total rate is the same regardless of which fragment ion is monitored. This is the situation expected when all dissociation paths originate from the same parent-ion structure, or at least if the various parent-ion structures are fully equilibrated. One notable exception has been the dissociation of CH3N02+ which was found to decay via two distinct rates to the products, NO+ + CH30 and CH30+ + NO,23 This observation is proof that at least two types of non-equilibrated precursor ions are involved in the dissociation.* In the case of the previously discussed C5H10 ions, the two parent ions correspond to the initially formed ion and its rearranged, lower-energy isomer.A similar situation apparently exists in the dissociation of C4HsOS and C4HsS2 ions except that the two parent ions produce different fragment ions so that each decay can be described by a single exponential. However, the rates (as shown in fig. 5) for the C4HsOS+ dissoci- ation are different. It corresponds to our rate model case (B), in which P1 and P, are different product ions. The rate data for C4HsS2 are very similar. The fragment C2H5S+ is produced at a rate which is ten times higher than that forming C3H2S2+ and C4H7S+. Very little information is available concerning the energies of the C4HsOS+ and C4HsS2+ isomers.However, we can learn something about them from a comparison of these ions with their oxygen analogues, C4Hs02+. A recent PEPICO study of dioxan 22 and two of its isomers, butanoic acid 24 and ethyl acetate 25 indicated that these three ions dissociate directly, withour prior isomerization. This is evident not only from the rate studies, but also from the fragments produced by each ion. Table 1 summarizes the major fragments from the three C4H802+ isomers, as well as those from C4HsOS+ and C4HsS2+. This table is very revealing. Consider the fragment ion C2H5X+, X = 0 or S. This is one of the three ions produced by dioxan near its dissociation limit. Although this ion is also formed in the dissociation of ethyl acetate, it is a minor product and the structures of the C2H50+ ions are probably not the same.It is significant that this product ion is formed by both C4HsOS+ and C4HsS2+ with a rate about ten times higher than the rate of parent-ion decay as * Meisels et al.23 analysed their results in terms of different electronic states. In view of our results and model presented here, it is likely that these electronic states refer to different CH,NO: structures, e.g. CH30-N=O+ and CH2-OH-NO+, the latter being more stable.52 ISOMERIZATION OF ENERGY-SELECTED IONS 3 I v) . 0 Y h 1 o6 1 o5 10.0 10.2 10.4 10.6 photon energyleV Fig. 5. Measured total C4H80S+ dissociation rates as a function of the ion internal energy. The solid line through the C2H5S+ points (0) is an R.R.K.M./Q.E.T. calculation assuming a 1,4-oxathian structure for both parent and transition-state ions (see tables 2 and 3).The solid line through the C4H6S+ (0 ) and C2H4S+ (0) and points is an R.R.K.M./Q.E.T. calcu- lation assuming an isomerized, lower-energy parent-ion structure (see text). measured by the appearance of the other fragments. Yet the fraction of C2H5S+ ions is only ca. 20%, and it is not energetically the first ion formed (table 2). All the more slowly formed fragment ions, such as C4H,S+, C3H2S2+ and C2H4SO+, are ones which are also produced in the butanoic acid and ethyl acetate ion dissociation. Table 1. Major fragments formed from C4H802+ isomers and C4H80S+ and C4H8SF+ n e u t r a l OH ocz H5 "The numbers in parentheses are the orders of the fragment ion appearance energies.The lowest is first.T. BAER, W. A. BRAND, T. L. BUNN AND J. J. BUTLER Table 2. 0 K onsets a for fragments from 1,4-oxathian (C4HSOS) product IE or AEo/eV C,H,OS + 8.53 i 0.02 C,H,jS+ + H20 9.73 * 0.05 C2H,SO+ + C2H4 9.79 & 0.05 C4H70+ + SH 9.78 f 0.08 CzH,S+ + CZHjO 9.85 * 0.05 C2H4S+ + CzH40 10.0 * 0.1 H2CS+ + C3H60 10.9 * 0.1 53 The 0 K onsets were obtained from the 298 K onset by adding 0.1 eV, the average C4H8US thermal energy, to the measured onset. The most stable of the three C4H802+ isomers investigated is the butanoic acid ion which, on the absolute (0 K) energy scale, has a heat of formation of 5.54 eV. In addition, both the ethyl acetate ion (5.68 eV) and its enol isomer (5 eV) are considerably more stable than the dioxan ion (6.23 eV).We can assume that a similar relation- ship holds for the sulphur analogues. (Such an analogy must, in fact, be made with considerable caution because the sdphur atom can cause the relative energies to vary considerably.) This analogy offers a simpIe explanation for the different rates observed in the dissociation of C,H,OS+ and of C,H,S,+ ions. The cyclic structure of these ions can be identified with isomer A in fig. 2. An as yet unknown isomer, which could be the thioacetate, the thiobutanoic acid or perhaps a combination of these, is isomer B. Whereas the barrier to isomerization in dioxan is large, it is evidently sufficiently small in the case of the sulphur analogues, so that direct dissoci- ation to C,H,S+ is in competition with rearrangement to the more slowly dissociating isomers.One test of this dissociation mechanism is the comparison of the experimental dissociation rates with those caIculated using the statistical theory of unimolecular decay (R.R.K.M.1Q.E.T.). We have done this for the case of C4H80S+. An activ- ation energy of 1.32 eV is obtained for the production of C,H,S+ from the data of table 2. The R.R.K.M./Q.E.T. calculation requires this activation energy, as well as the vibrational frequencies for the molecular ion and the transition state, as input. The vibrational frequencies of 1,4-oxathian 26 (table 3) were used for both the ion and Table 3. Molecular ion and transition-state frequencies (cm- ') for R.R.K.M.1Q.E.T. calculated dissociation rates of C&OS + Dissociation to C4H,S+ + SH ' 2980 2980 2980 2940 2910 2910 2880 2880 1720 1460 1460 1440 1410 1410 1370 1350 1260 1260 1180 1110 1090 1080 1000 950 940 770 760 590 545 460 413 378 226 226 210 106 Dissociation to CIHsS ' + C2H30 a 2970 2950 2950 2920 2970 2920 2860 2860 1460 1440 1420 1410 1390 1360 1320 1290 1270 1210 I200 1170 1110 1050 1010 1000 970 950 830 810 690 670 550 430 400 340 252 206 a The same frequencies were used for the molecular ions and the transition states.The 940 and 830 cm-' frequencies were the assumed reaction coordinates for the transition states of the two dis- sociation paths.54 ISOMERIZATION OF ENERGY-SELECTED IONS the transition state. This procedure gave a good fit for the previously studied dioxan dissociation, The solid Iine through the C,H,S+ points is the result of this calcu- Iation.A similar attempt was made to fit the C4H,S+ and C2H4SO+ points. Their slightly lower activation energies of 1-20 and 1.26 eV and the fact that the parent ion is reacting via two paths (a symmetry factor of 2) resulted in rates which are 2 to 3 orders of magnitude higher than experimentally observed. Clearly the C4H80S+ ion is isomerizing prior to dissociation to a lower-energy structure from which it wiIl decay more slowly, as suggested in case (B) of our model. A fitting of the calcuIated rates to our measured ones using the " loose " vibrational frequencies of table 3 and the parent-ion energy as an adjustable parameter suggests a value of 8.05 eV for the isomer (in contrast to 8.53 eV for 1,4-oxathian). CONCLUSION We have shown that under appropriate conditions two-component dissociation rates can be expected in sIowly fragmenting energy-selected ions.This complex behaviour can be ascribed to competition between isomerization and direct dissoci- ation. A11 of these cases are in accord with the statistical theory of unimolecular decay.. In fact if sufficient information concerning the dissociation path and energies were available, the rates could be calcdated with the theory. ExampIes of such dissociation/isomerization of energy-selected ions have now been found for the decay of the ions of C3H3Cl?g3 CH30N0,23 C4H8:p5 C5H10,16 and C4Hs- OS, C4H8S2. The Iatter data are presented here for the first time. We are grateful to the Department of Energy and the National Science foundation for financial support of this project.(a) J. P. Maier, Ace. Chern. Res., 1982,lS 1 ; (b) M. Allan, E. Kloster-Jensen and J. P. Maier, J. Chenz. Soc., Favaday Trans. I , 1977,73, 1417; (c) J. P. Maier and F. Thommen, Chem. Phys., 1980,51, 319. T. Baer, 3. P. Tsai and A. S. Werner, J . Chem. Phys., 1975,62, 2497. B. P. Tsai, A. S. Werner and T. Baer, J. Chem. Phys., 1975,63,4384. T. Baer, D. Smith, B. P. Tsai and A. S. Werner, Adu. Mass Spectrom., 1978,7A, 56. T. Hsieh, J. P. Gilman, M. J. Weiss and G. G. Meisels, J. Pkys. Chern., 1981,82,2722. chap. 5, pp. 153-196. J. Dannacher and J. P. Stadelmam, Cfiem. Phys., 1980,48, 79. J. H. D. Eland, J. Chem. Phys., 1979, 70,2926. I. Powis and C. J. Danby, Chem. Phys. Lett., 1979, 65, 390. lo R. Stockbauer, Int. J. Mass Spectrom.Ian Phys., 1977, 25, 401. I1 I. Nenner, P. M. Guyon, T. Baer and T. R. Gover, J. Chem. Phys., 1980,72, 6587. l2 A. S. Werner and T . Baer, J. Chem. Phys., 1975, 62,2900. l3 T. Baer, G. D. Willett, D. Smith and J. S . Phillips, J. Chem. Phys., 1979, 70, 4076. l4 D. Smith, T. Baer, G. D. WilIett and R. C. Ormerod, Inf. J. Mass Specfrom. Ion Phys., 1979,30, l5 G. D. WiIlett and T. Baer, J. Am. Chem. Soc., 1980, 102, 6774. l6 W. A. Brand and T. Baer, J. Am. Chem. Sac., to be submitted. l7 (a) D. P. Stevenson,J. Am. Chem. Soc., 1958,80,1571; (b) S. Meyerson, Appl. Spectrusc., 1968, l8 P. N. T. VanVelzen and W. J. Van der Hart, Chern. Phys., 1981,61, 335. l9 (a) T. Nishita and F. W. McLafferty, Urg. Mass Specfrom., 1977, 12, 7 5 ; (b) K. Levsen and J. 2o J. B. Pedley and J. Rylance, Sussex-NPL Compufer Analysed ThermochemicaI Ddia: Organic ti T, Baer, in Gas Phase Ion Chemisfry, ed. M. T. Bowers (Academic Press, New York, 1979), 155. 22, 30. Heimbrecht, Org. Mass Specfrom., 1977, 12, 131. and OrganarnetaIIic Compounds (University of Sussex, Sussex, 1977).T. BAER, W. A. BRAND, T. L. BUNN AND J. J. BUTLER 55 21 H, M. Rosenstock, K. Draxl, B. W. Steiner and J. T. Herron, J. Phys. Chern. Ref. Data, 1977,6, 22 M. L. Fraser-Monteiro, L. Fraser-Monteiro, J. J. Butler, T. Baer and J. R. Hass, J. Phys. Chem., 23 G. G. Meisels, T. Hsieh and J. P. Gillman, J. Chem. Phys., 1980,73,4126. 24 3. J. Butler, L. Fraser-Monteiro, M. L. Fraser-Monteiro, T. Baer and J. R. Hass, J. Phys. Chern., ’’ L. Fraser-Monteiro, M. L. Fraser-Monteiro, J. J. Butler and T. Baer, J. Phys. Chem., 1982, 86, 26 0. H. Ellestad, P. KIaboe and G. Gaegen, Spectvochim. Acta, Part A , 1972, 28, 137. 1. 1982, 86, 739. 1982,86,752. 747.

ISSN:0301-7249    

DOI:10.1039/DC9837500045

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1983, 75, 57-76 Kinetics of Ion-Molecule Collision Complexes in the Gas Phase Experiment and Theory BY MICHAEL T. BOWERS, MARTIN F. JARROLD, WINFREID WAGNER-REDEKER, PAUL R. KEMPER AND LEWIS M. BASS Department of Chemistry, University of California, Santa Barbara, California 93 106, U S A . Received 20th December, 1982 Essential elements of a transition switching model formulation of statistical rate theory are presented. The model is developed for use on rather complex polyatomic potential surfaces. Data on the energy dependence of the absolute unimolecular rate constants and branching ratios, thermal biornolecular rate constants and branching ratios including isotopic- substitution studies, and kinetic-energy distributions for the lowest-energy pathways are presented for the C4H$', CsH:' and C6H;' reaction systems. Detailed comparisons are made between experiment and the transition-state switching model.Comments are directed toward tests of the fundamental assumptions of statistical rate theory and toward detailed mechanistic interpretation of the specific systems studied. 1, INTRODUCTION In the past ten years great strides have been made in our understanding of the detaiIs of kinetic processes in the gas phase. Progress has been across-the-board in both theory and experimentation, in both neutral and charged systems, and in both small and large molecules. In this paper we will focus on the current state of affairs in reactions of rather complex gaseous ions. Emphasis will be on ground-electronic- state reactive properties.Tn the process, however, information about intramolecular energy flow will be deduced, both between different electronic states of the system and between different portions of the (complex) ground electronic hypersurface. First, a system is seIected that lends itself to experimental study by a variety of techniques that sample different aspects of the kinetics/dynamics occurring on the surface. In particdar, we are interested in systems where absolute unimolecular rate constants and branching ratios have been measured as a function of energy, where bimolecular reactions can be used to form the reactive intermediate with high angular momentum with a narrow internal energy spread, where kinetic-energy distributions for the low-energy pathways can be measured, and where isotopic substitution can lead to information on bimolecular reaction mechanisms.To date we have identified and studied three such systems, listed both in the order of study and in the order of complexity: C4Hs+',' C,H,+',Z and The second aspect of our approach is to model the chemistry of these systems using statistical rate theory. The foundations of the theory are the usual ones based on R.R.K.M.4/Q.E.T.5 concepts and on microscopic variational transition-state t h e ~ r y . ~ ? ~ Our approach to the problem is straightforward. C6HX6+'.358 ION-MOLECULE COLLISION COMPLEXES We have also borrowed heavily from the phase-space approach of Light * and Nikitit~.~ A recent summary has been writtenlo Our aim is to use rigorous methods (as " rigorous " as C6H,+' allows!) and one set of transition-state parameters to fit all of the experimental data simultaneously.Comparison between experiment and theory allows us to address important assumptions in statistical rate theory as well as our ad hoc assumptions about the details of any one reaction system. Some improvements in the theory applicable to polyatomic reaction systems will be noted. The paper will be organized as follows. First, a section on theory will be presen- ted. Emphasis will be on concepts with minimal details included. Next, examples will be taken from C,H,+., C4H,+' and C6HZ'. The examples will be chosen to high- light the various kinds of information the experiment/theory interplay yields. Finally, a brief summary section is included that collects the main points we hope to make in the body of the paper.2. THEORETICAL As noted in the Introduction, the goal of our research has not been to develop new theories of gas kinetics, but to adapt and improve existing theory for application to reactions of polyatomic ions. As such our primary role is one of interpreter rather than inventor. We are interested in tailoring statistical rate theory to fit the particular requirements of gas-phase ion-molecule reactions. One of the important aspects of ion-molecule collisions is that the long-range " physical " potential often dominates the encounter. For collisions between ions and non-polar neutral molecules a tractable analytical form for this potential can be written and used to develop useful expressions for the orbiting transition-state propertie~.~*'~ One direct consequence of this analytical potential is the development of simple expressions for capture-collision limiting rate constants and cross-sections and their dependence on energy and angular momentum.As we shall see, many properties of ion-molecule chemistry depend strongly on the orbiting transition state. As a consequence, one of our early goals was to extend the phase-space concepts of Light developed for atom-diatom systems to any pair of atomic or molecular partners interacting via the ion-induced dipole potential." Successfully accomplishing this goal has allowed us to apply phase-space theory to both unimolecular l2 and bi- molecular l3 reactions of general polyatomic ions. As the system moves through the orbiting transition state towards smaller r, the " chemical " potential begins to dominate. A series of flux minima, or transition states, is encountered as the system seeks to sample different parts of the hypersurface.The specific outcome of the encounter will be determined both by the properties of these transition states and the order in which the system encounters them. It is this aspect of the theoretical model that will be dealt with in some detail here. Our ground rules include treating the system classically (except for calculation of the vibrational densities of the states), rigorously conserving total energy and angular momentum (but ignoring quantum numbers of any kind), and assuming the basic assumptions of transition-state theory hold.Two aspects of the theoretical model will be dealt with in some detail here: (1) transition-state switching and (2) application of the theory to experimentally observed quantities. 2.1. THE TRANSITION-STATE SWITCHING MODEL For the general unimolecular reaction A , ~ " + B + cM.T. BOWERS et al. 59 the microcanonical transition-state theory rate constant can be written as where pA(E,J) is the density of states of reactant A at total energy E and angular momentum J, FS(E,J) is the local equilibrium flux through the first transition state the system encounters and cu(E,J) is the probability that the system will proceed to products once it has successfully traversed the first transition state. The factor w(E,J) was first introduced by Miller l4 to describe bimolecular reactions with mechanisms inter- mediate between " direct " and " intermediate complex".The extension of w(E,J) to cuver unimolecular reactions and complex reaction surfaces with many transition states was made by Chesnavich et a1.7*'o It is instructive to explore the form of w(E,J) for several kinds of reaction coordinate. In each case a probability branching analysis, similar to that developed by Miller,14 is used. It is also useful to note that for bimolecular reactions Ft (EJ) in eqn (2) is unchanged, but pA(E,J) corresponds to the density of states of the reactant pair. In the remainder of this section the (EJ) argument will be dropped for notational clarity. 2.1.1. SINGLE- WELL UNIMOLECULAR REACTIONS For this case the reaction coordinate is given in fig.I(a) and FORB F$ + FOR* - p S p R 3 p A X w = (3) where the fluxes through the various transition states are defined in fig. l(a). Since the flux is a local minimum at the tight transition state, F$, and at the orbiting tran- sition state, FORB, a local flux maximum, FMAX, must occur between them. Methods exist 137~10 for evaluating F# and FORB, but not for FMAX. Hence, if this procedure is to be useful, approximate methods for dealing with FMAX must be developed. For the case of a substantial configuration barrier along the reaction coordinate [the dotted line in fig. l(a)], Barrier along reaction coordinate F# < FORB FMAX - FORB W - t l (4) k = F:/pA which corresponds to the usual form of the microscopic rate constant for such cases. The assertion FMAX z FORB is reasonable because the properties of the configurational saddle point severely limit the flux at that point and completely dominate the dynamics in the system.The situation becomes substantially more complex when the reaction coordinate appears as the solid line in fig. l(a). In this case there is no clear-cut saddle point on the energy surface and hence F, PAX and FORB will have similar values for at least selected values of E and J. Before discussing how we deal with this case, it is useful to discuss the origin of Fz at an energy below the threshold for dissociation (i.e. at E < E"). Consider the system moving at constant energy from separated products toward the reactant well. When the system begins to experience the attractive forces, the potential energy60 ION-MOLECULE COLLISION COMPLEXES begins to decrease and the number of available states to increase, which leads to an increase in the flux.Simultaneously, however, translations and rotations in the separated products begin to become vibrations in the reactant molecule leading to wore widely spaced energy leiels and a decrease in the flux. These two effects offset each other and lead to a flux minimum somewhere between the point where the potential begins to significantly decrease and its minimum value. A set of such flux minima E P E reaction coordinate ( b ) F reaction coordinate Fig. 1. Schematic reaction coordinate diagrams for (a) a single-well unimolecular reaction with and without a barrier to the reverse reaction; (b) a double-well unimolecular reaction.exists for the accessible E,J manifold. For simplicity, we wilI assume onIy a single A ux minimum (transition state), i.e. an E,J " averaged " transition state. It wouId be interesting to test this concept on a set of realistic ion-molecule potential surfaces. Our procedure for dealing with this case is to choose a tight transition state based on its ability to model absolute and relative rate constants (more on this point later). Then for every value of E,J the flux, Fz, is calculated and compared with FORB. The minimum flux point is chosen as the transition state in accordance with the variational criteria. At values of E E E", the orbiting transition state will have the minimum flux since the number of available states is so small.At E > E" the tight transition state rapidly takes over, however, because of the rapid increase in FORB with E. A schematic plot of Fi and FORB against E is given in fig. 2. There is a region where F * z FORB and where PAX is also approximately equal to both. However, this '' transition " region is very small because FORB increases very much faster with E than with F t , Consequently we choose to ignore this region and simply test F# and FORB to see which one is the minimum and set FMAX equal to the larger flux. While thisM.T. BOWERS et al. 61 L E Fig. 2. Schematic plot of log F against E for a tight transition state (Fs) and an orbiting transition state (FORB). procedure is clearly not exactly correct, it does capture the main features and does allow for efficient transition-state switching along reaction coordinates of the type being discussed. In summary, No barrier along reaction coordinate FMAX = greater of {FS,FoRB) E -+ E"; co -+ FORBIF$; k + FoRB/pA E & E"; 03 + 1; k 4 FJlpA.In general FM"(E, J ) k(E,J) == PA(E, J , FMrN(E,J) = lesser of {FoRB(E,J), FJ(E,J)}. 2.1.2. DOUBLE DEEP-WELL POTENTIALS All of the systems of interest here have more complex reaction coordinates than the ones discussed above. Typically there are several competing exit channels and often several deep energy wells and corresponding transition states of kinetic import- ance. An example is given in fig. l(b) for a double deep-well potential. All of the arguments given above pertain to reaction of the outer well to products. The overall rate constant for molecules starting in the inner well is There are two cases to consider. p1 9 p2 and p1 z p2.In this case a probability branching analysis gives (1) P1 9 P2 FMIN + p M r N - ~:FMIN/FMAX u = (7)62 ION-MOLECULE COLLISION COMPLEXES where FMrN = lesser of {FoRB,Fi). Since FyAX is in a potential well, it is reasonable to assume FYAx $ FMIN,F; and hence FIFM" k = - PdFi + PMIN)' (2) P1 x P 2 In this case the detailed kinetics must be considered. C1 + Cz -+ products is assumed with C , in a steady state, then F:F'" = F: (PI + pz) + p i FM" (9) If a mechanism of the type (10) which is the general expression for calculating microscopic rate constants for uni- molecular reactions of double deep-well potentials. Finally, a word should be said about birnolecular reactions on double deep-well potential surfaces.In these cases, another pair of transition states (orbiting and tight) have to be considered in the incoming channel. The same process as outlined above can be used to develop expressions for w and k. Details are given elsewhere.l-'O 2.2. MODELLING EXPERIMENTAL SYSTEMS In order to apply the transition-state switching model to experiment, care must be taken to compute what is actually measured. Experimental conditions must be accurately reproduced by the model. Hence it is not useful to establish general equations for calculating experimental parameters. Such equations have been generated for each of the specific reaction systems discussed here (C4H;.,l C4Hi'2 and C6H$'3). In this section, several examples of what must be done will be given.2.2.1, ABSOLUTE UNIMOLECULAR RATE CONSTANTS In all three cases of interest here, the experimental unimolecular rate constant has been determined using the photoion-photoelectron coincidence (PIPECO) technique. Details are given by Baer l5 elsewhere in this Discussion. In this technique the total rate constant for loss of a parent ion is obtained by fitting a measured time-of-flight distribution of one of the product ions (usually the most abundant one). This experimental procedure requires the solution of the kinetic equations of the system if experiment is to be compared with the0ry.l For example, if parent ion P+' with internal energy E, is dissociating to product i+, then where [P+.J0 is the concentration of P"' at t = 0.Eqn (11) describes the experi- mentally observed kinetics. A similar equation can be written in terms of the theo- retical microscopic rate constants. Equating the two time derivatives allows theM.T. BOWERS et al. 63 experimental rate constant to be determined in terms of the theoretical rate constants so experiment and theory can be compared. The final relationship between the observed experimental rate constant, k:t,b,"(E,), and the theoretical microscopic rate constants is where the total theoretical unimolecular rate constant is and where the vibrational energy in the dissociating ion is E, = E - BJ2. (14) The J distribution, P(J), is given by the Boltzmann distribution for the neutral precur- sor that was photoionized to form the reactant ion.The ki(E,,J) are calculated using the transition-state switching model previously discussed. An interesting point is that the experimentally observed rate constant is time-dependent according to eqn (12) because the exponential term causes J-states with faster dissociation rates to have smaller populations as time progresses. In all systems studied to date, these effects have been small (<lo%) over the experimentally accessible time regime. 2.2.2. THE UNIMOLECULAR BRANCHING RATIO To calculate the branching ratios as a function of vibrational energy the details of the kinetics of the system must again be considered yielding ki(Ev'J) (1 - exp[-ktot(EV,J)z]} (15) - = k I J P ( J ) [i+'] * [P+~Io 0 ktot(Ev, J ) where z is the reaction time in the experiment.Eqn (15) can be directly compared to experiment. 2.2.3. KINETIC-ENERGY DISTRIBUTIONS In the work discussed here it is assumed that the kinetic-energy distribution is given by the orbiting transition state. Under conditions when the orbiting transition state dominates the dynamics of the system, such an assumption is clearly reasonable. However, for most reaction channels and for most values of E and J, the tight tran- sition state is of minimum flux. Hence, correspondence between theory and experi- ment in these cases implies that the dynamics " relax " the system after passing through the tight transition state, allowing the orbiting transition state to dictate the kinetic- energy distribution even though the tight transition state determines the kinetics. This circumstance only occurs in practice when the critical energy of the tight tran- sition state is less than or approximately equal to the dissociation energy (discussed later).The microscopic kinetic energy release is given by the ratio of orbiting flux at translation energy E, divided by the total orbiting flux at E,J FoRB(E,J;Et)dEt p(E, J; Et)dEt = FORB(& J )64 ION-MOLECULE COLLISION COMPLEXES In ion chemistry these distributions are almost always measured in a mass spectro- meter, usually a reversed-geometry instrument (magnet followed by electrostatic sector).16 Hence, the measured distributions correspond to unimolecular reactions of ions formed by electron impact that dissociate in the time window presented as the parent ion traverses the second field-free region (between the magnet and electrostatic analyser).The probability that the parent ion dissociates in the second field-free region, P,(E,J), depends on k,,,(E,J) and the time window sampled in the experiment. The probability of forming products i is R,(E,J) = k,(E,J)/k,,,(E,J). Hence the probability that the system will dissociate into channel i with kinetic energy Et is where P(J) is the parent ion J distribution given by the thermal J distribution of the neutral molecule and Pdcp is the nascent ion internal-energy distribution generated by electron impact. 2.2.4. BIMOLECULAR REACTIONS The details of the formalism necessary to calculate bimolecular reaction efficiencies, product distribution and collision-complex lifetimes wiJl not be given here. The basic procedure is to calculate the steady-state collision complex E, J distribution as deter- mined by the flux through the orbiting transition state in the incoming channel.The microscopic rate constants into each product channel are then averaged over this distribution to get the thermal-energy rate constants necessary for comparison with experiment. The lifetime of the collision complex is then where E l ( T ) is the thermally averaged unimolecular rate constant for reaction into channel i from collision complexes formed from reagents at temperature T. 2.2.5. FLUXES Details for calculating fluxes through the various transition states are given else- ~ h e r e . l * ~ J ~ Of importance here is the fact that both energy and angular momentum are rigorously conserved throughout.A few words on the procedures used to estimate transition-state parameters are also in order. For the stable species and orbiting transition states the parameters were taken from the literature or estimated on a rational basis. For the tight tran- sition states many of the parameters could be reasonably estimated; however, some parameters are not known (such as tight transition-state energies and rotational constants) and hence upper and lower limits were evaluated and the parameters were treated as adjustable within these limits. Once determined, all transition-state properties are left unchanged and used to calculate all rate constants and energy distributions for comparison with experiment.M.T. BOWERS et al. 65 3. COMPARISON OF EXPERIMENT AND THEORY In much of the elegant recent work in reaction kinetics the emphasis has been on determining as precisely as possible the reactant and product states.As a consequence, these experiments usually look very closely at events occurring in only one channel of a multichannel reaction, or in events occurring within several states within one channel. The approach we are taking in this work is the opposite. Instead of looking at a " single tree " carefully, we are looking at the " forest." It is our experience that the reaction kinetics of any one channel can often be fitted using statistical theory methods by a variety of mechanisms and transition states. However, if a simultaneous fit for many channels from both bimolecular and unimolecular per- spectives is required, then the choices usually dwindle to a precious few and a much clearer perspective of the kinetics is often obtained.In this section a few examples are selected from the C4Hi', C4H2. and C6H;' reaction systems. The systems increase in complexity as written with C4H i. having three low-energy, unimolecular exit channels, C4Hi. having four, and C6Hi. having six. In each system at least one of these exit channels consists of an ion and molecule that lend themselves to study as a bimolecular ion-molecule reaction (C2H t '/C2H4 in C4H ;f ', C2H t '/C2H4 in C4Hi' etc.). Being able to use bimolecular reactions to generate intermediate complexes that can also be directly generated by electron or photon impact on neutral molecules has several advantages.First, complexes formed by ion-molecule reaction have a narrow band of energies, but a broad angular momentum distribution peaked reaction coordinate energy scale is quantitative. Fig. 3. Schematic potential-energy surface used in niodelling the C4H8+ ' system. The at high J values. Second, isotopic substitution can be used to probe details of the reaction surface. Third, microscopic reversibility requires the same transition states be involved in both the unimolecular and bimolecular reactions. Hence, if an orbit- ing transition state is required to explain the rate constant for a bimolecular reaction, a tight transition state cannot be invoked in the same reaction channel to explain a unimolecular rate constant for the same E,J range the bimolecular reaction samples.Awareness of this fact was crucial in our initial development of the transition-state switching model, and helped to explain several paradoxes we had enc0untered.l The three pseudo-reaction-coordinate diagrams are given in fig. 3-5. These66 ION-MOLECULE COLLISION COMPLEXES 0 reaction coordinate energy scale is quantitative. Fig. 4. Schematic potential-energy surface used in modelling the C4H6 + system. The diagrams should be viewed as schematic. The various energies of the reaction channels and intermediate states are quantitative as far as possible. The transition- state energies shown are derived from modelling the kinetic data by theory. No mechanistic information should be implied by the dissociation assymtotes drawn from the benzene+' and fulvene'' structures in fig.5 . In the remainder of this section, some comments will be made about various aspects of the reacting systems. Unfortunately, space does not allow a thorough discussion of the literature, which is extensive in each case. Reasonably corn- A d reaction coordinate Fig. 5. Schematic potential-energy surface used in modelling the C6H6+. system. The energy scale is quantitative. There is no mechanistic information in the placement of the exit channels relative to the benzene" and fulvene+' wells.M.T. BOWERS et al. 67 prehensive summaries are found in ref. (1) for C,H:'; ref. (2) for C,Hi'; and ref. (3), (17) and (18) for C6Hg'. 3.1. ABSOLUTE UNIMOLECULAR RATE CONSTANTS The absolute unimolecular rate constants have been measured by Baer and co- workers 17719*20 using PIPECO techniques.The results are given in fig. 6. For the C,Hz' system the experimental data are for five different C4H, isomers.L9 The most stable are the but-2-ene+' isomers, and these are assumed the dominant structures in our calculations. In this case, the buta-1,3-diene cation is substantially more stable than the others and is used in the calculations. For the C4Hi' ions, three isomers are used to generate the experi- mental data.I7 The benzene parent ion is most stable, but the fulvene parent ion contributes ca. 15% to the phase space, and hence both ions were used in the calcula- tions. For all three systems (C,Hi', C,H:' and C,Hi'), both open-chain and cyclic isomers were studied experimentally. For the C,Hi' system six different isomers were used.2* Several things are apparent from the data.(1) In all cases, all neutral isomers give essentially identical plots of k(E) against E. Hence, isomerization barriers are substantially lower than the lowest energy dissoci- ation thresholds. (2) In essentially all cases, the original ionization event accesses an excited electronic state surface, often differing substantially from isomer to isomer. There is no evidence of isolated electronic state properties in these data. (3) The theoretical curves give an excellent fit to the magnitude of k(E). The theory assumes statistically complete E,J randomization on the ground-state surface. (4) The theory curves have a somewhat steeper slope than experiment. The most probable reason for most of this discrepancy is the failure to account for the thermal internal energy in the neutral molecule being ionized.Accounting for this effect would lower the low-energy data points somewhat, with a lesser effect as the available energy increases. Conversely, including the effect in the theory would raise the low- energy part with a decreasing effect as energy increases. Such a calculation has been done for the C6H: ' case in fig. 6(c). The dotted line has corrected the benzene parent ion for the thermal internal energy in the benzene neutral. Clearly the agreement is much better. [It is, in fact, nearly exact for the benzene points in fig. 6(c) designated by the open squares. Each isomer will have a slightly different correction due to the presence of internal rotors, etc.] 3.2.UNIMOLECULAR BRANCHING RATIOS The branching ratios for the low-energy pathways have been measured by Werner and Baer 2o for C4H:', Eland et a1.21 for C6Hi' and Hsieh et a1.22 for C4H$'. The data are summarized in fig. 7. In the C4Hi' data, all open-chain isomers gave essentially identical results (we have used these in fig. 7) with slight differences being observed for the two cyclic isomers. In C6Hi ' we have used the breakdown graph for the benzene parent ion." Baer et a1.l' point out that these curves are essentially the same for all isomers in the low-energy regime (< I .5 eV above threshold). In C4H: ' all breakdown graphs are essentially the same.2o Several points are immediately apparent: (1) In the C4H$' and C4H2' cases, excellent agreement between experiment and theory is evident over the entire energy range.Hence both the absolute and relative rate constants are fitted using the same set of transition states and assuming a1168 ION-MOLECULE COLLISION COMPLEXES 7 - n 3 6 . CI reaction occurs on the ground-state surface regardless of the initial state of formation of the parent ion. (2) In both the C,Hg' and C,Hi' cases, the formation of the C,Hi ' ion required a tight transition state at energies only slightly above threshold. The reverse bi- molecular reactions of this ion (with C2H4 and C2H2, respectively) occur at nearly the polarization potential collision Iimit,ls3 however, and hence an orbiting transition 5- 0.1 0.2 0.3 0.4 0.5 vibrational energy above threshold/eV a 4 0.1 0.2 0.3 0.4 0.5 vibrational energy above threshold/eV vibrational energy above threshold/eV Fig. 6.Plot of k::: against E for (a) C4Hi', (b) C4H:' and (c) C6H:.. theoretical. The solid lines are The dashed line in (c) includes the vibrational energy in the benzene neutral moleculc in the calculation(see text). state is required at energies near threshold. The transition state switching model is required to fit both pieces of information. (3) The branching ratio for C,HZ' is well fitted within 2.5 eV of threshold for all products. The C6H and C,H a- data begin to deviate from the statistically expected result above ca. 2.5 eV above threshold, however. Such a deviation could be evidence for participation of isolated electronic states. The 2A,, state of benzene has an onsetM.T.BOWERS et al. 69 energy above thresholdlev energy above thresholdlev (C 1 + + .+ I I I 1.0 2 .o 3.0 4.0 energy above thresholdlev Fig. 7. Plot of k l / z k r against E for (a) C4H Q ' (b) C4H 2 ' and (c) C6H $ '. The solid lines are theoretical. Symbols as follows: (a) 0, C3H:'; +, C4H8+' + C4H; a, CzHz'; (b) 1 C4Hz' + CdH, +- ; 0, C3H;'; +, CzH4"; (c) a, CsHf'; 0, CgH:'; +, C4HZ'; X , CsHZ.70 ION-MOLECULE COLLISION COMPLEXES at 16.84 eV and may be responsible for the observed behavio~r.~~ It is not possible to fit these data using statistical theory if the quasi-equilibrium hypothesis is invoked. The C,W:' data of Baer et al," however, do not appear to deviate as strongly as the data in fig. 7(c) (see next section). (4) The ratio C,H<'/C,Hi' as a function of energy in the C,Hi ' system has been given by Baer et a1.,I7 and is reproduced in fig.8. Baer et al. suggested the down-turn 0 .CI * =! +a X u" --.. 4-m X u" I 0.2 0 0 0 0 0.01 I f I 1.0 2 .o 3.0 4.0 energy above thresholdlev Fig. 8. Plot of C3H$/C4Hz' against E for the C6H:' system. The solid line is theoretical. in the ratio at ca 2 eV above threshold may be due to the onset of the linear C,Hi' isomer near this energy. Our calculations have this structure incorporated (see fig. 5 ) and accurately modelled the ratio over the entire energy range (solid line in fig. 8). A clear deviation in the ratio is observed for hexa-1 ,S-diyne at ca. 2.5 eV above threshold. The third photoelectron band of this molecule has an onset at that energy 24 (pre- sumably a 2B, and/or ' A , state).18 It appears this state does not obey the quasi- equilibrium hypothesis but dissociates without isomerizing to a structure common to benzene and hexa-2,4-diyne parent ions.(5) In the theoretical section it was pointed out that for exit channels without reverse activation energies, both a tight and orbiting transition state are involved and that the tight transition state has an energy Et < E". In each of the systems discussed here there is at least one such reaction channel, notably C&' -+ C& $. CH3 C4H:' -+ C3Hf + CH; and C,Hi' + C,H:' + C,H, Tn each case Ez is required to be less than E", usually by 0.1-0.2 eV. The data simply cannot be fitted if Et 3 Eo. Hence this appears to be a general property of this type of reaction coordinate.3.3. BiMOLECULAR REACTIONS In each of the three systems at least one of the unimolecular reaction channels produces products that can easily be used to study the reverse bimolecular reaction. These bimdecular reactions provide information that cannot be obtained from uni-M.T. BOWERS et al. 71 molecular reactions alone. While space does not permit a detailed presentation here, some highlights will be given. 3.3.1. ANGULAR-MOMENTUM EFFECTS ON BRANCHING RATIOS Bimolecular ion-molecule reactions form collision complexes with a large amount of angular momentum relative to a thermal distribution. That this angular momen- tum can have substantial effects on branching ratios was first pointed out by Meisels et al.,25 and by Klots.26 The branching ratios for the reactions f / C2Hi' + C2H4 - (C,H$)* \ I C4H3 + H' and C2Hi' + C2H, =+ (C4H:.)*- \ C4H3 + H' \ are given in table 1 along with PIPECO results at the same energy and with the theoreti- cal predictions of the model discussed here.It is evident that the H' loss channel is severely suppressed in the high-J bimolecular case. The results are quantitatively accommodated by the transition-state switching model. Meisels and Klots both attributed the observed suppression of the H' loss channel at high J to the low polari- zability and low mass of the H' atom neutral fragment. This interpretation implicitly Table 1. Experimental and theoretical branching ratios G H f ' + C2H4 PIPECO, C4Hz ' ionic bimolecular reaction unimolecular reaction experiment theory experiment theory product C3HZ 0.90 0.89 0.72 0.72 C4H 3 0.10 0.11 0.28 0.28 C2Ha' + C2H2 PIPECO, C4H $ * bimolecular reaction unimolecular reaction ionic product experiment theory experiment theory C3H: 0.74 0.77 0.50 0.49 0.26 0.23 0.43 0.48 - - 0.06 0.03 C2H f ' C4Hf a Ref.(1); ref. (22); ref. (2); ref. (20); theory as described in this work.72 ION-MOLECULE COLLISION COMPLEXES assumes the orbiting transition states in the two exit channels dominate the kinetics. The switching model clearly shows the tight transition state is rate Iimiting over almost the entire range of E and J, however. Hence, it is the lower rotational constant in the CH; loss channel that actually diverts the high-l complexes away from the H' loss channel. 3.2.2. ISOTOPIC SUBSTITUTION Different isotopic variants can be used to study reaction mechanisms.For example, the results of the reactions C,DZ' + C2H4 -+ products C2H;* + CzD4 -+ products are given in table 2. Table 2. Isotopic scrambling in the reactions C2H$ ' + CZDZ + products and C2D ' + C2H,+pruducts ion is0 topic product distribution experiment statis tical 22 f 4 54 L- 5 24 5 34 f 10 66 & 10 20 60 20 33 67 7 53 40 * The same results were obtained for C2H4+'/CZD~ and C2H:'/C2H2 within experimental error (i.e. for X = H, Y = D or X = D, Y = H); 'ref. (2). It is apparent that the C3H f / C H ; and C4H +'/H' channels exhibit complete scramb- ling. These reactions must pass through the deep C4HZ' we11 so this is the expected result. The charge-transfer product channel C2H t '/C2H2, however, is extremely non- statistical with little isotopic scrambhg observed, suggesting this channel does not pass through the deep well portion of the surface (see fig.4). 3.2.3. MECHANISMS OF CHARGE TRANSFER The question of whether charge-transfer reactions proceed via long-range electron jump or intimate collision complexes is a long-standing one in ion ~hemistry.~' The isotopic-labeling studies discussed in (2) above strongly suggest that the C4H $ * deep well is not involved. The shallow we11 (see fig. 4) may be involved, however. Stable van der Wads type cluster ions have been shown to exist for (C,H4): ' and (C,H,)$ ' 29 and hence could we11 exist for (C2H2)-(C2H4)+'. The predictions of the switching model incorporating the small we11 are compared with experiment in table 3.ClearIy the agreement is excelTent-booth for the bimolecular reaction and the PIPECO result at the same energy. The fact that the two sets of experimental data are so different and that the theory can fit both simultaneously with one set of transition states isM.T. BOWERS et al. 73 strong evidence that the charge-transfer reaction proceeds essentially exclusively via the loose complex and not via long-range electron jump. If one assumes no scramb- ling in the shallow well, the small amount of scrambling evident in table 2 suggests that 10-20% of the charge-transfer reactions actually sample the deep well in semi- quantitative agreement with theory.2 Table 3. Calculated and experimental branching ratios in the C4H;' system G H t ' + CZH~ PIPECO ionic bimolecular reaction unimolecular reaction product experiment a theory experiment theory C3H s 0.20 0.20 0.25 0.30 C4H : 0.16 0.17 0.56 0.55 C2H: ' 0.64 0.63 0.19 0.15 k t o t kcollision 1.06 1 .oo Ref.(2); ref. (20). 3.2.4. KINETIC-ENERGY RELEASE The energy distributions for products of unimolecular reactions have been measured for all low-energy pathways that have metastables using mass-spectrometric techniques. The theoretical model presented in section 2 calculates expected distri- butions generated by the orbiting transition state. Techniques have been presented in the literature for calculating product energy distributions from tight transition states in some instances, but we have not yet incorporated them into the model. A typical example is given in fig.9 for the reactions C4H:' -+ C3H3 + CH; C4H+B' + C4H3 + H'. The following observations can be made. (1) The C3H3 distribution is relatively well fit by the transition-state switching theory. Examination of the details indicate that the tight transition state is rate determining over the range of E,J states that contribute to the distribution. Hence, the dynamics of the system must be such that a statistical distribution is maintained between the orbiting and tight transition states. This result appears to be generally true for all reactions in which Et < E" (C4H2' + C3H; + CH; and C6H+6'+ C4Hz' + C2H4). (2) The He loss experimental distribution is much broader than the theoretical prediction. This seems to be a general case for H' loss reactions.In the C4H+6' system there is a reverse activation energy in the H' loss channel. In the C6H2' system the H loss distribution is broader than the theoretical, but not as dramatically so as for the C4Hg' system shown in fig. 9. A similar " broader than statistical " result has been obtained for the reaction F' + (CCH3)2C=CF2 + CH3FC=CH2 + CH;. Worry and Marcus3' obtained agreement between theory and experiment by con- sidering only the " transitional " modes that go from bending vibrations at the tight74 ION-MOLECULE COLLISION COMPLEXES transition state to rotations in the products. Perhaps a similar mechanism is operative here, although the method has not yet been successfully developed and applied for low-J unimolecular reactions. \ & I I J I 0.00 0.05 0.10 0.15 0.20 - 0.0 0.1 0.2 0.3 0.4 EJeV Fig.9. Theoretical (-) and experimental (- - - ) kinetic-energy distributions for metastable C4Hi * ion. The theoretical curves correspond to the orbiting transition-state predictions. (a) C4Hi'+C3Hz + CH;; (b) C,H:'+C,H: + Ha. 4. SUMMARY The main points of this paper are these: (1) A transition-state-switching statistical- theory model that rigorously conserves energy and angular momentum has been developed. (2) There are, in general, two functional transition states in exit channels without a reverse activation energy-a tight one at smaller values of r and an orbiting one at long range. (3) The energy of the tight transition state in cases described in (2) above is always Ez 5 E". (4) Transition-state switching is required to fit the energy- dependent branching ratios in all systems discussed here.(5) The quasi-equilibriumM.T. BOWERS et al. 75 hypothesis holds for C,H+,' and C4H+6' over the entire range studied. (6) The quasi- equilibrium hypothesis appears to be violated for both benzene and hexa-l,5-diyne parent ions at ca. 2 eV above the lowest dissociation threshold. (7) The dynamics appear to maintain a statistical distribution of states between the tight and orbiting transition state for reactions where E $ < E". (8) The charge-transfer reactions (C,H$'/C,H,, C,H;'/C,H, and C2Hz+ '/C,H,) proceed primarily via a loosely bound van der Waals complex and not by either long-range electron jump or by a full sampling of the system phase space. (9) The theoretical model quantitatively accounts for the angular-momentum effects apparent when bimolecular and uni- molecular branching ratios are compared at equal energies.The support of the National Science Foundation (grant CHE80-20464), the Air Force Office of Scientific Research (AFOSR-82-0035) and the donors of the Petroleum Research Fund of the American Chemical Society (PRF-12008-AC5,6) is gratefully acknowledged. Much credit should also go to Dr. Walter Chesnavich, who did much of the theoretical development discussed here. W. J. Chesnavich, L. M. Bass, T. Su and M. T. Bowers, J. Chem. Phys., 1981, 74,2228. M. F. Jarrold, L. M. Bass, P. R. Kemper, P. A. M. van Koppen and M. T. Bowers, J. Chem. Phys. 1983, 78, 3756. M. F. Jarrold, W. Wagner-Redeker, N. Kirchner, A. J. Illies and M.T. Bowers, to be published. R. A. Marcus and 0. K. Rice, J. Phys. Colloid Chem., 1951,55, 894; R. A. Marcus, J. Chem. Phys., 1952,20,4658; for useful reviews, see J. P. Robinson and K. A. Holbrook, Unimolecular Reactions (Wiley, New York, 1972); W. Forst, Theory of Unimolecular Reactions (Academic Press, New York, 1973). H. M. Rosenstock, M. B. Wallenstein, A. L. Wahrhaftig and E. Eyring, Proc. Nut1 Acad. Sci. USA, 1952,38, 667. J. C. Keck, J. Chem. Phys., 1960,32, 1035; Adu. Chem. Phys., 1967, 13, 85. For a recent review of our earlier work, see W. J. Chesnavich and M. T. Bowers, in Gas Phase Ion Chemistry, ed. M. T. Bowers (Academic Press, New York, 1979), vol. 1, chap. 4, pp. 119- 151. J. C. Light, J. Chem. Phys., 1964,40, 3221; Discuss. Faruduy Soc., 1967, 44, 14; P. Pechukas and J. C.Light, J. Chem. Phys., 1965,42, 3281. E. Nikitin, Theor. Exp. Chem. (USSR), 1965, 1, 83, 90,275. lo W. J. Chesnavich and M. T. Bowers, Prog. React. Kinetics, 1982, 11, 137. l1 W. J. Chesnavich and M. T. Bowers, J. Chem. Phys., 1977, 66, 2306. l2 W. J. Chesnavich and M. T. Bowers, J. Am. Chem. SOC., 1977, 99, 1705. l3 W. J . Chesnavich and M. T. Bowers, J. Am. Chem. SOC., 1976,98, 8301. l4 W. H. Miller, J. Chem. Phys., 1976, 65, 2216. l5 T. Baer, Faraday Discuss. Chem. Soc., 1983, 75, 45. l6 For a description of the technique see M. F. Jarrold, A. J. Illies and M. T. Bowers, Chem. Phys., 1982, 65, 19. l7 T. Baer, G. D. Willett, D. Smith and J. S. Phillips, J. Chem. Phys., 1979, 70, 4076. H. M. Rosenstock, J. Dannacher and J. F. Liebman, J. Rad. Phys. Chem., in press. l9 T. Baer, personal communication. See ref. (1). 2o A. S. Werner and T. Baer, J. Chem. Phys., 1975, 62, 2900. J. H. D. Eland, R. Frey, H. Schulte and B. Brehm, Int. J. Mass Spectrom. Ion Phys., 1976, 21, 209. 22 T. Hsieh, J. P. Gilman, M. J. West and G. G. Meisels, J. Phys. Chem., 1981, 85, 2722. 23 D. W. Turner, C. Baker, A. D. Baker and C. R. Brundle, Molecular Photoelectron Spectroscopy 24 F. Brogli, E. Heilbronner, J. Wirz, E. Kloster-Jensen, R. G. Bergman, K. P. C. Vollhardt and " G. G. Meisels, G. M. L. Verboom, M. J. Seiss and T. Hsieh,J. Am. Chem. SOC., 1979,101,7189. 26 C. E. Klots, in Kinetics of Ion-Molecule Reactions, ed. P. Ausloos (Plenum Press, New York, 27 For discussion of this issue see J. B. Laudenslager, W. T. Huntress and M. T. Bowers, J. Chem. (Wiley-Interscience, New York, 1970). A. J. Ashe 111, Helv. Chim. Acta, 1973, 56, 2171. 1979).76 ION-MOLECULE COLLISION COMPLEXES Phys., 1974, 61, 4600; R. Marz, in Kinetics of Gas Phase Ions, ed. P. Ausloos (Plenum Press, New York, 1979). S. T. Ceyer, P. W. Tiedemann, C. Y. Ng, B. H. Mahan and Y . T. Lee, J. Chem. Phys., 1979,70, 2138. 29 Y. Ono and C. Y. Ng, J. Chem. Phys., 1982,77, 2947. 30 G. Worry and R. A. Marcus, J. Chem. Phys., 1977,67, 1636.

ISSN:0301-7249    

DOI:10.1039/DC9837500057

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1983, 75, 77-88 Intramolecular Decay of Some Open-shell Polyatomic Cations BY JOHN P. MAIER, MARTIN OCHSNER AND FRITZ TWOMMEN Physikalisch-Chemisches Institut der Universitat Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland Received 16th December, 1982 Radiative and non-radiative relaxation processes of selected open-shell polyatomic cations in various vibrational levels of their lowest excited electronic states have been studied by two complementary techniques. Fluorescence quantum yields and lifetimes are obtained by photoelectron-photon coincidence measurements, and laser-excited fluorescence of rotation- ally cooled cations provides lifetime data at much higher resolution and-for further levels. The information forthc_oming on the intramolecular decay of isojated CO*, B2 Ei, ClCN+@rI, H-(-CEC-)~-H + A 211v and 1,2,4,5tetrafluorobenzene + B2Bl, is discussed.INTRODUCTION Investigations of the radiative and non-radiative relaxation behaviour of open- shell polyatomic cations in their lowest excited electronic states in a collision-free environment have been systematically carried out in recent years.x.2 The opportunity for such studies was provided by the detection of the radiative decay of certain types of open-shell organic cations from their 2z or 2B excited states to their ground states 22.3 As a consequence, several techniques have been applied to probe their spectro- scopic structure in the gas phase 4*5 and to follow the radiationless processes quan- ti tative1y.l~ For spectroscopic purposes, our most recent approaches involve the study of the emission spectra of rotationally cooled cations prepared by electron-impact excitation of seeded helium supersonic free jets, and of the laser excitation spectra of rotationally and vibrationally cooled cations.6 These techniques enable one to deduce vibrational frequencies of the cations in their ground and excited states to within 5 1 cm-l, or better.With the smaller cations the rotational structure can be probed using the laser excitation method with a resolution of 0.001 nm. In studies of radiationless processes the initial phase involved the measurements of the lifetimes of the cations in selected vibrational levels within the excited state using the appropriate emission bands.3 This was followed by quantitative determinations of the radiative and non-radiative rate constants for the pathways depleting state- selected cations from the absolute measurements of the fluorescence quantum yields and lifetimes by a photoelectron-photon coincidence technique.By this means the radiative, k,(u'), and non-radiative, knr(v'), rate constants of the selected level u' have been evaluated for most types of open-shell organic cations that decay radiatively. These studies and a summary of the findings have recently been reviewed.l In this contribution we present the results of such quantitative investi- gations of the radiationless processes of some selected cations using the photoelectron- photon coincidence apparatus with an improved electron energy resolution (ca. 2078 INTRAMOLECULAR DECAY OF POLYATOMIC CATIONS meV) as well as the laser excitation measurements of lifetimes at much higher resolu- tion (ca.0.1 meV). By combining the two sets of results the non-radiative rate dependence on the vibrational modes excited can be quantitatively established. To illustrate these aspects the triatomic cation CO t is considered, since non-unity fluorescence quantum yields are observed for the ’X: state even though the frag- mentation pathway is not accessible at these energies, and the ClCN+ species in the 21J state provides a further example of pronounced non-exponential decay which can be followed as a function of the vibrational excitation. The other cations chosen, of diacetylenes and of 1,2,4,5-tetrafluorobenzene, show characteristic decay in the statistical limit, and the non-radiative rates of the various modes could be determined using the two techniques in conjunction.EXPERIMENTAL The principle of the photoelectron-photon coincidence technique ’ is indicated by the scheme shown in fig. 1, where the experimental arrangement is schematically depicted. The apparatus has been described in detail elsewhere.’ An effusive beam of sample gas is ionised by a collimated beam of He(1a) photons. The resulting photoelectrons (eKeE.) are LENS SYSTEM TO PM He LAMP LIGHT BAFFLES ELECTRON LENSES H EM1 SPHERICAL ANALYSER IONISATJON REGION n n u u IM Fig. 1. Arrangement of the photoelectron-photon coincidence apparatus used for the measurements of fluorescence quantum yields and lifetimes of state selected cations.energy-selected by a 180” hemispherical analyser (5.5 cm diameter) and are then used as a time reference for the creation of cations in a specific excited state, e.g. 2J(u’). The time differences between these electrons and wavelength-undispersed photons originating from the ionisation region are measured and then stored in a multichannel analyser. If the excited cation in the selected state ‘A(u’) decays radiatively, the result is a curve of true coincidences superimposed on a uniform background of random coincidences. From such a curve the fluorescence quantum yield, qF(u’), can be evaluated using the relationship NTINe = VF(U’)h”(4J. P. MAIER, M. OCIISNER AND F. THOMMEN 79 @ where NT is the rate of true coincidences and IV, is the rate of photoelectrons defining the level v' of the excited cation.8 The wavelength-dependent probability of detecting photons, &(A), can be determined by coinc_idence measu5emen ts using cations with fluorescence quantum yields of unity [e.g.N f (B 'EL), CO: (A 211,)] and is found to be in the range of ( 3 4 ) x For the measurements presented in this article the electron band-pass has been reduced to 20-25 meV (for ca. 2-3 eV electrons) by means of an electron lens system as sketched in fig. 1. The probability of detecting energy-selected electrons, fe, remains ( 3 4 ) x lob3. Typical accumulation periods to attain sufficient coincidence statistics are 15-20 h with N, = 50-200 Hz, while the rate of undispersed photons is often between 5 and 10 kHz. The sample pressure is held in the range 10-4-10-3 Torr * to ensure collision-free con- ditions.The measurements of lifetimes of cations in selected levels of the excited state at much higher resolution were carried out by laser-excited fluoIescence using the arrangement sketched in fig. 2. This apparatus has been used previously for the recording of laser excitation spectra for 250 < R/nm < 600. DIGITIZER M I C R O - COMPUTER - INTER-- - FACE t-- 4 Ini H PENNING I IONISATION I Fig. 2. Scheme of the experimental arrangement for the lifetime measurements by laser excited fluorescence. of open-shell cations.l0 The cations are prepared in the ground states, 2z, vibrationally and rotationally cooled to liquid-nitrogen temperature by means of Penning ionisation with_argon metastables and collisional relaxation with the precooled rare-gas ~ a r r i e r .~ The 2 A ( ~ ' ) t 'f(u" = 0) excitation is produced by a dye-laser pumped by a nitrogen laser, operating with 0.02 nm bandwidth, typically at 30 Hz. The data accumulation was accomplished by a transient digitizer-microcomputer system as indicated in fig. 2. The number of individual decay curves added depended on the intensity of the respective excitation bands, and was typicalIy ca. 600. The lifetimes were extracted from a weighted least-squares fit to the decay curves and the overall time resolution was ca. 5 ns. Although the total pressure in the excitation region was ca. 0.2 Torr, for the short lifetimes (<lo0 ns) measu_red no significant differences were found to the values (e-g.for the zeroth levels of the ' A states) obtained * 1 Torr = 101 3251760 Pa.80 INTRAMOLECULAR DECAY OF POLYATOMIC CATIONS previously either by the photoelectron-photon coincidence or the electron-impact emission approaches operating at much lower pressures ( < 10- Torr). EXAMPLES AND DISCUSSION co; B2zc,' Of the triatomic open-shell cations decaying radiatively from their lowest excited electronic states, many have lifetimes in the microsecond range, i.e. CS t (2 211 .), H20+ (k 'la,>, H2S+ (2 2A1) and XCN+ (2 2Z+) with X = Cl,Br,I.3 These are consequently not suitable for lifetime or fluorescence quantum yield studies by either the photo- electron-photon coincidence or the outlined laser excitation techniques, because of the restricted time window.The cations with the shorter lifetimes are on the other hand used to calibrate absolutely the coincidence apparatus, since they often have fluorescence quantum yields of unity, e.g. C o t (J211U), N20+ [J2Z+ (v' = O)]. In two cases predissociation competes with the radiative relaxation and leads to less than unity yields, i.e. N20+ [2 'Z+ (u' # O)], COS+ [z 211 (u' In addition, fluorescence quantum yields less than unity were measured for C o t B" 2Ei l1 and l2 levels, although fragmentation channels are energetically inaccessible for these levels.8 The measurements have now been repeated using the coincidence apparatus with improved resolution. Compared to the previous determinations, the progressions of the vl vibration originating in the 2 211 state and in the 2ZC= state are now sufficiently discernible, so that cations formed initially in the 1 " n = 1-3 0)].8 14.0 17.0 18.0 I/eV Fig.3. He(1or) photoelectron spectrum of COz, recorded under coincidence conditions, with constant band-pass of ca. 20 meV.J. P. MAIER, M. OCHSNER AND P. THOMMEN 87 teveIs of the B' 2Xi state could be relatively cleanly selected. This can be seen in the photoelectron spectrum, shown in fig. 3, recorded under coincidence conditions. The measured photodectron-photon coincidence data are presented in table 1 and confirm the earlier results within the error limits. The lifetimes for the 1 p1 = 1-3 levels are the same as for the vibrational levels of the 2 TI state (ca,120 ns, 1 with n = &4), and reflect the radiationless coupling between the and A states.Table 1, Fluorescence quantum yields and lifetimes within the B' E: state of CO; O0 1.00 f 0.05 139 f 7 l 1 0.85 & 0.09 119 z t 12 l2 0.64 f 0.09 120 f 12 l 3 0.57 f 0.09 120A 12 This coupling phenomenon has been discussed in several articles.2b11*12 Most recently, it was estimated from branching-ratio measurements using optical and electron spectroscopies that 5&60% of the state population crosses over into the 2 state,I3 and a high-resolution study of the rotational structure yielded a further insight into the interaction of the The consequence of the coupling is that the emitted photons are distributed over the wavelength region. For the 8 2Et Oo Ievel it was shown that two thirds of the photons are emitted in the d < 330 nm region, one third in the 330-450 nrn region and no coincidences were detected for 3, > 450 nm.*J5 The fluorescence quantum yieIds for the 1" levels given in table 1 were evaluated assuming a similar wavelength intensity distribution of the emitted photons as for the 0' level and that they still all fall below 450 nm.That is not the case could now be shown by coincidence measure- ments where the photons were wavelength selected using filters. For the l2 Ievel, only 50% of the intensity was found in the region A < 330 nm and true coincidences were measured even for R > 450 nrn. Quantitative Coincidence determinations, which are extremely time consuming, yield the following qF(12) values in the indicated wavelength regions: 0.48 :& 0.05, 250-330 nm; 0.15 & 0.03, 330-400 nm; 0.11 0.02, 400-450 nm.Because the sensitivity of the photomultiplier falls off with in- creasing wavelength, and especialIy rapidly above 750 nm, only a lower limit qF( 12) z 0.08 for A > 450 nrn can be given. It therefore appears that with increasing vib- rational excitation (1" for JI = 1-3) progressively more intensity of the transition is red-shifted to regions beyond 450 nrn as a result of the coupling with the LT 211 11 state and leads to apparently decreasing fluorescence quantum yields (cf. table 1). ClCN+ B zn Non-exponential decay, under collision-free conditions, has now been established for a few of the smaller polyatomic open-shell cations, i.e. CS; (B %:), X-C=N+ (8 %) and X-C=C-Hf (A2n) with X = Cl,Br,X.16 This decay behaviour was appar- ent initially in the electron excitation approach and has been confirmed by the p h ot oelectron-phot on coincidence measurements, which exclude cascad ing processes.zfl states of X-C=C-H+ X = C1,Br have been discussed in recent articles in detail.lJ6 The main feature of the decay curves is that the short component decreases and the long component increases in time, and that the amplitude ratio of the short to Iong components The coincidence data for selected levels within the82 INTRAMOLECULAR DECAY OF POLYATOMIC CATIONS increases with the excess energy. - Corresponding measurements have now been carried out for CI-C=N+ in the B 211 state. Photoelectron-photon coincidences were accumulated for six levels within the 211 state (fig.4) and these correspond to the excitation of the v,, v(C-CI) stretching vibration. The non-exponential behaviour is evident in the decay curve shown in the inset of fig. 4. loo0 1 I 1 I I I 1 I 1 I 12.0 13.0 14.0 15.0 16.0 I/eV Fig. 4. He(1ct) photoelectron spectrum of ClCN and a semilogarithmic plot of a photoelectron- photon coincidence curve accumulated at location 0. The levels studied are indicated by the arrows above the spectrum and the bars show the energy band-pass (ca. 20 meV). The lifetimes, zi, and amplitudes, Ai, for the short (i = 1) and long (i = 2) com- ponents have been evaluated by fitting a model of two exponentials to the decay part of the coincidence curves. The fluorescence quantum yields, ( ~ l ~ , of the two com- ponents were then fitted to the relationship Ptot = Pl + v2 = A171 + &2.These results are summarized in table 2. At locations 0, @ and @ only mono- exponential decay curves could be observed. The reason for this becomes clear when the semilogarithmic plot of a coincidence curve accumulated at location 0 is con- sidered. Pronounced biexponential decay can only be observed with this method if the lifetimes of the two components differ at least by a factor of two. The uncertain- ties of qtot are &lo% and &20% for the estimated qi and zi values. Thus it is assumed that for location @ the lifetimes and amplitudes of the two components are nearly the same. However, for locations @ and @ the fluorescence quantum yield ql is expected to be too small to be observed. Nevertheless, the trend that with increasing vibrational excess energy the lifetimes z1 become shorter and z2 longer, respectively, is demonstrated (table 2).This decay behaviour can be rationalized using the " intermediate-case " descrip-J. P. MAIER, M. OCHSNER AND F. THOMMEN 83 tion of radiationless transition theory.” Isoenergetic vibronic levels of the lower states, 8 2n and 2Z+ provide both a quasi-continuum as in the statistical limit and discrete levels as in the resonance limit. Thus the prompt decay of the system, described by ql and zl, reflects the radiative decay to the ground state, &x, and the Table 2. Fluorescence quantum yields and lifetimes for the prompt (1) and long (2) decay components and their relative amplitude ratio, A1/A2, measured at selected ionisation energies I,within the n state of ClCN+. The labels correspond to the locations in fig.4. 0.99 280 (280) 0.84 0.31 210 0.53 400 1.1 0.63 0.18 140 0.45 510 1.5 0.55 0.1 1 90 0.44 590 1.6 0.49 0.49 650 0.41 0.41 780 15.16 0 15.22 0 15.29 0 15.35 Go 15.42 0 15.48 0 irreversible radiationless transition form the state levels to the quasi-continuum. The long component is the result of a resonant coupling of the initially populated level with discrete isoenergetic levels of the 2 and 8 states followed by radiative relaxation to the ground state. With increasing vibrational excess energy the width of the distribution of the oscillator strength of the state levels increases and therefore the lifetimes 22 become longer. This dilution of oscillator strength has already been observed for the halogenoacetylene ~ati0ns.l~ Although the lower-lying 2Z+ state carries oscillator strength to the x2n state and has a lifetime around 4.4ps,18 radiation- less or radiative decay from the to the 2 state levels followed by radiative decay to the 8 state would result in long lifetimes but not in the observed increase of z2 with excess energy. H-(-C=C-)-2H+ A nu The relaxation of diacetylene cations in the A 213u state has been investigated earlier using the photoelectron-photon coincidence approach, sampling energy slices of ca.600 cm-l centred on the bands due to the excitation of the v3, symmetric C-C stretching mode.19 Values of qF(t)’) of less than unity were measured {qF(Oo) = 0.72 and 0.80 for diacetylene and [2H2]diacetylene cations} in the 2 state, and these decrease with excess energy.The non-radiative pathway depleting the selected level is assumed to be internal conversion to the 8 state manifold. For both diacetylene cations the rate constants for the radiative decay are found to be constant over 2500 cm-’ excess energy, k,(u’) = 1 .O x lo7 s-l. The dependence of the non-radiative rates on the vibrational levels populated can be further investi- gated using the laser excitation technique. The excited level can be selected with a resolution of 1 cm-’ and from the lifetime data the rates, k,Ju’), for the radiationless transition can be inferred according to z(u‘)-’ = k,(d) + knr(v’). Such lifetime determinations have been carried out using the prominent bands in the 2 2n t 8 211, laser excitation spectrum of the two diacetylene cations.The narrowing of the vibronic bands by cooling the species to liquid-nitrogen temperature enables the transitions to be clearly selected. Lifetimes of the levels corresponding to the excitation of the three ZC,+ (Dmh symmetry classification) stretching vibrations,84 INTRAMOLECULAR DECAY OF POLYATOMIC CATIONS i.e. v1 (-.C-H), v, (CFC) and vj (C-C), as well as the overtone of the v7 bending mode could be measured. The evaluated data are presented in table 3 and are associated with uncertainties of *5%. The displacements, AC, of the levels relative to the zeroth level are taken from the gas-phase excitation spectrum and the assign- ments correspond to those made in the matrix study of this transition.20 Table 3.Lifetimes of H-CGC-CC-C-H+ and D-CIC--C=C-D+ in various vibrational levels of the 12rIn, state measured by laser excitation. The values in parentheses are from the earlier photoelectron-photon coincidence determination^.^' H-C=C-CEEC-H+ D-CsC-GC-D' level AF/crn - zH(u)'/ns AC/crn - rD(v')/ns O0 3l 72 32 3172 74 2l 3272 3174 l1 0 805 860 1 592 1 664 1 684 1 959 2 453 2 477 2 815 72 (72) 62 (62) 60 58 (59) 61 58 50 57 56 49 0 783 835 1551 1 625 1 650 1887 2 390 2 446 2 746 80 (79) 73 (72) 75 70 (66) 69 69 62 70 70 68 The lifetimes for the Oo band as well as for the 3n, n = 1,2 levels are in good agree- ment with the values obtained from the coincidence measurements l9 which are given in brackets. Overall, there_ is a decrease of the lifetimes with increasing vibrational excess energy within the A 21-Iu state.Nevertheless, the data show that the non- radiative rate not only depends on the internal energy but also on the vibrational mode excited. The shorter the lifetime the more active the excited mode for the radiation- less process and it seems that the high-frequency stretching modes v1 and v2 are the most active ones. However, in the case for the 1' levels the ratio knr(l1)/knr(O0) is significantly higher for the di hydro- than for the dideutero-diacetylene cation. This behaviour may reflect a deuterium effect or else is simply the result that less energy is required to excite the l 1 level in the dideuterodiacetylene cation. That the C-H stretching mode is the more active mode for the radiationless process than the C-D one had already been considered in the coincidence study.19 The ratios krr(u')/kFr(u') for the 0' and 3 ", n = 1-3 levels are ca.1.5, indicating such a deuterium effect. For the 3", 72n levels and their combination bands, which are coupled by Fermi resonance,21 similar lifetimes are expected. These are indeed equal for the levels with similar excess energy within the 2 211 states (table 3). 1,2,4,5-TETRAFLUOROBENZENE CATION: 'Blu Halogeno-substituted benzene cations in their excited states, B, also show decay features characteristic of the statistical limit; 22 monoexponential decay, qF(u') < 1 and k,,,(u') increase with excess energy. To gain further insight into the relaxation behaviour a higher-resolution coincidence and laser excitation study of 1,2,4,5- tetrafluorobenzene cation was undertaken.The latter species was chosen because of the relatively high symmetry (&) and that qF(B 2Blu Oo) < 1 *23J. P. MAIER, M. OCHSNER AND F. THOMMEN 85 In the coincidence experiment the fluorescence quantum yields and lifetimes of the levels associated with the totally symmetric v3 and v, vibrational modes, and their combinations, could be determined (cf. fig. 5) with a band-pass of ca. 160 crn-l. The results are given in table 4, and it is seen that a significant increase of the non- I 1 1 1 I I I I 1 9.0 10,o 11,o 12,o 13.0 IleV Fig. 5. Photoelectron-photon coincidence curve for the Oo level of the 2Blu state of 1,2,4,5- tctrafluorobenzene cation; N, = 110 Hz, NT = 0.2 Hz accumulated in 21 h and the corres- ponding He(1a) photoelectron spectrum showing the levels studied.radiative rates for the levels of the v3, vg vibrations is apparent. The radiative rate is found to be constant within the error limits; k,(v’) = 1.9 x lo7 s-l, allowing further non-radiative rates to be calculated from the z(u’) data obtained for various levels by laser excitation with ca. 1 cm-l bandwidth. The non-radiativc rate describes Table 4. Fluorescence quantum yields, pF(v’), lifetimes, ~ ( u ’ ) , and non-radiative rates, knr(v’), at selected ionisation energies, I, within the 281. state of 1,2,4,5-tetrafluorobenzene cation. The labels correspond to the numbering in fig. 5. f 2.35 0.63 f 0.05 33 & 3 1.1 + 0 . 3 0.53 0.05 29 & 2 1.7 f 0.4 12.44 12.52 0.48 i 0.05 26 + 3 2.1 i- 0.4 12.61 0.38 + 0.04 21 3 3 3.1 i O .6 12.70 0.34 f 0,04 19 2 3.7 i 0.7 12.79 0.24 i 0.03 t l 5 6.0 f 1.1 12.88 0.18 k 0.04 t15 8.7 f 2.4 0 Q 0 @ 0 8 0 the transition from the initially populated level u’ of the excited state to the quasi- continuum formed by isoenergetic levels of the 2 and 2 state^.^^.^^ The B” 2Blu t 2 2B2g laser excitation spectrum of rotationally cooled 1,2,4,5- tetrafluorobenzene cation is shown in fig. 6. The spectrum covers an excess-energy86 INTRAMOLECULAR DECAY OF POLYATOMIC CATIONS 51, I I I r I I I u I I I 365 370 375 380 385 390 405 410 A/nm Fig. 6. Laser excitation spectrum of the 2 B l , t ~ zBzg transition of 1,2,4,5-tetrafluorobenzene cation at liquid-nitrogen temperature in the gas phase (not corrected for laser intensity distribution).l;1 5 H I I I 0 1000 d o 0 3000 4600 V"/cm-' Fig. 7. Relative non-radiative rates, knr(u')/knr(Oo), of 1,2,4,5-te_trafluorobenzene cation plotted as a function of the vibrational excess energy within the B 'Blu state. The values from the laser experiment are marked by 0 and from the coincidence measurements by w.J. P. MAIER, M. OCHSNER AND F. THOMMEN 87 range of ca. 2000 cm-l; only very weak bands could be seen at higher energy. The detection of higher excited levels is difficult because the intensity of an excitation band not only depends on the pF(v’) value of the emitting level but also on the probability of populating this level in the absorption process from the cationic ground state. Lifetimes were measured using most of the prominent bands in the spectrum, and these correspond to levels of the A,, v, to v6, vibrations.The assignment is taken over from the excitation spectrum recorded in a neon matrix.25 The results and the evalu- ated non-radiative rates are presented in table 5 and are associated with uncertainties of &5-10%. Table 5. Lifetimes z ( d ) of 1,2,4,S-tetrafluorobenzene cation in various vibrational levels of the & 281. state measured by laser excitation. The non-radiative rate constants k,,(d) are calculated assuming k,(d) = 1.9 x lo7 s-l, obtained from the coincidence studies. level AF/cm - z(d)/ns k,,(v’)/107 s-l 0 272 462 545 733 926 1180 1 200 1388 1 528 1 662 1 852 32 32 28 27 28 26 21 25 27 20 23 25 1.2 1.2 1.7 1.8 1.7 2.0 2.9 2.1 1.8 3.1 2.5 2.1 In fig. 7 the relative non-radiative rates, log[kn,(v’)/kn,(OO)], from both experiments are plotted against the vibrational excess energy within the ’BlU state.By and large, the non-radiative rates increase monotonically. However, the differences found for the levels probed within the 1900 cm-l range, suggest a mode-specific internal con- version process. It seems that the higher-frequency modes v2, v., are more active than the low-frequency modes vs, v6. CONCLUDING REMARKS The decay behaviour of the open-shell cations discussed illustrate further know- ledge that could be gained either using photoelectron-photon coincidence measure- ments with improved resolution, such as for the triatomics, or from complementary studies of lifetimes using laser excitation of rotationally cooled cations.Although the excess energy accessible (ca. 2000-3000 cm-l) in the latter approach is less than in the photoionisation experiment, the much higher resolution enables more vibrational levels to be probed. However, the coincidence data provide the absolute calibration for the evaluation of the non-radiative rates and some mode specific trends are apparent. The studies described in this article have been supported by the Schweizerischer Nationalfonds zur Forderung der wissenschaftlichen Forschung (project no. 2.2- 0.17-0.81). Ciba-Geigy SA, Sandoz SA and F. Hoffmann-La Roche & Cie SA, Base1 are thanked for financial support.88 INTRAMOLECULAR DECAY OF POLYATOMIC CATIONS J. P. Maier and F. Thommen, in Ions and Light, vol. I11 of Gas Phase Ion Chemistry, ed.M. T . Bowers (Academic Press, New York, 1983). S. Leach, G. Dujardin and G. Taieb, J . Chim. Phys., Phys. Chim. Biol., 1980, 77, 705. J. P. Maier, Chimia, 1980, 34, 219; J. P. Maier, in Kinetics of Ion-Molecule Reactions, ed. P. Ausloos (Plenum Press, New York, 1979) and references therein. J. P. Maier, 0. Marthaler, L. Misev and F. Thommen, Faraday Discuss. Chem. SOC., 1981,71, 181; J. P. Maier, Acc. Chem. Res., 1982, 15, 18. T. A. Miller and V. E. Bondybey, J. Chim. Phys., Phys. Chim. Biol., 1980,77, 695. Chemistry, ed. T. A. Miller and V. E. Bondybey (North-Holland, Amsterdam, 1983). J. P. Maier and F. Thommen, Chem. Phys., 1980,51, 319. M. W. Ruf, Ph.D. Dissertation (Universitat Freiburg, W. Germany, 1976). ti D. Klapstein, J. P. Maier and L. Misev, in Molecular Ions: Spectroscopy, Structure and ’ M. Bloch and D. W. Turner, Chem. Phys. Lett., 1975, 30, 344. lo J. P. Maier and L. Misev, Chem. Phys., 1980, 51, 3 1 1 . l1 S. Leach, M. Devoret and J. H. D. Eland, Chem. Phys., 1978,33, 113. l2 S. Leach, P. S. Stannard and W. M. Gelbart, Mol. Phys., 1978, 36, 1119. l3 M. Endoh, M. Tsuji and Y . Nishimura, J. Chem. Phys., 1982, 77, 4027. l4 M. A. Johnson, J. Rostas and R. N. Zare, Chem. Phys. Lett., 1982,92, 225. E. W. Schlag, R. Frey, B. Gotchev, W. B. Peatman and H. Pollak, Chem. Phys. Lett., 1977,51, 406. l6 J. P. Maier and F. Thommen, in Intramolecular Dynamics, ed. J. Jortner and B. Pullman (Reidel, Amsterdam, 1982). l7 F. Lahmani, A. Tramer and C. Tric, J. Chem. Phys., 1974,60,4431; A. Tramer and R. Voltz, in Excited States, ed. E. C. Lim (Academic Press, New York, 1979), vol. 4, p. 281. M. Allan and J. P. Maier, Chem. Phys. Lett., 1976, 41, 231. l9 J. P. Maier and F. Thommen, J. Chem. Phys., 1980,73, 5616. 2o V. E. Bondybey and J. H. English, J. Chem. Phys., 1979, 71, 777. 21 J. H. Callomon, Can. J . Phys., 1956,34,1046; W. L. Smith, Proc. R. Soc. London, Ser. A , 1967, 22 K. F. Freed, Top. Appl. Phys., 1976, 15, 23; P. Avouris, W. M. Gelbart and M. A. El-Sayed, 23 J. P. Maier and F. Thommen, Chem. Phys., 1981,57, 319. 24 G. Dujardin, S. Leach and G. Taieb, Chem. Phys., 1980,46, 407. 25 V. E. Bondybey, J. H. English and T. A. Miller, J. Chem. Phys., 1979, 71, 1088. 300,519. Chem. Rev., 1977, 77, 793.

ISSN:0301-7249    

DOI:10.1039/DC9837500077

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Dr. R. Naaman (Weizmann Institute, Israel) said: I address my question to Prof. Parmenter: In your experiments with the pressure " timing ", how did you take into account the possibility of rotational relaxation in the excited electronic state. This relaxation could cause similar effects as you have been seen, because of the depletion of population from high-b states in the excited vibronic band. Prof. C . S. Parmenter (Indiana Uniuersity) said: I cannot envision how collisional rotational relaxation in our room-temperature experiments could cause the transformation from unstructured to structured emission that we observe. Our narrow-band excitation produces initiaIly a non-Boltzmann rotational distribution with limited J spread in the S1 state. Whether this distribution involves high J or low J or both, the result will surely be to produce fluorescence bands with rotational contours less expansive than those from 300 K Boltzmann distributions.' Thus relaxation towards a Boltzmann rotational distribution will broaden contours and produce an effect quite the opposite of the " pressure narrowing " that is actually observed.See, for example, R. A. Coveleskie and C. S. Parmenter, J. Chem. Phys., 1978, 60, 1044. Prof. C. S. Parmenter (Indiam University) said: Prof. Leach's conclusion from the non-radiative decay characteristics of the excited ions concerning the onset of intra- molecular vibrational redistribution (IVR) is quite consistent with the onsets in excited neutrals seen with the more direct fluorescence probes. The experiments that see IVR as fluorescence congestion in room temperature systems would place the onset of IVR somewhere near 2000 cm-l excess vibrational energy, just as is observed by the nun-radiative decay probes.See, for example, D. A. Dolson, K. W. Holtzclaw, S. H. Lee, S . Munchak, C. S. Parmenter, B. M. Stone and A. E. W. Knight, Laser Chemistry, in press. Prof. T . Baer (Uniuersity of North Carolina) said: In fig. 2 of Prof. Leach's paper the TPES shows a large continuum transition to high vibrational levels of the 3 or 3 states just below the How does one take into account this contribution above the origin of the state at 12.6 eV? There is evidence in small molecules that this resonant autoionization can be quite strong below a certain electronic state, but then suddenly cease above that state.Is there an a priori means of extrapolating this background for the present data? state onset. Dr. S. Leach (Universitb Paris-Sud, Orsay) said: In C,F$, the ground electron$ state 8 'ElP is doubly degenerate, so that we call the 2A2u first excited state the B state in order to be able to use the same notation in comparing analogous states in 1esE symmetrical poIyfluorobenzene cations. Thus only the 8 state lies below the B state. state region can have component: resulting from the following three processes : (i) direct ionization to form an ion B(u) state vibronic level, (ii) autoionization to form an ion &u) state vibronic level, (iii) autoionization to form a highly excited vibrational level f$ of the ion ground state. In order to determine the fluorescence quantum yield qF of a B(u) level, we require The threshold photoelectron signal at a particular excitation energy in the90 GENERAL DISCUSSION a knowledge of the branching ratio for the sum of components (i) + (ii). Our procedure involves separating the TPE signal corresponding to process (iii) from that of processes (i) + (ii).The TPE continuum in the region 1 1.8-13.8 eV was fitted by two straight lines (r. and p. In the 11.8-12.5 eV region, which is just below the state, the TPE con- tinuum coincides with the TPES itself and the a line was fitted to it and then extra- polated to the 12.5-12.9 eV region. The line p was drawn to fit the TPE continuum underlying the peaks in the 13-13.8 eV region. We consider that the TPE signal underlying these lines corresponds to the formation of 2; levels, i.e.high vibrational levels of the T2Elp ground state of C6Fi. The part of the signal above these lines is considered as being due to formation of the state by direct and/or indirect ionization. state of I ,3,5-C6F3H5, is discussed elsewhere.' The fluorescence quantum yields of energy-selected levels measured with the T-PEFCO technique for both C,Fi and 1,3,5-C6F3Hs are in good agreement with the values obtained by Maier and Thommen using the PEFCO method, which involves a fixed excitation waveIength (He I), so excluding autoionization processes in this energy region. Taking into account the difference in energy resolution in the two types of experiment, the agree- ment between the two sets of results on fluorescence quantum yields (and lifetimes) is good for comparable vibronic energies and j_ustifies, a posteriori, our method for determining branching ratios for forming the B(v) and fi ion levels.It should be further remarked that ion formation in Franck-Condon gap regions has been observed as discrete TPES peaks in N,O, COS, CS, and C6H6,3g4 but that our TPES appear to be continuous in energy (within the resolution of ca. 45 meV) in the Franck-Condon gaps of sym-trifluor obenzene and hexafluoro benzene G. Dujardin, S. Leach, 0. Dutuit, T . Govers and P. M. Guyon, J. Chem. Php., 1983, 79, 644. J. P. Maier and F. Thomen, Chern. Phys., 1981,57, 319. T. Baer, P. M. Guyon, I. Nenner, A. Tabcht-Fouhailk, R. Botter, L. F. A. Ferreira and T. R. Govers, J.Chem. Phys., 1979,70, 1585. P. T. Murray and T. 3aer, Int. J. Mass Spectrom. Ion Phys., 1979, 30, 165. The full procedure, which was also applied to the analogous case of the Prof. J, Troe (Lmiversitdt Giittingen) said: In the paper by Dr. Dujardin and Prof. Leach the analogy of radiationless transitions and unimolecular reactions was stressed. The calculation of non-radiative rates contains important statistical parts. It appears, therefore, that the authors could also express their non-radiative rates in the language of statistical theories of unimolecular reactions. A suitable transmission coefficient should then be introduced. Dr. E. CastelIucci (University of Florence), DK. 0. Braitbart (Hebrew Uniuersity of Jerusalem), Dr. G. Dujardin and Dr. S. Leach (Universitk Pans-Sud, Orsayj said: We have measured fluorescence decay curves of the molecular cations XCN+, where X =- C1, Br or I, using the photoion-fluorescence photon coincidence technique (PIFCO).I Quantum yields-and lifetimes of emissions from the lowest excited elec- tronic states, i.e.J2E+ and B 211, were obtained and were used to study the mechan- isms of their energy decay. According to current models for isolated molecules,2 the halogen cyanide cations should be classified in the smalI-molecule resonant or, possibly, intermediate case of radiationless transitions. This follows from their small number of vibrational modes [two stretching, v,(C-N) and v3(C-X), and one bending, vJ, so that these molecular cations should exhibit a rather low density of levels at the excited-state energies.The fluorescence decay curves we obtained have dear double-exponential shapesGENERAL DISCUSSION 91 - for ClCN+ and BrCN+ cations and apparently a single-exponential shape in the case of ICN+. Triatomic molecular cations have already been reported to exhibit either double-exponential, e.g. CS; , or single-exponential, e.g. CO; , COS" and N20+, decays.3 Previous measurements using the electron impact excitation tech- nique also showed double-exponential decay curves for the three XCN+ cations. The PIFCO technique used in the present measurements permits direct identifica- tion of the emitting cation. A variety of experimental conditions have been used to obtain the maximum of information from the results. Exciting V.U.V.radiation of 21.22 eV (He I) or 16.85 eV (Ne 1) was employed and photoions were accelerated with fields ranging from 150 to 20 V cm-l; various PMT and filters were used to isolate different emission transitions. A typical fluorescence curve which has been obtained is reported in fig. I, where the time-of-flight mass spectrum is also given so as to identify the emitting species. m 8 3 1000 .# 8 800 v) 600 2 3 d .C. 8 400 2 00 500 ns - 0 100 200 300 channels 2000 1 1500 J 5 0 0 400 500 Fig. 1. Ne I photoelectron-photoion (i.e. time-of-flight mass spectrum) (right orginate) and photoion-fluorescence photon (left ordinate) coincidence curves of ClCN+ : B 'II state emission, Inset : semilog plot of a reduced portion of the photoion-fluorescence photon coincidence curve, The parameters of the fluorescence decays, i.e.lifetimes, quantum yields and the amplitude ratios for different decay components, were obtained from a least-squares fitting to the decay curves with double-exponential functions. The resuIts are collected in table 1. Fluorescence quantum yields apparently less than unity have been obtained for the three cations. A fluorescence quantum yield less than unity could indicate the presence of dissipative channels which, on energy grounds, cannot be fragmentations, at least for CICN+ and BrCN+, although for ICN+ this possibility cannot be ruled out owing to the proximity of the threshold for the ICN+ -+ I + + CN reaction to the PES vibronic envelope of the The harmonic densities of levels at the B" 211 state energy were calculated by a direct TI state.'92 GENERAL DISCUSSION count using the cationic vibrational frequencies and are of the order of ca.lfcm-l for the three cations. Such low level densities are not compatible with the existence of non-radiative dissipative channels for the electronic energy and would not justify classification of the halogen cyanide cations to the intermediate case of intramolecular energy decay mechanisms in isolated molecule^.^ The real level densities could, however, be higher if one takes into account vibrational anharmonicity, doubling of Table 1. Fluorescence quantum yields, lifetimes and amplitude ratio f2r the prompt (1) and delayed (2) decay components of the fluorescence emission from the B 211 state of ClCN+, BrCN+ and ICN + , measured with the photoion-fluorescence photon coincidence technique ClCN+ 0.80 0.11 180 f 20 0.69 900 f 70 2/1 BrCN+ 0.70 340 f 40 770 f 70 2/1 ICN+' 0.83 390 ? levels due to spin-orbit coupling, vibrational-rotational coupling and degeneracy of the bending mode.In addition, we _have calculated an increase of about a factor of three in the level densities when the B state vibrational energy increased by 0.5 eV. Our Ne I excitation results for ClCN+ can be compared with the He I excitation results presented by Maier et a1.,8 which have been obtained by level-selected photo- electron-photon coincidence measurement of the B2 n - 2 211 emission. Our values for short and long lifetimes, quantum yields and amplitude ratio of the prompt and delayed fluorescence components, averaged over the vibronic levels of the 211 state, are reasonably compatible with the set of values obtained for selected ionization energies, if one takes into account that the fluorescence quantum yields reported in ref.(8) show a trend to decrease as higher vibronic levels are selected for excitation. G. Dujardin, S. Leach and G. Taieb, Chem. Phys., 1980,46, 407. P. Avouris, W. M. Gelbart and M. A. El-Sayed, Chem. Reu., 1977,77, 793. J. P. Maier and F. Thommen, Chem. Phys., 1980,51, 319. M. Allan and J. P. Maier, Chem. Phys. Lett., 1976, 41, 231. E. Heilbronner, V. Hornung and K. A. Muszkat, Helu. Chim. Acta, 1970,53, 347. J, M. Hollas and T. A. Sutherley, Mol. Phys., 1971, 22, 213. ' F. Lahmani, A. Tramer and C. Tric, J. Chem. Phys., 1974,60,4431. * J. P.Maier, M. Ochsner and F. Thommen, Faraday Discuss. Chem. Soc., 1983,75,77. Dr. S . Leach (Uniuersite' Paris-Sud, Orsay) said: I wish to put two questions to Prof. Baer. (a) Are the equilibrium and reversibility conditions necessary for a kinetic treatment valid in all the cases treated by Baer et al.? (b) Multiexponential decay of fluorescence from the excited electronic states of some molecular ions can be treated in terms of the intermediate case of radiationless transition theory.' The intermediate case can also be formulated in kinetic terms.2 It would be of interest to examine whether the two-component decay behaviour of internal energy-selected metastable ions studied by Baer et al. can be formulated in terms of intermediate case theory. If this is possible, an analysis of the physical meaning of the parameters in the metastable ion case might give new insights into the dissociation and rearrange- ment mechanisms.S. Leach, G. Dujardin and G. Taieb, J. Chim. Phys., 1980, 77, 705. ' F. Lahmani, A. Tramer and C. Tric, J. Chem. Phys., 1974, 60, 4431.GENERAL DISCUSSION 93 Prof, C. Lifshitz (Hebrew University of Jerusalem) said : Have equivalent activa- tion entropies, A S , and pre-exponential factors A *, been calculated for the adopted R.R.K.M./Q.E.T. models? Dr. Henry M I Rosenstock had demonstrated in a series of elegant prior to his untimely death, the usefulness of such calcuIations, in spite of the fact that these pertain to the canonical system and measurements are carried out by PEPICO for rate energy dependences in microcanonical systems. Neutral systems have the advantage over ionic systems in that the models can be checked against experimental pre-exponentia1 A cc factors.The degree of tightness or looseness of transition states in ionic systems has in the past been very often chosen quite arbitrarily. Two results are of importance in this respect: (a) For equivalent ionic and neutral reactions, A, factors are about equal. This has now been demon- strated for C6H,Br+* -+ C6H,+ + Br* and C,H,Br 3 C6H5 + Br in bromobenzene 2 * 5 as well as for C&+- -+ C,H,+- + C,H, and CBHS -+ C,H, + C,H, in cycio- octatriene; 6,7 (b) The rate-energy dependence, k(E), is independent of the exact frequency changes made in the transition-state model, as long as the activation entropy, A S , is the same.While the validity of high-pressure Arrhenius pre- exponential factors A co for the hypothetical equivalent thermal unimolecular reactions cannot be tested experimentally, the plausibility of the calculated values of A S and A , can be assessed by comparison with known values for similar neutral reactions. As more information becomes available for ionic reactions, compilations of A S values will become useful. Also, and perhaps most importantly, very few k(E) determinations have been carried out over a very wide internal energy range, there are uncertainties in Eb threshold energy (i.e. activation energy) values and, as has been demonstrated by Rosenstock et al.,1-4 there are energy-entropy trade-offs. As new and more accurate Eo values become available, for the 1,4-oxathiane system for example, transition-state models will presumably have to be varied; this can be done through a single parameter, the activation entropy, A S .H. M. Rosenstock, R. Stockbauer and A. C. Parr, J . Chem. Phys., 1979, 71, 3708. H. M. Rosenstock, R. Stockbauer and A. C. Pam, J. Chem. Phys., 1980,73, 773. H. M. Rosenstock, R. Stockbauer and A. C. Pax, Int. J. Mass Specfrom. Zon Phys., 1981, 38, 323. H. M. Rosenstock, R. Stockbauer and A. C. Parr, J. Chim. Plzys., 1980,77, 745, C. Lifshitz, Mass Spectrum. Rev., 1982, 1, 309. C. Lifshitz, M. Goldenberg, Y. Malinovich and M. Peres, Int. J. Mass Spectrom. Ion. Phys., 1983, 46, 269; C. Lifshitz and Y . Malinovich, to be published; A , (1000 K) = 1.1 x 1015 s-l. ' D. Dudek, K.Glanzer and 3. Troe, Ber. Bunsenges. Phys. Chem., 1979, 83, 776; R. Walsh, personal communication, April 1983. Dr. J. A. Laramke (Warwick University) said: Prof. Baer has presented a phase- space model for a competitive dissociation/isomerkation reaction (fig. 2 in his paper). The model also appears to describe the variation of ionic product intensity with scattering angle for a collision-induced dissociation. That is, low scattering angles can access a limited number of exit channels from the space, whereas at larger scatter- ing angles the ion may exit along new pathways. This assumes that the excess energy deposited by the collision can be represented as a line parallel to the reaction coordinate in the figure. Could Prof. Baer please comment on this? Dr.L. Carlsen (Risa, Denmark) said: I address my comments to Prof. Baer. In your paper you describe experiments on the possible isomerization of C,H,O$., C4H80S+m and C,H,S,+. ions, which is of interest in connection with our recent investigations on the electron impact induced decomposition of ethyl acetate and its sulphur analogues. You state in your paper that your isomer B (in the monothio94 GENERAL DISCUSSION case) could be the thioacetate. However, I do not from your paper understand if you actually studied the sulphur analogues of ethyl acetate in ordcr to establish their possible idcntity as B. In this connection I wish to add that we have studied reactions of ionized ethyl acetate and its sulphur analogues.' In contrast to ethyl acetate, where no intramolecular isomerization occurs, it has been shown that isomerizations take place in the corresponding monothio derivatives (Le.ethyl thionoacetate and ethyl thioloacetate). J t appears'that both esters exhibit isomerizations ; however these do not lead to the same thiono/thiolo ratio. Both esters also, as a dominant reaction, eliminate ethylene. It seems that in the case of the ionized monothioacetates we in fact have a system as described in your paper [eqn (2)-(4)], ethylthionoacetate and ethyl thioloacetate constituting the isomers A and B. At present no conclusive informa- tion on the dithio-derivative can be given. Studies on the possible isomerization of ethyl dithioacetate are in progress. H. Egsgaard, E. Larsen and L. Carlsen, Int. J. Mass Spectrom. Ion Phy?;., 1983,47, 359.L. Carlsen and H. Egsgaard, J. Chem. Suc., Perkin Tans. 2, 1982, 1081. Prof. T. Baer (University of North Carolina) said: I will deal with the various questions and comments in turn. First Dr. Leach: It is certainly the case that phenomenologically the two-component decay rates we observe are similar to those found in fluorescence. When both are treated in terms of a kinetic model, the parallel is complete: A&X++v These equations can be solved in a manner similar to those of eqn (2)-(4) in our paper. When it is assumed that kl k2 > k3, the rate is given by the two-component fluores- cence rate : fluorescence rate = C,exp( - y+ t ) + Czexp( - y - t ) in which y+ = k, + k, + k3 and y - = k,k,/(k, + k2 + k3). Now the difference in the two situations is that in the case of the fluorescence the two states A and X * are two different electronic states.In the model presented in our paper the states are two different isomers. It is of course true that in one sense two isomers are simply two different electronic states. However, when the connectivity relationships among the atoms differ, I suspect that it is more useful and practical to think of the two isomers as two separate molecules. As a consequence, it would appear to be very difficult to apply the intermediate-case theory developed for two-component fluorescence to our phenomena. However, there is a chance that this could lead to some very interesting new insights. Prof. Lifshitz is quite right: A S is a useful quantity to consider and report.Although there are a great number of vibrational frequencies to choose in describing the transition state and the precursor ion, in fact there are effectively only two para- meters, the geometric averages of the frequencies. The AS* appropriately combines these. It is certainly true that low-angle collisions tend to produce ions with little internal- energy content, while high-angle collisions impart a great deal of internal energy to the ion. As a result, the latter ions will dissociate via paths not open to the low- energy ions. Also, the low-energy ions may decay via two-component dissociation rates; however, this effect is probably not observable in the typical collisional dis- sociation experiment. It is not clear to me what Dr. Laramke means by his lastGENERAL DISCUSSION 95 statement.The reaction coordinate represents the evolution of the ion in time. It does so at a fixed energy, so that this coordinate is not related to energy content. However, we do plan on doing so. Dr. Carlsen’s studies, of which I was not aware, are very relevant in this regard. In view of the fact that thioacetates eliminate ethylene, Dr. Carlsen may wish to consider the possibility that the isomerized structure is the thiobutanoic acid, because ethylene loss is a dominant dissociation path for this ion. However, I must take exception to the statement that ethyl acetate does not rearrange prior to dis- sociation. Our dissociation rate measurements of energy-selected ethyl acetate ions indicates that this ion does isomerize to a lower-energy form prior to diss0ciation.l We have not yet studied the thioacetate compounds.T. Baer et al, J. Phys. Chem., 1982, 86, 752. Prof. T. Baer (University of North Carolina) said : In view of Prof. Bowers’ model it would seem to be exceedingly important to obtain lifetime measurements of cold molecules or ions near the dissociation onset. To observe the effect of the transition- state switching from a tight to an orbiting one, it is most likely necessary to use rotationally cold species. Is this a correct deduction? Prof. C. Lifshitz (Hebrew University of Jerusalem) said; We have recently observed non-statistical behaviour in unimolecular reactions demonstrating double deep-well potentials. In these reactions enol ions undergo a rate-determining isomerization to the keto form, followed by fast dissociation to an acetyl cation and a neutral radical.We view this isomerization as a chemical activation method, carried out at low pressures (<lov5 Torr), and therefore being free of possible collision-induced energy redistribution. Isomerized acetone ions are produced from their enol isomers with 1.0-1.3 eV excess energy above the dissociation limit. The two methyls are lost at unequal rates; the ratio of the “ new ” methyl being prefer- ably lost over the “ old ” methyl being ca. 1.3 : 1. The “ new ” methyl loss is also ac- companied with a higher kinetic-energy release. Both of the methyl losses demon- strate bimodal kinetic-energy release distributions (KERD) with a high-energy “ unrelaxed ” part and a low-energy “ relaxed ” part, the latter being in fair agree- ment with an R.R.K.M./Q.E.T.calculation. The enol ion of butan-2-one demon- strates similar behaviour upon ethyl loss, although it has a larger contribution of the relaxed distribution, presumably because of its higher number of degrees of freedom and its lower excess energy (ca. 0.5 eV) upon isomerization. s. The nuclear motions excited in the isomerization step of C3H60+* are more strongly coupled to the reaction coordinate than to other vibrational motions. The intermediate acetone cation is calculated to have a lifetime z < 5 x C. Lifshitz and E. Tzidony, Znt. J. Muss Spectrom. Zon Phys., 1981, 39, 181. G. Depke, C. Lifshitz, H. Schwarz and E. Tzidony, Agnew. Chem., Int. Ed. Engl., 1981,20, 792. C. Lifshitz, Int.J. Mass Spectrom. Zon Phys., 1982, 43, 179. C. Lifshitz, P. Berger and E. Tzidony, Chem. Phys. Lett., 1983,95, 109. C . Lifshitz, J. Phys. Chem., 1983, 87, 2304. Dr. J. M. Jasinski (ZBM, New York) said: I would like to address a question to Prof. Bowers regarding the transition-state switching model. Tight transi- tion states at energies below the centrifugal barrier have been used in conjunction with orbiting transition states for some time now in modelling various aspects of ion- molecule reaction kinetics, particularly by Brauman and his group at Stanf~rd.l-~ In those cases, the tight transition state corresponds to a local maximum in the potential-energy surface along the reaction coordinate. Its effect on the calculated96 GENERAL DISCUSSION reaction rates is a function of the height of the local maximum relative to the centrifugal barrier and the particular properties of the chemical system being modelled.On the other hand, a number of ion-molecule processes can be modelled quantitatively using only orbiting-type transition states, e.g. most exothermic bimolecular ion-molecule reactions and three-body association reaction^.^^^ It seems that whether or not there is a tight transition state along the reaction coordinate, and what its effect on the observable dynamics is, should depend sensi- tively on the particular reaction under consideration. Therefore, I wonder about the generality of your assumption, which forms the basis for the transition-state switching model, of both a tight and an orbiting transition state for all exit channels for all ion- molecule potential surfaces.To what degree and upon what evidence do you believe that this assumption is general and accurately describes physical reality in these systems ? W. E. Farneth and J. I. Brauman, J. Am. Chem. SOC., 1976,98, 7891. W. N. Olmstead and J. I. Brauman, J . Am. Chem. SOC., 1977,99,4219. J. M. Jasinski and J. I. Brauman, J. Am. Chem. SOC., 1980, 102, 2906. W. N. Olmstead, M. Lev-On, D. M. Golden and J. I. Brauman, J. Am. Chem. SOC., 1977,99, 992. J. M. Jasinski, R. N. Rosenfeld, D. M. Golden and J. I. Brauman, J. Am. Chem. SOC., 1979,101, 2259. Prof. M . T . Bowers (University of California) said: I turn first to Prof. Baer. In the limit of J-+ 0, in the " diatomic " approximation, the orbiting transition state vanishes and only the tight transition state remains.Hence, for ultra-cold molecules in which J + 0 the effects of the orbiting transition state will be minimized. If the reaction coordinate is a symmetric stretch, no effect will be observed. If the reaction coordinate is a bend, or some combination of motions, then a relatively small " rotational " barrier will exist along the reaction coordinate at large Y. In my view it will be difficult to observe these effects experimentally. What would be interesting would be the measurement of rate-energy curves for the same reaction for dramatically different J distributions. Here the main thing being varied is the rotational barrier and, hence, the relative importance of the orbiting transition state.Near threshold the orbiting transition state should become rate determining, and direct observation of the cross-over region becomes possible. All measurements in the literature begin at energies well above threshold, a practical requirement necessary for obtaining usable signals. Improving the technology of the experiment may be required before threshold data can be obtained and direct testing of the transition- state switching model effected. The systems mentioned by Prof. Lifshitz are very interesting in that they give bimodal kinetic-energy distributions that are not easily rationalized by statistical theory. It is not appropriate to give an in-depth discussion of these systems here. I would make three cautionary comments, however. First, I am concerned about the data themselves.I do not know the exact arrange- ment and sizes of the slits in the instrument used (a double-focusing MAT 311), but since the peak shapes were for metastables in the first field-free region, the metastable ionic products must travel of the order of a metre or greater to reach the detector. The instrument in question focuses ions only in the xy plane (x is the direction of ion motion and y the direction of the electric field in the electrostatic analyser) but not in the z direction. Hence, for metastables with relatively large energy release (>O. 1 eV) of relatively low mass, rather severe z discrimination can occur. This leads to the well known " dishing " of metastable peaks with very large releases and to more A better experiment is to try to push the measurements closer to threshold.GENERAL DISCUSSION 97 subtle effects on peak shape for intermediate energy releases.The z dimension of the slits in the instrument can be measured (especially the detector slit) and discrimination calculated for known beam profiles of reactions for various values of energy release. An experimental test for discrimination would include measuring the metastable peak shape at several values of the accelerating voltage and seeing if the same kinetic energy distribution is obtained. Is the reaction occurring entirely on the ground-state surface or are excited electronic states involved ? If two separate mechanisms were competing, then a bimodal KERD would be expected. It simply seems unlikely to me that a complex rearrangement such as enol to keto would take place along a reaction coordinate that was somehow decoupled from the rest of the phase space, if only ground-state surfaces were involved.Why should it be decoupled? Thirdly, I would be reluctant to use the theory of Haney and Franklin as repre- sentative of state-of-the-art statistical theory calculations. It would seem that recent phase-space methods are much better suited to the experiment at hand. These will not yield a bimodal KERD for the reactions in question, but will give a reasonable estimate of the distribution expected from the orbiting transition state. The papers of Brauman and coworkers cited by Dr. Jasinski dealt with isomeriza- tion barriers along the reaction coordinate. In our early work with C4H6+- systems we also postulated an isomerization barrier as the source of the transition state for CH; loss in an attempt to rationalize the k(E) curve measured by Werner and Bae~-.~ Subsequent collaborative work with Baer showed isomerization was not the source of the small slope of the k(E) against E curve, and the transition-state switching model (TSSM) needed to be invoked to fit the data.6 The TSSM is more in the spirit of work published by Hase on the C2H6 + CH; + CHj system. Experimentally it had been found that a “ loose ” transition state was needed to fit the methyl radial association reaction, but a relatively “ tight ” transition state was needed to fit the dissociation reaction.Hase’s solution was to postulate the transition state was a function of E and J and use the variational criteria to choose between the two.What we have done in TSSM is borrow this idea and generalize it by assuming all reaction coordinates without reverse activation energies contain two transition states : one orbiting (i.e. the loosest possible) and one tight. This is a more likely scenario for ion-molecule systems than for neutral-neutral systems because of the extreme long range of the ion-induced dipole and ion-dipole interactions. As a consequence, the “ physical ” centrifugal barrier occurs at much larger r for ion-molecule pairs than for neutrals, while the “ chemical ” configurational interactions occur at comparable values of r for both ion-molecule systems and neutral systems. Why two distinct transition states should result is dealt with briefly in the manuscript following eqn (4).It is dealt with in more detail elsewhere in the literature.8 Secondly, I am concerned about the mechanism of the reactions. M. A. Haney and J. L. Franklin, J. Chem. Phys., 1968, 48, 4093. See, for example, W. J. Chesnavich, L. M. Bass, T. Su and M. T. Bowers, J. Chem. Phys., 1981, 74, 2228. W. J. Chesnavich and M. T. Bowers, J . Am. Chem. Soc., 1977,99, 1705. A. S . Werner and T. Baer, J. Chem. Phys., 1975, 62, 2900. T. Baer, personal communication. We assumed [ref. (3)] isomerization to form 2-methylcyclo- propenium ion was rate determining followed by fast loss of CHj. We synthesized 2-methyl- cyclopropene and Baer found it reacted at the same rate as all other C4HB isomers. Hence the tight transition state must be further out along the reaction coordinate.M. F. Jarrold, L. M. Bass, P. R. Kemper, P. A. M. van Koppen and M. T. Bowers, J . Chem. Phys., 1983, 78, 3756. ’ W. L. Hase, J . Chem. Phys., 1972, 57, 730; 1976, 64, 2442.98 GENERAL DISCUSSION * W. J. Chesnavich, L. Bass, T. Su and M. T. Bowers, J. Chem. Phys., 1981, 74, 2228; W. J. Chesnavich and M. T. Bowers, Prog. React. Kinet., 1982, 11, 139. Prof. M. Quack (ETH, Zurich) said: Prof. Bowers and his coworkers have demonstrated in their paper how their transition-state switching model can account for many of the observations made in ion-molecule reactions. It may be useful to compare some of the points raised with previous observations and developments in radical-radical reactions involving neutral (i) There used to be a folklore in the field that radical-radical recombinations have loose (" orbiting ") transition states. Detailed calculations with the adiabatic channel model and extensive com- parison with experiment [reviewed in ref.(4) of this comment] showed that a better qualitative description involves a transition state which tightens as a function of increasing energy and angular momentum (the actual description with the full adiabatic-channel model does not involve the concept of a localised transition state at all). I would suggest that a similar situation may occur in many ion-molecule reactions and that an implementation of the adiabatic-channel model in this context would be useful. It would automatically also include transition-state switching, if it occurs.(ii) Bowers et al. also address the question of kinetic-energy release, or more generally of product-energy distributions after decomposition. It has been shown previously in the theory of product-energy distributions from the adiabatic-channel model, how considerable changes arise, compared with phase-space theory, from the tightening of the transition state. A number of model calculations have been presented in relation to molecular-beam experiments. Although there remain ambiguities, on the whole there seems to be consistency between experiment and From the general behaviour of the results of adiabatic-channel calcula- tions, the wide distributions for hydrogen-atom elimination reported in fig. 9 of the paper of Bowers et al. are not unexpected, although, of course, this point must be established by more detailed calculations.I should mention here in particular also the beautiful results by Beresford et al.,' who were able to show that the product-state distributions of CN from CF&N multiphoton dissociation can be accounted for by the adiabatic-channel model, although phase-space theory fails in this case. (iii) An interesting question would be, of course, whether the transition-state switching postulated for ion-molecule reactions is also of general importance in radical-radical reactions. In the framework of the adiabatic-channel model this would imply that one has two or more about equally important maxima (separated by minima) in the q-dependent Helmholtz energy (canonical language) or minima in the q- dependent number of locally open adiabatic channels (microcanonical language).In principle, one may even have two maxima of free energy with a minimum in between, just where one has a maximum in potential en erg^.^ Although I have done related calculation^,^^^ I have in no case found conclusive evidence for the importance of such behaviour. This also includes one ion-molecule reaction (H; -+ H2 + Hf),599 for which an ab initio potential surface exists." Therefore one might wish to reconsider the question of whether transition-state switching is really of general importance as suggested by Bowers et al. for ion-molecule reactions. This point must not be confused with the previously established fact that the dominant transition state is often strongly dependent upon energy and angular momentum.' M.Quack and J. Troe, Ber. Bunsenges. Phys. Chem., 1973, 77, 1020; Ber. Bunsenges. PhyJ. Chem., 1974,78, 240. M. Quack and J. Troe, Ber. Bunsenges. Phys. Chem., 1975, 79, 170. M. Quack and J. Troe, Ber. Bunsenges. Phys. Chem., 1977, 81, 329.GENERAL DISCUSSION 99 M. Quack and J. Troe, Statistical Methods in Scattering, in Theoretical Chemistry, Advances and Perspectives, ed. D. Henderson (Academic Press, New York, 1981), vol. 6B. M. Quack, J. Phys. Chem., 1979, 83, 150. M. Quack, Chem. Phys., 1980,51, 353. M. Quack, in Intramolecular Dynamics, ed. J. Jortner and B. Pullmann (D. Reidel, Dordrecht, 1982), p. 371. * J. R. Beresford, G. Hancock, A. J. McRobert, J. Catanzarite, G. Radakrishnan, H. Reisler and C . Wittig, Faraday Discuss.Chem. SOC., 1983, 75, 211. M. Quack, unpublished results. lo I. G. Csizmadia, J. C. Polanyi, J. C . Roach and W. H. Wong, Can. J. Chem., 1969, 47, 4097. Prof. M . Bowers (University of California) (communicated) : In response to Prof. Quack, I make the following observations. (i) One of the important differences between radical-radical reactions and ion-molecule reactions is the nature of the long-range potential. In radical-radical reactions V(r) varies as r - 6 while in ion- molecule reactions V(r) varies as r-4 or r-’, depending on whether or not the neutral collision partner has a dipole moment or not. One major consequence of this difference is that the “ physical ” or “ orbiting ” transition state in neutral systems occurs at a value of r not much different from transition states arising from the chemical part of the potential. This usually is not the case in ion-molecule reactions where the “ orbiting ” transition state occurs at large values of r relative to chemical- bonding distances.I have no idea what the adiabatic-channel model would yield for ion-molecule reactions and it would be interesting to do some calculations. How- ever, if it did not adequately account for the long-range ion-neutral part of the potential, it would have no physical meaning. (ii) All kinetic-energy distributions we calculate use the orbiting transition state only. These calculations are not designed to “ fit ” experiment but rather to see if the dynamics maintain the system at a “ statistical equilibrium ” as it passes through the orbiting transition state.Clearly more needs to be done to model these distribu- tions accurately. (iii) In my view, transition-state switching will have only limited application to radical-radical recombination reactions for reasons similar to those discussed in (i) above. I also feel, however, that at present it is the only available model that fits the very extensive data available on the three reacting systems discussed in our paper. There is also more to the transition-state switching model than simply dealing with one reaction channel. It provides a means for dealing with complex reaction surfaces with many competing reaction channels. Finally, I would suggest that Prof. Quack has developed his intuition on a statistical reaction rate theory by considering reactions in neutral systems.There are some aspects of these systems that carry over to ion-molecule reactions, but not, in general, those that deal with the long-range part of the potential. Hence his remarks on the possible validity of the application of the adiabatic-channel model to ion-molecule reactions are at this point speculative and do not benefit from trial calculations on real ion-molecule systems. I think the field would be better served if we deal in facts based on comparison of experiment and theory, as we have done in the main body of this paper, rather than on speculation based on what may not be a valid analogy. Prof. J . Troe (Universitat Gottingen) said : In Prof. Bowers’ paper the nature of the effective transition state of ion-molecule complexes is expressed by the mixing of a loose and a rigid transition state.The loose transition state can be represented by phase-space theory, the rigid transition state by rigid R.R.K.M. theory. The question arises whether this is a physically relevant model, i.e. whether the correct energy pattern of the individual channel maxima is reproduced.100 GENERAL DISCUSSION An alternative description of the intermediate character of the transition state between rigid and loose is provided by the statistical adiabatic-channel model (SACM) of Quack and Troe.' Here, individual channel potential curves are constructed and channel (state)-selected transition states are obtained. In the simplified version of ref. (2) [fork, ( T ) ] and (3) [for k(E, J ) ] , the corresponding energy pattern of activated complex states has been analysed.It was found that this energy pattern can be represented roughly in terms of one single-interpolation parameter (CLIP), being between an oscillator type (as in rigid R.R.K.M.) and a rotor type (as in phase-space theory). In particular, the classical number-of-states function for one activated complex coordinate was found to be of the form with E interpolated in terms of a/P between a reactant oscillator quantum and a product rotational constant, and x being between 1 and 0.5, nearly independent of the energy. It would be interesting to apply the statistical adiabatic-channel model to ion-molecule potentials in order to identify the corresponding relation between E and x and the characteristic potential parameters.Then one could decide whether a switching between two transition states is required as suggested in this paper, or whether the SACM provides already an adequate description for the intermediate semi-rigid semi-loose character of the transition state. W ( E ) z ( E / E ) ~ M. Quack and J. Troe, Ber. Bunsenges. Phys. Chem., 1974, 78, 240. J. Troe, J. Chem. Phys., 1981, 75, 226. J. Troe, J. Chem. Phys., in press. Prof. M. T. Bowers (Uniuersity of California) said : In the transition-state switching model we have proposed, there is not a mixing of transition states between loose (we call it " orbiting ") and tight. Rather, the assumption is that these transition states retain their individual character. The kinetics of the system, for a given E, Jstate, are determined by the variational principle; the transition state of lowest flux is assumed to be rate determining.The rationale for two transition states is based on the fact that the centrifugal barrier is at very large values of r for ion-neutral collisions at low values of relative translational energy. This transition state does, in fact, dominate the kinetics of simple association reactions ' including the kinetic-energy release of fragmentation of nascent (i.e. unstabilized) ion-molecule collision complexes.2 When the systems get more complex, like the three discussed in this manuscript, both orbiting and tight transition states in the same reaction channel appear to be required. Finally, I am not sure what Prof. Troe means by the " energy pattern of the individual channel maxima ".The key issue is how the flux varies along the reaction coordinate as a function of E and J . We propose that in most ion-neutral reaction coordinates, that do not exhibit reverse activation energies, there are two flux minima (free-energy maxima) along the reaction coordinate at low values of relative translational energy of the separated fragments. Some good trajectory calculations on a realistic surface may be required to clarify the issue further. L. Bass, W. J. Chesnavich and M. T. Bowers, J. Am. Chem. Soc., 1979, 101, 5493. A. J. Illies, M. F. Jarrold, L. M. Bass and M. T. Bowers, 1. Am. Chem. Soc., 1983, in press. Dr. M. J. Pilling (Oxford University) said: Prof. Quack and Troe have criticised Prof. Bowers' transition-state switching model from the standpoint of adiabatic- channel theory.I should like to make a related point and ask two questions, but using as a basis the minimum-sum-of-states criterion for locating transition states, since this is closer in spirit to the minimum flux model used by Prof. Bowers. In modelling radical recombination reactions, the position of the minimum sum ofGENERAL DISCUSSION 101 states depends on the electronic potential-energy function, on the centrifugal barrier and on the dependence of the vibrational frequencies on inter-radical distance. The position of the minimum changes with energy content and can move to significantly shorter bond distances as the energy increases. However, despite its complex origin, only one minimum sum of states is found at any given energy.In ion-molecule reactions, there is a long-range r - 4 contribution to the potential and the differences between radical-radical and ion-molecule reactions are often ascribed to this long- range potential. However, I should like to ask (i) has it ever been shown numerically that two flux minima occur in ion-molecule combination reactions and (ii) is it valid to separate rotational and vibrational effects, as in the transition-state switching model, since this i s not the case with radical recombination reactions. Prof. M. T. Bowers (University of California) said: The answer to Dr. Pilling’s first question is no, at least not to my knowledge. Such calculations need to be done. The answer to your second question is also no, at least if rigorous theory is used.However, it must be remembered that thc orbiting transition state, at energies near thrcshold, occurs at ion-neutral separations of the order of 10 A. This transition state is essentially purely rotational in character. In neutral systems this type of transition state woti€d usually occur at much smaller values of r. At translational energies signifi- cantly above threshold the meaning of an orbiting transition state begins to blur and the arguments made in support of transition-state switching are not valid. This is not the case if the energy is supplied in the form of vibrational energy, however, as vibrational energy has little effect on the potential surface. In radical-radical recombination reactions the value of r for the centrifugal barrier is comparable to r values expected for tight transition states.Hence, I would not expect two clearly separated transition states in these systems, except under exceptional circumstances. Prof. B. S . Rabinovitch (University of Washington) said : I wish to comment on an aspect of the lifetime of the collision complex formed in ion-molecule reactions and the type of reaction represented. Both of these features depend upon the potential surface involved. Examples have been known for some time, both the ion and neutral chemistry of various systems, that range all the way from short-lived long- range, direct stripping processes to intimate, long-lived “ complex ” formation preceding reaction. Indeed, such behaviour illustrates the continuity of the formal description of chemical reaction type as being bimolecular or unimolecular in character.But even for a single given reaction the observed behaviour may be drastically altered from one extreme to the other by variation of the relative trans- lational energy of the colliding partners. Bowers et al. have themselves illustrated such variation for relatively simple species1 1 also note some brilliant crossed-beam ion-molecule experiments of this type conducted a dozen years ago with complex species by Henglein and by Wolfgang and coworkers; the former dealt with the proton-transfer process, the latter with the C,H: + C,H, reaction. Moreover, rel- evant to the topic of this Discussion, both of these groups used their results to probe the time domain and extent of intramolecular vibrational-energy relaxation in the collision entity.Although aspects of their experiments and interpretations are subject to revision, Wolfgang and coworkers concluded that the extent of internal relaxation of energy varied with the lifetime of the collisional entity on a (sub-) pico- second time scale. M. T. Bowers, D. D. Ellerman and J. King, J , Chem, Phys., 1969, 50, 4787. A. Henglein, J. Chern. Phys., 1970, 53, 458. A. Lee, R. L. Jxroy, Z. Herman and R. Wolfgang, Chem. Phys. Lett., 1972,12, 569.102 GENERAL DISCUSSION Prof. M. T. Bowers (Uniuersity of California) said: I agree with the general flow of Prof. Rabinovitch’s comments. It does appear that for many systems an inter- mediate complex is formed at low relative translational energies which more or less smoothly goes over to a direct mechanism at higher translational energies. In recent years this concept has been used by Ferguson and coworkers to interpret some experiments on a flow-drift-type apparatus. Regarding the C2H:* + C2H4 reaction studied by Lee et a1.,2 it does indeed appear they may have misinterpreted their data. The contour diagrams clearly show the mechanism of the reaction is changing from complex to direct as the relative trans- lational energy increase^.^ However, Lee et al. based their conclusion that the extent of internal relaxation varied with collision energy (complex lifetime) on comparison of calculated and experimental product kinetic-energy distributions. They used the theory of Safron et aL4 and were only able to get agreement with experiment if they severely limited the number of oscillators contributing to the phase space in the C4H: intermediate. Subsequently, however, Chesnavich and Bowers5 and Klots6 showed that good agreement between experiment and theory was obtained for phase- space theory calculations using all oscillators in the C,Hi* intermediate. traced the problem to an incorrect assumption of Lee et aL2 when applying the theory of Safron et aL4 Small deviations between the phase-space results and experiments were observed at higher energies. One of the possible explanations Chesnavich and Bowers gave for these deviations is that orbital angular momentum may not be randomized at higher relative energies, but may be passing on preferentially to product translational energy. Hence, the slower rotational motions may be experiencing a lack of intramolecular relaxation prior to the onset of a similar effect, at shorter lifetimes, for vibrations. Klots D. L. Albritton, in Kinetics of Zon-Molecule Reactions, ed. P. Ausloos (Plenum Press, New York, 1979), p. 119; M. Durup-Ferguson, H. Bohringer, D. W. Fahey and E. E. Ferguson, J. Chem. Phys., in press. A. Lee, R. L. Leroy, Z. Herman, R. Wolfgang and J. C. Tully, Chem. Phys. Lett., 1972, 12, 596. Z. Herman, A, Lee and R. Wolfgang, J. Chem. Phys., 1969, 51, 452. S. A. Safron, N. D. Weinstein, D. R. Hershbach and J. C. Tully, Chem. Phys. Lett., 1972, 12, 564. W. J. Chesnavich and M. T. Bowers, J. Am. Chem. SOC., 1976,98, 8301. C. E. Klots, J. Chem. Phys., 1976, 64, 4269.

ISSN:0301-7249    

DOI:10.1039/DC9837500089

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1983,75, 103-1 15 On the Theory of Intramolecular Energy Transfer BY RUDOLPH A. MARCUS Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91 125, U.S.A. Receiued 21 st December, 1982 We consider the distinguishing features of two main types of classical anharmonic motion in molecules, their quantum parallels, and conditions that classical chaos also be sufficient for " quantum chaos". Implications are considered for experimental reaction rates, R.R.K. M. theory, spectra and a possible type of system for intramolecular laser-selective chemistry. A theory of intramolecular energy transfer between two ligands of a heavy atom is described for a system which may contain many coordinates. It is partly statistical and, for the modes of each ligand which communicate through the heavy atom, dynamical.1 . INTRODUCTION Quasiperiodic and chaotic motion in molecules, and more generally in anharmonic systems, has been the subject of considerable interest in recent years, and of many conferences and reviews [see e g . ref. (1)-(14)]* From a chemist's viewpoint several main questions are the following: (1) What are the main distinguishing features of the different types of dassical mechanical anharmonic motion (quasiperiodic versus chaotic) ? (2) What is known about the corresponding quantum-mechanical be- haviour? Does classical " chaos " imply a quantum " chaos " ? How might one define the latter? (3) Do quasiperiodic and chaotic behaviour always differ in their collision-free intramolecular energy redistribution, or is some other aspect of the anharmonic motion equally or more important? (4) What implications do the theoretical anharmonic studies have for experimental data, e.g.for the rates of uni- molecular processes, for spectra and for intramolecular laser-selective chemistry ? How would one differentiate experimentally between the various types of behaviour, taking state preparation into account ? (5) What does statistical (e.g. R.R.K.M.) behaviour imply in terms of the underlying dynamics? We address these questions in sections 2-5 from a unified viewpoint. We consider a possible type of system for intramolecular laser-selective chemistry (section 6) and present a theory of intramolecular energy transfer between two parts of a molecule joined by a heavy atom (section 7).A criterion fur overlapping avoided crossings, and conditions that classical chaos imply " quantum chaos " are given in section 3. Some of the material is presented here to set the background on some points about which there has been some confusion or misunderstanding in the liter- ature. We have recently reviewed the literature on quasiperiodic versus chaotic motion.' Extensive references are given there and we shall sometimes refer to that and other reviews instead of to the original articles.104 INTRAMOLECULAR ENERGY TRANSFER 2. CLASSICAL QUASIPERIODIC VERSUS CHAOTIC MOTION '-14 Quasiperiodic motion implies classically that for a system of N coordinates the system moves on an N-dimensional surface in 2N-dimensional phase space (or less than N dimensions if the motion is degenerate).The surface (a torus) is determined by N constants of the-motion (the action variables) and the trajectory uniformly covers that surface. The time dependence of any coordinate can be expressed as a Fourier series in the N phases (the angle variables) and thereby in terms of N funda- mental frequencies (less if degenerate), plus combinations and overtones. The power spectrum of any dynamical variable can be obtained by computing its autocorrelation function using a classical trajectory; Mathematically, classical quasiperiodic motion for a system of N coordinates exists if there are N integrals of the motion whose Poisson brackets with each other vanish. If this applies to all initial conditions the system is '' integrable." In general, most dynamical systems are not integrable, but for a large fraction of the initial conditions at small perturbations from an integrable system these N commuting integrals of the motion do exist, according to the KAM (Kolmogorov, Arnold, Moser) theorem.The resulting quasiperiodic motion for such an initial condition then exists stably for infinite time for these initial conditions. In a classical quasiperiodic trajectory there can, in the absence of internal reson- ances, be very unequal distributions of energy among the various modes of vibration for infinite time, a point to which we shall return later. Internal resonances (par- ticularly, but not restricted to, 1 : 1 resonances) cause extensive periodic sharing of the energy among the participating zeroth-order (harmonic) modes.Isolated reson- ances provide another example of quasiperiodic motion and have been so treated in the literature. Usually, as the energy of the system increases, or as the perturbation from an integrable system (one with N actions) increases, an increasing fraction of the trajec- tories becomes chaotic. In classically chaotic motion all of the N action variables no longer exist, although some may, and correspondingly instead of having a sharp-lined power spectrum the spectrum becomes " grassy". (It may become " broadened", but there are not enough cycles of the trajectory to make it a continuum.) Each trajectory becomes very sensitive to its initial condition, and initially neighbouring trajectories separate in phase space exponentially with time.In contrast, neighbour- ing quasiperiodic trajectories separate linearly in time. Nevertheless, chaotic behaviour of an individual trajectory need not be the same as microcanonically averaged behaviour. In a two-oscillator system, in particular, it is confined between adjacent tori which have persisted. Chirikov's approach to the onset of chaotic motion is the following, and involves an '' overlap of resonance^".^ A particular trajectory for some initial condition (specific actions) has a commensurable set of frequencies (a resonance) and nearby tori have almost commensurable frequencies near this ratio, and are within the '' width of this resonance " in action space: two or more of the independent angle variables (phases) in these trajectories can become correlated.Nearby there is another periodic trajectory with a different commensurability and SO the centre of a different resonance. Around it are tori with near-commensurability almost equal to this new commensurability and this resonance has its own " resonance width". Although each resonance is not in itself chaotic (one can define actions for any trajectory in this width), an overlap of these widths in phase space initiates the onset of chaos, according to Chirikov, a chaos which under stronger perturbations or higher energies becomes increasingly widespread. the above frequencies appear as sharp lines.R. A. MARCUS 105 In addition to quasiperiodic and chaotic l4 motion another type of motion has been found recently in hard-walled " rational billiards " problems.1s The system is unusual in that it lies on a special two-dimensional surface in four-dimensional phase space, termed a " multiply handled sphere," which is not a torus.The action integrals do not exist, and the surface has been termed pse~dointegrable.*~*'~ It is apparently not known whether any such surface applies to any molecular system. Initially neighbouring trajectories would presumably separate rapidly when one pro- ceeded along a handle and the other, on the other side of a bifurcation, along the main sphere. Such " bifurcations " would presumably cause irregularities in the classical power spectrum of a trajectory on that surface, Thus this special case has some properties of a chaotic system (even " overlapping avoided crossings "),15 although the trajectory lies on a surface of the same dimensionality as a torus. We omit consider- ation of this special case.3. QUANTUM BEHAVIOUR The semiclassical quantum analogue of quasiperiodic motion is now well under- stood. [It is reviewed in ref. (I), (6) and (ll).] As long as action variables exist for the trajectory, one can introduce an Einstein-Brillouin-Keller semiclassical quantiz- ation of them (a WKB approximation for solving the Schrodinger equation). Keller showed how to do this for non-separable systems with hard-walled potentials l6 and more recently Eastes l7 and Noid l8 showed how to determine action integrals from quasiperiodic classical trajectories from non-separable systems with smoothly varying potentials.Previously, thc only such systems successfully treated had been those in which one can separate variables or in which one can apply classical canonical perturbation theory to obtain the actions. A variety of perturbative or perturbative- iterative methods have since been presented. [For reviews see ref. (I), (6) and (I l).] When trajectory or perturbative methods are applied in their simplest form, one has a " primitive " semiclassical quantization. This quantization is satisfactory in most instances when the motion is quasiperiodic. Systems with and without internal resonances have now been quantized in this way. However, when the quantized trajectory occurs near a separatrix, i.e. near a surface which separates tori describing two different types of motion, a uniform semiclassical quantization is needed. Ex- amples include the one-dimensional double-well problern,l9 the pendulum problem, the Henon-Heiles system (the border between librating and precessing trajectories),20 local-mode-normal-mode description of triatomic molecules ABA,Z1 group local-mode treatment of polyatomic molecules ABC in which A and C are each a set of atoms and B is heavy,22 and avoided crossings (" anti-crossings ") in quantum mechanics.23 All of these problems are either or have been approximately reduced to that of one dimension.Some of the properties of the classical quasiperiodic trajectories have parallels in those of the corresponding quantum-mechanical states : the spectrum, which was relatively simple for the classical quasiperiodic system, is comparatively simple in the quantum-mechanical system.(Of course, when there are a number of coordinates there can be many regular sequences of overtones and combinations. Unravelling the spectrum may then become impractical.) One difference from a classical mechani- cal spectrum is that the latter is obtained from a trajectory at some given energy, and so the classical overtones ate exact multiples of the fundamentals at that energy. In the quantum case each spectral line involves a transition between two energies, and so is at best only centred on some energy. Since this mean energy increases with each overtone, the exact multiple relation is lost. Nevertheless there is a regular progression106 INTRAMOLECULAR ENERGY TRANSFER of spectral lines (overtones and combinations) with gradually changing spacing. Taking this effect into account in the comparison between classical and quantum spectra, so that the individual spectral lines are always compared at a mean action corresponding to the mean quantum number, there was good agreement not only of the positions of the spectral lines but also of their intensities for the cases studied in ref.(24). There is also a correspondence of the " shape " of the wavefunction and the corresponding semiclassical trajectory (i.e. the one whose actions match the quantum numbers via a semiclassical relation).' They display regular (rather than chaotic) contour patterns. Indeed, semiclassically calculated wavefunctions are expected to compare well with the quantum-mechanical ones, if any needed uniform approximation is introduced.Correspondingly, there is also a correspondence of the so-called PoincarC surfaces of section with Wigner distribution functions (calculated in a par- ticular way).25 One thus expects that, apart from tunnelling and interference contri- butions to any particular problem, there will be a close correspondence between various reduced properties calculated by averaging using a wavefunction or by phase averaging over the corresponding classical torus. Quantum-mechanically an integrable system could be defined, by analogy with the classical definition1' of commuting Poisson brackets, as a system for which the operators for the N integrals of the motion of an N-coordinate system commute with each other.A consequence of this definition is that there are sequences of quantum numbers (nl, . ',nN), just as there are individual actions (J1, - .,JN), which describe the state of the system, rather than just one quantum number ordering the states. Correspondingly there are expected to be " regular patterns " in the wavefunctions and regular progressions of spectral lines. It is quite likely, by analogy with the KAM theorem, that this behaviour continues to be true (at least approximately now) for small perturbations from an integrable system. However, only in the limit ti = 0 may it be exactly the case. In summary there is a close relation between quantum states and the related quasi- periodic trajectories, when the latter exist. The former have " regular wave patterns '' and " regular spectra." Since quasiperiodic motion for the general case, i.e.the case which is not integrable, has not been precisely defined for quantum systems, but only by semiclassical corres- pondence, it is not surprising that no generally accepted definition of quantum chaos exists. By analogy with the corresponding classical case one could compare two wave- packets initially neighbouring in some phase-space sense and examine the nature of their separation in time, exponential or linear, of values of some phase space operator in the two packets. In contrast to classical mechanics, however, the two packets would no longer each be at specified energies, and the results might depend on their width and on the states chosen for the packet, there being at least several quantum states rather than just one.A number of investigators, including Heller, Brumer and Shapiro, Weissman and Jortner, and Pechukas, have discussed the evaluation of wave- packets from various viewpoints [partly reviewed in ref. (l)]. Another criterion for distinguishing which quantum states are analogous to quasiperiodic and chaotic trajectories would be the spectrum-whether or not irregularities exist." Recently we described a way that irregularities can occur [ref. (1) and references cited therein]. States can mix wave patterns via avoided crossings of eigenvalues. Avoided crossings belonging to a certain sequence would show a regular progression of the spacings. (A single classical resonance can embrace within its width many, few, or no pairs of states which show avoided crossings.) Like an isolated classical re~onance,~ an avoided crossing would not in itself constitute cha0s.lR.A. MARCUS 107 We next consider overlapping avoided crossings. A criterion for this overlap is obtained below. Each avoided crossing has some " width " over which the two wavefunctions are changing rapidly in the plots of the two eigenvalues versus a perturbation parameter A. One can define a " width " for an avoided crossing in the following way. The two- state pair of eigenvalues are, using degenerate perturbation theory, given by where H,, and H22 depend on A and where, for the purposes of the present illustration only, H12 is constant. The difference of the slopes of the two " crossing " eigenvalue plots in the vicinity of the avoided crossing (i.e. at a value of A where H,, = H22) is denoted by s: s = a(H,, - H22)/aA at H,, = H22.(3-2) The " half-width " w, in A-space, of the avoided crossing is the distance in A, from the crossing-point, needed for 41Hl2I2 to be comparable to (H,, - H22)2. Hence w E 21H121/s. (3.3) 21H121 is the splitting of the two states at the A where H,, = H22. With the half-width defined as in eqn (3.3) one sees that overlap of two avoided crossings A and B occurs when the distance between their respective centres AA satisfies AA < W, + Wb (3.4) and when they simultaneously have a state in common. One can extend the argu- ments to a multi-parameter space. The avoiding curves become a special case of the well known conical intersection.1° One view of" quantum chaos "'*14 is that it corresponds not to an isolated avoided crossing but to overlapping avoided crossings. The latter would yield irregularities in the spectrum and a complex contour pattern for the wavefunction of a quantum state, reflecting the involvement in several regularly patterned wavefunctions in this state.' The generation of statistical (" scrambled ") wavefunctions by overlapping avoided crossings,26 and the properties of the former, have both been discussed recently.27 " Quantum chaos " does not 28-30 automatically follow when the corresponding classical trajectories are chaotic.If, for the particular actions of interest, the width of the classical resonance is less than h, it has little consequence quantum- mechanically.(More precisely, the splitting at the avoided crossing becomes very If it is much larger than h this classical resonance can support a number of the pairwise avoided crossings, each arising in a sequence of eigenvalues. If the overlap of two classical resonance widths is so small that it does not contain even a single quantum state, this overlap of the classical resonances is of little consequence quantum-mechanically. If there is a quantum state in this classical overlap width, it participates in the two overlapping classical resonances and hence in both sets of avoided crossings, and eqn (3.4) is fulfilled. That is, it participates in overlapping avoided crossings. The conditions that classical chaos (assumed to involve overlapping resonances) also yields quantum chaos (assumed to involve overlapping avoided crossings) are that (i) the width of a Chirikov resonance exceeds h, i.e.that it contains at least one quantum ~ t a t e , ~ ~ - ~ O (ii) that the overlap width of the two overlapping resonances also contains at least one quantum state,28 and (iii) that the centre of the resonance be close to a quantum ~ t a t e . ~ ~ ~ ~ ~108 INTRAMOLECULAR ENERGY TRANSFER The half-width of the resonance is, in the notation of section 6, Z.\/(AIR), where A and !2 are properties of the resonant Hamiltonian and where the width of the angle space is 271 (using the action-angle variables introduced in section 6). Thus the first condition for “ quantum chaos ” is * The second condition can be written as follows: If AJlj is any projection of the overlapping resonance zone on any two-dimensional action space (JiJj) then we require that there be at least one system of orthogonal axes in action space such that t dJij/h > 1, for all i,jpairs.[See also ref. (9) and (30).] One system which has been extensively studied is the 99-state Henon-Heiles system, an anharmonic coupled oscillator system with C3” symmetry and nearly 1 : 1 resonant. It becomes classically chaotic in part at the higher energies. In contrast, it has regular sequences of eigenvalue with slowly varying spacing^,^' its wavefunctions show regular patterns of their contour plots,32 and its eigenvalues are semiclassically very well approximated by those of an integrable H a r n i l t ~ n i a n . ~ ~ ’ ~ ~ (This second Hamiltonian is a perturbation approximation to the former, and so has well defined action variables.) Incidentally most semiclassical quasiperiodic states in this system show extensive redistribution of energy; they correspond to precessing t r a j e c t ~ r i c s .~ ~ We conclude that the 99-state system is quantum-mechanically ‘‘ regular ”, For this same system the PoincarC surface of section of a particular chaotic trajectory resembled in coverage the Wigner distribution function of a quantum state at the same energy.25 However, one cannot conclude that the latter quantum state is chaotic. Because there are only two coordinates the same chaotic trajectory is trapped between two residual tori, each of which would also be expected to have Poincark surface of sections of rather similar shape to the Wigner distribution func- tion.We are currently investigating this possibility. 4. INTRAMOLECULAR ENERGY REDISTRIBUTION We have already noted that one can have extensive classical-mechanical intra- molecular energy redistribution even in a quasiperiodic system, when there is an internal resonance, but that the redistribution is quasiperiodic rather than chaotic. The precessing trajectories in the Hhon-Heiles system provide an example. Simi- larly, in quantum systems where there is a low order (e.g. 1 : 1, 2 : 1, etc.) degeneracy or near-degeneracy (as in Fermi resonance) there is an extensive energy sharing 35 or, if one introduces a suitable wavepacket, energy redistribution among zeroth-order modes. Another example of equipartitioning of energy among zeroth-order (harmonic) modes, when one such mode was excited, was for an integrable system, the Toda lattice.Six-atom and fifteen-atom systems were i n ~ e s t i g a t e d . ~ ~ . ~ ~ The vibration frequencies were reported for the six-atom system 37 and several were commensurable. Thus internal resonances, rather than quasiperiodic versus chaotic motion per se, played a major role in the equipartitioning in that system, as well as in the precessing trajectories of the HCnon-Heiles system. This importance of internal resonances for energy redistribution is also seen in a study of a seven-atom system C-C-C-M-C-C-C, where M is a heavy atom and * Eqn (12) of ref. (28) actually gives the half-width and should be multiplied by a factor of 2.7 Eqn (3.6) replaces eqn (14) of ref. (28), which is correct only for a two-coordinate system.R. A. MARCUS 109 the oscillators are Morse ~ s c i l l a t o r s . ~ ~ When the energy of excitation of one of the bonds is fairly high (0.3 of the bond dissociation energy) there is relatively little transfer of energy from the first half of the molecule to the second, although there is extensive energy transfer among the bonds of the first half themselves. When the excitation energy is instead small, only 0.05 of the bond dissociation energy, there is complete transfer. Finally, when the Morse oscillators are replaced by harmonic ones, with 0.3 of the Morse bond dissociation energy, there is again complete transfer. We have since determined the spectrum of each trajectory.The motion in all the above cases is quasiperiodic. The above effects can be explained in terms of internal resonances. When the energy of the first part of the molecule is high, its frequencies are red-shifted from those of the second part and there is comparatively little energy transfer. Because of the quasiperiodicity the observed non-equilibrating behaviour should persist cIassically for infinite time. Transfer is complete in the other two cases because of resonance. It is essential for the effect that the two parts of the molecule be coupled by a small perturbation. For example, when the mass of M in C-C-C-M-C-C-C is reduced by a factor of two there is now extensive energy-sharing at the excitation energy of 0.3 dissociation energy.38 (The effect of C-M-C bond angle was also studied; it changes the effective mass of M.) A close analogy is the coupled two- pendulum problem.When the frequencies are not commensurate there is not com- plete energy transfer. Theoretical studies of resonant effects in triatomic molecules have been described r e ~ e n t l y . ~ ~ * ~ ~ - ~ ~ We have noted that in both the quasiperiodic and chaotic cases there can be extensive intramolecular energy transfer in the presence of internal resonances. (In the quantum case, the collisional or optical excitation is a " pulse " and generates a wavepacket which then shows little or extensive redistribution, depending on the nature and energies of the states it embraces.) However, although the redistributed energy may be equipartitioned, a single resonance is not " microcanonical " and so R.R.K.M.theory would not be automatically satisfied by it. This point is easily established by an example of a I : 1 resonance.* Incidentally, classically the vibration phases of the isolated-molecule phase average themselves in the non-degenerate case. In the case where any zeroth-order degenerate vibrations are anharmonicaliy coupled, they undergo a slow phase averaging also. There is no need for collisions to cause this phase averaging. Although isolated resonances do not create a microcanonical distribution even among participating modes, the resonant coupling of the resonant modes with other modes could yield a more microcanonical distribution. (The J; in the preceding footnote would no longer be approximately constant.) Thus R.R.K.M.behaviour is consistent with rather " chaotic " quantum states. What is not clear as yet is how chaotic the system has to be. Particularly interesting is that highly specific excitation, namely of CH overtones in molecules, may have produced largely statistical results [cf. discussion in ref. (l)], but very few systems have been studied thus far. Excitation of molecules typically involves a " pulse "-such as an optical pulse or a collision. The pulse will typically involve the excitation of a wavepacket, a group of states. After some initial dephasing period the system will then behave as an ensembIe of systems, each disappearing according to its own rate law, and not be a single * One can define34 an internal angular momentumJ,.At large values ofJ,, Jp2 is roughly constant. The conjugate phase There is a lower limit to the action .I, of the x mode, because Jv2 # 0. Thus, J, and Jy are not microcanonically distributed. The actual distribution function is omitted here for brevity. is (slowly) varying and so phase-averages itseIf.110 INTRAMOLECULAR ENERGY TRANSFER exponential (“ intermediate-level structure ”), if the exact eigenfunctions are not “ scrambled.” However, when the density of states coupled to the excited initial ‘‘ state ” is so high as to become essentially a continuum, one obtains a single exponen- tial decay, at least with the usual assumptions of radiationless transition theory, provided the pulse width is sufficiently large to embrace all zeroth-order coupled states [e.g.ref. (42) and references cited therein]. 5. MOLECULAR SPECTRA The spectrum will depend on the nature of the underlying dynamical motion. When that motion is the analogue of quasiperiodic, or if the wavefunctions are not too “ scrambled ” (i.e. if a relevant state does not participate in many overlapping avoided crossings) one expects regular sequences of spectral lines. (Even in the classical chaotic case, however, there is expected to be some propensity for the positions of the overtones.) If wavefunctions were considerably scrambled Franck-Condon factors in a vibronic excitation would yield an appropriate “ dispersed ” absorption of fluorescence band : many states of neighbouring energies would have comparable Franck-Condon factors. The sharpness of the low-lying excitation spectral lines corresponding to low vib- rational excitation in aromatics, and of the fluorescence, indicates little scrambling.However, at higher energies the dispersed fluorescence in Parmenter’~,~~ Levy’s 44 and in Smalley’s 45 work suggests some scrambling. 6. INTRAMOLECULAR LASER-SELECTIVE CHEMISTRY Some evidence that the intramolecular energy transfer between two ligands attached to a heavy atom may be slow has been given by Rogers et al., based on an interpretation of data of the chemical activation reaction of F with S n ( a l l ~ I ) ~ . ~ ~ The theoretical dynamics of energy interchange between two ligands in C-C-C- Sn-C-C-C were considered recently.38 These various results suggest the possi- bility of intramolecular laser-selective chemistry : one chooses a molecule in which one reaction can occur in the vibrationally-excited ligand and a distinguishable kind of reaction in the other ligand(s). The excitation could be induced by CH overtone excitation, infrared multiphoton absorption, or chemical activation.From the results we obtained earlier,3s the kinetic-energy coupling between the two ligands would be less the larger the mass of the metal atom and the more the ligand-metal- ligand bond angle differs from 180”. Other things being equal, the greater the excess energy per bond (in the quasiperiodic regime at least) the more the frequencies of the excited part of the molecule are anharmonically red-shifted. Thus if that part of the molecule and the remaining ligands are in resonance when each has zero energy, they should become off-resonant with increasing excitation energy, and there should be less energy transfer or (quantum-mechanically) a reduced rate of transfer.The calculations we have described thus far have been in the quasiperiodic regime. The onset of chaos would have several effects, one being to broaden the vibrational spectrum of the excited ligand and increase the possibility of some resonance, How- ever, further excitation in the chaotic regime would further red-shift the frequencies in the excited part of the molecule and so would dccrease the resonance (if they were in resonance at low energy). Some calculations in this chaotic regime are being attemp- ted, although numerical errors are large in this regime. Since the excess energy per bond is an important factor in the effect, and since it is not practical to make detailed calculations, it would be useful to combine a statisticalR.A. MARCUS 111 theory with dynamical calculations so as to treat larger systems. theory is described in the next section. Such a type of 7. THEORY OF INTRAMOLECULAR ENERGY TRANSFER ACROSS A HEAVY ATOM We consider that one ligand is excited by any one of several sources, such as those described in section 6. We divide the excited ligand into two parts, one of which (mode 1) is capable of promoting intramolecular resonant energy transfer to another ligand of the heavy atom M. The case where there may be more than one such mode is included later. The excitation energy is assumed to rapidly redistribute itself within the excited ligand, and so a statistical approximation will be used to calculate the chance that mode 1 has an approximate action J,.(1, here denotes the conventional action divided by 27c. The angle conjugate to it has a range 0-27~) To apply the resonance theory described by eqn (7.3)-(7.12) below it is necessary to know what value to use for J1 and, if the second ligand is at least thermally excited, what to use for J2. We do this first for J1, and an analogous argument would be used We first calculate the distribution function P(J,) for J1. We consider the excited ligand as having an energy in the range (E, E + dE), and the approximate energy of mode 1 as being EJl. Assuming a microcanonical distribution and counting quantum states we have for J2.P(J1)dJl = P’(E - E,,)dJl/P(E)fi (7.1) where p is the density of quantum states of the excited ligand, p’ is that for the part of the ligand not including mode 1, and the phase-space volume element for a given dJl is 27d dJ,. It, divided by 11, is the number of quantum states which it contains, i.e. dJi/h. The mean of Ji,Ji, in this isolated ligand and the mean-square deviation are, of course, JJIP(Jl)dJ1 and [(J, - ~l)2P,dJ,, integrated over all energetically allowed J1. If classical values are used for p’ and p, the h cancels. l/[(AJ1)’], e,g. Jl, but we recognize that there can be fluctuations in J, and in a more complete calculation will include them. If the modes 1 and 2 are off-resonance for J1 = Jl they may be closer to resonance for some Jl in the above range.The value of 11, the resonant centre Jlr, the resonant width (defined below) and the above range of J1 values defined above all become important factors in the energy transfer. We consider energy transfer between mode 1 , of action J1, to an accepting mode in another ligand of M, mode 2 with action J2. J2 and J1 are both conventional actions divided by 27c. Extensive energy transfer can occur if J1 and J2 fall within the “ width ” of an intramolecular resonance. To find the conditions for such a transfer we adapt the standard treatment of an isolated non-linear resonance. The Hamil- tonian for modes 1 and 2 and for the contribution of M is written as a sum We use as a value for J, some value in the above range, Ji where ITM contains, via kinetic- and possibly potential-energy coupling, momenta and/ or coordinates of modes 1 and 2, HIM is next expanded in a Fourier series of these angle variables for those modes.The general term contains cosine and/or sine functions of mlqi + m2q2, where ml and m2 are integers and q, and q2 are angle variables conjugate to J1 and J2.112 INTRAMOLECULAR ENERGY TRANSFER There is a resonance when the angular frequencies mi = aHi/aJi satisfy a relation n,w,(Jlr> - n2w2(J2r) = 0 (7.3) at some value (Jlr,J2') of (Jl,J2). This value is the '' resonance centre." n1 and n2 are positive integers. The angle n,y, - pt2p2, whose time derivative is approximately the frequency nlwl(J1) - ~ z ~ m ~ ( J ~ ) , is slowly varying when J1 and J2 are sufficiently in the neighbourhood of (Jlr,JZr).We consider the case where there is a resonartt term in the expansion of HM in the double Fourier series in qr and y2, i s . a term such as cos(nly7, - n2p2), which we denote by cos a. Coupling of modes 1 and 2 also occurs if there is cos 2a, sine terms, etc., but for simplicity of presentation we consider this cos x term only. Its coefficient will be a function of J1 and J2, which we replace as usual by their values at the centre of the resonance, Jlr and J2r. To treat the dynamical problem we proceed in a standard way, and introduce a transformation into new action-angle variables (Jx, 2, J , 8) such that a is nlpl - nzp2 and J is a constant of the motion. This change of variables is accomplished by a generating function S, such as s = Jor("lv71 - n2vJ 4- J k 2 P I + n1yl2).(7.4) Since D: = &S/8Ja 8 = aS/a J and J i = &S/8yi, we have a = %PI - n2V12, 0 = n2p1 t alp2 J2 = nIJ - n2Ja (7.5) (7.6) JI = nlJa -+ n2J, For molecules, Hl and H2 are typically approximately quadratic functions of J1 and J2. In any event, in standard resonance theory H - HM would be expanded as a quadratic function of J, - J i ; J,, obtained in terms of Jf and Jl from eqn (7.61, is (n,J: - PtJ~)/(n~ + n;). We now have from eqn (7.2) H = H,(J,') + H2(Ji) - +Sr(Ja - Ji)2 + A(JI,J;)cos a = E (7.7) where E is the energy of these two modes and --IR = n:a2Hl/i3Jf + n,a2H2/i3J& evaluated at (JF,Jg>. (7.8) (7.9) As a special case R = x;m; +- x;co;. Here &o$ and xzco; are the anharmonicity constants. Eqn (7.9) assumes the quadratic dependence of HI on Jl to be good at a11 J1 and to be independent of the energy excess of the other modes of the excited Iigand.(They may contribute to an effective potential for mode 1.) Similar remarks apply to H2 and J2. We denote Hl(Ji) $- H2(J3 - E by B. At the maximum of any curve in fig. I , +Q(J, - Ji)2 equals B + A , while on the separatrix one sees that J, - J i vanishes when a = &n, i.e. when B = A . Thus the height of the maximum of the separatrix is 22/(A/!2) and the depth of the minimum is -22/(A/i2). AccordingIy, if for any trajectory the maximum distance of Ja from J i satisfies max]J, - JAl < 2l/(A/Q) (7.10) the projection of the trajectory on the (JU,a) plane in phase space will describe an " ellipse '' in fig. I . There is a " phase locking" (the p1 and q2 are seen to be correlated, because the range of CL is now restricted), and Ja - Ji oscillates about zero.R.A. MARCUS 113 In terms of energy redistribution one sees from eqn (7.6) that it is extensive when For example, using eqn (7.10) we find at a given J maxlJ, - Jfl < 2n14(A/R) (7.1 1) maxlJ, - J$I < 2n2d(A/R). (7.12) 1.e. regardless of the initial values of J1 and J2, the modal actions J1 and J2 will oscillate about Jf and Jl, as long as J1 and J2 lie in the resonance widths, eqn (7.11)-(7.12). If Jf and Ji are equal (as they are in intrinsic 1 : 1 near-resonant systems), there is an J a - J i oscillates about zero. a. Fig. 1. Phase space plot of J, - JL against a for various values of E in eqn (7.7) and hence of the B defined later in the text.The separatrix consists of the two curves passing through a = +n. On the separatrix B equals the A in eqn (7.7). equipartitioning. On the other hand if maxlJ1 - Jil lies outside the halfwidth 2n14(A/R), there will be little or no tendency, classically, for such redistribution. The time required for resonant transfer of energy between modes 1 and 2 is readily obtained from eqn (7.7). Well within the separatrix cos a can be expanded about a = 0, yielding a harmonic-oscillator expression for the motion of a. The period is then found to be 2n/d(AR). The time for Ja - Ji to go from its maximum positive value to its maximum negative value is the half-period z, z = n/d(AR). (7.13) This is the typical time for the resonant intramolecular energy transfer when it is classically allowed.The conditions for the validity of resonance theory (e.g. the system must be sufficiently anharmonic) have been given by Chirikov [ref. (7), eqn (3.23)]. There may be other pairs of modes which involve motion of the bonds of M, and which may be resonant. They can readily be included when they are isolated reson- ances. Currently we are obtaining information about the various resonances from power spectra of the bonds linking M with the ligands, using power spectra of trajectories 24 of the isolated ligands and of the molecule as a whole. We turn next to quantum effects. Each classical resonance can embrace none, one or a number of pairs of resonant quantum states of modes 1 and 2. Extensive energy transfer among the two modes, and hence between the two ligands containing them, can occur quantum-mechanically between such pairs of resonant states even when it is classical forbidden, i.e.even when condition (7.10) is violated. The rate of transfer, reflected in the splitting of the quantum-mechanical levels of modes I and 2, is then114 INTRAMOLECULAR ENERGY TRANSFER relatively low. This splitting can be calculated semiclassically 21 based, in the present case, on eqn (7.7). The semidassical results for the splitting in triatomic ABA rndecules agreed with quantum-mechanicaI results very we11 when the splitting was However, when the splitting was small (corresponding to large values of IJ, - &I> the error was considerable [e.g. the 15,O } 10,s > and 14,O } & ]0,4 } cases in table JI of ref. (2111.In such cases a quantum-mechanical calculation would be needed. The error in these small-splitting cases is presumably due to replacing the J1 and J2 in A(J,,J,) by constants. In such cases it might be better to use quantum- mechanical perturbation theory to calculate the splittings rather than a semidassical formuIa [eqn. (7.7)]. 8. CONCLUSIONS AND SUMMARY The nature of anharmonic motion in moIecules, dassical and quantum, is discussed, and implications for a statistical theory (microcanonical) such as R.R.K.M., for uni- molecular processes, are considered. An approximate statistical plus dynamical theory for intramolecuIar energy transfer between ligands across a central metal atom is given, with possible consequences for intramolecular laser-selective chemistry. I should like to acknowledge the helpful collaboration (identified in the references) with D.W. Noid, M. L. Koszykowski, V. Lopez and T. Uzer on various portions of this research. The research was supported by a grant from the National Science Foundation. I am indebted to Drs. Vicente Lopez and David Wardlaw for their helpfuI comments. This is contribution no. 6775. D. W. Noid, M. L. Koszykowski and R. A. Marcus, Annu. Rev. Phys. Chem., 1981,32, 267. P. Brumer, Ado. Chem. Phys., 1981,47, 201. S. A. Rice, Adv. Chern. Phys., 1981, 47, 117. M. Tabor, M u . Chem. Phys., 1981, 46, 73. Nonlinear Dynamics, ed. R. G. Helleman, in Ann. N. Y. Acad. Sci., 1980, 357. N. C . Handy, Semiclassical Methodx in Moierular Scattering and Spectroscopy, NATO Adu.Stlcdy Inst. Ser. C: Math. Physical Sci., ed. M. S. Child (Reidel, Dordrecht, 1980), p. 297. 3. V, Chirikov, Phys. Rep., 1979, 52, 243. Topics in Nonlinear Dynamics. York, 1978) ; Various artides of Am. Corzf. i%eor. Chem., in J . Phys. Chem., 1982,86,2113ff. M. V. Berry, in ref. (8), p. 16. lo V. I. Amol’d, Mutheniarical Methods of Classicai Mechanics (Springer-Veriag, New York, 1978). l1 I. C. Percival, Adt.. Chem. Phys., 1977, 36, 1. l2 Stochastic Behavior in Classical and Quantum Hamiltmian Systems. Lecture Notes in Physics, Proc. Volta Mem. Conf., Corno, If&, ed. G. Casati and J. Ford (Springer-Verlag, New York, 1977), vol. 93. l3 1. Ford, Fundamental Prublems in Statistical Mechanics, ed. E. G. D. Cohen (North HolIand, Amsterdam, 1975), vol.3, p. 215; J. Ford, Adv. Chem. Phys., 1973,24, 15s. l4 M. V. Berry, Ann. Phys. N. Y., 1981, 131, 163. l5 P. J. Richens and M. V. Berry, Physica, 1981, 2D, 495. l6 J. B. Keller, Ann. Phys., 1958, 4, 180. l7 W. Eastes and R. A. Marcus, J. Chern. Phys., 1974, 61, 4301. l9 J. N. L. Connor, Chem. PAYS. Lett., 1969,4,419. 2o C. Jaffd and W. P. Reinhardt, d. Chem. Phys., 1982, 77, 5191. 21 E. L. Sibert, J. T. Hynes and W. P. Reinhardt, J. Chem. Phys., 1982, 77, 3595. 22 V. Lopez and R, A. Marcus, to be published. 23 T. Uzer, D. W. Noid and R. A. Marcus, to be published. l4 M. L. Koszykowski, D. W. Noid and R. A. Marcus, J. Phys. Chem., 1982, 86, 21 13; D, W. 25 J. S, Hutchinson and R. E. Wyatt, Chem. Phys. Left., 1980, 72, 378. AIP Conf. Proc. No. 46, ed. S. Jorna, (Am. Inst. Phys., New D. W. Noid and R. A. Marcus, J. Chern. Phys., 1975, 62,2119. Noid, M. L. Koszykowski and R. A. Marcus, J. Chem. Phys., 1977, 67, 404.R. A. MARCUS 115 26 R. B. Gerber and R. A. Marcus, J. Chem. Phys., to be submitted. ’’ R. B. Gerber, V. Buch and M. A. Ratner, Chem. Phys. Lett., 1982, 89, 171. ” K. G. Kay, J. Chem. Phys., 1980,72, 5955. 30 E. V. Shuryak, Sou. Phys. JETP, 1976,44, 1070. 31 D. W. Noid, M. L. Koszykowski, M. Tabor and R. A. Marcus, J . Chem. Phys., 1980,72,6169. 32 M. D. Feit, J. A. Fleck Jr and A. Steiger, J . Comput. Phys., 1982, 47, 412. 33 J. B. Delos and R. T. Swimm, Chem. Phys. Lett., 1977,47,76; R. T. Swimm and J. B. Delos, J . 34 D. W. Noid and R. A. Marcus, J. Chem. Phys., 1977, 67, 559. 35 D. W. Noid, M. L. Koszykowski and R. A. Marcus, J. Chem. Phys., 1979, 71, 2864. 36 N. Saito, N. Ooyama, Y. Aizawa and H. Hirooka, Prog. Theor. Phys., Suppl., 1970,45,209. 37 J. Ford, S. D. Stoddard and J. S . Turner, Prog. Theor. Phys., 1973, 50, 1547. 38 V. Lopez and R. A. Marcus, Chem. Phys. Lett., 1982, 93, 232. 39 R. T. Lawton and M. S . Child, MoZ. Phys., 1979,37, 1799. 40 C. Jaffk and P. Brumer, J. Chem. Phys., 1980, 73, 5646. 41 cf. D. W. Oxtoby and S . A. Rice, J. Chem. Phys., 1976, 65, 1676. 42 J. Jortner and R. D. Levine, Adu. Chem. Phys., 1981, 47, 1 . 43 R. A. Covaleskie, D. A. Dolson and C. S . Parmenter, J . Chem. Phys., 1980, 72, 5774. 44 J. E. Kenny, D. V. Brumbaugh and D. H. Levy, J . Chem. Phys. 1979, 71, 4757. 45 J. B. Hopkins, D. E. Powers and R. E . Smalley, J . Chem. Phys., 1980,73, 683. 46 P. Rogers, D. C . Montague, J. P. Frank, S. C. Tyler and F. S . Rowland, Chem. Phys. Lett., 1982, R. A. Marcus, Ann. N.Y. Acad. Sci., 1980, 357, 169. Chem. Phys., 1979,71, 1706. 89,9.

ISSN:0301-7249    

DOI:10.1039/DC9837500103

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Furaday Discuss. Chem. SOC., 1983, 75, 117-130 Pulsed Laser Preparation and Quantum Superposition State Evolution in Regular and Irregular Systems BY RONALD D. TAYLOR AND PAUL BRUMER * Department of Chemistry, University of Toronto, Toronto, Ontario M5S 1A1, Canada Received 20th December, 1982 The nature of the prepared state is crucial to an understanding of isolated-molecule intramolecular dynamics. A model is utilized to compare the created quantum superposition state in pulsed laser excitation from (1) a ground electronic state with regular nuclear wave- functions to an excited electronic state with regular nuclear wavefunctions with that from (2) the regular ground state to an excited state with irregular nuclear wavefunctions. All results are in the quantum-mechanical small-molecule limit.Visual inspection of the created state and its evolution shows distinct qualitative differences in these two cases, although various theoretically studied quantities do not reveal an obvious distinction. The differences are expected to be experimentally observable in tirne-resolved emission. 1. JNTRODUCTION The vibrational dynamics of isolated rnoIecuIes has historicaliy been formdated differently in two distinct energy regimes. At low energies motion is assumed separable and well approximated by a zeroth-order (normal, local, etc.) modes description. At higher vibrational energies an initial distribution is assumed to relax efficiently to a statistical result. In particular, the relaxation rate is expected to exceed the rates of competitive processes.This historical viewpoint has gained support from recent formal and computational developments in ergodic theory and in the clussical dynamics of N-degree-of-freedom non-linear Hamiltonian systems. The importance of these results to intramolecular dynamics is twofold. First, formal theory provides a concise definition of the ideal cases of regular quasiperiodic behaviour and of mixing systems.’ Secondly, COM- putational results on typical non-lincar systems show motion at low energies which is close to regular quasiperiodic ( N conserved isolating integrals of motion), whereas higher-energy dynamics shows relaxat ion dynamics characterized by fewer (but not necessarily conserved isolating integrals. Although formal techniques do not provide a useful route for identifying a system as ergodic or mixing, several com- putational diagnostics are available allowing one to identify dynamics which are not regular.Thus, although technical problems exist in classical intramolecular dynamics, the conceptual hasis for a transition in behaviour with increasing energy seems well established. The status of quantum intramolecular dynamics is quite different. In this case efforts to define formal, conceptually useful, ideal systems displaying finite time relaxation behaviour are in progress. Similarly, computationa1 diagnostics to identify a quantum analogue of regular and irregular motion are only in their earIy * I. W. Killarn Research Fellow,118 SUPERPOSITION STATE PREPARATION stages of development. A variety of different approaches, such as the examination of localized wavepacket dynamic^,^ eigenvalue distributions,s properties of Wigner functions,6 mapping dynamics etc., have been proposed to address these problems.Our approach, exemplified in this study, is to explore the exact quantum dynamics of model systems which are well characterized in the classical limit in an effort to ascertain similarities and differences between the quantum dynamics of systems which are (classically) regular as compared with those which are (classically) mixing. Further motivation for the study discussed in Section 4, on systems described in Section 3, is provided in Section 2 below. 2. MOTIVATION AND ORIENTATION Both classical quasiperiodic and statistical behaviour are observable in systems as small as N = 2.Thus it suffices to consider the quantum dynamics of isolated small molecules. An ideal experiment on the intramolecular dynamics of an isolated small molecule is readily envisaged. Here, a preparation device leaves the molecule, with bound molecular eigenstates {p,} and energy eigenvalues {Ej}, in a superposition state a v(0) = c CjPP i The subsequent time evolution, neglecting radiative emission, is given by i In this case intramolecular dynamics depends solely upon the molecular properties {p,}, {E,}, the interaction between the molecule and preparation device described by {c,}, and their cumulative effect in the sum [eqn (2)]. Observation of the evolving state entails projection onto states defined by the particular measurement. Such observations are necessarily dzpendent upon the coefficients {c,}.Previous efforts to explore regular versus irregular quantum behaviour have thus far focused on several aspects of eqn (2), such as the nature of the energy eigen- value spectrum,s the structural characteristics of p, in the semiclassical limit and the evolution of V(t) with the initial state taken as a localized wavepacket or mixed state.lo Little effort has, however, been directed towards the subject of this paper, the examination of similarities and differences in {cj} resulting from the response of " quantum-regular " and " quantum-irregular " systems to similar preparation devices, The importance of the preparation step to small-molecule dynamics is well recognized l1 and easily motivated. Consider, for example, an experiment which prepares, in two systems, one highly coupled and one uncoupled, a ~ ( 0 ) comprised of two exact molecular eigenstates.No matter what the extent of coupling between arbitrarily envisioned zeroth-order modes, an emission experiment, on either system, would only reveal qualitatively similar quantum beats or elementary properties of individual molecular eigenstates. Similarly,12 if experiments on uncoupled or highly coupled systems prepare a ~ ( 0 ) comprised of a large number of states, but with similar {c,} energy envelopes, then the ~ ( t ) evolve with qualitatively similar P(t) = 1(V(0)lV(t))12. Thus, if one performs similar experiments on two molecules and dctects characteristics of' regular behaviour in one system and irregular behaviour in the other, the origin of these differences can be attributed to the different nature of the created superposition states.These arguments motivate a comparison of state preparation and evolution in " quantum-mixing " and " quantum-regular " systems. In the absence of suitableR. D. TAYLOR AND P. BRUMER 119 definitions of these terms, and in the light of our expectation of an intimate relationship between systems which are classically mixing and " quantum mixing," we utilize below systems which are fully characterized in the classical limit. These systems (particles in boxes, circles and stadia) bear only a small resemblance to realistic N = 2 molecular vibrational systems, but they will display the distinct characteristics of regular and irregular quantum behaviour in preparation and evolution, should such differences exist.Indeed, we strongly advocate quantum studies on such sys- tems whose classical behaviour is fully characterized since regularity, mixing etc. is expected to be a property of the Hamiltonian. Below, we consider electronic excitation via pulsed laser irradiation, both as a model for these popular experiments and as a controlled means of preparing super- position states. Alternative means of preparation are under inve~tigation.'~ Our focus here is on ~ ( 0 ) and V ( t ) and not on realistic measurements of the evolution, to be reported e1~ewhere.l~ Hence, radiative emission, for example, is neglected. We emphasize that our interest is in quantum behaviour, with no effort being made to examine behaviour under semiclassical conditions.3. METHODOLOGY 3.1. MODEL SYSTEMS To explore differences in state preparation we model two cases: (I) a transition from a ground electronic state with regular vibrational eigenfunctions to an excited electronic state with regular vibrational eigenfunctions and (2) a transition from the regular ground state to an excited electronic state with irregular vibrational eigen- functions. Here the regular-irregular terminology is based on behaviour in the classical limit. Model N = 2 systems which are regular in the classical limit are readily constructed. This is not the case, however, for mixing systems, a condition proved for only a limited number of Hamiltonians. Of particular interest is the stadium billiard, i.e.a particle confined to apotential-free region by an infinite potential boundary comprised of two parallel lines and two circular caps (see fig. 1). Numerical eigensolutions for this system, regarded as a simple model for irregular N = 2 vibrational motion in an excited electronic state, are available. Regular vibrational motion in the excited state is modelled by a particle confined to a rectangle. Sides of the rectangle are chosen to be of irrationally related lengths to eliminate degeneracies. Finally, ground-electronic-state regular dynamics is chosen as that of the particle con- fined to a circle of radius rc, ensuring no symmetry relationships between ground- and excited-state eigenfunctions. In addition to the circle+rectangle (denoted the CR case) and circle-tstadium (denoted the CS case) systems an additional system, where the rectangle has a 2 : 1 ratio of sides, was studied.This system (denoted the CRD case), containing highly degenerate energy levels, is expected to be even further removed l5 from quantum ergodic behaviour. System potentials are shown in fig. 1. The Schrodinger equation in each of these cases is given by - h2 - (a2q/ax2 + a2q/ay2) = Ep; E = h2K2/2m 2m (3) with q = 0 on the appropriate boundary. Stadium eigenfunctions were determined numerically by an extension of the program of McDonald and Kaufman l6 to include states of all symmetry. Rectangle solutions are well known as is the ground-state solution of the particle-in-a-circle with eigenfunction qo(r,O) and eigenvalue h2K,2/2m given, with J,(Kr) the ith Bessel function, by qo(r,@ = J o ( ~ o ~ ) / ~ ~ ~ ~ c J l ~ ~ ~ ~ c ~ l for r < r,; qo(r,8) = 0 for r > rc (4)120 SUPERPOSITION STATE PREPARATION with Kor, = 2.404 825 5.This is the only state of the circle required below, since the system is assumed to initially reside in this ground state. In each of the circle, stadium and rectangle the numerical value of an energy eigenstate is inversely proportional to the (area)+ contained within the boundary, allowing for a scaling of energies into ranges appropriate to molecular problems. With rn chosen as a typical reduced mass (here m = 19 610 a.u. = 10.7 a.m.u.) the areas of the figures were set in the following manner. First, the area of the circle was G E S E E S c a s e - c s C R 0 Fig.1. Shape of vibrational potential for ground electronic state (GES) and excited electronic state (EES) for various cases. Dimensions utilized were a, = 1.1423 a.u. for the CS cases, 1, = 4.5692 a.u., 1, = 2.0394 a.u. for the CR cases and 1, = 21, = 4.3171 a.u. for the CRD cases. Dark circles within EES figures denote relative size and location of GES for 3 cases. adjusted so that Eo = k2Ki/2rn = 9.12 x a.u., i.e. the ground " vibrational " state of the circle is at a typical vibrational energy. The resultant Y, = 0.127 19 a.u. is typical of the size of probability density associated with a low-lying vibrational state of a vibrator. Secondly, the area of the stadium was determined by placing highly excited states of the stadium, known l6 to be characterized by highly convoluted nodal patterns, in an energy range such that they were accessible from Eo by visible radiation.In particular, the centre of the excited vibrational manifold (at K = 51.2989 for a stadium l6 whose area is z) was located 0.0735 a.u. above E,, yielding a readjusted stadium area of 9.3 185 a.u. The excited-state rectangle areas were set equal to that of the stadium. Once again the resultant dimensions are typical of the probability distribution associated with a highly excited vibrator. The final specification required to describe the model fully is the location of the centre of the ground-state circle with respect to the centre of the excited-state rectangles or stadium. For each of the CR, CRD and CS cases three relative locations were examined to allow a study of the dependence of the prepared state on circle location.This provides preliminary insight into whether statistical behaviour may be linked with the insensitivity of prepared state to aspects of the preparation. In particular, the circles were positioned, with respect to the centre of the excited-state potential, at C1R. D. TAYLOR AND P. BRUMER 121 = (1.2695,0.6984), C2 = (1.8407,0.4128) and C3 = (0.4128,0.4128). We distinguish these models as C1 R (ground-state circle Cl , excitation to the non-degenerate rectangle), C3S (ground-state circle C3, excitation to the stadium) etc. Relative positions of these circles are also shown in fig. 1 . The number of states q,, contributing to ~ ( 0 ) due to pulsed laser preparation is dependent upon the density of qn states and the laser pulse time.To observe quantum intramolecular relaxation, which for small molecules is dephasing on a timescale less than the density of states p, requires that hp exceeds the laser pulse time. An appropriate choice of laser characteristics allows, therefore, for a study with ~ ( 0 ) containing relatively few states. The relevant 16-23 states " picked up " by the laser pulse described later below consist of 16 states in the stadium between K,, = 29.6067 a.u. and K,, = 29.9456, 21 states of the non-degenerate rectangle in the range K,, = 29.5573-29.9966 and 23 states of the degenerate rectangle between K,, = 29.5509 and K,, = 30.0045. A wide variety of different (n,,n,) quantum numbers appear in the latter two sets. The resultant densities of states are 0.14-0.15 state cm-', a reason- able density in the small-molecule limit.This leads to a relatively short " recurrence time " of ca. 7.4 x 10-13 s, necessitating excitation pulses on the sub-picosecond time- scale. Less experimentally demanding pulses, i.e. in the nanosecond range, could have readily been used by altering the model to increase the state density. Such machinations are, however, unnecessary theoretically since the resultant excitation of 16-23 states would behave functionally the same but on a different timescale. 3.2. STATE PREPARATION AND TIME EVOLUTION The full Hamiltonian for the interaction of our " molecular " model with an external laser field is given by H = H,,, + V, where H, is the Hamiltonian for the model and V is its interaction with the laser field. Here " electronic " eigenstates of H,,, are circle + rectangle or circle + stadium with " vibrational '' eigenstates describcd above.The system is assumed initially to reside in the ground state of the circle, denoted y o ; {q,,,n # 0} denotes the set of " vibrational " eigenstates of the excited " electronic " state. A similar indexing convention applies to the energy eigenvalues. Matrix elements of the interaction potential V,, = (q,l Vlq,} are obtained within the following approximations: (1) all electronic transitions are assumed to be Franck- Condon in nature; (2) vibrational states within the excited electronic state are not radiatively coupled to one another; (3) radiative emission is neglected; (4) there is no interstate coupling between vibrational levels of the excited and ground electronic states.Under these circumstances : V," = 0 V,,= V,,,,-O; m#O,n#O ( 5 ) where pel is the electronic transition moment and ~ ( t ) defines the coherent, transform- limited, laser pulse. In particular, we choose ~ ( t ) as gaussian, to model a typical experimental environment; l8 i.e. where wL is the laser frequency, t, is the temporal location of the pulse maximum, 4 2122 SUPERPOSITION STATE PREPARATION is the field strength and the phase constant 6 is chosen, without loss of generality, as 6 = -w,t,. The full temporal width, at half maximum, of the pulse is 1.665~. Computations below utilized E = 109V m-l, t , = 22, wL = 0.0735 a.u., and pel = I .27 debye. Two pulse widths, at z = 8309 a.u.and 11 000 a.u., were studied. At any time t the wavefunction associated with H, is given by y(t) = co(t)poe-iEot/fi + 2 cn(t)p,e-iEntlfi. (7) n= 1 Substitution of eqn (7) into the time-dependent Schrodinger equation gives the stan- dard set of coupled equatioins for cJ(t) which, within the rotating wave approximation, is given by ikC,(t) = C c,(t) von(t) e-iwnot = pel e 2 c,,(t)f,,cos[m,,(t - t,)]e-(l- rm)2/72 e-%t (8) n=l n= 1 ikkn(t) = co(t) vn0(t) eiwnol = pel~~O(t)fnO cos[coL(t - t,)]e-(t- rJ2/r2 e i o n o r where mno = (En - Eo)/h. Below, eqn (8) is solved numerically to determine cJ(t), with co(0) = 1, hence by- passing approximations associated with perturbation treatments. Comparison with first-order perturbation-theory results, discussed below, indicates agreement between numerical results and first-order perturbation theory to 6% at large t .3.3 THE PREPARED STATE As described, preparation is confined to times when the laser field induces changes in the level populations. Hence the prepared state is comprised of cqn (2) with cl(t) asymptotically constant, The qualitative nature of this state is readily ascertained from perturbation-theory. Justification for using perturbation-theory results to gain this qualitative insight lies in its relative accuracy for the particular conditions dis- cussed above. After expanding cos[w,(t - t,)] in exponentials in eqn (8) and neg- lecting terms of the form exp[&i(o, + cono)(t - tm)] (rotating-wave approximation), a direct application of first-order perturbation theory yields co(t) = 1 cn(t) = (~~2/7~/4ih)p,~f,,~ e'wnotm e-(wt - 0n0)'~'14 equivalent to a form previously obtained by Rhodes.19 For ( t - t,) and t, $ z, eqn (9) approaches the asymptotic limit co(oo) = 1 c,(Q) = (qiel~dn/2ih)fno e - h --W,d2x2/4 e -i(Eo -En)tm/ft, (10) Practical studies show this approximation to be valid, for the systems studied, for t , z 22, t and is no wider in energy than permitted by the frequency distribution, and possibly narrower if the Franck-Condon factors at large IwL - wn,l are negligible. State populations Ic,(co)12 display the frequency distribution of the gaussian laser pulse modulated by Franck-Condon factors.Finally states are prepared with specific phases which may be interpreted as follows: the system is prepared as a coherent super- position state (Le.each state enters with the same phase) at t = t, which then evolves 42. Consistent with expectation, the prepared state is centred at mL,R. D. TAYLOR AND P. BRUMER I23 freely as e - W f - 'm)l'. Such a view is a useful mnemonic but does not, of course, provide a realistic description of the true preparation dynamics. 4. COMPUTATIONAL RESULTS Eqn (8) was solved numericaliy, with c,(O) = 1, for each of the nine cases described above. Analysis of the results was carried out by direct examination of P(t) = I< v/(O)I y(t)}12, c,,(oo), ly(t), etc. A technique to measure quantitatively quantum stocfiasticity in prepared pure states, proposed by Heller,20 was also applied. Space limitations prevent the presentation of more than a sampling of results.Further analysis is provided e1~ewhere.l~ 4.1. FRANCK-CONDON FACTORS As described above, the essential features of the prepared state, as viewed in the vrr representation, arise from the laser frequency distribution and the molecular Franck- Condon factorsf,,. Suggestions 2o have been made to the effect that the nature of the Franck-Condon factor distribution amongst pn states should provide a useful means of distinguishing excitation to regular as compared with irregular states. Similarly, substantially different analytic approximations to fo,, for excitation to regular and irregular states have been proposed,21 leading one to expect differences in fOn in excitation to regular or irregular states. Table 1 provides a summary of a11 Franck-Condon factors contributing to the nine cases under study.Several features are apparent. First, the fon do not display qualitative differences between excitation Table 1. Franck-Condon factors, fo, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 I5 16 17 I8 19 20 21 22 23 -0.012 0.032 -0.01 2 0.01 3 - 0.024 0.003 - 0.019 -0.005 0.01 8 0.006 -0.017 -0.005 0.003 0.014 0.01 8 -0.004 -0.001 -0.001 0.022 0.002 0,002 -0.023 -0.005 -0.019 -0.009 -0.006 -0.016 -0.003 -0.029 -0.019 0.002 -0.003 -0.020 -0.001 0.01 8 0.003 0.010 -0.023 -0.008 -0.008 -0.01 1 0.020 -0.01 3 0.028 -0.003 0.010 0.023 -0.005 0.001 0.032 -0.028 0.01 5 -0.01 6 0.0 -0.029 -0.001 0.002 0.02s 0.021 0.002 0.0 0.01 1 -0.01 6 0.017 0.018 0.024 0.0 -0.032 0.0 0.029 0.024 -0.001 -0.030 0.01 5 0.031 0.020 -0.025 -0.001 -0.017 0.002 -0.005 0.019 0.027 0.030 0.004 0.008 -0.001 -0.01 1 0.006 -0.025 0.030 0.028 0.004 0.014 - 0 .O26 --0.006 -0.01 2 -0.012 0.01 3 0.030 0.001 -0.005 -0.023 0.007 -0.030 0.01 3 -0.020 -0.006 0.024 -0.010 0.01 8 -0.034 0.008 -0.005 0.030 0.018 --0.007 0.001 0.008 -0.015 0.020 0.017 0.0 0.01 7 0.029 -0.006 0.021 -0.029 -0.006 - 0.023 - 0.01 5 -0.025 0.016 -0.008 -0.018 -0.009 0.003 --0.002 -0.032 -0.028 0.017 -0.012 -0.004 -0.012 0.026 -0.024 0.006 -0.010 0.004 -0.003 -0.015 0.016 -0.006 0.026 0.017 0.011 0.019 0.019 0.016 0.003 -0.002 0.026 0.032 -0.009 -0.008 -0.009 0.002 -0.030 0.013 0.002 -0.007 0.013 -0.002 0.006 -0.001 0.0 0.019 -0.030 0.006 0.017 States are indexed in order of increasing energy.124 SUPERPOSITION STATE PREPARATION to the rectangle and to the stadium states.For example, severalf,,, in both the rectangle and stadium cases are negligible compared to adjacent levels. Secondly, the fon values depend, for a given excited state, on the location of the ground-state circle. Thus an insensitivity to circle location in the CS cases is not observed in this detailed examination, via fon, of the excited-state wavefunctions. An immediate consequence of these results is that high-resolution spectroscopy measurements of the intensity distribution amongst neighbouring states, proportional to lfOnI2, will not display obvious qualitative features distinguishing excited states of the rectangle type from excited states of the stadium type. It is unnecessary, perhaps, to emphasize that such eigenstates are visually quite distinct, in displaying recognizable nodal patterns for the rectangles and erratic nodal patterns for the stadium.4.2. P(ly(v/) AND PSTo(v/lry) The survival probability P(t‘) = I(y/(O)ly/(t’))[’ has been used 430~12 as one measure of quantum dephasing. The behaviour of this function, quantitatively examined later below, is qualitatively similar, in all cases, to that seen elswhere.12 That is, P(t’) shows an initial falloff to values close to zero followed by recurrences of varying sizes. Studies of times up to t x 10hp show no recurrences prior to hp, major recurrences for the degenerate rectangle case at t x hp and prominent recurrences in other cases at t z 2hp. The quantity P(y/Iy/), the long-time average of P(t’), plays a major role in a proposed 20*22 means of quantifying quantum stochasticity .In particular, one com- pares P(y/ly/) with PgTO(y/yl y / ) , the latter corresponding to a statistical average which incorporates knowledge of system dynamics only up to time T. Specifically, = 2 (pnv)2 = 2 ( 1 ~ ~ 1 ~ ) ~ ( 1 la) T+w n n Application of eqn (1 lb) with Tchosen as the minimum of the initial P(t‘) falloff, which differs for each case, is discussed below. Adopting this choice ensures that knowledge about the energy envelope of y/ in each of the systems is restricted to the broadest envelope. Additional calculations, with T z hp 30 x lo3 a.u., i.e. a maximum time in Kay’s approach,1° were also carried out. For this T knowledge of the energy envelope is still confined to relatively broad features.In all cases t ‘ = 0 is defined at t = 22 + t,, i.e. where the sizable c,(t) are no longer changing, to within 0.1%. Two comparisons are possible, first of piTo*v with p r , fluctuations of the latter about the former being attributed 2o to non-stochasticity, and secondly of P(y/l v / ) with PtTo(v/Ily). Here P(y/Iy/) 3 PtTo(y/Iy/), deviations of P(y/Iy/) from P$To(y/Iy/) providing a suggested measure of non-stochasticity.20 Fig. 2 displays typical p , and pzTo for a stadium case (CIS) and a non-degenerate rectangle case ( C l R ) with 2 = 8309.5 a.u. and T set at the initial P(t’) minimum for each case. These figures, typical of those obtained, show no substantial qualitative difference between the two cases. Table 2 provides a comparison of P(yIy) and PSTo(y/Iy/) for three C R and CS cases for two different values of z.Examination of the table shows most states to be classified as highly non-stochastic using this criterion, the most nan-stochastic being one of the C2S cases. Although there is some indication that, on the average, CS cases show P(yIy) closer to PSTTo(y/1y/) than do the C R cases, this feature does not persist in comparisons with Tchosen as 30 000 a.u. (not shown) in which the opposite is the case. In light of these results no effort was made to extend the derivation of PsTo to cases involving degeneracies in order to examine CRD cases.R. D. TAYLOR AND P. BRUMER - - lbl - - - (10 O O h o - 0 (1 00 0 0 b O O 0 I 125 0.03 Pn 0.02 0.01 0 0.03 0.0 2 P n 0.01 0 E/ 1 0 - 2 a. u. Fig. 2. Ic,,12 (sticks) and p:To (circles) for (a) C1 S and (b) C2R.Note that the overall gaussian envelope is due to the laser profile. Circles clearly identify all energy eigenvalues. These results indicate that the direct cxarnination of Ic,12, or offon contained therc- in, do not identify strong qualitative differences in the states created in excitation to the regular rectangle wavefunctions or excitation to the irregular stadium wavefunctions. Explicit examination of Iv/(t)12 and of P(t') do, however, reveal differences, as described below. Table 2. P ( I , Y ~ ~ ) as compared with Psro(yilly) c1s c2s c3s ClR C2R C3R c1 s c2s c 3 s C1R C2R C3R z = 8309.5 a.u. 19.0 0.0636 15.5 0.049 1 20.0 0.091 1 13.5 0.0982 14.0 0.1597 23 .O 0.1756 z = 11 000a.u. 22.5 0.1685 21 .o 0.1238 26.5 0.1990 16.5 0.1949 22.0 0.5062 24.5 0.4030 0.1062 0.1062 0.1478 0.1967 0,3352 0.3530 0.2201 0.1975 0.2927 0.3510 0.8472 0.8153 66.9 116.3 62.2 100,3 109.8 101 .o 30.6 59.5 47.1 80.1 67.4 102.3 Dev = lOO(P-- PsTo)/PsTo, measuring thc percent deviation of P from PSTo.126 SUPERPOSITION STATE PREPARATION 4.3.THE SURVIVAL PRORABILITY P(t’) The quantity P(t’) was caIcuIated for each of the cases described above. Here, once again, t ’ = 0 corresponds to time t = 22 -1- t , when the laser pulse is no longer effective in changing state populations. We focus on the details of the initial de- phasing. Perturbation theory suggests, with A a cumdative constant, that We recailpi2 for comparison purposes? properties of P(t’) for the case where Ic,J2 are purely gaussian, i.e.where ]fnOl2 is constant in eqn (12). Under these circumstances P(t‘) = e--1‘21T2 for t’ < hp with a dephasing time t l j e = z [i.e. P(t‘ = tlJ = I/e]. This behaviour would result if the coefficients in the created state simply mimicked the laser frequency profile. Consider then the observed dephasing times, denoted ?l/e(calc,), shown in table 3. The results demonstrate that: (a) for all degenerate Table 3. Calculated dephasing times case tl,,(CdIC.) Dev‘ c1s c2s c3s ClRD C2RD C3RD CIR C2R C3R c1s c2s c3s Cf RD C2RD C3RD C1R C2R C3R T = 8309.5a.u. 8676 7860 8500 6000 5909 6992 6967 8783 8814 z = I 1 000a.u. 11 292 9784 10 546 7096 6985 8489 8865 b 11 384 4.4 -5.4 2.3 -27 - 29 -16 -16 16 6.1 2.6 -11.0 -4.1 -35 -36 - 23 -19 - 3.5 Dev = 100 [rll,(calc) - TIIT; all times in atomic units.’Initial dephasing faIloff not well defined due to interference. rectangle cases t,,,(caIc) deviates substantially from 2 ; (b) for all but one of the stadium cases t,,,(calc) = z to within 6%; ( c ) the nondegenerate rectangle case shows be- haviour characteristic of both the stadium and degenerate rectangle cases. There is, then, a clear propensity for the stadium P ( t r ) to behave, in its initial faIloff, as if If,,]” were constant; i.e. P(t ‘) adopts the falloff imposed by the laser frequency profile. This behaviour is shown more clearly in fig. 3 where the short-time falloff in several representative cases is compared with e--f”lT2, An analysis of the cosine transformR. D. TAYLOR AND P. BRUMER 127 0 5 10 15 t’/103 a.u.20 Fig. 3. Initial P(t’) falloff for C1S ( x ), C1R (0), ClRD ( A ) and C3R (0) with z = 11 000. Solid line is exp(-t’2/t2) with z = 11 000 a.u. P(co) of P(t‘) provides information complementary to that seen in the time represent- ation. That is, t,,,(calc) < z arises from contributions to P(m) at large frequencies, which exceed contributions expected if P(t ’) displayed a purely gaussian falloff. 4.4. THE PROBABILITY DENSITY I v/(t)12 The above techniques provide projected views of the created state and its evolution, best studied either by modelling realistic measurements or by examining 1 ty(t)12. Consideration of the latter does indeed display qualitative differences between states prepared in the rectangle case or the stadium case. Space limitations prevent but the briefest display of these differences, to be discussed in detail e1~ewhere.l~ Fig.4(a) and (b) provide a comparison, typical of those obtained, of the created state Ity(t = t, + 2z)I2 in coordinate space for CIS and ClR, z = 11 000 a.u. Several features are apparent. First, the density is spread, in both cases, throughout the allowed region, no enhanced density being observed in the region of the ground-state circle. This is hardly surprising since effective cancellation in the excited super- position state, to create a localized excited state, can only occur with a sufficiently large number of states. Secondly, the excited-state stadium wavefunction shows randomly distributed regions of slightly heavier probability whereas the excited-state rectangle wavefunction shows a heavy concentration of density in the upper half of the rectangle.Some recognizable aspects of the nodal character of the individually contributing eigenstates are also evident. Fig. 4(c) and (d), showing the same wavefunctions at a time approximately z later, display the principal observed qualitative distinction between excitation to regular as compared with irregular states. That is, the heavy probability in the rectangle moves together, in a rather regular way. In sharp con- trast, density in the stadium simply “ shimmers,” with local regions of higher proba- bility density moving randomly about. We further note that although the motion of128 SUPERPOSITION STATE PREPARATION 1 I I I I I 1 1 1 1 -3 c 0 .- I 0 * c - IR.D. TAYLOR AND P. BRUMER 129 Iv/(t)12 in the rectangle sometimes appears to display a pseudo-period, the P(t’) behaviour in such cases still indicates that v/(t) differs extensively from ~ ( 0 ) . This distinctive difference between excitation to regular and irregular eigenstates should be manifest in, for example, time-resolved emission to relatively localized states. Calculations to examine this feature are in progress.14 5. DISCUSSION It is well known that different dynamical behaviour can be manifest in the same system. For example, distributions display 23 finite time relaxation in classical regular systems and isolated quasiperiodic trajectories exist in classical mixing systems. Thus, in addition to knowledge of the formal system properties, knowledge of the nature of the prepared state is vital. This study has been designed to examine the relationship between system properties and the created state in a specific model experiment.The results make clear that qualitatively different types of states are created in the simple models examined, but that this difference is not readily reflected in several of the quantities examined. We have noted a propensity in the stadium system for P(t’) to assume dephasing characteristics of the imposed laser field, in contrast to the degener- ate rectangle system. This property can be shared, however, by the CR cases. Furthermore, we anticipate that time-dependent emission studies to localized states should be able to distinguish CR type from CS type cases. There are several issues central to current studies of quantum intramolecular dynamics : (a) establishing the quantum behaviour of molecular systems, (b) estab- lishing a formal quantum ergodic theory with ideal systems of some molecular utility and (c) understanding the classical-quantum correspondence in the irregular regime.The aim of this paper has been to contribute to the first two of these topics through a model with ideal classical properties and reasonably well characterized quantum states. The import of our results to realistic molecule systems [(a) above] depends on the extent to which the observed features carry over to real molecules. Barring the selected potentials, some concern might be expressed about the small number of eigenstates contributing to ~ ( 0 ) . In this regard we believe that this regime is, in fact, a useful one for experimental studies of small molecules.Here the number of states is sufficient to observe dephasing and experimental requirements are modest in demanding longer pulses as the density of states increases (e.g. z z 5 ns for p z lo4/ cm-I). Dephasing occurs, as is evident from the calculation, on timescales similar to z, or longer if they,,, distribution is narrow. The relevance of these results for a useful formal quantum ergodic theory is also clear. Any such theory, if it is to follow the spirit of classical ergodic theories, must identify a system as regular, ergodic, etc. Features, such as those reported in this paper, of the prepared states in model systems provide necessary input into such studies.Financial support, provided by N.S.E.R.C. and the Dreyfus Foundation, is gratefully acknowledged. We acknowledge extensive interaction with Prof. M. Shapiro and helpful conversations with Prof. K. Kay and thank Profs. Kaufman and McDonald for providing the stadium eigenfunction program. L. S. Kassel, Homogeneous Gas Reactions (Chem. Cat. Co, New York, 1932). For reviews see M. V. Berry, in Topics in Nonlinear Dynamics, ed. S. Jorna (American Institute130 SUPERPOSITION STATE PREPARATION of Physics, New York, 1978); P. Brumer, A h . Chem. Phys., 1981, 47, 201; M. Tabor, A h . Chem. Phys., 1981,00,000. G. Contopoulos, L. Galgani and A. Giorgilli, Phys. Reu. A, 1978, 18, 1183. E. J. Heller, in PotentiaZ Energy Surfaces and Dynamics Calculafions, ed. D. G . Truhlar (Plenum Press, New Yurk, 1981); J. S. Hutchinsan and R. E. Wyatt, Phys. Rev., 1981, A S , 1567. M. V. Berry and M. Tabor, Puoc. Roy. Soc. London A , 1477, 356, 375; D. W. Nuid, M. L. Koszykowsky and R. A. Marcus, Chem. Phys. Lett., 1980, 73, 269. M. V. Berry, Philos. Trans. R. Soc. London, Ser. A , 1977,287,237; J. S . Hutchinson arid R. E. Wyatt, G e m . Phys. Letr., 1980,72, 384. Utilizing the more general density matrix approach would allow for mixed states which include more averaging than desirabIe for studying pure quantum intramolecular relaxation. ’ The possibility that statistical behaviour implies less sensitivity to details of the creation step is not excluded. lo K. G. Kay, J. Chem. Phys., 1980,72, 5955. l1 See, e.g., J. Jortner and R. D. Levine, Adv. Chem. Phys., 1981, 47, 1 ; E. J. Heller and W. M. l2 f. Brumer and M. Shapiro, Chem. Phys. Left., 1980,72, 528; 1982,90,481. l3 P. Brumer, work in progress. l4 R. D. Taylor and P. Brumer, work in progress ; D. Gruner and P. 3rumer, work in progress. l5 S. Nordholm and S. A. Rice, J . Chem. P h y ~ . , 1975, 61, 203. S. W. McDonald and A. N. Kaufman, Phys. Rev. Lett., 1979,42, 1189. l7 K. F. Freed, C h m . Phys. Lert., 1976,42,600. l8 The coherent transform limited pulse is, of course, an experimental ideal. l9 W. Rhodes, in Radiationless Transiriom, ed. S. G. Lin (Academic Press, New York, 1980). 2* E. J. HelIer, J. Chem. Phys,, 1980, 72, 1337. 21 W-K. Liu and D. M. Noid, Chem. Phys. Lett., 1980,74, 152. 22 A criterion similar in appearance to this equation has been proposed by Kay.” However, as See K. G. Kay, Towards a ’ H. J. Korsch and M. V. Berry, Physica, 1981, 30, 627. Gelbart, J. Chern. Phys., 1980, 73, 626. formulated, it is not expected to be applicable to pure states. Cumprehensiue Semiclassical Ergodic Theory (pr e pr in t ) , 23 C. Jaffe and P. Brumer, to be published.

ISSN:0301-7249    

DOI:10.1039/DC9837500117

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraduy Discuss. Chem. Soc., 1983, 75, 131-140 A Quantum-mechanical Internal-collision Model for State-selected Unimolecular Decomposition BY BRUCE K. HOLMER Theoretical Chemistry Institute, 1 I02 University Avenue, Madison, Wisconsin 53706, U S A . AND MARK S. CW~LD Theoretical Chemistry Department, University of Oxford, 1 South Parks Road, Oxford OX1 3QZ Receiued 29th December, 1982 A quan turn-mechanical theory for state-selected unimolecul ar decomposition is developed in terms of the cumulative effect of a succession of internal collisions. An application to the model decomposition of C 0 2 to CO('Z+) + O('D) is developed within the restrictions of a forced harmonic oscillator approximation for the internal scattering matrix. Preliminary results over a restricted energy range yield a distribution of lifetimes over two orders of magnitude, 0.640 ps, with evidence of a statistically determined behaviour on average but with wide fluctuations and with significant quantum effects on the branching ratios to different product channels.1. INTRODUCTION poses the question as to whether the statistical outcome arises from a statistical distribution of decomposing states or whether the decomposition of individual states is itself statistical or rnode-specifi~.~~~ Tn other words, are there close similarities or sub- stantial differences between the lifetimes and fragment state distributions arising from different quantum states in a narrow total energy region? Our approach to the problem is similar to that of Waite and Miller: although there are some differences in the formdation of the theory and the chosen applications are quite different.The central idea for both theories is the physically appealing one that decomposition arises from a sequence of internal collisions between the latent decomposition fragments. The quantity that expresses this idea is an internal S matrix which is introduced in such a way as to allow the imposition of different boundary conditions for open and closed channels. This results in fact in a unified description of bound, resonant and scattering states, so that one might hope in the future to extrapolate from knowledge of the bound states into the continuum. The present computational advantage of the scheme is that one can take over known and tested approximations for single-colIision processes and apply them in the multiple collision or resonant context. Waite and Miller adopt a semiclassical perturbation 10s approximation to describe leakage from the Hhon-Heiles potential.We adopt a forced harmonic oscillator model closely akin to the TTFlTS approxi- mation,1° and apply it to a system chosen roughly to mimic the decomposition of Results are presented for total energies between the CO(u = 2) and CO(u = 3) The success of existing statistical theories of unimoIecular decomposition co* to CO(%+) + O('D).132 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL dissociation thresholds, over which range there are 3 open and 20 accessible closed channels. 28 resonances with lifetimes between 0.6 and 60 ps were detected, but there may be as many as 15-20 others with much longer or much shorter lifetimes.In analysing their behaviour we ask first whether the magnitude of the lifetime is consistent with free-energy transfer between all channels. The answer is yes for the levels with widths in the middle of the range, but not for all levels because the range of lifetimes covers two orders of magnitude. Secondly do the resonant eigenstates show strong interchannel mixing? The answer is definitely yes, to the extent that it is difficult to give any meaningful assignment to most resonances. Finally, are the branching ratios between open channels statistically determined ? At first sight the answer appears to be no, but closer inspection suggests a pattern consistent with quantum-mechanically determined fluctuations about a statistical classical mean.The conclusions are that the decomposition dynamics for this model are statistically determined only in order of magnitude and to the extent that quanta1 interferences are ignored. Test calculations with the mean internal energy transfer reduced by an order of magnitude show clearly mode-specific behaviour. 2. THE INTERNAL COLLISION MODEL It is assumed that the dominant internal collisions occur within a radius R,, which lies inside the outermost classical turning point for all relevant closed channels (see fig. 1 later). This justifies the introduction of an internal S m a t r i ~ , ~ s, defined such that thejth component of the ith internal scattering state behaves as y/ji)(R)RzR, k j - *(Si je - i ~ j ( R ) - S.CJ .eiV,(R)) (1) where qj(R) is defined as the JWKB phase integral R qj(R) = kj(R)dR + n14 a/ a, being the inner turning point for channel j and [ glj[ therefore gives the probability of transition from channel i to channel j in a single collision. Multiple collisions are introduced by constructing linear combinations of the scattering solution + ( i ) with components ylf) and choosing the coefficients B a i consistent with the existence of closed channels, denoted collectively (yk(R)). The correct JWKB behaviour for such channels is Vk(R) Z Ckkk' Sin (&(R) where, with bk as the outer turning point, R @k(R) = lk kk(R)dR - ~ 1 4 . This is conveniently written for later manipulation as (5)B. K. HOLMER AND M. S . CHILD 133 where ctk is the complete JWKB phase integral Consistency between eqn (I), (4) and (5) now implies the following conditions on the coefficients B a i , Bake-i,"k = - 2 B,, s,, ei% (9) i The number of such equations is the number of closed channeIs.The remaining conditions are imposed by requiring that different solutions +(") are incoming in particular channels, a = I say. Thus with x replaed by l, the solution @ is defined such that BZi = &Ii (10) and the sum in eqn (9) is conveniently divided into open (i = l') and closed ( I = k') channel parts : (11) BIke-jak = - 2 BI,, skrkeiak - 2 61,. S,,eiak. El (I + C) = k' I' This is equivalent to the matrix equation (12) (1 3) where b, and Gl have components - b - B e-ia I k - [k k ) g l k = - SIkeiuk and C is a closed-channel matrix with elements ZVk = eiak'gk&ak.It follows from eqn (12) that bI = ml(i + E)-'. yif)(R) Rz" kF+(6,,. e b P ) - SI,.ebi(W} (15) (16) (17) This means on back-substitution from eqn (I), (13) and (15) in eqn (4) that where the true scattering-matrix elements S,, are given by sIl, = ,!Tile - 2 ei4 T',. (I + ~ ) - 1 ,JlkO eirk. hk' The first term corresponds to direct and the second to resonant scattering. This is the form appropriate to a scattering application of the theory. For analysis of the decay properties of the resonant states it is more convenient to replace the incoming boundary conditions [eqn (lo)] by the Seigert I1 boundary condition Bmj = 0 for all open channek i = 1. det (e2ir + see) = 0 (18) (19) The terms dli, in eqn (1 1) then vanish, leaving an eigenvalue equation where 9' is the closed-channel part of s.energies l1 The solutions of eqn (19) are complex E = E, - i r,l2 (20)134 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL specifying the positions, En, and widths, r,, of the resonant states. Furthermore the corresponding eigenvectors, conveniently labelled B,, with components Bnk, determine the outgoing amplitude in open-channel wl, because according to eqn (1) and (4), with Bli = 0, yln)(R) - kl - +TnleiW) (21) with The partial decay probability into channel I is therefore giving a partial decay width It may also be noted that if all channels are closed, sCc includes the entire internal S matrix, the unitarity of which ensures that the eigenenergies are real, because the eigenvalues of a unitary matrix are complex numbers with modulus unity and the complete phase integrals ctk are real at real energies.Thus the theory encompasses bound states as well as direct and resonant scattering. 3. UNITARISED FORCED HARMONIC OSCILLATOR APPROXIMATION As noted by Waite and Miller the strength of present formulation lies in our ability to approximate the internal scattering matrix s, because an exact numerical solution for 5: would be only slightly less time consuming than for the full S matrix given by eqn (17). Waite and Miller use a semiclassical perturbation 10s approxi- mation, but an attractive alternative to handle interactions between the stretching modes of a molecule is the impulse approximation9Pl0 whereby the motion of one oscillator (treated as harmonic) is linearly forced by motion of the other.The s matrix is given in this approximation by where the two parameters E and S are given by E = (2phc0)" F(t ')e'wt'dt '1 I I-, m t and S = (2phco)-' ,/ F(t)F(t')sinco(t - t')dt'dt --co -03 p and co being, respectively, the reduced mass and frequency of the oscillator and F(t) the time evolution of the forcing term. A practical problem in implementing the theory is that s must be unitary in order that the eigenvalues should be real for a fully closed-channel problem. s given by eqn (25) is, however, an infinite matrix, so some truncation is required. Furthermore in the way that the theory is applied the forcing function F(t) differs for each transition, because the time dependence of F(t) arises from the channel velocity which may differ widely from one channel to another.B.K. HOLMER AND M. S. CHILD 135 A convenient unitarisation procedure appropriate to the search for complex eigen- values by solution of eqn (19) is to recognise that the unitary of S for real E generalises to S(E)S(E*)~ = I (28) when E is complex. Thus if so@) is the approximation obtained by truncating s ( E ) , an appropriate form for the theory is Sapp = w(E*)] [T(E*)S(o)(E)T(E*)]3[T(E*)] (29) where T(E*) = (fS'"(E*f]")*. This is preferable to the R matrix procedure adopted by Waites and Miller,B in that it more accurately preserves the magnitudes of individual s matrix elements. 4. MORSE-DRIVEN HARMONIC-OSCILLATOR SYSTEM Any system chosen for implementation of eqn (25)-(27) in the context of uni- molecular dissociation must give rise to both open and closed channels and must involve coupling to a harmonic oscillator.A simple example is the ccupled harmonic1 Morse oscillator sys tern, with hamiI tonian where = r - y(x + xeq); y = m c l h + m,). This reduces under the substitutions x = (h*k-*pBC-*){, r = q/a to (33) where or equivalently where s = '7 - pt. The extraction of an approximate linearly forced harmonic oscillator from this Hamiltonian folIows established The first step is to ignore the cross-term pspc in eqn (35) and to soIve the resulting Hamiltonian equations for the time evolution of136 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL The result may be written u(t) = (1 -!')/{I -f3cos[(l - f ' ) + f i t ] } for O <f< 1 = (f- l)/{f%osh[(J'- l)%t] - 1) forf > 1 (37) where ij2 = 2dP2 (m -t l)/m, f = (E,/D) (38) Em being the Morse energy, measured from the minimum of the channel potential in question.The ranges 0 < f < 1 andf > 1 therefore correspond to closed and open channels, respectively. The next step is to linearise the potential in eqn (34) and to substitute for u(t) from eqn (37) to find F(t); thus V/hm z - l( $) = - (F(t) (39) where F(t) = - 2dD[u2(t) - ~ ( t ) ] . (40) This is the function that determines the parameters E and 6 given by eqn (26) and A cruder approximation (27). F(t) = - 2dPG(l - It]/z), T < It] = 0, z > Itl. was, however, adopted for the present exploratory purposes, with G =f+(1 +f3) z = arc cosf'/[fi(l -f)+] 0 < f < 1 = h[f+ + (f- l)+]/[fi(f- I)+] f'> 1 (42) and f + = ($2 +hW f" = E,"/(m + 110.(43) (44) Heref, and Emv refer to the vth channel The resulting expressions for E and 6, relevant to the calculation of &, are E == 8d2p2(G/T)' (1 - COST)' (45) and 6 = b2pZG2[72 3 + 12(cosz - sinT/z) - 3sinzI. (46) 5. NUMERICAL APPLICATION Parameter values for a test calculation were chosen roughly to model the uni- molecular dissociation of CO, to CO (%+) + O('D) : D, = 43 980 ern-', kw, = 1868 cm-I and h ~ o o - 2 1 7 0 ern-'. The first two imply a Morse exponent a = 3.461 and anharmonicity hcu,x, = 19.84 cm'l. Thc corresponding dimcnsion- less parameters appearing in section 4 are d = 20.3, p = 0.0772, d = 0.860 and m = 0.485.B. K. HOLMER AND M. S . CHILD 137 Detailed calculations were performed for 4340 < < 6510, with E measured from the dissociation threshold for CU (v = 0).Over this range there are three open channels, u = 0, 1,2, and 19 accessible closed channels, i.c. with potential minima < E. The closed-channel potentials and zero%-order eigenvaIues are shown in fig. 1 and typical internal collision probabilities lS,,,l* for u = 0, 1, 2 are illustrated in fig. 2. The latter vary very little over the energy range, because its span of 2000 cm-l is small compared with the maximum kinetic energy of 35 000-40 000 crn-l. Mean 6500 6000 5 500 5000 -4500 -0.25 0 0.25 0.5 0.75 1.0 1.25 20 40 R - RJA 3 I E -+ k Fig. 1. ChanneI potentials and zeroth-order eigenvalues between the CO ( u = 2) and (u = 3) dissociation thresholds. Calculated resonance energies €, and widths r,, are indicated to the right of the diagram.The symbol * indicates levels for which information is given in table 1 and fig. 3. energy transfers of roughly 10 CO vibrational quanta are therefore typical for tran- sitions to and from the open channels. This corresponds to E E 10 in eqn (25) and (26). The maximum kinetic energy in the highest channels is, however, reduced by an order of magnitude; eqn (42) and (45) show that this reduces E to roughly unity, with a corresponding reduction in the mean energy transfer to one CO vibrational quantum. Solutions of eqn (19) for the resonance positions and widths were obtained by scanning the real energy rangein steps of 2 cm-' followed by a Newton-Raphson search in the neighbourhood of the apparent resonances, a procedure that yielded the 28 resonances with energies En and widths I?, indicated to the right of fig. 1.The range of widths is 2.6-55.6 cm-' with a mean width r z 9 cm-'. It is evident, however, that the number of detected resonances is considerably less than the number of zeroth- order states. Further resonances with either substantially larger or substantially smaller widths may therefore be expected. Detailed resuIts for six channels are given for illustrative purposes in table 1 and fig. 3. Table 1 shows the resonance energy En, with r,, and branching ratios Pno,138 0.1 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL - I.= 2 I I I I I , t k Fig. 2. Internal single-collision transition probabilities I Slkl * for scattering into the open channels 1 = 0, 1 , 2 at E = 5577.22 cm-'.The pattern varies very little with energy. 1 and k refer to the CO vibrational label u. Pnl and Pn2 between the three open channels; fig. 3 gives the square moduli IBnkI2 of the Seigert state eigenvector components. The first point of interest is that the lifetimes 7, x hjr,, (47) consistent with the linewidths shown, span the physically plausible range 0.1 < zJps < 10. This gives some confidence in the practical relevance of the model. Secondly the magnitude of I' given by the crude formula r x ( ~ Q ~ E ) P (48) where 6 is the classical Morse oscillator frequency which governs the rate of internal collisions, may be used to estimate the probability P of predissociation per collision. With hG taken as 600 cm-l, which is the mean zorse energy separation relevant to the states shown in fig.1 and r taken as the mean r = 9 cm-', this results in P x 0.09, which may be compared with the statistical probability of 0.13 for scattering from one of a total of 23 channels into one of three which are open. Thus the mean behaviour is approximately statistical, although there is a substantial spread from P x 0.006 to P x 0.6 implied by the range of observed level widths. TabIe I. Decomposition characteristics of selected resonances a 4378.88 22.36 0.41 3 0.300 0.288 b 5218.89 55.68 0.463 0.016 0.536 C 5577.22 2.54 0.286 0.099 0.616 d 5828.94 11.72 0.837 0.059 0.104 e 6249.15 4.92 0.091 0.275 0.634 f 6503.67 0.57 0.062 0.593 0.345B. K. HOLMER AND M. S. CHILD 139 0.2 0.2 0.2 N - U c Qi 0.2 Further evidence of broady statistical behaviour is shown by a roughly uniform distribution over the closed channels for a11 calculated resonances, as illustrated in fig.3. A few cases [e.g. fig. 3(e)] show a predominance of one particular channel, but a more uniform pattern is the norm when all 28 calculated resonances are included. By contrast, the pattern of branching ratios given in table 1 appears to deviate - - ( f 1 I 1 1 L.1 I 1 . 1 1 - ( e l 1 1 . t , l . . . ! . . I - ( d 1 1 . . I I . . ! I 1 1 . 1 . IC) - I , 1 t l I 1 I , 0.2 1 k Fig. 3. Closed-channeI eigenvector components lBnk12 for the selected resonances (a)-(f) detailed in table 1 . markedly from the statisticaI result P,, = P,, = Pn2 = 1/3. This may, however, be a quantum effect. If ITnl12 in eqn (23) is replaced by lL12 = 2 p n k I 2 l L 1 * (49) k which involves ignoring cross-terms in BnL3*nk.&kS1*nk‘, this has a classical interpre- tation as the decomposition probability ISkr\* to channel I, weighted by the strength lBnkI2 of channel k in the eigenvector.Branching ratios calculated by means of eqn (49) are much closer to the statistical values. For the resonance at En = 5577.2 cm” for example, the values Pno = 0.29, P,, = 0.10, Pn2 = 0.62 given in table 1 are re- placed by 0.37, 0.35 and 0.28. The overall conclusion is that the resonances investigated have physicaIly realistic lifetimes and decompose at the mean in a broadly statistical manner. The range of140 QUANTUM-MECHANICAL INTERNAL-COLLISION MODEL lifetimes spans two orders of magnitude however, and non-statistical quantum corrections to the open-channel branching ratios are apparent.Finally it should be noted that owing to resolution difficulties the calculation revcaled only 28 out of possibly 47 resonances and the behaviour of the missing resonances may well be difl'erent. A few test calculations were also performed with much smaller energy transfers, by reducing the coupling parameter p from 0.0772 to 0.007 72. The resulting resonances, being only slightly shifted from the zeroth-order eigenvalues, were then much more easily detected. Correspondingly the resonance eigenvectors showed one or two dominant components. Their widths were roughly comparable with the stronger- coupling model for states in the lowest channel, D = 3, (0.4 < . T,/crn-l .: 23.6) but fell off sharply for more buried resonances. Finally the decomposition ratios typically gave a weight of 0.9 to the highest open channel, u I- 2. This shows that an order-of- magnitude change in the mean energy transfer per internal collision causes a complete change from statistical to highly selective decomposition. J. P. Robinson and K. A. Holbrook, Unimolecular Reactions (Wiley-Interscience, New York, 1972). W. Forst, Theory of Unimoleculur Reurtions (Academic Press, New York, 1973). K. A. Reddy and M. V . Berry, Chem. Phys. Lett., 1979, 66,223. R. Naaman, D. M. Lubman and R. N. Zare, J. Chem. Phys., 1979, 71, 4192. R. B. Hall and A. Kaldor, J . Chem, Phys., 1979, 70,4027, B, A. Waite and W. H. Miller, J , Chern. Phys., 1980, 73, 3713; 1981, 74, 3910, ' B. D. Cannon and F. F. Crirn, J + Chem. Phys., 1981,72, 1752. ' B, A, Waite and W. H. Miller, J. Chem. Phys., 1982, 76, 2412. M. S. Child, Molecular Collision Theory (Academic Press, London, 1974). lo F. E. Heidrich, K. R. Wilson and D. Rapp, J. Chem. Phys., 1971,54, 3885. I i A. J . F . Siegert, Phys. Rev., 1939, 56, 750.

ISSN:0301-7249    

DOI:10.1039/DC9837500131

出版商:RSC    

年代:1983

数据来源: RSC