Picosecond Dynamics of Intramolecular Vibrational-Energy Redistribution (IVR) in Jet-Cooled Molecules Citation Felker, Peter Mark (1985) Picosecond Dynamics of Intramolecular Vibrational-Energy Redistribution (IVR) in Jet-Cooled Molecules. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/x48p-n096. https://resolver.caltech.edu/CaltechETD:etd-04072008-152253 Abstract Theoretical and experimental results concerning the dynamical manifestations of intramolecular vibrational-energy redistribution (IVR) in the temporally resolved fluorescence of jet-cooled molecules are presented. A first concern is with the derivation of the characteristics of the beat-modulated fluorescence decays which arise from multilevel vibrational coupling within a molecule. Relations connecting quantum beat frequencies, phases, and modulation depths to coupling parameters are presented. Likely sources of deviations of experimental results from theoretical predictions are considered. And, the direct link between IVR and time-resolved fluorescence experiments is discussed with emphasis placed on the physical interpretation of vibrational quantum beats and the nature of IVR as a function of vibrational energy in a molecule. Secondly, the results of picosecond time-resolved measurements of IVR on jet-cooled anthracene at different excess energies are reported and analyzed. The nature of IVR as a function of vibrational energy, the relevant timescales for the process, and the details of pertinent vibrational couplings are determined. Thirdly, experimental and theoretical results pertaining to the role of molecular rotations in IVR are presented. Finally, experimental results on jet-cooled t-stilbene are reported and analyzed with the aim of assessing the generality of our theoretical treatment and the anthracene results. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemistry) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Awards: The Herbert Newby McCoy Award, 1983 Thesis Availability: Public (worldwide access) Research Advisor(s): Marcus, Rudolph A. Thesis Committee: Marcus, Rudolph A. (chair) Zewail, Ahmed H. McKoy, Basil Vincent Anson, Fred C. Defense Date: 4 December 1984 Funders: Funding Agency Grant Number NSF DMR-8105034 NSF CHE-821135 IBM UNSPECIFIED Record Number: CaltechETD:etd-04072008-152253 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-04072008-152253 DOI: 10.7907/x48p-n096 Related URLs: URL URL Type Description https://doi.org/10.1063/1.448246 DOI Article adapted for Chapter 3. https://doi.org/10.1063/1.448247 DOI Article adapted for Chapter 4. https://doi.org/10.1063/1.448248 DOI Aritcle adapted for Chapter 5. https://doi.org/10.1063/1.448249 DOI Article adapted for Chapter 6. https://doi.org/10.1021/j150669a011 DOI Article adapted for Chapter 7. https://doi.org/10.1016/0009-2614(83)87375-8 DOI Article adapted for Appendix I. https://doi.org/10.1103/physrevlett.53.501 DOI Article adapted for Appendix II. https://doi.org/10.1016/0009-2614(84)85196-9 DOI Article adapted for Appendix III. ORCID: Author ORCID Felker, Peter Mark 0000-0002-4907-7231 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1297 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 08 Apr 2008 Last Modified: 20 Nov 2025 00:43 Thesis Files Preview PDF (Felker_pm_1985.pdf) - Final Version See Usage Policy. 8MB Repository Staff Only: item control page

PICOSECOND DYNAMICS OF INTRAMOLECULAR VIBRATIONAL-ENERGY REDISTRIBUTION (IVR) IN JET-COOLED MOLECULES Thesis by Peter Mark Felker In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1985 (submitted December 4, 1984) ii ACKNOWLEDGEMENTS Being a believer in a creative, loving God, I also believe that all that is of worth in this thesis has come into being through His power. For this and for the joys of scientific research, I am thankful. I am greatly indebted to many people who have contributed directly and indi- rectly to the work reported on in these pages. It has been a pleasure to be under the supervision of Professor Ahmed Zewail. I thank him for his sound direction, his contagious enthusiasm, his drive to attack difficult scientific problems, and his willingness to treat his students as colleagues. If I had to do it over again, I would gladly choose to work under him. I have been substantially aided by interactions with many members of the Zewail research group. In particular, I thank Bill Lambert, whose experimental guidance has been invaluable to me. I also thank Tina Wood for typing numerous manuscripts. I have benefitted in many ways from friends and institutions at Caltech. I thank my cronies at the Lambert memorial table: Dean Neikirk, James Garvey, Wyman Williams, and Don Caldwell for uncountable fun-filled hours. I thank my old roomate Rich Durand for being such a self-sacrificing friend. And, I acknowledge the importance of the Caltech Men’s Glee Club in providing a valuable escape from the everyday pressures of research. . My family has never failed to be supportive of my career as a perpetual student. For this and for more things than can be written down or expressed, I thank my wife, my parents, my grandparents, and all the members of my family. Finally, I thank again my wife Patti for her love and for her patience, even in the times when I slip from doing science, to being obsessed with science. Financial support for this research was made available through National Sci- ence Foundation grant nos. DMR-8105034 and CHE-8211356 awarded to Professor Ahmed H. Zewail. I am grateful to IBM for their support in the form of a fellowship. To my parents, David and Mary Anne Felker iv ABSTRACT Theoretical and experimental results concerning the dynamical manifestations of intramolecular vibrational-energy redistribution (IVR) in the temporally resolved fluorescence of jet-cooled molecules are presented. A first concern is with the deriva- tion of the characteristics of the beat-modulated fluorescence decays which arise from multilevel vibrational coupling within a molecule. Relations connecting quan- tum beat frequencies, phases, and modulation depths to coupling parameters are presented. Likely sources of deviations of experimental results from theoretical predictions are considered. And, the direct link between IVR and time-resolved fluorescence experiments is discussed with emphasis placed on the physical inter- pretation of vibrational quantum beats and the nature of IVR as a function of vibrational energy in a molecule. Secondly, the results of picosecond time-resolved measurements of IVR on jet-cooled anthracene at different excess energies are re- ported and analyzed. The nature of IVR as a function of vibrational energy, the relevant timescales for the process, and the details of pertinent vibrational couplings are determined. Thirdly, experimental and theoretical results pertaining to the role of molecular rotations in IVR are presented. Finally, experimental results on jet- cooled ¢-stilbene are reported and analyzed with the aim of assessing the generality of our theoretical treatment and the anthracene results. v TABLE OF CONTENTS 1, Introduction ........... cc cece cece cece e eee eee e enone eee ee tees cece eeesenenes 1 References ...... ccc c ccc trace eee r cece e eee nner ee ete eereeee sees eeeceerees 7 2. Experimental ......... 0... cece cect cece ete n eee e teen een e ee te ee eneeeeeaneees 8 I. Cooling by Free-jet Expansion ............ cece ccc ce cece ceeceecteceeeeees 9 II. Measurement of Fluorescence Decays by Time-Correlated Single Photon Counting (TSPO) ........... ccc cece cece eee cnet een eeeeeeeerenees 17 IIL. Experimental Apparatus .......... cece cece cece eee te cece eeeenee cence 22 References .......cccccccc sete cece cen e eee ee en cece eee ebe eee ce eeesneeeees 28 Figure Captions and Figures ............. cscs cecec ence ce cteeeeteeeeeeeeee 29 3. Coherence Effects and IVR ..... 2... ccc cece cece c cree c eee e eee ee ence eeeeaee 32 L. Introduction 2.0.0.0... cc cece ccc cece eee renee eee cere neces eee ee seeees 33 TL. Theory 2.0... . cece ccc cece cece ee eee cece cece eect eee e eee tent eesneee 35 IIL. Specific Results for N <5 ..... cece ccc cece cece teen ee eeeeceees 42 IV. Quantum Beat Parameters and the Determination of Ho ............. 45 V. Experimental Results vs. Theoretical Predictions ...................++- 48 VI. Phase-Shifted Quantum Beats and IVR .............. ccc cece eee eee 57 References ....... ccc ccc cece cece ener e ence eee be ne ee eee t et eeneeeeenes 65 4 9 3 (a 67 Figure Captions and Figures ............. 0. ccc cece cece cece eee e eer eeeeees 70 4. Energy Dependence of IVR in Anthracene ............... cece eee eee e ee eee 75 I. Introduction 2.2.2.0... cc cece cece cece eee eter e tee ee beets sence eeenees 76 IL. Experimental ............ 0c cece cece cece cence cece cence eee n eee eteenee 77 IIL. Low Energy Region ........... cc cece cece cence cece eevee ete eeeereeeeers 82 IV. Intermediate Energy Region .............. cece eee cece ee eee cece eee ceeee 84 V. High Energy: Ey = 1792 cm7! ... oe eeece ee cee eee eee neces 99 VI. Discussion and Conclusions ............... ccc ce eee e cece en ee ne eeenees 103 References 2.2.0.0... cece ccc eee ence cence eee esate cece ee see eee eeeeees 109 6 5) = lil Figure Captions and Figures .............. cece eee eee cece cence cee ceeeee 113 5. Rotational Effects on IVR ........... eee ccc cece cece cece eee e nee eeeceees 143 T. Introduction .......... cece cece cece cece eect eee ee te ence ee ee nese eeeeees 144 Il. Theoretical Simulations ..............0cc cece cece eect eee e ee eee teenies 146 TIT. Experimental .......... 00... c cece cece cece eee eect enero ete ce eeeeees 151 IV. Results 12.0.0... . ccc cece cece eee eee eae e cece ce ee ec cesses cece eeeeees 153 V. Discussion .......... cece ccc cece cert eee e ence ec ee tence eeeeeeeeeeeesees 155 References 2.0... ccc ccc cert cece ee tee eee eee eee e eee ee tee tense eens neeees 161 Figure Captions and Figures ..............-. cece cece cece ee ec er eeeeeeees 163 6. Energy Dependence of IVR in #-Stilbene .............. 0. cece cece eee eens 175 I. Introduction .......... cece ccc cece ence ener e eee e teen teeter eneeees 176 Il. Experimental 2.0.0.0... .. 0c cece c cece cece cece ee tec nee teen eee e ee ennes 178 THT. Results 2.0.0... . cece e cece ete eee tee e ee eee ce tee tee eeneeenes 179 TV, Discussion 2.0.0.0... cece cece c cece cece eee e eee ee eee ne eens sees eneees 185 V. Comclusions .......... ccc cece cece ect e cece reece ence ce eeeeeeteneeeenees 191 APPendix 20... . cece cece cee cee cere eee e tenet ene ee ee ee eee eeeeees 192 References 0.0.0... cece cece cece cece eee eee e eee ee eee eee eee eeateeeeees 193 09 1) Cc 194 Figure Captions and Figures .............. cc cece ccc e cece tect nceaeteees 196 7. Mode-Specific IVR: Direct Picosecond Results ............. ccc cece e cee eees 206 References 2.0... . cece cece bee ee cnet eee nett ee ceeeeeetnseeeeees 211 Figure Captions and Figures vi Appendix I — Observation of Restricted IVR in Large Molecules: Quasi-Periodic Behavior, Phase-Shifted, and Non-Phase-Shifted Quantum Beats 0.2.0... e ccc nec c cece teen cece cvencetescenceceseeeeees 215 Appendix IT — Direct Observation of Nonchaotic Multilevel Vibrational Energy Flow in Isolated Polyatomic Molecules .................-. 223 Appendix III — Direct Picosecond Time Resolution of Dissipative Intramolecular Vibrational-Energy Redistribution (IVR) in Isolated Molecules ............cccceccccctecceccucecccucteestecuveeucsucerees 228 CHAPTER 1 INTRODUCTION 2 Among the most fundamental of current problems in chemical physics is the characterization of dynamical processes in individual molecules effectively isolated from external perturbations. Such intramolecular processes, which arise following the excitation of a gaseous molecule, may be expected to be markedly dependent on the nature of the initially excited state. In large part, it is this expected dependence of dynamics on initial state that provides the justification for detailed studies of these processes, for it implies that one may be able to intimately control the physics and/or chemistry of a molecule simply by varying excitation parameters. Thus, for the past twenty years experimentalists and theorists have been interested in developing propensity rules by which one can predict intramolecular dynamics given knowledge of molecular parameters and the initially prepared state. Without question, and perhaps not surprisingly, the greatest success in this effort has been at the coarsest level of molecular level structure—that is, at the level of the electronic state. The reasons for this success are many, but for the most part are related to: 1) the relatively small number of relevant electronic states, 2) the success which theories of electronic structure have had both in characterizing these states and their couplings to other electronic states, and 3) the fact that the rates of the relevant processes can be directly measured by experiment. For the next coarsest regime of molecular level structure, the vibrational regime, the characterization of intramolecular processes is not nearly as complete. This rela- tive lack of knowledge of vibrational effects compared to the knowledge of electronic effects on intramolecular dynamics is due primarily to the larger number of relevant states, less precise knowledge of relevant couplings, and experimental difficulties in measuring vibrational state-specific rates. But it is precisely at the vibrational level where one might expect initial state-dependent processes to yield the greatest Measure and variety of control over subsequent dynamics. For chemical processes this expectation derives from the nature of molecular motions as distortions of chemical bonds, which implies that different vibrational motions lend different re- activities to the molecule. For photophysical processes, Franck-Condon factors and 3 promoting vibrational modes are expected to be intimately involved in determining rates. Numerous processes such as multiphoton dissociation, laser-selective chem- istry, radiationless transitions, and unimolecular reactions may therefore be strongly dependent on the vibrational state initially excited. At the heart of the question of vibrational state specificity in intramolecular dynamics is the matter of intramolecular vibrational energy redistribution (IVR).? This is the process by which energy that is initially localized in a particular -vi- brational motion redistributes in time such that different vibrational motions gain energy at the expense of the initial motion. One can readily see that the extent to which this redistribution occurs and the timescale on which it happens has direct bearing on vibrational state-specific processes. For if IVR spreads energy over a large number of vibrations in a time short compared to other intramolecular decay processes, then memory of the initial state will be lost and with it the possibility of state-specific dynamics. Thus, insofar as vibrational character influences molec- ular dynamics, the characterization of IVR is essential to the understanding of the dynamics. It is with the aim of contributing to this characterization that the work comprising this thesis is presented. Experimental approaches to the study of IVR typically have tended to fall into one of two classes of experiments;?* the measurement of unimolecular reaction rates or the steady-state measurement of molecular emission upon excitation. In the former approach information concerning the IVR process is obtained by com- parison of the observed rates of gas phase reactions with rates calculated assuming instantaneous and complete vibrational energy randomization subsequent to exci- tation of the molecule. In many cases the predictions of such “statistical” theories (e.g., RRKM theory*) closely match observed behavior, which agreement implies rapid and extensive IVR. The latter approach relies on the fact that the spectral ‘characteristics of emission from a molecule will depend intimately on the vibra- tional character of the molecular excited state. If one prepares a gaseous molecule in a well-defined vibrational state and subsequently observes emission bands which 4 would not be expected to arise from this initially prepared state, then some IVR process can be inferred. Moreover, a “rate” can be calculated for this process by comparing the intensities of expected emission bands (so-called vibrationally unre- laxed emission) and unexpected emission bands (vibrationally relaxed emission). Until the development of seeded supersonic jets for molecular spectroscopy both of the above-mentioned general methods of studying IVR processes were seri- ously handicapped by the requirement that the gaseous samples be at temperatures on the order of, or greater than, room temperature just to achieve vapor densities large enough to render experiments possible. For reasonably large molecules (e.g., napthalene and anthracene) at such temperatures, the thermal distribution of rovi- brational levels is composed of a large number of states with significant population. The situation makes it all but impossible to excite well-defined initial states, even with narrow band light sources. Thus, experiments done in the gas bulb generally suffer severe ambiguities in interpretation due to this thermal congestion effect. The use of ultra-cold gaseous samples generated by free-jet expansion‘ can eliminate much of the ambiguity associated with thermal effects while still providing a sample in which the molecules of interest are effectively isolated. In the past decade or so, these features of jet-cooled samples have been extensively exploited in the study of IVR by both unimolecular reaction rate measurements® and emission spectroscopy” (or combinations of both). The results of these studies have been very fruitful in contributing to the characterization of [VR processes. Nevertheless, it has been apparent that deficiencies are present in these jet experiments as well. These deficiencies arise from the fact that the measurements made do not directly monitor the IVR process in time, but instead are time-integrated quantities that can only be related to IVR indirectly. In an effort to directly study the IVR processes of large molecules in the time domain, we have been led to use picosecond spectroscopic techniques to make mea- surements on molecules cooled by free-jet expansion (the so-called “picosecond-jet” technique*). In this manner we have hoped to eliminate both the ambiguity asso- 5 ciated with thermal congestion (although it must be noted that even in the jet at very low temperatures, significant rotational thermal congestion persists for large molecules) and the deficiencies of time-integrated experiments. The key features, therefore, of this experimental technique are the relative absence of thermal conges- tion in the molecular samples (and the concomitant ability to prepare well-defined initial states) and the capability of observing dynamical processes on a picosecond timescale. The specific work which is reported on in this thesis involves the use of the picosecond-jet technique to measure the time- and frequency-resolved fluorescence of two molecules as function of initially prepared vibrational level in their first excited singlet electronic states (S,) and in this manner to directly probe IVR within the S, manifolds. As a prototypical example of IVR in large aromatics, results for the anthracene molecule are first presented. The intent is to examine the detailed nature of vibrational energy flow in this rigid, “unreactive” molecule and to discover the ways in which the amount of vibrational energy and the rotational level structure affect this flow. Particular emphasis is placed on the phenomenon of phase-shifted quantum beats,® which occurs for excitation energies within a small energy range of the anthracene S, manifold and which represents the manifestation of IVR amongst a small number of vibrational levels (restricted IVR). Emphasis is also placed on how the character of the energy flow changes at higher vibrational energies to encompass a larger number of levels and to become dissipative in nature.” The second molecule studied is trans-stilbene. It differs in significant ways from the anthracene molecule in that it is reactive in the S, state® (it undergoes a trans- cig photoisomerization), is much less rigid, has many low frequency vibrational modes, and has reduced symmetry. It therefore represents an interesting test of the general applicability of IVR trends deduced from the anthracene results. Moreover, its study sheds some light on the interplay between IVR and reactive dynamical processes. The structure of the main body of the thesis takes the following form. In 6 Chapter 2 a brief discussion of the principles underlying the picosecond-jet technique is presented. Chapter 3° deals with the theory of how the dynamics of IVR is manifested as coherence effects in fluorescence decays. Chapter 4!° is a presentation of results pertaining to IVR in the S; manifold of anthracene. The ways in which the rotational level structure of molecules can influence the manifestations of IVR in fluorescence decay measurements is the subject of Chapter 5.11 Finally, Chapters 6!? and 7!5 deal with results pertinent to IVR in S, tstilbene. All of the Chapters 3-7 were written as manuscripts to be submitted as papers to journals. This might have the effect of rendering the thesis somewhat disjointed to the reader. For this I apologize and ask the reader’s indulgence. The thesis also has four appendices, three of which are published reports®” of our initial work on restricted and dissipative IVR. The fourth appendix consists of the FORTRAN computer program used in the Fourier analysis of fluorescence decays. REFERENCES 1. See, for example, N. Bloembergen and E. Yablonovitch, Phys. Today, 31, 23 (1978); A. H. Zewail, Phys. Today 88, 27 (1980). Nn . For reviews see: (a) R. H. P. Rettschnick, in Radéationless Transitions, edited by S. H. Lin (Academic, New York, 1980); (b) R. E. Smalley, J. Phys. Chem. 86, 3504 (1982); (c) C. S. Parmenter, Faraday Discuss. Chem. Soc. 75, 7 (1983). R. A. Marcus, J. Chem. Phys. 20, 359 (1952). - R. E, Smalley, L. Wharton, and D. H. Levy, Accts. Chem. Res. 10, 139 (1977). o For example, see: P. M. Felker and A. H. Zewail, in Applications of Picosecond Spectroscopy to Chemistry, edited by K. B. Eisenthal (Reidel, Dortrecht, 1984); and references therein. o> (a) P. M. Felker and A. H. Zewail, Chem. Phys. Lett. 102, 113 (1983); (b) P. M. Felker and A. H. Zewail, Phys. Rev. Lett. 58, 501 (1984). P. M. Felker and A. H. Zewail, Chem. Phys. Lett. 108, 303 (1984). eo N J. A. Syage, W. R. Lambert, P. M. Felker, A. H. Zewail, and R. M. Hochstrasser, Chem. Phys. Lett. 88, 266 (1982); and references therein. 9. P. M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. 10. P. M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. 11. P. M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. 12. P. M. Felker, W. R. Lambert, and A. H. Zewail, J. Chem. Phys. - submitted. 13. P. M. Felker and A. H. Zewail, J. Phys. Chem. - submitted. CHAPTER 2 EXPERIMENTAL 9 In this chapter the intent is to present a discussion of the principles underly- ing the picosecond-jet technique.! As has been stated in Chapter 1, the utility of this experimental approach to the study of IVR derives from the combination of two features—the use of a free-jet expansion to cool the gaseous molecular sample and the use of picosecond spectroscopic techniques to probe dynamical processes in the time domain. Thus, we shall be concerned with elucidating the mechanism by which a freely expanding gas is cooled, and explaining the fundamentals of the spectroscopic technique by which we measure fluorescence decays on a picosecond timescale. At the end of the chapter an overview of the experimental apparatus and a discussion of the sensitivity of the method are given. Detailed information concerning the design of the apparatus may be found elsewhere.” Specific experi- mental conditions relevant to the results dealt with in subsequent chapters will be presented in those chapters. I. COOLING BY FREE-JET EXPANSION A. Useful relations The expansion of a high pressure gas through an orifice of diameter D into a region of low pressure, with D being large in comparison to the mean free path of the gas molecules, is termed a free-jet expansion.*® The effect of such an expansion is to convert, via collisions, the random thermal velocity of the gas before the orifice into directed flow and, in so doing, to reduce the translational temperature of the gas. To the extent that the translational degrees of freedom communicate with any internal degrees of freedom that exist in the gas (#.¢., the rotational and vibrational motions of any polyatomic constituents) these are cooled as well. Probably the most convenient method of calculating the thermodynamic prop- erties of a free-jet expansion is that which involves the treatment of the gaseous flow in the hydrodynamic limit and the expansion taken to be isentropic.‘ In this manner one can obtain a number of useful equations relating gas expansion param- eters to jet translational temperature and density. For instance, for a monatomic gas in the hydrodynamic regime, the translational temperature T as a function of 10 D; z, the distance from the orifice; and nozzle temperature Tp, is given by:* ae 1.09(2)4/3" (1.1) The gas density p(z,9,D) for an ideal monatomic gas at distance z, off-expansion axis angle 9, and orifice diameter D, is found to be:? 29; P\2 (2, 8, D) = 0.1699 cos a(—) ; (1.2) where po is the density of the gas before the orifice. Going beyond the continuum approximation, as is necessary for large z, is somewhat more complicated since it involves treating the expansion on a micro- scopic level. Nevertheless, analyses have been made and have produced results® of interest to the spectroscopist. In particular, unlike that which is suggested by (1.1), T does not continue decreasing as z — oo. Instead, a terminal translational temperature T; is reached at large z, and is a function of the product of Py) D, where Po is the nozzle gas pressure. For argon,‘ 1 T = Ty 5000(PpD)*" (1.3) with Po in atmospheres and D in centimeters. This equation, besides indicating the influence of nozzle pressure in the cooling process, also suggests the importance of collisions in translational cooling, being that the total number of two body collisons in the expansion is proportional to P)D.5 Besides translational temperature and gas density, a third quantity is of interest to the spectroscopist; the two body collision rate in the jet. One would expect this rate to decrease with density and therefore to be a function of z/D. Lubman, e¢ al.,” have calculated hard sphere collision rates for a variety of expansion parameters and z/D values. Their findings indicate that for typical conditions and reasonably large t/D values (> 25), collision rates in a jet are about two orders of magnitude less than typical large molecule fluorescence decay rates (~ 107 — 10° sec—'). This, of course, is important to those who undertake to study intramolecular decay rates ll in a supersonic jet. The steep dependence of collision rate on z/D is also of use in a practical sense. It suggests an empirical assessment of collisional influences on experimental observables via the measurement of such observables as a function of z. 12 B. A microscopic picture of cooling The results surveyed above concerning the parameters which affect cooling, density, and collision rate in a free-jet expansion are of obvious importance to the free-jet spectroscopist. They were presented to aid the unfamiliar reader in assessing the experimental results presented later in the thesis. Unfortunately, they reveal little of the mechanism by which gases are cooled in the process of expansion. To try to provide a qualitative picture of this process is the aim of this section. We start with the assumption that translational cooling is the result of col- lisional interactions.’ But obviously, all collisions between gas molecules are not cooling collisions. Thus, the task is to derive the requirements that must be ful- filled if a collision between two hard sphere gas molecules is to be a “cooling” collision, and then to argue that there is a bias towards such collisions in a freely expanding gas. Once an intuitive feel for translational cooling is established, it is then a relatively easy matter to give a qualitative explanation of the rotational and vibrational cooling experienced by a polyatomic molecule seeded in a monatomic carrier gas. Consider the identical hard spheres 1 and 2 of radius a heading on a collision course with velocities in the laboratory frame #,; and i;. Choose the z direction in space as the direction of the relative velocity #, = 0; — 02;, and the y direction such that the centers of the two spheres initially lie in the zy plane. With this axes choice, the centers of the two spheres at any instant of time are at the same 2 value. Thus one can analyze the collisional interaction in two dimensions. Keeping the z and y spatial directions for the z’ and y’ directions, respectively, in the center of mass (cm) frame (see Fig. 1), the velocities of the spheres are 7” = (|%|,0,0) and oe” = (—|%|,0,0), and they approach each other moving parallel to the z/-axis. An impact parameter b is defined by the quantity y, — y,, where yj is the constant y’ value that sphere 2 travels along and y}, is the analogous value for sphere 1. The value of 6 can range from —2a to +2a. 13 After colliding, the velocities of the spheres in the cm frame become é= ~ oF (1.4) where é is the unit vector (cos 20, — sin 20,0) and 6 = cos~!(). In the laboratory frame the final velocities are given by ~ oom aypaVt ae (1.52) and . mm i, =V - a. (1.50) where V is the cm velocity a (aH pte) = (ti + %2;). We are interested in determining those collisions which change the magnitudes of the velocities of the two spheres such that they are closer together after collision than they were before. For such “cooling” collisions to occur lois] — [Sez|| < [lee] — [eel (1.6) Squaring both sides and noting that lois)? + |@as|? = ltl? + |i’, (1.7) (1.6) reduces to #17 |l#7| > [Hill ei]. (1.8) Now, isle] = (it + EE BE ginny aityt (a9 and VPP let drs |lee| = (IV |* + 3 a3 IV: a,|2)3. (1.98) Thus (1.8) amounts to la, |2|V - a]? < |V- a, 2. (1.10) 14 Since #, = (|é,-|,0,0,) this becomes |#,|? (Vz cos 20 — V, sin 26)? < |#,|?V2, (1.112) which can be reduced to 2 2 cos 20 Vy -Ve < VeVy 99" (1.118) where V, and V, are the z and y components, respectively, of V. Now if we av- erage the right side of (1.11b) over all possible values of the impact parameter by integration over b: 2a 0 . cos 26 sin 6 cos 28 db = —2a——————d@ = 0 . [. sin 20 [ O oin 20 , (1.12) we finally obtain the b-averaged condition for a cooling collision: Vi < V2. (1.13) This condition can be expressed in terms of the angle (= ¢) between the projections of V and 3; in the zy scattering plane by using cos ¢ = 1.14 inv? + V3 n4) and (1.13) to obtain | cos ¢| > 73 or 7 T 37 57 “Z<O<G and 7 <9S< | (1.15) the condition for cooling collisions. Qualitively this means that collisions between two spheres travelling in similar directions at different speeds will tend to be cooling collisions, while collisions between spheres travelling in very different directions at similar speeds will tend not to be cooling collisions. In a confined gas there is no geometrical preference for any given values of ¢. One can see this by noting that the Boltzmann factor for the probability that the vectors V and v, characterize the sphere 1-sphere 2 two body problem depends on | * > “ v Tye > Diy + Fay - Toy = AV)? + er (1.16) 15 and is independent of ¢. Thus, one does not expect a predominance of either cooling or noncooling collisions in a gas at equilibrium. A freely expanding gas is different, however. As gas atoms travel from the expansion orifice into the evacuated chamber, those atoms with velocities pointing in similar directions will tend to travel along the same streamline emanating from the expansion orifice. This geometrical sorting effect spatially localizes atoms travelling in the same direction. In turn, it provides an environment in which cooling collisions are more likely to occur than noncooling ones, for a given atom’s most likely collision partners are those atoms which are travelling in the same direction as it is. This qualitative picture of the translational cooling process is treated more rigorously and quantitatively by Toennies and Winkelmann.® By assuming a spher- ically symmetric expansion and by invoking Maxwellian velocity distributions for atoms travelling along a given streamline, they solve, as a function of radial distance, the Boltzmann equation for the translational temperatures parallel (7) and per- pendicular (T_) to the streamline. The treatment allows them to deduce that the effect of collisions in an expansion is to reduce Tj at the expense of increasing ran- dom motion in the perpendicular direction, where geometrical sorting then serves to reduces T.. Thus, geometrical sorting is the means by which an environment is created in which cooling collisions predominate. C. Rotational and vibrational cooling Translational cooling is not the only cooling which occurs in a free-jet expan- sion when the expanding gas is composed in part of polyatomics. In such cases the rotational and vibrational degrees of freedom of the polyatomic gas molecules are also cooled.*> This happens as a result of the translational cooling. To see this, imagine what will happen if a small amount of some polyatomic is mixed in with gaseous atoms (t.¢., the carrier gas) undergoing expansion. Collisions will occur during expansion which will tend to equalize the velocities of the various species in the jet. In addition, collisions between atoms and molecules will induce energy flow to and from the internal degrees of freedom of the molecules. However, as the 16 translational velocity distribution sharpens, the average collision energies decrease. As a result, less collisions between the atomic gas and the polyatomic will be able to excite internal degrees of freedom. No comparable conservation of energy con- straints exist on de-excitation collisions, however. Collision induced energy flow involving the internal degrees of freedom of the polyatomic will therefore become one-sided toward de-excitation and vibrational-rotational cooling will result. In general, the rotational and vibrational temperatures achieved in an expan- sion are not as low as that for translation. For the most part rotational temperatures have been found to follow translational temperatures (~ 0~5 K) fairly closely* and to be on the order of 10 K or less, while vibrational temperatures tend to have somewhat higher values* (< 50 K) owing to the generally smaller cross sections for vibrational-translational energy exchange compared to rotational-translational energy exchange. The degree of rotational and vibrational cooling depends on the particular polyatomic involved and the particular carrier gas, as well as on the expansion parameters. 17 Il. MEASUREMENT OF FLUORESCENCE DECAYS BY TIME-CORRELATED SINGLE PHOTON COUNTING (TSPC). A. General Scheme The TSPC method?® of observing fluorescence decays has proven very useful in application to samples with lifetimes of ~ 10 psec to hundreds of nanoseconds. Detection does not require strongly fluorescing samples or a very intense excitation source. Moreover, excellent signal-to-noise ratios are obtained with data collection often lasting less than ten minutes. In short, TSPC detection can be applied with excellent results to a large variety of samples. The method of TSPC depends on the assumption that by individually timing many single photon emission events from an ensemble of molecules and by conse- quently producing a histogram of events vs. time, one can faithfully reproduce the fluorescence decay of an “average” molecule in that ensemble. This is accomplished as follows (see Fig. 2): Light from a pulsed source is split into two directions. One part serves to produce an electrical pulse via a fast photodiode (PD). This electrical pulse, after appropriate shaping by timing electronics (PSE), starts a timing clock (TAC). The other part of the light pulse, meanwhile, is directed toward the fluo- rescent sample (S). Optics collect and focus a portion of the resulting fluorescence on the photocathode of a fast photomultiplier (PMT). If a fluorescence photon is detected, the resulting PMT electrical pulse serves to stop the clock started via the photodiode. (The TSPC scheme can be used in a reverse configuration also, in which the PMT pulse starts the clock and the PD pulse is delayed and then stops the clock.) If both start and stop events occur, the time on the clock is read by an analyzer and the event is stored in a memory slot designated for its particular clock time. If no stop pulse arrives at the clock for a given time after its start, the clock is automatically set to zero to await restarting. Also, a stop pulse has no effect if no start pulse has triggered the clock. The storage locations of the analyzer add up all the start-stop events characterized by like clock times. In doing so, a histogram of events vs. time is produced. The result is a reconstruction of the fluorescence 18 decay of the sample. Consideration of the above reveals that there is a limitation of the TSPC scheme. Thai is, it is possible to detect only one photon from any given exci- tation pulse. If the excitation intensity is such that two or more photons reach the photocathode an appreciable fraction of the times that only one photon does, the observed decay will reflect this as anomalous intensity at early times. In other words, due to biased data collection, the decay will appear faster than it is. This effect can be made negligible simply by reducing the fluorescence intensity such that the number of multiple photon detection events is only a small fraction of sin- gle photon events. In practice, this is accomplished in this laboratory by keeping the photon counting rate at less than about one percent of the excitation source repetition rate. B. Excitation Source The nature of the TSPC scheme dictates the ideal characteristics for the exci- tation source to be used in its application. Firstly, of course, it should be pulsed and should produce pulses with widths that do not limit the temporal resolution of detection. Secondly, it should be able to produce pulses at repetition rates high enough that, despite the necessary limitations imposed on the photon counting rate, decays can be collected in a reasonable amount of time. Thirdly, the pulse repetition rate should be variable to allow operation at the highest repetition rate consistent with the requirement that the sample be allowed to relax fully between pulses. Finally (and independent of the mode of detection), the source should be tunable. | All of these requirements are met well by laser systems known as synchronously pumped, mode-locked, cavity dumped dye lasers.!° These systems, which typically consist of a mode-locked argon ion laser pumping a cavity length-matched dye laser, produce tunable ~ 15 psec pulses at repetition rates variable from about 80 MHz down into the few KHz range. Although the peak pulse powers of such lasers are typically smaller by a factor of one thousand or more compared to other, low 19 repetition rate sources, this is compensated for by the high laser repetition rate and the inherent sensitivity of TSPC detection. C. Temporal resolution The temporal resolution available with TSPC using a picosecond laser is typ- ically not limited by the excitation source. In most systems, the limit arises from timing jitter in the photomultiplier. This jitter is primarily the result of the ther- mal distribution of photoelectrons in the photocathode of the detector and of the distribution of distances over which photoelectrons must travel before reaching the gain portion of the device. These two factors introduce a spread in the interval be- tween the collision of a photon with the photocathode and the ultimate production of an electrical pulse by the photomultiplier. Photomultiplier products designed to reduce this transit time jitter allow one to limit it to approximately 80 psec. Given the typical jitter of ~ 20 psec in the other timing electronics and a 15 psec laser pulsewidth, this still represents a limiting value for the overall temporal resolution of a TSPC system. Finite temporal resolution has the effect of distorting measured fluorescence decays away from “true” molecular fluorescence decays. The effect can be succinctly expressed mathematically by means of an integral that relates the observed decay F(t) to the convolution of the true decay I(t) with the “system response function” R(é):4 , | F(t) = i R(t!)I(t — t*)ae’. (2.1) The system response function represents that decay which would be observed if the true molecular fluorescence decay were a 6-function in time. It thus takes into account the overall temporal response of the measurement apparatus. Response functions can be accurately measured for a given experimental system by measuring with that same system, the temporal profile of excitation light scatter. D. Data analysis A fluorescence decay obtained via TSPC consists of a 1 x N dimensional array of integers, the positions (channels) in the array representing clock times and the 20 values of the integer representing the number of start-stop events recorded for each of the clock times. The inherent “signal to noise” in a particular of the decay is determined by counting error and follows Poisson statistics:'! the count N(é;) in channel i differs from the “true” count with a standard deviation of JN (ti). Thus, the signal to noise goes as JN (ti). In analyzing TSPC decays, it is often useful to fit a measured decay to an assumed molecular decay function with adjustable parameters. The digital form of the data and the well-defined source of noise in the decay allow this to be done quite readily with a small computer (e¢.g., a PDP 11/23) using a nonlinear least squares fitting algorithm.!? In such a procedure the sum of the squares of the weighted residuals given by = > wea (2.2) (where N(t;} is the number of measured counts in the it channel, F(t;) is the number of counts calculated to be in the i*” channel using an assumed decay func- tion, and W(t;) is the weight!? for a given data point and is determined by Poisson statistics to be N(t;)) is minimized by adjusting parameters used in the calculation of the F(t;). The goodness of the fit can be judged by the final value for y?. For many cases in which the temporal response of detection must be accounted for via the convolution integral {sum)!* (2.1) in the calculation of F(é;) of (2.2), the nonlinear least squares fitting procedure can be rendered more than an order of magnitude faster by using recursion relations in calculating the F(t;). Such relations are applicable if the true molecular decay I(t;) that is to be convoluted with the system response function R(é;) to yield F(t;) can be expressed as a sum of exponentials of the form e* ti where K does not depend on ¢;. This is the case not only for single and multi-exponential decays, but also for decays containing sinusoidal modulation terms, in which case K is imaginary. In these situations, instead of having to calculate the convolution sum ty F(t) = D> R(t)I(t; - At (2.3) t?=0 21 every time for all relevant f;, one notes that ty+At tj F(t) = DY Re EA MM Ag = eK At S™ Rit eK EM At + R(t, + ANAL. t’=0 t'=0 (2.3) Thus, f(t, + At) = f(tje*4! + R(t; + AdjAt (2.4) and F(t; + Aé) can be expressed simply in terms of F(t;). The savings in compu- tational time that this represents is a factor on the order of N/2, where N is the number of channels comprising a given decay. Since N is typically greater than 500, this is a significant savings. Besides fits to measured decays, much of the data analysis in this thesis relies on Fourier transform techniques to extract information from decays. A discussion of this appears in Chapter 4, section II-B. The FORTRAN computer program used in this regard is contained in Appendix IV. 22 Ill. EXPERIMENTAL APPARATUS A. Overview Excitation source The experiments described in this thesis were performed using a Spectra- Physics synchronously pumped, cavity dumped dye laser (rhodamine 6G perchlor- ate—2 x 10-° M in ethylene glycol, or DCM—5 x 10~* M in 60/40 volume percent ethylene glycol/ benzyl alcohol as dye). The laser produces ~ 15 psec pulses with frequency bandwidths in the red varying from about 1 cm~! to 3 cm~! depending on the tuning element used (the 1 cm! value corresponds to the use of a three- plate birefringent filter with ultra-fine tuning etalon, the 3 cm~! value to using just the three-plate filter). The laser pulse train repetition rate is variable from 4 MHz down to 0.4 KHz, although for the work reported herein, it was operated almost exclusively at 4 MHz. At this repetition rate, the average power is ~ 50 mW or about 0.12 erg per pulse. To obtain excitation light in the ultraviolet with this laser, it was necessary to frequency double the red output. This was done by focusing it into a 1 cm long LilOs crystal. The conversion efficiency of red to UV with this set-up was found to be about ten percent.2* Frequency doubling has the effect of increasing the bandwidth of the light (and decreasing the temporal pulse width). In the UV, the bandwidth varied from ~ 2 to 5 cm™!, depending on the tuning element used in the laser. After frequency doubling, the UV was directed through quartz windows into the vacuum chamber where, after focusing, it intersected the free-jet expansion at right angles to the expansion axis. Free-jet apparatus and fluorescence collection The detailed design of the free-jet apparatus has been dealt with elsewhere.” Briefly, the continuous free-jet expansion was formed by applying a high pressure of carrier gas (typically He, Ne, or Ng) to a heated pyrex nozzle containing the compound of interest. The high pressure gas expanded into an evacuated region (< 10—* Torr) through a 100—150 ym orifice in the tip of the nozzle. The evacuated 23 region was maintained at low pressure by a ring-jet booster pump (Edwards 18B4A) backed by a mechanical pump (Kinney KT150). The nozzle apparatus has flexibility for alignment in three directions, which flexibility is extremely useful in the course of doing experiments. Fluorescence was collected and collimated by an f = 1 quartz lens through a quartz window parallel to and about 1.5 inches from the expansion axis. A second quartz lens with a focal length of 50 cm focused the collimated fluorescence onto the slit of 0.5 M Spex monochromator (dispersion 16 A per mm) located about 50 cm away. Photon counting equipment The measurement of fluorescence decays was done by time-correlated single photon counting (see section II and Fig. 2 above). The fast photodiode used was a Hewlett-Packard 5082-4203 modified as in Ref. 14. Typically, +100 mV pulses (~ 1 nsec FWHM) are obtained from it. These pulses were amplified by a Hewlett- Packard 8447E inverting amplifier, directed through a delay line and into a constant fraction discriminator (either an Ortec 473A or, more recently, a Tennelec 455) which produces a uniform output timing pulse for every suitable input pulse. The output pulse served to trigger the stop channel of a time-to-amplitude converter (TAC, Ortec 457). The reverse configuration TSPC scheme was used because this enhances the overall detection efficiency. Fluorescence photons were detected (typ- ically with about twenty percent quantum efficiency) by a Hamamatsu R1564U multichannel plate photomultiplier run at —3300 V. The output pulses from the detector (250 psec FWHM, 80 psec transit time jitter) were amplified in one or two stages with Comlin CDC100 noninverting video amplifiers. The pulses then went to a constant fraction discriminator (Ortec 473A, or Tennelec 455), the output of which triggered the start channel of the TAC. Finally, the output of the TAC was read by a multichannel analyzer (MCA, Tracor-Northern TN-1706) run in the pulse height analysis mode. A typical system response function was 80 psec FWHM. In general, such a narrow response can be obtained only by raising the lower limit voltage level of the discriminator in the photomultiplier channel to > 50 mV for the 24 Tennelec 455. Thus, a narrow response function is obtained at the cost of losing some signal. The data stored by the MCA is ultimately read, stored, and analyzed by a DEC PDP 11/23 computer system. The measurement of dispersed fluorescence spectra is similar to the measure- ment of fluorescence decays except that the MCA is used in multichannel scalar mode wherein it counts the number of pulses reaching it for a given interval of time, stores this number, and then steps on to the next channel and does the same thing, ete. The stepping of the MCA is controlled by the same microprocessor (Spex CD2) used to drive the monochromator grating. Thus the wavelength setting of the monochromator and the channel setting of the MCA step in synchrony. The pulses counted by the MCA can be from either the TAC or the discriminator in the photomultiplier channel. Using each has its own advantage (a lower baseline when the TAC is used, a greater overall signal when the discriminator output is used), but both methods yield essentially the same spectra. B. Sensitivity of the method As a final point of this chapter, it is pertinent to derive an expression for the signal level to be expected in a picosecond-jet experiment utilizing photon counting as the detection scheme. To do this, it is useful to start with several definitions: Eo = the energy per pulse of excitation light WY = the frequency of the light € = the molar extinction coefficient (in liter/(mole-cm)) of the molecular sample for light of frequency vp (multipying « by a constant factor of 1.7x 1072, which is 1000 divided by Avogadro’s number, gives the absorption cross-section in cm?) 5 = the path length of the light through the free-jet (in cm) C = the concentration of the molecular species in the jet (in moles/liter) ©, = the quantum yield of fluorescence of the molecule excited with light at frequency 1%. Upon impinging upon the free-jet sample, a pulse of excitation light will excite 25 a total of n* molecules. Assuming (1) the absence of stimulated emission, (2) that spontaneous emission is negligible during the pulse, and (3) that the number of excited molecules equals the number of photons absorbed by the sample, then using the Beer-Lambert law E - n* = in (t ~ eC), (2.5) which in the weak absorption limit becomes E n* = Fang (COCs (2.6) where we have assumed that the frequency bandwidth of the laser is on the order of, or less than, the rotational bandwidth characterizing the jet-cooled sample. Of the n* excited molecules, only @;n* will fluoresce. The total number of fluorescence photons is simply this number. In most experimental set-ups there is a geometrical constraint to the collection of fluorescence photons. Let us assume that the fluorescing sample is located at the focal point of a lens with a focal length-to-aperture ratio of f, and that fluorescence is collected via this lens. Such an arrangement samples a fraction (= g,) of the total fluorescence GpF? | (2.7) Further fluorescence is lost for geometrical reasons if a monochromator is used in 1 n= 5 (1 - detection. Let us assume that a second lens focuses the fluorescence onto the slit of a monochromator (we also assume that the slit of the monochromator is parallel to the beam of the excitation light). The image of the sample fluorescence on the slit will have a width equal to wM, where w is the width of the excitation light beam at the free-jet, and M is the overall magnification of the detection optics. If the slit of the monochromator has a width s, then the fraction (= gz) of fluorescence which is focused onto the slit and actually enters the monochromator is ; g2 = DA’ (2.8) 26 The total number of fluorescence photons (mm) getting into the monochromator is Mim = gigadsn*. (2.9) Now the use of a monochromator in detection also implies that only fluorescence photons of the proper wavelength will be detected. Thus, the photon counting signal will ultimately depend on the spectral characteristics of the fluorescence and the detection resolution used. Let us lump these together into one factor (= gs), representing the fraction of photons entering the monochromator and having the proper wavelength to make it to the detector. Finally, the detection quantum efficiency gz of the photomultiplier must be taken into account. The total number of electrical pulses (signal counts) produced by fluorescence photons can now be expressed as Nt = gigagsDaD;n* (2.10) or _ $93DgOy f Eo r= OOM Gap F hg (2) (2.11) It is instructive to calculate a typical value of n; for the experimental apparatus de- scribed in III-A. For this system and assuming UV excitation (u/c = 30000 cm—'): f=l1we-l1mm, M ~5, 6 ~ 0.2, Ey ~ 0.01 erg, huy = 6 x 10-!? erg, and b=0.5 cm. Thus, ny ~ 5 x 10°cags OC, | (2.12) with s in mm. C can be calculated by (1.2) of section I-A (averaging over 6), if one assumes a partial pressure (Po) for the sample in the nozzle. Let us assume Po =1 Torr. Then po of (1.2), which is equivalent to the concentration of sample molecules in the nozzle, is -_!t _ —5 : po = 760(22.5) = 5.7x 107° moles/liter, (2.13) and by (1.2) _, 5.7 x 10-5(0.16) -D p=Ce 2 (=), (2.14) 27 which for ¢ = 30 gives CG ~5.1x10-*moles/liter (or 3 x 10!*molecules/cm’). (2.15) Using (2.12), nt = 2.5 x 107% eagg By (2.16) with s in mm. This expression is relatively sample-independent, since C can be adjusted by changing the temperature of the nozzle. If now one assumes reasonable values for « and ®, of 10° liter/mole and 0.1, respectively, and also assumes a moderate detection resolution of 1.6 A so that s = 0.1 mm and gs ~ 107?, then ny =2.510-* counts/laser pulse. (2.17) Given that the laser pulse repetition rate is 4 x 10° Hz, the photon counting signal is 10° counts per second. Such a signal is more than enough to collect a decay in a reasonable amount of time (< 1 hr). For example, for a 1 KHz count rate, a one- hour collection time, and for collection over 1000 channels spanning the fluorescence lifetime, one obtains an average number of counts per channel of about 2000, and a corresponding signal to noise of about 45. 28 REFERENCES 1. (a) A. Zewail, Faraday Discussions Chem. Soc. 75, 315 (1983); (b) P. M. Felker and A. H. Zewail, in Applications of Picosecond Spectroscopy to Chemistry, edited by K. B. Eisenthal (Reidel, Dortrecht, 1984), p. 273. 2. (a) W. R. Lambert, Ph. D. thesis, California Institute of Technology (1982); (b) W. R. Lambert, P. M. Felker, and A. H. Zewail, J. Chem. Phys. 81, 2217 (1984). i) . A. Kantrowitz and J. Grey, Rev. Sci. Instrum. 22, 328 (1951). mm . R. E. Smalley, L. Wharton, and D. H. Levy, Accts. Chem. Res. 10, 139 (1977); and references therein. 5. D. H. Levy, L. Wharton, and R. E. Smalley in Chemical and Biological Applica- tions of Lasers, Vol. II, edited C. B. Moore (Academic, San Francisco, 1977) - and references therein. . J.B. Anderson and J. B. Fenn, Phys. Fluids 8, 780 (1965). 6 7. D. M. Lubman, C. T. Rettner, and R. N. Zare, J. Phys. Chem. 86, 1129 (1982). 8. J. P. Toennies and Klaus Winklemann, J. Chem. Phys. 66, 3965 (1977). 9 . For example, C. C. Lo, B. Leskover, P. R. Harteg, and K. Sauer, Rev. Sci. Instrum. 47, 1113 (1976). 10. For a simple discussion of mode-locking by synchronous pumping see: C. K. chan, Laser Technical Bulletin no. 8, Spectra-Physics Laser Products Division, eb. 1978. 11. For example, C. Lewis, W. R. Ware, L. J. Doemeny, and T. L Nemzek, Rev. Sci. Instrum. 44, 107 (1973). 12. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969). 13. We use the terms convolution integral and convolution sum interchangably. Obviously, in treating real data, the convolution sum is that which is evaluated. 14. G. H. McCall, Rev. Sci. Instrum. 48, 865 (1972). 29 FIGURE CAPTIONS 1. Schematic illustrating the collision of two hard spheres in their center of mass frame of reference. The axes are defined such that the centers of the two spheres are always in a plane parallel to the z’ = 0 plane. a) shows the two spheres approaching each other with relative velocity |d,| along the z’ direction and with impact parameter b. b) shows the moment of impact and the angle @ = cos~(b/2a). 2. Schematic of the TSPC scheme for detection of fluorescence decays. The filled pulses denote light pulses and the empty pulses denote electrical ones. PD stands for photodiode, PMT for photomultiplier tube, S for sample, PSE for pulse shaping electronics (discriminator), TAC for time to amplitude converter, and MCA for multichannel analyzer. 30 Figure 1 31 Figure 2 posy owl | 4$Sd 90uU99Sa/0N| 4 Y4ASV 1 $4D}S }¢$—— ASd nl LWd TH 1 dd as 32 CHAPTER 3 COHERENCE EFFECTS AND IVR* * Submitted to the Journal of Chemical Physics, as “Dynamics of intramolecular vibrational-energy redistribution (IVR). I Coherence Effects,” by Peter M. Felker and Ahmed H. Zewail 33 I. INTRODUCTION The fundamental process of intramolecular vibrational energy redistribution (IVR) in isolated molecules involves the transfer of energy from a prepared ini- tial state to other isoenergetic states. This collisionless redistribution of energy is the result of intramolecular couplings (anharmonic and/or Coriolis interactions) - between rovibrational states. Spectral manifestations of IVR have been discussed previously and will not be addressed extensively here!. That which is of concern in this work involves the temporal characteristics of dispersed fluorescence on the picosecond timescale?. Our interest in these characteristics derives from the fact that the temporal behavior of a given fluorescence band is a direct view of the evolu- tion of vibrational energy in the excited state vibrational manifold; from time- and frequency-resolved fluorescence measurements one can obtain information relating to the timescale of IVR, the nature (restricted or dissipative) of the process, and the coupling matrix elements involved.°-5 Recently, we have presented experimental results*—* on anthracene which il- lustrate the use of the picosecond technique in revealing IVR dynamics. There are, however, several relevant points that have not been addressed in detail con- cerning the dynamics of IVR in general, and its manifestations in the fluorescence decays of large molecules in particular. Firstly, there has been no comprehensive consideration of the consequences of multilevel vibrational coupling in IVR. Such consideration is especially essential to the understanding of restricted IVR (1.e., IVR amongst a few vibrational levels), as manifested in fluorescence decays as phase- shifted quantum beats.*:* Secondly, one would like to understand more about the changes in the nature of IVR as a function of vibrational energy in such multilevel systems. The parameters governing transitions from absent to restricted to dissi- pative IVR, as well as the sharpness of these transitions, are of interest. Thirdly, one might legitimately ask how it is possible to observe vibrational coherence ef- fects (phase-shifted quantum beats) even though any given coupled vibrational level has associated with it a rotational level structure. With the large number of rovi- 34 brational states involved in the coupling between just two vibrational levels, one might think it likely that the superposition of the many different resulting beat frequencies would give rise to fluorescence decays with coherence effects completely smeared out. This is clearly not always the case. Thus, it remains to consider why this is so and what manifestations of the rotational level structure are present in a beat-modulated decay. Finally, it would be desirable to assess the generality of the concepts of IVR, which have been derived from the experimental observations made first on anthracene, to other molecules. Elsewhere,** we have presented initial theoretical and experimental studies of IVR in what we have termed the restricted limit. In particular, the dynamics of vibrational coupling amongst two or three vibrational levels has been considered. For these cases the manifestations of [VR in the temporal behavior of spectrally resolved fluorescence is relatively easy to recognize and interpret. In view of the expectation that vibrational coupling in molecules may involve any number of levels (especially at high energies) and that many such physical situations are amenable to study by picosecond spectroscopy, it would be useful to have some theoretical guidelines of the dynamical manifestations which a general system of N coupled vi- brational levels might exhibit. In this chapter we derive and present such guidelines, discuss some possible deviations from the predicted behavior, and, finally, discuss the physical significance of the results. The results presented herein will be used in accompanying publications to interpret experimental observations made in this laboratory on the molecules anthracene (Chapters 4 and 5)*” and trans-stilbene (Chapter 6)° in the restricted and dissipative limits of IVR. 35 Il. THEORY A. General framework We consider the case of N coupled vibrational levels in the S; manifold (Fig. 1). (Here we ignore the presence of the rotational level structure, the effects of which are discussed in Chapter 5.") These zero-order states, which we take to be products of one-dimensional harmonic oscillator wavefunctions, are denoted as la), |b), je),... (In denoting a generic zero-order state, |) shall be used.) The molecular Hamiltonian matrix H, when expressed in this zero-order basis, is not diagonal because of coupling terms involving anharmonicity, etc. Diagonalization of H yields the molecular eigenstates |1), |2}, |3),..., (with energies E,, Ee, Es,..., respectively), which are linear combinations of the zero-order states: I= S° aly), F=1,...,N; (2.1) 7=4,),... where the a; form a real,? orthonormal N x N matrix of the form : Q1e Aaa aoe aNea C=] %b Wp ..- Ong |]. (2.2) Now consider that a molecule in vibrational state |g) of the Sp manifold is excited by a linearly polarized 5-function laser pulse (polarization vector ¢) at the proper wavelength to create a coherent superposition of the molecular eigenstates \I), J =1,...N. Such a coherent superposition of states will fluoresce to the levels {|f}} im the So manifold. The fluorescence intensity os. time to one of these So levels | f,) (assuming no analysis of fluorescence polarization) is given by!° N 2 : T(Q=K DY) DE Pela Pala) PM Lr PRL Leer (2.3) [,J=1m=1 where K is a constant, P,(g, I) = (g|é- fi|I) is the excitation transition matrix ele- ment connecting |g) and |I), P"(f,,I) = (f,|ém - aI) is a fluorescence transition matrix element connecting |I) and |f,), é, and é, are the two perpendicular flu- orescence detection polarizations, wry = (Ej - Er) /h, and I is the decay rate of 36 the excited eigenstates (taken for simplicity to be equal for all J'1). We call such a fluorescence band a y-type band. In general, because of Franck-Condon factors and symmetry restrictions, only a limited number of the entire manifold of S, vibrational levels will have absorption strength from |g), as evidenced by the sharp and uncongested absorption and fluo- rescence excitation spectra of large jet-cooled molecules.!? Thus, it is reasonable to take only one of the coupled zero-order levels as having any appreciable absorption strength from |g). We take this level to be |a). This leads to the following expression for P;(g, I): Pz(g,I) = araPz(g,a)' for 2 =1,...,N. (2.4) Similarly, one expects the Franck-Condon overlap for emission to the So level |f.,} to be dominated by one zero-order level |y). In other words, PR (fas DT) = aryPP (fay) for P= 1,...,N. (2.5) Substituting these expressions into (2.3) yields the following for the intensity vs. time of a y-type band: N 2 L(t)=K DY Do area seers y{Ps(9,a) PP (fae rth’ (2.6a) £,f=1m=1 N N 1,() = K'|” a7, 07, +2 > Apa JaMryA sy coswyytle Fe, (2.6) f=1 I>J where K’ is a constant. (Note that if the fluorescence were analyzed for polarization, i.e., the sum over m were restricted in (2.6a), exactly the same time dependence as (2.6b) would obtain for I(t). Only K' would change.'*) Now |) represents a generic zero-order state. Thus, one expects N different types of fluorescence bands in the fluorescence spectrum, each distinguished by the zero-order level which contributes to the emission transition moment. The time- integrated intensity of a given band type will scale with the quantity ~ 07,045. Moreover, from (2.6b) each of these bands will be modulated by the same N(N — 37 1)/2 cosine modulation terms (quantum beats), the frequencies of which will be determined by the energy differences between eigenstates, and the phases of which are determined by the particular band type via the coupling coefficients a. (One may further note that wry +wyxK = wrx. In general, N(N — 1)(N — 2)/6 such triplets of Fourier components are present for an N-level system when N > 3.) In what follows, we shall be interested in making generalizations concerning the amplitudes and the phases (¢.e., signs) of quantities of the form M _ 20a JaA1 yO Z, (Wrz) = N 2.2 vera ATe@Ty which from (2.6b) represents the modulation depth (normalized Fourier amplitude), for a y-type band, of the Fourier component at frequency wy). B. Quantum beat modulation depths There are several generalizations that can be made about the amplitudes of the modulation depths for a given band type. In making them it is necessary to distinguish between two types of fluorescence bands. The first type is the a-type band, which, using the conventions and nomenclature outlined above, represents a fluorescence band that derives its absorption and emission strength from the same zero-order state (|a)). The second type of band is the non-a-type band, which by definition derives its absorption and emission strength from two different zero-order states ( |a) and |7), respectively). For an a-type band (2.7) reduces to N . ist ate Since the a’s are real, then 0 < 207, a2, < at +a4_. This places a restriction on ’ la“Ja fa Ja Ma(wrs) = (2.8) the magnitude of each Fourier component contributing to an a-type band : 0 < M,a(wrs) < 1 for all Wrj. (2.9) In addition to restrictions on individual components, it is also possible to place bounds on the sum of the modulation depths of an a-type band : N N > M,(wrz) = 2 i> sai CT PEP al\Wrs) = rN 4 . I>J=1 t=1 “Ia (2.10) Such bounds derive from N N N (do 7)? => o%+2 S> 03,03, =1, (2.11) =i f=1 I>J=1 which yields > 1 M,(wrz) ==on 7 TL. (2.12) I>J=1 ei=1 OF, And, since )-7_, a4, is fixed by normalization such that UN < Vil, at, <1, then N 0<¢ SS Mw) <N~-1. (2.13) I>J=1 For non-a-type bands one must treat the general form for M,(w;7) given by (2.7). Similar to a-type bands, one can place bounds on the magnitudes of individual modulation depths by noting that |2asaasaaryasy| < aj,07, + 03,07, ( this follows from [araary + &JaQJ_|? > 0). Therefore, from (2.7) 0<|M,(wrz)| <1 for all wy, y 4a. (2.14) Also, since N N 2° —) rat saAsyAy » Mw) = See (2.15) I>J=1 vr=1 Ate tT, and N N o> QfaQp_)? = > Of 4 OT +2 > Apadsaayasy =0, (2.16) I=1 I=1 I>J=1 then N > M,(wrs) = -1, if y fa. | (2.17) I>J=1 That is, the sum of the modulation depths for any non-a-type band must equal —1. C. Quantum beat phase distributions Having considered some general restrictions on the magnitudes of the modula- tion depths of the Fourier components modulating the decay of a given band type, it is now useful to consider the phases of these components, or, equivalently, to 39 consider the signs of the modulation coefficients. In particular, we are interested in generalizations that can be made concerning the possible distributions of quantum beat phases that can occur in a given decay. To address this task succinctly it is convenient to start with several definitions. Firstly, we define 8; to represent the sign of a given element of the orthonormal C matrix: 81 = sign(a;,) = +1. (2.18) Secondly, we define &, to be an array of such quantities representing the signs of the 7" row of C: Gy = (814; 824) ---, 8Ny) = Sign(Q1y, A94,.-., Ny). (2.19) In similar fashion we define & to be a column array of signs of the I‘ column of C. Fourthly, we define s,(w,,) to be the phase of the Fourier component at frequency wyyz which modulates a y-type band: 8,(wrs) = sign[M.,(wzz)| = +1. ; (2.20) Finally, we define A, to be the 1 x [N(N — 1)/2| dimensional array of quantum beat phases characterizing a 7-type band : Ay = (84(w2), s(w1s), . ++) 8q(Win), 84(Wa3),-.-, 84(WN—1,N))- (2.21) Next it is useful to make three observations concerning the arrays &, and &. The first observation is that one is free to choose all the values for one and only one of the arrays @,. This freedom derives from the freedom to choose the phases of the eigenvectors composing the columns of C. It is convenient to use this to choose @, = (+1,4+1,41,...,4+1). (2.22) Secondly and thirdly, we note that Gy fry iy #7, (2.23) 40 and that Ate if IAS. (2.24) These conditions follow from the orthogonality of the rows and columns of C, re- spectively. We are now ready to consider the possible values of A, for an N-level system. Clearly, Aq = (+1, +1,+1,...,+1), (2.25) for any system, since by (2.10) 8a(wrz) = sign(a},0%,)=+1 for all I, J. (2.26) In other words, a-types bands are modulated only by positive cosine terms (ef, For non-a-type bands 8y (Wry) = sign(areayear4a74) = sign(ay,as4) = 814874 (2.27) by our choice of eigenvector phases. Thus, the distribution Ay is entirely determined by @, for this choice of phases. One can now derive several conditions on the non- a-type phase distributions. Firstly, it is evident that all the values occurring in the quantum beat phase distribution array A, are not independent. If the values 84(wiyz), J = 2,3,...,N are known, then since 8y(wir)8y(wis) = (814)? 827844 = 84(wr3), (2.28) it is clear that all other values in the array can be found, too. Thus, there are N —1 independent phases for a given band of an N-level system, and one can specify a phase distribution just by specifying the 1 x (N — 1) dimensional array: 5, = (s,(wiz), 84(w1s),-.-,84(Win)). (2.29) Instead of working with the larger arrays Ay, we shall equivalently consider the 5, arrays henceforth. 4] Secondly, it is evident from (2.27) that if #, is nontrivially different from then 5, is different from 6, and vice versa. That is, Gy Ftdy 3 5, ATy. (2.30) Thirdly, this one-to-one correspondence between nontrivially different f, and the resulting quantum beat phase distributions 5, can be used along with (2.23) to show that for a given N-level system by, #6y = ify #4. (2.31) That is, each type of the N possible types of fluorescence bands that arise from an N-level system will have a unique quantum beat phase distribution. Fourthly, one can use the correspondence between &, and 5, to calculate the total possible number of different phase distributions for an N-level system. This can be done by calculating the number of nontrivially different #,, which is 1 N(N-1) | N(N -1)(N — 2) _1 N _ oN=1 ptt N+ >t 5 te)= Sti) = 21. (2.32) Thus, there are 2%—! possible values of 5, for N coupled levels. (We enumerate these possibilities for N < 5 below.) Note that not all these possibilities are realized for a given N-level system (N > 3), since only N bands occur in any given spectrum. Note also that we have not considered the restriction of (2.24) on the #. It will be shown below that, although this does not restrict the possible values of 5x, it can restrict the possible combinations of 5, that can occur together in any given spectrum. Fifthly, if one notes that the total number of distinct ways of distributing two values over N — 1 variables is simply 2%~!, then it is clear that the 2"—! allowed by for an N level system represent all the concetvable ways in which +1 and —1 can be distributed in these 1 x (N —1) dimensional arrays. As a consequence of this, it is very easy to write down all the possible 5, for a given system. 42 Finally, one can derive a general restriction on the distribution of quantum beat phases over the triplet of beat frequencies wry, wx, wrk (= wry + wi). We define bry * to be such a distribution : f7* = (8q(w1a), 84 (wae), 8x(wre)), (2.83) and recall the expression for the s,(w) (2.27). Now either 87, = 834 = 8K, OF one of these three signs is different from the other two. The first possibility gives 6ivK = (+1,+1,+1), while the second can result in one of the three distributions (+1, -1,-1), (-1,4+1,-1), or (-1,-1,+1). Thus, out of the eight conceivable distributions, only the one with all positive phases and the three with two negative- one positive phase are actually possible. Ill. SPECIFIC RESULTS FOR N <5 As an illustration of the general concepts derived above, and as a source of information on the coupling cases most amenable to study by quantum beat spec- troscopy, we present in this section specific results for N < 5. A.N=2 This case has been treated previously in part®*. We include the results here for the sake of completeness. According to the theory presented above, one expects two types of bands (a-type and b-type), and one beat frequency (w 12) modulating both types of bands. The phase of the modulation of the a-type band will be +1 (s.e., 8a(wi2) = +1), while that of the b-type band will be —1 (#.¢., s(wi2) = —1). Furthermore, while by (2.17) My(wi2) = —1 and is coupling independent, M, (w12) is dependent on the coupling between the zero-order states. In fact, it is a simple matter, given the following definition for the Hamiltonian matrix in the zero-order basis : Ea Vab Hy = ' 3.1 (vs Ei, ) (8.1) to show that the a-type modulation depth is 1 Ma(w12) = ope py" (3.2) 43 where R = |(£a — Ey)/2Vae|. Because hug = 2|V.,|(R? +1)°, (3.3) then the measurement of w 2 and M,(w,2) allows one to obtain the zero-order energy difference |Z, — E,| and the magnitude of the coupling matrix element \Vaol. B. N =3 For N = 3 one expects three types of bands (a- ,b-, and c-type) in the dispersed fluorescence spectrum. Each will be modulated by the three beat frequencies Wa, Was, and wy3(= wie + ws). One also expects four possible quantum beat phase distributions for these bands. Unlike the case of N = 2, all M,(wz ) values are coupling dependent (i.¢., ay, dependent) for N = 3 {and for N > 3). It is only the sum of modulation depths for any given non-a-type band which is coupling independent (¢.f., (2.17)). A summary of results for N = 3 appears in Table 1. (Note that in Tables 1, 2, and 3 we have labelled specific phase distributions. Note also that in Tables 1 and 2 we have included the dependent quantum beat phases as well as the independent part 6,.) C.N=4 A summary of information for N = 4, including possible beat phase distribu- tions, is contained in Table 2. For this case and for N > 4 there are additional restrictions on the phase distributions, which do not appear for N < 3 and which were not addressed in the general treatment of section II-C. These restrictions de- tive from the condition that &! 4 #7, for I # J, and they limit the combinations of 5, which can appear in a given spectrum. As an example, consider the phase distributions i, 5, 57, and 58 from Table 2, and assume that they correspond to the a-, b-, c-, and d-type bands, respectively, in the spectrum of a 4level system. Using these distributions and (2.17), one obtains 1 41 1 1 . 1 1 -1 -1l sign(C) = 1-1 1 oLt (3.4) - -l1 1 1 1 44 where each row of the matrix is determined up to an overall sign. Now, although G, # Fy in (3.4), s = &, contrary to the restrictions on the g's (2.24). Thus, the combination of i, 52, 7, and 5, cannot correspond to bands in the same spectrum. Five other such forbidden combinations of phase distributions exist for N = 4. We shall not deal any further with the specifics of these forbidden combinations except to note, again, that they are present for all V greater than three. D.N=5 A summary of results for N = 5 is presented in Table 3. 45 IV. QUANTUM BEAT PARAMETERS AND THE DETERMINATION OF Ho Just as observed quantum beat parameters for NV = 2 can be used to calculate parameters associated with the Hamiltonian matrix expressed in the zero-order basis (#.e., Ho), so too can this be done for N > 3. Unlike N = 2, however, expressions for the parameters of Hy cannot be conveniently presented for N > 3. Nevertheless, a general method can be outlined by which a calculation of Hy can be performed given measured beat frequencies and modulation depths. We present here such an outline for N = 3. Extension of the method to larger values of N is straightforward. The calculation of Ho using real data taken on anthracene will be presented in the following chapter. The basis of the method of calculation is the relation CEC’ = Hy, (4.1) which relates the eigenvalue matrix E, the eigenvector matrix C, and the trans- pose of the eigenvector matrix C', to Hy. Now the eigenvalue matrix is par- tially determined by the beat frequencies. If one labels the frequencies such that |w12| + |wes| = |wis|, then the following possibilities exist for E - 0 860 0 where Zo is an unknown constant and I is the unit matrix. The matrix C can be determined from relative values of the M,(wy7)’s and from the orthonormality of its rows and columns. Firstly, it is convenient to calculate the ay, via the relations 46 and the normalization relation a ay, =1. The signs of the ay, can be arbitrar- ily chosen to be positive. Next, the values a7, can be calculated by the relations My (wes)a14 26 Mp (wis )a2 1b ( ) Mz (wes Jara a3, = ——————a 4.45 3 Mo (wiz )ase 1b (4.45) and the normalization condition a a}, = 1. Note that the relative signs of the a7, are fixed by the beat phases of the b-type band (e.g., if 6 = (-1,+1), then &, = +(+1,-1,+1)). Thus, the second row of C is determined up to an overall sign. Finally, the values a7, can be calculated up to an overall sign by using several relations, including the normality of the columns of C and the orthogonality of its rows. One may note that the values of the elements of C are overspecified if one has experimental knowledge of all the quantum beat modulation depths for all three types of bands. The extra parameters can be used as self-consistency checks on the calculated C. For instance, absolute modulation depths were not used at all in the calculation outlined above, nor were the modulations depths and phases of the c-type band. These experimental values should, however, be reproduced if one calculates them starting with the derived C. Similarly, a number of orthogonality and normality relations (e¢.g., an Garp = 0) need not figure in the calculation of C. These relations should, nevertheless, obtain if the calculated matrix accurately represents the physical situation. Now, its being clear that one can determine C and E up to various additive and multiplicative constants, what can be determined for Hy ? Firstly, by (4.1), (4.2), and the orthonormality of C 0 0 0 Hp =EoI£C{ 0 fuye| 0 | Ct. (4.5) 0 0 lwis| Next, the effect of the ambiguities in the overall signs of the rows of C must be 47 considered. Since hy, a general matrix element of the second term on the right- hand side of (4.5) is haya = hay = O24 09-4" |wW12| + 3,03" |W13], (4.6) it is clear that changing the overall signs of the y** row of C will leave the diagonal element h., unchanged, but will change the signs of those off-diagonal elements involving 7. From this fact and (4.5), it is evident that by using quantum beat parameters, one can obtain Ea Vas Vac Ho = Eol+| Vas Ey Vie |, (4.7) Vae Voc E. where Eq and the signs of the V,. are unknown, but everything else is known. 48 V. EXPERIMENTAL RESULTS VS. THEORETICAL PREDICTIONS To handle real data it is useful to have some idea of the way in which the theoretical results, which have been derived using a set of ideal assumptions, can be modified by less than ideal conditions. In this section we discuss effects which the experimental excitation bandwidth, the temporal response of detection, the spectral resolution of detection, the excitation of more than one optically active level, and the presence of sequence congestion can have on experimental observations. Since in many cases these deviations can lead to errors in interpretation, especially in the interpretation of the number of levels involved in the coupling, we suggest ways to avoid such errors. No attempt will be made to be exhaustive in the enumeration of all the causes of possible deviations. Instead, several cases will be presented in the expectation that they represent a significant fraction of those cases likely to be encountered in practice, and in the hope that they will illustrate methods of analysis by which other cases may be treated. Reference to a number of the cases presented here is made in the following chapters in discussing real experimental results. A. Case 1: Narrow excitation bandwidth and state preparation | In this case we assume that the bandwidth of the experimental source is not large enough to coherently excite all the N eigenstates which arise from the coupling of the N zero-order levels (Fig. 3). This situation is likely to occur in real molecules, since anharmonic coupling matrix elements may be expected to be as high as several wavenumbers and therefore to give rise to significantly coupled states with splittings on this order. For simplicity, let us say that one state, |), lies outside the laser bandwidth. In such a case, N band types are still possible, but the temporal behavior of a 4-type band will be given by N-1 I(t) ~ > Oat zadsya se Eth t (5.1) I,J=1 instead of by (2.6). Thus, although the system involves N coupled zero-order levels, each band will be modulated by the number of beat frequencies expected for an 49 (N — 1)-level system. (In the general case wherein n eigenstates lie outside the laser bandwidth, the number of band types will still be N, but the number of beat frequencies will correspond to that expected for an N —n level system.) Fortunately, another deviation from theoretical predictions can be used to give one an idea of the actual number of coupled zero-order levels. If one calculates the total modulation depth of a 7-type band (non-a-type) from (5.1), one obtains = 2 tore AraAsJaAyAsy > (wr) = 7 (5.2) I>J=1 Lire 07,055 Given that ane Ora, # 0, then clearly, N-1 N-1 (>> Qraay4)? = > OF, 0, +2 > Qed saAy~Asy > 0, (5.3) = [=1 I>J=1 and, therefore, N-1 N-1 25 my Aah gApyAys > M,,(wrs) = i2tei— — Tt > -l. (5.4) 2 I>J=1 I=1 OTe *1y (Note that the relation vr j=1 My(wrsz) > —1, for No < N, holds in general.) Equation (5.4) contrasts with the rule given by (2.17). Thus, the measured total modulation depth of a non-a-type band provides a measure by which one can decide if the number of levels indicated by the number of beat frequencies is, in fact, the total number of coupled levels. (This, of course, assumes the ability to accurately measure modulation depths.) This same information would also be obtained if the beat phase distributions indicated that more than N — 1 types of bands occurred in the spectrum. In terms of the physics of IVR, case 1 corresponds to the incomplete preparation of the |a) state at ¢ = 0. In other words, the initially prepared state is not purely |a) as it would be if the bandwidth of the laser spanned all the eigenstates arising from the coupled system of zero-order states (see section VI-A), but it contains contributions from the other zero-order states, as well. The deviations resulting 50 from the case 1-type situation are therefore fundamental in nature and underscore the fact that the IVR process is intimately dependent on the preparation of the initial state. B. Case 2: Finite temporal resolution In this case we assume that the temporal resolution of the detection system is not sufficient to resolve some of the beat frequencies present in the decays. Obvi- ously, under all circumstances the inability to resolve some beat components reduces the number of observed beat frequencies from the N(N — 1)/2 expected for N cou- pled levels. In many cases this will not lead to an incorrect assignment of the number of coupled levels involved. For instance, if for a four-level system only five beat frequencies were resolvable, then despite the missing component, one would not be led to assign the behavior to a three-level system, since in that case there would be only three beat components. Similarly, if one observed three beat compo- nents at say 5, 7, and 8 GHz, it would be immediately apparent that the number of coupled levels involved would have to exceed three, since 5+ 7 4 8 GHz. On the other hand, errors in interpretation can be made if the only beat frequencies that are unresolvable are all those involving one of the eigenstates. For example, say all wyn, for J = 1,...,N —1, are too large. In such a case each band will be apparently modulated as if it arose from an (N — 1)-level system : N N=-1 I(t} ~ > at, at, +2 > Apa saA pA yy coswyyztle*. (5.5) T=1 [>J=1 Nevertheless, here as with case 1, errors in interpretation can, in principle, be avoided by consideration of the total modulation depths of non-a-type bands pints 2 yet AreaQslegai~aszy » M,,(wrs) = 2S 22 (5.6) [>J=1 Lr=1 “Fat Since (5.6) is the same as (5.2) except for one extra positive term in the denominator, then the sum of modulation depths in (5.6) is > —1. Thus, this experimentally measurable quantity can again be used to ascertain whether the apparent number of coupled levels is, in fact, the actual number of coupled levels. ol C. Case 3: Finite spectral resolution Similarly to effects found due to finite temporal resolution, one might expect spectral detection resolution problems to be an especially prevalent source of experi- mental deviations from the theoretical predictions of sections II and III. Even single, nonvibrationally coupled $1 vibrational levels of large molecules (for instance, the 5; vibrationless level) have rich spectra with many bands. Since an N-level system is a superposition of such spectra, there is bound to be some spectral overlap of bands, even for N as small as three. This overlap can result in a number of appar- ent deviations from theoretical predictions. For example, if an a-type band overlaps with a non-a-type band, then the measured decay may appear to be non-a-type be- cause of Fourier components with —1 phases, yet the total modulation depth of the decay will be found to be > ~—1. Or, if two different non-a-type bands overlap, then it is possible that the measured decay may exhibit forbidden phase behavior; ¢.g., a quantum beat triplet may appear to have a phase distribution STIK 65°" = (+1, +1, -1). Clearly, spectral overlap can also be an especially serious hindrance if one wishes to obtain accurate modulation depths for a calculation of Ho. This must be kept in mind in choosing the experimental results to be used in making such a calculation. D. Case 4: More than one optically active level, uncoupled As a fourth case, consider the situation wherein the eigenstates formed by the coupling of the M zero-order states |a),|b),... and the eigenstates formed by the coupling of the N — M zero-order states |a’), |b’},... are all spanned by the finite bandwidth of the laser pulse. Assume that the unprimed zero-order states are only coupled to other unprimed states and give rise to the eigenstates |I), J =1,...,M, while the primed states are only coupled to primed states and in so doing give rise to the eigenstates |Z), J = M+1,...,N; (see Fig. 4). In such a situation C is block-diagonalized into two blocks such that all the elements a;, are zero if 1) 7 is unprimed and J = M+1,...,N; or 2) ¥ is primed and I = 1,...,M. Now, 52 further assume that only one zero-order state in each set (say |a) and |a’)) has any absorption strength from the populated Spo level. Using (2.3) one can obtain the following for the temporal behavior of the fluorescence band associated with the transition to the So level | f,) : L(t) =I, +15+55, (5.74) where M 2 Te~ YY ara saPl (Sy IPM (fa, De Gest, (5.76) i,J=1m=1 N 2 ty ~ > > Arar sa PE" (fy, J)PP (fy, T)eW Gert Tt, (5.7c) I,f=M+1m=1 and M N 2 B~230 SM area sa PM (fy S)PP (Fy, L) cos(wrztheF*. (5.74) f=1 J=M+1m=1 As before, we assume that the transition to |f,) gains its emission strength from only one of the zero-order levels. Unlike before, however, one sees from (5.7) that not all N band types are modulated by all the same beat frequencies. In fact, three different classes of band types can be distinguished based on the frequencies which modulate the bands. Firstly, if the band gains its emission strength from one of the unprimed zero-order states, then 53 = S$ = 0 and Sj reduces to what one would expect for a band of an M-level system. One expects as many as M types of bands exhibiting such behavior. Similarly, if the band gains its emission strength from one of the primed zero-order states, then S} = S$ = 0 and S? reduces to what one would expect for a band arising from an (N — M)-level system. As many as N — M bands of this type may occur. Finally, there will be at least one band (é¢., the resonance fluorescence band) which gains its emission strength from |a) and |a’). I,,(t) for such a band will have contributions from S!, S?, and S$. Thus, all N(N — 1)/2 beat frequencies of the N-level system may modulate the band. 53 Moreover, all these Fourier components will have +1 phases since a;, and /or ara only enter as squared terms in S}, S?, and Ss. The situation corresponding to case 4 is not particularly unlikely to occur. For example, one can easily imagine two sets of states at the same energy having different symmetry transformation properties. While inter-set anharmonic coupling in such a case would be symmetry forbidden, absorption could still occur to states in both of the sets (e.g., in anthracene, S, vibrational levels of both ag and by, symmetry have been observed to be optically active”). BE. Case 5: More than optically active level, coupled Here, we consider a situation which gives rise to deviations from theory, not because of any experimental limitations, but because the physical situation does not correspond to the basic assumptions of the theory. In particular, we consider the case wherein there are no restrictions on the absorption transition matrix elements P2(g,I) or on the emission transition matrix elements P’"(f,,I) in (2.3); that is, (2.4) and (2.5) do not hold. Since this general case is difficult to treat for any but the simplest systems, we only present results for N = 2. As in section II-A, it is useful here to express the transition matrix elements in terms of the zero-order states |a) and |b). We also neglect the effects of detection polarization. Thus, P,(g, I) = ara Pz(g, a) + aypPz(g,), (5.8) and Pa( fy, 2) = araPa(fy, 4) + are Pa( fy, 6). (5.9) For the sake of clarity we define X, = P,(g,y) and D, = Pi(f,,7). Using the facts!® that aig = —a@qy = @ and dog = yy = B along with (2.3) to find the decay of the fluorescence band with final state | f}, one obtains the following for the modulation depth of the beat component wy9: M(wy) = 2(aXq + BX4)(8Xq -— aXs)(aDz + BDy)(8Da — aDs) (aX, + AXp)*(aD, + BDs)* + (8X, —aX;)*(BD, aD,» (49) 54 It is apparent from (5.10) that the general case differs from that of section II in that modulation depths do not depend solely on the coupling coefficients but are dependent on zero-order transition matrix elements, as well. Even in a two-level system, one might therefore expect many different fluorescence bands, each with different M(w 2) depending on the particular values of D, and D, for the band. In addition, the possibility exists, despite strong coupling between |a) and |b), that no modulations will appear in any decays if X, and Xp, or Dz and Dy, take on certain values. | It is instructive to consider two limiting cases of (5.10). In particular, one wonders what happens if either D, or D; is equal to zero, but both the X, have nonzero values. This corresponds to the case of two coupled optically active levels which have different fluorescence spectra. If one detects the band for which D, equals zero, one has what is analogous to an a-type band, the modulation depth of which is 2(aXo + 8Xs)(BX_ — aX,)ap Malwa) = oe oxen (5.11) With D, equal to zero one obtains . Ms (wg) = —22ie%e + 8Xo)(BXa — aXs)as (5.12) (aX + 9Xy)? 8? + (8X, — aXp)?20?' Comparing (5.11) and (5.12), one can see that the decays of the two types of bands have opposite quantum beat phases (+1 ve. —1) just as in the two-level case of sec- tion III-A. Now, however, that decay which has the —1 phase cannot be ascertained without specific knowledge of the coupling coefficients and the X.. The same holds true for the magnitudes of M,(w1g) and M,(wig). In general, one expects the mod- ulation depths to decrease with both weaker coupling and with |X,| more nearly equal to |X,|. One also expects no decay for which the modulation depth equals —1 (in contrast to (2.17)). It is implicit to the relevance of this chapter that in many real situations the approximations of (2.4) and (2.5) be good ones and that case 5 be an overcom- plication. As we have argued, theoretical considerations and experimental results 55 support this. Given cases for which (2.4) and (2.5) are not valid, one sees from (5.10)-(5.12) that even for N = 2 significant deviations from the results of sections II and III can occur. F. Case 6: Thermal congestion A primary assumption made in the treatment of section III involves the pres- ence of only one initially populated So state (|g)). However, even in jet-cooled sam- ples there exists a distribution of initial states due to incomplete cooling of both rotational and vibrational degrees of freedom. This distribution of initial states can in turn give rise to a distribution of excited states upon laser excitation. The effects that a distribution of excited state rotational levels has on beat-modulated decays are dealt with in Chapter 5.” The influence of the vibrational distribution, although not as pervasive as that of the rotational distribution, nevertheless can be substantial. For instance, consider a vibrational distribution characterized by some molecules in the Spo level |g) and all others in the level |g.). Excitation will project the two types of molecules up into entirely different regions of the excited state level structure given that |g,} and |go) are well separated in energy relative to the bandwidth of the laser pulse. Spectrally, the effect this has is to introduce more congestion in the observed fluorescence spectrum than would otherwise be present if all molecules were initially in |gi}) or |g2). Temporally, if excitation from |g;) co- herently prepares an Nj-level system and excitation from |go) prepares an No-level system, then there will be bands in the observed fluorescence spectrum reflecting coupling amongst Nj levels, and different bands reflecting coupling amongst N2 levels. The point is that two different and independent dynamical processes arising from two incoherently related excitations are manifested in the observed spectrum and fluorescence decays. Thus, in cases where observations indicate deviations from theoretical predictions regarding the number of beat components and the number of beat triplets, it is possible that the reason for this is vibrational thermal congestion. In the extreme situations where many Sp vibrational states are initially populated, thermal congestion can completely mask any coherence effects that may be present. 86 Of course, this is at the heart of the rationale to use jet-cooled molecular samples in the study of large molecule vibrational dynamics. Experimental results pertaining to the effects of thermal congestion are presented in Chapter 5.” 57 VI. PHASE-SHIFTED QUANTUM BEATS AND IVR A. Direct view of IVR Being that phase-shifted quantum beats occur as one of the dynamical manifes- tations of S, vibrational coupling in molecules and that the specific characteristics of observed beats can be used to obtain specific vibrational coupling information about a molecule, then the study of phase-shifted vibrational beats amounts to the study of IVR. However, the connection between beats and IVR is even closer than this might lead one to believe. Far from being simple dynamical manifestations of vibrational coupling, beat-modulated fluorescence decays are direct pictures of the distribution in time of vibrational energy in a molecule. This point may be seen by considering the contribution ( g,(é) ) of the zero-order state |y) to the excited state |W(t)) created by the laser pulse: p(t) = |(7|¥(¢)))?. (6.1) Using the assumptions of §-function excitation and the exclusive absorption strength of |a) one obtains for the normalized excited state N N |W(t)) = > ya|T)e~CertT/2)t > > Oradpy|ye He EtE ANE, (6.2) f=1 f=1 y=a,6,... And using (6.2), N’ . . N . g(t) = | > Opaapye HIF /2)t 2 > Apatyatryagye “HIDE (6,3) [=1 [,J=1 (Actually, the requirement of a 6-function excitation pulse is somewhat restrictive. A more realistic condition is that the inverse correlation time of the excitation source—the coherence width—be much greater than the maximum value of wy,. This same condition, which ensures that all of the relevant eigenstates are excited and locked in phase at f = 0, also applies to (2.6). Comprehensive discussions on the question of initial state preparation can be found in Refs. 10 and 16.) 58 By comparison of (6.3) with (2.6), it is evident that p(t) ~ I(t). (6.4) The temporal behavior of a y-type band is exactly the same as the temporal behavior of the |y) content of the excited state. In another manner of speaking, a y-type decay provides a direct picture of the oscillatory flow of energy into and out of the vibrational motion described by |y). Thus, the two fundamental aspects of the IVR process can be addressed by time- and frequency-resolved fluorescence. The extent of IVR (the number of coupled vibrational levels) is given by the number of beat frequencies which modulate the decays of the bands in a spectrum, and the tsmescale for energy redistribution is given by the inverses of the beat frequencies. There are other ways in which the correspondence between I(t) and p(t) can be useful in the study of IVR processes. For example, (6.4) allows one to carry all the theoretical results on the modulation phases and modulation depths of fluorescence decays (sections IT and III) over to the behavior of g,(t). Conversely, if one first considers the behavior of g.(t), which has a ready interpretation in terms of IVR, then it is possible to come to a fuller appreciation of the physical significance of some of the results derived for I,(t). One example of this point is (2.17), which holds that the total modulation depth of a non-a-type decay is —1. What does this mean physically ? To answer this note from (6.3) that for 7 # a, N N N : p+(0) => AratsJatiryasy = 2 wr Qyaasy) areasy =O (6.5) I,J=1 f=1 J=1 by orthogonality. That is, |y) does not contribute to the initially prepared excited state. Now by (6.4) this implies that I,(0) = 0, which in turn, by (2.6) and (2.7) implies that the total modulation depth of the y-type band must be -1. Thus, the modulation depth sum rule (2.17) for non-a-type bands is a reflection of the vibrational content of the initially prepared excited state. Similarly, the fact that an a-type band is modulated exclusively by beats with positive phases is a reflection 59 of the vibrational content of |W(t)) at t= 0. By (6.3) and the normalization of the Qa, N N N pa(0)= D7 a7,03, = (D> a3,)a3, = 1. (6.6) I,J=1 f=1 Jal That is, |(0)) = |a). Therefore, g(t) < p,(0) and Z,(t) < I,(0). But this can only be true in general if all the a-type phases are +1. Thus, the qualitative difference in phase behavior between a-type and non-a-type bands has clear meaning in terms of IVR. The presence of —1 phases in the decays of y-type (7 # a) bands reflects the fact that vibrational energy will redistribute into the |y) states as time increases. On the other hand, the fact that the amount of vibrational energy in |a) is at a maximum at ¢ = 0 (by virtue of the exclusive absorption strength of |a)) is reflected in the entirely positive phases of a-type decays. A final utility of the correspondence between I,,(é) and g(t) is that it provides a means by which one can experimentally search for quasi-periodic and chaotic vibrational behavior in molecules. Being that such behaviors have been shown to be manifested in characteristically different ways in g+(t),1” then measurement of I(t) would be expected to provide information on the nature (s.e., quasi-periodic or chaotic) of the vibrational dynamics of a given molecule at a given energy. B. Dependence of decay behavior on total vibrational energy A number of published accounts!!418 have dealt with the effects of excess vibrational energy on large molecule IVR, as manifested in dispersed fluorescence spectra. From these studies, trends toward increasing spectral congestion and de- creasing intensity in the unrelaxed region of the fluorescence spectra were observed as the vibrational energy was increased. These have been interpreted as being manifestations of faster and more extensive IVR. at higher energies. Physically, this is reasonable since the average spacing of vibrational states decreases, and, thus, the opportunity for coupling increases with energy. However, the information available from purely spectral results is limited and can be misleading®. On the other hand, we have seen that fluorescence decay measurements can directly reveal 60 a large amount of information about IVR. Thus, it is of interest to consider how time-resolved results might be expected to be dependent on excess vibrational en- ergy and how they might correlate with the trends in dispersed fluorescence spectra. As mentioned above, the principal expected modification of vibrational cou- pling with increasing vibrational energy is an average increase in the number of levels coupled to one another. The consideration of the changes in decay behavior with excess energy is, therefore, tantamount to the consideration of the changing nature of decays as N increases. Guided by experimental results®, we have found it useful to classify decays according to three ranges of N: (1) VN =1, (2) V =2 to ~ 10, and (3) N > 10. These ranges of N may be identified with low, intermediate, and high vibrational energy regimes, respectively, in real molecules. No IVR The N = 1 case obviously corresponds to no IVR. The spectral manifestations of this are uncluttered spectra consisting of sharp vibronic bands assignable in terms of optically active vibrational intervals. The decays of all the bands in such a spectrum are identical and in most situations (.e., barring any unusual nonradiative channels such as small energy gap intersystem crossing) will be single exponential and unmodulated. Restricted IVR The N = 2 to ~ 10 case corresponds to restricted IVR. A dispersed fluorescence spectrum belonging to this case consists of an uncongested, vibrationally unrelaxed region (see section II-D) composed of weak intensity a-type bands, and a congested, vibrationally relaxed region consisting primarily of non-a-type bands. The decays of the various bands in the spectra adhere to the characteristics presented in sections Il and III. They differ from the N = 1 case in that the decay behavior depends on detection wavelength, and that modulations are always present. The feature that distinguishes the decays of this case from those arising from systems of greater N is the presence of a number of essentially full recurrences on the timescale of the fluorescence lifetime. That is, each intensity I,(t) (and each g,(¢)) has a number 61 of maxima, all of which approach the global maximum of that intensity. IVR is restricted in the sense that energy does not irreversibly flow from one vibration into others, but instead oscillates between a small number of levels. Dissipative IVR Finally, the case where N is large corresponds to dissipative IVR. Spectrally, this case is characterized by very weak a-type fluorescence bands in the unrelaxed region, and very congested, predominantly non-a-type fluorescence in the relaxed spectral region. Due to the large number of interference terms modulating fluores- cence decays belonging to this case, the overall contribution of these terms is at a maximum at ¢ = 0, after which a dephasing occurs which renders the contribution small for all subsequent times. For a-type decays, in which all the interference terms have +1 phases, this dephasing gives rise to a rapid initial decay component®!® fol- lowed by a long component of comparatively small intensity. For non-a-type decays, in which the initial intensity of zero arises from the —1 value for the sum of the modulation depths, the dephasing results in a risetime followed by a long decay. In both cases the possibility exists that some small modulations may occur on the long decay components; t.¢., some partial recurrences may occur on the timescale of the fluorescence lifetime. However, the dephasing is sufficiently complete to exclude anything close to a full recurrence on this timescale. This type of decay behavior represents the irreversible flow of energy out of the initially prepared |a) excited state into the other coupled zero-order states. Lahmani, e¢ al.!° have shown (in the context of singlet-triplet coupling) that the decays corresponding to large N may be reproduced by assuming a kinetic rate equation model for the values g,(t). For an a-type decay this model predicts a double exponential decay of the form In(t) ~ pa(t) = S{(N — eT ETA 4 eM}, | (6.7) i N where I’ is the same as in section II, and A is a measure of the width of the distribution of beat frequencies and corresponds to the dephasing contribution to 62 the decay. For non-a-type decays taken together the model gives LHW ~ YO walt) = tere et tay, (6.8) Ta ya These equations, by virtue of their very simplicity, serve several useful purposes. Firstly, they provide at a glance the general temporal behavior of IVR in the dis- sipative regime. Secondly, they serve as convenient functions with which one can fit experimental decays and thereby obtain IVR parameters. Thirdly, they relate in a simple way the parameter A to the rate of dissipative IVR. A, in turn, can be related?° to molecular parameters such as the vibrational density of states and vibrational coupling matrix elements. Finally, their derivation by means of a kinetic model makes apparent the close analogy between IVR. and kinetic behavior. In par- ticular, one can meaningfully speak of 1 /A as an equilibration time during which a molecule, initially jarred out of equilibrium by the laser, evolves to a “steady-state” in which no further evolution in the distribution of vibrational energy occurs. One must, however, beware of taking (6.7) and (6.8) too literally. Firstly, as pointed out in Ref. 19, the equations exclude the possibility of quantum beats. Secondly, (6.7) predicts that the ratio of the pre-exponential factors of fast to slow fluorescence for an a-type decay will be equal to N — 1, the number of states coupled to ja). However, by (2.13) one can see that the value N — 1 is actually an upper limit to this ratio, and it holds only if |a;,| = 1/N, for all J. 63 VII SUMMARY AND CONCLUSIONS In this chapter we have considered the dynamics of molecular vibrational en- ergy flow between an arbitrary number of coupled vibrational levels. In particu- lax, the concern has been centered on the dynamical manifestations of vibrational coupling in the temporal characteristics of spectrally resolved fluorescence. These manifestations take the form of coherence (phase-shifted quantum beats) in fluores- cence decays, whose number of beat frequencies is related to the number of coupled . levels, and whose beat phases and modulation depths are dependent on the fluores- ceuce band detected. It has been shown that these phases and modulation depths are functions of the coefficients that connect the the basis set of the zero-order, coupled vibrational wavefunctions to the molecular eigenstate basis set. As such, they are not arbitrary, but adhere to certain rules and restrictions ultimately based in the orthonormality of the mixing coefficients. Conversely, it has been shown that with knowledge of beat phases and modulation depths, it is possible to deter- mine these coefficients, which, when so determined, can then be used along with the beat frequencies to construct the Hamiltonian matrix describing the coupling between the zero-order vibrational states. Thus, the measurement of time- and frequency-resolved fluorescence can 1) indicate the presence of vibrational coupling in a molecule, 2) indicate the number of coupled levels, and 3) provide information leading to the determination of the details of the coupling. It has also been shown in this chapter that, more than just providing parameters pertinent to vibrational coupling, the temporal behavior of dispersed fluorescence has an immediate and simple interpretation in terms of IVR. The intensity ve. time of a given band in a fluorescence spectrum is a direct picture of the evolution of vibrational energy in the excited state. Quantum beat frequencies are related to the IVR timescale. Beat phases and modulation depths have straightforward interpretations in terms of the vibrational energy content of the excited state. And, most important, the nature of IVR at a particular vibrational energy is apparent from the decays of fluorescence bands. Thus, one can readily assess the transition 64 from absent to restricted to dissipative IVR in a given molecule. The results presented in this chapter have been derived to facilitate the in- terpretation of real experimental data. In the following chapters these theoretical results will be applied to observations made on the molecules anthracene® and trans- stilbene.® 65 REFERENCES nv fo ene oT Py 1c. 11. 12. 13. 14: 15. 16. For a review see: C. S. Parmenter, Faraday Discuss. Chem. Soc. 75, 7(1983). P. M. Felker and A. H. Zewail, in Applications of Picosecond Spectroscopy to Chemistry, edited by K. B. Eisenthal (D. Reidel, Dortrecht, 1984), p. 273. P. M. Felker and A. H. Zewail, Chem. Phys. Lett. 102, 113(1984). P. M. Felker and A. H. Zewail, Phys. Rev. Lett. 58, 501(1984). P. M. Felker and A. H. Zewail, Chem. Phys. Lett. 108, 303(1984). P. M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. P. M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. P. M. Felker, W. R. Lambert, and A. H. Zewail, J. Chem. Phys. - submitted. We assume here that all of the matrix elements of Ho, the Hamiltonian matrix in the zero-order basis, can be expressed as real numbers, in which case all the elements of C can be taken as real. See, for example: S. Haroche, in High Resolution Laser Spectroscopy, edited by K. Shimoda (Springer, New York, 1976), p. 254. Since all the excited eigenstates are 5, levels of about the same energy, one does not expect I to vary much from level to level. Taking I to be equal for all levels is therefore a reasonable approximation. It is pertinent to note that even if [ were not the same for all eigenstates, this would not affect the beat phases or modulation depths in a given decay. Examples are the spectra of anthracene appearing in: W. R. Lambert, P. M. Felker, J. A. Syage, and A. H. Zewail, J. Chem. Phys. 81, 2195(1984). This is a different situation from that which obtains in, for example, Zeeman quantum beats, where beat phases and modulations depend on the polariza- tions of the excitation source and detection analyser, as well as on the mag- netic field direction. A discussion of polarization effects on molecular quantum beats may be found in: M. Bixon, J. Jortner, and Y. Dothan, Mol. Phys. 17, 109(1969). For experimental results on Zeeman quantum beats see, for exam- ple, Ref. 10; and R. Wallenstein, J. A. Paisner, and A. L. Schawlow, Phys. Rev. Lett. $2, 1333(1974). For example, S. M. Beck, J. B. Hopkins, D. E. Powers, and R. E. Smalley, J. Chem. Phys. 74, 43(1981). G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules (Van Nos- trand, Princeton, 1945), p. 216. W. Rhodes, in Radiationless Transitions, edited by S. H. Lin (Academic, New York, 1980), p. 219. 17. 18. 19. 20. 66 See, for example: (a) G. Hose and H. S. Taylor, Chem. Phys. 84, 375(1984); (b) M. D. Feit and J. A. Fleck, Jr., J. Chem. Phys. 80, 2578(1984). For example, (a) P. S. H. Fitch, L. Wharton, D. Levy, J. Chem. Phys. 70, 2018(1979); (b) A. Amirav, U. Even, and J. Jortner, ibid. 74, 3475(1981); (c } C. Bouzou, C. Jouvet, J. B. Leblond, Ph. Millie, A. Tramer, and M. Sulkes, Chem. Phys. Lett. 97, 61(1983); (d) M. Fujii, T. Ebata, N. Mikami, M. Ito, S. H. Kable, W. D. Lawrance, T. B. Parsons, and A. E. W. Knight, J. Phys. Chem. 81, 2209(1984). F. Lahmani, A. Tramer, and C. Tric, J. Chem. Phys. 60, 4431(1974). C. Tric, Chem. Phys. 14, 189(1976). 67 Table 1. A summary of results for N = 3. No. of band-types No. of beat frequencies No. of beat triplets 3 3 1 Four possible quantum beat phase distributions: Distribution Independent phases Dependent phase S,,(wia) S,(w1s) S.;(wes) 51 +1 +1 +1 52 +1 ~1 -1 <3 _ 65 -1 +1 1 ie -1 -1 +1 68 Table 2. A summary of results for N = 4. No. of band-types No. of beat frequencies No. of beat triplets 4 6 4 Eight possible quantum beat phase distributions: Distribution Independent phases Dependent phases Sy(wia) Sy(wis) Sy(wis) Sy (was) Sy(wea) Sy(wsa) 51 +1 +1 +1 +1 +1 +1 5? +1 -1 -1 -1 -1 +1 53 1 +1 -1 -1 +1 1 54 -1 —1 +1 +1 -1 —1 55 +1 1 +1 —1 +1 ~1 58 +1 +1 -1 +1 -1 -1 5° —1 +1 +1 ~1 -1 +1 4 i=) 2 oO | — i fot { j—_ + pot + t— + re 69 Table 3. A summary of results for N = 5. No. of band-types No. of beat frequencies No. of beat triplets 5 10 10 Sixteen possible quantum beat phase distributions: Distribution Independent Phases : Sala) Salis) Salus) Salons) 2 +1 +1 +1 -1 3 +1 +1 —1 +1 4 +1 —1 +1 +1 5 ~—1 +1 +1 +1 6 +1 +1 ~1 -1 7 +1 -1 +1 ~-1 8 —1 +1 +1 —1 9 +1 ~—1 —1 +1 10 —1 +1 -1 +1 11 -1 —1 +1 +1 12 +1 -1 —1 —1 13 ~1 +1 ~1 —1 oh oS TF | J | ht / | | a eo | 1 + pt | o+ | nn a 70 FIGURE CAPTIONS 1. Level diagram illustrating the general situation to be treated herein. The zero-order harmonic vibrational states on the left are coupled by anharmonic interactions. Only one of these states, the optically active |a) level, can be reached from the ground state via the excitation process. All of the other levels are optically inactive (dark). On the right are pictured the vibrational eigenstates that result from the interactions among the zero-order states. The different lengths of the lines representing these levels are meant to indicate that each eigenstate has a different contribution from |a), and hence has a different absorption strength from the ground state. Level diagram and fluorescence spectrum representing the dispersed fluores- cence characteristics to be expected from a set of coupled vibrational levels in S,. Excitation prepares the zero-order |a) state (indicated by the asterisk) which then undergoes IVR. Emission gaining its strength from |a) is termed vibrationally unrelaxed and tends to occur in the blue region of the spectrum. Emission gaining its strength from |b), |c),..., is termed vibrationally relaxed and occurs near and to the red of the 0$ transition energy of the molecule. Schematic diagram of the situation treated as case 1 (section V-A), wherein it is assumed that the bandwidth (Aw,) of the excitation source is not large enough to span all the eigenstates {|1),...,|N)) resulting from the coupling of a set of N zero-order states. Schematic diagram of the situation treated as case 4 (section V-D), wherein it is assumed that the laser bandwidth is large enough to excite the eigenstates (right) corresponding to two independent (noninteracting) sets of coupled zero- order levels (left). Only two zero-order levels are assumed to have absorption strength from the ground state—the |a) and |a’) states. 71 Figure 1 _ ZERO-ORDER STATES i 7 OPTICALLY, ; © C b y ACTIVE | LEVEL ° | OPTICALLY INACTIVE LEVELS > > VIBRATIONAL EIGENSTATES MV“ Ee | een =e eee oa -~eecee35we3ewe \“ 7 Figure 2 |a>* IVR — , Wn b>, |c>, |dd, .. Y ‘AN \ i "Unrelaxed" “Relaxed U-emission R-emission “oN “™N <—_eeeeee¢©66h6(7)lhC<‘“‘:SD ! ie | 1 | WW 74 Figure 4 ZERO-ORDER STATES | 0) ee [o> ee e a WW, X EIGENSTATES ®eeeeg? 75 CHAPTER 4 ENERGY DEPENDENCE OF IVR IN ANTHRACENE* * Submitted to the Journal of Chemical Physics, as “Dynamics of intramolecular vibrational-energy redistribution (IVR). Il Excess energy dependence,” by Peter M. Felker and Ahmed H. Zewail 76 I. INTRODUCTION In the context of the theory presented in the previous chapter this chapter provides an account of experimental results pertaining to IVR in anthracene. The results were obtained with temporal resolution five times better than that used in previous studies.'~* This has allowed the resolution of beat frequencies up to ~ 12 GHz, and the resolution of decays with time constants on the order of 10 psec. A primary emphasis of the work is on elucidating the changes which take place in IVR processes as the vibrational energy of the molecule increases. In addi- tion, using the theoretical results of Chapter 3 on multilevel systems we show the utility of using quantum beat spectroscopy to reveal parameters associated with vibrational coupling in anthracene for excess vibrational energies ranging from 0 to ~ 2000 cm7?. The organization of the chapter is as follows. Firstly, the experimental tech- niques and data analysis are discussed. Of principal importance in this section is the presentation of the Fourier analysis methods which allow one to extract quantum beat frequencies, phases, and modulation depths from fluorescence decays. Sec- ondly, the results for various single vibronic levels (SVL’s) in S, anthracene are presented in the order of increasing SVL energy. These results are analyzed to determine the number of coupled levels giving rise to the observed behavior and, where possible, to determine elements of the Hamiltonian matrix in the zero-order vibrational basis set. Finally, the overall trends of the results and the timescales for IVR in the molecule are discussed. 77 Il. EXPERIMENTAL A. Apparatus The picosecond-jet apparatus used in the experiments reported herein is de- scribed in detail elsewhere”. The nozzle conditions were much the same as for pre- vious work on anthracene!—*. The anthracene sample (Aldrich gold label, 99.9%) was heated to ~ 180° C and expanded through a ~ 100 ym pinhole with 30 —40 psi helium. The laser-to-nozzle distance was typically 3mm. The excitation source con- sisted of the frequency-doubled (LilO3) output of a synchronously pumped, cavity- dumped dye laser (DCM as dye), the bandwidth of which was ~ 2 cm~! FWHM and the temporal pulse width of which was 15 ps FWHM. Fluorescence was col- lected with right angle geometry and directed through a 0.5 M monochromator. The dispersed fluorescence spectral resolution varied with the particular measurement being made and is noted with the data. Fluorescence photons were detected with a Hamamatsu R1564U multichannel plate photomultiplier, the output of which was amplified before encountering the timing electronics. Decays were measured using time-correlated single photon counting. The total temporal response of detection (response function) was typically 80 ps FWHM as measured by scattering laser pulses off of the nozzle. Care was taken to monitor any major temporal shifts in the peak of the response function since such shifts affect the apparent phases of quantum beat components. Generally, large shifts were not a problem and are not relevant to any of the results included herein. B. Extraction of beat parameters by Fourier analysis It is well known that a measured fluorescence decay F(t) may be related to a true molecular decay I(t) by the convolution integral® F(t) = [ R(t )r(é — t') de’, (2.1) where R(t) is the response function of the experimental apparatus. From theoretical considerations’ pertaining to vibrational quantum beats the molecular decays are 78 expected to be of the form N T(t) =Re[Ae“™ (1+ S$ M(wry)e“#™*)], (2.2) [>J=1 where A is a constant, I is the excited state decay rate, N is the number of coupled vibrational levels, and M (wr s) is the modulation depth of the beat frequency wyy. Now, since one is interested in extracting the values for w;; and M(wrz) from a Measured decay, Fourier analysis is the natural method by which to accomplish this. In particular, one can make use of the theorem! for the Fourier transform of a convolution integral to obtain F(u) = Tw) R(w) (2.3) or I(w) = F(w)/R(w), (2.4) where F(w), I(w), and R(w) are the Fourier transforms of F(t), I(t), and R(é), respectively. This procedure of dividing the transform of the measured decay by the transform of the response function automatically compensates for two defects in F(w): (1) it enhances the amplitudes of the higher frequency components which tend to be washed out by the finite response function width and (2) it establishes the proper phases for the beat components. By measuring F(t) and R(é), and then calculating F(w) and R(w), one can obtain I(w) by (2.4). For this paper we have calculated F(w) and R(w) by using the Cooley-Tukey fast Fourier transform algorithm® to transform data zero-filled! to 4096 points. After finding I(w), the wyy are easily obtained by the positions of the bands in the Fourier spectrum. The values of the beat frequencies reported herein are actually: the values wy; /22. The accuracy of these numbers is ultimately limited by the resolution of the Fourier spectra and the accuracy of the time base. A conservative estimate of the error in quoted beat frequencies is +5%. 79 The M (wry) are available from the Fourier amplitudes of components in I (w), as is evident from Re(I(w)) = K(—- — + > TM (wr) na tT? t (ors - al (2.5) In principle, the heights of the bands of Re(I(w)) normalized to the zero frequency band height should give the various absolute modulation depths M(w,,). In prac- tice, it is more accurate to integrate over the individual band shapes in I(w) and then to normalize these sums to the integrated intensity of the zero frequency band. This is better for two reasons. Firstly, small variations in vibrational coupling with rotational level!! tend to broaden the nonzero frequency bands in I(w) relative to the breadth [ of the zero frequency band. That which is pertinent, therefore, are not the heights but the integrated intensities of the bands. Secondly, it is often nec- essary to collect data in such a manner that decays are truncated at a significant fraction of their initial intensities. This has the effect of introducing sinusoidal oscil- lations about zero into the derived I(w). Summing over a band averages out these oscillations and, by so doing, gives a more accurate value for the Fourier amplitude. A number of factors contribute to errors in the derived values for the modula- tion depths (see section V of Chapter 3), The first is noise in the Fourier spectra. The second factor arises from the inability to completely spectrally isolate bands in the fluorescence spectrum. The influence of this factor varies from band to band. It can be significant even for detection resolution better than 1 A. Thirdly, small drifts in the response function during the course of the measurement of a decay can make a large difference in the values derived for the modulation depths of high frequency beat components. This is the case because the width of the measured response function R(t) will, in general, be narrower than the effective response function R.;7(t) corresponding to the decay. (Typically, the time needed to meas- ure a response function is a small fraction of the time needed to measure a decay. This is why one expects R(t) to be narrower than R,;;(t).) Thus, in (2.4) R(w) will be larger than it should be. A specific example assuming a Gaussian response 80 function will illustrate the point. If the measured R(t) is 77 psec FWHM, while Rezz(t) is 85 psec FWHM (only a drift of one channel under typical experimental conditions) then R(w) ~ Res;(w) = 0.975 for w/2r = 1 GHz, but R(w) = 0.218 while Rez7(w) = 0.158 for w/2r = 8.5 GHz. By using R(w) in (2.4) essentially no error will be introduced for the 1 GHz modulation depth, but the modulation depth derived for the 8.5 GHz component will be only S188 = 72% of its actual value. One expects this source of error in modulation depths to increase in influence as the time needed to measure a decay increase. It is likely that it is the major source of error in the quoted modulation depths of high frequency components. Despite the value that I(w) has in the analysis of a beat-modulated fluorescence decay, the large zero-frequency component and the marked, truncation-induced os- cillations of a typical [(w) spectrum render it relatively unattractive for display purposes. On the other hand, the Fourier spectrum I,,(w) derived solely from the transformation of the modulated portion of a decay multiplied by a suitable shaping function provides almost as much information, as the corresponding I(w) (absolute modulation depths are not available from I(w)), but is rid of the zero-frequency peak and the spurious oscillations. For this reason most of the Fourier spectra which appear as figures in this chapter are I,,(w) and not I(w) spectra. To obtain [,,(w) for a given decay, the decay is first fit to a single exponential function by using a nonlinear least-squares fitting algorithm.!? Convolution effects are not considered in this initial portion of the analysis since, in general, 1 /T is much greater than the width of R(t). The fitted function is then subtracted from F(t) and the residual F,(é) is multiplied by the function!’ (t; — t)>, which causes it to go to zero smoothly as ¢ — ty, t; corresponding to the point in time at which F(t) is truncated. This manipulation substantially eliminates any spurious oscillations in the Fourier spectrum. The function (f; — £)°F,,(t) is then Fourier transformed and divided by R(w) to give Im(w). Now, as we have stated, Re(Im(w)) preserves the phase behavior and the relative values of the M(w; ) of a given decay. One should, however, note two points about the spectra. Firstly, the application of 81 the shaping function (ty — é)> to Fy, (t) results in artificial widths for the Fourier bands in In(w). Secondly, the shape of the bands appearing in I,,(w) depend on that portion of F(t) which is chosen initially to be fit to the exponential function. The band shape distortions that arise from this second cause become more serious, the farther a decay is fit from its peak. For all practical purposes neither of the above artifacts is important enough to substantially modify the values for the beat frequencies, phase behavior, or relative modulation depths derived for a given decay. C. Analysis of fluorescence decays in the dissipative regime In the limit where the number of coupled vibrational levels is large enough to give rise to dissipative IVR,° the utility of Fourier transform techniques is limited due to the large number of beat components in any given decay. For such cases, the extraction of IVR parameters from decay measurements can be accomplished by using equations (6.7) and (6.8) of Chapter 3” to fit observed decays. Experimentally, it is easier to obtain a “pure” a-type (vibrationally unrelaxed) decay than it is to obtain vibrationally relaxed fluorescence that is uncontaminated by a-type emission. Thus, only fits to a-type (i.e., (6.7) of Chapter 3) dissipative decays are reported herein. These fits were accomplished by the method of nonlinear least squares.!? The convolution of the true molecular decay with the temporal response of the detection system was taken into account. 82 Ill. LOW ENERGY REGION The fluorescence spectroscopy of anthracene SVL’s having energies from E,;, = 0 to 1200 cm! has already been reported on!4:15, Almost all of the spectra consist of distinct bands which can be assigned exclusively in terms of transitions arising from an optically active SVL (see section II-D of Chapter 3 for what we mean by an optically active level). These spectral characteristics are consistent with the mani- festations of uncoupled vibrational levels. Previous studies in this laboratory have corroborated this interpretation in that the fluorescence decays of each of the bands in a given spectrum have been found to be identical single exponentials. These for- mer studies, however, were done with a detection temporal resolution that only allowed beat frequencies of less than ~ 2 GHz to be resolved. It is now apparent that in most cases beats arising from vibrational couplings in anthracene are of fre- quencies greater than this*. Thus, we have remeasured with better time resolution some decays corresponding to excitation of SVL’s in the low energy regime. Fig. 1 provides an example of the fluorescence spectrum and decay upon excitation of the 12? (766cm—!) SVL. The spectrum is analyzed in Ref. 14. The decay, which cor- responds to detection of the 390 cm~? fluorescence band, is clearly unmodulated, and is the same as the decays of the other bands in the spectrum. (The apparent risetime of the decay is a convolution effect, as has been verified by calculating the convolution of an 18 nsec exponential with our measured system response func- tion. Such apparent risetimes become longer, the longer the fluorescence lifetime becomes.) Decays have also been measured for other excitation bands in this low energy regime. No beats have been observed despite the temporal resolution. Both the spectral'*:15 and decay results indicate, therefore, that vibrational coupling leading to IVR does not occur to an appreciable extent in anthracene SVL’s up to ~ 1200 cm—?. In saying this we do not exclude the possibility that strong Fermi resonances, resulting in splittings of several wavenumbers or more, are not present in this energy regime. Indeed, other results!®!® have provided evidence for such coupling in the higher energy portion of the region. The difficulty in preparing such 83 coupled systems coherently, however, argues against considering them in terms of dynamical energy flow. More discussion on this point is presented in section VI-C to follow. 84 IV. INTERMEDIATE ENERGY REGION Without exception, the excitation of jet-cooled anthracene at each of the five prominent vibronic bands in its excitation spectrum!* 5 between 5S; + 1880 cm~! and S; + 1514 cm! results in quantum beat-modulated fluorescence decays, the beat patterns of which depend on the particular fluorescence band being detected. We have already reported* preliminary results for two of these excitation bands: Eyin = 1380 cm! and 1420 cm7!. In this section we analyze in detail the results for these two bands in terms of the theory presented in Chapter 3, and obtain parameters pertaining to IVR and vibrational coupling at these energies. For the three other bands, we present representative decays, but do not provide any exten- sive analysis of the results. A. E,i, = 1380 em~!: The a,(6) mode We have previously reported results!:? pertaining to the characteristics of the quantum beats which arise from excitation of this prominent band in the excitation spectrum. (The nature of the vibrational level has been assigned in Ref. 14.) In fact, it is this excitation which gives rise to those beats which we reported on three years ago and which represent the first example of a quantum interference effect in a large molecule. It turns out that these “old” results actually represent manifestations of only a minor aspect of the vibrational dynamics associated with this excitation energy; increased temporal resolution has revealed that fast beats* modulate many of the dispersed fluorescence bands which appear to be unmodulated under 400 psec resolution. As shall be seen below, the new data lend themselves to a relatively straightforward interpretation in terms of IVR. Results The fluorescence spectrum resulting from excitation to S$, +1380 cm! is shown in Fig. 2 along with a higher resolution scan of a portion of the spectrum. A previous attempt!® at assigning this spectrum made note of the fact that while a number of the bands are assignable in terms of optically active intervals (most notably those bands at shifts of 390, 755, 780, 1168, and 1408 cm™! from the excitation energy), 85 most of the bands, including some of the most intense ones, are not. This mix of assignable (vibrationally unrelaxed) and unassignable (vibrationally relaxed) bands, and the congestion in and to the red of the S, origin region indicate (see section II-D of Chapter 3) that the spectrum arises from a relatively small number of coupled vibrational levels. Confirmation of this is provided by time-resolved results, which reveal beat- modulated fluorescence decays that depend on detection wavelength. It is conven- ient to consider the decays of various groups of these bands separately. Fig. 3 presents typical decays of some of those bands (group I) in the spectrum that can be readily assigned. All of the decays have a similar appearance and have roughly the same Fourier spectra. Fig. 4-top, which is the Fourier spectrum that results from transformation of the decay of the 390 cm7! band, is representative of these spectra. In this spectrum, eight beat components are evident at 0.2, 1.8, 2.1, 2.6, 3.5, 4.3, 4.9, and 8.4 GHz. Note that all of these components have positive phases. The Fourier spectra of Figs. 4-middle and 4-bottom illustrate the changes that take place in the 390 cm! decay as the excitation source is tuned ~ 2 cm! . to the red and to the blue of the excitation peak, respectively. Clearly, there is an enhancement in the components at 1.8, 2.6, and 4.3 GHz for the red excitation relative to the blue. On the other hand, the blue excitation results in a decay which is almost exclusively modulated by the 3.5, 4.9, and 8.4 GHz components (with some contribution from the 2.1 GHz component). For blue excitation one obtains absolute modulation depths of 0.12, 0.16, and 0.75 for the 3.5, 4.9, and 8.4 GHz components, respectively. The decays of the bands at 1460, 1460 + 390, and 1460 + 1408 cm~! (group II) are presented in Fig. 5. All three decays look similar, as is confirmed by Fourier analysis. The apparently decreased modulation depths in the lower two decays are attributable to congestion in the spectral regions in which their corresponding fluorescence bands occur. Fig. 6 presents the Fourier spectrum which results from transforming the modulated residual of the 1460 cm~! decay. Only three prominent 86 beat components at 3.5, 4.9, and 8.4 GHz appear. Note that the phases of the 4.9 and 8.4 GHz components are negative, while the 3.5 GHz component has a positive phase. The absolute modulation depths of the three components, in order of increasing frequency, are found from Fourier analysis to be 0.14, —0.63, and —0.26. The decays of the bands at 1429, 1528, 1528 + 390, and 1528 + 1408 cm-! (group III) are presented in Fig. 7. Note the similar appearance of the decays. The Fourier transform spectrum of the 1528 cm! decay is presented in Fig. 8 and is representative of the other Fourier spectra. Again, the 3.5, 4.9, and 8.4 GHz components are the only discernible components. Now, however, the phase of the 4.9 GHz component is positive, while the other two components have negative phases. The derived absolute modulation depths for the decay of this band are quite sensitive to detection wavelength and resolution, as is reasonable considering its position in the spectrum (Fig. 2). The values for the absolute modulation depths derived from the decay of the 1528 cm™! band are —0.45, 0.16, and —0.41, for the three components in order of increasing frequency. The fourth group of bands which shall be considered (group IV) are the three which occur in the congested 1480 to 1520 cm~! region of the spectrum. In this region the ability to spectrally isolate the various bands to measure a decay, while still maintaining a reasonable signal level, was limited. Nevertheless, modulated decays were observed, as shown in Fig. 9. For some of these decays, Fourier analysis (not shown) reveals some contribution from the 3.5, 4.9, 8.4 GHz triplet of beat components. Also present, however, unlike for the decays of groups II and III, are some of the other beat components that appear in the group I decays. In particular, all or some of the components at 1.8, 2.6, and 4.3 GHz appear prominently in these decays. Moreover, they exhibit phase behaviors that are different from that of the 390 cm—! decay: the 1490 cm~! decay has the 1.8 and 4.3 GHz components with negative phases and the 2.6 GHz component with positive, while the 1505 cm~! decay has the 4.3 GHz component with positive phase and the other two with 87 negative phases. Finally, we consider the decay of the band at 1125 cm~!, which has been extensively studied previously’? using ~ 400 ps temporal resolution. The decay of this band using 80 ps temporal resolution is presented in Fig 10. The decay was taken under expansion conditions? which favor the appearance of the 0.2 GHz beat component ( a 0.5 GHz component predominates under less efficient rotational cooling conditions?). This component is clearly present in the decay. Moreover, Fourier analysis confirms our earlier assertion? that the component has a negative phase. While higher frequency components also appear to modulate the 1125 em~! decay, the difficulty in obtaining relatively noiseless data for this weak fluorescence band has precluded any detailed consideration of these components. Interpretation As we have stated above, the characteristics of the dispersed fluorescence spec- trum and the spectral dependence of the fluorescence decays corresponding to ex- citation at E,,, = 1880 cm—! in anthracene both indicate that the experimental results are the dynamical manifestations of vibrational coupling in the S, state. One would, however, like to characterize this coupling more fully. The major prob- lem is to determine the number of levels involved. According to Chapter 3 this can be done if the number of beat components and the number of beat triplets is known. (By a beat triplet? we mean the three beat components wy, WyK, and wyx. Note that two of the frequencies of such a triplet sum to the third.) The number of com- ponents (8) and the number of triplets (2) evident from the 390 cm~! decay do not correspond to the theoretical expectations for any number of coupled levels. This could be due to at least two causes, as pointed out in sections V-B and V-D of the Chapter 3. Either some beat components are too weak or of too high a frequency to be observed, or the finite excitation bandwidth spans at least two independent sets of coupled levels. Here, there is strong reason to believe that the latter situation is the one that obtains. This conclusion arises from the fact that the relative Fourier amplitudes of the various beat components are dependent on excitation energy; the 88 triplet of frequencies at 1.8, 2.6, and 4.3 GHz is enhanced relative to the triplet at 3.5, 4.9, and 8.4 GHz when the excitation is detuned to the red and vice versa when the excitation is detuned to the blue (Fig. 4). Moreover, the other groups of bands in the dispersed fluorescence spectrum have decays in which one or the other beat triplet dominates. This is consistent with the manifestations of two sets of three coupled levels, the energy separation of the two sets being small enough to be within the excitation bandwidth, but large enough to preclude the observa- tion of any beat frequencies corresponding to energy differences between two states belonging to two different sets!?. In the nomenclature of section V-D (case 4) of Chapter 3, the bands of group I would correspond to overlapping a- and a’-type bands. The bands of groups II and III would then correspond to b-type and c-type bands, respectively, since the phase behaviors of the two types of decays are con- sistent with theoretical expectations for the two non-a-type bands of a three-level system: two —1 phases and one +1 phase for each decay. The 1487 and 1502 cm~! bands would be assigned as b’- and c'-type bands. For these, too, the beat phase behaviors of the decays are consistent with the expectations of theory for the two non-a-type bands of a three-level system. The sums of the modulation depths for the non-a-type bands, by (2.17) of the previous chapter, should be equal to —1. However, the b-type 1460 cm! decay gives a value of —0.75, and the c-type 1528 em~! decay gives a value of —0.70. As was pointed out in section II-B, this could be due to a number of sources of experimental error, particularly problems with spectral resolution and with response function drifts. Alternatively, this could be due to unobserved eigenstates (see sections V-A and V-B of Chapter 3). For example, the |a), |b), and |c) zero-order levels may be just three of four coupled levels, with the coupling being such that three of the eigenstates are closely spaced, and the fourth being widely separated from these three. It is likely that both types of influences play some role in the less than ideal values derived from experiment. In any case, the values —0.75 and —0.70 are close enough to —1 to warrant tentative treatment of the unprimed system in terms of 89 three coupled levels. One may also note that two of the beat frequencies modulating the 390 cm~! band remain unexplained by the scheme, presented above, of two sets of three cou- pled levels; the components at 0.2 and 2.1 GHz. We have previously shown? that for the E,,, = 1380 cm~! excitation there exists a marked rotational level depen- dence to the coupling which gives rise to the 0.2 GHz component, the coupling only occurring significantly for certain higher energy rotational levels. This combines with the facts that (1) the 0.2 GHz modulation of the 390 cm~! band is relatively weak and (2) any other bands exhibiting decays with significant modulation at this frequency (e.g., the 1125 cm~! band) appear only with weak intensity in the flu- orescence spectrum, to indicate that the 0.2 GHz component is a manifestation of a coupling that figures only in a minor way, and at higher rotational energy lev- els, in the IVR dynamics at this excess vibrational energy. Whether or not this coupling is associated with the |a), |b), and |c) states, the |a’), |b’), and |c’) states, or with neither, cannot be answered without higher resolution excitation contour experiments. As for the beat component at 2.1 GHz, one is handicapped by the fact that no bands have been observed, other than those of group I, which exhibit in their decays a prominent component at this frequency. This suggests that the component does not arise from coupling involving the unprimed or primed set of states. It is possible that it does arise from the same coupling that gives rise to the 0.2 GHz component or that it represents the manifestations of two more coupled levels spanned by the bandwidth of the excitation pulse. Calculation of Ho As we have shown in section IV of the preceding chapter, it is possible to use observed quantum beat parameters to calculate the vibrational Hamiltonian matrix in the zero-order basis set (i.e., Ho). In this section we do this for the three-level system composed of the |a}, |b), and |c) coupled levels which give rise to the fluorescence bands modulated at 3.5, 4.9, and 8.4 GHz. Our choice of this coupled system is based on the quality of the data associated with it and the fact 90 that these levels contribute most prominently to the fluorescence arising from the 1380 cm~! excitation band. We start by labelling the beat frequencies such that |wi2/2z| = 3.5, |wes/2a| = 4.9, and |wi3/2z| = 8.4 GHz. In so doing, the following is obtained for the diagonal Hamiltonian matrix: 0.0 0.0 0.0 E=E£,1+] 0.0 3.5 0.0 0.0 0.0 8.4 where Ep is an unknown constant, and where the energy values are expressed in GHz. ; (4.1) Next, we calculate the a7, from the relative modulation depths of the 390 cm=! decay. Using Ma (was) M, (wes) = 0.22 and ———— = 1.31, Ma, (w13) a M, (wig) (4.3a) and (4.3b) of Chapter 3, and the normalization condition, one obtains Q1e = 629; ag, = .295; a3, = .720, where we have exercised the freedom to choose all these values to be Positive.” Next, we may calculate either the ay, or the az, values. We choose the aTy because the 1460 cm! b-type band is very strong and relatively isolated. Therefore, one expects that the values for the relative modulation depths for this band are fairly accurate. Using My (wes) Mz (wes) —————. = 2.47 and —_—— = ~—4,42, M, (wis) M,(wi2) (4.4a) and (4.4b) of Chapter 3, and the normalization condition for the a7, one obtains O14 = 0.152; ag, = 0.799; a3, = —.582, where our choice of a1, as positive fixes the signs of agp and agp. Finally, we calculate the |a;¢| using the relations e,=1 for! =1,2,3, y=4,6,.., 91 and then determine the signs of the coefficients using the quantum phase behavior of the c-type bands (see Fig. 8). This yields ac = —.762; age = 0.525; asc = 0.378. As a whole, the eigenvector coefficient matrix is 0.629 0.295 0.720 C=] 0.152 0.799 ~—.528 |, (4.2) —.762 0.525 0.378 with the columns labelled by the molecular eigenstates and the rows labelled by the zero-order states. As pointed out in Chapter 3, a number of tests can be made on this calculated matrix. Firstly, the orthogonality of its rows and columns may be checked. Calculating the six possible cross-products, one finds no deviations from orthogonality of over 5.3° and an average deviation of about 4°. Secondly, the absolute modulation depths can be calculated from C and compared with experi- mental values. Such a comparison appears in Table 1. One sees that the calculated magnitudes are generally ~ 20 — 30% higher than the experimental ones, probably due to experimental limitations or unobserved eigenstates. Using the values for C and E, one can calculate Ho up to various constants by (4.1) of Chapter 3. Doing so, one obtains 4.66 -2.70 2.83 Hy**° = Holt | -2.70 5.08 —.38 |, (4.3) 2.83 —.38 2.16 where the three rows and columns are both labelled by a, 6, and ec. Note that the relative signs of the off-diagonal elements are not unique®. This is because the rows of C are only fixed up to an overall sign by the quantum beat results. Thus, instead of the distribution (—1, +1, —1) for the signs of the interaction matrix elements (Vas, Vac, Vie), one also could have obtained (+1, ~—1,-1), (+1, +1, +1), or (—1, -1,+1) by different choices for the overall signs of the rows of C. 92 B. Eviy = 1420 em™': The a,(5) mode The 1420 cm™* excitation band, which has been assigned in Ref. 14, is the one for which phase-shifted quantum beats were first discovered.® At that time the beating behavior appeared to arise from a coupled two-level system. Since then, with better time resolution it has become evident that the vibrational dynamics at this energy involve at least three coupled levels.* In this section we present and analyze the results for this band in a manner similar to the treatment of the 1380 cm—! band in the previous section. Results The fluorescence spectrum resulting from the excitation of the 1420 cm! band is shown in Fig. 11 together with a high resolution spectrum of the region near the S; origin. The general characteristics of the spectrum are quite similar to the 1380 cm~! spectrum; i.¢., there is a mix of assignable and unassignable bands, there is some resonance fluorescence, and there is resolvable spectral congestion in and to the red of the 5; origin region. Also similar to the 1380 cm—! results, the time-resolved results reveal fluores- cence decays which are beat-modulated and which depend on detection wavelength. Again, we shall discuss the various types of decays separately. Fig. 12-top presents the decay of the 390 cm~! band. Fourier analysis of the decay (Fig. 13-top) reveals three prominent beat components at 1.0, 9.7, and 10.7 GHz, all with positive phases and with absolute modulation depths of 0.55, 0.36, and 0.52, respectively. The 390 cm~! decay is representative of a number of other decays, as well; those of the 0, 780, and 1168 cm! bands. (Note that all of these bands have counterparts in the group I, a-type bands of the 1380 cm—! spectrum and that they are all assignable in terms of intervals associated with optically active modes.1*) There are also other bands in the congested region of the spectrum near the S, origin which appear to exhibit decay behavior similar to the 390 em~' band (e.g., the 1487 cm~! band). Fig. 12-middle presents the decay of the 1750 cm~! band in the dispersed 93 fluorescence. As we have reported,** this decay shows an obvious 180° phase shift of the 1.0 GHz beat component relative to the +1 phase of the same beat component in the 390 cm™* decay. Fourier analysis (Fig. 13-middle) reveals that besides this beat component, the decay is also modulated by at least four others: the 9.7 and 10.7 GHz components found in the 390 cm! decay, and two more components at 3.5 and 4.5 GHz. The absolute modulation depths for these components in order of increasing frequency are —0.74, —0.06, 0.04, —0.11, and 0.08. The sum of the modulation depths is —0.79. Although we have not measured the decays of the bands at 1750+ 390 and 1750+1408 cm7! with 80 ps temporal resolution, previous measurements® of the decays showed that the decay behavior of these bands is consistent with that of the 1750 cm~ band (i.e., the 1 GHz component was found to have a large modulation depth and a negative phase). A third band at 1500 cm~! exhibits different decay behavior than both the 390 and 1750 cm~! bands. Fig. 12-bottom presents the decay of this band and Fig. 13-bottom the Fourier spectrum of the residual. Despite the noise in the Fourier spectrum, three Fourier components are evident at 1.0, 9.7, and 10.7 GHz, with phases of +1, —1, and —1, respectively. Nominal values for the relative modulation depths derived from the 1500 cm~! decay are 0.2, —0.9, and —1.0, in order of increasing beat frequency. Finally, in Fig. 14 the decays for many of the prominent bands that appear in the region of the S, origin are presented. We shall not analyze these decays here extensively, since the limitations of spectral and temporal resolution prohibit any very quantitative analysis. It is evident that all of the decays are modulated to some extent by the 1 GHz beat component. Moreover, some, and perhaps all, are modulated prominently with the 9.7 and 10.7 GHz components, as well. Interpretation As with the results for the 1880 cm~! band, it is desirable to come to an understanding of the number of coupled levels being manifested in the results for the 1420 cm™' band. Just considering the Fourier spectrum (Fig. 13) for the 94 390 cm! decay, which exhibits one triplet of beat components, one would come to the conclusion that only three levels were involved. However, the 1750 cm~! decay, with its five beat components (Fig. 13) is not consistent with this. Instead, the latter decay indicates that the 1.0, 9.7, and 10.7 GHz beat frequencies are just three of the six energy splittings expected for a four-level system. Considering the five beat components of the 1750 cm~! decay as whole, one sees that they are consistent with the manifestations of a four-level system, if it is assumed that one beat component, at either 6.2 or 14.2 GHz, is undetectable due to weak modulation or (in the case of the 14.2 GHz component) an unresolvable frequency. In both cases there would be six beat components and four beat triplets, as expected” for a four-level system. ( The former case would have the triplets (1.0, 3.5, 4.5}, (1.0, 9.7, 10.7), (3.5, 6.2, 9.7) and (4.5, 6.2, 10.7), whereas the latter would have the triplets (1.0, 3.5, 4.5), (1.0, 9.7, 10.7), (3.5, 10.7, 14.2), and (4.5, 9.7, 14.2).} Note that no other values for the sixth beat frequency would be consistent with a four-level system. Note also how the 1420 cm™! case differs from the 1380 cm~! case, for which six beat frequencies were assigned to two independent three-level systems: unlike in the 1420 cm! case, in the 1880 cm! case the beat frequencies are such that there is no way to form four triplets. All in all, the results indicate that the 390 cm~! band is an a-type band and the 1750 cm~! band is a non-a-type band (say b-type) of a coupled four-level system. The existence of a situation wherein four vibrational levels are coupled implies the existence of four types of fluorescence bands, each with different quantum beat phase behaviors. Obviously, we have observed two of these band types. We can also identify a third one: the band at 1500 em~!, which we call e-type. Our inability to identify a d-type band is probably due to problems associated with the spectral overlap of fluorescence bands. Calculation of Ho To calculate a four-by-four Ho matrix for E,;, = 1420 cm™!, an initial as- sumption must be made involving the value for the unknown sixth beat frequency. 95 We shall choose the lower value 6.2 GHz, since using it seems to give values for the elements of C which more nearly approximate those of an orthonormal matrix than the values calculated using 14.2 GHz for the sixth beat frequency. Hence we label the beat frequencies such that |w12/2x| = 1.0 GHz, |wes/2z| = 3.5 GHz, ws, /2z| = 6.2 GHz, |wis/2x| = 4.5 GHz, |we4/2x| = 9.7 GHz, and |wi4/2m| = 10.7 GHz. This gives an eigenvalue matrix 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 45 0.0 |° (4.4) 0.0 0.0 0.0 10.7 E=E£oI+ Next the eigenvector coefficient matrix C must be considered. We start with the calculation of the a;, from the relative modulation depths of the three detectable Fourier components modulating the 390 cm7! a-type band. These values give af, =-60007, and a2, =.609a?,. Because the Fourier amplitudes of the wis, wes, and W34 Components are so weak as to be undetectable, there is no way to calculate as,. One only knows that the value must be small. Thus, we make an assumption and fix the value of Qgq at 0.05. Then, with the normalization condition one finds the following values for the aya: O1q = 0.672, aeq = 0,521, a3, = 0.050, a4, = 0.525. The next step is to calculate the a4. Using the relative modulation depths of the beat components of the 1750 cm! b-type band and the normalization condition on the ay, one obtains Qin= —.503, a4 = 0.699, a35 = —.498, 4p = ~.095. The a;- could conceivably be calculated using the relative modulation depths of the 1500 cm™!, c-type band, but for the fact that some of the beat components are lost in the noise of the pertinent Fourier spectrum. Nevertheless, one can use M.(w14)&2a et ae = A M,(wes)o10 3e re 96 to obtain a1, = 0.86a2-. Then using the orthogonality of the a7. with the ay, and the aya, and, finally, the normalization of the ay,, one obtains Qi¢ = 0.316, age = 0.367, a3, = 0.350, O4¢ = —.802. Lastly, the ja ;g| can be calculated using the relations > af,=1 forl=1,4. y=a,65,... The relative signs of the azz can then be obtained from orthogonality requirements. The calculated eigenvector matrix is 0.672 0.521 0.050 0.525 ~.503 0.699 —.498 —.095 C=! 0316 0.367 0.350 —s02 | (4.5) 0.442 -.324 —.792 —.269 The largest deviation from orthogonality for any of the rows or columns is 3.0°, with the average being about 1.5°. In addition, the square root of the sum of the squares of the fourth row, which sum was not fixed by the calculation, is found to be very close to 1 (#.¢., 1.003). For each of the four types of the fluorescence bands the absolute modulation depths predicted from the calculated C are presented in Table 2 along with the available experimental values. Again the experimental values are consistently lower (by 20—30%) than the calculated ones, which deviation probably arises from the same causes as those which give rise to the less than ideal total modulation depth of the 1750 cm~! band. One may note that the calculated values in Table 2 can serve to aid in the identification of the d—type bands. In particular, the calculated value of —0.73 for Ma(w1g) suggests that perhaps the band at 1521 cm™! is d-type (see the decay in Fig. 14). Just as for E,;, = 1380 cm~!, Ho for Ey;, = 1420 cm7! may be calculated using (4.1) of Chapter 3. This results in 3.23 —.28 -4.24 —-1.86 -.28 1.70 0.29 1.82 (4.6) —4,24 0.29 7.57 0.94 |° ° -186 1.82 0.94 3.70 Ho*”° = EI + 97 Again it must be noted that the distribution of signs for the off-diagonal elements in Ho is not a unique one’ and that other distributions are not precluded by the quantum beat results. In any case, the magnitudes of the coupling matrix elements are of the most interest, so this sign ambiguity is not particularly limiting. C. Ey% = 1409, 1501, and 1514 em=! The 1380 and 1420 cm~! excitation bands occur in a region of the excitation spectrum which contains a number of other prominent bands,!* the ones at 1409, 1501, and 1514 cm~!. Given that the former two excitations give rise to dynamical behavior that is characteristic of restricted IVR, one might expect the latter three excitations to do so, as well. Previously published spectral evidence!® has indeed indicated that restricted IVR influences the molecular dynamics at these energies. However, time-domain evidence has been lacking. Figs. 15, 16, and 17 provide such evidence for the 1409, 1501, and 1514 cm~! excitations, respectively. In each figure the uppermost decay corresponds to detection of the 390 cm—! band, which from spectroscopic considerations!* and in analogy with the decay behavior of this band for E,;, = 1380 and 1420 cm~!, should be a-type. Consistent with this, Fourier analysis of the three 390 cm! decays reveals that all beat components enter into the decays with +1 phases. On the other hand, each of the spectra arising from the three excitations also contain bands with different decays than the corresponding 390 cm~! decay, the most notable differences being that some of the beat components appear with —1 phases. Examples of such decays are shown in the lower portions of Figs. 15-17. It is pertinent to note that the order of magnitude of the beat frequencies appearing in the decays of these figures is the same as for the beat components associated with 1380 and 1420 cm~! excitations. It is clear from the experimental results that the decay behaviors of the 1409, 1501, and 1514 cm™! excitations are qualitatively similar to the behaviors of the 1380 and 1420 cm—! excitation bands, and, moreover, are consistent with the mani- festations of restricted IVR. Since the vibrational density of states (see section VI-A below) does not change much more than a factor of two in going from E,;, = 1380 98 Evin = 1380 to Ey: = 1514 cm™!, this similarity is not particularly surprising. The significance of the results is that they show the general occurrence of phase-shifted beats in the energy region corresponding to restricted IVR. 99 V. HIGH ENERGY: £5 = 1792 em=! Above Ey, = 1514 cm! in S$, anthracene, there are only a small number of excitation energies which yield dispersed fluorescence spectra with resolvable (or partially resolvable) structure®!5 and which are thereby amenable to the direct study of IVR through the measurement of the temporal evolution of individual fluorescence band types. One such excitation corresponds to Ey; = 1792 cm—!, which is most likely a combination band of the 385 cm~! (5, 12) mode!* with - the 1409 cm~! band. As may be seen from Fig. 18, the dispersed fluorescence spectrum which arises from this excitation is quite congested compared with lower energy spectra, but nevertheless has at least two discrete bands of weak intensity in the blue portion of the spectrum (the bands at 390 and 780 cm~!) and also some incompletely resolved structure with overall strong intensity in the region near the 5S origin. The 390 and 780 cm~! bands (vibrationally unrelaxed or a-type bands), being assignable in terms of optically active modes, correspond to transitions from the optically prepared zero-order state. The bands in the congested region near the S, origin arise predominantly from zero-order states populated by the IVR process (vibrationally relaxed bands). Given that the two types of bands arise from different zero-order states, one expects differences in their decays to provide a picture of vibrational energy flow at this energy (see section VI-B of Chapter 3). Fig. 19 presents the decays of the 390 and 780 cm-! bands. It is evident that they are very similar. Fitting each according to the procedure of section II-C gives a short component lifetime of 22 psec and a long component lifetime of 6.4 nsec. Fig. 20 shows a typical fit to the data. We would make two points about the lifetimes obtained from the fits. Firstly, the short lifetime differs significantly from the value of ~75 psec that we have quoted in a preliminary work.> This discrepancy arises from the fact that in the previous work we fit only the initial portion of a given decay to a single exponential. Evidently, the contribution of the long-time component at early time is substantial enough to render such a procedure inaccurate. One expects the double exponential fitting procedure to yield a much more accurate fast 100 time constant. Secondly, the long-time component has a lifetime that matches the lifetime’* of the total fluorescence decay at this energy, as one would expect from (6.7) of Chapter 3. Finally, it is worth noting that Fourier analysis (see Fig. 21) reveals that all of the large number of beat components which contribute to the decays of Fig. 19 have +1 phases as expected for an a-type band. (For comparison we include in Fig. 21 the uncluttered Fourier spectrum of the a-type 390 cm7! decay corresponding to 5; + 1380 cm! excitation. ) Since the 390 and 780 cm~! bands are weak in intensity and occur in a re- gion close to the excitation wavelength, several checks were made to ensure that the measured decays of these bands represent molecular decays and not scattered laser light. These checks included measuring decays at a detection wavelength of 3410 A(Fig. 22a), a wavelength which is spectrally closer to the exciting light but at which there is no fluorescence, and measuring decays without any anthracene in the free-jet expansion (Fig. 22b). These checks yielded no evidence for any signifi- cant scatter contribution. Some leakage light was found to contribute to the decays. However, this spurious component was determined to be due to molecular fluores- cence and to contribute to the long-time intensity of the decays. This is evident from Fig. 19-bottom, which shows an increased intensity of the long-time compo- nent of the 390 cm~! decay for decreased spectral resolution of detection relative to that used to detect the decay of Fig. 19-top. Laser-to-nozzle distance and carrier gas dependences were also checked in order to assess the influence of intermolecular interactions on the observations. No significant effects of this kind were observed (see Fig. 23). Fig. 24 presents decays corresponding to three fluorescence detection wave- lengths in the region of the S, origin. There are several notable features of these decays. Firstly, each has a time constant of ca. 6 ns. Secondly, all of the decays are modulated, but exhibit different beat patterns. Finally, the decays all have finite risetimes. As may be seen from Fig. 24-bottom, which is a decay measured at the same detection wavelength as used for the decay above but at lower spectral 101 resolution, there is a pronounced spectral resolution effect on the decays measured in this region. Given a number of different types of fluorescence bands in the same spectral region, this is not surprising’. The apparent risetimes of the decays were examined in several ways to determine whether they were real risetimes. For exam- ple, all of the decays were compared with decays of the 390 cm! band measured immediately before or immediately after. In all cases the decays of Fig. 24 peaked after the 390 cm—! decay did and appeared to rise at the same rate at which the short component of the 390 cm~! transient decayed. Also, the smooth decay of Fig. 24-bottom, and others like it, were compared with calculated decays of 6 ns exponentials convoluted with the measured system response function. In all cases a significant difference between the calculated and measured decays was found, with the measured decays rising more slowly than the calculated ones. (Notably, this dif- ference was not found for fluorescence at the same detection wavelength when the excitation was to Eyi) = 2750 cm~‘.) Finally, Fourier analysis of the decays reveals Fourier components with negative phases. Such components must be present for a decay with a risetime. Notably, no components with negative phases modulate the 390 or 780 cm~! decays. We interpret the results for Ey;4 = 1792 cm~! in terms of dissipative vibra- tional energy flow. The 390 and 780 cm™! a-type decays, being direct representa- tions of the |a) content of the excited state (see section VI of Chapter 3), show that vibrational energy redistributes dissipatively from |a) to the states coupled to |a) on a timescale of 22 psec. This initial redistribution of energy is also reflected in the risetimes of the decays of the bands near the S, origin, decays which are di- rect representations of the contributions of the states other than |a) to the excited state’. The presence of beats in the fluorescence decays is a manifestation of the fact that although the initial energy flow is substantially irreversible, it does not reach a steady state on the 22 ps timescale. Rather, the vibrational content of the excited state continues to evolve to some extent throughout the timescale of the fluorescence lifetime. 102 Regarding the number N of coupled vibrational levels being manifested in the fluorescence behavior of the S; + 1792 em7! excitation, it is more difficult to determine an accurate value than it is in the lower energy regime. This is a result of the large number of beat components in, and complicated Fourier spectra of, the 1792 cm~! decays. Nevertheless, it is reasonably certain from the Fourier transform of the 390 cm~! decay (Fig. 21) that greater than ten coupled levels give rise to the observed results. This is supported by the parameters derived from the fits to the decays of Fig. 19, since from these it is found that the ratio of fast to slow fluorescence in the decays is ~ 17. This ratio represents a lower limit to the number of coupled states in the dissipative regime (see section VI-B of Chapter 3). It is notable that if one assumes that the number of coupled levels at a particular Evi scales with the vibrational density of states (see section VI-A below), then the roughly fivefold increase in this density in going from 1400 to 1800 cm~!, along with the values of N at 1380 and 1420 cm~!, would suggest that there are fifteen to twenty states coupled at Fy;, = 1792 em7!. The results for Ey;, = 1792 cm~! are also interpretable in terms of the dense intermediate case of radiationless transitions.1® The vibrational coupling at this energy is such that the number of effectively coupled zero-order states N is larger than in the lower energy regime of restricted IVR (sparse intermediate case), but smaller than would be the case if the intrinsic linewidths of the states exceeded their energy spacings (statistical limit). This conclusion is reached by consideration of two facets of the fluorescence decays. Firstly, the short-time component of the 390 and 780 cm~! decays and the rise times of the other decays are indicative of an intramolecular dephasing between the eigenstates composing the optically pre- pared state. Secondly, the presence in all of the decays of a long-time component with beats indicates that the dephasing does not preclude partial recurrences on the timescale of the fluorescence lifetime. The short-time behavior distinguishes the dense from the sparse intermediate case, while the presence of long-lived intramolec- ular coherence distinguishes the intermediate case from the statisical limit.1° 103 VI. DISCUSSION AND CONCLUSIONS A. IVR vs. vibrational energy The experimental results presented above allow one to make a number of differ- ent conclusions concerning IVR in anthracene. The first pertain to the nature of the process as a function of vibrational energy in the molecule. It is clear in light of the discussion in section VI-B of Chapter 3 that the low, intermediate, and high energy results for anthracene represent manifestations of absent, restricted, and dissipative IVR, respectively. In the low energy regime (< 1200 cm"), vibrational mixing is not extensive, if present at all. The excitation process prepares just one vibrational eigenstate which undergoes no further vibrational evolution. For somewhat higher energies (i.¢., 1380 to 1520 cm~!) the vibrational coupling becomes appreciable; systems of coupled states involving two to approximately ten levels characterize the regime. Upon preparation of an optically active zero-order state, IVR occurs but is restricted in the sense that essentially full recurrences in the vibrational energy dis- tribution occur on the timescale of the fluorescence lifetime. The timescale of IVR in the restricted regime is given by the order of magnitude of the inverses of the beat frequencies which modulate dispersed fluorescence decays. For anthracene the beat frequencies are on the order of 5 GHz, leading to a timescale for restricted IVR, of several hundreds of picoseconds. Finally, at high energies (e.g., Eyiy = 1792 em!) the extent of vibrational coupling is great enough (systems of > 10 levels) that dissipative IVR occurs. In these cases, IVR is such that nothing close to a full recurrence occurs on the fluorescence timescale. The timescale of IVR in the dissi- pative regime is given by that of the initial, approximately exponential dephasing of the optically prepared state. For the E,;, = 1792 cm7! band, ri», ~ 22 psec. It is of interest to correlate changes in the nature of IVR in anthracene with the vibrational density of states (p,;,) in the S, manifold. We have calculated Pvid at various energies using a direct count procedure and the calculated frequencies of Refs. 19. The results are shown in Fig. 25. In the region of no IVR, yj» is less than ~ 10 per wavenumber. For E,,;, = 1380 to 1520 em~!, py goes from ~ 25 104 to 40 percm™+. And for Ey:, = 1800 cem~!, py ~ 120 percm=!. These values of py for the different regimes of IVR in anthracene are useful for two reasons. Firstly, they suggest that vibrational coupling in the molecule is selective. For instance, at Ey;, = 1380 cm7! there are three strongly coupled vibrational states within an 8.4 GHz energy interval. But, if there were mixing between all the states in this interval, then since pig = 25 per cm~! one would expect at least twice as many coupled levels. A similar point can be made for the Eyis = 1420 and 1792 cm~! excitations; the number of coupled levels is less than would be expected from consideration of the total density of vibrational states. This selectivity in coupling is not unreasonable. For vibrational states to be coupled by anharmonic interactions, they must have the same vibrational symmetry. In Doz anthracene this means that a given vibrational state can couple anharmonically with at most one-eighth of the total number of other states in the molecule. Thus, selectivity in coupling is expected on the basis of symmetry considerations. Secondly, the specific values for py, are useful for their predictive value. Using these values, symmetry considerations, and knowledge of py;i5 ve. energy one can make predictions of the nature of IVR as a function of energy for other molecules. A last point of interest concerning IVR as a function of energy in anthracene derives from the connection between IVR behavior and quasi-periodic or ergodic vibrational dynamics in molecules.?° For a model Hamiltonian (Henon-Heiles), the transition with increasing energy from classical mechanical quasi-periodic to ergodic. vibrational dynamics has been found”! to be correlated with quantum mechanical quantities derived from the same Hamiltonian. Regular, periodic behavior of the quantity x(t) = |(U(2)/y)/?, (6.1) which is the contribution of the harmonic state |y) to the time dependent vibrational wavefunction |W(¢)), correlates with classical quasi-periodic behavior. On the other 105 hand, irregular behavior of (é) correlates with classical chaotic behavior. Since p4(t) ~ I(t), (6.2) where I,,(t) is the intensity vs. time of a y-type fluorescence band (see section VI-A, Chapter 3), then the decays of fluorescence bands are indicators of quasi-periodic or ergodic vibrational dynamics. The upshot is that the regular, periodic decays characterizing the restricted IVR regime in anthracene suggest quasi-periodic quan- tum dynamics, while the less regular, high energy dissipative decays suggest quan- tum chaotic behavior. Thus, time-resolved measurements of dispersed fluorescence promise to be very useful in the study of the nature and onset of quantum chaos. B. Coupling matrix elements Besides providing information concerning the nature and timescale of IVR in anthracene, the experimental results also provide information pertaining to the details of the couplings which give rise to IVR. Thus, for the restricted IVR. that occurs for Ey, = 1380 and 1420 cm™', one can see from (4.3) and (4.6) that the magnitudes of the coupling matrix elements between zero-order states range from 0.3 to 4.2 GHz. For the dissipative 1792 cm—! case, the variance in coupling matrix elements, V, can be calculated?? by using the golden rule-type expression hao 22V 7p and by assuming that because of symmetry selectivity in coupling, p = ®t = 15 percm™! (0.5 per GHz). This gives |V| = 3.8 GHz. It is clear therefore that coupling matrix elements on the order of several GHz heavily influence both restricted and dissipative IVR in the molecule. Another point concerning the vibrational coupling elements in anthracene per- tains to the distribution of the coupling between zero-order states. One can see from (4.3) and (4.6) that significant coupling occurs between all the coupled zero-order states. Thus, for instance, one would be mistaken in viewing the IVR process as arising from the coupling of an optically active state with a manifold of uncoupled bath states. There ie coupling within the bath manifold, and this coupling affects the IVR dynamics. 106 C. The purity of optically prepared states As we have discussed in section V of Chapter 3, various experimental limita- tions can prevent the complete characterization of IVR processes by time-resolved spectroscopy. One limitation of a fundamental nature involves the inability to op- tically prepare all the eigenstates that arise from the coupling of a given set of zero-order states. In cases where this limitation applies, the optically prepared state is not a pure zero-order harmonic vibrational state. At the very outset the vibrational energy is distributed amongst a number of vibrational motions. Under the conditions of our experiments, we are able to coherently prepare eigenstates within a range of approximately 2 cm—!. Therefore, any coupling inter- actions on this order will prevent the preparation of a pure zero-order state. There is experimental evidence that vibrational couplings of this Magnitude are present in anthracene. Firstly, the sums of the absolute quantum beat modulation depths of the non-a-type fluorescence bands for E,,;, = 1380 and 1420 cm—! are not -1. This kind of behavior can be the result of the inability to prepare a pure zero-order state (see section V-A of Chapter 3). A second point related to the first is that the abso- lute modulation depths calculated from the eigenvector matrices (4.2) and (4.5) are consistently greater in magnitude than observed modulation depths. This, too, can be due to an impure initial state. Finally, evidence from excitation spectra indicates that Fermi resonances that give rise to splittings on the order of several cm—! are prevalent in anthracene. All of this evidence points to the fact that one must be somewhat cautious in determining the purity of the |a) state. For anthracene, it probably is the case in the restricted regime that impure initial states are prepared by the laser, although the sums of absolute modulation depths for non-a-type bands are close enough to ~1 to suggest that the degree of impurity is not particularly great. D. Spectral vs. temporal measurements In light of the previous section, it is pertinent to discuss the merits of time- resolved vs. time-integrated approaches to the experimental study of IVR and 107 vibrational coupling. On the one hand, we have seen that the limitations of finite excitation bandwidth and finite temporal resolution can restrict the utility of time- resolved techniques to the regime of relatively small magnitude matrix elements. In the cases of large magnitude matrix elements, time-integrated fluorescence exci- tation spectra can provide valuable information about long range coupling that is unavailable from fluorescence decays. On the other hand, time-resolved measure- ments have many advantages over fluorescence excitation and dispersed fluorescence studies. Medium resolution (~ 0.05 cm~') excitation spectra are only useful for the study of couplings with matrix elements on the order of, or greater than, rotational contour widths (one to several wavenumbers for jet-cooled large molecules). For smaller couplings, rotational congestion necessitates taking Doppler-free excitation spectra to obtain the number and distribution of the eigenstates resulting from the coupled system of zero-order levels. Whatever the magnitudes of the matrix ele- ments, it is never certain from excitation spectra alone that the structure in them arises from a coupled system of levels. This is in contrast to time-resolved measure- ments which, by virtue of quantum beat phases behavior, can differentiate between the manifestations of coupled vs. uncoupled states. Time-integrated dispersed fluorescence measurements also have limited util- ity in the study of IVR. Such measurements cannot provide detailed information pertaining to the evolution of vibrational energy in a molecule. For instance, the dis- persed fluorescence spectra arising from excitations into the restricted IVR regime look very much the same as the spectra resulting from excitation into the dissipa- tive IVR regime. It is only with knowledge from time-resolved studies that one can confidently make conclusions about the nature of IVR behavior. Also, in deter- mining timescales for IVR, dispersed fluorescence measurements are limited. There is no way by such measurements to determine the recurrence times characterizing vibrational energy flow in the restricted regime. In the dissipative regime, one can in principle determine the timescale for IVR by measurements of the relative in- tensities of vibrationally relaxed and unrelaxed bands in a given spectrum. The 108 problem with such measurements is that both types of fluorescence occur together in the congested region at and to the red of the 09 energy. There is no accurate way to separate out the two types of contributions in this region unless one has precise knowledge of the emission Frank-Condon factors for the optically prepared state. E. Future directions The results and conclusions of this chapter point to several areas wherein fur- ther work could add significantly to the understanding of IVR in anthracene in particular, and large molecules in general. One such area is the role of rotations. Herein and in Chapter 3, we have neglected rotations and, thereby, have assumed implicitly the prevalence of anharmonic coupling interactions in IVR. In the fol- lowing chapter we show how the rotational level structure can affect anharmonic interactions. More detailed work, both experimentally and theoretically, is needed, however, to assess the influence and manifestations of Coriolis coupling. A second area involves the question of the significance of large magnitude ma- trix elements in the vibrational coupling of large molecules. As we have mentioned earlier, work directed toward this problem is in progress.!® A final area involves the determination of the S, vibrational normal modes of anthracene, with the aim of ultimately trying to make the connection between observed coupling matrix elements and the natures of the vibrational states that are coupled. One wonders if all zero-order states of a given symmetry within a given energy interval are significantly coupled, or if, on the other hand, there are propensity rules governing the magnitudes of the couplings between various types of vibrations. Such knowledge is of obvious importance to the understanding of both IVR and vibrational level-specific dynamics. 109 REFERENCES eo Fe YY SP sn 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. W.R. Lambert, P. M. Felker, and A. H. Zewail, J. Chem. Phys. 75, 5958(1981). W.R. Lambert, P. M. Felker, and A. H. Zewail, J. Chem. Phys. 81, 2217(1984). P. M. Felker and A. H. Zewail, Chem. Phys. Lett. 102, 113(1983). P. M. Felker and A. H. Zewail, Phys. Rev. Lett. 58, 501(1984). P. M. Felker and A. H. Zewail, Chem. Phys. Lett. 108, 303(1984). See, for example: C. Lewis, W. R. Ware, L. J. Doemeny, and T. L. Nemzek, Rev. Sci. Instr. 44, 107(1973). P. M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. Le Kaplan, Advanced Calculus (Addison-Wesley, Reading, Mass., 1973), pp. 12-14. (a) J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297(1965); (b) E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, NJ ; 1974). E. Bartholdi and R. R. Ernst, J. Magn. Reson. 2, 9(1973). P. M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969). J. Chaiken, M. Gurnick, and J. D. McDonald, J. Chem. Phys. 74, 106(1981). W. R. Lambert, P. M. Felker, J. A. Syage, and A. H. Zewail, J. Chem. Phys. 81, 2195(1984). | W.R. Lambert, P. M. Felker, and A. H. Zewail, J. Chem. Phys. 81, 2209(1984). B. W. Keelan and A. H. Zewail - submitted. This interpretation is further supported by results reported in Ref. 16, which show that there are indeed two overlapping rotational band contours separated by about one wavenumber in the 1380 cm! region of the anthracene excitation spectrum. K. F. Freed and A. Nitzan, J. Chem. Phys. 73, 4765(1980). (a) D. J. Evans and D. B. Scully, Spectrochim. Acta 20, 891(1964); (b) B. N. Cyvin and S. J. Cyvin, J. Phys. Chem. 78, 1430(1969). Note that the values reported in these papers represent those calculated for Sp modes. One expects that the S; values are lower on the average than these. Therefore, in reality the pyy vs. Ey curve of Fig. 25 probably is shifted somewhat toward lower energies. 110 20. For a review see: D. W. Noid, M. L. Kos Rev. Phys. Chem. 32, 267(1981). zykowski, and R. A. Marcus, Ann. 21. For example, (a) G. Hose and H. S. Taylor, Chem. Phys. 84, 375(1984); (b) M. D. Feit and J. A. Fleck, Jr., J. Chem. Phys. 80, 2578(1984). 22. C. Tric, Chem. Phys. 14, 189(1976). 111 Table 1. Calculated and observed absolute modulation depths for bands in the E,;, = 1380 cm~! spectrum of anthracene. Band-type Individual modulation depths Sum of 3.5 GHz 4.9 GHz 8.4 GHz modulations a-type: calculated +0.16 +0.21 +0.95 +1.26 observed +0.12 +0.16 +0.75 +1.03 b-type: calculated +0.22 —0.86 —0.35 —0.99 observed +0.14 —0.63 —0.26 —0.75 c-type: calculated —0.45 +0.26 —0.80 —0.99 observed —0.45 +0.16 —0.41 —0.70 112 Table 2. Calculated and observed absolute modulation depths for bands in the Ey, = 1420 cm~! spectrum of anthracene. Band-type Individual modulation depths Sum of 1.0 3.5 4.5 6.2 9.7 10.7 GHz modulations a-type: calculated +0.69 +0.004 +0.006 +0.004 +0.42 +0.70 +1.82 observed 0.55 — _ _ +0.36 +0.52 +1.43 b-type: calculated —0.98 -0.07 +0.07 +0.01 -0.15 +0.13 —0.99 observed -0.74 -0.06 +0.04 — -0.11 +0.08 —0.79 c-type: calculated +0.31 +0.03 +0.03 -0.06 -0.62 —0.68 —0.99 d-type: calculated -0.73 +010 -0.17 +0.08 +0.34 —-0.61 —0.99 113 FIGURE CAPTIONS 1. Dispersed fluorescence spectrum and fluorescence decay resulting from ex- citation of jet-cooled anthracene to 5, + 766 cm~! (127). The fluorescence spectrum was obtained with 1.6 A monochromator resolution (= R). An ar- row marks the excitation wavelength. The decay corresponds to detection of the 390 cm~! band in the spectrum with R = 3.2 A. Dispersed fluorescence spectra resulting excitation of jet-cooled anthracene to S, +1380 cm~'. The upper portion is a high resolution (R = 0.5 A) trace of the region in and about the wavelength of the 0} transition of the molecule. The lower portion taken with R = 1.6 A covers a wider Tange and includes the excitation wavelength (arrow). Various bands in the spectra are marked with their shifts in cm~! from the excitation energy. Fluorescence decays of group I bands (see text) in the Ey,, = 1380 cem~! spectrum of anthracene. The shifts of the bands from the excitation energy are given in the figure. Since these bands are relatively isolated spectrally, low detection resolution was used to maximize the signal. From top to bottom R = 24, 16, and 5 A. Fourier spectra of decay residuals (see section II-B) for the 390 cm~! band in the £,;, = 1380 cm~! spectrum of anthracene for three slightly different excitation energies. All other experimental conditions were the same for the decays. The top spectrum corresponds to excitation at the band maximum, the middle spectrum to excitation about 2 cm~! to the red of this, and the bottom spectrum about 2 cm~! to the blue of the maximum. Prominent Fourier bands are labelled with their frequencies in GHz. Note that some bands change in relative intensity from spectrum to spectrum. Note also that all bands have positive amplitudes (+1 phases). Fluorescence decays of group II bands in the Eyip = 13880 cm—! spectrum of anthracene. The shifts of the bands from the excitation energy are given in the figure. From top to bottom R = 0.5, 1.0, and 1.6 A. 10. 11. 12. 13. 114 Fourier spectrum of the residual of the decay of Fig. 5-top. Fourier bands are labelled with their frequency values in GHz. Note that two of these bands have —1 phases and one has a +1 phase. Fluorescence decays of group III bands in the Evie = 1380 cm7! spectrum of anthracene. The shifts of the bands from the excitation energy are given in the figure. From top to bottom R = 0.8, 1.0, 1.0, and 1.3 A. Fourier spectrum of the residual of the decay of Fig. 7-top. Bands are labelled with their frequency values in GHz. Note that two of the bands have —1 phases and one has a +1 phase. Note also, however, that the phase behavior is different from that in Fig. 6. Fluorescence decays of group IV bands in the Eyiy = 1380 cem—! spectrum of anthracene. The shifts of the bands from the excitation energy are given in the figure. For all decays R = 1.6 A. Fluorescence decays of the 1125 cm~! band in the Ey,, = 1380 em-! spectrum of anthracene on long (top) and short (bottom) timescales. The decays are modulated principally at 0.2 GHz. The top decay was taken with R = 1.6 A and the bottom with R = 3.2 A. Dispersed fluorescence resulting from excitation of jet-cooled anthracene to S; + 1420 cm7!. The upper portion was taken with R = 0.6 A and the lower with R = 1.6 A. Various bands in the spectra are marked with their shifts in cm~' from the excitation energy. Representative decay types for fluorescence bands in the Eyiy = 1420 cm! spectrum of anthracene. The wavenumber shifts of the bands from the exci- tation energy are given in the figure. From top to bottom R = 16.0, 1.6, and 1.6 A. Fourier spectra of the residuals of the decays of Fig. 12. Bands in the middle spectrum are labelled with values in GHz. Although there appear to be more than three components in the lower spectrum, only the ones at 1.0, 9.7, and 10.7 GHz are reproducible. Any other apparent bands in this spectrum are 14. 15. 16. 17. 18. 19. 115 therefore probably noise. Fluorescence decays of bands in the E,;, = 1420 cm! spectrum of anthracene that occur near the 0 transition energy of the molecule (see the spectrum of Fig. 11-top). The wavenumber shifts of the bands from the excitation energy are given in the figure. From top to bottom R = 1.6, 1.6, 0.8, 1.6, and 1.6 A. Fluorescence decays of two bands in the Ey, = 1409 cem-! spectrum of jet- cooled anthracene. The wavenumber shifts of the bands from the excitation energy are given in the figure. For the upper decay R = 16 A, for the lower R=1.6 A. Note that the 390 cm~! band is very isolated spectrally compared with the 1640 cm—? band (see Ref. 15). Thus it was possible to use a relatively large value of R in measuring the 390 cm~! decay. The same holds true for the 390 cm—! decays in Figs. 16 and 17. Fluorescence decays of two bands in the E,;, = 1501 em~! spectrum of an- thracene. The wavenumber shifts of the bands from the excitation energy are given in the figure. For the upper decay R = 24 A, and for the lower one R=16A. Fluorescence decays of two bands in the Z,;, = 1514 cm~! spectrum of an- thracene. The wavenumber shifts of the bands from the excitation energy are given in the figure. For the upper decay R = 32 A, and for the lower one R=24A. Dispersed fluorescence resulting from excitation of jet-cooled anthracene to S, +1792 cm~!. The upper portion was taken with R = 0.5 A. For the lower portion R = 1.6 A for the main spectrum and R = 3.2 A for the inset, which corresponds to the vibrationally unrelaxed region. Various bands in the spectra are labelled with their wavenumber shifts from the excitation energy. Fluorescence decays for the 390 and 780 cm~! bands in the spectrum of an- thracene excited to S,; +1792cm7!. The top two decays were taken with R = 3.2 A. For the bottom decay R= 32 A. A slight increase in the relative intensity of the long-time component is apparent in the lower decay compared 20. 2. - 22. 24. 116 with the upper two. Double exponential fit (line) to the measured decay of Fig. 19-bottom. The weighted difference (residual) between the observed decay and the fitted curve appears at the top of the figure. Best fit values of the two lifetimes and the ratio of fast to slow fluorescence are given in the figure. One notes that the best fit fast to slow ratio is greater than what one would judge by eye from the decay because the finite temporal response of detection tends to reduce the apparent magnitude of the fast component. Top and middle: Fourier spectra (I(w)—section II-B) of the decay of Fig. 19- bottom. The top spectrum is the same as the middle one but is expanded along the ordinate. Bottom: The Fourier spectrum of the decay of the 390 cm—! band in the E,;, = 1380 cm~! spectrum (see Fig. 4-bottom), presented to provide an idea of the kind of Fourier “structure” (note the oscillations) to be expected as a result of the truncation of measured decays. (a) Decays measured for similar collection for on-resonance (top) and off- resonance (bottom) detection, and for excitation of anthracene to Evin = 1792 cm~*. Conditions for both decays (R = 3.2 A) were identical except for detection wavelength. (b) Decays measured for similar collection times and under the same experimental conditions (excitation to Ey;, = 1792 em}, detection of the 390 cm~! band with R = 3.2 A) except that the bottom de- cay corresponds to a situation wherein the free-jet expansion was devoid of anthracene. Fluorescence decays of the 390 cm! band in the Ey;5 = 1792 cm~! spectrum of anthracene for (a) different carrier gas conditions and (b) different laser to nozzle distances. For the decays in (a), R = 32 A. In (b), R=16 A. Fluorescence decays of bands in the vibrationally relaxed region of the Eyi, = 1792 cm! spectrum of anthracene (see Fig. 18-top). Wavenumber shifts of the detected bands from the excitation energy are given in the figure. From top to bottom, R = 1.6, 3.2, 1.6, and 16 A. 117 25. Vibrational density of states (p,i9) ve. vibrational energy (Ey) in anthracene calculated using a direct count method and the frequencies of Refs. 19. Given in the figure are characteristic times for IVR in the absent, restricted, and dissipative regimes. 118 Figure 1 E Vip = 766 crn! 390 LY { 3500 | i I I Wavelength (A) i ee VY; = 390 cni'! = 4000 | i | | q { OTL. Time (nsec) 119 Figure 2 E,j» = 1380 ent! 1490 1460. | 1505 1528 514 | 158! 1429 1403 | \ I 3600 3650 3900 Wavelength (A) 120 Figure 3 E Vip = 1380 cnr! ¥, = 750 and 780 cm! Vy = 1162 cri! | ~ Time (nsec) 121 Figure 4 0) E ip = 1380cm"! Y, = 390cm! 84 b) 3 = @ oO 0 c) 0 beh 4 T T ‘ — Frequency (GHz) 122 Figure 5 E 21380 cm™ V4 = 1460 cm™ V4 = 1460 +390 cm" V4 = 1460+1408 cm" _ -_ 1 t q t 4 q ~Time (nsec) Re [In (w)] 123 Evin = 1380 cm! UJ, = 1460 cm! 35S O- 8.47 4. oN a T 1 0 2 4 é 5 Frequency (GHz) 124 Figure 7 E i, !380 em™ V2 '1528cem™ V4 = 1528 +390 cm™ V4 = 1528+ 1408 cm" ¥421429cem" Time (nsec) Re [Im (w)] 125 E Vip = 1380 cm! | YZ = 1528cm"! Oana sf ew 0 2 | T 4 6 Frequency (GHz) 126 Figure 9 E ip = 1380 em"! ¥521490 cm" y= oa | UY, = 1514 cm Time (nsec) 7 127 Figure 10 E n= 1380 cm"! V2 1125 cm"! 0 | ] { t t | 35 Time (nsec) | ] ' i q q ‘ “I Time (nsec) 128 Figure 11 Ein = 1420 env! 1521 i500 | 1566 1480 1598 As I 3600 1" Lad ' | Nl l T 3400 3900 Wavelength (A) 129 Figure 12 E vip = 1420 cm"! Y= 390cm" V2 1500 em™ t i ! | { t ft Time (nsec) Ref In (w)] 130 Figure 13 Ew 21420 em"! %, = 390 cm"! %, = 1500 em'! O- q T T Frequency (GHz) 131 Figure 14 Eat 420 em™ ¥,* 462 em" Time (nsec) 132 Figure 15 EY, = 1409 cm™! V4 = 390 cm" Time (nsec) 133 Figure 16 = =| Vy = 390cm! ¥, = 1600 cmv! ~ —_— on md a — — Time (nsec) 134 Figure 17 Ey = '514enr! V, = 390cnr! V, = 1680 crt! 135 Figure 18 E Vin = 1792 env! 1880 I915 I970 . q 3600 3650 Vibrationally Vibrationally “Unrelaxed" "Relaxed" | 390 780 Laser i =. - T T T T a 3390 3890 Wavelength (&) 136 Figure 19 E Vib = 1792 cm V, = 780 cnt! tear vated Vy =390cmi! Wide Slits Time (nsec) 137 Figure 20 Time (nsec) Re [! (al) 138 Figure 21 Ein? !792 em! % = 390cm"! Eine 1792 em"! % = 390 em"! Evib= 1380 em”! 8, = 390 cm"! Ly T T 1 ; 8 Frequency (GHz) 139 Figure 22 a) dg 734358 (390 err") Ag? 34108 f T T T T T T T 0 7 Time (nsec) With Anthracene 7 Hien Without Anthracene t qT U q U q q ~~“ Time (nsec) 140 Figure 23 a) Pe235psig He P= 35 psig Np — 0 Time (nsec) b) x» = 3mm _ PA aie eet x 210mm Time (nsec) 141 Figure 24 E ip * 1792 ent! Dy = 1880 cn? ¥, = 1970 cm! Vg = 1970 cn! Wide Slits “TL Time (nsec) 142 Figure 25 OS Oo! (wd) qiAd OSI 002 143 CHAPTER 5 ROTATIONAL EFFECTS ON IVR* * Submitted to the Journal of Chemical Physics, as “Dynamics of intramolecular vibrational-energy redistribution (IVR). Ill Role of rotations,” by Peter M. Felker and Ahmed H. Zewail 144 I INTRODUCTION In the theoretical treatment of vibrational quantum beats (Chapter 3!) and in the interpretations of experimental results?~® pertaining to this phenomenon that have been reported on by this laboratory, little attention has been directed toward one molecular characteristic that might be expected to play a large role in influenc- ing the observed consequences of vibrational coupling: the fact that each vibrational level is actually an entire manifold of rotational levels. In some sense, our neglect of this aspect of molecular level structure can be justified by the close correspondence between the theory constructed without its inclusion and experimental results?—*6, This argues that the observed results are, on the whole, manifestations of vibra- tional couplings which are weakly dependent on rotational level. On the other hand, we have reported’ some results which can only be interpreted by taking into account the rotational level structure. Morever, a comprehensive description of vibrational quantum beats should come to terms with the manifestations that can arise from rotational effects on vibrational coupling. This is a particularly pertinent task since it has been widely believed that such effects preclude the possibility of observing vibrational quantum beats in the fluorescence of large molecules, even molecules cooled by supersonic expansion; the number of populated ground state rotational levels and the variation in vibrational coupling with rotational level in the excited state are thought to be large enough that any coherence effects due to rovibration-rovibration coupling will be washed out. In actual fact, the number of superimposed incoherent excitations due to rotational level structure is large for large molecules (like anthracene), even at very low rotational temperatures (e.g., 2 K). It is clear, however, that vibrational quantum beats can be observed for such molecules. Therefore, it must be that rotational effects on vibrational coupling need not be so marked as to render quantum beats unobservable. Yet, it may also be true that such effects are large enough to manifest themselves more subtly in beat-modulated decays. Since the first observation? of phase-shifted quantum beats in anthracene, one 145 characteristic has been noted which cannot be reconciled with theoretical predic- tions. While theory! predicts essentially equal decay rates for the amplitudes of the modulated and unmodulated portions of a given decay, it has been observed in general that the amplitude of the modulated portion of an experimental decay {3.¢., the quantum beat envelope) decays faster than that of the unmodulated por- tion (i.¢., the lifetime-type decay). Two aspects of this discrepancy point toward a rotational involvement in vibrational coupling. Firstly, the theory in question does not account for rotations, and it is plausible to attribute the errors in theoretical predictions to the neglect of such a factor. Secondly, and more significantly, an increased beat decay rate implies that the widths of the beat components in the frequency domain are greater than the width of the zero frequency component cor- responding to the unmodulated exponential. An increased width is just what one would expect, given vibrational couplings which are dependent on rotational level. In an effort to establish a firmer link between quantum beat envelope decays and rotational influences we have performed a number of experiments and theoretical simulations. At the heart of both the experimental and theoretical approaches is the fact that, if vibrational coupling is dependent on rotational level, then this will be manifested as a dependence of beat-modulated fluorescence decays on the rotational temperature of the sample. By varying this temperature and observing the changes in decays, one can assess the rotational influences on beats. In this chapter, we present experimental results which show that the decay rates of quantum beat envelopes of jet-cooled anthracene increase significantly as the rotational temperature of the sample increases. This behavior is found to be consistent with the results of theoretical simulations for which it is assumed that 1) vibrational levels are coupled by anharmonic coupling and 2) rotational effects on this coupling arise solely from the differences in rotational constants between coupled vibrational levels. The results lead us to conclude that rotational effects on vibrational coupling need not wash out the manifestations of vibrational coherence in large molecules, but may be subtly manifest in observed beat-modulated decays. 146 Il. THEORETICAL SIMULATIONS A. Outline of calculations In this section the results of calculations designed to elucidate the effects which rotational temperature may have on vibrational quantum beats in anthracene will be presented. First, however, it is pertinent to outline these calculations and point out the various approximations inherent in them. A basic assumption involved in the computations is that any coherence effects due to the coherent excitation of rotational levels within the same vibrational state are negligible. This assumption, which can be justified on experimental and theoretical grounds’, has the conse- quence of allowing one to calculate a fluorescence decay simply by summing the individual decays associated with individual rotational levels. In particular, if one adopts the definitions and nomenclature of the previous two chapters!’®, and con- siders the fluorescence intensity [,(t) of a y-type fluorescence band, then L()=30 > W(J, Ka, T)I4(J, Ka, t), (2.1) J=0 Ka=-J where J and K, are the usual rotational quantum numbers of an approximate prolate symmetric top® and refer to rotational levels in the manifold of the |a) (optically active) zero-order vibrational level, T is the rotational temperature of the sample, W (J, Ka,T) is a weighting factor for each rovibrational level |a, J, Ka), and I,(J, K.,¢) is the y-type fluorescence decay which arises from the coupling of \a, J, K,) with the rotational levels of the other zero-order vibrational states. To calculate I.,(é) from (2.1) it is very useful to make several simplifying ap- proximations. The first has already been mentioned; the molecule is treated as an approximate symmetric top. This allows one to calculate the rotational energy of a given rotational level |J, K,) as® E(J,K,) = =(B +C)I(J +1) +(A- =(B +0))K?, (2.2) where A, B, and C are the rotational constants of the molecule and for anthracene are taken to have mean values® of 2155, 453, and 375 MHz, respectively. A second 147 useful approximation is the assumption that the thermal distribution of ground state rotational levels that exists prior to the excitation of the sample is projected into the excited state upon excitation. One then obtains the following expression® for the weighting factor in (2.1): W(J,Ka,T) = gw (J, Ka)(2I +1) exp(-E(J, Ka) /kaT), (2.3) where gy(J,Kq) is the nuclear spin statistical weight (which shall henceforth be taken equal to 1 for all rotational levels!°), and kg is Boltzmann’s constant. A third approximation involves limiting the sum over J in (2.1). This is justified by the expectation that W(J,K.,T) will become small for high enough J,K,. The calculations reported here for anthracene were made for J < 30. No significant qualitative deviations from the results of these calculations were noted when the J range was increased to J = 60 and beyond. The factors I,(J, Ka,t) in (2.1) represent the effects of rovibrational coupling in the decay. The form which these terms take depends wholly on the assumptions made concerning the coupling. Once these assumptions are made, I,(J, Ka, t) can be calculated by first diagonalizing the rovibrational Hamiltonian for la, J, K,) and all the levels coupled to it, and then by using the eigenvalues obtained to get the beat frequencies and the eigenvectors obtained to get the modulation depths (magnitudes and phases) of the beat components.!4 Knowing both W(J,K,,T) and 1,(J, Ka, t) for all pertinent rotational levels, I(t) can be calculated. The calculations of I(t) presented in this chapter have been performed in such a way as to allow direct comparison with experimental decays. In particular, it is not actually the I,(t) which are shown in the figures but convolutions of typical temporal response functions with I,,(t) (see eqn. (2.1) of Chapter 4). Also, the simulated decays have been calculated for one thousand discrete points in time to match the number of channels that were used for collecting experimental decays. 148 B. Anharmonic coupling Anharmonic coupling between zero-order rovibrational levels is characterized by two general features (see Fig. 1). Firstly, angular momentum selection rules and symmetry restrictions® limit this type of coupling to rovibrational levels of |a), |b), |c),... having the same J, K, values. Thus, for a coupling between N vibra- tional states, one need only consider an N x N Hamiltonian matrix for each value of J, Ka, the rows and columns of such a matrix being labelled by the zero-order states |a, J, Ka), |b, J, Ka), |\c, J, Ka).... Secondly, the off-diagonal elements of the Hamiltonian matrices do not change with rotational level® (to first order). As a consequence of this, I,({J,K.a,#) only depends on J, K, through variations in the energy spacings of the zero-order levels; i.e., variations in the differences between diagonal elements of the Hamiltonian matrix. Such variations with rotational level can occur if some of the coupled vibrational states have different rotational con- stants, in which case the zero-order energy differences between states |y) and |+’) can be expressed as B+C Bid Eyy(J, Ka) = E4(0,0) + Ang (—-)I(F +1) + A yw(A- ° )K?, (2.4) where A, (24°) and A,y(A — (2£2)) represent the mismatches in rotational constants between |) and |7’). The perturbations which contribute to rotational constant mismatches between vibrational states are small ones’. Thus, these mismatches are expected to be small fractions of the mean rotational constants. Since data pertaining to the rotational constants of vibrational states of aromatics are hard to come by, an order of mag- nitude for the effect has been assumed after consideration of more readily available measurements of the changes in constants between electronic states®. Such changes are on the order of several parts in a thousand. For the results presented herein it has been assumed that rotational mismatches between vibrational states are roughly one part in one thousand. For anthracene, this means that Ay (Bt) ~ 0.4 MHz and A,~"(A — BYC) ~ 1.7 MHz. Actual values for mismatches for a given cal- 149 culation were generated by a random number generator constrained to produce rotational constants within a set range. Two-level coupling is conceptually the easiest case since there are only pairwise interactions up the rotational level structure of the coupled vibrational states (see Fig. 1). One expects the overall coupling between manifolds to give rise to a large number of different beat frequencies, each corresponding to the interaction of a pair of rovibrational levels, but all being fairly close to some average value. At finite rotational temperatures one expects that the superposition of all of these different fluorescence decays will result in a total fluorescence decay in which only one beat component appears to be present and in which the rate of decay of the beat envelope will be greater than the decay rate of the unmodulated portion of the transient. Moreover, this envelope decay rate should increase with increasing rotational temperature. In Fig. 2 we show calculated results of both a-type and b-type decays for a model two-level system. For these calculations E,,(0,0) of (2.4) and V,,, the anharmonic coupling matrix element, have been chosen to be 2.24 and 1.0, respectively, to give typical values for both the beat frequencies (3 GHz) and the modulation depth (0.28) of the a-type band. The other calculational parameters are included in the figure caption. One can clearly see from the figure that the beat envelope decay rate and its behavior as a function of temperature match our intuitive expectations of this behavior. To aid direct comparison with experimental data, calculations have also been performed using the 4 x 4 and 3 X 3 Hamiltonians which were derived from experi- mental beat data, and which correspond to the S$; +1420 cm~! and S, +1380 cm7! excitations of anthracene, respectively. These two Hamiltonians appear in the previous chapter®. Fig. 3 presents calculated decays of the b-type bands of the S; + 1420 em! system for different rotational temperatures. The b-type band was chosen because good experimental data for it are relatively easy to obtain and be- cause the decay is dominated by one beat component (1 GHz with a —1 phase). The rotational constants used for the calculation appear in the figure caption. Sets of 150 other constants yielded qualitatively similar results. The system response function used was 150 psec FWHM (two channels). Clearly, there is a marked rotational temperature effect on the calculated decays. Although all of the decays are modu- lated similarly at early time, at later times the beats are increasingly washed out as the rotational temperature increases; that is, the apparent decay rate of the beat envelope increases at higher rotational temperatures. Behavior similar to this also occurs in calculated decays of other types of bands in the fluorescence spectrum. Figs. 4 and 5 present calculated decays of the b-type bands in the fluorescence spectrum corresponding to 5S; +1380 cm! excitation. Such decays® are modulated by beat components at 3.5, 4.9, and 8.4 GHz with phases of +1, —1, and —1, respectively. The two figures correspond to two different sets of rotational constants. Both sets of decays were calculated using a response function of about 90 psec FWHM (six channels). Again, and in both figures, one can see an increase in the beat envelope decay rate as the rotational temperature increases. One may notice an additional point from the decays of Fig. 5. Not all beat components decay at the same rate—one component persists at long time, even for T,o¢ = 10 K. Evidently, those components which turn out to be more (or less) sensitive to rotational temperature are determined by the mismatches in rotational constants. As with the S; +1420 cm! case, calculated decays for the other band types in the S; + 1380 cm~! spectrum show trends similar to the decays of the b-type band. 151 Ill. EXPERIMENTAL The experimental apparatus and conditions have been reported in the preced- ing chapter®. The dependence of beat parameters on rotational temperature was assessed by measuring fluorescence decays as a function of carrier gas and carrier gas pressure. The carrier gases used were helium, neon, and nitrogen. A small laser-to-nozzle distance (x) dependence was observed for the beat-modulated de- cays; the greater z, the slower the decay rate of the beat envelope. Because of this, data for different carrier gas parameters are only compared for the same values of z. An excitation wavelength and bandwidth effect was also observed on the decays. In general, decays collected for excitation slightly off a given band maximum were less modulated than for excitation at a band maximum. Similarly, the greater the bandwidth of excitation, the smaller the observed modulation depths. Again, those decays collected for different carrier gas parameters are compared herein only if they were measured under identical excitation conditions. Most data were taken for the S$; +1420 cm™! excitation in anthracene, although some are presented for the S; + 1380 cm! excitation. There are two reasons for the preponderance of S; + 1420 cm! data. Firstly, it is relatively easy to measure decays of the intense, spectrally isolated 1750 cm! band in the S; + 1420 cm~! 6 fluorescence spectrum”. Secondly, the decays of this band are dominated by one beat component. Thus, they may be fit to functions having a relatively small number of adjustable parameters. Experimental decays that were dominated by one beat frequency were fit to the convolution of the measured system response function with a function of the form: I(t) = Ayfe7P*! + Ageb?! cos wt] + As, (3.1) where A,, Ag, As, [ 1, and I, are parameters. Such fits were accomplished via a nonlinear least-squares algorithm!?. The fit parameters obtained in this way are useful indications of the effects which rotational temperature has on modulation depths (Az) and quantum beat envelope decay rates (I'g). 152 The Fourier analysis of several experimental decays was also undertaken. The method used is the same as that reported on in the preceding chapter. The real parts of the Fourier transforms of entire decays compensated for response function effects are presented. Unlike those in Chapter 4, the decays that were subjected to analysis were truncated at a late enough time to eliminate substantially any truncation artifacts in the transforms. 153 IV. RESULTS A. S; + 1420 em7! Fig. 6 presents decays of the 1750 cm—! fluorescence band of anthracene excited to S; + 1420 cm! for three different carrier gas conditions. One point which is immediately obvious from the figure is that the amplitude of the 1 GHz component, which modulates all three decays, decays at an increasingly faster rate as the carrier gas changes from 75 psig neon to 40 psig helium to 40 psig nitrogen. A more quantitative measure of this trend can be obtained from the fits of these decays. The fit parameters corresponding to the modulation depth of the 1 GHz component are —0.7, —0.67, and —0.61, and the decay rates of the beat envelopes are 0.13, 0.18, and 0.31 GHz for the three decays from top to bottom. While the values for the three modulation depths do not vary very widely and are close to the values previously reported, the values for '2 change from being very close to the unmodulated decay rate (0.11 GHz) to a value which is a factor of ~ 3 greater than this. This trend is matched by the width of the 1 GHz band in the Fourier spectra of the decays of Fig. 6. These spectra, which appear in Fig. 7, show a clear increase in the bandwidth of this component as the carrier gas changes from Ne to He to Ne. | The changes which occur in I’, as the carrier gas is varied also occur when the carrier gas pressure is varied. In general, as the carrier gas pressure increases, I’, decreases. This effect on Tz is not as marked as that which arises when the gas is changed from Ng to Ne (for instance, a decay taken under the same conditions as for those corresponding to Fig. 6-top, except that P = 50 psig rather than 75 psig Ne, was fit to a function with T, = 0.15 GHz) but it is reproducible for all three carrier - gases. In the extreme case of no carrier gas, the quantum beats are completely washed out, as shown in Fig. 8. Besides the 1750 cm~! band, the carrier gas dependences of the decays of other fluorescence bands arising from S, + 1420 cm—! excitation have also been measured. In particular, the 390 and 780 cm—! a-type fluorescence bands® have been studied. Fig. 9 shows results for the 780 cm—! band. The overall beat modulation depth 154 for the decays is less than that of the 1750 cm~! decays for two reasons; (1) the inherent modulation depth for the 1 GHz component is less for the a-type bands in the spectrum than for the non-a-type 1750 cm~! band, and (2) the large detection bandwidth needed to measure decays of the weak 780 cm~! band made it impossible to spectrally isolate the band. The values for I'2 derived from fits to the decays of Fig. 9 are in good agreement with those found for the 1750 em~! decays for similar carrier gas conditions. B. 5, + 1380 em™! Extensive studies of the effects of carrier gas parameters on the decays of the bands in the 5; + 1380 cm~! fluorescence spectrum have not been performed with 80 psec time resolution. Nevertheless, the limited studies which have been per- formed reveal behavior entirely consistent with that observed for the 51 +1420 cm7! excitation. Fig. 10 provides an example. The two decays correspond to the de- tection of the 1460 cm~! (b-type) band® in the spectrum for two different carrier gases. Although the presence of three beat frequencies in the decays makes fitting them difficult, it is clear without any fits that the modulations wash out much more quickly in the nitrogen than in the neon decay. Similar behavior has been observed for the 390 (a-type) and 1528 cm™! (c-type) bands® in the spectrum. 155 V. DISCUSSION It is clear from the simulated decays of Figs. 2-5 that a marked rotational temperature effect on the beat-modulated decays which arise from vibrational states coupled by anharmonic interactions can exist, and that this effect is primarily on the decay rates of quantum beat envelopes. One would now like to directly relate this trend to experimental results. The analysis of the experimental results of section IV in terms of the effects of rotational temperature on beats requires some prior knowledge of the way in which | different experimental expansion parameters affect the rotational temperature of the sample. Knowledge of this sort is available from experimental and theoretical sources. Experimentally, we have reported? the measurement of 0° B-type rota- tional contours of anthracene for a variety of expansion conditions. The band shape was found to change noticeably with changes in expansion conditions (see Fig. 20, Ref. 9). These changes match the changes that occur in calculated B-type contours as a result of variations in the rotational temperature (see Fig. 21, Ref. 9). In par- ticular, the measured contours match the calculated behavior if it is assumed that the rotational temperature follows the trend T(Ne) < T(He) < T(No) for changes in carrier gas. Although no fits to the experimental contours have been performed, rough estimates of the rotational temperature for various expansion conditions can be made by comparison of the experimental and simulated contours. These esti- mates go from 1 K for 50 psig Ne to 10 K for 20 psig No. Theoretically, an idea of the rotational temperature of a free-jet sample can be obtained if it is assumed that the rotational temperature follows the terminal translational temperature of the expansion. Terminal translational temperatures have been calculated!* for typical expansion conditions used in this laboratory. The general trend and the absolute magnitudes of these temperatures for different carrier gases and pressures are consistent with the conclusions derived from the contour measurements. For example, for a nozzle temperature of 450 K and a pinhole diameter of 150 xm, 40 psig expansions of Ne, He, and Nz were calculated 156 to have terminal translational temperatures of 0.9, 1.2, and 3.9 K, respectively. One expects the rotational temperatures of such expansions seeded with anthracene to be somewhat higher than these values, being that rotational degrees of freedom do not cool as efficiently as translational degrees of freedom. Having established a semi-quantitative link between expansion parameters and rotational temperature, one can now consider the changes in quantum beat decays with rotational temperature. As was noted in section IV, the major trend which emerges from the experimental results is that the decay rate of quantum beat en- velopes increases as the carrier gas is changed from Ne to He to No. Interpreting this trend strictly in terms of the changes induced in rotational temperature with changes in carrier gas, one may say that as the rotational temperature increases, the beat decay rate increases, as well. Now, it may be argued that some carrier gas dependent collisional interaction between the carrier gas and anthracene is respon- sible for the observed trend, instead of changes in rotational temperature. However, if this were the case, the beat decay rate would be expected to increase as the pres- sure of a given carrier gas were increased. This is the opposite of what is observed. On the other hand, this kind of behavior with pressure is consistent with rotational temperature changes since the temperature would be expected to decrease with increasing pressure!*. Therefore, one arrives at the conclusion that increasing the rotational temperature of anthracene increases the quantum beat decay rates of the fluorescence decays corresponding to the S$; + 1380 cm~! and S, + 1420 cm7! excitations. Complementary to this effect of rotational temperature on decay pa- rameters is the effect of temperature on the widths of the beat component in the Fourier spectra (Fig. 7). Just as one might expect, the increase in beat decay rate with increasing rotational temperature is directly linked to an increase in bandwidth in the frequency domain. The changes with rotational temperature that occur in measured decays are quite similar to those changes which occur in simulated decays calculated assuming anharmonic coupling and rotational constant mismatches. This similarity is readily 157 seen in a comparison of Figs. 3 and 6, and Figs. 4 and 10. We take this as strong evidence that the observed beat envelope decay behavior arises from some anhar- monic coupling-rotational constant mismatch mechanism. The operation of such a mechanism in enhancing beat decay rates is perhaps best explained by reference to what happens in the frequency domain. Given small variations with rotational level in the Hamiltonian matrices (Ho(J,K.)) describing the coupling of a set of rovi- brational levels, one expects similar variations in the eigenvalues of these matrices. Given also a situation wherein Ho(J,K.) tends to deviate more from Ho(J’, K!) when J and K, differ more from J’ and Ki (c.f., (2.4}), then one expects the eigenvalues of the two matrices to differ more for more widely separated J, K, and J', K!.. Now at low rotational temperatures, there is a narrow distribution of pop- ulated rotational levels. Thus, the relevant values of J,K, are all close to one another and the variations in eigenvalues of the relevant Ho(J, Kz) are small. As T increases, however, more widely varying values of J,K, become relevant, and the variations in eigenvalues increase. Small variations in eigenvalues translate directly into beat frequency bandwidths. Thus, it is clear that these beat frequency band- widths will increase with increasing rotational temperature given the hypotheses about Ho(J, K.). Moreover, since the rate of decay of a beat component can be directly related to the bandwidth of that component, it is also clear that beat decay rates should increase with increasing rotational temperature. Note that in the preceding paragraph we have emphasized the smallness of the variations in Ho(J,K.). Large variations do not, in general, give rise to the sort of quasi-continous variations in eigenvalues which result in bandwidths for beat components. It is the smallness of the variations in Ho(J,K.) that would be expected given anharmonic coupling and rotational constant mismatch that renders such a mechanism a plausible explanation for the observed behavior. A connection between beat envelope decay rates and the rotational constants of zero-order vibrational states could turn out to have very useful consequences in the study of intramolecular vibrational energy redistribution (IVR). Because 158 different classes of vibrational motions in a molecule would be expected to have rotational constants with characteristicly different deviations from average values, then knowledge of these deviations could facilitate the assignment of vibrational states in terms of the normal modes of the molecule. The ability to do this for the coupled vibrational states giving rise to an IVR process is of tremendous and obvious value to the full understanding of the process. To help accomplish this with beat envelope decay rates, one would require a more quantitative theoretical connection between these rates and rotational constant mismatches. Experimentally, more precise measurements of rotational temperatures and decay rates (especially for decays modulated by a number of components) would be needed. Such advances are certainly not inconceivable and, indeed, may be at hand. At this point, a few remarks concerning the uniqueness of the anharmonic coupling-rotational constant mismatch mechanism in explaining the trend of the experimental results should be made. One can certainly envision other coupling mechanisms which would give rise to distributions of beat frequencies dependent on rotational temperature. In particular, one might consider Coriolis coupling®-!*. Coriolis coupling, however, has matrix elements which are steeply dependent on J,K,. Given a Hamiltonian matrix with off-diagonal elements solely determined by Coriolis coupling, one would not expect the simple beat patterns and small numbers of beat components observed in the experimental decays®. Indeed, even decay simulations done by us using a 2 x 2 Hamiltonian matrix have been found to produce extremely complicated beat patterns. Nevertheless, this is not proof positive that Coriolis coupling does not bear any influence on the experimental results. We say this because Hamiltonian matrices with interaction matrix elements solely determined by Coriolis coupling are not the only ones in which such coupling may be realized. In fact, it is more probable that most real Hamiltonians consist of some states coupled anharmonically and other ones coupled (probably weakly) by Coriolis interactions. This kind of situation could give rise to subile rotational temperature effects such as have been reported on herein. And, they may explain 159 other, less subtle effects reported on elsewhere®. The large number of possibilities for such mixed coupling has precluded, however, any detailed consideration here. The theoretical and experimental results of this chapter on the role of rota- tions in IVR can be related to temperature effects on intramolecular dephasing. At 0 K rotational temperature, IVR occurs with the rotational “bath” being es- sentially empty. ['2 approaches the fluorescence decay rate T', in this limit. Or, in the language of the density matrix formalism, Tz approaches T;. As the rota- tional temperature increases, however, rotational dephasing, brought about by the many overlapping incoherent excitations from thermally populated J, K, levels, is possible. This pure dephasing causes Tz to become less than T,. In a sense, the rotational reservoir in this case is similar to the phonon reservoir in solids, for which Tz has been measured to be less than T; at temperatures above ~ 2 K.'5 160 VI. CONCLUSIONS We have considered theoretically the effects which the rotational level struc- ture can have on beat-modulated fluorescence decays that arise from anharmonically coupled vibrational states. For small mismatches in rotational constants between coupled vibrational states, these effects primarily take the form of increased decay rates for quantum beat envelopes relative to the overall fluorescence decay rates. Experimental results of decays as a function of carrier gas parameters reveal be- havior in anthracene which is entirely consistent with the theoretical results. Taken together, all the results argue convincingly that 1) vibrational coherence effects need not be washed out by the rotational level structure, 2) anharmonic coupling is the primary coupling interaction giving rise to phase-shifted beats in anthracene, and 3) the effects of rotations on IVR can be assessed by time-resolved experiments. 161 REFERENCES P. M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. P. M. Felker and A. H. Zewail, Chem. Phys. Lett. 102, 113(1983). P. M. Felker and A. H. Zewail, Phys. Rev. Lett. 58, 501(1984). 1 2 3 4. P.M. Felker and A. H. Zewail, Chem. Phys. Lett. 108 303(1984). 5. W.R. Lambert, P. M. Felker, and A. H. Zewail, J. Chem. Phys. 81, 2217(1984). 6 P. M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. 7. Experimentally, the justification derives from the fact that no quantum beats have been observed for a large number of low energy (Evie < 1200 cm?) excitation bands in anthracene. Given observable coherence effects involving rotational levels in the same vibrational manifold, one would expect to observe beats for excitation of essentially every band. Theoretically, one notes selection tules which limit both the number of levels which can be prepared coherently usa excitation and which restrict the number of beating fluorescence transitions from these coherently prepared levels. Further, one notes the large distribution of beat frequencies and the large number of high frequency components to be expected from the coherent preparation of rotational levels within the same vibrational state. 8. See G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules (Van Nostrand, Princeton, 1945). 9. W.R. Lambert, P. M. Felker, J. A. Syage, and A. H. Zewail, J. Chem. Phys. 81, 2195(1984). | 10. The actual values for the four types of levels are not all equal (they have relative values of 1.12, 1.00, 0.94, and 0.94). Nevertheless, the number of thermally populated rotational levels is large enough, and the weights near enough to one another, that one expects that the differences in weights do not give rise to any readily observable effects. 11. See section II of Ref. 1 (Chapter 3). Note that I,(J,K,,t) used herein corre- sponds to I,(t) in Ref. 1 in the sense that both are associated with a single N x N Hamiltonian matrix. [,(é) in this chapter, however, is associated with the large number of independent matrices describing the vibrational coupling interactions as a function of rotational level. 12. P. R. Bevington, Data Reduction and Error Analysis for the Pysical Sciences (McGraw-Hill, New York, 1969). 13. W. R. Lambert, Ph.D. thesis, California Institute of Technology (1982). 14, E. B. Wilson, J. Chem. Phys. 4, 313(1936). 162 15. M. J. Burns, W.-K. Liu, and A. H. Zewail, in: Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, Vol. 4, edited by V. M. Agranovich and R. M. Hochstrasser (North-Holland, Amsterdam, 1983), p. 301. 163 FIGURE CAPTIONS 1. Schematic diagram of the anharmonic coupling between the rotational levels of two zero-order vibrational states—ja) (left) and |b) (right). Only those rotational levels having the same rotational quantum numbers (J, K,) are coupled. Moreover, the coupling matrix element V is constant for each pair of coupled rovibrational states. Coupling can vary up the rotational manifold only through differences in energy spacings between coupled states. (Such differences are not shown in the figure.) Note that there has been no attempt to draw the spacings between levels to scale. 2. Simulated decays as a function of rotational temperature for the a-type and b-type decays of a coupled two-level system. The lowest rotational states of the two zero-order levels were taken to be spaced by 2.24 GHz (the |b) state being at higher energy) and the coupling matrix element was taken to be 1 GHz. The rotational constants used, (2°) and (4— 2°), were 0.4119 and 1.7396 GHz for the |a) state; and 0.4116 and 1.7385 GHz for the |b) state. For other details see the text. 3. Simulated b-type decays as a function of rotational temperature for the coupled ~ four-level system described by the Hamiltonian matrix of eqn.(4.6), Chapter 4, which represents the coupling situation at Ey;, = 1420 cm! in S, anthracene. The randomly generated rotational constants (2°) and (4 — 24°) were for the |a), |b), |e), and |d) states, respectively: 0.4119 and 1.7396; 0.4124 and 1.7417; 0.4120 and 1.7401; and 0.4122 and 1.7407 GHz. 4. Simulated b-type decays as a function of rotational temperature for the coupled three-level system described by the Hamiltonian matrix of eqn.(4.3), Chapter 4, which represents the coupling situation at E,;, = 1380 cm~! in S, anthracene. The rotational constants (2°) and (A — 2£©) used were, for the |a). |b), and |e) states, respectively: 0.4127, 1.7428; 0.4119, 1.7395; and 0.4117, 1.7389 GHz. 5. Same as Fig. 4 but using the rotational constants: 0.4116, 1.7383; 0.4119, 1.7397; and 0.4122, 1.7409 GHz for the |a), |b), and |c) states, respectively. 6. 164 Measured fluorescence decays of the 1750 cm! (b-type) band in the Ey, = 1420 cm! spectrum of jet-cooled anthracene as a function of carrier gas pa- rameters. Decays were measured under identical conditions except for carrier gas. For each decay z = 6 mm, the monochromator resolution R = 3.2 A, and the laser bandwidth BW ~ 2 cm™}. Fourier spectra (I(w)—see section II-B of Chapter 4) of the decays of Fig. 6. The peaks are negative because the 1 GHz component in the decays has a —1 phase. 8. Measured fluorescence decays for detection of the 1750 cm~! band in the E,;, = 1420 cm~! spectrum of anthracene for no ¢arrier gas (top) and for 30 psig neon (bottom), all other conditions being the same (BW ~ 3cm7!, R = 8.0 A, z = 3mm). The relative lack of modulation in the neon decay compared to the decay of Fig. 6-top is primarily due to the poorer detection spectral resolution used in obtaining the former decay. 9. Measured fluorescence decays for detection of the 780 cm~* (a-type) band in the 10. Eviy = 1420 cm! spectrum of jet-cooled anthracene as a function of carrier gas parameters. For each decay z = 3 mm, R=16 A, and BW ~ 2cm7!. Measured fluorescence decays for detection of the 1460 cm7! (b-type) band in the Ey; = 1880 cm~! spectrum of anthracene as a function of carrier gas parameters. For each decay z = 3mm, R = 0.8 A, and BW ~2cm7?. 165 Figure 1 ZERO-ORDER ROVIBRATIONAL STATES : ven - 1,+1 ° syn 1,41 ro 4,0 yo ’ 1,0 f 0,0 wr J,K, = 0,0 [a> [b> MANIFOLD MANIFOLD 166 Figure 2 THEORETICAL THEORETICAL a- type Decoys b-type Decoys i Trot # IK Trot # 1K 5K SK 1OK OK oO O- Time (nsec) Time (nsec) 167 Figure 3 THEORETICAL Trot ® 2.5 K ; a 5K = ha OK 0 a | a] T 1 1 i] 70 Time (nsec) Figure 4 THEORETICAL 170 Figure 6 = =| V5 = 1750cnr' EXPERIMENTAL 70 psig Ne 40 psig He _ 40 psig N, f T T T T T T T 0 70 Time (nsec) 171 Figure 7 = =I v, = 1750 cn’! EXPERIMENTAL 0 ete cael 70 psig Ne _~ 3 L440 ® 7 40 psig He 0 | T T 1 0 0.5 1.0 15 Frequency (GHz) 172 Figure 8 E vib = 1420 cm”! Y= 1750 cm”! Experimental No carrier gas 30 psig Ne “Lh | q L q | Time (nsec) 173 Figure 9 - | E vib = 1420 cni v= 780cni! EXPERIMENTAL 50 psig Ne 40psig He 40psig Np t q in t q J . 70 Time (nsec) 174 Figure 10 Evin = 1380 cm! Y, = 1460 cm! EXPERIMENTAL | 50 psig Ne J SO psig N, a ! ] T T T T 1 1 0 14 Time (nsec) 175 CHAPTER 6 ENERGY DEPENDENCE OF IVR IN t-STILBENE* * Submitted to the Journal of Chemical Physics, as “Dynamics of intramolecular vibrational-energy redistribution (IVR). IV Excess energy dependence, t-stilbene,” by P. M. Felker, W. R. Lambert, and A. H. Zewail 176 I. INTRODUCTION In this chapter we present experimental studies of IVR in jet-cooled t-stilbene. The aim is to assess the generality of the picture of IVR which has emerged from the preceding three chapters.1—* For several reasons this picture of IVR might be considered primarily to be representative of “well-behaved” molecular systems such as anthracene. These reasons are rooted in the theoretical treatment of Chapter 3 and the assumptions made therein. It was shown in Chapters 4 and 5 that, for the anthracene molecule, the theory of Chapter 3 applies rather well, especially when ac- count is taken of the possible rotational level dependence of anharmonic vibrational coupling. The uniqueness of the anthracene system arises from several molecular properties: 1) anthracene is a relatively rigid molecule and consequently less vul- nerable to vibrational coupling interactions having matrix elements dependent on rotational level, 2) the molecule is of high enough symmetry (D2,) to render the great majority of vibrational modes optically inactive and to effectively limit vibra- tional interactions, and 3) no ultra-fast radiationless process other than IVR occurs in the pertinent energy region of the S, state. IVR in molecules which undergo sig- nificant Coriolis vibrational coupling, which have a large fraction of optically active vibrational modes, or which are subject to photophysical or photochemical processes on timescales comparable to or more rapid than IVR, might not be expected to be well described by the theoretical treatment of Chapter 3. With the above points in mind, one notes that t-stilbene has several character- istics that could render it less than well-behaved when compared with anthracene. The molecule has a number of low frequency, presumably large amplitude vibra- tional modes*:> (e.g, the phenyl torsions) which could induce large anharmonic coupling matrix elements or couplings with marked rotational level dependences. It has low symmetry (Cox, or less®*), which leads to rich spectral structure and, possi- bly, to complex coupling interactions. And, for sufficiently great vibrational energy in its S, state, it undergoes photoisomerization on a sub-nanosecond timescale.” It is possible, therefore, that IVR in the molecule need not be manifested experimen- 177 tally in the ways that are described in Chapter 3 and have been shown to apply to anthracene. Herein, we report the results of measurements of the temporal behavior of dis- persed fluorescence as a function of excess vibrational energy in S, é-stilbene. The results show behavior which is qualitatively similar to that observed for anthracene. In the low energy region, decays are predominantly single exponential. In an inter- mediate energy region, excitations exist which give rise to beat-modulated decays with spectrally dependent quantum beat phases. (The beats are observed at excess energies below the barrier energy for photoisomerization.) At higher energies, de- cays which show behavior typical of dissipative IVR processes are observed. While close scrutiny of the results reveals some deviations from the theory of Chapter 3 and the anthracene results of Chapter 4, the trend with energy argues for the gen- eral nature of the transition, first shown to occur in anthracene, from absent to restricted to dissipative IVR. in large molecules. 178 Il. EXPERIMENTAL The experimental apparatus, procedures, and data handling techniques were primarily the same as used in Chapters 4 and 5. trane-Stilbene (Eastman, scintilla- tion grade; or Aldrich, > 96%) was heated to ~ 125° C and expanded with 30-50 psi He through a 100 4m pinhole into the vacuum chamber. The frequency doubled output of a synchronously pumped, cavity-dumped dye laser (DCM or R6G as dye) intersected the jet at z = 3 mm downstream from the expansion orifice. The deter- mination of the particular absorption bands being excited in any given experiment was made by correlating the monochromator measurement of laser scatter with the normalized excitation spectrum of Ref. 5. The laser source was operated at all times with a three-plate birefringent filter as a tuning element. In many cases a fine or ultra-fine tuning etalon was also used. In the former tuning configuration the exci- tation bandwidth (BW) in the UV was ~ 5 cem™?, while in the latter configurations it was 2—3 cm —!. Fluorescence was collected through a 0.5 M monochromator and detected with a fast photomultiplier. Decays were measured using time-correlated single photon counting. Data obtained with both 80 and 300 psec FWHM temporal resolution are presented here. Those decays obtained at the lower temporal resolu- tion are indicated. The data were analyzed using the Fourier transform techniques and fitting procedures outlined in Chapter 4. 179 II, RESULTS The UV spectroscopy of jet-cooled é-stilbene is quite complex®:3-9 owing to the large number of optically active and low frequency vibrational modes in the molecule. Nevertheless, the dispersed fluorescence spectra’ of the molecule as a function of vibrational energy show trends that have come to be regarded as charac- teristic of jet-cooled large molecules (see Figs. 1 and 2). For low energy excitations the spectra (for example, those of Fig. 1) are structured and contain significant amounts of vibrationally unrelaxed fluorescence (see section II-D of Chapter 3). At higher energies the spectra (Fig. 2) become very congested and consist pri- marily of vibrational relaxed emission. From the spectral trends one can come to some idea of the energy at which dissipative IVR appears to become significant. In ¢-stilbene this energy is about 1230 cm7! (an energy near the threshold energy for photoisomerization’ in the molecule). However, the transition from absent to restricted to dissipative IVR cannot be accurately identified without temporally re- solved measurements, as discussed in Chapters 3 and 4. In the following, we present results divided into three energy groups, with the divisions based on temporally re- solved data. A. Low energy region Many bands occur in the excitation spectrum®*° of ¢-stilbene in the region from S$; + 0 to 752 cm7!. While we cannot claim to have checked the temporal behavior of dispersed fluorescence for all significantly intense excitation bands in this region, we have made such measurements for the most prominent bands below Eye = 592 cm~! and for most of the significantly intense excitation bands in the Ey» = 592 to 752 cm7! region. With one exception, the decays were found to exhibit no quantum beats. (Of course, it must be remembered that the observation of beats relies on sufficient temporal resolution.) In contrast to the other excitation bands in its vicinity, excitation of the band at S| +663 cm! gives rise to fluorescence bands, the decays of which are beat- modulated. Fig 1-lower left shows the fluorescence spectrum corresponding to this 180 excitation and Fig 3-lower left the decay of the band (marked by an asterisk in the spectrum of Fig. 1) shifted 800 cm~' from the excitation energy. This decay is representative of the decays of most of the other bands in the spectrum. By Fourier analysis it is found to be modulated by a 780 MHz beat component (FWHM of about 250 MHz) which has a +1 phase (see Chapter 3). In addition to this type of decay, however, at least one weak band (e.g., the one at a ~ 585 cm~') was found to exhibit a decay modulated at the 780 MHz frequency but with a —1 phase. (It should be noted that this phase-shifted decay is not particularly well- modulated. This is discussed in section IV.) The presence of quantum beats and the dependence of their phases on detection indicate that, unlike levels of similar energy, the Ey:, = 663 cm~! level undergoes restricted IVR. 181 B. Intermediate energy region In the region from E,,;, = 789 to 1170 cm! a total of seven excitation bands have been observed to give rise to quantum beat-modulated decays. These bands are the ones at E,;, = 789, 821, 852, 860, 987, 1048, and 1170 cm~!. The fluorescence spectra arising from excitation of these bands are given in Fig. 1. These excitations represent roughly one-half of the total number of prominent bands in this region of the excitation spectrum.* Dispersed fluorescence decays for the other excitation bands in this region have been measured with 80 psec temporal resolution and have been found to exhibit no obvious beats. Fig. 3 shows representative decays for the seven beating excitations. The ex- citation energies are given in the figure. The respective detection wavelengths are given by asterisks in the spectra of Fig. 1. It is important to point out that the degree of detection resolution was found to be critical to the observation of these beat-modulated decays. (Thus, for example, no beats were observed at any excita- tion energy when the detection resolution was the worst the monochromator would allow, t.¢., 50 A.) This fact implies that there is some dependence of decay behavior on detection wavelength, as is the case for anthracene.” Now, Fourier analysis of the decays of Fig. 3 reveals that all of the prominent beat frequencies in each of the decays occur with +1 phases. However, for three of the beating excitations, other dispersed fluorescence bands have been found to exhibit beating decays with com- ponents having —1 phases. The best example of phase-shifted behavior is shown in Fig. 4. The two decays arise from excitation to the E,,;, = 789 cm—! band. The top decay corresponds to detection of the 610 cm~! band in the spectrum, while the bottom decay corresponds to detection of the 700 cm! band. Clearly, the dominant 1.3 GHz beat component is phase-shifted 180° in the lower decay relative to the upper one. Similar phase-shift behavior has been found for bands in the spectra of the E,y;, = 860 and 987 cm! levels as well. For the four other beating excitations at E,,, = 821, 852, 1048, and 1170 cm™!, the combination of weak fluorescence intensity and congested fluorescence spectra render very difficult 182 all but the most cursory experimental surveys of decays vs. detection wavelength. Thus, the fact of our not having observed phase-shifted beats for bands in these spectra does not at all mean that they do not exist. The beat frequencies determined by Fourier analysis to be prominent in the decays of Fig. 3 are given in Table 1. It is pertinent to mention that the resolution in the Fourier spectra was 30 MHz and the typical beat component bandwidths were ~ 250 Mhz. Thus, the values in the table reflect these sources of uncertainty. 183 C. High energy region Excitation of t-stilbene to energies above E,;, ~ 1230 cm™! in Sj gives rise to fluorescence spectra which bear the characteristics of spectra of levels that undergo dissipative IVR, as may be seen from the the spectra of the Ey;, = 1246 and 1249 cm~' vibrational levels shown in Fig. 2. (For details of the excitation spectrum in this region see Ref. 5.) Inset in the figure are the blue portions of these spectra. One notices in the insets the weak band that occurs in each spectrum at a shift of 205 cm~! from the excitation energy. Similar bands are also present in the spectra of the 1237, 1241, and 1332 cm~! vibrational levels. These bands in each spectra can be confidently assigned in terms of transitions from the optically prepared state to the vibrational level in So characterized by one quantum of the optically active 95 vibrational mode. They can therefore be assigned as vibrationally unrelaxed bands (see Chapter 3, section II-D). In turn, the broad, intense emission in the red parts of the spectra has the characteristics of vibrationally relaxed fluorescence. Fig. 5 shows the decays that are observed when the 205 cm~! bands in the Evyio = 1237, 1241, 1246, 1249, and 1332 cm™! spectra are detected with 3 A spec- tral resolution. Each decay, although being of the same, approximately double exponential form, is different from the others. This is apparent in the different modulation patterns that occur in their long-time components. We emphasize that these individual patterns are reproducible. Double exponential fits with convolution also reveal differences in the decays. Of particular pertinence are the short- and long-time decay constants and the ratios of fast to slow fluorescence, the values of which are given in Fig. 5. The values for the E,;, = 1246 and 1249 cm~! decays are quoted quite confidently, since they have been reproduced many times. The other three excitations are less well-studied, especially the 1332 cm~! one, and the values for them are quoted primarily to give an idea of approximate magnitudes from decay to decay. Two experiments were performed to check for possible scatter artifacts in the decays of Fig. 5. The first experiment compared the decays observed (for the same 184 collection times) upon excitation to the 1249 cm~' band and detection (3 A res- olution) at 3004 (Fig. 6a) and 2999 A (Fig. 6b). The first detection wavelength corresponds to the 205 cm™! band in the spectrum, while the second corresponds to no emission band and is closer to the laser wavelength. The second experiment in- volved tuning the excitation source to 10 cm! to the blue of the Ey;5 = 1249 cm~! band, a region of the excitation spectrum that has little intensity, and detecting at a 205 cm! red shift from this. The “decay” that was observed is shown in Fig. 6c and should be compared with the decay corresponding to on-resonance 1249 cm! excitation and detection of the 205 cm" band (Fig. 6a). Clearly, the results of both experiments represented in Fig. 6 argue against scatter artifacts in the measured decays. While the double exponential form characterizes the decays of the 205 cm7! in the high energy spectra, similar decays have not been observed when detecting in the vibrationally relaxed region of the spectra. Here, decays devoid of any early time spike and with lifetimes matching those of the long components of the Fig. 5 decays are characteristic of the transients observed for these detection wavelengths. An example is shown in Fig. 7. Given that spectral evidence and the decays of Fig. 5 are indicative of dissipative IVR, one might expect decays such as that shown in Fig. 7 to exhibit finite risetimes matching the initial decay time of the pertinent transient in Fig. 5 (see Chapter 3, section VIB). In fact, the decay of Fig. 7 does show a risetime. However, it should be noted that this behavior, at least in the best-studied case of excitation to Eyi5 = 1249 cm7!, is not the same for every detection wavelength in the vibrationally relaxed region; the apparent risetimes of decays change with detection wavelength. This is probably the result of underly- ing vibrationally unrelaxed emission in the vibrationally relaxed region. (We note that this behavior is somewhat different from that observed for dissipative IVR in anthracene,” for which risetimes in the relaxed spectral region are less sensitive to detection wavelength and apparently match the initial decay of the unrelaxed emission.) 185 IV. DISCUSSION A. IVR vs. vibrational energy The general trend of the experimental results as a function of vibrational en- ergy in é-stilbene is very similar to that of the results of jet-cooled anthracene (Chapter 4). In the low energy regime (< 752 cm~') the data (except for one vi- brational level) indicate the absence of IVR on the timescale of the fluorescence lifetime. For intermediate energies (789-1170 cm~*), the presence of phase-shifted quantum beats points to the existence of restricted IVR and vibrational coherence effects. Finally, at high energies (> 1230 cm—1), the decay behavior is consistent with dissipative IVR. One may note that the onsets of restricted and dissipative IVR in t-stilbene oc- cur at vibrational energies significantly lower than the onset energies in anthracene.” There are two plausible explanations for this. It is first instructive to consider the vibrational density of states (p,;,) vs. vibrational energy (E,,») for the molecules. A plot of p,i vs. Eyiy for t-stilbene appears in Fig. 8. (A listing of the frequen- cies used and an outline of the calculation of py, is given in an appendix to this chapter.) Comparing Fig. 8 to the p,,» for anthracene (Fig. 25, Chapter 4), one can see that the ¢-stilbene curve is shifted to lower energy. (This is primarily due to the larger number of low frequency vibrational modes*® in the molecule.) It seems likely that this shift of p,;5 to lower energy is responsible for the lower energy IVR onsets in t-stilbene. Indeed, if one directly compares the ranges of p,;, for the restricted and dissipative IVR energy regions, respectively, in the two molecules, it is evident that the values for ¢-stilbene (~ 10 to 50 per cm7', and > 160 per cm~?) are very close to those for anthracene (~ 25 to 40 per cm™! and 120 per cm‘). A second possible reason for the lower energy onsets of IVR in t-stilbene involves its reduced symmetry relative to that anthracene. While in anthracene, symmetry restrictions are expected to limit the vibrational states available to couple with one another, these restrictions are significantly reduced in ¢-stilbene. Thus, the effective density of states available for coupling in the latter molecule is a greater fraction 186 of the total density of states. This factor may be manifest in the fact that even in terms of pyin, t-stilbene IVR seems to “turn on” sooner than anthracene [VR. 187 B. IVR timescales and coupling matrix elements Besides the trend in the nature of IVR as a function of energy, the IVR timescales and vibrational coupling matrix elements manifested in t-stilbene de- cays are similar to those characterizing the process in anthracene.” In the restricted regime, recurrence times on the order of several hundreds of picoseconds seem to be the rule. In the dissipative regime, the timescales for initial vibrational dephasing are similar to that of the E,i» = 1792 cm~! decays in anthracene. One point of in- terest in this regard, that is dealt with more explicitly in Chapter 7, is that there is a vibrational mode dependence to the dissipative IVR rate in the F,;, ~ 1240 cm! region.!° Further characterization of this mode specificity awaits work pertaining to the assignment of the vibrational levels involved. Regarding the vibrational coupling matrix elements manifested in the modu- lated decays, the data for ¢-stilbene are not as amenable to quantitative analysis as those for anthracene (section IV of Chapter 4). Reasons for this are dealt with in the next section. Nevertheless, the order of magnitude for the coupling matrix elements can be obtained fairly accurately from the beat frequencies. From Ta- ble 1, one sees that restricted IVR in t-stilbene arises from vibrational couplings of similar magnitude to the couplings in anthracene. In the dissipative regime one can calculate!! the variance in coupling matrix elements V, analogous to the 3.8 GHz value calculated for anthracene (see section VI-B of Chapter 4), by mak- ing use of a golden rule-type relation: a ~ 2rV2p. If one assumes C2 or C; Ti symmetry for the molecule and symmetry restrictions on vibrational coupling, then p= Pit = 80cm! for Ey» ~ 1240 cm~!. Using this, V is found to range from about 1.1 to 1.7 GHz for the vibrational levels in the 1240 cm~! energy region. 188 C. Vibrational coherence in ¢-stilbene vs. anthracene Although there is much about the experimental results on ¢-stilbene which is similar to the anthracene results (Chapter 4) and which is readily interpreted in the context of the theory of Chapter 3, there are also significant points of difference that must be addressed. A first such aspect is the fact that for the observed phase- shifted (non-a-type) decays of t-stilbene the sum of modulation depths of the beat components are not nearly as large as they are in anthracene or as they are predicted to be (see (2.17) of Chapter 3). This is evident from Fig. 4 for Ey, = 789 cm=! and occurs to an even greater degree for the 663, 860, and 987 cm=! excitations. We have treated possible reasons for such behavior in section V of Chapter 3. Aside from “trivial” reasons (e.g., insufficient temporal or spectral resolution) there are at least two other causes which would seem to be plausible given the molecule’s characteristics. Firstly, owing to the reduced symmetry of t-stilbene, one expects a large fraction (relative to anthracene) of its vibrational modes to be optically active. In turn, the probability of encountering two or more coupled optically active vibrational levels is higher. Such a situation runs counter to the basic assumptions of Chapter 3 and has been shown (section V-E, Chapter 3) to give rise to undermodulated non-a-type decays. Secondly, the presumed nonrigidity of the molecule would be expected to lead to some large anharmonic coupling matrix elements which could couple widely separated zero-order vibrational states. As has been shown in section V-A of Chapter 3, this can lead to less than ideal modulation depths. A second point of difference between the ¢-stilbene and anthracene results is the fact that many prominent excitation bands in the energy region corresponding to restricted IVR in ¢-stilbene are nonbeating excitations, while in anthracene, beating excitations seem to be much more prevalent. Reasons for this could be the same as those given above for the poorly modulated non-a-type decays. In addi- tion, one may note that the relative nonrigidity of the molecule might be expected to lead to a large rotational influence in vibrational coupling compared to more rigid 189 molecules. For instance, vibrational levels involving a large amplitude phenyl tor- sional motion could conceivably have very different rotational constants than levels involving benzene-type vibrational motions. Now we have shown in Chapter 5 that if vibrational levels with small differences (i.¢., one part in a thousand) in rota- tional constants are coupled, then the vibrational coherence effects arising from the coupling are partially washed out. Large differences in rotational constants might be expected to completely mask these effects. Also, it may be that the “nonbeat- ing” excitation bands correspond to S, vibrational levels that undergo significant Coriolis interactions. Finally, it is possible that there is some mode selectivity for vibrational coupling in the restricted IVR regime, in which case some of the non- beating levels may be nonbeating simply because they are not effectively coupled to any other levels. We note that there is evidence! for such mode selectivity in the dissipative regime of the molecule. A third aspect of t-stilbene quantum beats which should be addressed is the fact that for four excitation energies, beating decays have been observed to be modu- lated, at least in part, by a component at about 780 MHz. It seems unlikely that four unrelated vibrational levels would each be coupled to a nearby level such that all four coupled systems would give rise to pairs of eigenstates separated by the 780 MHz energy interval. In response to this we would make several points. Firstly, spec- troscopic evidence indicates that two of the pertinent excitations, E,;, = 663 cm7! and E,,;, = 860 cm, are related. The 860 cm~! band is a combination band of the 663 cm! interval and the optically active vz5 interval.® Since this is so, one can certainly imagine that the vibrational coupling situation (#.e., the zero-order Hamiltonian matrix) at the 663 cm~! energy might be very similar to that at the 860 cm~! energy. In such a case, the resulting beat-modulated decays would also be very similar. A second point is that the resolution in the Fourier spectra and the inherent breadth of the Fourier components in these spectra (probably due to rotational level effects*) render it difficult to determine beat frequencies to better than +100 MHz. Thus, there may be real differences in the beat frequencies which 190 are not apparent from Fourier analysis. Further, there is other evidence that not all the decays are modulated in the same way, in that the decays corresponding to the levels at ys = 987 and 1170 cm™* have other beat components. Finally, it is possible that the prevalence of the 780 MHz splitting is not fortuitous but instead represents a vibrational splitting of the kind that would arise between eigenstates of a multiple-well potential. The pheny! torsional potential surface in ¢-stilbene is of this kind. This problem, however, is outside the scope of this chapter. In ending this section it is pertinent to note that the energy corresponding to the onset of dissipative IVR in t-stilbene is very similar to the experimentally deduced threshold energy for photoisomerization. It is likely that this is not fortuitous and that there is a strong connection between the two processes. The question is what this connection is. One can certainly see how IVR might be necessary for the photoisomerization to occur. But, it may also be true that the molecular changes associated with isomerization influence the redistribution of vibrational energy. It is notable that IVR at 1250 cm™! is faster than the fastest photisomerization rate that has been measured thus far (~ 7.5 x 10° sec—! at Ey:y = 2950 cm‘). This is at least partial justification for the treatment of the isomerization rates in terms of a statistical method (i.e., RRKM).7° 191 V. CONCLUSIONS IVR in S, é-stilbene follows much the same trend with energy as does the (in many ways) dissimilar molecule, anthracene. At low energies there is no apparent vibrational evolution of optically prepared states on the timescale of the fluores- cence lifetime. At intermediate energies (~ 760 to 1170 cm~') restricted IVR is manifested as phase-shifted quantum beat-modulated fluorescence decays. At high excitation energies (> 1230 cm~1) IVR becomes dissipative in nature. The occur- rence of this trend in more than one molecule suggests its generality and allows for comparison of details of the IVR in different species. Such comparison does, in fact, show differences between vibrational coupling and IVR in #-stilbene vs. that in anthracene, which differences can be attributed to differences in molecular characteristics. 192 APPENDIX The density of states results presented in Fig. 8 were calculated using a direct count procedure.!* The frequencies used in the count were obtained from three sources: 1) our assignments (Ref. 5) of S, modes from the fluorescence excitation spectrum, 2) assignments of modes from IR and Raman work (Ref. 12), and 3) calculated frequencies (Ref. 4). The frequencies used appear in Table 2. Note that only frequencies less than 1500 cm~! are pertinent. This is why all 72 normal modes of the molecule are not represented in the table. We would note that the calculated py» is sensitive to the grouping of the frequencies for the direct count and to the values used for the lowest frequency modes, as illustrated in Ref. 7b. 193 REFERENCES 1. P.M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. 2. P.M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. 3. P.M. Felker and A. H. Zewail, J. Chem. Phys. - submitted. 4. A. Warshel, J. Chem. Phys. 62, 214(1975). 5. J. A. Syage, P. M. Felker, and A. H. Zewail, J. Chem. Phys. (in press). 6 . M. Traetteberg, E. B. Frantsen, F. C. Mijlhoff, and A. Hoekstra, J. Mol. Struct. 26, 57(1975). _ (a) J. A. Syage, W. R. Lambert, P. M. Felker, A. H. Zewail, and R. M. Hochstrasser, Chem. Phys. Lett. 88, 268(1982); and references therein; (b) J. A. Syage, P. M. Felker, and A. H. Zewail, J. Chem. Phys. (in press) 8. A. Amiray and J. Jortner, Chem. Phys. Lett. 95, 295(1983). 9. T.S. Zwier, E. Carrasquillo M., and D. H. Levy, J. Chem. Phys. 78, 5493(1983). 10. P. M. Felker and A. H. Zewail, J. Phys. Chem. - submitted. ll. C. Tric, Chem. Phys. 14, 189(1976). 12. Z. Meic and H. Gusten, Spectrochim. Acta 34A, 101(1978). 13. L. R. Khunkar, R. A. Marcus, A. H. Zewail, J. Phys. Chem. 87, 2473(1983). 194 Table 1. Prominent beat frequencies modulating the decays of Fig. 3. E,i, in cm~! Frequencies (w/2z7) in GHz 663 0.78 789 1.3 ; 1.754 821 1.35 ; 6.1; 7.0 ; 7.4° 852 1.6 860 0.78 987 0.75 ; 1.4 1048 0.55 ; 3.3 ; 3.9 1170 0.75 ; 0.9¢ ; 1.8 “ This frequency does not appear in the decay of Fig. 3 but does prominently modulate the decays of some of the bands in the E,,, = 789 cm! spectrum. ® Other higher frequency bands also appear to be present, but noise in the Fourier spectrum obscures them. ° The 0.75 and 0.9 GHz components may be parts of one very wide band in the Fourier spectrum. 195 Table 2. trans-Stilbene normal mode frequencies (in cm~+) used in the density of states calculation (Fig. 8). 83° 88° 95° 108 198° 233 239 280° 338 410 455 467 526 540 591% 612 622° 675 691 735 763 840 842 845 846° 907 916 963 969 969 973° 985 995? 998 999 1027 1059 1069 1153 1156 1181 1187 1218 12497 1297 1315 1321° 1329 1336 1336 1339 14307 1450 1463* 1494 “ From Ref. 5. ® From Ref. 4. All other values are from Ref. 12. 196 FIGURE CAPTIONS 1. Dispersed fluorescence spectra of jet-cooled ¢-stilbene for excitation energies that have been observed to give rise to beat-modulated fluorescence decays. Excess energies in S; for the excitations are given in cm~! in the figure and the excitation wavelengths are marked with arrows in the spectra. The asterisks refer to detection wavelengths for the decays of Fig. 3. All of the spectra were obtained using similar experimental conditions (see text): te, BW = -2-3 cm7!; and monochromator resolution R = 0.5 A for most spectra, except R = 0.65 A for the E,;, = 821 and 860 cm™! spectra and R = 0.32 A for the Evin = 789 cm—! spectrum. Dispersed fluorescence spectra of jet-cooled t-stilbene for excitation to S, + 1246 cm7! (top), and S, + 1249 cm—! (bottom). Inset are the blue portions of each spectrum. In all spectra the positions of the excitation wavelength and the 205 cm~! band are marked. Both the main spectra were obtained with BW =~2 cm and R = 0.64 A. For the insets BW ~ 2 cm! and R=3.2 A. Quantum beat-modulated fluorescence decays observed for excitation of various bands (the excess S$, vibrational energies are given in cm~! in the figure) of jet-cooled ¢-stilbene . The particular fluorescence band detected for each decay is given by an asterisk in the appropriate spectrum in Fig. 1. All decays were obtained with 80 psec temporal resolution and BW ~ 5 cm™!, except the ones corresponding to the S; + 852 and 860 cm™! excitations, which were measured with 300 psec resolution and BW ~ 2cm7!. R for the decays was 1.6 A, except the S, + 821 and 987 cm~! decays for which R = 3.2 A and 16.0 A, respectively. 4. Fluorescence decays of two bands in the E,;, = 789 cm~! spectrum of t-stilbene. The wavenumber shifts of the bands from the excitation energy are given in the figure. Note the clear phase-shift of the prominent 1.3 GHz beat component in the lower relative to the upper decay. For both decays BW ~ 5 cm7~!. For the upper decay R = 1.6 A, and for the lower one R = 1.2 A. 197 5. Decays and double exponential fits corresponding to the 205 cm—! bands in the (from top to bottom) Eyi = 1237, 1241, 1246, 1249, and 1332 cm~! spectra of jet-cooled ¢-stilbene. Given in the figure are the best fit parameters for both the fast and slow lifetimes, and the ratio (F/S) of pre-exponential factors of fast vs. slow fluorescence. All decays were obtained using BW ~ 2 cm7! and R=3.2 A. 6. Comparison of decays observed for similar collection times for: a) on-resonance excitation and detection of the 205 cm~* band in the Eyiy = 1249 cm=! spec- trum of ¢-stilbene, b) on-resonance excitation of the S; +1249 cm~! band and off-resonance detection (Ag ~ 2999 A or 9g ~ 150 cm™!), and c) off-resonance excitation (10 cm~' to the blue of the 1249 cm~! band) and “on-resonance” detection in the region of the 205 cm! band. All other experimental condi- tions were the same for all three decays: BW ~ 2 cm~! and R=1.6 A. 7. Measured fluorescence decay for detection in the blue part of the vibrationally relaxed region of the Ey;, = 1249 em! spectrum of ¢t-stilbene. The detection wavelength is given in the figure. The decay has a risetime similar to the initial decay time (47 psec) of the 205 cm~? band in the same spectrum, and a long component having the same lifetime (2.0 nsec) as the long component of the 205 cm~! decay. The decay was measured with BW ~ 2 cm~! and R =3.2 A. 8. Vibrat.onal density of states (pyin) vs. vibrational energy (Ey) in Sy ¢- stilbene, calculated as outlined in the appendix to this chapter. In the figure are given the characteristic times for IVR in the absent, resticted, and dissipative regimes. 198 Figure 1 Wovelength chy Wavelength (A) 199 Figure 2 ib = 1246em"! Laser Loser oO scm"! ae! Evib= 1249cm7! Laser 205cm! : Loser 205em! f 2980 : 3180 Wavelength (A) 200 Figure 3 , Pe | 1170 em’! 789 cm’! A 987 em! — — ° 6 T T T T T + t Time (nsec) Time (nsec) 201 Figure 4 E ib = 789.cm™! Vg =~ 610 cm™! (In-phase) Vg = 700 cm! (Out - of - phase) Time (nsec) 202 Figure 5 203 Figure 6 G) | Evib = 1249cm7! V4 = 205cm7! saad Weim. b) Off-Resonance Detection c) | Off -Resonance Excitation Warner echt hamden isd aati elon AA eae byl tvtnaflnt cr T T T T T fe) 7 Time (nsec) 204 Figure 7 E vib =1249cm | dg= 3060A Time (nsec) Pvib (cm) 600 205 Figure 8 SOOr 400 300;-- 200+ lOO- 100's of ps 1500 206 CHAPTER 7 MODE-SPECIFIC IVR: _ DIRECT PICOSECOND RESULTS* * Submitted to the Journal of Physical Chemistry, as “Moede-specific IVR: Direct picosecond results,” by Peter M. Felker and Ahmed H. Zewail 207 The characterization of intramolecular vibrational-energy redistribution (IVR) in the high energy regime is presently one of the most challenging problems in intramolecular dynamics.’ An issue of particular importance concerns the depen- dence of IVR dynamics on the character of the initially excited vibrational motion. One would like to know whether certain types of levels couple more efficiently than other types to the intramolecular bath of vibrational states. Such dependences, if they were to exist in the high energy regime, would show that dissipative IVR need not be solely governed by excess vibrational energy and would help to pinpoint the molecular parameters influencing the IVR process. Recently, (time-integrated) dispersed fluorescence measurements have spectro- scopic features that are consistent with mode-dependent vibrational coupling in para-difluorobenzene.? However, the timescales for the selective IVR of different modes have not been reported yet, since this requires direct picosecond temporal resolution of the IVR. In this chapter, we report on the direct measurement of mode-dependent, dis- sipative IVR rates in jet-cooled ¢-stilbene. It is found that excitation of different bands in the S; + 1250 cm! energy region of the molecule gives rise to different IVR rates, as determined by the temporal evolution of vibrationally unrelaxed flu- orescence. The results are discussed in terms of the spectroscopic assignments for the excitation bands. The experimental system has been described in detail elsewhere.® trans-Stilbene (Eastman scintillation grade, or Aldrich > 96%) at 125° C was expanded with 30 to 40 psig He through a 100 ym pinhole into an evacuated chamber. The frequency- doubled output (Av ~ 2 cm!) of a synchronously pumped, cavity-dumped dye laser (Rhodamine 6G as dye) excited the jet at z = 3 mm from the pinhole. Flu- orescence was detected at right angles through a monochromator. Decays were measured by time-correlated single photon counting with a system response func- tion of about 80 psec FWHM.‘ The decays of vibrationally unrelaxed fluorescence were fit to a double exponential decay function. Various checks of carrier gas de- 208 pendences and the effects of off-resonance excitation and detection were made to verify the intramolecular natures of the decays (as opposed to their being the result of intermolecular interactions or laser scatter). In the S; + 1250 cm™! region of the ¢-stilbene excitation spectrum,® there exist four closely spaced bands at 1236, 1240, 1246, and 1249 cm~!. Here, we are primarily concerned with the two bluest bands. Excitation to these bands gives rise to very similar fluorescence spectra. An example is the spectrum of Fig. 1, which corresponds to the 1249 cm™! band. One notices the weak, structured blue portion of the spectrum near the excitation energy, and the broad, intense portion near and to the red of the S; — 0$ transition energy of the molecule. Fig 2a shows the decay and double exponential fit of the weak 205 cm=! band in the Eyiy = 1246 cm—! spectrum and Fig. 2b shows that of the same band in the E,,;, = 1249 cm7! spec- trum. (Similar (approximately) double exponential decays also occur for detection of the 205 cm~! bands in the E,;, = 1236 and 1240 cm~! spectra.) For neither excitation have. decays like those of Fig. 2 been observed for detection wavelengths greater than 3060 A (t.¢., the broad intense region in Fig. 1). In this region, only decays with long (~ 2 nsec) components, or long components with risetimes similar to the initial decay times of the Fig. 2 transients®, have been observed. Inset in Fig. 2 are the early time portions of the two transients. One can tell from the insets that the initial decay of the 1246 cm! transient is faster than that of the 1249 cm! one. This is verified by the double exponential fits, which give lifetimes of 27 psec and 2.0 nsec for the 1246 cm—! decay and 47 psec and 2.0 nsec for the 1249 cm7! decay. While the long lifetimes are similar and match the lifetimes for detection in the red region of the spectrum, the short decays are measurably and reproducibly different. The fluorescence spectra of the Ey,, = 1246 and 1249 cm! excitations show the characteristics of spectra which arise from optically prepared levels that undergo significant vibrational state mixing. The blue regions of each consist of weak, readily assignable bands (the 205 cm~! band can be assigned as a transition from the 209 S; optically prepared state to the So level characterized by one quantum of the strongly optically active vgs mode*) which represent vibrationally unrelaxed (U) transitions, while the broad, intense region to the red of about 3100 A corresponds to the superposition of a large number of predominantly vibrationally relaxed (R) bands. The two decays shown in Fig. 2 are consistent with the two types of decays expected for U and R fluorescence in the dissipative IVR regime. As we have discussed elsewhere,’ the double exponential decay behavior of U bands can be interpreted in terms of a fast dissipative IVR process (initial decay) that leads to an approximate steady-state in energy flow (long component). The fact that dissipative IVR occurs in the E,,;, ~ 1250 cm™! region of the t-stilbene level structure also is consistent with the facts that 1) the density of vibrational states at this energy (~ 160 per cm~?) is similar to the density of states at which dissipative IVR occurs in anthracene (t.¢., ~ 120 per cm~'), and 2) the energy region below 1170 cm™! in the molecule has been found® to consist of levels, some of which show manifestations (phase-shifted quantum beats) in fluorescence decays of restricted IVR.° That which is novel, however, is that the levels at 1246 and 1249 cm—! have significantly different IVR rates. Moreover, the higher energy level has a slower IVR decay rate than the lower energy one. In a simple golden-rule picture such behavior could be due to either inhomogeneities in the density of “bath” vibrational states (pyi») in the 1246-49 cm! region or to a factor of ~ 1.5 difference in the average coupling matrix elements (V) that connect the optically prepared levels to the bath states. Since py,» at this energy is fairly high, one does not expect inhomogeneities of sufficient magnitude to effect a factor of two difference in rates. On the other hand, spectroscopic considerations® have led to the assignment of the 1246. cm—! interval as a combination band and the 1249 cm~! interval as a fundamental vibration. Since the two levels are of significantly different vibrational character, one might expect them to be characterized by different V. Thus the 210 mode-dependent dissipative IVR, mantfested as different decay rates, seeme likely to have origins in the different natures of the vibrational motions initially excited. A full account of this work will be published later when the picosecond IVR dynamics of other levels in t-stilbene is characterized. 211 REFERENCES 1. For a review see: C. S. Parmenter, Faraday Discuss. Chem. Soc. 75, 7(1983). 2. M. Fujii, T. Ebata, N. Mikami, M. Ito, S. H. Kable, W. D. Lawrance, T. B. Parsons, and A. E. W. Knight, J. Chem. Phys. 88, 2937(1984). 3. W.R. Lambert, P. M. Felker, and A. H. Zewail, J. Chem. Phys. 81, 2217(1984). 4. In Ref. 3 the photomultiplier tube that was used was considerably slower than the tube used for the obtaining the results reported in this paper. This is the reason for the much faster response function quoted herein. 5. J. A. Syage, P. M. Felker, and A. H. Zewail. J. Chem. Phys. (in press). 6. It should be noted that apparent risetimes for decays in this region vary with detection wavelength from 50 psec to faster than this. This is probably due to contributions from vibrationally unrelaxed emission. 7. P.M. Felker and A. H. Zewail, Chem. Phys. Lett. 108, 303(1984). 8 P.M. Felker, W. R. Lambert, A. H. Zewail, J. Chem. Phys. - submitted. 9. P.M. Felker and A. H. Zewail, Chem. Phys. Lett. 102, 113(1983); P. M. Felker and A. H. Zewail, Phys. Rev. Lett. 58, 501(1984). 212 FIGURE CAPTIONS 1. Dispersed fluorescence spectrum of the 5S; +1249 cm! level of t-stilbene. Inset in the figure is that portion of the spectrum (the vibrationally unrelaxed region) near the excitation wavelength. The monochromator resolution R for the inset was 3 A (~ 30 cm™'). For the main spectrum R = 0.6 A. Other experimental conditions are given in the text. 2. Measured decays (dots) with double exponential fits (lines) of the 205 cm~! flu- orescence bands in the 5; + 1246 cm~' spectrum (top) and the 5, +1249 cm-! spectrum (bottom) of f-stilbene. Inset are the expanded early time portions of the two measured decays, and the fast lifetimes obtained from the double exponential fits (convolution of the system temporal response function was ac- counted for). Both decays were measured with R = 3.2 A. Other experimental conditions are given in the text. 213 Figure 1 E vib = 1249 cm™! “Unrelaxed" "Relaxed" =~ Laser. 205 cm7! [ I Lf | 4 2980 ° 3180 Wavelength (A) 214 Figure 2 Evib =1246cm7! ‘a Time (nsec) 215 APPENDIX I* * Published paper: Peter M. Felker and Ahmed H. Zewail, Chem. Phys. Lett. 102, 113 (1983). Volume 102, number 2,3 CHEMICAL PHYSICS LETTERS 18 November 1983 OBSERVATION OF RESTRICTED IVR IN LARGE MOLECULES: QUASI-PERIODIC BEHAVIOR, PHASE-SHIFTED AND NON-PHASE-SHIFTED QUANTUM BEATS Peter M. FELKER * and Ahmed H. ZEWAIL ** Arthur Amos Noyes Laboratory of Chemical Physics *, California Institute of Technology, Pasadena, California 91125, USA Received 19 August 1983; in final form 22 August 1983 The concept of phase-shifted quantum beats in large molecules is developed. Measurements are reported for jet-cooled anthracene excited by a 15 ps pulse to S, + 1420 cm™!. The results represent the direct observation, in the time domain, of vibrational energy flow within specific sets of modes in the molecule. The importance of the results to intramolecular vibrational-energy redistribution (IVR) and quasié-periodic behavior is discussed. 1. Introduction Quasi-periodic versus ergodic behavior in large systems with many degrees of freedom has been, and still is, of great current interest [1]. Of particular in- terest are two questions: (1) At what energy region does a particular molecular system undergo a change between these two limits of behavior, and (2) what are the observable manifestations (e.g. spectroscopic) of the two types of behavior in large isolated mole- cules, with their complicated level structures and couplings? Lambert et al. [2] have reported the observation of quantum beats in the fluorescence decay of jet- cooled anthracene excited by picosecond pulses to an excess vibrational energy of 1380 cm™! in S}. This observation of a coherence phenomenon in a large, isolated molecule led us to examine the role that rovibrational coupling might have in creating this dynamical effect at this excess energy. However, a clear link between the coherence (or the recurrence) and the dynamics of vibrational energy flow was not established. In fact, until this report, a direct view of energy flow between vibrational modes in the time do- main has not been seen. Both adequate temporal and * IBM Research Fellow. ** Camille and Henry Dreyfus Foundation Teacher—Scholar, + Contribution No. 6899. 0 009-2614/83/0000—0000/$ 03.00 © 1983 North-Holland spectral resolution are needed to observe the dynamics of initially prepared and redistributed states. In this letter, we present experimental evidence for restricted IVR (quasi-periodic behavior) in anthracene. Temporal and spectral manifestations of this dynamic- al effect are illustrated by the novel observation of phase-shifted quantum beats and the time evolution of dispersed fluorescence spectra of jet-cooled mole- cules. The concept of phase-shifted quantum beats can be illustrated using the scheme of fig. 1, which repre- a) —__ v,,_* ==) {ie} a Het Bor, 2 Ho f If) _ If,> lod —— Fig. 1. A schematic level structure diagram pertaining to an experiment involving two zero-order excited state vibrational levels, a) and Ib), coupled by an interaction Vp: Im is active in absorption from |g and emission to tf), while |b) is not active in absorption but is active in emission to fp). See the text (section 2) for further details. Volume 102, number 2,3 sents a fluorescence experiment involving two nearby S; vibrational levels; la) and (b), where |a) is active in absorption and |b) is not. After the laser pulse excites la), the coupling between |a) and |b) will manifest it- self through quantum-beat-modulated fluorescence decays from the coupled levels. But, because of dif- ferences in the emission Franck—Condon factors be- tween |a) and |b), there will be two types of emission bands. Those that derive their strength from |b) will exhibit beating decays that are phase-shifted by 180° from the modulated decays of bands that derive their strength from |a). If such behavior is actually observ- ed, one can attribute the phenomenon to a restrict- ed IVR process, obtain the characteristic time con- stant for this process, and observe a time evolution in the emission spectrum. The experiments presented here on jet-cooled anthracene, excited to S; + 1420 em! bya 15 ps pulse, demonstrate this idea, and provide the abovementioned information concerning IVR. 2. Theoretical In the following section, we derive the fluorescence signal that arises from the situation represented by fig. 1. First, consider the two zero-order S, vibrational levels, |a) and |b), which are separated by energy Ey, and coupled via an interaction matrix element Vp: The resulting vibrational eigenstates, ley) and lep , are separated by an energy £5 = (E3, +4iV 31 yv2 and take the form: ley) = ay, la) + @y, 1b) (la) and \€2) = a2, 1a) + ay, ib), (1b) where A, = Ay (=a), (2a) @9_ = a1, (= 8). (2b) a and f are real, and a? + 62 =1. Expressions for a and § are well known and involve V,, and Ey [3]. Now consider a process by which {e,) and je.) are, first, prepared coherently by a delta-function light pulse through absorption from the ground-state level |g), and, subsequently, emit to the manifold of CHEMICAL PHYSICS LETTERS 217 18 November 1983 levels |f). The resulting fluorescence intensity versus time [4] (assuming the same relaxation rate, I’, for le,) and je,), although generalization to the case of dif- ferent I is straightforward) is 2 Ke)=K a 2 igh yl ijl X exp(—I't) exp(—i£t/M), (3) Where tn, = (n\i:|m) is the transition electric dipole moment matrix element between |) and |m) t, and where K is a constant. In terms of the zero-order states 2 Kt)=K x Py (iatag * %Hbg) X (Qjatiga + MyMppKQjattas + Apts) xX (Qgury + Gy upy) exp(—Pr) exp(—-i£ t/A). (4) If one assumes that, because of Franck—Condon fac- tors, only |a) can combine with the initial state \g (i.e. | Mag #0, | Hyg! = 0), then one may see that two limiting types of bands (type a and type b), differen- tiated by the zero-order state that provides the emis. sion strength, may occur in the dispersed fluorescence spectrum (see fig. 1). Type-a bands are defined as those for which the fluorescence transition involves a final state |f,), such that lay, | ¥ 0 but lu, fa! = 0. The decay for type-a bands may be calculated by using eq. (4) and restricting the f summation to just one f, term: 1,09) = Klidgy,|?luyglexp(—Pr) X [1 — 2026? + 2026? cos(E,>2/m)], (5) where we have substituted for the Gmn- For type-b bands, the fluorescence transition is taken to involve a final state |f,), for which [Mps,| #0 and Ju, i = 0. From eq. (4) t Note that we have suppressed the influences of excitation and detection polarization in eq. (3). This neglect is justi fied for consideration of our experimental results being that no polarization analysis is made in the experiments re ported here, yet beats are still observed. For a full discus sion of polarization effects and. molecular quantum beats, see ref. [5]. Volume 102, number 2,3 By (t) * Kl g!? lpg? X exp(—Ir)a?6? [1 — cos(E;21/h)]. (6) Comparing the decays of the two types of bands, it is evident that both types exhibit beat-modulated decays at an angular frequency of E,/k However, two sig- nificant differences occur between the forms of the two types of decays. Firstly, the type-a and type-b beat patterns are 180° out-of-phase. Secondly, the degree of modulation of type-a decays depends on and £ (i.e. depends on the vibrational coupling) and in general is less than 100%, whereas type-b decays are 100% modulated, independent of a and §. Note that eq. (5) and eq. (6) represent observables associated with the dynamics of a limited IVR process: The op- tically prepared state, ja), evolves in time into some mixture of |a) and ib). The decays of type-a bands probe the |a) content of the excited non-stationary state, those of type-b bands, the |b) content. Hence, phase-shifted and non-phase-shifted beats can sort out the vibrational subsets involved in the IVR. Significantly, the spectral dependence of beat pat- terns, which is predicted by the vibrational coupling scheme above, is different for cases where beats arise from coupling wherein (a) is an S, level coupled to a triplet or Sp level [6,7], {b). In such cases [Mp =0 for all f corresponding to optical fluorescence transi- tions to Sp levels, and all fluorescence bands in the spectrum decay in exactly the same way, i.e. ina fashion similar to type-a bands. Eqs. (5) and (6) provide a basis upon which the dynamical manifestations of vibration—vibration coupling within an electronic state manifold may be analyzed. In what follows, we present quantum-beat results for anthracene which, owing to their very close fit to the predictions of this section, provide clear evidence that we have observed the time evolution of a limited intramolecular vibrational energy redistribu- tion (IVR) process. 3. Experimental A short account of the experimental apparatus has appeared [2]. Full details are published elsewhere {8]. Briefly, He, Ne, or N, at 30 psi, seeded with anthracene at *180°C, was expanded through a 100 CHEMICAL PHYSICS LETTERS 218 18 November 1983 um pinhole into an evacuated chamber (<1073 Torr). Both anthracene (Aldrich 98%) and purified anthracene (recrystallized from CCl,) were used. No differences in results were observed between the two. Also, using these materials we reproduced the results for the 1380 cm~! excitation {2] obtained using zone refined anthracene. The 6870 A output of a syn- chronously pumped, cavity-dumped picosecond dye laser (DCM as dye, a three-piate birefringent filter and fine-tuning etalon as tuning elements) was frequency- doubled (Li]O,) to produce UV pulses (=3 cm~! bandwidth) which crossed the jet expansion at dis- tances (= X) of 2-8 mm from the nozzle pinhole. No effect on the results reported here was observed as X was varied over this range. Fluorescence was col- lected with f/1 optics, spectrally dispersed and detect- ed by a fast photomultiplier. Fluorescence decays were measured by time-correlated single-photon counting. The influence of magnetic fields on beat-modulat- ed decays was assessed by using the arrangement de- scribed in ref. [7]. To make certain that there was a field under our anthracene experimental conditions, published results [7] for pyrazine were reproduced. 4. Results Upon excitation of the prominent, jet-cooled an- thracene band at S, + 1420 cm7! in the excitation spectrum [9] (3435 A), phase-shifted quantum beats (995 + 30 MHz) have been observed (fig. 2). The ex- perimental decays that appear in fig. 2 correspond to detection of the dispersed fluorescence bands appear- ing at 390 and 1750 cm7! shifts, respectively, from | the excitation energy (see fig. 3a). The two decays show a clear 180° phase difference between them. As is apparent from the figure, the 390 cm7! decay may be simulated * by convoluting a decay function of the form A exp(—I't)[1 + 0.4 cos(wr)] with the sys- tem response function; where w = 22 X 10° s~! and * Our prime motivation in performing these simulations is not to obtain extremely accurate values for decay param- eters, but instead to clearly illustrate that the 390 cm™! decay involves a plus cosine term and the 1750 cm™! decay a minus cosine term. The decay of the beat envelope and its relation to differences in I's will be dealt with later. 219 18 November 1983 CHEMICAL PHYSICS LETTERS Volume 102, number 2,3 . Theoretical : : 3 : : : : : ; . Experimental of “sgl ‘ *% a OY SS. wes $ é i 8 F iy 2% ° poi yi & Py Ser re ~% t i i / — N i ! \ N 1 ! Q 1 2 3 4 5 6 7 Time (ns) Fig. 2. Experimental and theoretical fluorescence decays. The calculation of the theoretical decays is outlined in the text. The top theoretical decay corresponds to a plus cosine modulation term, the bottom decay to a minus cosine term. X = 3 mm, and P = 30 psi He (the effect of X and carrier gas are discussed in the text). Beats were observed when X varied from 2 to 8 mm. The top ex- perimental decay corresponds to detection of the 390 cm™! dispersed fluorescence band with detection bandwidth (R) of 16 A. The bottom decay corresponds to detection of the 1750 cm=! fluorescence band with R 2 1.6 A (see footnote *). Nore that the peaks in the top experimental decay correspond to the troughs in the bottom decay, The time window indicated in the figure is pertinent to fig. 3b. Volume 102, number 2,3 3400 3900 Wavelength (A) 1750 env! All Time Lo J 3600 3670 Wavelength (&) Fig. 3. (a) Dispersed fluorescence spectrum for anthracene excited to S, + 1420 cm™!. X¥ = 3 mm, P = 30 psi He, R = 1.6 A: The 390 and 1750 cm™! bands are marked as is the exci tation wavelength. (b) Dispersed fluorescence spectra corre- sponding to anthracene excited to S, + 1420 cm. The top spectrum detected emission at all times. The detection for the bottom spectrum was time-gated to the window indicated in fig. 2. For both spectra ¥ = 2 mm, P = 30 psiHe,R =1.1 A. The 1750 cm™! band is marked. The line drawn across the top of the 1750 cm™! serves to illustrate the decrease in rela- tive intensity of this band in the time-gated spectrum. T= 1(8.6 X10? s~!), The 1750 cm! decay is reproduced if a decay function of the form A X exp(—Ft)(1 — 0.9 cos(wr)] is used to calculate a convoluted decay. The difference in modulation be- tween the 390 and 1750 cm7! decays is very apparent in fig. 4. Clearly, the decay of the 1750 cm! band is better modulated than that of the 390 cm=! band for the experimental conditions given in the caption of the figure *, CHEMICAL PHYSICS LETTERS 220 18 November 1983 1750 em! Residual otk 4 J 1750 em! Band 390 em! Band I 1 i L l ie) 5 fs) Is 20 25 30 35 Time (ns) Fig. 4. Beat-modulated fluorescence decays of the 1750 and 390 cm™! bands for excitation of anthracene to S, + 1420 em™!. X = 3 mm, P = 30 psi Ne. For the 1750 cm7! decay R = 0.5 A; for the 390 cm! decay R = 1.6 A (see footnote ». At the top of the figure is the residual which results from the subtraction of a best fit exponential decay function from the experimental decay. Note the modulation of the 1750 cm7! decay relative to that of the 390 cm! decay. In addition to the beating decays of the 390 and 1750 cm7! bands, other bands exhibit modulated de- cays. The resonance fluorescence band (0 cm7! shift), the band at 1482 cm! , and others have been found to show beats similar to those of fig. 2a (i.e. +cos(wr) beats, 0° phase), while decays of the bands at 1750 + 390 and 1750 + 1408 cm~!, as well as others, beat in a manner similar to fig. 2b (i.e. * One may note that the monochromator resolution for the 1750 cm” decay of fig. 4 is better than that for the 390 cm™ decay. No bandwidth effect on the modulation of the 390 cm™! decay has been observed, consistent with the fact that it appears in a congestion-free region of the spectrum. On the other hand, the modulation of the 1750 cm7! band has been found to be detection bandwidth sensitive (modu- lation increases with increasing resolution). Thus, if anything the 1750 cm™! may be more modulated than it appears to be in the figure, while the 390 cm™! decay modulation is accurately represented. Volume 102, number 2,3 —cos(wr) beats, 180° phase). The difference in beat phase between the 1750 cm~! band and other bands that appear in the 3600—3670 A region of the dispers- ed fluorescence spectrum is quite apparent in fig. 3b. The top spectrum of fig. 3b corresponds to collection of emission at ail times. The lower one corresponds to detection of the emission occurring in the narrow time window or gate indicated on the experimental decays of fig. 2. Clearly, the relative intensity of the 1750 cm! band decreases in the gated spectrum. (Al- though the spectra of fig. 3b are not corrected for laser power drifts, the trend of the spectra regarding the relative intensity of the 1750 cm7! band is quite reproducible.) A slight carrier gas effect on the beats has been observed. As the gas is changed from Ne or He to N2, decreases in beat modulation occur. However, no new beat frequencies appear. This behavior contrasts with results for other beats, e.g. those that appear for S + 1380 cm7! excitation in anthracene [8]. No magnetic field effect on the beats was observed for detection of the well-modulated 1750 cm~! band, suggesting that there is no involvement of S, triplet coupling in the beats, in contrast to results published on pyrazine [7]. §. Discussion The experimental results presented above clearly match the theoretical predictions of section 2 pertain- ing to the manifestations of vibration—vibration coupling in quantum-beat-modulated decays. In the nomenclature of section 2, the zero, 390, and 1482 1 bands are “type a”. They are not fully modulat- ed and involve a phase of 0°. The 1750, 1750 + 390, and 1750 + 1408 cm7! bands, being more fully mod- ulated and having phases of 180°, are assigned as “typeb”. These assignments are quite reasonable in terms of anthracene spectroscopy. The 0, 390, as 1482 em7! bands can be assigned as tay fn} tare 3128, and {A} frist {9} *§ respectively 4 and because of this * The notation {a }in } represents an excited-state vibration- al level consisting of a set of vibrational modes (ie. {A }) characterized by a set of vibrational quantum numbers {n}. Similarly, {A}tq}: represents a level of the ground state. fale {m} tepresents a fluorescence or absorption transition. CHEMICAL PHYSICS LETTERS 221 18 November 1983 are expected to be type a. In contrast, the 1750 cm~! interval does not correspond to obvious fundamental, overtone, . combination bands of known optically active (oj t tT ground- state modes. This suggests that the band is of the form {C} {i} {A'}{™), where {C} is some set of optically inactive modes and {A’} some set of optically active ones. Note that the level {C} {1} {A’ yim} cannot be populated by absorption from the Og level, and thus must gain absorption strength (under jet conditions) by coupling to some optically active level t Ti. e. {A} {m} ). Thus, the 1750 cm! band is expected to be type b. The 1750 + 390 and 1750 + 1408 om * bands are ve assign able [9] as (C} {7} (a'} {4129 ana (cyt cay fmt respectively, and are, therefore, also expected to A type b. The above results represent the first clear observa- tions in the time domain of vibrational energy flow in a jet-cooled large molecule. They may be expected to yield much new information pertaining to both IVR processes in anthracene and such processes in general. Regarding the particular characteristics of the vi- brational dynamics of anthracene in its S, + 1420 cm! level, it is clear (owing to the close fit of our experimental results to the predictions of a two-level model) that only a very limited vibrational energy re- distribution process is being observed, and that it is characterized by a finite recurrence time (1 ns) and a vibrational coupling matrix element of less than 0.5 GHz. The lack of a pronounced carrier gas effect, in contrast to other beats in anthracene [8], implies that rotational levels only minimally influence the IVR dynamics of the optically prepared level at this particular excess energy. Besides the immediate implications our results have to energy flow in anthracene, it is very im- portant, with regard to the study of IVR processes in general, that the dynamical manifestations of vibra- tional coupling have now been observed and can be studied in further detail. A first aspect of importance is the establishment of the utility of using the concept § Evidence in ref. [9] indicates that the S, + 1420 cm™! level is the 5! level. tt By an optically active leve] we mean a level that can be reached in absorption from the vibrationless level of the ground state. By an optically active mode we mean a mode that can undergo transitions involving a change of vibra- tional quantum number. Volume 102, number 2,3 of quantum-beat phase shifts as a diagnostic for limit- ed IVR processes. One may expect quantum beats arising from vibrational couplings to occur in many molecules. The analysis of phase shifts could prove to be essential to a full understanding of IVR dynam- ics in such cases. A second implication of our work pertains to the spectral methods widely used to study IVR processes in large molecules [10]. Being that we now are able to assign so-called vibrationally “relaxed” (type-b) and “unrelaxed” (type-a) fluorescence bands via mea- surements in the time domain, it will be possible to examine the extent of the validity of using the charac- teristics of dispersed fluorescence spectra (i.e. broaden- ing, red-shifts, etc.) as a means of arriving at IVR parameters. Finally, the limited nature of the IVR process for the S; + 1420 cm7! anthracene level, provides a clean example of quasi-periodic behavior in a large system. By further studies of phase-shifted and non-phase-shifted quantum beats in anthracene and other molecules, it may be possible to arrive at some knowledge of the characteristics of the transi- tion between quasi-periodic and ergodic vibrational behavior in this large molecule. We plan to present a full account of this work later. Acknowledgement This material is based upon work supported by the National Science Foundation under Grant No. DMR- 8105034 and Gmat. No. CHE-8211356. We wish to thank Professor R. Marcus for many enlightening dis- cussions, especially on the quasi-periodic behavior. CHEMICAL PHYSICS LETTERS 222 18 November 1983 References [1] D.W. Noid, M.L. Koszykowski and R.A. Marcus, Ann. Rev. Phys. Chem. 32 (1981) 267. [2] W.R. Lambert, P.M. Felker and A.H. Zewail, J. Chem. Phys. 75 (1981) 5958. [3] G. Herzberg, Infrared and Raman spectra of polyatomic molecules (Van Nostrand, Princeton, 1945). [4] S. Haroche, in: High resolution laser spectroscopy, ed. K. Shimoda (Springer, Berlin, 1976) pp. 254-313. [5] M. Bixon, J. Jortner and Y. Dothan, Mol. Phys. 17 (1969) 109. {6] J. Chaiken, M. Gurnick and J.D. McDonald, J. Chem. Phys. 74.(1981) 106; H. Henke, H.L. Selzie, T.R. Hays, S.H. Lin and E.W. Schiag, Chem. Phys. Letters 77 (1981) 448; M. Ivanco, J. Hayes, W. Sharfin and S.C. Wallace, J. Chem. Phys. 78 (1983) 6531; S. Okajima, H. Sarbusa and E.C. Lim, J. Chem. Phys. 76 (1982) 2096; B.J. van der Meer, H.Th. Jonkman, G.M. ter Horst and J. Kommandeur, J. Chem. Phys. 76 (1982) 2099. {7} P.M. Felker, W.R. Lambert and A.H. Zewail, Chem. Phys., submitted for publication. [8] W.R. Lambert, P.M. Felker and A.H. Zewail, J. Chem. Phys., submitted for publication. [9] W.R. Lambert, P.M. Felker, J.A. Syage and A.H. Zewail, J. Chem. Phys., submitted for publication. {10} P.S.H. Fitch, L. Wharton and D. Levy, J. Chem. Phys. 70 (1979) 2018; S.M. Beck, J.B. Hopkins, D.E. Powers and R.E. Smalley, J. Chem. Phys. 74 (1981) 43; A. Amirav, U. Even and J. Jortner, J. Chem. Phys. 74 (1981) 3745; C. Bouzou, C. Jouvet, J.B. Leblond, Ph. Millie, A. Tramer and M. Suikes, Chem. Phys. Letters 97 (1983) 161. 223 APPENDIX II* * Published paper: Peter M. Felker and Ahmed H. Zewail, Phys. Rev. Lett. 53, 501 (1984). 224 Direct Observation of Nonchaotic Multilevel Vibrational Energy Flow in Isolated Polyatomic Molecules Peter M. Felker and Ahmed H. Zewail Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 92215 (Received 11 May 1984) With Picosecond spectroscopy and molecular beams it is shown that nonchaotic multilevel vibrational energy flow is present in large polyatomic molecules. This Letter reports on this novel observation and its probing of the fundamental process of energy redistribution in molecules. PACS numbers: 82.20.Rp, 33.50.Hv The fundamental process of collisionless in- tramolecular vibrational energy redistribution (IVR) in polyatomic molecules has been extensive- ly studied for at least two major reasons. Firstly, IVR plays a central role in a wide variety of molecu- lar processes such as multiphoton dissociation, laser-selective chemistry, radiationless transitions, and unimolecular reactions.! Secondly, classical and semiclassical treatments? of vibrational dynam- ics in a coupied two-mode system (Henon-Heiles Hamiltonian) have revealed interesting changes in vibrational motion (quasiperiodic or chaotic) which depend on total energy and coupling parameters. The results have stimulated research aimed at de- fining and discovering analogous quantum behavior in molecules. For these reasons two fundamental questions concerning IVR in real molecules are im- portant: Given the impulsive excitation of an iso- lated molecule to some nonstationary excited state, how rapidly does this initial state evolve to encom- pass other vibrational motions, and to what extent do other vibrational motions become involved in the vibrational motion as a whole? An intuitive feel for the concept of IVR can be gained by consideration of the classical case of cou- pled oscillators represented in Fig. 1. With pendu- lum a initially set in motion and pendulum 6 at rest, the effect of the coupling V,, is to transfer energy from a to 6. In the quantum analog to this situation (Fig. 1, middie) a harmonic oscillator zero-order state |a) is assumed to be prepared by a light pulse at :=0. Due to the coupling V,, to the zero-order state |b) (via anharmonicity, for instance) this ini- tially prepared state evolves in time to contain some contribution from |). It is this wave-function evolution which describes the flow of energy from the vibrational motion represented by |a) to that - represented by |b). In the situation where the ini- tially prepared |a) is coupled to more than one oth- er level (Fig. 1, bottom) multilevel IVR occurs. Time-integrated experimental measurements’ concerning IVR are somewhat indirect in that the observables measured depend on IVR but are re- moved from the primary process itself. Recently, however, the combination of ultracold, seeded su- personic molecular beams with picosecond laser spectroscopy has been shown to be useful to the study in real time of a number of aspects of isolated-molecule dynamics.‘ In this Letter we re- port on the direct observation, using these tech- niques, of nonchaotic multilevel TVR in the large molecule, anthracene. Classical-2 Vob a b Quantum-2 Vv loo ib» {iF fy fp Quontum-N le> Ic> ) 3s” — | Ls fh f fy FIG. 1. Schematic representations of coupled oscills- tors in classical and quantum systems. In the quantun: cases the probing of IVR by fluorescence spectroscopy is indicated. Note that other coupling schemes for quas- tum N are also possible. © 1984 The American Physical Society We shall be concerned with IVR in anthracene at ~ 1400-cm~! excess vibrational energy in the first excited singlet electronic state $;. At these ener- gies the density of 5S; vibrational states is ~25/em~'. Excitation to S, from an So level selectively prepares the (ultracold) molecule in one well-defined vibrational state (harmonic state) while the remainder of the states are not excited be- cause of symmetry and/or Franck-Condon re- strictions. The scheme of our experiment (Fig. 1) is to prepare such a well-defined optically allowed vibrational state, |a), and then to probe the tem- poral evolution of the state by temporally and ‘spec- trally resolving the fluorescence from the sample. In what follows, we shail show that the flow of vi- brational energy from an initially prepared state to other vibrational states is manifested in fluores- cence decays as modulations (quantum beats), the Phases of which are spectrally dependent, and the number of Fourier components of which are depen- dent on the number of levels involved in the energy flow. Consider the case of N coupled vibrational levels (Fig. 1, bottom). It is convenient to initially deal with the molecular eigenstates, |J), 7=1,N, which result from the coupling of these levels: = J a,ly). (1) 7oeahec,... where the a’s are elements of a real orthogonal ma- trix. Delta function excitation of a molecule from the vibrational level |g) of So will prepare these N eigenstates coherently, given that at least one of the zero-order states |y) has a nonzero transition mo- ment with |g). The wavelength-integrated fluores- cence versus time to the manifold of Sp vibrational leveis (| /)) is given by*® ()=c> 5 ul(Lg) pls J u(JS) f idel where C is a constant, the 4 (/,g)’s are dipole ma- trix elements between (/| and |g), I is the excited-state decay rate (taken for simplicity to be the same for all N excited states), and wy ™=(E,—E,)/", E, and E, being the energies of the eigenstates |/) and |J), respectively. Now, as noted above, usually only one harmonic vibrational level, say |c), within a given small energy region has nonzero absorption from |g). Thus, # (1g) = a, (a.g) for all J. Similarly, for fluores- cence into state [f,),u(1.f,) =aryu(y.s,) for all 225 i. We shall call the corresponding fluorescence band a y-type band. Given this it is clear from Eq. (2) that the fluorescence spectrum from an A-level system may Contain as many as N types of bands, ail with different temporal behaviors determined by the zero-order level which contributes to the emis- sion dipole of the band. Furthermore, since the ex- cited state formed by 8-function excitation is (0) ~ Say lt)expl—1E, +1/2)), iwi it is clear with use of Eqs. (1) and (2) [restricting f to f,] that the intensity of a y-type band, which is proportional to $ jg Bl jg XjyAj, exp[ - tliayt+ r)], ideal is also directly proportional to the contribution of ly) to the excited state: {[(ylw(1)){?. Thus the measurement of the decays of different band types pro- vides @ direct picture of the vibrational content of the excited state. in addition, being that the “[(yle(2)) |? are measures of vibrational chaos,’ the fluorescence decays are related to this, too, as we shail discuss later. The simplest case is for N= 2. Here, two types of bands can arise, a type and 5 type.* The tem- poral characteristics of each of the bands can be derived from Eq. (2) by using the eigenstate resuits for two-level coupling: |1)=ala)+f{b) and |2)=—gla)+alb). Both types of bands have have fluorescence decays modulated by a cosw)2/ interference term. However, the cosine term has a negative coefficient for the b-type band, whereas for the a-type band is it positive. These phase-shifted beats are manifestations of the oscillatory flow of energy between the |a@) and |b) vibrational motions. (Note the analogy with the oscillatory dis- tribution of energy that occurs in the coupled pen- dulums of Fig. 1.) The spectral dependence of quantum beat phases also applies to N #3. Herein, we only consider results for N=3. In this case, a-, b-, and c-type bands can arise in the fluorescence spectrum. As is evident from Eq. (2) all three types will have de- cays modulated at the frequencies w)., #23, and @12+@2;. The three cosine terms, however, will have different phases depending on the band type. In particular, it can be shown, with use of the orthonormal properties of the a’s, that (1) for a- type bands all cosine terms are positive, (2) for b and c-type bands two of the terms are negative and the third positive, and (3) the cosine term that is positive for the b-type band is different from that for the c-type band. The experiments reported here show that behavior matching these predictions oc- curs. Our experimental apparatus is described in detail elsewhere.** Briefly, an anthracene beam is formed by expanding anthracene vapor at 180°C with 35-50-psi helium through a 100-4m pinhole into a chamber at < 107? Torr. A picosecond uv pulse (Ar ~ 15 ps, Av <2 cm~') derived from fre- quency doubling a synchronously pumped, cavity- dumped (4 MHz) dye laser intersects the expansion 3 mm downstream from the pinhole. Fluorescence is collected with right-angle geometry, wavelength analyzed by a 0.5-m monochromator and detected by a fast multichannel plate photomultiplier (Hamamatsu R1564U). Decays are measured with time-correlated single-photon counting. Unlike our previous measurements,‘ the total time response of the detection system is now ~ 80 ps full width at half maximum. Decays are fitted by a single ex- ponential function which is then subtracted from the data to yield the modulated portion (residual) of a given decay. One can analyze the residual for beat frequencies and phases by Fourier transforma- tion of the residual, followed by division by the Fourier transform of the system response function. This automatically establishes the proper phases for - the various beat components. Thus, positive cosine components appear as positive peaks in the real part of the Fourier transform and negative cosine ones Anthracene beam; To; ~ 3K | Vexent' 1420em"| ty : \Spicoseconds "In phase” (390 cm!) “Out of phase" (1750 em!) Fluorescence Intensity (a.u.) i 6 0 4 Time (ns) FIG. 2. The fluorescence decays of two bands in the spectrum of jet-cooled anthracene excited to Ey, = 1420 cm~'. The arrow marks the temporal position of the ex- Citation pulse. 226 as negative peaks. A clear illustration of the concept of the spectral dependence of quantum beat phases is presented in Fig. 2, which consists of two decays taken for vibra- tional excitation of anthracene to 1420 cm~! in S). The upper decay corresponds to detection of the fluorescence band at a red shift of 390 cm~! from the excitation energy, and the lower decay to detec- tion of the band at 1750 cm7~!. The slow modula- tion (~ 1 GHz) is clearly a positive cosine in the upper decay and a negative cosine in the lower de- cay. Two other prominent beat frequencies at 9.7 and 10.7 GHz also modulate the decays of Fig. 2. Fourier analysis reveals that all three beat com- ponents have 0° phases in the upper decay (a type), whereas in the lower decay (not a type) the 1- and 9.7-GHz components have 180° phases, and the 10.7-GHz components a 0° phase. Moreover, a third type of band has been observed, the decay of which has the 1-GHz component with 0° phase and the two fast components with 180° phases. The quantum beat behavior for this excitation therefore matches that expected for three coupled vibrational levels. Another anthracene S, vibrational level which manifests multilevel coupling through spectrally dependent quantum beat phases is that which is populated by excitation to E,;,= 1380 cm~!. Figure 3 displays the Fourier transforms of. beat patterns for three of the bands that appear prominently in the fluorescence spectrum resulting from this exci- tation. The phases of the three beat components (3.5, 4.9, and 8.4 GHz) clearly match the predic- +) (1460 cm!) + a g - - 2 4 E q S +, (390 em!) + 3 O-— — = - - e +; (528em!) + Oor— fe) 0 2 4 6 8 0 Frequency (GHz) FIG. 3. Fourier-transform spectra of the quantum beat residuals of three bands in the fluorescence spectrum of anthracene excited to Ew, = 1380 cm~!. tions for three coupled vibrational levels. Having directly observed vibrational energy flow in anthracene it is pertinent for one to consider the implications of these results. Firstly, it is evident that for excitations to E,;, = 1400 cm™', vibrational couplings from 1 to 10 GHz (~0.3 cm™') figure prominently in IVR processes. meaningful that at these energies, energy flow in- volves approximately three levels spaced within 10 GHz. From direct counts of S, vibrational levels using calculated frequencies one would expect ap- proximately ten levels in a 10-GHz region at Eye= 1400 cm='. This implies selectivity in the extent of IVR for these levels.’ Such selectivity could reasonably be the result of symmetry re- strictions on vibrational coupling. Thirdly, the quantity |(a|~(+)) |? has been shown’ to be a use- ful measure of vibrational chaos in the quantum re- gime. Regular, periodic variations in this quantity indicate nonchaotic behavior. In this sense, then, the herein reported a-type decays, which are regular and periodic, represent the observation of non- chaotic IVR. Having demonstrated a technique for probing molecular states that are the quantum ana- logs of quasiperiodic regular motion we shall, in a future publication, show how the observation of ra- pid decay describes the analog of irregular motion and the rapid spreading out of energy deposited in a local region.'° This work was supported by the National Science Foundation through Grant No. CHE-8211356. One 227 Secondly, it is . of us (P.M.F.) is an IBM Predoctora! Fellow and one of us (A.H.Z.) is a Camille and Henry Dreyfus Foundation Teacher Scholar. 'See, e.g., N. Bloembergen and E. Yablonovitch, Phys. Today 31 (5S), 23 (1978); A. H. Zewail, Phys. Today 33 (11), 27 (1980). 2For a review see D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Annu. Rev. Phys. Chem. 32, 267 (1981). 3For a review see C. S. Parmenter, Faraday Discuss. Chem. Soc. 75, 7 (1983). 4A. Zewail, Faraday Discuss. Chem. Soc. 75, 315 (1983). 5§. Haroche, in High Resolution Laser Spectroscopy, edited by K. Shimoda (Springer-Verlag, Berlin, 1976), pp. 254-313. 6p, M. Felker and A. H. Zewail, Chem. Phys. Lett. 102, 113 (1983); the lack of influence of laser and detec- tion polarizations on beats is discussed in this reference. ™. D. Feit and J. A. Fleck, Jr., J. Chem. Phys. 80, 2578 (1984), and references therein. SW. R. Lambert, P. M. Felker, and A. H. Zewail, to be published. SHere, we have not considered the role of rotations in vibrational coupling. Experimental results to be pub- lished elsewhere show that the effect of rotations on modulated decays is primarily to enhance the rate of de- cay of quantum beat envelopes. 10For theoretical discussion see, e.g., G. Hose and H. S. Taylor, J. Chem. Phys. 84, 375 (1984). 228 APPENDIX II* * Published paper: Peter M. Felker and Ahmed H. Zewail, Chem. Phys. Lett. 108, 303 (1984). | 229 Volume 108, number 4 CHEMICAL PHYSICS LETTERS 13 July 1984 DIRECT PICOSECOND TIME RESOLUTION OF DISSIPATIVE INTRAMOLECULAR VIBRATIONAL-ENERGY REDISTRIBUTION (IVR) IN ISOLATED MOLECULES Peter M. FELKER * and Ahmed H. ZEWAIL ** Arthur Amos Noyes Laboratory of Chemical Physics t California Institute of Technology, Pasadena, California 91125, USA Received 22 May 1984 Direct picosecond time resolution of dissipative IVR is reported for the first time. For jet-cooled anthracene with Evin = 1792 cm ~!, the unrelaxed fluorescence displays a fast decay of 75 ps, while the relaxed fluorescence displays a corre- sponding rise. The onset for dissipative IVR is related to the effective density of states and coupling matrix elements, and reflects the importance of time-resolved measurements. 1. Introduction Using picosecond techniques in supersonic jet beams [1] we reported last year on the observation of restricted intramolecular vibrational-energy redistri- bution (IVR) in the large molecule anthracene [2]. The experimental results on jet-cooled anthracene at intermediate excess vibrational energies [2,3] (Lyin = 1400 cm—!) in S; have revealed the existence of IVR amongst a small number (3) of vibrational levels and have indicated magnitudes of vibrational coupling matrix elements (10-2—0.3 cm—!) that significantly influence this process. This information on multilevel vibrational coupling was obtained from the observation of spectrally dependent phase-shifted quantum beats (frequencies ranging from 0.2 to 11 GHz) in fluores- cence decays, One important aspect of the IVR prob- lem that has not been addressed, until now, is the ex- perimental characterization in real time of IVR be- tween a larger number of vibrational levels at higher energies. This characterization is necessary to establish the role that molecular parameters (total vibrational * IBM Predoctoral Fellow, ** Camille and Henry Dreyfus Foundation Teacher—Scholar. t Contribution No. 7028, energy, molecular structure, vibrational density of states, etc.) play in affecting both the time scale and the extent of dissipative (irreversible) IVR. Time-integrated fluorescence measurements [4], while having led the way in the study of IVR in this energy regime, are limited in that they provide no di- rect view of the process. Instead, from a given spectrum a redistribution rate is inferred from the time-inte- grated intensity ratio of vibrationally unrelaxed fluo- rescence (U) to vibrationally relaxed fluorescence (R), and the extent of IVR is inferred from the degree of congestion in the spectrum (see fig. 1). Such inferences can be ambiguous since it is not always apparent which bands in a spectrum are U and which are R, nor is it certain that spectral congestion and complexity are manifestations solely of dynamical IVR processes. In this letter we report on the picosecond time res- olution of U and R fluorescence in the emission spec- trum of jet-cooled anthracene excited to Ey, = 1792 cm—! in the S, manifold. The decays taken in the U region exhibit an initial fast component (=75 ps) fol- lowed by a modulated long time decay. In contrast. the R-region fluorescence transients exhibit a finite tise time, the time scale of which matches the +75 ps component of the U-region decays. and subsequent long decay. We interpret these new results as direct evidence for the onset of dissipative [VR in anthracene Volume 108, number 4 joy” IVR Ip>, led, lo>, \ ik | 0 ALAN LL 4 ‘ “Unrelaxed” "Relaxed" U-emission R-emission —c8 Fig. 1. An energy level diagram representing the dissipative flow of vibrational energy from the laser-prepared ja) vibra- tional level to the manifold of coupied levels |b), 1c), |@,... . As indicated in the figure this process can be probed by fluo- rescence measurements made in the vibrationally unrelaxed (U) and relaxed (R) spectral regions. and for its dynamical time scale at this E, ;,. For com- parison we present results for lower £,:,s where we observe no fast decay or buildup in the dispersed fluo- rescence. 2. Theory Elsewhere [2,3] we have discussed the theory for restricted IVR and its predictions regarding experi- ments done in real time. Here, we shall be concerned with the dynamical behavior of the N harmonic (zero- order) S, vibrational levels |a), |b), |c),..., which are mutually coupled by anharmonic and/or vibration— rotation terms in the molecular Hamiltonian. In gener- al, only one of the zero-order levels, say |a), will have an appreciable Franck—Condon factor for absorption from the vibrationless level of Sp, |0) (i.e. we take only |a) as being optically active, see fig. 1), Consequently, 5-function excitation of a molecule from {0) will give rise to the excited state CHEMICAL PHYSICS LETTERS 13 July 1984 N v(t) ~ zu o,,i/) exp(—iE;t/f) exp(—Pr/2) , (1) where the |D= Zy=a,b,..y!Y are the molecular eigenstates resulting from the coupling of the zero- order levels |), the as form a real, orthonormal, NX N matrix, the E7s are the energies of the |/), and is the decay rate of the eigenstates (taken for simplic- ity to be equal for all /). The contribution of a given zero-order level |y) to ¥(r) is P(t) = Krol? N ~ B2 tay) exp[—(ia,,+T)e] , = |a) by the orthonormality of the as. As? increases, however, the phase terms in eq. (1) evolve and other zero-order states contribute to y(r). This evolution is the quantum mechanical description of vibrational en- ergy flow from the motion represented by |a) to those motions represented by |b), Ic), etc. The details of this IVR process (the number of levels involved, the time scale) depend on the particular details of the coupling between zero-order levels. The evolution of (1) can be fruitfully probed by the temporal and spectral resolution of the fluores- cence arising from (7). The usefulness of this experi- mental approach derives from the fact that emission to a given Sg final state If), will generally gain its strength from just one zero-order excited state level \'y) (again for reasons of Franck—Condon overlap). The intensity of such a fluorescence band (which we call y-type) is L@~ z (07 ,027,,07,),) exp[—(iewyy + Pr] and is, therefore, directly proportional to P(t). Thus, knowledge of the temporal behaviour of all the bands in the fluorescence spectrum gives one a direct indica- tion of the evolution of the vibrational content of the excited state. On the other hand, if the decay of spec- trally integrated fluorescence is measured this infor- mation is masked or lost. In the limit of small N, restricted IVR occurs. Such a process is characterized by oscillatory energy flow Volume 108, number 4 between the coupled vibrational levels. This is mani- fested in fluorescence decays as cosine modulation terms, the phases of which (0° or 180°) are dependent on fluorescence detection wavelength [2,3]. The evolution of y(t) for larger NV has been treated in the literature within the context of both the inter- mediate and statistical cases of radiationless transitions [S—8]. In such treatments, it is convenient to consider two quantities P,(t) = ( Lal, +2 z aot, costenyt)) exp(—I'r) (2a) and uo P= & ( Do2.a}, y=b,c,.. y=b,¢,.. I +2 paCA 707,07.) COS( wy ))exp(—T) . (2b) Eq. (2a) is the contribution of the initially prepared vibrational state |a) to W(r) and eq. (2b) is the overall contribution to ¥(r) of those states coupled to [a). Now for large NV, the number (i.e. 4 M(N — 1)) of co- sine terms with different frequencies in eqs. (2) is suf- ficient to lead to a dephasing process whereby the sums over interference terms in eqs. (2a) and (2b) are large only for t < A-!, where A~! is a measure of width of the distribution of wy, values. (A is normal- ly approximated by the quantity 222 [5,7], where V is a measure of the variance for vibrational cou- pling [7] and 9 is the density of interacting vibration- al states.) Lahmani et al. [5] have shown that the quantum mechanical decays that are represented by eqs. (2) can be approximately reproduced by treating the kinetic scheme illustrated in fig. 1 if one makes the correspondence k; + k, + A and k;/k, + N, where k; is the forward IVR “rate constant” from |a) to other modes and k, is the reverse rate constant. The P(t) val- ues are treated as populations. Solving the population equations for this kinetic system yields P,(1) = (ky +k,)7} X {kj exp[—(k; + k, + Pr] + k,exp(—I'1)} + N-1{(N — 1) exp[4I + A)e] + exp(—I'2)} , (3a) CHEMICAL PHYSICS LETTERS 13 July 1984 2s Pi(t) = [ki/(K; + &,)] Y X {exp(-Is) — exp[—(k, +k, +P)r]} ((N — 1)/N] {exp(-P't) — exp(-(P + A)r]} (3b) Thus, with the aid of this kinetic analogy, one can in- terpret the initial dephasing in eqs. (2) and (3) as giv- ing rise to a dissipative IVR process by which vibra- tional population flows irreversibly out of ja) into the other vibrational states on a time scale A-! | after which time a steady state is reached and the relative vibrational populations do not appreciably change. Of course, a kinetic model cannot describe completely a quantum dynamical process, except in the statistical limit. In particular, one might expect two deviations from eqs. (3a) and (3b). Firstly, the initial decay of P,(t) (or buildup of xP, (t)) need not, in general, be exponential. Secondly, the long time decays of these quantities need not be monotonically decreasing func- tions but may, instead, exhibit quantum beats. Bearing this in mind, however, the simplicity of the kinetic results renders the kinetic model an important interpre- tive tool. Now, as we have noted above, the values P,(t) can be directly monitored by the temporal and spectral res- olution of fluorescence. For large N, spectral conges- tion usually precludes the possibility of measuring de- cays for each type of band. Nevertheless, the optical- ly active property of the |a) level, which allows its se- lective excitation as the initially prepared state, also effects a spectral separation of a-type fluorescence bands from the other types of bands [4]. In general, only a-type bands, which are called vibrationally unre- laxed bands (U), appear as discrete bands in the un- cluttered region of the fluorescence spectrum just to the red of the excitation energy (fig. 1). For these bands, one expects decays of the form of eq. (3a). The other bands, which are called vibrationally relaxed (R), in general, appear further to the red in the fluores. cence spectrum, in a congested region starting rough- ly at the wavelength of the S$) + Sy 0O—0 transition [4]. Without resolving individual bands, one expects the fluorescence decays in this region to approximate the expression given in eq. (3b). By measuring decays in these two regions one can derive parameters charac- Volume 108, number 4 terizing the flow of energy from |a) to other |7). As a final point, it is worth noting that eqs. (3a) and (3b) provide a basis by which the rate of IVR (i.e. A) can be deduced by time-integrated fluorescence mea- surements of U intensity (/y) versus R intensity (/p). This derives from the fact that Illy = f P(o ari f Z. P(t) dt = T/(P +d) (4) in the limit of fast IVR (i.e. 4 >I). Thus, in this limit Jy /Tp becomes the ratio of the fluorescence decay rate to the rate of IVR. By obtaining the former (T) from conventional fluorescence measurements one can there- fore deduce A. As we shall discuss later, this procedure could lead to erroneous rates for IVR. 3. Experimental Details of the experimental apparatus and tech- niques may be found elsewhere [9]. Briefly, a free-jet expansion of 35 psi He and anthracene vapor (180°C) through a 100 um pinhole was intersected 3 mm down- stream from the orifice by the frequency doubled out- put (Ar = 15 ps, Av = 2 cm~!) of a synchronously pumped, cavity dumped (4 MHz) dye laser. Fluores- cence was detected at right angles, wavelength ana- lyzed by a monochromator, and detected by a fast multichannel plate phototube. Fluorescence decays were measured by time-correlated single photon count- ing. The temporal response of the detection system (typically 80 ps fwhm) is now capable of resolving de- cays of ~10 ps time constant. This feature is impor- tant for the reported experiments and for other recent measurements [3] of “very fast” quantum beats. By letting the anthracene sample run out and then mea- suring a “decay’’, the laser scatter contribution was found to be negligible for typical detection conditions. To assess the influence of intermolecular collisions on measured decays the laser-to-nozzle distance (X) was increased up to 10 mm (X/D = 100). No changes in decays were observed upon this variation. 4. Results Fig. 2 presents the dispersed fluorescence spectrum CHEMICAL PHYSICS LETTERS 232 13 July 1984 Vibrationally Vibrationally “Unrelaxed” "Relaxed" id 390 780 * Ny Laser 3390 : 3890 Wavelength (A) Fig. 2. The fluorescence spectra which result from the excita- lion of jet-cooled anthracene to Eyin = 1792 cm™! inS,. Spectral resolution was 1.6 A for the main spectrum and 3.2 A for the inset. The spectral position of the lascr is marked by an arrow, Laser scatter contributes snificantly only to the resonance fluorescence band. The asterisks mark the de- tection wavelengths for the decays of fig. 3. “Unrelaxed” “Reloxed” R T 4 6 Time (ns) o- N lig. 3. Fluorescence decays of U and R emission from jet- cooled anthracene excited to £yip = 1792 cm7!.. The inset shows the early time behavior of the decays. Detection wave- lengths are indicated by asterisks in fig. 2. The detection spec tral resolution was *20 A. The apparent ratio of fast to slow fluorescence in the U decay isless than the actual value, which relates to NV in the text, duc to the finite temporal response of detection and spectral resolution. Quantum beats were ob- served for R fluorescence decays when the spectral resolution was better (e.g. 3.2 A). 233 Voiume 108, number 4 CHEMICAL PHYSICS LETTERS 13 July 1984 which results from excitation of jet-cooled anthracene resolved bands at shifts of 390 and 780 cm~! from to Ey i, = 1792 cm! in S, (for spectroscopic data on ¥ex [9,10], and the congested region to the red of this, jet-cooled anthracene, see refs. [9,10] ). One may note wherein almost all of the fluorescence intensity occurs. two spectral regions: the one just to the red of the ex- Knowing that the 390 cm~! interval represents one citation energy (¥,,) which contains two weak, well- quantum of the strongly optically active a,(12) So T= 18ns Evie: 766cm" * Evie = 1420 cm"! Eve 21792 cm" —— t~75ps “e J | 1 qT U 1 qT qT q qT T 4 ie) 2 4 6 13400 3500 3600 3700 3800 3300 Time (ns) Wavelength (A) Fig. 4. U fluorescence decays and dispersed fluorescence spectra of jet-cooled anthracene for three values of Evip. All decays were obtained by detection of the 390 cm! bands (marked by astcrisks) in the various fluorescence spectra. The apparent rise of the decay for Eyip = 766 cm™! is a result of the convolution of a long singic exponential function with the system response function (width ~80 ps), All fluorescence spectra were obtained with 1.6 A resulution, Arrows mark the excitation wavciength for each spectrum. Volume 108, number 4 mode {9,10}, it is clear that the 390 and 780 cm7! bands are U bands. On the other hand, the congested region has the characteristics typical of R fluorescence. In fig. 3 are the fluorescence decays obtained by de- tection of emission in the two regions (the detection wavelengths are marked by asterisks in fig. 2). The in- set of the figure displays the early time behavior of the two decays. The relative temporal position of the two decays is real and has not been adjusted for display purposes. Neither the initial portion of the U decay nor the rise of the R decay are response function limit- ed, as judged by decay simulations incorporating re- sponse function convolution effects. Decays at other wavelengths have also been mea- sured. The decay of the 780 cm—! band was found to be the same as that of the 390 cm~! band. Other de- cays measured in the congested region from 3620 to 3640 A were found to vary with detection wavelength and to exhibit quantum beats. All such R decays, however, had finite risetimes comparable to that of the R decay in fig. 3. As a point of reference, we display in fig. 4 typical temporal and spectral behaviors of anthracene fluores- cence as a function of vibrational energy in S,. §. Discussion 5.1. IVR and intramolecular dephasing The temporal behaviors of the U and R emissions reported herein are consistent with previous theoreti- cal predictions derived from the quantum mechanical treatment of many level IVR. As such, they are inter- preted as manifestations of dissipative IVR in anthra- cene at £,;,, = 1792 cm~!. The fast (#75 ps) decay of the U fluorescence and the rise of the R fluores. cence correspond, respectively, to the intramolecular dephasing of interference terms in eqs. (2a) and (2b). Further comparison with theory is aided by use of the simple kinetic expressions, eqs. (3a) and (3b). Clearly, the U decay of fig. 3 has the approximate func- tional form of the sum of two exponentials (eq. (3a)). The initial 75 ps component is the irreversible IVR decay (dephasing) of the laser-prepared zero-order vi- brational level. The R fluorescence decay, in turn, is approximated by the difference of two exponentials (eq. (3b)). The rise time of this decay represents the CHEMICAL PHYSICS LETTERS 13 July 1984 flow of energy into the vibrational levels coupled to the initially prepared state. The long time components of the U and R decays correspond to an intramolecu- lar steady state wherein the distribution of vibrational energy remains approximately constant. In this regard, it is notable that the long time components are modu- lated for both U and R fluorescence (for good enough spectral resolution). This quantum effect, which can- not be handled by the kinetic model, indicates that the vibrational coupling at Ei, = 1792 cm~! is best clas- sified as being in the strong-coupled intermediate case {6-—-8]. Moreover, it shows that it is possible for IVR to exhibit partial recurrences and, yet, also be dissipa- tive in nature! 5.2. Temporal versus spectral measurements of IVR There are several points to be made concerning the =75 ps time scale for IVR (A © 13 X 10? s~!) de. duced from the U decay of fig. 3. Firstly, it is instruc- tive to compare this value to that which one obtains from eq. (4) by using /p//y = 600 (from fig. 2) and the lifetime at this energy, 1/f = 6 ns [9,10]. These numbers lead to Typ * 10 ps. This order of magni- tude discrepancy between time-resolved and time- integrated results is due to a number of factors includ- ing the presence of U fluorescence in the R spectral re- gion, approximations involved in the kinetic model leading to eq. (4), and/or experimental difficulties in measuring relative fluorescence intensities. In any case, it points up the difficulties involved in extracting A values from time-integrated measurements. 5.3. Homogeneous broadening and IVR Since A is related to the bandwidth over which the |a) level is spread through coupling with other vibra- tional levels, it could, in principle, be obtained by ab- sorption or fluorescence excitation experiments. ASso- ciated with such measurements, however, are linewidth contributions from rotational inhomogeneous broad- ening, Doppler broadening, and overlapping vibronic transitions, all of which obscure the dynamically im- portant broadening. Thus, a spectral linewidth mea- surement of 4 must be interpreted with some care. In cases where care is taken to consider or eliminate “trivial” spectral broadening the A value obtained can be related to an actual temporal measurement only if Volume }108, number 4 the initial dephasing of U fluorescence or the rise time of R fluorescence can be measured. That is, temporal and spectral resulution of fluorescence is necessary: & is not related to the long time decay of U or R fluores. cence. Since the apparent lifetime at this excess ener- gy is 6 ns, the homogeneous broadening [11] (ina two-level approximation!) is vy, = 1/77 = 1/2ar = 26 MHz. assuming a Lorentzian line shape. In our case, however, the fast component alone will translate to a 2 GHz width. Therefore it is apparent that in the case where there is multilevel IVR the apparent width and the apparent lifetime might not correspond; only if the early time behavior is resolved would one know about such correspondence. This important point is also crucial to the dynamics of benzene in the channel- 3 region [12]. 5.4. Effective density of states for IVR The measured A can be related to average molecu- lar parametersvia A = 2nV2p. Now, py, (1792 cm— 1) = 120/cm~!. However, one would expect symmetry restrictions [13] on vibrational coupling to give a den- sity of interacting states p = p,,,/8, assuming that anharmonic coupling is the dominant coupling mecha- nism. (The factor of 8 comes from the number of irre- ducible representations (D4,,) using the formula given in ref. [13].) Using p = 15/cm~! and A= 0.4 cm7! (13 X 109 Hz) one obtains V = 0.07 cm! . This value is Quite consistent with the values for the matrix ele- ments manifested in quantum beat modulated decays at intermediate energies in anthracene [2,3]. However, caution should be taken when discussing V since we already know from our quantum beat data that V can vary from 107? to at least 0.3 cm=!. Also, if the spec- tral distribution of eigenstates is irregular such as it might be for moderate N (e.g. N = 10) it is difficult to relate A to a well-defined spectral width. A fuller treat- ment of this problem will be dealt with elsewhere. 5.5. Excess energy behavior It is significant that the decay behavior exhibited by anthracene at Fy, = 1792 cm~! does not occur at lower energies in the molecule (fig. 4). In fact, two re- gimes at lower energies may be identified and classi- fied by the decay behavior of U bands. At the lowest energies (e.g. Ey, = 766 cm~!) all bands are U bands CHEMICAL PHYSICS LETTERS 13 July 1984 and the decays are single exponential. This energy re- gime corresponds to one in which IVR is negligible. U decays at intermediate energies [2.3] (e.g. Fi, = 1420 cm7!), on the other hand, are modulated by small numbers of Fourier components (1, 9.7, 10.7 GHz), but do noe display irreversible initial decays, This en- ergy regime corresponds to one in which IVR is re- stricted in nature. The overall trend of U decay behav- ior with energy is paralleled by spectral changes in flu- orescence [9,10] (fig. 4) which reflect the increasing influence of IVR. Of course, this change with energy from negligible to restricted to dissipative IVR is what one would expect given the behavior of the vibration- al density of states ' with energy (~I/cm7! at 766 em~!; =25/cm~! at 1400 cm~!; =120/em7! at 1792 cm~!), assuming V does not change drastically with energy in this energy range of interest. Acknowledgement This material is based upon work supported by the National Science Foundation under Grant No. CHE- 8211356. * Our treatment here has not taken account of the influence of rotational level structure on observed fluorescence de- cays. In a subsequent publication we will present exper+ mental and theoretical evidence which show that changes in vibrational coupling with rotational level are manifested in fluorescence decays as enhanced decay rates of quantum beat envelopes. References [1] A. Zewail, Faraday Discussions Chem. Soc. 75 (1983) 315; P.M. Felker and A.H. Zewail, in: Proceedings of the 1983 NATO Advanced Research Workshop, Acquafredda di Maratea, Italy, 5-11 Junc, 1983, to be published. [2] P.M. Fetker and A.H. Zewail, Chem. Phys. Letters 102 (1983) 113. [3] P.M. Fetker and A.H. Zewail, Phys, Rev. Letters, sub- mitted for publication. [4} C.S. Parmenter, l'araday Discussions Chem. Soc. 75 (1983) 7; R.E. Smalley, J. Phys. Chem. 86 (1982) 3504. [5] Il. Lahmani, A. Tramer and C. Tric, J. Chem. Phys. 60 (1974) 4431. Volume 108, number 4 [6] S. Mukamel and J. Jortner, in: Excited states, Vol. 3, ed. E.C. Lim (Academic Press, New York, 1977); S.A. Rice, in: Excited states, Vol. 2, ed. E.C. Lim (Aca- demic Press, New York, 1975). [7] C. Tric, Chem. Phys. 14 (1976) 189. [8] A. Tramer and R. Voltz, in: Excited states, Vol.4. ed, E.C. Lim (Academic Press, New York, 1979), K.F, Freed and A. Nitzan, J. Chem. Phys. 73 (1980) 4765; S. Mukamel and R.E. Smalley, J. Chem. Phys. 73 (1980) 4156. [9] W.R. Lambert, P.M. Felker and A.H. Zewail, J. Chem. Phys. (September, 1984), to be published. CHEMICAL PHYSICS LETTERS 13 July 1984 [10] W.R. Lambert, P.M. Felker, J.A. Syage and A.H. Zewail, J. Chem. Phys. (September, 1984), to be published. {11] M.J. Bruns, W.-K. Liu and A.H. Zewail, in: Spectroscopy and cacitation dynamics of condensed molecular sys tems, Vol. 4, eds. V.M. Agranovich and R.M. Hochstrasscr (North-Holland, Amsterdam, 1983) ch. 7, p. 301. [12] E. Riedie, H.J. Neusser and E.W. Schlag, J. Phys. Chem. 86 (1982) 4847; E, Riedie and H.J. Neusser, J, Chem. Phys., to be pub- lished. [13] S.M. Lederman, J.H. Runnels and R. Marcus, J. Phys. Chem. 87 (1983) 4364. 237 APPENDIX IV FOURIER TRANSFORM COMPUTER PROGRAM 238 The printout below is the FORTRAN computer program used in analyzing data for this paper. It relies on the fact that the the Fourier transform of the convolution integral of two functions is equal to the product of the Fourier transforms of the two functions, to give information concerning the modulation depths and phases of beat components in decays for which the finite detection temporal resolution cannot be ignored. Further information and references are given in Chapter 4, section II-B. The program was written to run interactively on a PDP 11/23 computer. ANANAANAANANAANAAN an a AaAAM 15 239 PROGRAM FMAIN DESIGNED TO OBTAIN THE FOURIER TRANSFORM OF A DECAY OR RESIDUAL SUCH THAT IT IS FREE OF CONVOLUTION EFFECTS ONE SUBROUTINE: "FFT" A FAST FOURIER TRANSFORM ROUTINE USING UP TO 4996 POINTS CIN POWERS OF TWO) FILES INPUT ARE THE RESPONSE FUNCTION FILE AND EITHER A DECAY OR A RESIDUAL FILE THE PROGRAM DOES THE FOLLOWING: 1. THE RESPONSE FUNCTION IS FOURIER TRANSFORMED 2. THE DECAY (OR RESIDUAL) IS FOURIER TRANSFORMED 3. THE DECAY TRANSFORM (COMPLEX NUMBERS) IS DIVIDED. BY THE RESPONSE TRANSFORM (COMPLEX NUMBERS} 4. EITHER THE REAL SPECTRUM (ABSORPTION) ARISING FROM THIS PROCEDURE OR THE ABSOLUTE VALUE SPECTRUM (POWER) ARISING FROM THIS IS OUTPUT INTO THE DATA FILE NU. DEPENDING ON USER CHOICE DIMENSION XR(4896),X1(4996) ,FILEI(3),FILE2(3) N24896 NU@12 TYPE *,° INPUT THE RESPONSE, AND DATA FILE NAMES” ACCEPT S,FILE1 ACCEPT 5,FILE2 FORMAT (3A4) TYPE *,°IS THE DATA FILE A DECAY OR RESIDUAL FILE ?? ACCEPT 7.,ANS FORMAT(A1) TYPE *,°ENTER THE NO. OF POINTS IN THE DECAY/RESIDUAL FILE.’ ACCEPT *,NPTS TYPE *, ENTER THE HIGHEST FREQUENCY (IN CHANNELS) TO BE CONSIDERED.” ACCEPT *,NFREQ TYPE *,°DO YOU WANT A POWER OR AN AMPLITUDE SPECTRUM 7?” ACCEPT 7,ANS2 INPUT THE RESPONSE OPENCUNIT#=4,NAME@FILE1,TYPE©’OLD* ,ACCESS=’DIRECT’, RECORDSIZE=1822) READ(4°1) NNN,CT,ACC,(XRCI),T=1,NPTS) NNN AND ACC ARE UNUSED, CT IS THE EXPERIMENTAL VALUE OF PSEC PER CHANNEL, AND XR(I) ARE THE DATA POINTS CLOSE(UNIT#=4,DISP=°’SAVE* } ZERO FILL DO 6 I=NPTS+1,N XRT) 8. FOURIER TRANSFORM THE RESPONSE CALL FFT(XR,XI,N,NU) TYPE *,°THE RESPONSE TRANSFORM IS DONE’ TEMPORARILY STORE THE RESPONSE TRANSFORM OPEN(CUNIT=4,NAME=°SAVE.DAT’,TYPE=’NEW’ ,ACCESS=’DIRECT’ RECORDSIZE#=2) DO 18 Is1,N WRITEC4°I) XRUT),XICD) XRT) =e. XIC1T) #8. CLOSE(UNIT#=4,DISP="SAVE’) INPUT THE DECAY OR RESIDUAL FILE IF(ANS1.EQ.’R’*) GO TO 15 OPENCUNIT#=4,NAME=FILE2,TYPE=’OLD’ ,ACCESS=’DIRECT’, RECORDSIZE#1829) READ(4°1) NNN,CT,ACC,(XR(I),T®1,NPTS) CLOSECUNIT=4,DISP=°SAVE?) GO TO 16 OPENCUNIT=4,NAME=FILE2,TYPE=°OLD’* ,ACCESS= DIRECT’, 22 22 2s 27 28 29 240 FOURIER TRANSFORM THE DECAY OR RESIDUAL FILE CALL FFT(XR,XI,N,NU) TYPE *,°THE DECAY TRANSFORM IS DONE’ DIVIDE THE DECAY (OR RESIDUAL) TRANSFORM BY THE RESPONSE OPENCUNIT=4,NAME="’SAVE.DAT’ ,TYPE=°OLD*° ,ACCESS=’DIRECT’, RECORDSIZE#2) DO 28 I#1,N READ(4°I) XR1,XT1 Fis(XR1**2 +XI1**2) XRZ=XRCT) X12=XIC1T) XROT)=CXRI*XRZ+XT 1 *XI12)/F 1 XTCLT SCXRI*XIT2-XT1*XR2)/F 1 CLOSECUNIT=4,0I1SP=°DELETE® } WRITE THE NORMALIZED FOURIER SPECTRUM TO FILE NU. IF(ANS.EQ.°A’) GO TO 25 DO 22 I#1,NFREQ XR CT) sSQRTCXRCL)**2+XR CNT +L) **2 +X TCT **Z24XTCN=L 41) **2) GO TO 28 CONTINUE DO 27 I=1,NFREQ XRCI)SXRCI}+XRCN-T+1) CONTINUE DO 29 I=NFREQ+1,N XR(1) 28. FIND FMAX IN GHZ FMAX=FLOATUNFREQ) *1999./(4896.*CT) OPEN (UNIT=24,NAME=’NU.’°,TYPE=°*NEW’ ,ACCESS=°DIRECT’, RECORDSIZE#=1829) WRITE(4’1) 1888,8.,FMAX,(XR(1),1[#1,1888) CLOSE(UNIT#=4,DISP*="SAVE® ) RETURN END ietekekelel se] 182 181 1gg 183 282 241 SUBROUTINE FFTCXREAL .XIMAG,N,NU) FAST FOURIER TRANSFORM ROUTINE DESIGNED To TRANSFORM UP TO 4896 COMPLEX NUMBER INPUTS WHICH ARE DESTROYED AND TAKEN THE C PLACE OF BY THE TRANSFORM. THE FREQUENCY RESOLUTION IS (1/T) € WHERE T IS THE TOTAL TIME-RANGE OF THE INPUT FILE DIMENSION XREAL(1),XIMAG(1) EYE={@.,1.) N2=N/2 NUI=NU-1 K=g DO 198 Le#l,NU DO i181 I=1,N2 NP=IBITR(K/2**NU1 , NU} P=FLOATCNP } ARG=6.283185*P/FLOAT(N) C=COS(ARG) S*SINCARG) Kl=Ket KIN2Z=K1+N2 TREAL =XREAL (K1N2)*C+XIMAG(KIN2)*S TIMAG=-XREAL (KINZ) *S+#XIMAG(IIN2) #*C XREAL (K1N2)=XREAL (K1)-TREAL XITMAGCKIN2)=XIMAG(K1)=-TIMAG XREAL(K1)*XREAL(K1)+TREAL XIMAG(K1)=XIMAG(K1)+TIMAG K=K+i K=K+N2 IFCK.LT.N) GO To 182 K=% NUI1=NU1-1 N2=N2/2 DO 183 K=1,N TeIBITR(K-1,NU)41 IF(I.LE.K) GO To 183 TREAL=XREAL(K) TIMAG=XIMAG(I} XREAL(K)*XREALCI) XIMAG(K)*XIMAG(T) XREAL(1)=TREAL XIMAG(I)=TIMAG CONTINUE N22=N/2 GO TO 153 DO 152 I=N22+1,N XREAL(1)=XIMAGCLI-N22) CONTINUE CONTINUE RETURN END FUNCTION IBITR(J,NU) JleJ IBITR=g DO 298 I#1,NU J2=2J1/2 IBITR=IBITR*24(J1-2*J2) Jl=J2 — RETURN END