J . Phys. Chem. 1991, 95, 7973-7993
7973
Femtosecond Real-Time Probing of Reactions. 7. A Quantum and Classical Mechanical Study of the ICN Dissociation Experimentt Gareth Roberts and Ahmed H. Zewail* Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91 125 (Received: March 21, 1991) A comparison is presented between experiment and quantum and classical simulations of the time-dependent dynamics of the dissociation reaction of ICN via the A continuum. To illustrate the approach, second-order perturbation theory and a classical model (Ber.Bunsen-Ges.Phys. Chem. 1988,92,373) are employed to calculate the real-time behavior of dissociating [I-CN]** species during the course of the reaction. In this way, the abilities of the two methods to describe accurately the reaction dynamics as revealed experimentally are evaluated: it is found that both quantum dynamical and classical mechanical treatments are capable of reproducing the essential temporal features probed by experiment and that, in this instance, classical equations of motion offer an adequate description of the process of nuclear separation. The difference between the relevant excited-state potential functions (over the long-range region) employed in the calculations is recovered from the simulated data by means of an inversion procedure (J. Chem. Phys. 1989, 90, 829) that relates absorption of the probing laser pulse to interfragment distance along the reaction coordinate. In addition, the variation in calculated transient behavior resulting from changes in the parameters describing the functional forms of the two potentials governing the observed reaction dynamics is examined in terms of their effect on characteristic reaction dissociation times, the lifetimes of transition-state configurations, and inversion to the difference between potential energy curves. Finally, comparison is made with analogous experimental and theoretical investigations of the fragmentation of the heavier Biz molecule.
I. Introduction The dissociation of cyanogen iodide was one of the first gasphase reactions to be studied'-' in real time in the femtosecond This reaction has received considerable attention throughout the last twenty or more years, having been the object of detailed investigation by means of a broad range of experiand theoreticaPM techniques. It is now well established'2J3J9.21-25that following electronic excitation via the A continuum, ICN dissociates through two exit channels, leading to ground-state and spin-orbit-excited iodine atoms respectively: ICN
+ hv (210 I X I350 nm)
-
[I.-CN]**
-
-
-
[I...CN]** I('P,/z) + CN(XZZ+) ( l a )
+
I(2Pl/2) CN(XZZ+)
(1b)
Time-integrated studies of the dissociation have provided many detailed features of the dynamics of both exit channels, (la) and (1 b). Such an approach has involved a variety of methods, including measurements of photofragment polarization and timeof-flight distributions,12 product quantum-state-resolved distributions,1520subDoppler laser-induced fluorescence,2'*22absolute quantum yields of I(2P1,2)atoms,22-25*29J'and the spatial anisotropy and relative a-l'ignment of the separating photoproducts .21,22,2632 Real-time investigationsof reaction 1 have utilized the technique by which of femtosecond transition-state spectroscopy (FTS),59&'0 [I-CN] ** transition-state configurations in the process of separating to form I(2P3/z,l/2) + CN(X22+) products have been monitored as a function of time.'" The pumplaser pulse initiates reaction within the long-wavelength tail of the A continuum at A = 306 and 285 nm: most experiments have employed a however, and have pump-laser wavelength of X = 306 nm,1-3,5*6 thus been directed toward probing reaction la in which only ground-state iodine atoms are energetically feasible. When the probe laser is tuned to X = 388.5 nm, commensurate with the band head of the P branch of the well-known CN(B22+ X2Z+ (0,O)) band ~ y s t e m , ~ ~ J ~an - ~induction '-'~ period lasting about 200 fs after time zero (pump excitation) is observedzd followed by a monotonic
-
'Contribution 8409 of the Arthur Amos Noyes Laboratory of Chemical Physics. The investigations reported in this p p e r were begun while Professor R. B. Bernstein was a Sherman Fairchild Distinguished Scholar at Caltech: we have benefited greatly from his contribution to the initial stages of this work and from many hours of stimulating discussions.
increase in laser induced fluorescence (LIF) signal intensity to some final asymptotic value attained at T 500 fs. From these (1) Scherer, N. F.; Knee,J. L. Smith, D. D.; Zewail, A. H. J . Phys. Chem. 1985,89, 5142. (2) Dantus, M.; Rosker, M. J.; Zewail, A. H. J . Chem. Phys. 1987, 87, 2395. (3) Rosker, M.J.; Dantus, M.; Zewail, A. H. Science 1988, 241, 1200. (4) Rosker, M. J.; Dantus, M.; Zewail, A. H. J . Chem. Phys. 1988, 89, 6113. ( 5 ) Dantus, M.; Rosker, M. J.; Zewail, A. H. J . Chem. Phys. 1988, 89, 6128. (6) Dantus, M.; Bowman, R. M.; Baskin, J. S.;Zewail, A. H. Chem. Phys. k i t . 1989, 159, 406. (7) Zewail, A. H.J. Chem. Soc., Faraday Trans. 2 1989,85, 1221. (8) Zewail, A. H. Science 1988,242, 1645. (9) Zewail, A. H.; Bernstein, R. B. Chem. Eng. News 1988.66 (Nov 7), 24. (IO) Gruebele, M.;Zewail, A. H. Phys. Today 1990, 43 (9,24; Eer. Bunsen-Ges. Phys. Chem. 1990, 94, 1210. (1 1) Khundkar, L. R.; Zewail, A. H. Ann. Rev. Phys. Chem. 1990,41, 15. (12) Ling, J. H.; Wilson, K. R. J . Chem. Phys. 1975, 63, 101. (13) Baronavski, A. P.; McDonald, J. R. Chem. Phys. Lett. 1977,45, 172. (14) Sabety-Dzvonik, M. J.; Cody, R. J. J . Chem. Phys. 1977, 66, 125. (15) Krieger, W.; Higer, J.; Pfab, J. Chem. Phys. Lerr. 1982, 85, 69. (16) Baronavski. A. P. Chem. Phvs. 1982.66. 217. (17) Fisher, W. H.;Carrington, T.iFilseth, S. V.; Sadowski, C. M.; Dugan, C. H. Chem. Phys. 1983, 82,443. (18) Fisher, W. H.; Eng, R.; Carrington, T.; Dugan, C. H.; Filseth, S.V.; Sadowski. C. M. Chem. Phvs. 1984.89.457. (19) Marinelli, W. J.; Siiakumar, N.; Houston, P. L. J . Phys. Chem. 1984, 88, 6685. (20) Nadler, I.; Reisler, H.; Wittig, C. Chem. Phys. Leu. 1984, 103,451. (21) Shokoohi, F.; Hay, S.;Wittig, C. Chem. Phys. Len. 1984, 110, 1. (22) Nadler, I.; Mahgerefteh, D.; Reisler, H.; Wittig, C. J . Chem. Phys. 1985,82,3885. (23) Amimoto, S.T.; Wiesenfeld, J. R.; Young, R. H. Chem. Phys. Lett. 1979, 65, 402. (24) Pitts, W. M.;Baronavski, A. P. Chem. Phys. Leu. 1980, 71, 395. (25) Hess, W. P.; Leone, S.R. J. Chem. Phys. 1987,86, 3773. (26) Hall, G. E.; Sivakumar, N.; Houston, P. L. J . Chem. Phys. 1986,84, 2 120. (27) Joswig, H.; OHalloran, M. A,; Zare, R. N.; Child, M.S.Faraday Discuss. Chem. Soc. 1986,82, 79. (28) OHalloran, M. A.; Joswig, H.; Zare, R. N. J. Chem. Phys. 1987,87, 303. (29) Hasselbrink, E.; Waldeck, J. R.; Zare, R. N. Chem. Phys. 1988,126, 191. (30) Black, J. F.; Waldeck, J. R.; Hasselbrink, E.;Zare, R. N. 1.Chem. SOC.,Faraday Trans. 2 1990.85, 1044. (31) Black, J. F.; Waldeck, J. R.; Zare, R. N. J . Chem. Phys. 1990, 92, 3519. (32) Black, J. F.;Hasselbrink, E.;Waldeck, J. R.; are,R. N. Mol. P h p . 1990, 71, 1143. (33) Shapiro, M.; Levine, R. D. Chem. Phys. Leu. 1970,5,499; 1970, 7, 156. (34) Mukamel, S.;Jortner, J. J . Chem. Phys. 1974, 60, 4760.
0022-365419 1 /2095-7973%02.50/0 0 1991 American Chemical Society
1914 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 data the time taken for the C N radical to separate from the force field of the iodine atom (so as to be spectroscopically identifiable in its final internal quantum-state distribution) has been measured as ~ ~ =/ 205 2 f 30 fs fo: this pump w a ~ e l e n g t h . ~ By, ~detuning the probe laser to wavelengths to the red of the B2Z+ X2Z+ band, a rise and decay in the LIF signal is monitored, from which it has been determined that [I.-CN] ** transition states persist for times of the order of 20-50 fs.4*5From data of these types, various radial characteristics of the potential energy surface (PES) controlling dissociation have been d e d u ~ e d . ~ ~ ~ ~ ~ ~ ~ Experiments that involve detuning the probe laser to the blue of the X = 388.5 nm line permit the formation of C N in different rotational levels to be examined, a corresponding increase in T~ being monitored with increasing product rotational ex~itati0n.d~ These results have been discussed in terms of the time development of angular momentum during the dissociation reaction, leading to an asymptotic (long-time) distribution of rotational states of C N final p r o d ~ c tthe ; ~ observed variation of T~ is too small to be simply explained by a decrease in the availakle translational energy with increasing product rotation but rather reflects the degree of torque acting on bentSO*56757 ICN in the excited A state and the angular topology (and centrifugal effects) of the dissociative PES. By advantageous use of the inherent polarization properties of the laser pulses, the time dependence of coherence and alignment in reaction 1 have also been investigated6 and the observations related to the angular part of the excited-state PES7
-
~~
~
(35) Morse, M. D.; Freed, K. F.; Band, Y. B. Chem. Phys. Lerr. 1976,44, 125. (36) Halavee, U.; Shapiro, M. Chem. Phys. 1977, 21, 105. (37) Beswick, J. A.; Jortner, J. Chem. Phys. 1977, 24, 1. (38) Morse, M. D.; Freed, K. F.; Band, Y. E. J. Chem. Phys. 1970, 70, 3604, 3620. (39) Bcswick, J. A.; Gelbart, W. M. J . Phys. Chem. 1980, 84, 3148. (40) Morse, M. D.; F r d , K. F. J . Chem. Phys. 1981, 74, 4395. (41) Heather, R. W.; Light, J. C. J . Chem. Phys. 1983, 78, 5513. (42) Williams, S. 0.; Imre, D. G. J . Phys. Chem. 1988,92,6636,6648. (43) Heather, R.; Metiu, H. Chem. Phys. Leu. 1989, 157, 505. (44) Lee, S.-Y.; Pollard, W. T.; Mathies, R. A. Chem. Phys. Lerr. 1989, 160, 531. (45) Henriksen, N. E.; Heller, E. J. J . Chem. Phys. 1989, 91,4700. (46) J a m , M.; Atabeck, 0.;Leforestier, C. J . Chem. Phys. 1989, 91, 1585. (47) Vigud, J.; Girard, B.; Gou€dard, G.; Billy, N. Phys. Reu. Leu. 1989, 62, 1358. (48) Guo, H.; Schatz, G . C. J . Chem. Phys. 1990, 92, 1634. (49) Beswick, J. A.; Jortner, J. Chem. Phys. Lerr. 1990, 168, 246. (50) Yabushita, S.;Morokuma, K. Chem. Phys. Lerr. 1990, 175, 518. (51) Krause, J. L.;Shapiro, M.; Bersohn, R. J . Chem. Phys., submitted
for publication. (52) Pattengill. M. D. Chem. Phys. 1983, 78, 229. (53) Pattengill, M. D. Chem. Phys. 1984, 87, 419. (54) Waite, B. A.; Helvajian, H.; Dunlap, B. I.; Baronavski, A. P. Chem. Phys. Letr. 1984, I I I , 544. (55) Goldfield, E. M.; Houston, P. L.; Ezra, G. S.J . Chem. Phys. 1986, 84, 3120. (56) Dugan, C. H.; Anthony, D. J . Phys. Chem. 1987, 91, 3929. (57) Dugan, C. H. J . Phys. Chem. 1988, 92, 720. (58) Benjamin, I.; Wilson, K. R. J . Chem. Phys. 1989, 90, 4176. (59) Yan, Y. J.; Fried, L. E.; Mukamel, S.J. Phys. Chem. 1989,93,8149. (60) Mukamel, S. Ann. Reu. Phys. Chem. 1990, 41, 647. (61) Fried, L. E.; Mukamel, S.J . Chem. Phys. 1990, 93, 3063. (62) Bcrsohn, R.; Zewail, A. H. Ber. Bunsen-Ges. Phys. Chem. 1988,92, 373. (63) Bernstein, R. B.; Zewail, A. H. J . Chem. Phys. 1989, 90, 829. (64) HOW, K. E.; Klotz, L. C.; Wilson, K. R. J. Chem. Phys. 1970, 52, 4588. (65) Simons, J . P.; Tasker, P. W. Mol. Phys. 1973, 26, 1267. (66) Brown, R. C.; Heller, E. J. J . Chem. Phys. 1981, 75, 186. (67) Jevons, W. Proc. R. Soc. A (London) 1926, 112, 407. (68) Kiess, N. H.; Broida, H. P. J . Mol. Specrrosc. 1961, 7, 194 (69) Engleman. R., Jr. J . Mol. Specrrosc. 1974, 49, 106. (70) Jackson, W. M. J. Chem. Phys. 1974, 61, 4177.
Roberts and Zewail Theoretical studies of reaction 1 have also been n u m e r o u ~ . ~ ~ Many (earlier) investigations have concentrated on the calculation of absorption cross s e ~ t i o n s , product-state ~ ~ ~ ~ ~ branching ~ ~ ~ , ~ ~ ~ ratios,aJ5 frequency-resolvedabsorption, fluorescence, and Raman s p e ~ t r a , ~ ~ and - ~ ~C- N~ ~internal-state * ~ ~ distributions33-36d8-41,45.46.48,52-56~~,65from quantum mechanics3341 and by means of c l a ~ s i c a l ~ and ” - ~~~e ~m~i c l a s s i c a l ’procedures. ~*~~~ More recently, the application of ultrafast laser methods to the investigation of reaction 1 has prompted a range of calculations to be carried out that aim to predict the observed real-time behavior from q u a n t ~ m ~ ~or- ~(semi)classical -~l Most such studies in this vein have adopted a time-dependent a p p r o a ~ h , ” ~with ~ ~ a~view ~ ” ~to providing a rigorous conceptual framework within which a quantitative understanding of the many dynamical features of reaction 1 probed experimentally in the time-domain may be obtained. Bersohn and Zewail have offered a classical description of reaction 1, in which the time dependence of absorption of the probing laser pulse was related to an analytical form of a onedimensional repulsive PES for The results of this model reproduced the major features of the experimental FTS observation^.^*^,^ This work was extended by Bernstein and Zewho outlined a classical method for inverting the data obtained from FTS experiments to the difference between the excited-state PESs connected by the probe laser. These authors applied their procedure to reaction la, recovering an exponential repulsive form of the parametrized potential energy curve over which dissociation occurs when the upper PES accessed by the probe laser is assumed to be constant over the range of interfragment separations inv0lved.6~ In addition, consideration was also given to the treatment of one-dimensional PESs that were not purely repulsive functions of the reaction coordinate, for example when one or both of the excited-state PESs possesses a long-range van der Waals minimum. Williams and Imre42 carried out the first time-dependent quantum calculation of reaction la, computing the motion of wave packets over a repulsive PES that involved a single radial coordinate and obtaining FTS spectra in good qualitative agreement with those determined e ~ p e r i m e n t a l l y . ~Heather ~ ~ , ~ and Metiu have calculated time-dependent rotational coherence effects for reaction la, employing a similar method by including the effect of rotational motion of the C N which was previously treated by Zewail in terms of the time-evolution of C N rotational alignmente7 A quantum dynamical approach has likewise been employed by Lee et al. to calculate transient absorption cross-sections for ICN dissociation from the temporal overlap of the two moving wave packets created by the pump and probe-laser pulses.44 Henriksen and Heller45and Jacon et aleM have applied a quantum-mechanical wave packet treatment to derive Raman emission spectra and absorption cross sections for dissociation of ICN. From the time-evolution of classical trajectories over model PESs, absorption spectra as a function of time for the dissociation of ICN in both the gas phase and in Ar and Xe solvents have been computed by Benjamin and Wilson,s8 who obtained excellent agreement with experimental results.f3” Mukamel and co-workers have examined the ICN dissociation reaction, using a correlation function description of nonlinear optical processes in molecules based on a quantum phase-space representation of the density matrix,59@ further extending their approach to derive classical formulas for transition-state absorption of the probe-laser pulse.60*61 In this way, these authors were able to delineate in detail the various criteria necessary for successful application of classical models of molecular dissociation in the time domain in terms of the different temporal regimes for nuclear separation, wave packet dephasing, and laser pulse length.5w1 Density matrix methods have also been utilized by other groups of workers to compute femtosecond time- and frequency-resolved spectra of molecular A theoretical species in both the gaseous and solution (71) Lin. S. H.; Fain, B. Chem. Phys. Leu. 1989, 155, 216. (72) Fain, B.; Lin. S. H.; Hamer, N . J . Chem. Phys. 1989, 91, 4485.
Femtosecond Real-Time Probing of Reactions description of FTS investigations of reaction 1 in terms of CW frequency-resolved bound-free and free-free spectra has recently been given by Krause, Shapiro, and Bersohn using quantummechanical and semiclassical techniq~es.~'Beswick and Jortner have sought to clarify the time evolution for motion of localized wave packets prepared on a dissociative PES by absorption of an ultrashort laser photon, pointing out that the time scales for dephasing of the initial wave packet and for bond rupture to form spectroscopically identifiable products differ by some two orders of magnit~de.4~ In this work, experimental r e s ~ l t are s ~ compared ~ with quantum and classical calculations of reaction 1. The quantum-mechanical simulation involves computation of [I-CN] ** wave packets for various times following laser excitation in a manner analogous to the earlier work of Williams and Imre.42 The classical description employed here makes use of the approach developed previously by Bersohn and Zewai1.62 From the transient behavior so derived in both sets of calculations, the inversion procedure of Bernstein and Z e ~ a i isI ~applied ~ to deduce the functional forms of the model potential energy curves at large interfragment s e p arations, or more accurately, the difference between PESs connected by the probe laser. In this way, the sensitivity of calculated FTS transients as a function of the nature of the two PESs involved is examined by variation of the parameters appearing in the analytical forms for the relevant potential energy functions. Possible experimental ramifications of the presence of a centrifugal barrier on the reaction path leading to final products are also examined, and preliminary consideration is given to the type of dynamical behavior that might be observable in the time domain. Finally, the validity of the different theoretical approaches employed here is further examined by comparison of the results obtained for dissociation of ICN with analogous data75 for fragmentation of the heavier Biz molecule. The paper is organized as follows. In section I1 the theoretical framework underlying the quantum-dynamical and classical calculations of FTS behavior is briefly reviewed, and an outline is given of the main results of the potential inversion method developed by Bernstein and Z e ~ a i Iin; ~addition, ~ our selection of PESs for calculations on ICN photodissociation is described. The results of quantum and classical simulations of FTS spectra for the dissociation process (la), together with inversion of these data to the reactive potential and variation of the functional form of the PESs are presented in section 111. Summarized in section IV are experimental and theoretical results pertaining to the dissociation of Biz, which are contrasted with those for ICN dissociation. Section V comprises a summary and conclusions and attempts to offer a broader perspective of the real-time dynamics that characterize direct dissociation over repulsive PESs. 11. Theoretical Methodology Figure 1 displays schematically a simple pump-probe experiment in terms of three noninteracting potential energy curves (though the general designation "PES" is retained throughout), the simulation of which is to be carried out. Our discussion throughout this paper is confined to dissociation of a rigid quasi-diatomic molecule involving a single translatory degree of freedom (reaction coordinate); i.e., rovibrational excitation of the C N photofragment is considered to be negligible in comparison to the available translational energy for product recoil. It is therefore assumed that the short-time dynamics of the reacting system is dominated by one-dimensional motion over a PES that is repulsive in nature, here labeled Vl(r);for dissociation of ICN, this motion occurs along the I-CN stretching coordinate. Measurements of nascent product-state populations at selected photolysis wavelengths over the range X = 220-350 nm have indicated that the C N photofragment of reaction l a (giving rise to ground-state I(2P3,2) atoms) is produced vibrationally (73) Pollard, W.T.;Lee, S.-Y.;Mathies. R. A. J . Chem. Phys. 1990, 92, 4012. (74) Lin, S.H.;Fain, B.; Yeh, C. Y. Phys. Reu. A 1990, 41, 2718. (75) Bowman, R. M.; Gerdy, J. J.; Roberts, G.; Zewail, A. H.J . Phys. Chem. 1991, 95, 4635.
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 7975 60
I
I
3
3.5
I
I
I
4.5
5
50
40
,
B
n
2
30
3 20
10
2.5
r7h Figure 1. Potential energy diagram for ICN. Schematic diagram of ground- and excited-state potential curves for ICN at energies E I 60 X lo-' cm-l above the electronic minimum of the molecular ground state Vo(r),illustratingthe concept of FTS investigations. V,(r)and V2(r)are given by the functional form V,(r)= exp(-a,r) + 8, with VIo= 2.061 x loIOcm-', aI= 5.1813 A-I, = 26 130 cm-I, a2 = A-I, and B2 = 50760 cm-I. For presentation purposes only a single reactive potential V l ( r )is shown; omitted from the diagram are the several excited-state PESs correlating with CN(X22+)+ I(2P3/2)and I(2Pl/2,)products at infinite separation. The arrow labeled AI represents a vertical transition from the u = 0 level of the ground state at the equilibrium 1-CN separation re for a pump-laser wavelength of XI = 306 nm, which results in preparation on V,(r)of a wave packet comprising a distribution of continuum eigenstatesdetermined by the spectral profile of the laser pulse. (Note that at this photolysis energy, the wave packet prepared by the pump pulse initially samples classically forbidden regions of configuration space.) The vertical transitions to the highest lying potential V2(r)labeled X2- and X2 represent probe-laser pulses for on-resonance (final product) and off-resonance (transition-state)detection, respectively. A Gaussian profile for the pumplaser pulse with fwhm = 118 cm-l is depicted schematically on the left ordinate. Dol (= 81) and 0 0 2 (= 82) are dissociation limits for Vl(r)and V2(r)measured from the u = 0 level of Vo(r).
-
~ o l d l ~but J ~rotationally J ~ ~ ~ quite h0t,'sm9u38*31 and more or less exclusively in the ground electronic state X'2+.'3J4 (In contrast, the rotational distribution of CN product formed in reaction l b is peaked at low quantum numbers.'f16.'8-20~22.28.31) By use of a supersonic molecular beam source to achieve internal-state cooling, experiments carried out by Houston and c ~ - w o r k e r s ~and ~ J ~by Nadler et aLzo have demonstrated the unimportance of initial parent rotation as a source of the rotational excitation of C N p r o d ~ ~ t ; ' ~ in, any ~ ~ case, . ~ the ~ , rotational ~ ~ , ~ ~period ~ ~ of ~ ICN ~ ~ ~ in its J = 3 1 quantum level, for which room-temperature population is maximal, is longer than the time scale for recoil of the I C N photofragments of reaction 1 to form spectroscopically identifiable dissociation p r o d ~ c t s . ~ * ~It* ~has * ~been * * ~suggest~ ed5~7.'7-20,222428d1,35,38.40,52-~7 therefore that the ij2vibration of ICN is of considerable importance as it dissociates, since a bent configuration of the parent molecule correlates adiabatically with rotational motion of C N a t large interfragment separations.22,37,38.54,76,77 Experiments that make use of the polarized nature of laser beams have permitted the directionality of the transition moment involved in the initial absorption process from the ground state to be deduced.12*22*26v28 For example, a t the experimentally convenient photolysis wavelengths X = 266 and 248 nm, measurements of the alignment of C N angular momentum22*26,28 indicate that the dominant absorption transition is predominantly parallel in
+
(76) Ashfold, M. N. R.; Macpherson, M. T.; Simons, J. P. Top. Curr. Chem. 1979, 86, 1. (77) Felp. W.S.;Rupnik, K.;McGlynn, S.P. J. Phys. Chem. 1991, 95, 639.
1916 The Journal of Physical Chemisrry, Vol, 95, No. 21. 1991
Roberts and Zewail
character and occurs to a PES of R = O* symmetry in Hund's were based upon the same premise, being carried out on a model for (4s) coupling), together with case c (R,,w) notation (3110+ dissociative PES with a single minimum about zero bending angle. a smaller perpendicular contribution arising from excitation to Our (simplified) choice of PESs for the ICN system is described a PES of 52 = 1 symmetry, especially at the shorter in more detail in section 1I.C. wavelength.12v2z26*28*29s31,47 The o b ~ e r v e d ~ * ~ Jhigh ~ ~ rotational 3~' Referring to Figure 1, we note that the pump laser at waveexcitation of C N arising from reaction l a has been postulated length XI (= 306 nm as depicted) excites the ground-state wato result from a mechanism that involves nonadiabatic mixing vefunction from the potential Vo(r)to a single repulsive excitbetween an initially populated linear ($2 = O+) state connecting where a superposition of continuum eied-state potential Vl(r), with spin-orbit-excited I(2P;/2) atoms at infinite separation and genstates is formed constituting a wave packet. For ICN, the one or more bent excited-state surfaces that correlate with pump laser induces absorption via the A continuum as indicated ground-state I(2P3/2)+ CN(X2Z+) final products via the lower by (la); Vl(r)may possess R = 1 or O+ (3111or 3110+)symmetry energy exit channel.19~2r2e29,31,32,3s,38,47,48,ss Simultaneous optical and is populated by means of a perpendicular or parallel absorption excitation to more than one upper state of linear or bent molecular transition, respectively, from the R = O+ (IZ+)ground state as geometries with subsequent evolution of the dissociating system noted above. The wave packet so prepared, moving under the over essentially noninteracting PESs cannot be ruled out as a influence of the repulsive force directed along the reaction copossible alternative, however. ordinate r, then propagates over V l ( r ) to form dissociation Of the numerous3s~38-41~48~s2~s5~64 theoretical investigations that products, during which process it is subject to the perturbation been carried out with the aim of accounting for the observed C N due to the probe laser at wavelength X2 delayed with respect to rotational energy distributions arising from both reaction channels the pumplaser pulse by a time T. The probe laser excites the wave (la) and (1 b), the following studies may be highlighted. Goldfield packet on V l ( r )by means of a vertical free-free transition to a et al. have modeled the A-state dissociation by means of computing second excited-state potential V2(r),the topology of which at this classical trajectories over model PESs, assuming initial absorption stage may be assumed to be independent of radial coordinate r to a linear PES and taking into account the effect of nonadiabatic for the sake of simplicity (see section 1I.C). Those transition-state transitions between excited-state surfaces.s5 These authors find configurations [I-CN] ** that allow for absorption of probe-laser that while the results of their calculations reproduce CN rotational light define a detection Uwindown,or optically-coupled regi0n,"2~~ energy distributions, average rotational energies and branching centered about X2 and of spectral width determined by the temratios for reaction via both exit channels ( l a ) and ( lb)55 in a poral duration of the probe laser. qualitative fashion, they are unable to describe quantitatively all The PESs V l ( r )and V2(r)correlate with I(2P3/2) CN(X2Z+) aspects of the dissociation dynamics. A similar conclusion was and I(2P3/z)+ CN(B2Z+) asymptotic product states, respectively. reached by Guo and Schatz,48 who have performed an exact The influence of additional low-lying repulsive PESs Vl(r),for time-independent, quantum-mechanical investigation using the instance those of il = 2 (311z)and 0- (3110-)symmetry leading to coupled-channel model to calculate product rotational-state disground-state I(2P3/2) product via reaction la, are omitted from tributions, I-atom branching ratios, and total absorption cross Figure 1, since possible nonadiabatic interactions involving such sections. Satisfactory qualitative agreement with experimental surfaces and VI( r ) have been considered~9~22~2e29~31~32~3s~38~47~ss to r e s ~ l t s ~ ~ was - ~likewise ~ ~ ~obtained ~ - ~ by~ these ~ ~ workers, ~ ~ ~ ~ ~ ~occur ' ~within ~ the Franck-Condon region a t interfragment sepathough it was noted48 that closer quantitative accord awaits the rations shorter than those accessible with an experimental pump availability of accurate information concerning the detailed towavelength of XI = 306 nm. In this context, however, it may be pologies of the PES(s) controlling reaction. noted that the existence of a conical intersection between the $2 The results of recent ab initio calculations% of the excited-state = O+ (31&,+) and il = 1 ( I l l l ) potential curves (that correlate with potentials of ICN suggest that all relevant excited-state PESs of 1(2Pl/2)and I(2P3/z)atoms at infinite separation, respectively) at ICN involved in ultraviolet absorption via the A-band continuum an I-.CN distance of r = 3.32 A has recently been predicted,% may in fact possess nonlinear geometries in the Franck-Condon which would play an important role in governing the dissociation region such that a bent configuration is initially adopted by the dynamics at shorter photolysis wavelengths where reaction channel excited parent molecule, with equilibrium ICN angles between l b is accessible (see above). 40° and 6 0 ° , thereby exerting a strong torque on the recoiling The final wave packet on V2(r)can be monitored by emission C N product. The presence of a conical intersection between permitting elucidation of resonance fluorescence back to Vl(r),I-l' excited-state surfaces located well beyond the Franck-Condon of the real-time dynamics of the dissociation process by detection region, in addition to the strongly bent geometries of the PESs of such fluorescence as a function of the pumpprobe time delay at shorter distances, has permitted Yabushita and MorokumaSo T : this is the essence of the FTS experiment. Variation of the to reinterpret earlier experimental r e s u l t ~ l in ~ -light ~ ~ of these probe-laser wavelength allows the time-evolution of the wave predictions and to propose a model for photodissociation of ICN packet propagating over Vl(r)to be interrogated at different spatial that differs substantially from those previously advocatcoordinates, since in general the energy separation between VI(?) ed.19~22~2e29~31~32~35~38~47,ss By analogy with their theoretical study and V2(r)will not be constant as a function of dissociation coof excited-state PESs for ClCN, Reisler and colleagues have ordinate r. offered further insight into the ordering and geometries of the A. Quantum Dynamics. The quantum dynamical treatment electronic states involved in ICN dissociation, including the employed in this paper to simulate the FTS pumpprobe experpossibilities for avoided crossings between PESs and nonadiabatic iment is that of second-order time-dependent perturbation theory.79 transition^.'^ Dugan has also presented a model calculation that This method has previously been invoked and developed by accounts for the observed CN rotation in terms of the torque Williams and Imre,42Metiu and co-workers,8*86 and others to generated by a repulsive force developed between the I and C N study the behavior of gas-phase dissociation reactions in the time photofragments acting in a bent configuration on the excited-state domain. The time-dependent approach pioneered by Heller and potential.56.57 co-workers, using Gaussian wave packets, illuminated the analysis The angular part of the PES is neglected in this simple treatment, however; as pointed out by Williams and Imre,42 the (79) Loudon, R. The Qumfum Theory of Light;Clarendon Press: Oxford, dominant force in the Franck-Condon region is along the I-CN 1973; Chapter 1 1 , p 279. stretching coordinate, since both excited states V l ( r )and V2(r) (80)Engel, V.; Metiu, H.; Almeida, R.;Marcus, R. A.; Zewail, A. H. Chem.'PhG. Lett. 1988, IS2, 1 . display a maximum or minimum along the bending coordinate (81) Engel, V.; Metiu, H. J . Chem. Phys. 1989, 90, 6116. for a linear molecular geometry. Radial and angular topologies (82) Engel, V.; Metiu, H. J . Chem. Phys. 1989, 91, 1596. of the excited-state PESs of ICN have also been discussed by (83) Engel, V.; Schinke, R.;Hennig, S.; Metiu, H. J . Chem. Phys. 1990, Benjamin and Wilson,s8and their classical trajectory calculations 92, 1 . (39'n1)
+
(78) Bai, Y . Y . ;Segal, G. A.; Reisler, H. J . Chem. Phys. 1991, 94, 331.
(84) Engel, V.; Metiu, H. J . Chem. Phys. 1990, 92, 2317. (85) Metiu, H.; Engel, V. J . Chem. Phys. 1990, 93, 5693. (86) Metiu, H.; Engel, V. J . Opt. SOC.Am. B 1990, 7, 1709.
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 7977
Femtosecond Real-Time Probing of Reactions of dynamics in real time,"*66*87*88 and Kinsey and co-workers have elegantly applied the method to examine molecular fragmentation.89 The nuclear wave function I$2(t)) created on the potential V2 by absorption of pump and probe photons is calculated from the expression79
Here (cp,) represents a nuclear eigenstate of the uth vibrational level of the ground electronic state with a phase factor exp(-iwJ,). Wave functions are propagated by the operators Tiwhose explicit form is
Ti(r) = exp(-iHit)
(3)
with Hamiltonian operators Hi given by Hi = li)[K + V J ( i l
(4)
where K and Vi are the kinetic and potential energy operators, respectively. Ui,(tk)represents the time-dependent perturbation caused by the kth laser pulse, which optically couples electronic states li) and b): U,,(tk) = C(i,(r)'Ek(tk)
(5)
where pi,@) is the transition dipole moment operator connecting state to l i ) . In the absence of information concerning the variation of cci,(r) with coordinate r, the Condon approximation is assumed throughout this work; Le., I.ci (r) = p&) (it. Ek(tk) is the electric field vector of the radiation keld due to the kth laser pulse:
v)
Ek(tk) = gk(tk)Ekoexp(iwktk) (6) the temporal lineshape gk(tk)of which is taken to be Gaussian for both laser pulses.
= exP[-(h
-~ k 0 ) / d 1
(7)
The time delay T between pump- (k = 1) and probe-laser (k = 2) pulses is defined in the usual way as the difference between the times at which the intensities reach their maximum values:
= t20 - tl? Equation 2 indicates that at time t , the perturbation due to the pump photon, Ulo(tl), promotes the ground-state eigenfunction IIP~)exp(-iw,t,) to the electronic state 11). The wave function I+&)) so created is then propagated over Vl(r)by action of the operator T1(f2-tl) for a time duration t 2 - rl. At time t2, the perturbation U2,(t2)excites a transition between electronic states 11 ) and 12), corresponding to absorption of a probe-laser photon. The resulting wave function 1$2(t)) is subsequently propagated by T2(f- t 2 ) until some final time t. Following our earlier calculations of Hg12 wave packets,g0both Fourier-transform-limited laser pulses are arbitrarily truncated at *2uk so that the operators Vi ( t k )act on the appropriate wave function over a total period ottime 4uk; the length of time over which the propagation operators Ti(t)are applied in eq 2 is thus reduced proportionately. The complete promotion-propagation sequence therefore begins at time t = -20, (start of pumplaser pulse) and finishes at time t = +2u2 (end of probe-laser pulse). In what follows, we shall be concerned with only the total emission intensity I(r;X2)from the electronic state 12). This is proportional to the norm (&(f)J&(?)) of the final wave packet, representing the probability oflocating the system in state (2) at a time t immediately following the action of the probe-laser pulse. 7
(87) Heller, E. J. J . Chem. fhys. 1975, 62, 1544. (88) Heller, E. J. Acc. Chem. Res. 1981, 14, 368. (89) Imre, D.; Kinsey, J. L.; Sinha, A.; Krenos, J. J . fhys. Chem. 1984, 88, 3956. (90) Gruebele, M.; Roberts, G.; Zewail, A. H.fhilos. Trans. R. Soc. London, A 1990, 332, 223.
Wave packet propagation was carried out by using the splitoperator technique developed by Feit and c o - w ~ r k e r s . ~ ~Nu~' merical calculations were carried out such that eq 2 could be implemented with pump and probe pulses overlapped in time and were performed on a grid of 2048,4096, or 8 192 points, depending upon the kinetic energy of the wave packet and the distance range involved. As noted in a previous publication," care was taken to sample above the Nyquist limit in both momentum and coordinate space so as to mitigate against aliasing and edge effects of the discretized wave functions. Convergence was tested for in the usual manner by having the number of discretizations and the time step. For a propagation period of 300-500 fs, converged wave functions were obtained for time steps At < 1/(50AJi'), where AE is the characteristic energy scale of the problem. In all calculations, the initial wave function exp(iw,tl)lp,) was taken to be the Gaussian form of the u = 0 level of the ground electronic state. B. Classical Mechanics. The classical-mechanical analysis of ICN dissociation adopted in this paper employs the method prescribed by Bersohn and Zewail to derive an expression for the explicit time dependence of the probe-pulse absorption A ( t ) for classical motion of the reactive system over Vl(t).62 In this treatment, the first (pump) absorption step results in localized excitation of ICN by means of an initial laser pulse that is a 6 function in time, while the finite spectral bandwidth of the "temporally short" probe-laser pulse is considered to project a detection window (or optically coupled region) onto the potential for fragment separation such that maximum probe absorption occurs at some configuration r* along the dissociation coordinate. Fried and Mukamel have pointed out that spectral convolution of the absorption profile by the probe pulse is appropriate in those instances where the pump and probe pulses are short in comparison to the time dependence of the nuclear dynamics; temporal convolution is valid when the pulses are longer than the time scale for dephasing (given by the inverse line width of the overall absorption).6' This dephasing has a direct influence on the coherence effect discussed and ruled out by experiment:5 in these experiments, the coherence (and dephasing) time is much shorter than the laser pulse widths. For any distance r along the reactive potential VI@), absorption of the probe laser pulse may be writted2 A ( r ) = C exp(-(In 2)[(V2(r)- V , ( r ) )(V2(r*) - Vl(r*))I2/(AFm2l (8) where C is an arbitrary constant (set equal to unity) and the spectral profile of the probe laser is taken to be Gaussian with fwhm = A;. Equation 8 represents spectral convolution of the probe-pulse absorption A(r). Bersohn and ZewaiF2 have noted that AF for femtosecond laser pulses will exceed the natural line width of free fragments by several orders of magnitude, so that an intrinsically sharp absorption would appear to be Gaussian in shape. Transient behavior is determined by the classical time evolution of a particle over the PES Vl(r)governing dissociation, the a p propriate equation of motion for which is (P/2)(dr/dt?2 = E,, - VlW
(9)
where E,, is the kinetic energy available to separating fragments of reduced mass P in the center-of-mass frame relative to the dissociation limit of Vl(r).The distance traveled by a point mass as function of time is then given by r, = ro
+ u J ' [ l - Vl(r)/Eav]1/2dt'
(10)
Equation 10 can be integrated analytically in the case of a repulsive potential Vl(r)given by the simple exponential form of eq 16, yielding the expression (91) Fleck, J. A., Jr.; Morris,J. R.; Feit. M.D. Appl. fhys. 1976. IO, 129. (92) Feit, M. D.; Fleck, J. A., Jr.; Steiger, A. J. Compur. Phys. 1982,47, 412. (93) Feit, M. D.; Fleck, J. A., Jr. J . Chem. fhys. 1983, 78, 301.
7978
r, = ro - ( l / a l ) In [sech2 (vra,/2)]
(1 1)
Here, I is the time taken by a reactive trajectory on V l ( r )to reach the position r measured from t = 0 at the initial coordinate r, determined by classical Franck-Condon absorption from the ground state; t* is therefore the time corresponding to r = r*. The terminal velocity u = (2Eav/p)1/2is the classical counterpart of the asymptotic (long-time) group velocity of the quantum-dynamical wave packet. l/alis the so-called length or range parameter of the exponential potential function V l ( r )(see eq 16), and represents the distance by which potential energy has decreased to l / e of its value at the position r,. A straightforward manipulation of eqs 8 and 11 leads to an analytical solution for the time dependence of absorption of the probe-laser pulse given by A(t)
= C expl-(ln 2)Mt,t*)/(A'v/2)]21
(12)
where flt,t*)
Theoretical expressions for A(7) applicable for arbitrary pulse conditions have recently been derived by Fried and Mukamel.6' C. Choice of PESs. In this section, we indicate the sources from which the simple parametrized forms of the potential energy functions for the electronic states IO), Il), and 12) were selected. The ground-state potential Vo(r)is assumed to be harmonic, since optical excitation of the thermal population in the lowest vibrational level only is treated in this work, as appropriate for the room-temperature experimental conditions employed.'" The generally accepted value of re = 2.745 AIoSdating back to 1936" was taken to be the equilibrium interfragment separation between the 1 atom and center of mass of CN,IMbwhile the value we = 45 1.5 cm-' determined spectroscopically by Penney and Sutherland'& was employed as the fundamental oscillation frequency of the NC-I stretching vibration. As a simple characterization of the Vl(r)and V,(r) potentials of ICN we invoke a repulsive exponential curve:
vio exp(-a,r) + & Values of al = 5.1813 A-' and vio = 2.061 X V,(r) =
~ ~ ( t ) [ E , v / V l ( f ) ] sech2u2/ul u2/u~ x V ~ ( t * ) [ E , v / V l ( t * ) ] usech2u2/ul ~ / u ~ x* Eav(sech2x - sech2 x*) (13)
VI(?)and V2(t)are the potential energy curves of Figure 1 expressed as a function of time, while x = vtal/2 and x* = vr*al/2 when t = t*. If it is assumed that the high-lying potential V2(r) populated by the probe laser is independent of position along the reaction coordinate, meaning that a2 = A-1 (see eq 16), then eq 12 reduces to the simpler form
-
A ( t ) = C exp(-(ln 2)[Eav(sech2x*
- sech2 x)/(AZ/2)I2}
(14)
The distributions of both trajectory starting times and finishing (probing) times incurred by the use of laser pulses of finite temporal duration can be accounted for by convolution of the absorption A(t) of eq 12 with the temporal lineshapes gk(rk)of the pump and probe lasers: A ( r ) = A(t2'-
?I0)
= 11A(r)g12(7- t) dt
(15)
where gI2(?) represents the cross-correlation function of pump and probe pulses. Thus the initially imposed criterion of &function laser pulses is removed, which provides a more valid comparison with experiment. A ( T ) is taken to represent the observed LIF signal, which we designate 1(7;A2) in accordance with an earlier n0tation.6~ This temporal convolution procedure is of course important as the duration of the laser pulse widths becomes comparable to the time scale of the dynamics of dissociation, as pointed out previo~sly.~*~ Conversely, if the pulses are much shorter than the time required for nuclear recoil (but longer than the dephasing time) then convolution becomes unimportant. Examples of situations where the relevant reaction dynamics occur over much longer time scales than the laser pulse widths have been reported for dissociation of Bi2,5' NaI,9"Io2 and Hg12.103Jar (94) Bowman, R. M.; Dantus, M.; Zewail, A. H. Chem. Phys. Len 1990, 161, 297. (95) Dantus, M.;Bowman, R. M.; Zewail, A. H.Nature (London) 1990, 343. 737. (96) Gcrdy, J. J.; Dantus, M.; Bowman. R. M.;Zewail, A. H.Chem. Phys. Lerr. 1990. 171. 1. (97) Bdwman, R. M.; Dantus, M.; Zewail, A. H. Chem. Phys. Letr. 1990, 174, 546. (98) Rose, T. S.;Rosker, M.J.; Zewail, A. H.J . Chem. Phys. 1988,88, 6672. (99) Rosker, M. J.; Rose, T. S.;Zewail, A. H. Chem. Phys. Lett. 1988, 146, 175. (100) Rosc. T. S.;Raker, M. J.; Zewail, A. H. J. Chem. Phys. 1989, 91, 7415. (101) Cong, P.; Mokhtari. A.; Zewail, A. H. Chem. Phys. Letr. 1990,172,
__
-Koberts . . ana. Lewaii - ..
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991
109. - .
(102) Mokhtari, A.; Cong, P.; Herek, J. L.; Zewail, A. H. Nature (London) 1990, 348, 225. (103) Bowman, R. M.;Dantus, M.; Zewail, A. H.Chem. Phys. Let[. 1989, 156, 131.
(16)
1O'O cm-' were employed in this work, being the parameters obtained by Bersohn and ZewaiF2 from a consideration of spectroscopic investigations of the A X band system of ICN."."' From an empirical point of view, and following these same it is initially considered that JVl(r)- Vl(r*)l>> IV2(r)- V2(r*)lfor all values of r* accessed by tuning the probe-laser wavelength, i.e. that the shape of V2(r) is independent of reaction coordinate a t distances greater than the Franck-Condon region for absorption of pumplaser light. Accordingly, a value of a2= A-' was employed for the majority of the calculations reported in this work (so that V2(r)is simply P2). More realistic (finite) values of a2were also tried, however, as described in section 1II.D. & and fl2 represent, of course, the dissociation energies Dol and DO2shown in Figure 1; values of = 26 130 cm-' and B2 = 50 760 cm-I were used in the present calculations.62J12JI 3 In order to obtain the correct absolute values of X2 employed in the experiments of Dantus et al.,s it was found necessary to include the effect of the rotational energy content of the diatomic photoproduct. Commensurate with on-resonance (A2-) probing at the band head of the P branch of the B2Z+ X2Z+ (0,O)band
-
-
(104) Dantus, M.; Bowman, R. M.;Gruebele, M.; Zewail, A. H. J . Chem. Phys. 1989,91,7437. (105) Herzberg, G. Molecular Spectra and Molecular Structure II: Infrared and Raman Spectra of Polyatomic Molecules; Van Nostrand Reinhold New York, 1945; Chapter 2, p 174. (106) (a) Penney, W. G.; Sutherland, G.B. B. M. Proc. R . Soc. London, A 1936,156, 339. (b) The value re = 2.745 A has been employed by several research groups during recent years (see. for example, refs 3, 5 , 19, 26-32, 35,38,40, and 55 among others); its current usage appears to stem from the early theoretical study of ICN dissociation presented by Holdy, Klotz, and Wilson in 1970." This value was obtained by these workers from the structural parameters for ICN listed by Hmberg in 1945,Iw which were themselves taken from the gas-phase spectroscopic study of Penney and Sutherland reported in 1936.'061 In order to derive force constants for the vibrational modes of ICN, these authors employed an estimate of re(lq) = 2.12 A as the equilibrium internuclear separation between 1 and C atoms, calculated by adding the difference between the ionic radii of I and C1 to an earlier determination of the C C I separation in CCl, published in 1935," while for the distance between C and N atom, the value fdc-N) = 1 .I5 A was taken" from infrared absorption measurements on HCN. also camed out in 1935." We note that somewhat more recent estimates of r$,*) = 1.9921 0.0002 A and fdc-N) = 1.1604 0.0003 A determined by microwave spcctroscopylog lead to the value re = 2.616 A as the I-CN equilibrium separation. This result is in agreement with the spectroscopic parameters for ICN listed in the latest edition of the JANAF thermochemical tables.I1° (107) Sutton. L. E.;Brockway, L. 0. J . Am. Chem. Soc. 1935,57,473. (108) Bartunek, P. F.; Barker, E. F. Phys. Reo. 1935,18, 516. (109) Cazzoli, G.; Degli Esposti, C.; Favero, P. G. J . Mol. Srruct. 1978. 48, 1. (110) Chase. M.W., Jr.; Davies, C. A.; Downey, J. R.. Jr.; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. J . Phys. Chem. Ref. Data, Suppl. 1985, 14,601 (JANAF Thermochemical Tables, 3rd ed., Part I). (1 11) King, G. W.; Richardson, A. W. J. Mol. Spectrosc. 1966,21, 339. (112) Davis, D. D.; Okabe,H.J . Chem. Phys. 1968,49, 5526. (1 13) Huber. K. P.; Herzberg, G. Molecular Spectra and Molecular Structure IY: Constants of Diatomic Molecules; Van Nostrand Reinhold: New York. 1979.
*
*
Femtosecond Real-Time Probing of Reactions
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 7979
citation process transfers population from the ground state to the system,14.15*67-70 N" = 25 is adopted as the most populated roexcited PES V l ( r )over a range of times and consequently with tational level of the CN(X2Z+) ground electronic state1618 rea spread of kinetic energies corresponding to different initial sulting from the torques experienced by the parent molecule on starting coordinates on the upper-state PES. dissociation. Furthermore, rotational excitation of the radical is In applying eq 17 to deduce V l ( t )from FTS data therefore, considered to be independent of the NC-I stretching coordinate. it is assumed that the initially created wave packet describing the (That is, the time development of angular momentum during the separation of products is highly localized and remains so during dissociation process is neglected.) the course of the dissociation process. That is to say, the width Figure 1 displays the minimum number of PESs required for of the wave packet Ar (= 1/2[(h/Ap)2 ( 2 r A p / ~ ) ~for ] ~ a/ ~ description of the FTS experiment. It was pointed out in section Gaussian packet, where Ap is the corresponding momentum I that at a pump wavelength of XI = 306 nm the exit channel (1 b) distribution) stays essentially constant and that the degree of is effectively closed, with only a negligible fraction of reactive of the wave packet as a function of time dAr/dt (d2trajectories proceeding to I(2Pl/2)p r o d u ~ t , ' ~ , ' ~ , ~whereas ~ , ~ ~ 3 ~ - ~ ~ ,spreading ~~ (Ar)2/dt2 = ( 2 / ~ ~ ) [ ( A p )(Ar)2pVl"(r)]'15) ~ is negligible over at the shorter wavelengths XI = 295 and 285 nm, reactions l a the time period during which the system evolves to final product and 1b leading to formation of both spin-orbit levels of atomic states (or alternatively, that changes in the distribution of kinetic iodine are energetically fea~ible6~"Z"~ Most calculations reported energies of the initial wave packet with time are small). The width in this paper were carried out for a pump wavelength of XI = 306 Ar of the wave packet may or may not, however, be narrower than nm, well to the low-energy side of the absorption maximum for the range of spatial coordinates on the PES viewed-by the the A continuum located at X = 251.3 nm,77 though a number probe-laser pulse. of calculated transients for reaction l a were also obtained for Furthermore, the transition dipole moment ( 2 1 ~ ( ~ ~ ( rbetween )ll) higher photolysis energies using the same potential curves described the electronic states optically connected by the probe laser is above. From the laser excitation energies and dissociation limit assumed to be independent of spatial coordinate r (see section PI given earlier, optical excitation at XI = 306, 295, and 285 nm 1I.A). The vertical probe transition is also presumed to conserve is found to correspond to total available kinetic energies for the momentum of the separating products; Le., that no change fragment separation via the lower-lying exit channel ( l a ) of E,, in the internal motion of the probed fragment is induced as a result = 6550, 7769, and 8958 cm-I, compared to the spin-orbit splitting of the action of the probe laser pulse. A more complete description in atomic iodine of E(2P,12)- E('P312) = 7603 cm-I.lI4 The of the asymptotic LIF intensities would take into account the effect terminal velocities for I(2P3/2)+ CN(X2Z+)separation are thus of probing a (nonstatistical) rovibrational energy distribution of readily determined as u = 0.0269, 0.0293, and 0.0315 A fs-I, CN(X2Z+) product species. respectively, for the three photolysis wavelengths. Finally, it is stressed that application of eq 17 to FTS data D. Inversion to PES and the Classical Limit. Inversion to the strictly yields a difference in potential energy lAV21(r)l = IV2(t) PES from computed FTS data was carried out following the - vl(r)lbetween the two electronic states connected by the probe methodology put forward by Bernstein and Z e ~ a i l Recovery .~~ laser as a function of time (or interfragment distance), rather than of the functional form of the potential energy function V l ( t )for the potential controlling the dissociation dynamics alone. FTS the first excited state expressed as a function of time from deexperiments that involve systematic variation of the pumplaser convolved ITS transients of r(r;X2) versus r is readily affected by wavelength independently of that of the probe, thus permitting a straightforward analysis of long-time LIF signal intensities for control of the excess energy above threshold for dissociation, do different probe laser wavelengths X2. Detection of transient allow separation of V l ( r )and V2(r) to a certain degree of self[1.-CN] ** species evolving over different regions of configuration space on Vl(r) leads to an asymptotic fluorescence signal that is c o n ~ i s t e n c yas , ~ ~has been undertaken in a previous FTS study of the topologies of the low-lying excited-state PESs involved in determined by the spectral intensity of the (blue) wing of the probe the predissociation of NaI for i n ~ t a n c e . ' ~An ~ , ~analysis ~ ~ prolaser; LIF signals monitored at long delay times for different X2 cedure for extracting the topologies of bound-state potential energy thus project into the frequency domain the effective intensity curves for electronically excited 12(B31&-,+u)91*95 spectrum of the probe-laser p ~ l s e . 6It~ may be pointed out in this and ICI(A3111)118 regard that asymptotic values of ($2(r)l$Z(r)) calculated by eq from FTS measurements has recently been adopted that makes 2 instead reflect the spectral profile of the time-dependent electric use of Fourier transformation and RKR t e c h n i q u e ~ . ~ The ~~J~~ field vector EZ(r2)of the probe-laser pulse (see eq 6), rather than inversion procedure outlined above has also been applied to different functional forms of both excited-state PEss Vl(r) and V2(r), its intensity IZ(r2) per se (which in any case is trivially related to examples of which have been discussed in the paper of Bernstein E2(t2)). For both quantum and classical simulations, therefore, and Z e ~ a i 1 . ~ ~ the temporal potential V,(r) can be related to the spectral distribution of the probe laser about its nominal frequency I, and 111. Results and Discussion the detuning increment A = 12-- F2 with respect to the on-resA. Quantum Calculations. Figure 2 shows snapshots of the onance probe frequency bzm. time-dependent wave packet Irll(t)) as it evolves over the PES For a Gaussian probe-laser profile and an upper-state PES V2(r) V l ( r )at various times following laser excitation of ground-state accessed by the probe pulse that is assumed to show no dependence ICN at XI = 306 nm with a Gaussian pump pulse of fwhm = 125 on interfragment separation r, considerations of this nature lead fs. The potential energy functions Vl(r)and V2(r) used in these to an expression for Vl(r)of the form simulations are given by eq 16 with the parameters given in section V , ( t ) = A f (At/2){ln [C/I(r;X2)]/ln 2)'/2 (17) 1I.C. As indicated in Figure 1, and discussed previously by Fisher et al.,'7918optical excitation of ICN at XI = 306 nm is a classically where now f(t;X2) represents deconvolved FTS data points. forbidden process ( Vl(r)- tl = 7525 cm-' for r = re), so that initial Equation 11 permits Vl(r) to be conveniently transformed to Vl(r) motion of the wave packet along the dissociation coordinate is and vice versa. Application of eq 17 to yield a faithful repredominated by tunneling through the inner repulsive limb of the sentation potential energy function governing reaction from experimental measurements necessitates that various assumptions are made concerning the nature of the time development of the (1 15) Messiah, A. Quuntum Mechanics; North-Holland: Amsterdam, 1966; Vol. I, Chapter VI, pp 216-222. dissociative process. The inversion procedure outlined above (116) Zewail, A. H. J . Chem. Soc., Furuduy Truns. 2, submitted for required3 that the time evolution of the reacting system is governed publication. by the laws of classical mechanics so that the photofragmentation (117) Cong, P.; Herek, J. L.; Mohktari, A.; Zewail, A. H. Unpublished process can be viewed as the motion executed by a heavy particle results. (118) Janssen, M. H. M.; Bowman, R. M.; Zewail, A. H. Chem. Phys. traversing the controlling PES. In reality, the first (pump) ex-
+
Letr. 1990, 172, 99.
( 1 14) Moore, C. E. Atomic Energy Levels, NurI. Stund. Ref. Datu Ser. (US., Nurl. Bur. Srund.) 1971, 35, 106.
(119) Gruebele, M.; Roberts, G.; Dantus, M.; Bowman, R. M. Chem. Phys. b i t . 1990, 166. 459. (120) Bernstein, R. B.; Zewail, A. H. Chem. Phys. Len. 1990, 170, 321.
7980 The Journal of Physical Chemistry,Vol. 95, No. 21, 1991
Roberts and Zewail
I
I
1.0
0.1
~
IS
1-54 IS
1.150
1-300 I S
Is
0.a
h 0.4
0.2
2
4
6
e
10
12
14
rlh
Figure 2. [I.-CN]** wave packets. Time evolution of the wavefunction I q 1 ( t ) )over the PES V l ( r )of Figure 1 following excitation at AI = 306 nm by a Gaussian-shaped pump-laser pulse of fwhm = 125 fs centerd
at I = 0 fs.
t
/
0L-J
-100
1 '
I
I
I
0
100
I
I 200
I
300
t/fs
Figure 3. Time-dependent [I.-CN]** population on V l ( r ) . Plot of the population ( # , ( f ) l $ q ( f ) ) on the first excited PES Vl(r)of Figure 1 as a function of time following laser photolysis at XI = 306 nm with a Gaussian pump pulse of fwhm = 125 fs centered at t = 0 fs.
excited-state potential. Some 50 fs after the perturbation due to the pump laser begins, the expectation value (r)for I#l(t)) is (r) = ro = 2.9 A, where ro is the classical turning point on Vl(r)for this excitation energy (see section 1I.B). The wave function grows during the period (300 fs) over which the first laser pulse pumps more population to the excited state, achieving an approximately Gaussian shape at the end of this time. It then propagates out to larger values of r as time proceeds, retaining its shape with very little spreading for t 5 300 fs, at which time the wave packet samples regions of Vl(r)corresponding to an average separation of (r)= 10.7 A along the 1-CN interfragment coordinate, approximately 4 times longer than the ground-state equilibrium d i s t a n ~ e . ~Noteworthy ~~,'~ in our view is the fact that I#l(r)) begins to sample the asymptotic region of the PES, where Vl(r)is I50 cm-' above the dissociation limit, .4 after only some 50 fs, i.e. while the pump laser is still interacting with the system. By the time the perturbation due to the pump laser is complete at t = 150 fs, I#l(t)) is centered about an average position (r) = 6.7 A on the reaction path and extends over a range of radial coordinates Ar = 3.4 A (measured at fwhm). Up until this time, the wave packet experiences a large variation in potential energy over a small change in r (VI(r)/cm-l: = 13717 at r = -100 0 100 200 300 re = 2.145 A; = E,, = 6550 a t r = ro = 2.89 A; and = 1.73 X r/is at r = 6.7 A), giving rise to severe broadening of the abFigure 4. Transient behavior calculated by quantum dynamics. Graphs sorption spectrum. The greater part of the nuclear recoil process of the norm of the wavefunction 1&(1)) created on V,(r) by action of the leading to final product states, however, occurs over the long-range probe laser, representing the total fluorescence signal, versus pump-probe. part of the potential at distances r 1 6.7 A, and such motion takes time delay T for a pump wavelength of XI = 306 nm and various probe wavelengths (A,, nm): (i) 388.5; (ii) 388.9; (iii) 389.8; (iv) 390.4; (v) a much longer time to reach completion. Beswick and Jortner 391.4. Both transform-limited laser pulses have Gaussian temporal realized this point" and Fried and Mukamel have shown how the profiles of fwhm = 125 fs, and values of (+,(t)l$,(t)) are normalized with "tail" of the wave function can be monitored at long respect to the long-time asymptote of the on-resonance signal (i). ParAs noted in section 1I.B and shown in Figure 2, the finite pulse ametrized forms of V l ( r ) and Vz(r) are those given in the caption to width of the pump laser and its convolution with the early-time Figure 1. Note the different long-time values of (&(f)1&(f)) compared dynamics clearly play an important role in determining the initial to A ( T ) shown in Figure 5 (see text for discussion). This difference form of the wave packet I # , ( t ) ) on a purely repulsive potential but the reflects only the choice of the temporal width of E,@,) and l2(t2), such as Vl(r), and can be handled properly both theoretically and results to compare with experiments are not, of course, subject to this experimentally as described below (see also section IV). The choice. (See Figure 6 for inversion of FTS transients to the PES.) ~,~~ important point, however, is that a detection ~ i n d o wis~prothe FTS technique,l-II are displayed in Figure 4 for a pump jected onto the reactive potential by the probe laser (in this case wavelength of A I = 306 nm and five probe wavelengths, chosen of width A3 = 118 cm-I for a Gaussian pulse of fwhm = 125 fs in the time domain) that enables the evolution of the wave packet to match those employed in the previous experimental measureto be monitored in real time. ments reported by our group.5 The shapes of these curves have Displayed in Figure 3 is a graph of the excited-state population here we wish been discussed in detail elsewhere;2-11~42-44~51~58~~6263 ($l(t)l$l(t)) on the potential V,(r)as a function of time following to highlight only a few points. XI = 306 nm pumping. An initial rise in population is followed First, the calculated transients of Figure 4 are in general accord by a slight decrease to some steady-state level, indicative of with both the experimental measurements of Dantus, Rosker, and off-resonance excitation to the upper electronic state in relation Z e ~ a i l and ~ - ~with the results of earlier quantum-dynami~al~~*~*~I to the absorption maximum at X = 25 1.3 nm," as noted previously and c l a s ~ i c a l ~calculations. * ~ * ~ ~ ~ ~We note that the calculated by Williams and Imre.42 on-resonance signal, labeled (i) in Figure 4, reaches half-maximum Plots of (#z(t)1#2(t)) versus time delay T , representing the type intensity after approximately 60 fs. According to the classical of time-resolved laser-induced fluorescence spectra obtained by description of section II.B, such a time represents the period
'I
/
Femtosecond Real-Time Probing of Reactions required for the dissociating system to reach that point on the exponentially repulsive PES where Vl(t)= AP/2 cm-1,4,62previously de~ignated'-~.~-'~ as the clocking time 7l/2. As discussed in section 11I.D, 7 1 / 2 is the dissociation time for bond rupture within the framework of real-time measurements; as the spectral width of the probe-laser pulse is made increasingly narrow, the rise to the asymptotic value of an on-resonance FTS measurement becomes consequently sharper, cumulating in an instantaneous stegfunction response in the limit that A? tends to zero (&function frequency spectrum) for probe pulses of infinite duration in the time domain. At a time delay of 7 = 7112 = 60 fs, the expectation value ( r ) of the wave packet I#l(t)) on V l ( r )is (r) = 4.1 A, corresponding exactly with the classical value rm = 4.1 A given by eq 11. Calculations analogous to that displayed as Figure 4,curve (i), carried out for pump wavelengths XI = 295 and 285 nm, resulted in on-resonance transients characterized by values of 7l/2 H 57 and 55 fs, respectively, reflecting the influence of increasing recoil velocity for separation of products at shorter photolysis wavelengths on the calculated dissociation time (see eq 18 below). That values of 7I/2 decrease only by some 5 fs when the total available kinetic energy E,, for fragment separation is increased by some 2410 cm-I from E,, = 6550 cm-l (XI = 306 nm) to E,, = 8960 cm-l (XI = 285 nm) is merely a reflection of the small range parameter al-I = 0.193 A employed to construct the model PES Vl(r),which results in a large PE gradient dVl(r)/dr at distances smaller than about 3 A (and which determines the early time dynamical behavior of the reaction). The dissociation times reported above are much shorter than the values 7 1 / 2 = 205 f 303" and 160 f 305fs observed experimentally at pump wavelengths XI = 306 and 285 nm. This may be expected given the fact that the length parameter of our model V l ( r )potential is some 4 times shorter than the value aI-I = 0.8 A derived by Dantus et al. from their on-resonance "clocking" measurement^."^ Indeed, when the parameters of eq 16 for Vl(r) are changed to aI= 1.25 A-1 and VIo = 2.4272 X lo5cm-' (the latter chosen such that the classical turning point of the PES at an energy E,, above the dissociation limit for XI = 306 nm excitation is coincident with that for Vl(r)shown in Figure I), values of 7 , / 2 = 185, 175, and 165 fs are determined for pump wavelengths XI = 306, 295, and 285 nm, respectively. Taking due account of the different CN(X22+) product rotational energy distributions at different photolysis wavelength^,'^-^^-^^ the experimental values of 71/2 reported by Dantus et aLS at XI = 306 and 285 nm were found to scale with the same value of al-l = 0.8 A, indicating that fragment recoil velocity is the predominant factor in determining 71/2 rather than the competing effect of increased separation between ro and the optically coupled region provided by the probe laser when the pump energy is increased. These data were therefore considered5 to be consistent with the fact that (strong) interactions between the PES of s2 = O+ ()TI,+) symmetry leading to I(2Pl 2) atomic product and those of 51 = 0- (3r10-),1 (3111),and 2 (I TI2) symmetry giving rise to I(2P3/2) atoms occur at interfragment distances shorter than ro for XI = 285 nm excitation (where I(zP1/2)atoms are also energetically accessible), i.e. close to the Franck-Condon region. Such a conclusion would be in agreement with the model potential curves recently proposed by Black et aL31 as a result of their subDoppler LIF measurements at XI = 249 nm (though no 1.-CN distance scale is given in Figure 20 of that work3I). In contrast, recently reported ab initio com utationsS0 reveal the presence a t I - C N separations r = 3.32 of a conical intersection involving the PESs of R = O+ and R = 1 (In,) symmetry (the latter connecting with ground-state I(2P312)atoms), symmetry as discussed in section 11. The PES of R = O+ (%,+) has also been computed to possess a substantial attractive well some 3630 cm-I in depth with a minimum situated at about r = 3.22 Asso The region of such an intersection between surfaces could be traversed by [I-.CN]** wave packets prepared by pumplaser light at XI = 306, 295,and 285 nm, though on strictly energetic grounds, final I(2P1,2)+ CN(XZZ+)products can only be reached from u = 0 ground-state ICN by using pump wave-
K
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 7981 lengths of XI = 295 and 285 nm (see section 1I.C). At this point, we feel able only to recall that no direct evidence supporting the presence of one or more surface intersections at distances beyond the Franck-Condon region has been obtained from FTS meas u r e m e n t ~ ; ~delays - ~ in the appearance of the free C N product of reaction 1 b and/or resonance-type behavior (such as has been observed hitherto in the dissociation of NaI,98-102Hg12,103.1M and CH3112'for example) might be expecteds to result from reactive trajectories that sample those regions of configuration space where diabatic surfaces intersect en route to dissociation products. The implications of the possible location of a potential minimum along the dissociation coordinate in relation to FTS measurements have been discussed in detail elsewhere.63 The off-resonance fluorescence signals of Figure 4, denoted (ii)-(v), exhibit peak intensities that shift to shorter delay times 7 with increasing probe-laser wavelength X2. Within the framework of the classical model outlined in section II.B, this behavior has previously been interpreted2.4,5,7,11,62,63 in terms of detection of [I-CN] ** transition-state configurations across a range of coordinate positions on the reactive PES. Deconvolution of the combined pump- and probe-laser intensity profiles from the data shown in Figure 4 indicates that the time taken for the [I-CN] ** moiety to traverse the fwhm of the off-resonance spectra is about 5-1 5 fs, depending upon the off-resonance probe-laser frequency. Experimentally, [I.*CN] ** configurations have been reported to persist for approximately 20-50 fs.23495 In terms of the aforementioned classical description, Bersohn and Zewai162have shown that such "lifetimes" are related to the gradient dVl(r)/dr at the central point of probing r = r* on the dissociative PES (corresponding to maximum absorption of the probe pulse), and hence to the velocity of fragment separation at r*. The deconvolved fit to curve (v) of Figure 4 achieves maximum intensity at a time t* = 35 fs, corresponding tor* = 3.58 A, where V l ( t )= 185 cm-I (= A = 191 cm-', as expected6Z6'); it may be noted therefore that even the most off-resonant probe-laser frequency e m p l ~ y e d ~ - ~ interrogates the long-range tail of the reactive potential close to the dissociation limit (Vl(ro) = E," 6550 cm-' for XI = 306 nm). Figure 4 also shows a decreasing maximum signal intensity for the five transients as the probe-laser frequency is successively detuned to the red of the on-resonance wavelength, in agreement with the findings of other workers.2-5,42-44.5'~s*~' Williams and Imre42have invoked a curve-crossing mechanism, reminiscent of the well-known Landau-Zener model, to explain such an observation: at higher probe-laser frequencies, commensurate with long-time detection of [I-CN] ** at larger values of the reaction coordinate (see Figure l), the difference between the gradients dVl(r)/dr and dV2(r)/dr of the two PESs connected by the probe-laser pulse is less than at small values of r, thus enhancing transfer of population to the upper potential V2(r) from Vl(r)and giving rise to a more intense LIF signal. We find that maximum LIF intensity is determined by convolution of the time-dependent reaction dynamics with the temporal widths of the pump and probe lasers: for &function pulses, the peak intensity is identical for all transients, irrespective of probe-laser frequency, as has been noted e l ~ e w h e r e ; increasing ~ ~ . ~ ~ the temporal pulse widths of the two lasers up to fwhm = 125 fs both broadens and diminishes the maximum peak intensity of the calculated signals, the effect of which becomes more pronounced as the probe-laser frequency is detuned from the long-time resonance value. 2 4 5 1 1 1~52-63 B. Classical Simulations. Figure 5 depicts FTS spectra for reaction la calculated from eqs 14 and 15 for the same laser frequencies and pulse widths and model potential curves VI(?) and V2(r)as employed in the quantum-dynamical treatment of the previous section. Comparison of Figures 4 and 5 reveals the striking similarity between the quantum and classical calculations of FTS data for reaction la. The on-resonance transient labeled (i) in Figure 5 is characterized by a dissociation time of 71/2 = 44 fs, in reasonable accord with the value obtained from the quantum-dynamical calculation (see below). From eq 1 1 we may calculate that at a time t = 44 fs after impulsive bond cleavage, 9
(121) Dantus, M.; Janssen, M.
1
9
1
H.M.; Zewail, A. H. To be published.
Roberts and Zewail
1991
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Figure 5. Transient behavior calculated by classical mechanics. Graphs of probe laser absorption A(r)/arbitrary units versus pump-probe time delay T for a pump wavelength of AI = 306 nm and various probe wavelengths A? Times P for input into eq 11 are calculated on the basis of the distance traveled by a classical particle of reduced mass p on VI@) so as to attain maximum absorption A,,,&*) at the different values of A1: (i) f * = 300.0 (E O J ) fs (A, = 388.5 nm); (ii) t* = 49.5 fs (A, = 388.9 nm); (iii) t* = 40.9 fs (A, = 398.8 nm); (iv) f* = 38.2 fs (A, = 390.4 nm); (v) f* = 35.2 fs (A, = 391.4 nm). Both laser pulses are transform-limited Gaussians with fwhm = 125 fs (correspondingto spectral widths of fwhm = A3 = 118 cm-I), and values of A(r) are normalized with respect to the long-time asymptote of the on-resonance signal (i). Parametrized forms of V,(r)and V2(r)are those given in the caption to Figure 1. Note the different long-time values of A ( r ) compared to ($,(t)I$,(t)) shown in Figure 4 (see text for discussion).
the (classical) NC-01 bond distance is r, =I 3.8 A. Further onresonance simulations carried out at initial energies Epycorresponding to AI = 295 and 285 nm excitation yielded transient spectra characterized by dissociation times of 71/2 41 and 39 fs, respectively. That the classically determined values of r 1 are shorter than those arising from the quantumdynamical model of the preceding subsection by some 15 fs may be considered to result from the spreading of I $ , ( t ) ) during its preparation on the excited-state potential; we note that for XI = 306 nm and fwhm = 125 fs as before, ( r ) = 3.2 and 6.7 A at the temporal midpoint ( t = 0 fs) and end ( r = 150 fs) of the pumplaser pulse, respectively, whereas ro = 2.89 A marks the initial coordinate of a classical trajectory at the turning point of V , ( r )following Franck-Condon (vertical) absorption from the extended ground-state species by means of a &function laser pulse. It may be noted that the asymptotic values of ($2(t)l$2(t)) displayed in Figure 4 pertaining to off-resonance probe-laser frequencies (0.9349, 0.4789,0.2111, and 0.0265 for the curves labeled (ii)-(v), respectively) are larger than the corresponding data for A(T) shown in Figure 5 (0.8741, 0.2293, 0.0445, and 0.0007 for the transients denoted (ii)-(v)). As pointed out in section II.D, this simply reflects the squared dependence of 12(F2) on &(E2): the differences between the long-time signals calculated by quantum dynamics and classical mechanics a t a given probe-laser frequency are consistent with a spectral half-width for the electric vector E2(u2) (given by the asymptotes of ($2(r)1$2(t)))of the probe laser that is a factor of 4 2 higher than that for the corresponding frequency distribution of the probepulse intensity (determined by the long-time values of A(r)). The different asymptotic values of the graphs of Figures 4 and 5 naturally have an effect on the overall shapes of the calculated FTS transients (so that, for instance, the maxima of the offresonance transients of Figure 5 peak at earlier times than the analogous data shown in Figure 4). Figures 2,4, and 5 together therefore indicate that the longtime (t L 50 fs) dynamics of ICN dissociation according to reaction
-
l a can be treated satisfactorily by classical mechanics for the particular choice of temporal laser intensity profiles and potential functions V l ( r )and V2(r)used here.6z.63 This latter aspect is a reflection of Ehrenfest's theorem; for t 1 50 fs the derivative dV,(r)/dr of the potential function is negligible over the dimensions of the wave packet Iql(t)),implying that its time-evolution may be adequately described by Newton's Second Law.115For reaction la studied under the experimental conditions employed in the FTS measurements,2-5 it does not appear unreasonable to conclude therefore that the assumption of a one-to-one correspondence between the frequency of the probe-laser light and the distance apart of the separating photofragments represents an adequate working approximation. The similarity between FTS transients predicted by quantum dynamics and classical mechanics reflects the fact that in this case the intrinsic spreading of the wave packet I$l(t)) created on the reactive potential V l ( r )is negligible over the time scale for dissociation, such that the observed FTS behavior closely mimics the experimental response function (defined as the integral of the pumpprobe correlation function4). As noted the use of laser pulses with temporal widths 100-150 fs prevents an analysis of spreading of I$l(t)) on V , ( r )in this instance; investigations of the predissociation of NaI by FTS, on the other hand, which occurs over a longer (picosecond) time scale, permit the spreading of [Na-I]'* wave packets to be clearly res ~ l v e d . ~ ~ ~InJ ~section J ~ ' 1I.D it was pointed out that it is necessary to invoke both the above assumptions for successful application of the potential inversion method developed by Bernstein and Zewai16' and utilized below for analysis of the present data. C. Invelsion to the PES. Inversion of data of the type portrayed in Figures 4 and 5 to generate the radial potential V , ( r )may be carried out routinely using eqs 17 and 11 following deconvolution of the transform-limited pump- and probe-laser pulses. This is illustrated in Figure 6a,b, which displays selected data points (chosen to be approximately equidistant along the abscissa) representing the inverted potential obtained from simulated data of the type depicted in Figures 4 and 5 superimposed upon the potential function Vl(r)from which the FTS transients themselves were constructed, shown as a solid curve. Application of eq 17 to the time-resolved data calculated by quantum dynamics of course requires prior conversion of the frequency dependence of the electric vector E2(t2)to the spectral intensity distribution IZ(F2). The (expected) agreement between the inverted potential and V,(r) given by eq 16 is sufficiently satisfactory in our view so as to render curve fitting to the inverted data points unnecessary in this instance; both the quantum and classical simulations of reaction l a yield FTS spectra from which the (same) functional form of V , ( r ) may be readily recovered by using the method of section 1I.D. Figure 7 shows graphs of ( $ 2 ( t ) l $ 2 ( r ) ) against time delay for different probe spectral bandwidths A3 a t a single central wavelength A2 = 391.4 nm, and clearly illustrates the effect of varying the width of the optically coupled region provided by probe pulses of different temporal duration. In all cases, the frequency spread of the pump laser was kept constant at A t = 118 cm-I, corresponding to a pulse width in the time domain of 125 fs fwhm. Similar calculations were carried out for these values of A'v at probe-laser wavelengths A, identical with those employed in section 1II.A for probe pulses of fwhm = 125 fs temporal duration. Application of the potential inversion procedure outlined in section 1I.Dto data generated with different probe-laser bandwidths yields results identical with that displayed in Figure 6a. As a conclusion to this section, we illustrate the reconstruction of the dissociative potential energy curve from presently available experimental FTS data on ICN photodissociation.2d Figure 8 displays experimental transients for reaction l a obtained at five different probe laser wavelengths. If we assume that IVl(r) Vl(r*)l>> IV2(r)- V2(r*)land employ eq 11 to convert from interfragment separation r to dissociation time t, eq 8 may be readily rearranged to yield an expression for the temporal potential V l ( t )in terms of the deconvolved FTS data I ( t ; A 2 ) , the spectral ) t = f*. The bandwidth of the probe laser and the value of V I ( ?at
-
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 1983
Femtosecond Real-Time Probing of Reactions
300
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l
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Figure 7. Transient behavior calculated by quantum dynamics for different probe-laser pulse widths. Graphs of the probability density (+2( t ) l $ 2 ( t ) ) versus pump-probe time delay T as a function of the temporal duration of the probe-laser pulse.. Transient (ii) displayad here is identical with transient (v) shown in Figure 4; values of (+2(t)($2(t))for all curves are normalized with respect to the maximum peak intensity of transient (ii). Parametrized forms of Vl(r)and V,(r) are those given in the caption to Figure 1. Nominal (central) wavelengths are hi = 306 nm and A, = 391.4 nm in all cases. Both pump- and probe-laser pulses are Gaussian in shape; for the pump laser, the pulse width is fixed at fwhm = 125 fs (A2 = 118 cm-I). Probe pulse durations and spectral bandwidths are (i) fwhm = 259 fs (A3 4 9 cm-l), (ii) fwhm = 125 fs (A2 = 118 cm-l), (iii) fwhm = 62.5 fs (A2 = 236 cm-l), and (iv) fwhm = 12.5 fs (A3 = 1180 cm-I). TABLE I: Parameters Characterizing Different Potential Functions V&), Given by V , ( r ) = VI? exp(-alf) with j = a, b, or c, Employed To Investigate ETS Behavior for Different Exponential Forms of the Dissociative Potential ( V k ) = B2 - 8, = 24630 cm-') alj/A-' Vijo/cni1 3.5
4.5
4
5
4 Figure 6. Inversion to PES of FTS transients calculated by quantum dynamics and classical mechanics. Graphs of Vi@) against 1.-CN separation r obtained by inversion of simulated FTS transients in the form of plots of (a) ($2(r)l$2(r)) as a function of delay time T shown in Figure 4 and (b) A ( T ) as a function of delay time T shown in Figure 5. The points represent values of Vi@) determined by inverting FTS data at selectad values along the dissociation coordinate using eqs 14 and 15: (0, 0) X2 = 388.5 nm (A = 0 cm-l); (e,0)A, = 388.9 nm (A = 26 cm-l); (a0 ) X2 = 389.8 nm (A = 86 cm-I); (A,A) X2 = 390.4 nm (A = 125 cm-l); (+, 0 ) X2 = 391.4 nm (A = 191 cm-I). The solid line represents the functional form Vi(r). = Vio exp(-a,) relative to the dissociation limit pi, from which the transients shown in Figures 4 and 5 were calculated: Via = 2.061 x 10'0 cm-1; ai = 5.1813 A-I.
result of considerations of this type carried out by Dantus et al.5 is displayed in Figure 9, in which values of V l ( t )determined from transient (iii) of Figure 8 are shown as data points at selected values oft, while the potential function V l ( t )calculated from eqs 11 and 16 by assuming a Lorentzian distribution of probe-laser frequencies5 is represented by a solid curve. D. Sensitivity of Calculated FIS Data to the P a " d* PES. In this section, we consider the sensitivity of calculated FTS spectra, obtained from both classical mechanics and time-dependent quantum dynamics, to variation of the input values of q,V$, q k and v2ko (j = a, b, c; k = a, b) in the parametrized forms for Vi(r) and V2(r)given by eq 16. 1. Variation of Slmpe of Vl(r). For a given functional form of V2(r),here assumed independent of reaction coordinate (a2= = A-i), FTS transients were computed by the quantum and classical procedures outlined in sections 1I.A and 1I.B for three
Vdrl
v&j
VI&)
5.1813 1.2500 21.4161
2.0610 X 10'O 2.4212 X 10' 5.6239 X IO3O
dissociative potentials Vlj(r) (j = a, b, c), all of the general form given by eq 16. The various parameters cy1. and Vi? employed to construct the different Vlj(r) are listed in fable I, while Figure 10 illustrates the resulting potential curves. Vle(r)is identical with that employed in the calculations reported in sections 1II.A and 1II.B and comprises the parameters for the excited dissociative state of ICN leading to ground-state (CN(XZZ+) + I(*P3,J photofragments selected by Bersohn and Zewai162(see section 1I.C). The value of q b is that derived by Dantus et al.3-5 from investigations of the dissociation time of reaction l a by FTS (see section IILA), while alCwas chosen to be greater than a l eby a factor identical to that by which (Ylb is smaller than cyle. Vi: and Vl,O were determined such that the classical turning points of the three PESs occurred a t the same 1.-CN separation ro = 2.89 A for initial wave packet preparation at an energy Ea"above PI (using XI = 306 nm pump radiation). For all three potential functions Vlj(r) therefore, the wave packet initially created on the excited-state surface by the pump laser will achieve asymptotically the same group velocity, although of course the acceleration of the packets will differ in each case. Figure 1 la,b shows time-dependent LIF signals calculated by quantum dynamics and classical mechanics, respectively, for the potential energy curve designated Vlb(r); transients obtained for the potential VI&) are likewise displayed in Figure 12a,b. Analogous data for Vle(r)have been presented previously as Figures 4 and 5 . A number of points bear comparison with regard to the FTS transients presented in these diagrams. First, it may be noted that the transients calculated by classical mechanics resemble qualitatively those obtained from the full quantal
Roberts and Zewail
1984 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991
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Figure 8. Experimental transient behavior of ICN dissociation. Graphs of LIF signal intensity (arbitrary units) versus pumpprobe time delay 7 recorded at a pump wavelength of XI = 306 nm and various probe wavelengths (A2, nm): (i) 388.5;(ii) 388.9;(iii) 389.8;(iv) 390.4;(v) 391.4. Observed intensities are normalized with respect to the long-time asymptote of the on-resonance transient (i). The solid line through each set of data points represents the result of an optimized fit according to the classical model of section 1I.B convolved with the Lorentzian profile of the measured cross correlation of the pump- and probe-laser pulses (adapted from ref 5 ) .
3
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F w 9. Comparison of PES with result derived from FTS experiments. Graph of Vl(t)against classical dissociation time t obtained from experimental FTS data. The points represent values of V l ( t )as selected values oft determined from transient (iii) of Figure 8 (A, = 389.8nm (A = 86 cm-I)) following deconvolution of the experimental response function and assuming a Lorentzian profile for probe-pulse absorption with fwhm = 204 cm-I. The solid line represents the potential function Vl(t)= E scch2 (ora,/2) derived from eqs 13 and 15 with the value aI = 1 .I9051-lobtained from the optimized fit through the observed data shown as Figure 8(iii) (adapted from ref 5 ) . treatment, with best quantitative agreement obtained for the potential function Vlj(r) = Vlb(r) that is constructed from the smallest value of a I j(see below). As may be readily anticipated, it is found that as the value of aljis decreased (i.e., as the potential curve becomes less sharply repulsive), the maximum LIF intensity for a probe-laser pulse of given wavelength X2 shifts to longer delay times (compare especially Figure 11 with Figures 4 and 12), indicating that the region of Vlj(r) sampled by the probe-laser
3
3.5
r/ii
Figure 10. Potential energy curves relevant to examination of the variation of FTS signal intensities with modification of the shape of V l ( r ) .
Diagrams of excited-state potential functions relative to dissociation limit &, illustrating the different Vlj(r) employed to determine the effect of variation in shape of the reactive potential on calculated FTS data (shown in Figures 11 and 12). Potential parameters are taken from Table I.
pulse occurs at increasingly large 1.-CN distances; in terms of the classical mechanical description of section II.B, r* and t* increase with decreasing alj for a given off-resonance probe wavelength. The dissociation time r1/2for the on-resonance signals of Figures 11 and 12 also reflect the different gradients dVlj(r)/dr Over those regions of r where Vlj(r) changes significantly. The three onresonance transients calculated by perturbation theory, denoted (i) in Figures 4,1 la, and 12a, are characterized by values of r1/2 N 60, 185, and 25 fs; the corresponding classical data, shown in Figures 5 , 11b, and 12b, give rise to dissociation times of rl/z= 44, 181, and 11 fs. As alluded to above in general terms, it may be noted that the closest agreement between quantum and clas-
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 1985
Femtosecond Real-Time Probing of Reactions 1.2
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200
300
T/fS
Figure 11. Transient behavior calculated by quantum dynamics and classical mechanics for the set of potential functions shown in Figure lob. (a) Graphs of (+,(t)l+,(f)) versus pumpprobe time delay T for a pump wavelength of A, = 306 nm and various probe wavelengths (A,, nm): (i) 388.5;(ii) 388.9;(iii) 389.8; (iv) 390.4;(v) 391.4. (b) Graphs of probe-laser absorption A(s)/arbitrary units versus pumpprobe time delay T for a pump wavelength of AI = 306 nm and various probe wavelengths A,. Times t* for input into eq 1 1 are (i) t* = 500.0(E -) fs (A, = 388.5 nm), (ii) t* = 205.1 fs (A, = 388.9 nm), (iii) I* = 169.5 fs (A, = 398.8 nm), (iv) t* = 158.3 fs (X, = 390.4 nm), and (v) ?* = 145.6 fs (A, = 391.4 nm). Both pump. and probe-laser pulses have Gaussian temporal profiles of fwhm = 125 fs (corresponding to spectral widths of fwhm = A3 = 118 cm-' for transform-limited pulses), and values of ($,(t)I+,(f)) and A(T) are normalized with respect to the long-time asymptote of the on-resonance signals labeled (i).
Figure 12. Transient behavior calculated by quantum dynamics and classical mechanics for the set of potential functions shown in Figure 1Oc. (a) Graphs of (+,(t)(+,(t)) versus pumpprobe time delay 7 for a pump wavelength of AI = 306 nm and various probe wavelengths (A,, nm): (i) 388.5;(ii) 388.9;(5)389.8;(iv) 390.4;(v) 391.4. (b) Graphs of probe laser absorption A(r)/arbitrary units versus pumpprobe time delay T for a pump wavelength of AI = 306 nm and various probe wavelengths A2 Times t* for input into eq 11 are (i) t* = 300.0 (= -) fs (A, = 388.5 nm), (ii) t* = 11.9fs (A, = 388.9 nm), (iii) t* = 9.9fs (A, = 389.8nm), (iv) t* = 9.2fs (A, = 390.4nm), and (v) t* = 8.5 fs (A, = 391.4 nm). Both pump- and probe-laser pulses have Gaussian temporal profiles of fwhm = 125 fs (corresponding to spectral widths of fwhm = A3 = 118 cm-' for transform-limited pulses), and values of (+2(t)l+z(t))and A ( T ) are normalized with respect to the long-time asymptote of the on-resonance signals labeled (i).
sically calculated values of r1 is achieved for the dissociative potential Vlb(r)with the smaliest value of alj,since in this case the (long-time) dynamics of the wave packet prepared on the excited-state potential will most closely resemble the motion of a classical particle. (From an alternative perspective, the distance traveled by a classical particle within a time period t = 71/2 is significantly greater than I/alj,an assumption required for derivation of eq 18 below.) This variation of 71/2.may be placed on a more quantitative basis by means of the classical expression
probe-laser pulse where V,(r) = A?/2 cm-I. (From an experimental point of view, 7 l / 2 is measured as the time at which the on-resonance fluorescence signal attains 50% of its long-time inten~ity.)~~.'*~J~ From the values reported above for transients calculated by classical mechanics, values of aIj= 5.14, 1.25, and 20.57 A-1for j = a, b and c may be recovered, in overall agreement with the input parameters listed in Table I. The off-resonance transients of Figure 1la,b (labeled (ii)-(v), as before), when deconvolved from the temporal response of the combined effect of the pump- and probe-laser pulses, indicate that [I.-CN]'* species persist for times of the order of 18-52 fs measured at fwhm of the deconvolved signals; such lifetimes are in good agreement with the experimentally derived v a l ~ e s ~(see -~,~ section 1II.A). In contrast, [I-.CN] ** transition states propagating along the potential curve Vlc(r),giving rise to the transient behavior shown in Figure 12, are found to lead an extremely evanescent existence, lasting for durations less than approximately 6 fs as the dissociating intermediates traverse the fwhm of the
71/2=
(l/aljU) In [8Eav/A?'v]
(18)
previously obtained by Rosker et a1.: which relates the dissociation (or clocking) time to the length parameter I/crlj of the exponential PES Vlj(r). As noted in section M A , f i I 2 represents the time taken by a classical trajectory starting from initial coordinate r, on the potential energy curve V , ( r )to arrive a t that position within the detection window defined by an on-resonance
7986 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991
5001\ 6oo
t 4
i
400
(a)
25
-
-
V&>
200
1 00 0 1 4
'E
8
8
4
10
V n
12
-5
El
250
3
-
40 45
35 200
-
150
-
100
-
50
-
25
L\
1I-
4
0-5 3
3.1
3.2
3.3
3.4
3.5
3.8
r/A Figure 13. Inversion to PES of FTS transients calculated by quantum dynamics for the dissociative potentials Vi&) and Vi&). Graphs of Vij(r) against I-CN separation r obtained by inversion of simulated FTS transients in the form of plots of ($2(t)l$2(t))versus delay time T : (a) Vij(r) = Vlb(r);,(b)ylj(r) = VI&). The points represent values of Vl,(r) determined by inverting the FTS data shown in Figures 1la and 12a at selected values along the dissociation coordinate using eqs 14 and 15: (0) h2 = 388.5 nm (A = 0 cm-l); (e)X2 = 388.9 nm (A = 26 cm-'); (M)X2 = 389.8 nm (A = 86 cm-I); (A) X2 = 390.4 nm (A = 125 cm-'); (e) X2 = 391.4 nm (A = 191 cm-I). The solid line represents the functional from form Vi.@) = VI? exp(-alj) relative to the dissociation limit which the transients shown in Figures 1 la and 12a were calculated. probe-laser frequency spectrum en route to asymptotic product states. Once again, the longer lifetimes enjoyed by [I.-CN]'* in the former case simply reflects the shallower gradient of the potential compared to Vlc(r).The group velocity of the wave packet monitored by the laser pulse (dr(t)/dt = u[l Vlj(r)/EJl/2 in the classical mechanical description, where u is the terminal recoil velocity) at a particular wavelength h2 is identical in both cases, even though (r) (r*) is different, so that the evolving system samples transition-state configurations for a longer period of time a t larger I-CN distances when au is smallest; i.e., for a given off-resonance probe wavelength, the second laser pulse spans a larger optically coupled region connecting the two surfaces Vlj(r) and V2(r)for smaller values of Q1j.
Inversion of the FTS transients calculated by quantum dynamics shown in Figures l l a and 12a in a manner analogous to that carried out in section 1II.A for Vla(r) yields the results displayed in Figure 13, which shows as usual the inverted data points superimposed upon the potential functions VI&) and Vlc(r) from which the FTS signals were originally constructed. The same
2
3
4
5
6
7
8
r/A Figure 14. Potential energy curves relevant to examination of the variation of FTS signal intensities with modification of the shape of V2(+ Diagrams of excited-state potential functions relative to dissociation limit PI, illustrating the different V,(r) employed to determine the effect of variation in shape of the second-excited potential on calculated FTSdata (shown in Figures 15 and 16). Potential parameters are taken from Table 11.
procedure was also carried out for the classical results of Figures 1l b and 12b, from which and Vlc(r) were similarly obtained. That different probe-laser frequencies span the largest range of interfragment separations and are shifted to larger r values for the smallest value of au (= is clearly illustrated by comparison of Figure 13a with Figures 13b and 6; for Vlc(r)for example, it is only possible to recover the potential function over a range of I - C N distances approximately some 0.5 A wide. In all cases, however, the probe-laser pulse samples the long-range part of the dissociative potential. In addition, the more widely separated deconvolved FTS data (in terms of the times at which maximum signal is achieved at different probe wavelengths) obtained for VI&) permit a more precise determination of the potential than is possible for more steeply repulsive curves for which the peak intensities of the resulting transients are bunched closely together at short(er) times. Finally, Figures 6 and 13 clearly illustrate the point made by Bernstein and ZewaiP concerning the redundancy in the deduced V,.(r)data obtained from eq 17, which relies upon the spectral width of the probe-laser pulse. For the probe frequencies used here, broadly overlapping values of Vlj(r) can be determined in each case, though of course, the longest wavelength X2 extends the range of Vlj(r) values to shorter interfragment distances. 2. Variation of Shape of V 2 ( r ) .A study of the effects on the calculated FTS transients obtained by varying the nature of the
Femtosecond Real-Time Probing of Reactions
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 7987
TABLE II: Parameters Characterizing Different Potential Functions V&), Given by V , ( r ) = Vae exp(-aar) + (& - &) with k = a
or b, Employed To Investigate FlS Behavior for Different Exponential Forms of the Potential Accessed by the Probe-Laser Pulse ( V , ( r ) = VI&) = 2.4272 X 10, exp(-1,2S00r))
-I
// uppermost PES V2(r) accessed by the probe-laser pulse has also been carried out. Figure 14 shows the two sets of excited-state potential energy curves V l ( r )and V2k(r)(k = a, b) employed in this aspect of our investigations; Vza(r)displayed in Figure 14a is independent of reaction coordinate as before, while Figure 14b illustrates an exponentially repulsive form V2b(r). Table I1 lists the various parameters (Y2k and v 2 k 0 used to construct the two potential functions V2k(r). V l ( r )here is the same as VI&) of the preceding section, the parameters for which are given in Table I. The transient behavior resulting from the potential curves illustrated in Figure 14b (i.e., for exponentially repulsive V&)) is displayed in Figure 15a,b, which shows FTS signals calculated by means of the quantum and classical treatments of sections 1I.A and II.B, respectively. In the case where the second-excited potential is independent of dissociation coordinate ( V2a(r) = p2 - b , ) , analogous data are displayed in Figure 1 1. Comparison of either Figures 1 I a and 15a or Figures 11b and 15b reveals the consequences of adding a repulsive term to the upper excited potential energy curve; as v2k(r) is made increasingly repulsive (i.e., as u2 of eq 16 is decreased from ah = = A-' corresponding to a linear potential curve to a finite value of the order of (Y2b = (rib), a probe laser pulse of given off-resonance frequency samples regions of the reactive potential at ever decreasing interfragment separations, and hence also time delays; otherwise the shapes of the calculated FTS transients are broadly similar in shape for both potential energy curves V2k(f). Again, in the language of the classical model presented in section II.B, for a given off-resonance probe-laser wavelength, r* and t* decrease in value as the range parameter ff2k is reduced. The dissociation times that characterize the two on-resonant FTS spectra depicted in Figure 15a,b are 71/2 = 88 and 74 fs, respectively, considerably smaller than the results obtained for V&), where values of 71 185 and 181 fs were obtained from the quantum and classics[ treatments. When (Y2k = (Y2b # = ~ - 1 , eq 12 does not readily lend itself to development of a classical expression for 7l/2 in the same way that eq 14 leads to eq 18 in the limit that t =; nevertheless, the much smaller values of 71/2 determined in this case are consistent with a simple picture (namely the detection w i n d o ~ ~in~which , ~ ~ )the dissociative potential is probed by means of a vertical transition of wavelength x2 that connects V l ( r )and V2k(r) at ever shorter interfragment distances along the reaction coordinate as v2k(r) becomes increasingly repulsive. Deconvolution of the off-resonance transients shown in Figure 15 from the cross correlation of pump- and probe-laser pulses permits estimation of transition-state lifetimes for [I-CN] ** measured at fwhm of approximately 50-70 fs. That these times are longer than the values between 18 and 52 fs reported above for V,(r) is a manifestation of the dual effects of the larger optically coupled region at shorter internuclear separations generated by the probe laser when v2b(r) is no longer independent of reaction coordinate and the smaller local group velocity of the accelerating wave packet ( u ( t ) for a classical particle) interrogated at shorter average distances ( r ) (r* or t* corresponding to maximum absorption of a probe pulse of given frequency). Figure 16 displays the results of carrying out inversion to the potential V&) by means of the procedure outlined in section 1I.D on the calculated FTS data presented in Figure 15. In contrast to previous examples of the method given earlier in this paper, application of eq I7 in this case generates a difference in potentials IAV~I(~)I = lV2b(r) - Vlb(r)l, as pointed out in section 1I.D; in the standard manner adopted throughout this paper, this function is
-
"
12
,,,,
)
,
,
,
,
,
(
,
,
,
(
,
,,,, , , , ,
VI
2
E
7
-100
0
100
200
300
400
500
T/fS
Figure 15. Transient behavior calculated by quantum dynamics and classical mechanics for the set of potential functions shown in Figure 14b. (a) Graph of (J/,(t)(J/,(t))versus pumpprobe time delay T for a pump wavelength of AI = 306 nm and various probe wavelengths (A2, nm): (i) 388.5; (ii) 388.9; (iii) 389.8; (iv) 390.4; (v) 391.4. (b) Graphs of probe-laser absorption A(s)/arbitrary units versus pumpprobe time delay T for a pump wavelength of AI = 306 nm and various probe wavelengths A? Times t* for input into eq 11 are (i) I* = 500.0 (s 0 ) fs (Az = 388.5 nm), (ii) t* = 137.0 fs (A, = 388.9 nm), (iii) r* = 86.6 fs (A2 = 389.8 nm), (iv) t* = 65.8 fs (A2 = 390.4 nm), and (v) r* = 22.5 fs (A, = 391.4 nm). Both pump- and probe-laser pulses have Gaussian temporal profiles of fwhm = 125 fs (corresponding to spectral widths of fwhm = At = 118 cm-l for transform-limited pulses), and values of (J/,(r)1J/,(r)) and A ( T ) are normalized with respect to the long-time asymptote of the on-resonancesignals labeled (i).
shown as a solid line upon which are plotted the inverted data points derived from the FTS transients themselves. In the case when the second-excited potential V2(r)(= V&)) is independent of reaction coordinate, the result of inverting to the potential function VI&) has been presented earlier as Figure 13a. Owing to the broad spectral width of the probe-laser pulse in relation to the maximum detuning increment A = - 1, = 191 cm-l employed in this study, each of the transients shown in Figure 1 5 generates an essentially identical data set of lAV21(r)lyalues irrespective of probe-laser wavelength (we note that the maximum value of lAVzl(r)1, attained at r = I/((Ylb - (Y2b) In ((YlbVI!/ (YZbV2b0) = 2.6 A, is only about 206 cm-1): for presentation purposes, we have chosen to illustrate in Figure 16 the results of applying eq 17 to the calculated quantum and classical data obtained for A, = 389.8 nm, corresponding to a detuning increment of A = 86 cm-l with respect to the on-resonance probe wavelength AZm = 388.5 nm. For the same reason, a broadly similar range of coordinate positions along VI&) can be interrogated for both ij20D
7988 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991
Roberts and Zewail equated withf(r,t*) (and [AVzl(r)1withf(r,r*)), Le. thatf(t,t*) = k(A'v/2)[1n [C/f(t;X2)]/ln 2]Il2 (see eq 17). Since the method of carrying out the inversion to ~ A V z 1 ( rembodied )~ by eq 17 is strictly a classical procedure, it is not possible to obtain values of lAVzl(r)1from the FTS transients themselves at interfragment distances smaller than ro = 2.89 A, the initial coordinate (corresponding to t = 0) for classical absorption of pump-laser light at A, = 306 nm by ground-state ICN to the excited-state potential
th
Vl b(r).
I
I
I
1
I
200
150 4
' 0E 1
100
VtCff(r) = V , ( r )
W 4 4
d
The results of subsections III.D.l and III.D.2 show that changes to parameters in the exponential expressions for the potentials Vlj(r) and VZk(r) can result in considerable variation in the time dependence of calculated FTS spectra. It is clear that clocking of a direct photodissociation reaction in FTS experiments (Le., determination of r I l 2using probe-laser light that is on resonance with a free-fragment transition) is sensitive to the shape of both excited-state potential energy curves. E. Centrifugal Effects on Calculated FTS Data. Inclusion of an additional term in the dissociative PES V l ( r )describing the centrifugal b a r r i e P to reaction might be expected to give rise to dynamical effects that manifest themselves in the time domain in such a way as to be readily discernible from FTS spectra. In order to assess the possibilities from the classical perspective of section K C , a preliminary analysis of the variation of probe-pulse absorption along the reaction coordinate has been carried out for ICN dissociation via the less energetic reaction channel la leading to ground-state CN(X2Z+) + I(2P3,z) products. To take account of the centrifugal effect in the potential function controlling reaction, eq 16 is recast in the formlzZ
50
1
0
I 100
I
I 200
I
I 300
I
400
t/fs Figure 16. Inversion to potential difference of FTS transients calculated by quantum dynamics and classical mechanics. (a) Graph of lAV21(r)l against reaction coordinate r obtained by inversion of FTSdata simulated by quantum dynamics. (b) Graph of lAVzl(f)1against classical dissociation time f obtained by inversion of FTS data simulated by classical mechanics. The points represent values of lAVzl(r)1 and lAV21(f)lat selected values of r and f determined by inverting the transients labeled (iii) of Figure 15a,b using eqs 14 and 15: h2 = 398.8 nm (A = 86 cm-I). The solid curves shown in both graphs represent the functional forms lAV2](r)1= mr,r*)l and ~ A V z I ( f=) ~mr,t*)l given by eq 13, from which transients (iii) of Figure 15a,b were calculated.
pairs of potentials Vlb(r)/VZa(r)and VIb(r)/Vzb(r) even though the times r* and distances r* for maximum absorption of the probe pulse at a given frequency are shifted to smaller values in the latter case. (For example, for A = 86 cm-I: t* = 169 fs and r* = 6.4 A when 1/Zk(r) = Vz,(r);t* = 87 fs and r* = 4.2 A when V2k(r) = VZb(r).) Figure 16a displays the result of inverting to the potential difference lAVz1(r)l from the calculated timedependent FIX signal labeled (iii) in Figure 15a, where the known function IAV21(r)l is plotted against dissociation coordinate as before. The result of inverting the corresponding classical data of Figure 15b is shown in Figure 16b, in which by way of contrast lAVzl(r)l is recast as a function of the time t that measures the evolution of a classical trajectory over VI&) to give lAv,,(t)l (see section 1I.B). Values of lAVzl(r)l and lAVzl(r)lobtained from FTS data displayed in both Figures 16a and 16b are identical, however. We note that the inversion procedure of Bernstein and ZewaiP3 requires that lAV2I(t)l be related to the observed FTS signal by an expression of the form f ( t ; A z ) = Cexd- (In 2)[Vz(f) - Vl(t)+ A]2/(AZ/2)2); ) ~ therefore be comparison with eq 12 indicates that ~ A V z l ( tmay
+ 1(1 + l)h2/2pr2
(19)
where I is the quantum number corresponding to the orbital angular momentum L of relative motion during the half-collision, and the radial or central potential term VI@) = VIo exp(-qr) + &, as before. Figure 17a,b display two potential energy curves Vleff(r)relative to the dissociation limit PI = 0 for values of the orbital angular momentum quantum number I = 50 and 200. In each case Vlcff(r)= V l ( r )for 1 = 0 is shown for comparison, as is the centrifugal term 1(1+ l)h2/2p9. The parameters VIoand a1required for calculation of VI(?) according to eq 16 are those = employed in section I1.D for V , ( r ) = V I b ( f ) ;Le., VIo = 2.4272 X 10' cm-' and a I = a l b = 1.2500 A-'. Immediate inspection of these two diagrams reveals how the PES is shifted to longer distances along the radial direction with increasing centrifugal force. For the different potential energy curves depicted in Figure 17, the dependence of probe-pulse absorption A ( r ) on the distance apart of the recoiling photofragments has been calculated using eq 8. The pump-laser wavelength is fixed at XI = 306 nm in all cases, giving rise to different initial coordinates ro for classical motion over each of the potential energy curves. Displayed in Figure 18 are the resulting graphs of absorption A ( r ) versus interfragment distance for the same on-resonance frequency and four off-resonance frequencies employed throughout section 111. It should be pointed out that since these data are not displayed in the time domain, convolution of the dynamics with the finite temporal widths of the pump- and probe-laser pulses has not been taken into account: the graphs of Figure 18 represent 'deconvolved transients" of probe-laser absorption as a function of 1.-CN separation. A forthcoming pub1icationlz3will include integration of the classical equations of motion for reaction over PESs of the form given by eq 19 in order to predict the expected FTS behavior as a function of time and thus to offer a more valid comparison with experiment. Nevertheless, some interesting points do emerge from the preliminary analysis carried out so far. When centrifugal effects are neglected, the resulting absorption A ( r ) as a function of radial distance r corresponds exactly to that (122) Levine, R. D.; Bernstein, R. B. Molecular Reocrion Dynamics and Chemical Reoctioity; Oxford University Press: New York, 1987; Chapter 2,
pp 43, 44, 58. (123) Roberts, G . ; Zewail, A. H. To be published.
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 7989
Femtosecond Real-Time Probing of Reactions 25
I
I
I
I
12
I
t
2o
b
1
8-
E-
4-
5 2-
n
I
2 1.2
4
8
4
-5 2
3
4
5
8
7
6
E
IO
12
I
12
18
20
32
40
8
r/A Figure 17. Effective potential energy curves consisting of a central potential and centrifugal barrier. Diagrams of the excited-state potential function Vleff(r)(-) relative to dissociation limit PI, showing the inclusion of a centrifugal barrier term l(1 + l)h2/2jd (-.-) relative to the radial potential V , ( r ) (--) for different values of the orbital angular momentum quantum number I: (a) I = 50; (b) I = 200.
expected from the temporal absorption A(?)discussed previously in section 1II.B. Figure 18a shows plots of A ( r ) versus reaction coordinate for zero orbital angular momentum; when transformed into the time domain using eq 11 and convolved with the temporal widths of Gaussian-shaped p u m p and probe-laser pulses of fwhm = 125 fs using eq 15, these data yield FTS spectra that are identical with those displayed in Figure 5 . Presented in Figure 18b,c are analogous graphs of probe-pulse absorption versus interfragment separation for values of I = 50 and 200, respectively. Comparison of any two plots displayed in these figures for a given probe-laser wavelength, together with the corresponding data shown in Figure 18a, reveals that as orbital angular momentum I is increased, the onset of absorption commences at larger values of interfragment separation (and hence longer delay times), as may be anticipated from the form of the potential energy curves shown in Figure 17. At a given photolysis wavelength therefore, the dissociation times resulting from the on-resonance curves denoted (i) in Figure 18 would accordingly be expected to increase as the centrifugal contribution becomes larger. Similarly, the delay times at which the off-resonance curves labeled (ii)-(v) attain maximum intensity would also be expected to increase as orbital angular momentum is increased. Figure 18 further indicates that the peak widths of off-resonance data become correspondingly broader as the centrifugal effect becomes increasingly predominant. This too is readily explainable in terms
e
24
Figure 18. Coordinate-dependentprobe-laser absorption predicted for dissociation over the effective potential functions displayed in Figure 17. Graphs of probe-laser absorption &)/arbitrary units as a function of 1.-CN separation r along the dissociation coordinate of the potential energy curves shown in Figure 17 for a pump wavelength of XI = 306 nm and various probe wavelengths (A?,, nm): (i) 388.5; (ii) 388.9; (iii) 389.8; (iv) 390.4; (v) 391.4. Distances r* (A) for input into eq 8 are as
follows. (a) I = 0 (V,C"(r)= VI(?)):(i) 15.250 (E m); (ii) 7.315; (iii) 6.355; (iv) 6.055; (v) 5.720. (b) I = 5 0 (i) 40.000 (E m); (ii) 9.200; (iii) 6.890; (iv) 6.445; (v) 5.990. (c) I = 200 (i) 40.000 (E m); (ii) 34.745; (iii) 19.105; (iv) 15.845; (v) 12.820. The probe-laser pulse is Gaussian of spectral width A I (fwhm) = 118 cm-' (correspondingto a temporal profile of fwhm = 125 fs for transform-limited pulses).
of the potential energy curves displayed in Figure 17: as the centrifugal term of eq 19 is increased, the repulsive short-range becomes less severe-that is to say, part of the potential VICff(r)
7990 The Journal
of Physical Chemistry, Vola 95, No,21, 19191
the decrease in potential energy to the dissociation limit extends over a larger distance as orbital angular momentum is increased-with the result that the absorption A(r) extends over a larger range of coordinate positions for a probe-laser pulse of given spectral width AF. As a broad conclusion to the initial study of centrifugal effects on the reactive potential presented via Figure 18, it may noted that FTS measurements of the real-time dynamics of dissociation reactions afford the opportunity of elucidating the effective potential V l e r ( r )from measurements of the dissociation times associated with final product formation and the lifetimes of transition-state configurations, etc. Such experiments should further permit the separation of the centrifugal term, thus enabling the form of the resultant radial potential VI(?) to be ascertained, provided that the orbital angular momentum associated with a given half-callision process can be determined. For dissociation of ICN according to reaction 1, this can be achievedF7 by monitoring production of CN radicals in specific rotational states (though the broad spectral bandwidth of femtosecond pulses in effect means that only a distribution of closely-spaced rotational levels can be probed at a given frequency), which may be identified with the orbital angular momentum of the reaction as outlined below. Comparison of the C N rotational-state distributions of reactions la and l b determined by experiments carried out with a supersonic molecular beam source,19~20~27 characterized by a rotational temperature for 1CN of a few kelvins, with analogous room-temperature measurements where the parent molecule is in thermal equilibrium,'2-20,26,2g,3t,32 indicate that initial parent rotation is of relatively minor importance in determining the subsequent dissociation dynamics that lead to substantial rotational excitation of the C N photoproduct of reaction 1a.17,39,47,J2~53,56,57 Following the line of reasoning presented by Hall et a1.,26it is assumed that the initial rotational angular momentum JlCN of the parent species is zero before the half-collision (Le., that ICN resides solely in its ground rotational level). Conservation of angular momentum during the reactive process requires that JICN
=0=L
+ J C N + JI
(20)
where L denotes the orbital angular momentum of the half-collision, JCN = N f 1/2 is the total angular momentum of the CN radical (excluding nuclear spin) and Jt = 3/2 or 1/2 represents the total angular momentum of the I-atom fragment. For large values of N, the approximation JCN = N and JCN, L >> Jt can be made,26 in which case eq 20 collapses to the simple equality
L = -N
(21)
The vector describing rotational angular momentum of the C N product is therefore equal in magnitude but opposite in direction to that of the orbital angular momentum of relative motion during the dissociation. Measurements of the C N product in different rotational quantum states thus constitute a direct handle on the orbital angular momentum of the half-collision via the equality (21) and, since 1(1+ 1)t/2h= pub,122the impact parameter b also. Following ICN excitation at XI = 306 nm (where only ground-state I(2P3/2)atoms can be produced) and 285 nm (where both exit channels ( l a ) and (1b) are energetically accessible), Dantus et aL5 have determined dissociation times T , / ~for reaction 1 over a range of probe-laser wavelengths, corresponding to formation of C N radicals in different rotational states. Probing at the P-branch band head of the (0,O)band of the CN B2Z+ X22+ transition at X2 = 388.5 nm, where C N in rotational quantum levels N = 25 is monitored, these workers obtained dissociation times in the ranges T ~ =/ 175-205 ~ and 145-160 fs for pump wavelengths of XI = 306 and 285 nm, respectively (with a standard error of about 30 fs associated with each measurement); near the onset of the P-branch lines at X2 = 387.5 nm, however, commensurate with lower product rotational energies (N I 5), 1 were ~ much shorter than those for N = 25 were values of ~ 'that determined. Qualitatively therefore, it may be noted that the variation of dissociation time with product rotational state is consistent with the discussion presented above, though the observed
-
Roberts and Zewail increase in T t / 2 with rotational energy is remarkably large for a change in orbital angular momentum quantum number from I = 5 to = 25. It is stressed that a detailed quantitative appraisal of these data, together with the elucidation of VISff(r)and Vl(r) by means of 19, has yet to be carried out and awaits the completion of our classical calculations.t23 As noted in section I, Zewail has previously considered the variation of ~ ~ with / 2 probe-laser frequency in terms of initial bending motion of the ICN precursor and subsequent motion of [ I 4 N ] * * intermediates over an angle-dependent PES,in addition to possible centrifugal effect^.^
IV. Dissociation of Biz: Application to a Heavy-Particle System
In order to test the theoretical approaches of section I1 on a system of kinematic characteristics grossly dissimilar to that of reaction 1, both quantum dynamical and classical treatments described therein have recently7s been applied to the ultraviolet photofragmentation of diatomic bismuth, which is also considered to proceed over PESs that are repulsive in character at distances greater than the Franck-Condon Following absorption of a femtosecond laser pulse a t A, = 308 nm, the dissociation reaction results in production of ground-state Bi('S03/2)atoms and the two spin-orbit components of the first-excited 2DoJ term: Bi2(X'Z+*)
-
[Bi-.Bi]'*
-
Bi(6p3 ?30312)
+ Bi(6p3 2D03/2) (224
+
Bi(6p3 4S03/2) Bi(6p3 2DoS,,) (22b) The first subpicosecond real-time investigationsof reaction 22 were carried out by Sorokin and co-workers, who recorded broadband frequency-resolved absorption spectra in the vicinity of atomic resonance lines originating from the 2D03 and 2D05/2levels as a function of pump-probe time delay T . The ~results ~of their~ investigations led these workerst2' to propose an exponential dependence on internuclear separation (of the form given by eq 16) for two noninteracting potential energy curves Vl(r)and VI@) that govern the dynamics of reactions 22a and 22b. Application of the FTS technique to the investigation of Biz dissociation has resulted in the acquisition of information about the reaction dynamics over a range of internuclear coordinates on the two PESs controlling reaction via both exit channels.75 In contrast to the experimental scheme for probing the ICN dissociation reaction illustrated in Figure 1, a probe-laser pulse of given wavelength that monitors the evolution of reaction 22 can result in the population of several closely-spaced high-lying PESs V2(r) that correlate with high-lying electronic levels of atomic Bi. Owing to the coincidence of the probe-laser frequency with energy s e p arations within the electronic structure of Bi atoms and between the perturbed levels of the [Bi-Bi] ** moiety, it is therefore possible to interrogate the dynamics of reactions 22a and 22b individually at both final-product and transition-state configurations on the dissociative PESs V l ( r )and VI@) by monitoring the transient behavior at different fluorescence wavelengths Xdet.'lS To illustrate the nature of the data7s that can be obtained from FTS measurements in this case, just two examples of experimental transients that probe the dynamics of dissociating Bi molecules over the lower-lying PES Vt(r)leading to %'3/2 2Di3product levels are displayed in Figure 19a. Further examples oiFI'S data pertaining to this and the higher-lying exit channel (22b), together with details of their interpretation, are given in the recent publication of Bowman et aL7$ Both transients shown in Figure 19a were recorded at a single probe wavelength of X2 = 298.9 nm (and pumplaser wavelength of XI = 308 nm): the uppermost transient [Bi--Bi]**
+
(124) Glownia, J. H.;Misewich, J. A.; Sorokin, P. P. J. Chem. Phys. 1990, 92, 3335.
(125) Misewich, J . A.; Glownia, J. H.; Walkup, R. E.; Sorokin, P. P. In LIIrrafasr Phenomena VI& Harris,C. B., Ippcn, E. P., Mourou, G. A., %wail, A. H.,Eds.; Springer Series in Chemical Physics; Springer Verlag: Berlin,
1990; Vol. 53, pp 426-428. (126) Walkup, R. E.; Miscwich, J. A.; Glownia, J. H.; Sorokin, P. P. Phys. Rev. Lett. 1990, 65, 2366.
~
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 7991
Femtosecond Real-Time Probing of Reactions
i
L
I
I
nmr Delay
I
I
290
20
3.0
4.0
5.0
310
300
Wavelengthlnm
6.0
Figure 20. Frequency-resolved spectra of atomic Bi at different p u m p
rih
Figure 19. Experimental transients and correspondingexcited-state potential energy curves appropriate to experiments that monitor the timeevolution of Bi2 dissociation over Vi(r). (a) Graphs of LIF signal intensity (arbitrary units) versus pumpprobe time delay T recorded at pump and probe wavelengths of XI = 308 nm and X2 = 298.9 nm: the upper transient depicts off-resonancebehavior monitored at X , = 289.8 the lower transient shows on-resonance nm (6p27s2Plj2 6p3 2D03,2); behavior monitored at be,= 298.9 nm (6p27s'P312 6p3'D03/2).(b) Schematic diagram of excited-state potential energy curves at energies E S 50 X IO3 cm-I above the 'SO312 ground atomic level that may be invoked to explain how both on- and off-resonanceFTS data for reaction 22 can be obtained by using a single probe-laser wavelength. Potential parameters are given in ref 75 (adapted from ref 75).
-
probe delay times. Spectra of dispersed fluorescenceintensity (arbitrary units) against detection wavelength belarising from excitation of Bi vapor at XI = 308 nm and X2 = 298.9 nm at two pumpprobe delay times: (a) T < 250 fs; (b) T > 1000 fs (adapted from ref 75).
-
-
of Figure 19a, recorded a t bet= 289.8 nm (6p27s 2PI/2 6p3 2D03/2),exhibits off-resonance behavior characterized by a rise and decay in the observed LIF signal; displayed in the lower portion are typical on-resonance data monitored at ,A = 298.9 nm (6p27s 4P3/2 6p3 2D03/2). The origin of the different time dependences of these two transients may be explained in terms of a simple model that involves two high-lying surfaces V2(r)of Bi2populated by the probe laser.7s Figure 19b represents a potential energy diagram, similar in spirit to Figures 1, 10, and 14, which may be invoked to interpret the real-time data displayed in Figure 19a; shown are the dissociative potential V l ( r )and two V2(r)curves correlating with the 6p27s 2P1/2and 6$7s 4P3/2 6p3 5'312 asymptotic atomic levels, respectively. As the wave packet prepared by the pump pulse propagates over V,(r)toward 6p3 'DO312 6p3 4S03/2 dissociation products, the probe-laser energy may become resonant with transitions to one of the two surfaces Vz(r)at two distinct internuclear configurations: once in the transition-state region corresponding to early-time detection of [Bi-.Bi] ** species; and also at larger separations, resulting in long-time detection of final Bi(2D03/2)product. Figure 20a,b illustrates the early ( T < 250 fs) and long-time (T > lo00 fs) dynamics of reaction 22 in the spectral regime and shows a region of the ultraviolet emission spectrum from Xdet = 280 to 329 nm recorded at two different time delays when Bi vapor is irradiated by pump- and probe-laser pulses a t wavelengths XI = 308 nm and X2 = 298.9 Both spectra exhibit monochromator-resolution-limited peaks corresponding to atomic fluorescence and broader maxima resulting from scattered laser light. Figure 20a indicates that, a t early times, four transitions of atomic Bi can be detected at the following wavelengths bet: 290 nm (6p27s 2P1/2 6p3 2D032); 294 nm (6p27s 'P3/2 -.6p3 2D05/2);303 nm (6p27s 4P5/z bp3 2D05/2);307 nm (6p27s 4P1/2
-
+
+
-
-.
.e h cl
t=800fs
Y
c
t=1200fs
t=l800fs
t=2400fs
.4
.2
0
0
5
10
?A:
20
25
30
Figure 21. (Bi-Bi] ** wave packets. Time-evolution of the wavefunction II)~(~)) prepared on the dissociative PES Vi(r) by optical excitation at XI = 308 nm with a Gaussian-shaped pump-laser pulse of fwhm = 100 fs centered at t = 0 fs (adapted from ref 75).
-
-. 6p3 4S03/2). As shown in Figure 20b, an additional peak corresponding to the 6p27s 2P1/2 6p3 ZD03/2atomic transition on-resonance with the probe laser at X, = X2 = 298.9 nm becomes evident at later times, while the remaining atomic transitions have diminished in intensity. Thus at pumpprobe delay times T < 1000 fs, while Bi2 has not yet completely dissociated, there is no evidence of the unperturbed Bi atom transition, but several other emission lines arising from [ Bi-Bi] ** transition-state confrgurations at various stages along the reaction paths leading to final product states are discernible. For time delays T > 1000 fs, the separation between constituent atoms of the [Bi-Bi] ** moiety is sufficiently great such that any interaction between them is negligible, resulting in essentially free-fragment fluorescence from atomic Bi. Figure 21 is analogous to Figure 2 and shows snapshots of the wave function IJll(t))propagating over Vl(r)as a function of time
Roberts and Zewail
7992 The Journal of Physical Chemistry, Vol. 95, No. 21 1991 I
I2
,
-
,
,
,
~
1
,
,
,
~
1
,
,
,
(
,
,
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-
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1000
500
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2000
2500
-
d 3000
-
Figure 22. Transient behavior of [Bi-Bi]'* calculated by quantum dynamics. Graphs of (+2(t)1+2(r)) as a function of pumpprobe time delay 7 for a pump wavelength of A, = 308 nm and various probe wavelengths (A2, nm): (a) 298.9 (6p27s'Pj/2 6p3'Do,/,); (b) 302.5 (6p27s'P5 6p3W 5 / 2 ) ; (c) 298.9 ( V2(r)corresponds to the excitation energy of Bi(6p27s'PJI2));(d) 298.9 (V2(r)corresponds to the excitation enegy of Bi(6p/7s 'P312)). Both transform-limited laser pulses have Gaussian temporal profiles of fwhm = 100 fs, and values of (+2(r)1+2(r)) are normalized with respect to the longtime asymptote of Figure 22b. Functional forms of the potentials V , ( r ) , Vlr(r),and V2(r)are given in ref 75 (adapted from ref 75).
after optical excitation of Bi2 by a pump-laser pulse at AI = 308 nm centered at t = 0. In a manner analogous to that for dissociation of ICN, the wave function initially spreads throughout the duration of the laser pulse as more population is transferred to the upper-state potential but thereafter remains highly localized, with only minimal spreading, for times as long as 2400 fs.75 At this time, Iltl(t)) samples regions of the reaction coordinate corresponding to an average Bi-Bi separation of ( r ) = 26.0 A, nearly 10 times longer than the equilibrium bond length of re = 2.66 For reaction 22a, the wave packet may be considered to have reached the asymptotic region of the dissociative PES, where Vl(r)is only some 50 cm-* above the dissociation limit, after 1200 fs. Similar behavior has also been calculated for the wavefunction IJ.l,(t)) initially prepared on the surface Vlt(r)leading to spin-orbit-excited Bi(6p3 2D05/2)atoms, though in this case the group velocity of the time-evolving wave packet is much smaller, with expectation values of ( r ) = 12.9 8, after 2400 fs and about 15.1 A at t = 3000 f ~ . ' ~ Graphs of ( J.2(t)l$2(r)),representing observed LIF signals calculated by the quantum-dynamical model of section I1.B as before, are displayed in Figure 22 for reaction according to both equations 22a and 22b. Pump- and probe-laser frequencies were chosen to match those employed e~perimentally.~~ Figure 22a,b depicts on-resonance behavior corresponding to detection of Bi and 6p3 2D05/2spin-orbit levels at long atoms in their 6p3 2D03/2 times following dissociation via exit channels 22a and 22b. These plots are characterized by dissociation times r I l 2of = 990 and 1480 fs, respectively, in reasonable quantitative agreement with ~ f 50 the corresponding experimental estimates of T ~ / 1090 and 1500 f 50 fs.75 At delay times T = 1090 and 1500 fs, the
-
average positions of the wave packets and on the potentials Vl(r)and Vl,(r)are ( r ) = 11.6 and 7.5 A, corresponding closely to the classically determined values of rIw = 1 1.7 A and rlSm= 7.5 8, obtained from eq 11 with range parameters aI-I = ( Y 1 t - l = 2.0 A.124 Figure 22c,d displays off-resonance transients for the two exit channels leading to the 2DoJterm. These data were computed for a probe-laser wavelength of A2 = 298.9 nm detuned by 3625 and 577 cm-I, respectively, from the appropriate resonance transitions for free atomic absorption (A2 = 269.7 (6p27s 4P5/2 6p3 'D03/2) and 293.8 nm (6p27s 2P,/2 6p3 2D05/2)), and correspond to early-time detection of [Bi--Bi]'* wave packets a t transition-state regions of configuration space.75 Both transients show a rise and decay in signal intensity as the wave packets on Vl(r)and VI@) pass through the optically coupled region defined by the spectrum of the probe-laser pulse, with maxima located at delay times T of approximately 220 and 670 fs. At these delay are times, the average positions sampled by I G l ( t ) ) and ( r ) = 0.34 and 3.9 A along the dissociative potentials V l ( r )and V ] , ( r ) .A comparison of these results with the appropriate FTS data and their interpretation in terms of the topological characteristics of the reactive PESs has been given previ~usly;'~here we simply note that such data are in broad qualitative, though not quantitative, agreement with experimental results. The difference between calculated transients and those recorded by FTS experiments has been a t t r i b ~ t e dto~a~ lack of sophistication in the selection of model potentials V l ( r ) and VI,(?), which were ' ~ obey ~ the simple originally chosen by Sorokin and c o - ~ o r k e r s to exponential dependence on interfragment distance given by eq 16 such that the long-time (asymptotic) behavior predicted from
-
+
Femtosecond Real-Time Probing of Reactions trajectory calculations was in approximate quantitative accord with experiment. The overall shapes of the on- and off-resonance transients for reaction 22 obtained both experimentally and theoretically are qualitatively similar in form to analogous data for the dissociation of ICN given in section 111: this fact in itself suggests that fragmentation of Biz via the two exit channels giving rise to the ground-state term and zD03 and *Do,lz products proceeds over PESs that are predominant& repulsive in character at distances beyond the Franck-Condon region. It may be noted that values of the dissociation times T ~ for/ reactions ~ 22a and 22b are approximately 5-7 times longer than those computed and determined experimentally for ICN dissociation and that the off-resonance transients are correspondingly broader in the former case. These effects may readily be anticipated on simple kinematic and dynamical grounds (assuming little internal excitation of the I and CN products of reaction I); the reduced mass of Biz is p = 104.490 a.u compared to p = 21.591 a.u. for ICN while the translational energies available to dissociation products are E,, rc: 6550 and 1053 cm-' for reactions la and Ib (initiated by photolysis radiation at XI = 306 nm) and E,, 5102 and 1083 cm-' for reactions 22a and 22b (with XI = 308 nm), coupled with the assumed functional forms of the reactive potentials in both
V. Summary and Conclusions The femtosecond dynamics of systems involving repulsive,'d*7s,94*'03Jw b o ~ n d , 9 ~ ~and * " *q u a s i b o ~ n d ~ *PESs - ' ~ ~ have now been studied in a number of specific cases. Each of these potentials give rise to dynamical effects that are characterized in real time by the following phenomena: localization of the initial wave packet on the reactive PES prepared by the pumplaser pulse; subsequent spreading and decreasing amplitude of the wave packet as a function of time; and, for bound and quasi-bound PESs, oscillatory motion. In the bound and quasibound systems studied thus far94-1a'18the wave function is highly localized (of angstrom or subangstrom width) and spreading of the time-evolving wave packet is quite slow; dephasing and quantum rephasing can be observed on the picosecond time scale, while the nuclear dynamics occurs over periods of femtoseconds. For dissociative processes that take place over purely repulsive PESs such as the fragmentation of ICN reported in this paper, and also the dissociation of Biz,'S the initial wave packet samples a larger region of configuration space; not because of spreading effects inherent in the time-evolution of the wave but merely because of the finite pulse width of the pump-laser pulse and the very nature of the dissociative potential controlling reaction. From an experimental perspective, this entails the deconvolution of the temporal response function of the pump- and probe-laser pulses in order to determine the time constants of the nuclear separation dynamics, as explained elsewhere.4J Theoretically, the preparation and probing effects of the laser pulses are explicitly taken into account in solving the Schrodinger equation (section 1I.B) or the classical equations of motion (section 1I.C) for the reaction dynamics. In this article, quantum- and classical-mechanical calculations of the time-dependent dynamics of the dissociation of ICN have been presented and compared with experimental measurements obtained by the FTS technique; a previous study7s of the atomization of Biz has also been summarized and appropriate comparisons made. We consider that our calculated FTS transients are in reasonable qualitative agreement with both experimental data2-s and previous theoretical ~ ~ r The ability k of classical mechanics to describe the reaction dynamics of such
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 7993 processes over ultrashort time scales has been tested and evaluated, apart from failing to account for the purely quantum effects of rephasing and spreading of the dissociating wave packet, a remarkable degree of correspondence between quantum dynamics and the "single-trajectory" classical model62in describing nuclear motion over these time scales is exhibited. It should be mentioned that the effect of wave packet spreading could be taken into account by carrying out a multitrajectory classical treatment with a distribution of initial velocities and positions of the dissociating system. By variation of the parameters describing the simple exponential form of the dissociative potential function V,(r)invoked throughout this study, we have further investigated the effect on the expected FTS transients of changing the nature of the force field experienced by the initial wave packet prepared on the reactive potential. In this way, the acceleration of the wave packet as it propagates away from the Franck-Condon region of the reactive potential is altered, and its effect on the bond-breaking time and lifetimes of transition-state configurations examined. Similarly, inclusion of a repulsive component in the form of the high-lying potential curve V2(r)populated by the probe-laser pulse, in addition to that of Vl(r), has made it possible to investigate how changes in the dimensions and position of the detection window projected onto the reactive potential V l ( r ) are manifested in terms of the time-resolved "snapshots" of the reaction. By fixing the probe wavelength and changing the total energy available to dissociation products, the dynamics on V,(r)may be isolated and mapped by keeping the available energy constant and varying the probe wavelength, the separation AV2, = IVz(r) - V,(r)l between potential energy curves can be determined, as discussed in section 1II.D. Such an approach has been applied experimentally to isolate the dynamics of reaction 1 by variation of the pump-laser wavelength while the shapes of both PESs V2(r)and Vl(r) governing the observed dynamics of the predissociation of NaI have been examined in detail by independent variation of both pump- and probe-laser energie~.~~J@'.'~~ As confirmed by ex~eriment?-~ the results of section 1II.D demonstrate the sensitive nature of FTS measurements to changes in the characteristics of the two potentials V,(r) and V2(r). The addition of a centrifugal barrier to the PES controlling dissociation has been briefly considered and preliminary results communicated. Finally, the effect of changing the kinematics of direct dissociation was investigated by analysis of the fragmentation of the behavior Biz molecule. The predictive power of classical mechanics for both ICN and Biz systems is remarkable and consistent with quantum dynamical calculations.
Acknowledgment. We are grateful to the National Science Foundation and Air Force Office of Scientific Research for financial support. G.R. thanks SERC for the award of a NATO Postdoctoral Fellowship, during the tenure of which this work was carried out. We also thank the referees for a thorough review of this manuscript and for useful suggestions. Note Added in Proof. In addition to considering the effect of centrifugal energy in the dissociation of ICN, we have also carried out an analogous investigation involving the inclusion of an attractive term to the reactive potential due to the dipole/induced dipole interaction between C N and I fragments. Addition of this = attractive term to the potential modifies eq 19 to give Vleff(r) V , ( r ) [/(I 1)h2/2p9] - C6/P where C, = pCN2ar/(4rto)2 = 53 3 15 cm-' A6 (pCN is the dipole moment of C N and aIis the . polarizability ~ ~ will be~ detailed~ of ~I). The ~results of~ this study elsewhereIz3 once our calculations have been completed.
+
+
~
