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J. Phys. Chem. 1982, 86, 1184-1192
broadband, low power density). One example, of mode a, should suffice to indicate the methodology. Figure 9 shows resultsz3of a two-color experiment on tert-butylbenzene. The first pulse is a LIweak” UV pulse at 266 nm, which produces essentially only the parent ion (panel d). The second pulse, in the visible (at 560 nm), is absorbed by the tert-butylbenzene cation and causes photodissociation. For a 9.5-mJ pulse, more than 90% of the original parent ion is destroyed via the loss of a CH3 radical (panel a). A series of e ~ p e r i r n e n t scarried ~ ~ out in mode b served to elucidate the fragmentation pathways relevant to the MPI (in the visible) of the benzene molecule. A quantitative “pathway diagram” has been developed showing the relative rates of photodissociation of the following eight ion frgments relative to that of the C6H6+parent ion: C6H5+,C&+, C6H3+,C6H2+, C4H4+,C4H3+,C4H2+, and C3H3+.Some two dozen branching ratios were established for the fragmentation of these benzene-derived ions. From a large variety of experiments using the different modes of operation as described above, it has been possible to observe all of the processes designated e, f, g, h, and i in Figure 7. The results show the prevalence of multiphoton dissociation (MPD) of ions (vis-a-vissingle-photon fragmentation) at the typical laser pulse power levels used in MPIMS experiments.
Concluding Remarks The systematics of the multiphoton ionization-fragmentation phenomenon have been emerging over the past 2 or 3 years. The overall process can be interpreted in terms of three separate steps: (1)MPE (to a resonant intermediate state of the neutral), for the REMPI situation, (2) ionization of the excited neutral (or the groundstate molecule, for the NRMPI case) to the continuum forming the parent ion and an electron, and (3) uni- or multiphoton dissociation (usually MPD) of the parent ion yielding a distribution of daughter ions comprising the observable MPIMS fragmentation pattern. The roles played by energetics, spectroscopy, and dynamics in the overall MPI-fragmentation process are becoming clearer, as well as the conditions for applicability of the statistical theory of MPI-fragmentation. Acknowledgment. The author acknowledges the many significant contributions of his co-workers (named in the abstract), and to the National Science Foundation for financial support of this research. This research was supported by the National Science Foundation Grants CHE 77-11384 and CHE 78-25187. In addition, fond memories of a decade of close encounters with and inspiration from Professor J. 0. Hirschfelder at the University of Wisconsin are deeply appreciated.
Laser Probing of Vibrational Energy Redistribution and Dephasingt A. Zewail;
Wm. Lambert, P. Felker, J. Perry, and W. Warren’
Artbur A m Noyes Laboratory of Chemlcal Physlcs, CalHornie Institute of Technology, Pasadena, Callfornie 9 1125 (Received: September 8, 198 1)
This paper addresses questions important to the origin of optical dephasing and vibrational energy redistributions in molecules. Several laser techniques are discussed and three major findings are presented. These findings are related to (a) optical dephasing of molecules in the gas phase and in beams, (b) dephasing of high-energy vibrational overtone states of large molecules, and (c) energy randomizationand quantum beats in large molecules (anthracene) excited by picosecond pulses and cooled by supersonic jet expansion.
When molecules interact with coherent laser fields, they either rupture and produce new chemical species or they redistribute the deposited energy among the many internal degrees of freedom. The primary processes that govern the reactive chemical changes-laser selective chemistry1-or the nonreactive radiationless relaxation are of great current interest in at least three major areas of research: (a) multiphoton absorption, dissociation, and ionization by intense or weak laser fields; (b) chemical activation by selective laser pumping of certain states in molecules; and ( c ) intra- and intermolecular dephasing following selective and coherent excitations in molecules. Lasers can probe many of these important events that are not amenable to conventional spectroscopic methods. It is the purpose of this article to show how lasers with short ‘Based on a lecture given by A. Zewail in honor of Professor J. 0. Hirschfelder at the International Symposium: New Directions for the Molecular Theory of Gases and Liquids, Madison, WI, 1981. *Alfred P. Sloan Fellow and Camille & Henry Dreyfus Foundation Teacher-Scholar. National Science Foundation Postdoctoral Research Fellow. *Contribution No. 6517.
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duration (picoseconds) or narrow bandwidth ( -lo4 cm-’) can be used to probe some of the dynamics that describe “what goes on between and inside molecules”. Focus will be on (a) the origin of dephasing, (b) the relaxation of selective bond states following the preparation by light, and (c) vibrational energy redistribution in isolated large molecules. There will be no discussion of the multiphoton effects or the influence of lasers on chemical reaction yields (see companion articles in this issue by Professors Bernstein and Zare on these topics). Optical Phase Coherence and Energy Relaxation The coherent interaction between a laser and molecules can be handled in a variety of ways including the rigorous method developed recently by Professor Hirschfelder’s group. On the molecular level one would like to know, in a general sense, the answer to the following questions: (a) What is the nature of the states that we excite with light? (1) See the articles by A. H. Zewail; V. S. Letokhov; R. Zare and R. Bernstein; and Y. Lee and Y. R. Shen in the special issue on Laser Chemistry, Phys. Today, 33, No. 11 (1980).
0 1982 American Chemical Society
Laser Probing of Vibrational Energy Redistribution
(b) Are the different channels that lead to redistribution of energy separable? (c) What determines dephasing in large molecules? It is perhaps illustrative to consider first the most simple case, namely, the case of two Born-Oppenheimer states in contact with a bath and interacting with a coherent radiation field. A. The Two-Level-Bath Problem. To describe the origin of optical dephasing in such a system, we shall consider only the semiclassical approach-the molecule is treated quantum mechanically and the laser field classically. Semiclassically, we describe the process as follows: the laser field E interacts with an ensemble of molecules to produce a timedependent polarization, P(t),which in turn changes as the molecules dephase. So, our task is to find out how the polarization is related to dephasing and what dephasing means on the molecular level. Consider two vibronic states; a ground state $a(r)and an excited state gb(r). The laser field is simply a wave (propagation direction, z ) of the form
E(z,t) = e cos (ut - kz) = 1/2[eei(ot-k2) + ee-i(wt-kz)1 (1) where e is the amplitude and u is the frequency of the radiation. The state of the molecule (driven by the laser) at time t may be represented as $(r,t) = a(t)e-"J$,(r) + b(t)e-'"b$bb(r) (2) We now can calculate the time-dependent molecular polarization: P d t ) = ($(r,t)lfil$(r,t))= ab*pbae-i(%Wb)t+ a*bpabe-i(ob*& (3) where fi is the dipole-moment operator, and pb and pab are the transition moment matrix elements. Taking these matrix elements to be equal (Ip ) and setting ub-0, = uo, the transition frequency, we obtain P,(t) = p[ab*e+iwot+ a*be-&d] (4) Hence, the polarization, which is related to the radiation power, is zero if there is no coherent superposition or, in other words, if the molecule is certain to be in the state a or b. From eq 4 the total polarization for N molecules in the sample (assuming equal contribution and ignoring propagation effects) is therefore P ( t ) l/z[Pebd + Pe-iwd]= NplPab + &a] (5) where P ( t ) NP,(t) and P is its complex amplitude, i.e., P = Pred+ iP-. We chose the notation pab and Pba for the cross terms ab* e+"@and a*be-iod because they are indeed the off-diagonal elements of the ensemble density matrix p . Equation 5 shows that in order to create a polarization or optical coherence we need a nonvanishing interference term or equivalently the off-diagonal elements of p must be nonzero in the zero-order basis set. From eq 2 we see that the probability of finding the system in the excited (ground) state is simple lb21 (laI2). These probabilities decay by time constants, say Tlband TI,, respectively. Such phenomonological decay is the result of the Wigner-Weisskopf approximation, i.e., an exponential decay of the amplitudes a and b; a or b e-t/2Tl.Also, the cross terms of eq 2 will decay possibly by a different rate from the diagonal terms. This decay constant is T2or the dephasing time and is given by2
(2) For a review see A. H. Zewail,Acc. Chem. Res., 13,360 (1980), and references therein.
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982 1185
The Tl term in eq 6 comes from the diagonal elements and represents an average rate for the loss of population in the ab levels. Physically, the T,' term represents the additional decay caused by phase changes in the cross terms. In other words, the random and rapid variation in wo(t), the transition frequency, causes the off-diagonal elements to decay faster than the diagonal ones. One can show that the line width of the transition a-b is l/(aT2)if the band profile is Lorentzian. The total dephasing rate is therefore l/T2 It contains the rate for phase coherence loss (pure dephasing), and T1-l, the rate for irreversible loss of population (energy relaxation or randomization) in the two levels. The phenomenology described here is the optical analogue of magnetic resonance Tl and T2 of Bloch's equations, but the physics is different. B. Bath-Independent Dephasing in Multilevel Systems. It is desirable to use laser pulses that make the coefficients in eq 2 equal (the a/2 pulse limit). In a macroscopic system a single a/2 pulse might excite 10l8 molecules, each of which is coupled to some extent to all others. In large molecules, such excitation and coupling cannot be handled in a manner similar to that of the two-level limit we described before. This is simply because the eigenstates of such a system are very difficult to ascertain. Warren and Zewai13 have recently used the method of moments to obtain expressions for dephasing in multilevel systems. We considered two-level "subsystems" coupled together by an interaction Hamiltonian (e.g., the dipole-dipole interaction) to produce the real system (impurity molecules in a lattice) that we excite with light. Unlike the two-level case where the pure dephasing rate goes to zero at zero temperature, the multilevel system case shows a nonzero dephasing even at 0 K. The implications of these findings to isolated large molecules with overlapping resonances are quite interesting and will be detailed elsewhere.
Experimental Probing of Dephasing and Energy Relaxation: Techniques ( i ) Simple Pulse Sequences for Coherent Optical Transients. Laser techniques are now available to directly separate optical Tl and T2, energy and phase relaxation time constants. In diatomic molecules, which approximate very well the two-level limit, this has been demonstrated experimentally. In Figure 1, we depict the coherent transients2 obtained by using a CW laser and switch. A single-mode laser (
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Flgure 10. The timeresolved spectra of anthracene in the jet, excited at 235 cm-' of excess energy. The spectrum exhibiting a buiklup and decay is for X, = Ao,o, while the other spectrum is for emission into 1 the 390-cm-I So mode. The carrier gas is N, at 40 psi and x mm.
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regions in the level structure where the effective density of states is small even though the overall density of states reflects the statistical limit. Furthermore, the large density of rotational states does not in fact obscurea the coherence effects. The study of these beats is directly relevant to the mechanism of intramolecular dephasing and possibly to the threshold of energy randomization. In summary, we can conclude the following: (a) Excitation of anthracene in the beam at certain low energies above Sl0 shows no evidence of time evolution to other modes after the coherent excitation. (b) At about 1400 cm-I of excess energy there is definite phase coherence among a small number of eigenstates that are spanned by the coherence width of the laser. The energy redistribution time among these eigenstates cannot be shorter than our time resolution. (c) At high-excess energy (e.g., -4 X 1400 cm-I), vibrational dephasing is complete within our time resolution but the red-shifted emission is long lived, only a factor of three shorter than the 0,Oemission. We believe that we will know more about the energy threshold for dephasing and energy redistribution when the analysis of these new spectra is completed, and the origin of the beats is associated (if possible) to certain molecular states and with certain coherence widths.25i26Finally, (d) for a fixed excitation energy but at different laser-nozzle distances we observe changes in the time- and frequency-resolved spectra (different from the isolated moelcule spectra). For example, close to the nozzle and at 235 cm-' of excess energy, the spectrum depicts a buildup and a decay (Figure 10) when the detection wavelength was set at the 0,Oenergy. On the other hand, a decay was observed when the detection wavelength was varied to the emission energy corresponding to the 390-cm-' mode. These results clearly indicate the effects of "controlled" collisions on selective mode relaxation. In a forthcoming publication we hope to be able to analyze the results in terms of specific symmetry modes and collisional models like that advanced recently by Sulkes et a1.27
14.0
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Time Flgure 9. Fluorescence decays of anthracene obtained for the foilowing detection, excitation, and molecular beam conditions: (a) A,, (exdtatkn wavelength) = 3439 A, X, (detection wavelength) = 3577.5 A, x (noulellaser distance) N 5 mm, p (nozzle pressure) = 40 psi N,; (b) A, = 3439 A, X, = 3620 A, x N 5 mm, p = 40 psi N,; (c) A,, = 3439 A, b, = 3577.5 A, x N 1 mm, p = 40 psi N,; (d) A, = 3439 A, X, = 3577.5 A, x N 5 mm, p = 40 psi He. The insets show the residual obtained upon subtraction of the best exponential fit to the observed decay. The time scale is the same as for the ordinate. We have also obtained these spectra at 50 psi carrier gas pressure and observed no change. The beats reported here were also observed on a number of bands in the resonance fluorescence and "relaxed" (around 3820 A) regions.
in the beam, although further work is needed to identify the nature of states involved. Upon first consideration, the very high density of molecular states at energies corresponding to SIexcitation of molecules as large as anthracene might be expected to completely mask any coherence effects in their decay. Corroborating this premise is the fact that beats in fluorescence decays have, until now, only been observed by the McDonald and Schlag groupsB in smaller molecules with comparatively sparse level structures (the density of states, including rotational states, is relatively small). Our observation of beata in the fluorescence decay of anthracene indicates that in large molecules there exist energy (23) J. Chaiken, M. Gurnick, and J. D. McDonald, J. Chem. Phys., 74, 106 (1981), and references therein. E. Schlag, private communication.
(24) For a recent and detailed discussion see the article by K. Freed and A. Nitzan, J. Chem. Phys., 73, 4765 (1980). For a recent formal treatment, see S. Mukamel and R. Smalley, ibid., 73, 4156 (1980). (25) See the article by W. Rhodes in "RadiationlessTransition",S. H. Lin, Ed., Academic Press, New York, 1980, for an important discussion concerning preparation of states by coherent light sources. (26) Wm. R. Lambert, P. M. Felker, and A. H. Zewail, to be. submitted for publication. (27) M. Sulkes, J. Tuss, and S. A. Rice, J. Chem. Phys., 72, 5733 (1980), and references therein.
J. Phys. Chem. 1982, 86, 1192-1200
1192
Concluding Remarks This paper addresses the questions we raised about dephasing and vibrational energy redistribution in small and large molecules. We outlined the laser techniques that can be used to probe these dynamical processes. The relavance of optical TI and Tz to selectivity is emphasized; the success of laser-selective chemistry must depend on (28)Further recent experiments have revealed a dependence of the beat pattern as well as the modulation amplitude upon the pressure and nature of the carrier gas. These resulta and their interpretation will be discussed in a forthcoming paper.
knowledge of the time scales for irreversible vibrational redistribution and the loss of phase coherence. As evident from the anthracene (isolated molecule) results, these time scales depend on the excess energy in the molecule and the degree of internal cooling. Acknowledgment. This research was supported, in part, by grants from the National Science Foundation (CHE79-05683and DMR81-05034). Some portions of this paper have already appeared in other publications. I thank Professor Ph. Kottis and Dr. J.-P Lemaistre for their hospitality at the University of Bordeaux, Talence, France, where parts of this paper were written.
Resonance Energies and Llfetlmes from Stabllization-Based Methods Zlatko BaElZt and Jack Slrnons’ Chemistry Department, Unlverslty of Utah, Salt Lake Clty, Utah 84 112 (Received: September 9, 198 1: I n Final Form: September 18, 198 I )
A practical, computationallysimple procedure is presented for calculating energies and widths of resonances in atom-diatom complexes. It combines the stabilization method and a “golden rule” formula, employing only square-integrablebasis functions. The utility of the procedure is tested on rotationally predissociatingmodel atom-diatom van der Waals complexes. In addition, a procedure for performing coordinate rotation in small, selected subspaces of stabilization eigenvectors is proposed and applied to a two-open-channel model potential problem. A perturbation-basedscheme is developed for systematic selection of those stabilization eigenvectors which should be included in the subspace.
Introduction In low-energy atom (molecule)-molecule collisions, part of the relative kinetic energy of motion may be temporarily converted into excitation of the internal (rotational and/or vibrational) degrees of freedom of either partner. For sufficiently attractive interactions, the additional kinetic energy gained may enable states otherwise energetically inaccessible to be excited. When this excitation occurs, the atom-excited molecule system has insufficient energy, in its relative motion, to separate. The transient complex thus formed is referred to as a Feshbach or compoundstate resonance. Eventually, the internal energy is transferred back into the relative translational energy, leading to breakup of the complex. Hence, these resonances correspond to predissociating, metastable states characterized by total energies, E,, and widths, rr,the latter being related to the lifetime 7 by the uncertainty relationship 7 = h/F. Another class of resonances are the orbiting or shape resonances in which the colliding partners are temporarily held together by a centrifugal barrier. Literature devoted to theoretical and experimental investigations of both kinds of resonance states is extensive and has been summarized recently in two articles by Toennies and his collaborators.’J In a time-dependent picture, resonances can be viewed as localized wave packets made by superposition of continuum wave functions, which for a time 7 = h / r qualitatively resemble bound state^.^.^ During the time 7 , the amplitude of the scattering wave function at a resonance University of
Utah Graduate Research Fellow.
* Camille and Henry Dreyfus Fellow, John Simon
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energy is much larger in the region where the interaction potential is significant than in the asymptotic region. The localized nature of the resonance wave function has motivated the development of several purely L2methods for calculating resonance energies and widths. In such methods, scattering wave functions are expanded in terms of square-integrable basis functions. Here we can mention the secular equation method of Grabenstetter and Le Roy: the complex coordinate rotation method: the stabilization method pioneered by Hazi and Taylo9 (on which we focus our attention in this article), and the closely related truncated orthogonalization procedure of Holerien and Midtal.’ In the stabilization method3~*pe the wave function is expanded in a square-integrable basis and a finite dimensional matrix representation of the relevant Hamiltonian is constructed. The resonance eigenvalues (one or more) are identified as those which are “stable” (relatively insensitive) to variations of the basis, such as increasing the (1)G. Drolshagen, F. A. Gianturco, and J. P. Toennies, J. Chern. Phys., 73,2013 (1980). (2)J. P. Toennies, W. Weiz, and G. Wolf, J. Chem. Phys., 71,614 (1979). (3)A. U. Hazi and H. S. Taylor, Phys. Rev. A, 1, 1109 (1970),and references therein. (4)H. S.Taylor and A. U. Hazi, Plzys. Reu. A, 14,71 (1976). (5)J. E.Grabenstetter and R. L. Le Roy, Chern. Phys. 42,41(1979). (6)Proceedings of the 1977 Sanibel Workshop on Complex Scaling, Int. J. Quantum Chem., 14,343-542 (1978). (7)E. Hol~rienand J. Midtal, J. Chem. Phys., 45,2209 (1966),and references therein. (8)H. S. Taylor, Adu. Chern. Phys., 18,91 (1970). (9)R. K.Neebet, “VariationalMethods in Electron-Atom Scattering Theory”, Plenum Press, New York, 1980,Chapter 111.
0 1982 American Chemical Society
