Determination of real time dynamics in molecules by femtosecond

May 5, 1988 - experiments in real time on systems with extremely fast dynamical behavior. ..... spectra, and thermodynamic quantities fix the shape of...
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J . Phys. Chem. 1988, 92, 6648-6654

6648

Determination of Real Time Dynamics in Molecules by Femtosecond Laser Excitation Stewart 0. Williams and Dan G. Imre* Department of Chemistry, University of Washington, Seattle, Washington 98195 (Received: May 5, 1988)

A quantum simulation of the femtosecond pump-probe experiment on ICN photodissociation is presented. A I-D model

is constructed and used in a quantum mechanical time-dependent calculation to simulate two laser fields and three electronic states. The pumpprobe experiments are equated with a curve-crossing process with a time-dependent coupling between the two crossing curves. The probe frequency determines the p i n t at which the two curve cross, while the probe laser intensity determines the strength of the coupling. We find that the reaction for a &pulse excitation ICN I(2P,/2) + CN(X2Z+) on the lowest excited state of ICN takes on the order of 60 fs. With the laser pulses used in the experiment of Zewail et al., the quantum simulation reproduces the experimental results very well.

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Introduction Advances in laser technology have made it possible to perform experiments in real time on systems with extremely fast dynamical behavior. Laser pulses are short enough to be able to resolve nuclear motion during a photodissociation reaction for many systems. Pioneering experiments were performed by Zewail and co-workersI4 on the photodissociation reaction of ICN and NaI. We are going to concentrate here on the ICN experiments only. The experiment involves two short laser pulses (- 125 fs), one serving as a pump and the other as a probe, time delayed and at a different frequency. The pump laser transfers the molecules to the dissociative state which produces I CN(X2Z+). The probe laser catches the molecule on its path from ICN to I + C N and transfers it to a second excited state which results in production of I CN(B28+). The CN(B2Z+) CN(X2Z+) total fluorescence is the detected signal. In a recent paper,5 we presented a study for two photon transitions with short pules. Here we apply the methodology as well as some of the ideas in ref 5 to the ICN reaction. We will present a quantum mechanical simulation of the experiment. The numerical calculation is done in the same way as the experimental procedure. Thus we include three surfaces, ground and two excited electronic states, and two pulsed laser fields, connecting the three states. We follow the dynamics on the surface under study by looking at snapshots of the wave function at different times following the pump pulse. At the same time the probe laser is used in the calculation, as it is in the experiment, to probe the dynamics by transferring the molecule during the reaction to the second excited surface. Since the population on the second state is responsible for the fluorescence signal, we monitor the amount of population on the second excited state as a function of delay time between the two lasers. For simplicity ICN is treated as a diatomic, since most of the "action" in this case is along the C-I stretch. This should be a reasonably good representation of the system. The potentials are constructed to fit known experimental parameters. Despite the fact that many of these are poorly knovrn, we obtain fairly good agreement with experiment. Within the framework of our model, it is possible to improve this agreement by changing the potentials. It is not our aim here to fit experimental data as well as possible, but rather to present a frame of reference and a simple picture of what is happening in the lab. Our results substantiate the interpretation of the data as presented in ref 2 . Once one gets to reaction times which are on the order of a single vibrational period, it is not clear how to define reaction times.

+

+

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(1) Scherer, N . F.;Knee, J. L.; Smith, D.D.; Zewail, A. H. J . Phys. Chem. 1985,89, 5141. (2) Dantus. M.; Rosker, M. J.; Zewail, A. H. J . Chem. Phys. 1987, 87, 2395. (3) Rose, T.; Rosker, M. J.; Zewail, A. H. J . Chem. Phys. 1988.88.6672. (4) Rosker, M. J.; Rose, T.; Zewail, A . H . Chem. Phys. Lett. 1988, 146, 175.

(5) Williams, S. 0.;Imre, D.G. J . Phys. Chem., preceding paper in this issue.

0022-3654/88/2092-6648$01S O / O

TABLE I: Parameters Used in Calculation (atomic units) ( V = V o exd-a(x)I

+ C)

surface 1 surface 2

V,

a

0.0498

1.6" 1.5

0.03

C 0.117b

0.2343e

"Fit to absorption spectrum. bReference6 . cOkabe (ref 22) In ref 6, Bersohn and Zewail discuss at length a definition for reaction times and provide a classical model for the description of the experiment. Instead of trying to pinpoint a definition, we will present the time evolution for the system assuming the pump laser is 6 functions in time. This will serve as a reference point. Then we look at the effect a broader laser pulse has on the dynamics. For laser pulses narrower in frequency than the total absorption spectrum, we also have to consider the effect of the pump frequency on the system. Here we use a quantum mechanical wavepacket propagation study and present results for laser pulses of 40 and 125 fs. We will show that the pumpprobe experiments can be. viewed as a curve crossing between the two excited states, the one on which the reaction occurs and the state that is used for the probe. The probe frequency defines the crossing point, and the coupling between the two states is determined by the probe. laser intensity. Our simulation reproduces the experimental effects of the probe frequency very well.

Theory and Numerical Methods a. Setting up the System. To simulate the pumpprobe experiment requires at least three potential energy surfaces. We assume that most of the contribution to the time of evolution of the observables in the experiment is due to the dynamics in the C-I stretch coordinate. This is a reasonable assumption because both excited states are repulsive along the C-I coordinate.'** The C N product is not highly excited, indicating only minor activity in the C N stretch during the dissociation. The only other coordinate which might be important and is being ignored here is the bend.I0 By symmetry two options exist for the topology of the excited states along this coordinate-a minimum or a maximum at the linear geometry. In either of these two cases, the major force in the Franck-Condon region must then be along the C-I stretch. The width of the absorption spectrum is then mostly determined by the short time dynamics in the C-I (6) Bersohn, R.; Zewail, A. H. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 313. ( 7 ) Holdy, K. E.; Klotz, L. C.; Wilson, K. R. J. Chem. Phys. 1970, 52, 4588. (8) Morse, M.; Freed, K. F.; Band, Yehuda B. J . Chem. Phys. 1979, 70, 3620. (9) Morinelli, W. J.; Sivakumar, N.; Houston, P. L. J . Phys. Chem. 1984, 88, 6685. (10) Goldfield, E. M.; Houston, P. L.; Ezra, G. S. J. Chem. Phys. 1986, 84, 3120. ( 1 1 ) Williams, S. 0.;Imre, D.G. J. Phys. Chem. 1988, 92, 3363.

0 1988 American Chemical Society

Real Time Dynamics in Molecules

The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6649

\\

Figure 2. Schematic representation of laser excitation from a curvecrossing point of view for (a) resonant excitation and (b) off-resonance

excitation. I

00

I

I

I

20

40

Displacement in C-l (au) Figure 1. Two excited states used in calculations. Also shown is the

pump frequency (306 nm) and the probe frequencies we will use in this paper. This immediately defines many of the parameters for the first excited-state surface. The absorption spectrum has been shown to be composed of three separate absorption band^.^*^,'^-'^ Since the experiment was performed at 306-308 nm,l*2only one state contributes to the spectrum at this f r e q ~ e n c y .This ~ state correlates to CN(X2Z+) I(2P3/2).We assume an exponential form for both excited states. The parameters for the first excited-state curve were chosen to fit the dissociation limit6 as well as the center and width of the lowest band in the absorption spectrum as given in ref 9. There is much less information about the second excited state. From the experiment we know it also dissociates to I CN, and the final products seem to be CN(B2Z+) and I(2P3/2).The shape of this state is very important in determining the experimental observations. We have chosen the parameters as follows; the separation between the two excited states in the asymptotic limit is determined by the CN(X2Z+)to CN(B2Z+)transition, and we assume this curve should be less steep than that of the first excited state. Table I gives all the parameters for the two excited states. The ground state is assumed to be harmonic; it determines only the initial wave function which in this case is u" = 0, since the experiment is performed in a supersonic jet. Figure 1 shows the two excited states and indicates the pump and probe frequencies which we will use later. b. Theory. The time-dependent Schradinger equation is given by ( h = 1):

+

+

where +o is the ground-state eigenfunction; and q2are the wave functions on the first and second excited states, respectively, with L)~(O) and 1)~(0)= 0; Ho, H I , and H2 are the Hamiltonians for the ground and two excited states; the off diagonal terms Hloand H I 2 represent the pump and probe lasers, respectively; H l o = k l d l ( t ) ~ W lH t ;1 2 = p12A2(t-7)eiWzf; plo and p12are transition moments between the ground and first excited state, and first and second excited states, respectively; we assume both are constants; A , ( t )and A2(f-7) are pulse shapes and are taken to be Gaussian in time; w1 and w 2are the central pump and probe frequencies; 7 is the delay between the pump and probe. The numerical calculation was performed by solving the above equation on a grid using the FFT method as developed by Kosloff et al.l5*I6 When the two laser pulses are used, a grid of 1024 points (12) Hall, G. E.; Sivakumar, N.; Houston, P. L. J . Chem. Phys. 1986,84, 2120. (13) Hess, W. P.; Leone, S. R. J . Chem. Phys. 1987, 86, 3713. (14) Pitts, W. M.; Barononski, A. P. Chem. Phys. Lett. 1980, 71, 395.

with spacing of 0.03 au is required to ensure convergence and prevent edge effects. We test for convergence by halving the time step and grid spacings. To calculate C N fluorescence for a given probe frequency vs delay time between the two lasers, the following procedure is used: The pump laser pulse shape, A , ( t ) ,is set and one delay time, 7, is selected. This fixes A2(t7). With these two fields on, we solve for ql(t) and &(t) for a time long enough for both lasers to be completely off. At that point the norm, (+2(t)lL)2(t)),is calculated. This norm is the total CN(B2Z+) produced, which is proportional to CN* fluorescence. This procedure yields one point on the curve, CN* fluorescence vs delay. We then repeat the above for different delays. This treatment is exact for the model we constructed and will hold for all laser powers. We make sure throughout the calculation that the two laser fields, AI@)and AZ(t-7),are weak to prevent saturation effects. Equation 1 provides very little insight into what is happening during the experiment. A simpler picture can be obtained by examining the weak-field limit. In this limit, we can derive integral equations for each of the wave functions as follow^:^^*^^ We choose Eo = 0, then It+bo(t))= (+o(0)). So the ground-state wave function remains unchanged. The wave function on the first excited state is

The pump pulse transfers I+o) to the first excited state as long as A l ( t )> 0. The excited-state Hamiltonian, HI, propagates each piece of as it reaches the excited state. Note the laser frequency in this expression controls the phase of I+o) as it reaches the excited state. To calculate Iql(t)) then requires calculating the integral in eq 3 which is a rapidly oscillating function due to the phase factor BWlf(wlis a UV photon). Instead we can rewrite eq 3 in the form

where H1' = H I - wl. This amounts to shifting the excited-state surface by an amount wl,the laser energy. Besides being computationally convenient, it presents the excitation processes from a slightly different point of view.19~20We can think about the electronic transition from the ground state to the first excited state as a curve crossing between these two states. The laser frequency determines the energy shift and thus the crossing point. Figure 2 illustrates the two points of view for two laser frequencies. Note that, when the (15) (16) (17) (18) (19) (20) (21) (22) 19176.

Kosloff, D.; Kosloff, R. J. Comput. Phys. 1983, 52, 35. Kosloff, R.; Kosloff, D. J . Chem. Phys. 1983, 79, 1823. Tannor, D. J.; Kosloff, R.; Rice, S. A. J . Chem. Phys. 1986,85,5805. Rama Krishna, M. V.; Coalson, R. D. Chem. Phys. 1988,120,327. Thomas, T. F. J . Chem. Phys. 1982,86, 10. Bandrauk, A. D.; Turcotte, G. J . Phys. Chem. 1985,89, 3039. King, G. W.; Richardson, A. W. J . Mol. Spectrosc. 1966, 21, 339. Okabe, H. Photochemistry of Small Molecules; Wiley: New York,

6650 The Journal ofPhysical Chemistry. Val. 92. No. 23, 1988

I

Williams and Imre

on

80

40

00

3 265

120

4080

4 895

5710

Frequency (eV)

Displacement in C-l (au) Figure 3. Evolution of l$l(t)), the wave function on the first excited state, for a &function pulse.

Figure 4. Absorption spectrum for excitation into the first excited state. The arrow indicates the pump frequency (306 nm).

laser is tuned to the center of the absorption spectrum, the curve crossing is a t an energy very close to that of u " = 0, and when the laser is tuned to the red the curve crossing is at higher energy and appears as tunneling. This results in reduction of the absorption crossection. The coupling between the two states is determined by IA(t)I2,the laser intensity. In this case it is time dependent. The same picture emerges for the probe laser. This laser prepares l $ 2 ( 0 ) : I$2(f))

= J'dt

A2(f-r)e'W''e""2'I*l(t))

./.

(5)

Again, we can rewrite eq 5 incorporating a shift of the second excited state by amount wz + wlr yielding Hz' = H2 - w2 - w I 00

I$2(t)) = S . ' df Az(f'-

40

The probe laser transfers the wave function, I$,(t)). from the first excited state to the second excited state, where U2 propagates it and CN' is produced. I$l(t)) is moving on the first excited state; for a given probe frequency it reaches the crossing between curves 1 and 2 a t some time 61. CN* is produced only if crosses from state 1 to state 2. It can do so only if the coupling, A2(f-T), between these two states is turned on. The time evolution is very simple; it starts in the Franck-Condon region of I$l(f)) and proceeds from there to large C-I displacement, never to return. It gets only one chance a t the curve crossing. If the probe laser is on while I$l(f)) is near the crossing, CN* will be produced, will remain on state 1 and no fluorescence will be if not, I$l(f)) observed. This point of view makes it clear that the transfer of population from state 1 to state 2 depends on both the probe frequency and the time delay. Choosing the frequency fixes the crossing point, which in turn determines when l$l(f)) will "pass" over the crossing. 7 , the time delay between pump and probe, determines whether the coupling between the two states is on or off when I$,(t)) is over or near the crossing. We have presented the weak-field limit pumpprobe experiments in a curve-crossing framework. This point of view holds for all laser powers; we have used the weak field only to simplify the equations.

Results and Discussion To get a Feel for the dynamics on the first excited state, we calculate the time evolution of the ground-state wave function on this excited state assuming a &function pulse. Figure 3 shows snapshots of Ifil(f)) at various times following the laser pulse. We note that the wavepacket reaches the asymptotic region in =60 fs. The time evolution is very simple; Ifil(t)) remains fairly

80

120

Displacement in C-l (au)

~)8"'('-r)l$l(f)) (6)

Figure 5. Evolution of IJ.,(t)) for a 40-fs pulse. The excitation is onresonance with first excited state.

localized for a long time and simply proceeds to larger displacements. To define a reaction time unambiguously for ICN photodissociation is not a simple task. One can always claim that the potential is never exactly constant, or that a small tail of the wave function is still in the interaction region. What we find is that, 60 fs following the &function pulse in time, the wave function in momentum space remains constant to within graphic resolution. We can use this propagation to obtain the absorption spectrum using the equation"

44

=

~ J o ~.w-Y $ o l $ d ~ ) ) dl

(7)

Figure 4 shows the calculated absorption spectrum. A good fit to be experimental deconvoluted spectrum is obtained. The arrow indicates the pump frequency (306 nm) for the pumpprobe experiment. Since the experiments were performed with laser pulses of approximately 125 fs and the dissociation time from Figure 3 is found to be around 60 fs, we decided to first present a study for shorter laser pulses. We chose a Gaussian pulse of 40 fs fwhm. a. 40-fsPump and Probe. Figure 5 shows the time evolution for I$l(f)) for excitation with a 40-fs pulse tuned to the center of the absorption spectrum. f = 0 corresponds to the peak of A,(t). In comparison with Figure 3 , I$l(f)) is spread due to the width of the laser pulse. Note that, as in the case of the 6 pulse, by 60 fs the center of I$l(f)) has reached the asymptotic region. Figure 6 shows the population on the first excited state vs time. To test the basic idea of the pumpprobe experiment, we chose the pump frequency on resonance (the laser is tuned to the center

The Journal of Physical Chemisfry, Vol. 92. No. 23, 1988 6651

Real Time Dynamics in Molecules

50

0.0

Probe pulse delay (fs)

Time (fs) Figure 6. Total population on the tint excited state due to resonant laser excitation with a 404s pulse.

Figure 8. Total fluorescence for 40-fs probe frequency on-resonancein the F-C region, i.e., probe frequency = wb.

4.051

0.0 00

20

40

Displacemenl in C-I (au) Figure 7. Two repulsive excited states showing crossings for two probe frequencies wb and wb. wZr represents a probe frequency that is onresonance in the F-C region and shows the second excited State (---) crossing the lint excited state (-) at 0 displacement. Also shown is the crossing for probe frequency on resonance in the asymptotic region (wb), and the wave function, l$l(f)), for the &function pulse presented in Figure 3. of the absorption band) to avoid off-resonance effects. Two probe frequencies are used. They were chosen such that the first, wZr (Figure 7), is on-resonance in the F-C region. In other words, it amounts to curve crossing at the ground-state equilibrium geometry. The other, wZa(figure 7), is on-resonance in the asymptotic region. Figure 7 shows the potential energy curves for this study as well as the wave function for the 6 function pulse calibration. Inspection of this figure makes it clear what the two experiments should yield. For probe laser a t wZr,the two curves cross right a t the F-C region. We expect then that turning on the coupling, A2(i-T), between these two curves would be most effective a t short times. Once IGl(i)) moves away from the crossing, no population will transfer to state 2. Figure 8 shows the results for this laser frequency, showing that indeed for short delays the population is transferred most efficiently. When the probe is tuned to wzS,Ifil(i)) fully passes over the curve crossing after =IO0 fs. These two curves remain degenerate from here on. Figure 9 shows the results for the pumpprobe calculation at this frequency. Note that it reaches half-height a t about 50 fs, consistent with the center of the wavepacket reaching the crossing 50 fs following the pump pulse. As the rest of IJ.,(i)) makes it into the asymptotic region, more population is transferred to curve 2. At later times, IJ.l(i))is in a region where the two curves are degenerate and the population transfer to state 2 becomes constant.

0.0

25

50

75

1W

125

Probe pulse delay (fs) Figure 9. Total fluorescence for 404s probe frequency an-resonance in the asymptote, Le., probe frequency = wb. This study shows that indeed the basic idea of the experiment and the interpretation of its results are consistent. However, we find that the line shapes in Figures 8 and 9, which represent the experimental observables, are mostly determined by the laser pulse shape. This is due to the fact that even the 40-fs pulse is long on the time scale of the molecular dynamics. For example, when the probe frequency is a t w2,, it probes the time spent by the wavepacket in the Franck-Condon region. From Figure I, it is clear that the time is much shorter than 40 fs. Before we go on to present the results for the simulation of the experiment using wider pulses (125 fs) and pump frequency of 306 nm, we should look a t the effect of detuning on the time evolution. The laser energy is lower, and thus we would expect slightly slower motion on the excited state. Figure 10 shows snapshots of IJ.,(f)), for a 40-fs pump tuned to 306 nm. The time evolution is only slightly slower. More importantly, the population on the excited state seems to be decreasing as time goes on. Figure 11 shows the time evolution of the population on surface 1. Note, as the laser is turned off, a large fraction of the population makes a rapid return to the ground state? This introduces time-dependent behavior on surface 1 which is not associated with what one may nonnally mnsider the dissociation dynamics. when the experiment pro& the products such as here, this transient will not be observed. However, in a Raman experiment most of the contribution to the spectrum might be due to those molecules that make a rapid return to the ground state. b. 125-fs Pump and Probe. Here we present our results for conditions simulating the experiment.' The pump and probe pulse width is 125 fs and the pump frequency is 306 nm. Four different probe frequencies are used.

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The Journal ojPhysical Chemistry, Vol. 92, No. 23, 1988

Williams and lmre

0s

00

I; 40

SO

0.0

120

Displacement in C-l (au)

a

130

Time (is)

Figure 10. Evolution of I # > ( t ) ) for a 40-fs pulse and pump frequency of

306 nm. 0 01

Figure 13. Total population on the first excited state for 125-fs pump pulse tuned to 306 nm.

I

db

I

20

00

40

Displacement in C-l (au) Figure 11. Total population on the first excited state with a 404s pump laser pulse of 306 nm. Note not only the difference in shape of the population profile compared to Figure 6, but also the fact that the 306nm pump laser transfers 1% less population than the resonance pump laser.

. b

3.371

2-

I

A

c

Lu

3.238 -

1.0

20

3.0

40

Displacement in C-l (au)

00

40

SO

120

Displacement in C-l (au) Figure 12. Evolution of

I#,(t))

for a 1254s pump pulse at 306 nm.

Figure 12 shows I#l(f)) for the 125-fs pump laser. Note the large spread induced by the wider laser pulse and that, as before, the center of the wavepacket is well into the asymptotic region by 100 fs following the pump pulse; t = 0 once again represents the center of the pump pulse. Also note the decrease in population with time. This is more evident in Figure 13.

Figure 14. (a) Rcpulsive excited states showing crossings for four probe frequencies; the shaded circle indicates area enlarged in part b, and (b) enlargement of part a showing crossings for probe frequencies 390.5. 389.5, and 388.9 nm.

Figure 14a,h shows the curve crossings created by the three experimental probe frequencies, 390.5, 389.5. and 388.9 nm, as well as 433 nm which is a probe frequency we have added. W e chose to add this probe frequency since the potentials we have constructed for all three experimental probe frequencies cross at large displacements as evident from Figure 14a,h. It is very interesting that the experimental results show large variation with a small change in probe frequency. The laser width is at least 40 cm-I and the probe frequency differences are 66 and 40 cm-'.

Real Time Dynamics in Molecules

The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6653

1.2XlOd

E a)

v)0

E!

-

I

zs 1

0. =

390 5 nm

0.0

-100

00

100

200

300

Probe pulse delay (fs) Figure 16. Total fluorescence for the probe excitations shown Figure 15 plotted on one scale.

.1

w

00

1W

200

300

Probe pulse delay (fs)

i -100

0,=389.5 nrn

1

Figure 17. Experimental results for ICN at probe frequencies 3905, 3895, and 3889 %.,reproduced from ref 2.

1w 200 Probe pulse delay (fs)

0.0

3w

0.0 -lW

0.0

1M

2w

3w

Probe pulse delay (fs) Figure IS. Total fluorescence for 125-fs pulse showing results for probe frequencies (a) 433, (b) 390.5, (c) 389.5, and (d) 388.9 nm. In each case, the pump is a 125-fs pulse with central frequency at 306 nm.

A probe at 433 nm results in crossing at the point where the energy of the first excited state is the same as the pump frequency; thus we expect it to probe the short-time dynamics. Despite the fact that the other three probes are very close in energy they probe slightly different regions, from -2.0 to 3.5 bohr, with the lowest frequency probing the shorter displacement region. Figure 15a-d shows the results for the two-laser experiments. The trend in these figures for all probe frequencies is as one would predict based on the dynamics and the position of the curve crossings. These results are in fairly good agreement with the experimental results which are shown in Figure 17. Figure 16 shows fluorescence yield for all four probe frequencies on the same scale. We note a large change in signal intensity as a function of probe frequency. The experimental results are shown in Figure 17; no vertical scale is shown. However, the signalto-noise ratio is definitely deteriorating as we go to longer probe wavelength, suggesting a decrease in signal. This is consistent with our findings in Figure 16. The origin of this phenomenon can be understood in the framework of the Landau-Zener model for curve crossings. According to this model, the crossing probability is dependent on the difference in the slopes of the two potentials at the crossing

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The Journal of Physical Chemistry, Vol. 92, No. 23, 1988

point. The more parallel the two curves at the crossing point, the higher the crossing probability. Figure 14a,b shows that, for the higher frequencies, the two curves cross at shallow angles as compared to the lower frequencies. What this implies is that besides the behavior of the signals vs time there is important information in the strength of the signal as well. In a careful examination of Figure 17, two bottom traces show that there actually may be more than one maximum in each of these traces. If this indeed can be substantiated, it may imply that the two curves may cross twice, at two separate points. Each bump may be representing passage over one of the crossings. Indeed we found that many of the model potentials we attempted produced multiple crosses. In choosing our model potentials, we purposely avoided such topography. A comparison between Figures 8,9, and 15a-d reveals the effect of laser pulse width on the experimental observables. Since we are in a regime where the laser pulses are much longer than the dynamics they are probing, the experimental observables, line shapes, rise times, widths, etc. are mostly determined by the laser. We note however that the position of the curves in these figures do shift to longer times with increase in probe frequency. This is to be expected since higher probe frequency corresponds to curve crossing at larger displacement. We were hoping to find a simple correspondence between delay times for which the fluorescenceyields peaks and the time at which It)l(t)) passes over the curve crossing. We observe the right qualitative trends, the fluorescence curves shift to longer delay times with probe frequency, but no simple relationship is found. We feel that this is partly due to the fact that the pump frequency is far from the center of the absorption band. This introduces a transient on the population on the first excited state which may contribute to the overall observations. We cannot overemphasize the fact that the experimental observables are strong functions of both excited states. If we were to change the shape of the second excited state, the fluorescence vs delay curves would change. In this case, as long as we remain within the 1-D model we feel that the available data, absorption spectra, and thermodynamic quantities fix the shape of the first excited state. The second excited state, on the other hand, aside from the asymptotic region, is not known as well. We have assumed throughout the calculation a constant transition moment. This is a fairly good approximation for pl0, since it multiplies only which spans a very narrow range of , other hand, multiplies I$q(t)) and, coordinate space. M ~ on~ the as we pointed out throughout this paper, the experiment probes at a variety of displacements. The dependence of p12on the coordinate may play an important role in determining the observables. The probe of the overall dissociation time in the asymptotic limit would still give the right dissociation time.

Conclusion We have presented here a study for a two-photon pump-probe experiment on a rapidly photodissociating molecule. A fully

Williams and Imre

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quantum mechanical modeling for the photodissociation reaction I C N in the lowest excited state of ICN with a 6 ICN function laser pulse as well as laser pulses of 40 and 125 fs is made. We find very good agreement with experimental results presented in ref 2. A very simple and intuitive picture of the pump-probe experiment is obtained when the electronic excitations are cast in a curve-crossing language. The quantum simulation results show qualitatively the same behavior as expected based on classical ideas and Landau-Zener modeling of curve crossings. Our results are in good agreement with the experimental findings. A much better fit is possible by modification of the two excited-state potentials. We have not attempted to do so. It is also interesting that the quantum results are in good agreement with the simple classical model in ref 6. The experiment* which we have simulated here is a pioneering effort of the study of fast molecular processes in real time. There is a connection between the two-photon “pump dump” selectivity scheme as proposed by Tannor et al.” In this scheme, two short laser pulses are used to control the outcome of a unimolecular dissociation reaction. The delay between the pump and dump lasers is used to change the ratio of two possible products. The ICN experiment can also be presented from that point of view, where, by timing and choosing the right frequency of the pump and probe pulses, one can change the course of the one-photon reaction to the point where a different product is produced-in this case C N * rather than the CN. As we have shown here, the probability of producing C N * is crucially dependent upon the frequency and delay of the probe pulse. One can imagine similar schemes in the future where the proper sequence of pulses will control the outcome of reactions to the point where the chemical composition of the final product may be altered at will. We feel that the time-dependent point of view as presented here provides a very natural frame of thought for describing and designing experiments of this type. In these experiments where moving wavepackets are actually produced in the laboratory, their motion is often almost classical in nature making it very simple to interpret the experiment and its outcome. The quantum mechanical description of the experiment is a little more complex, but as we examine more cases, the moving wavepackets become more familiar and we can predict their qualitative behavior just as we can the dynamics of a rolling ball on a potential energy surface. In that respect, we attempted to provide a very pictorial interpretation, both here and in ref 5 and 11, of what is prepared by a photon.

+

Acknowledgment. This work has been supported by NSF Grant CHE-8507168 and by the donors of the Petroleum Research Fund, administered by the American Chemical Society. We thank Professor Eric Heller and his group, as well as Jinzhong Zhang, Melanie Domagala, and Sandra Tang, for helpful suggestions and lively discussions. Registry No. ICN, 506-78-5.