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classical hybrid dynamics of I2(A) photodissociation and recombination in matrix Ar, linear chain model
12May 1995 .%. ~ "
7~
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 237 (1995) 299-307
Quantum/classical hybrid dynamics of I2(A) photodissociation and recombination in matrix Ar, linear chain model Li Liu, Hua Guo Department of Chemistry, University of Toledo, Toledo, OH 43606, USA Received 9 February 1995
Abstract
A quantum/classical hybrid model is developed for the photodissociation and recombination dynamics of an 12 molecule in low temperature matrix Ar. This simplified model consists of an 12 molecule embedded in a linear chain of Ar atoms. The dynamics of 12 on its A state is treated quantum mechanically while the Ar matrix is described classically. The quantum system is self-consistently coupled with the classical bath. Our calculations show substantial energy transfer and retention of coherence in the early stages of the recombination process. The time-delayed pump-probe transients are calculated and compared with experiment and other theoretical simulations.
I. Introduction
With the development of ultrashort laser techniques, more and more chemical reactions have been investigated experimentally with femtosecond resolution [1]. However, exact quantum mechanical treatments for systems with more than three degrees of freedom are still formidable, despite recent advances in time-dependent quantum theory of molecular dynamics [2]. It is even more difficult, if not impossible, to provide a complete quantum mechanical description for chemical processes in condensed phases which involve a much larger number of modes. Alternative methodology in treating dynamical processes in these situations is thus highly desirable. Much effort has been devoted in this direction in the last few decades. It has long been realized that the motion of solvent is usually much slower than that of solute. Hence, it is advantageous to partition a
system into two interacting parts: the reactive (sub)system and the surrounding bath. For example, reduced density matrix approaches to condensed phase problems are almost uniformly based on the assumption that the system and bath modes are separable [3]. By assuming thermal equilibrium for the bath, the dynamics of the system can be characterized by the reduced density matrix which is governed by the Liouville-von Neumann equation. Similar ideas have been implemented without the density matrix formalism. In collision processes, for example, it can be shown [4] that the slow translational motion can be described by trajectories in the classical limit while vibrational modes are treated quantum mechanically. Such hybrid quantum/classical approaches have found wide applications in many dynamical calculations [4-13]. In this work, a quantum/classical hybrid model for the photodissociation and recombination of an 12
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 0 0 9 - 2614(95 ) 0 0 3 1 8 - 5
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L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
in matrix Ar is presented. The I2/liquid system has been extensively investigated theoretically and experimentally [14] and it serves as a prototypical model for studying the so-called 'cage effect' in condensed phases [15]. Here, we will focus on the A state dynamics of 12 in low temperature Ar. Femtosecond experiments indicated prompt recombination (caging) following photodissociation of 12 in both clusters [16] and matrices [17,18]. With a standard pump-probe scheme, these experiments revealed retention of coherence despite substantial energy transfer in the early stages of the dissociation. Classical trajectory calculations [16-20] have been reported for the dissociation and recombination processes and the results are generally consistent with the experimental observations. Although the iodine system is rather heavy and presumably behaves more or less classically, a quantum description is still desirable because some important dynamical features can only be treated quantum mechanically. For instance, the excited state iodine wave packet created by an ultrashort laser pulse is a coherent superposition of several vibrational states. The coherence of the wave packet and its destruction (dephase and relaxation) is a quantum mechanical phenomenon, which has no classical analog. Although many studies have shown that the center of a wave packet can sometimes be described by a classical trajectory [21], it is difficult to describe the complete wave packet dynamics classically. This is particularly conspicuous at turning points where the wave packet may exhibit significant interferences between the incident and reflected waves. Another important quantum feature in the recombination process is non-adiabatic transitions between the excited manifold and the ground electronic state (not treated here). The quantum/classical hybrid method allows one to incorporate an accurate quantum treatment of the system into a classical description of the environment and it may provide a different and more accurate viewpoint of the dynamics.
Ar I
Ar2
Arn. 1
Ar n
[
2. Linear chain (LC) model In this work, we impose a rather oversimplified approximation in our calculations. We assume that the dynamics of photodissociation and recombination of I 2 o c c u r in one dimension. Particularly, our model consists of an 12 molecule embedded in a linear chain of Ar atoms. The iodine molecule is coupled with an Ar atom at each end with a Lennard-Jones 6 - 1 2 potential. Each Ar atom in the bath is connected to its nearest neighbors with the same type of potentials. The last Ar atoms in the chain are connected to stationary walls. The LC model is schematically depicted in Fig. 1. Since the interaction between adjacent atoms is not harmonic, this one-dimensional model represents a non-trivial prototype for impurity system in condensed phases. Because the symmetry of the LC model and the heavy mass of the iodine, the center of mass of the molecule is removed from the model. The A r - I and A r - A r interaction parameters are adapted from previous work [17,22-24], and the intramolecular potentials (for both ground and excited states) of the iodine molecule are those that Potter et al. [24] and Zadoyan et al. [17] used in their molecular dynamics simulations. This LC model is similar to that of Messina and Coalson [25] in studying the emission spectrum of I 2 in Ar, and to that of Roitberg et al. [13] in calculating vibrational relaxation of an impurity in clusters. Admittedly, the LC model is a rudimental one which may not be able to convey a complete picture of dynamics. However, it is our believe that such a simple model should shed light on some general features of the dissociation/recombination as well as energy transfer processes. Since the LC model allows no dissociation, our simulations are only relevant to the recombination process. However, it is noted that no cage escape has been observed in low temperature rare gas matrices [17,18]. In addition, the solute would have little time to rotate during the
I
Am+ 1
Arn+2
Ar2n. 1
Ar2n
Fig. 1. Geometry of the linear chain (LC) model for I2Ar2n. The diatomic bond of 12 is described by the strong Morse potential; the interactions of the I - A t and A r - A r pairs are weak Lennard-Jones potentials.
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first few picoseconds and the LC model should be a reasonable approximation to the early dynamics. More importantly, the LC model presents an idealized and tractable approach to reveal the fundamental physics. It should be pointed out that the extension of the quantum/classical method from the LC model to a full three-dimensional simulation is straightforward.
Table 1 Potential parameters for the 12/mr system Lennard-Jones parameters
Ar-Ar Ar-I
O" (m)
84 130.24
3.40 3.617
Morse parameters
X A
3. Quantum/classical hybrid method
• (cm- 1)
D (cm- l )
Qe
/3 (,~- l)
V0 (cm- 1)
12547.2 1840
2.656 3.10
1.875 2.147
0 7605
Within the LC model described above, the total Hamiltonian can be written as follows:
H(Q, q) =H~ + H
b +Hsb 2n p2
p2
= --
2.
+ V~(Q) + Y'~ i=1
+ Vb(q)
Tmm
+ V~b(Q, q),
(1)
where m and /x are the mass of Ar and the reduced mass of 12, respectively. The quantum degree of freedom Q is the 12 vibrational coordinate while the collective bath coordinate q denotes the classical displacements of the 2n Ar atoms in the matrix. There are three electronic states of 12 involved in the model: the ground state, the A state and the ion-pair state. The ion-pair D state, in its Rittner form, is adapted from Tellinghuisen's work [26]. The X and A states are represented by Morse functions of the form Vs(Q) = O{exp[ - 2/3(Q - Q e ) ]
-2exp[-fl(Q-Qe)]}+V o.
(2)
The bath potential Vb is the sum of the Lennard-Jones type potentials between two adjacent Ar atoms, [[ O-~12
VLj(r)=4,[[r)
6]
--(~)
.
(3)
A similar form of potential is used for the systembath interaction V~b between an iodine atom and an adjacent Ar. All the potential parameters are defined in Table 1. The quantum/classical hybrid method is formally based on the time-dependent Hartree (TDH) approximation [27,28]. The TDH assumes that the total wave function can be factorized into the product of
the system part and the bath part. Such factorization allows one to treat the system and bath separately, with either a full quantum or a quantum/classical description. Because of the time-dependence of the wave function, energy transfer between the system and bath is allowed in the TDH scheme. The wave functions of the system and the bath are subjected to the following effective Hamiltonians: p2 H ~ " = - - + Vs(a ) + (V~b(O, q))q, (4a) 2/x 2n p2 n~ ff= ~ ~ + Vb(q) + (Vsb(Q ,
q))Q.
(4b)
i=1
The coupling terms between the system and bath are introduced by the mean-field potentials. The above approach is based on two considerations, namely, the interaction between the system and the classical environment is usually weak and the time-scale for the primary dynamics is much faster than the solvent motion. The quantum/classical hybrid method goes one step beyond the quantum TDH approximation in that the bath modes are treated classically. Under such circumstances, the effective potential for the quantum mode is modified by the time-dependent coupling which is a function of the solvent coordinate q. On the other hand, the dynamics of the system exerts forces on to the bath atoms through the averaged coupling potential. Specifically, the average over the system coordinate is made by substituting Q with its expectation value ( Q ) . The dynamics of the two subsystems is thus self-consistently coupled. For the quantum degree of freedom, the equation
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
302
of motion can be written as follows (atomic units are used) [29,30]:
/ Ha
0 i - - ~02 = Ot ~3
0 /
H21
H2
H23
q~2 ,
0
H32
H3
¢~3
(5)
where ~t)i (i = 1, 2, 3) are the nuclear wave functions on three electronic surfaces corresponding to the X, A and D states, respectively. The diagonal terms in the Hamiltonian matrix should be understood as effective Hamiltonians for each of the electronic states (Eq. (4a)). Note that the potentials are functions of the solvent coordinates and the dynamics of the system is influenced by the time-dependent fluctuations of the solvent. The off-diagonal terms in the Hamiltonian matrix represent radiative couplings between the electronic states, HI2 =/.~12Al exp('yl t2) exp(itolt),
(6a)
H23 =/XE3A 2 e x p [ - y z ( t -
(6b)
r ) 2] exp(ito2t),
where /~12 and /./,23 are the transition dipole moments for the X ~ A and the A ~ D transitions, respectively; to, A and y are the central frequency, laser intensity and the width of the pulses. The probe pulse is delayed relative to the pump pulse by r. Experimentally, the transient signal is collected by measuring the fluorescence from the probe state (D) after pump and delayed probe pulses. The laser-induced fluorescence (LIF) yield for a given time delay is thus [29,31] P(7-) = lim (q~3(t, r ) [ tP3(t , 7-)). t ---~oo
(7)
The above integral approaches a constant as the probe pulse vanishes. In our calculations, the probe process is treated perturbatively. That is, we assume A 2 is so small that no population trickles down from the D state to the A state during probe pulse duration. This weak-field approximation is identical to assuming H23 = 0 (but H32 ~ 0) in Eq. (5). Computationally, the probe and pump processes are thus separable and the CPU time is reduced by a factor of four. The equations of motion for the classical bath are given below in the Hamilton's form,
d qi dt
0H~ff Opi '
d Pi
0H~ff
dt
Oqi
(8)
The right-hand side of the second equation represents the forces exerted on this atom. Since only the nearest neighbor interactions are included, there are only two force terms for each solvent atom. For the two Ar atoms directly connected with the iodine, the forces are from both the I atom and from an adjacent Ar. For the rest of the Ar atoms, the action from the quantum system is indirect and all the forces are due to the nearby solvent atoms. The time-dependent Schr6dinger equation (Eq. (5)) is solved numerically on a grid by using an efficient fast Fourier transform (FFT) method [2]. Basically, the potential and kinetic energy operators are evaluated in their own diagonal representations and transformation of the wave packet between these representations is accomplished via FF£. We use a short iterative Lanczos method [2,32] to propagate the wave packet. The spatial grid has 512 uniformly distributed points from 2.35 to 7.6 .~ for 12. The widths of the pump and probe pulses are 100 fs (fwhm). For the classical motion of Ar, the Hamilton's equation (Eq. (8)) is solved by using the second-order differencing method. The propagation of the trajectory and the wave packet must be carried out in parallel. The time step for propagation is 0.2 fs. 100 Ar atoms are found to give converged results for dynamics up to 4 ps.
4. Results
4.1. Wave packet dynamics The potential curves of the I2/Ar2n system are given in Fig. 2. The potential energy curves for an isolated 12 are plotted in solid lines while the matrix modified effective potentials are given in either dash-dotted or dashed lines. The potential curve of probe state V3 is vertically downshifted and its equilibrium position is stretched outwards by = 0.2 A from its gas phase counterpart, because the solvation of the dipole stabilizes this state [17,33]. It can be readily seen that there is a repulsive wall at large I - I distances due to the presence of the solvent. This solvent wall is responsible for the recombination of the dissociated fragments (cage effect). Also note that the wall is pushed outwards as the dissociation proceeds towards larger I - I distances.
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
The snap shots of the wave packet are displayed in Fig. 3. The photodissociation of 12 near 734 nm is initiated through a vertical transition from the ground state to the A state at an initial excess energy of ,~ 1 0 0 0 c m - 1 above the dissociation limit. The high initial energy ( = 1500 cm -1 at t < 0 ) is due to transient species that will eventually return to the ground state when the pump pulse finishes. Since the wave packet experiences a strong repulsive force near the Franck-Condon region, it moves rapidly downhill and 12 dissociates. The elongation of the bond is then slowed down by the solvent wall and the I fragments start to recombine geminately. The average I - I distances for the wave packet at two pump wavelengths are plotted against time in Figs. 4b and 4c. It starts with the ground state equilibrium distance and reaches at = 4.2 A at the first encounter with the solvent cage. During the collision, most of the kinetic energy of the wave packet is transferred to the solvent. The energy transfer is extremely efficient since the energy density of the wave packet is very high (nearly in a continuum) and coincides well with the Ar phonon spectrum. The total energy of the 12 system is shown in Figs. 4a and 4d to decrease rapidly: more than 1000 cm-1 of
303
t=0fs
t =80fs
t= 160fs
t = 240 fs
t = 320 fs
t = 480 fs
820 fs
~
t= 1120fs i t
t = 128o fs
] ~ ~ , . ~ . 3
4
5
1 6
7
I-I Distance (/~) Fig. 3. Snap shots of wave packet motion on the A state of 12 at the pump wavelength of 734 nm.
4i A.............. D
.....
3
i
t.
o -1
X ,
,
i
3
. . . .
i
4
. . . .
i
5
. . . .
i
6
. . . .
J
,
7
I-I Distance (/~) Fig. 2. Potential energy curves for an isolated 12 (solid lines) and its effective potential energy curves in matrix Ar. The dashed-dot line ( . . . . . ) corresponds to the A state potential in a frozen Ar matrix before the photo-excitation. The short dashed line ( . . . . . ) and the dotted line ( . . . . . . ) represent the time-dependent potential at 800 and 1900 fs, respectively. The pump-probe scheme is depicted in the figure and the two probe windows are indicated by the two arrows.
energy is lost in the first = 500 fs. The significant energy transfer in the first few hundred femtoseconds is also noted in recent classical simulations in three dimensions [17,18]. Concurrent with the 12 energy loss, the energy of the Ar matrix increases (not shown here) and the total energy of the system is conserved. Before the photoabsorption of the molecule, the solvent is at equilibrium with the ground state 12 and its temperature is assumed to be 0 K for the sake of simplicity. Once the 12 is excited, the solvent atoms start to gradually move outwards due to the expanding 12. At t = 240 fs, the dissociating 12 molecule feels strongly the presence of the solvent cage since the solvent is lagging behind the rapid dissociation of 12 . At this point, part of the wave packet is reflected back by the solvent wall while the remaining part continuously pushes the solvent. As a result,
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
304
the wave packet exhibits a strong interference structure (see Fig. 3, t = 240 fs). At the same time, the solvent itself starts to yield to the highly energetic wave packet and acquire energy from the dissociating 12. The kinetic energy is propagated outwards in the solvent, i.e. sequential excitation of Ar, as shown in Fig. 5a. The Ar atoms are sequentially pushed from their initial equilibrium positions like a shock wave (see Fig. 5). Similar phenomenon has be observed in molecular dynamics simulations of photodissociation of impurity in matrices [34]. The position of the solvent wall at -~ 800 fs is more than 6 A (see Fig. 2, the long-dashed line). Because of the light mass compared with iodine, the Ar atoms, particularly the ones close to the I2, show certain extent of coherent motion. However, the motion of Ar is delayed somewhat from the 12 oscillation and has a slightly lower frequency. As results, the sol-
.........< V > of 12
2 f ;
6
vent wall expands and contracts with the 12 vibration. Subsequent to the initial collision with the solvent wall and the energy loss, the wave packet oscillates in the bound well of the A state. The oscillation has a much smaller amplitude and higher frequency than the first oscillation, as shown by the average I - I distance in Figs. 4b and 4e. The oscillation, which indicates the vibrational coherence, is clearly seen at the 734 nm pump wavelength, but not so well defined at short wavelengths, e.g. 705 nm. As indicated by the total energy of the 12 system in Figs. 4a and 4d, the energy loss after the initial collision is very slow, presumably due to the mismatch between the energy gap among the I - I vibration and the solvent phonon spectrum. This is in contrast to the classical simulation of Zadoyan et al. [17,18] who assigned a 12 ps decay time for the relaxation. It is noted that
~
........ < v > of 12
. . . . . < K > of 12
a
"; 1
. . . . . < K > of 12
- -
\~
- -
Total Energy of 12
d
Total Energy of 12
t
">~2
o -1 5 ,
,
,
e
4.5
3.5
, ,,
, ,
: ,
~
3 2.5
c
"i
f
°
0
1000
2000
3000
Time Delay (Is)
4000
1000
2000
3000
4000
Time Delay (Is)
Fig. 4. Upper panels are the averaged total energy (solid line), kinetic energy (dot-dashed line) and potential energy (dotted line) of 12 . The middle panels are the averaged I - I distance as a function of time. The simulated LIF signals are given in the lower panels. The results at 7 3 4 / 3 6 7 nm are plotted in (a)-(c) while the results at 7 0 5 / 3 5 2 nm in (d)-(f). The dotted lines in (c) and (f) are the simulated LIF signals with the outer probe window switched off.
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307 350
'~
200
,
,,~ !
- . . . . . . Ar~
If~ ;~ . . . . . . . . . .
0
'.~
:
o.6
Ar
,
il
i o., /i!i
.~
0.2
/
b
0 0
1000
2000
3000
4000
Time (fs) Fig. 5. The kinetic energies (a) and displacements (b) of the A r a t o m s w h i c h are nearest to the I a t o m at the right-hand side (see Fig. 1, n = 50) at the 734 nm excitation.
the phonon spectrum of the LC model is much more sparse than a three-dimensional simulation. Thus, the long time dynamics in our 1D model tends to be less reliable. 4.2. L I F transients
The experimental detection of the A state dynamics of 12 is made by using a frequency-doubled probe pulse and collecting the resulting fluorescence from the ion-pair states [17,18]. As indicated by the two probe arrows in Fig. 2, there are two windows for the probe pulse, which correspond to the resonant match between the A and D state potentials in the electromagnetic field. At the pump/probe wavelength of 734/367 nm, the inner and outer windows are approximately at 3.5 and 4.5 .~. Thus, the LIF signal is the probe of the dynamics occurring on the A state near these windows. The transition dipole moment for the A ~ D transition is that of Perrot et al. for the gas phase molecule [35]. The coordinatedependent dipole moment is fitted to a functional form similar to that proposed by Tellinghuisen [26]. The dipole at large I - I distance (Q > 4.2 ,~) is unknown and is extrapolated based on the data in
305
smaller distances. As noted earlier, the D state potential of 12 is adjusted in the calculations, following Zadoyan et al. [17]. However, no dynamic solvent modifications for this state have been considered. The calculated LIF transients are presented in Figs. 4c and 4f for the two wavelengths. Below, we focus our discussion on the dynamics at 734/367 nm. Similar conclusions can be drawn for the 705/352 nm results. As indicated by the first vertical bar in Figs. 4b and 4c, the first peak in the LIF transient can be unequivocally assigned to the first passage of the inner window (Q --- 3.5 ,~). When the wave packet moves to larger I-I distances, the LIF signal decreases. Immediately after the first peak, a small peak emerges at approximately 300 fs. This peak is due to the wave packet entering the outer probe window. At this point, most kinetic energy of the fragments has been lost and wave packet spends a long time at the turning point (see Fig. 3). Since the transition dipole at the outer window is much smaller than that at the inner window [26,35], this peak is not as strong as the first peak. A careful inspection of the LIF signal near 500 fs reveals a third peak (more conspicuous for the 705/352 nm excitation in Fig. 50, which was not seen in earlier classical simulations [17]. This peak can be identified as the inner window passage of the wave packet reflected from the solvent wall after the initial elastic collision. To verify this argument, we carried out simulations which switch off the outer window by carefully damping the transition dipole. The resulting LIF transients are given in the same figure in dashed lines. Since all the peaks here are due to the inner window, the origin of the peak at 500 fs is thus apparent. The reflection of the wave packet by the solvent wall is clearly seen in Fig. 3 near 500 fs. This peak raises some questions regarding the assignment of the experimental data near this region. First of all, our calculations indicate that the recurrence of the LIF signal due to the inner window is within 500 fs, instead of more than 1 ps in the classical study [17,18]. This early reappearance of the population at the inner window is apparently due to the partial reflection of a relatively broad wave packet by the solvent wall. The simultaneous dissociation and recombination are possible because of the
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L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
finite width of the pump pulse which creates a wave train on the excited state. The trailing part of the packet experiences a much farther solvent wall as a result of the energy transfer from the leading part of the packet to the solvent. In a classical simulation, the wave packet is approximated by a swarm of trajectories and each trajectory interacts individually with the solvent. This may be responsible for the absence of this peak in the classical simulations by Zadoyan et al. [17,18]. In addition, experimental data have shown that the peak-to-peak time elapse between the first two LIF peaks at 734/367 nm is approximately 400 fs [17,18]. It is thus difficult to interpret this as the sole consequence of the outer window probing because both the classical and our quantum/classical simulations indicate that such a peak occurs much earlier (~< 300 fs). Here, we propose to interpret the experimental peak at = 500 fs as the combined dynamics at both the inner window (reflected wave packet) and the outer window (outgoing wave packet). Although the exact value of the transition dipole moment at the outer window is unknown, it is generally agreed that it is much smaller than that near the inner window [26,35] (note that the same transition dipole is used for both windows in the classical simulation of I 2 / A r [17]). It is conceivable that the transition dipole at the outer window is so small that the inner window dynamics dominates the LIF transients (dashed lines in Figs. 5c and 5f). The concept of single window probing of the 12 photodissociation/recombination dynamics has been advanced by a number of groups [16,19,24]. This argument is partially supported by the high resolution experimental transients for the direct dissociation on the B state of an isolated I 2 at the pump wavelength of A1 < 505 nm (and a probe wavelength of 310 nm) [16,36,37], which show only a single peak. If the outer probe window were optically active, there would have been two peaks in the LIF signal. Moreover, the higher resolution LIF signal of photodissociation of I e in matrix Kr fails to show double peaks (or fork) near t = 400 fs [18]. Such a double-peak is a signature of double passage (dissociation and reflection) of the wave packet at the outer window. Another weakness of all the simulations is the lack of dynamic solvent modification of the D state potential. It is expected that the potential of the probe state is strongly influenced by the
solvent due to the large charge separation of the ion pair. Of course, an unequivocal assignment of this peak requires more reliable information on the position of the probe windows and the transition dipoles near these windows in a matrix or cluster environment. Following the peaks in the first picosecond, there is a sequence of peaks. Comparison between the single and dual window LIF transients (solid and dashed lines in Figs. 4c and 4f) indicates that the signal from the outer probe window is usually insignificant. This is because the wave packet hardly reaches the outer window after losing all its excess energy. These peaks can be grouped together and the time interval between two nearby given groups is roughly 900 fs (dashed vertical bars) for the pump/probe wavelengths at 734/367 nm. This frequency is clearly indicative of the coherent wave packet passing through the inner probe window. Each group has two strong peaks, corresponding to wave packets moving outwards and inwards. These peaks are usually more intense than the first peak because the kinetic energy of the recombined wave packet is much smaller than that of the initial dissociating wave packet. The former thus spends more time near the probe window, resulting in a larger peak intensity. Our calculated oscillation frequency for the coherent I - I motion is lower than the classical simulations in three dimensions. Again, it can be attributed to the inefficient relaxation of the wave packet after the initial energy loss. In addition, the classical transients exhibit much less structure. It is expected that the removal of the 1D restriction in the simulation should 'wash out' most of the sharp peaks seen here. Because of limitations of the LC model as discussed above, it is not possible for us to extract the long time behavior of the system, which has been shown to be closely related to the relaxation of the recombined 12 [17,18]. Since the energy loss in our 1D quantum/classical simulation is much slower than the experimental rate, a direct comparison has to be deferred to a full three-dimensional treatment. 5. Conclusion
We have shown in this work that the photodissociation and recombination dynamics of molecular
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
iodine can be treated w i t h a q u a n t u m / c l a s s i c a l hybrid method. The d y n a m i c s o f system (12) is described with a t i m e - d e p e n d e n t q u a n t u m w a v e packet and the bath (Ar) is treated classically. This m o d e l allows a proper treatment o f the w a v e packet m o t i o n on radiatively coupled electronic states, with the influence of a classical bath. Our m o d e l has b e e n able to reproduce the large energy transfer and the retention of c o h e r e n c e in the early photodissociat i o n / r e c o m b i n a t i o n d y n a m i c s o f 12 in matrix At. It p r o v i d e s a m u c h m o r e solid basis for the detailed analysis o f the q u a n t u m w a v e packet m o t i o n in c o n d e n s e d phases. This m e t h o d appears to be efficient and accurate, and it has the potential to be e x t e n d e d to three d i m e n s i o n s and to d y n a m i c s inv o l v i n g m o r e than one electronic state.
Acknowledgements This w o r k was supported by the donors o f P e t r o l e u m R e s e a r c h Fund, administered by A m e r i c a n C h e m i c a l Society; and by the National S c i e n c e F o u n d a t i o n ( C H E - 9 4 1 1 9 3 4 ) . H G w o u l d like to thank A.H. Z e w a i l for the w a r m hospitality during his visit to Caltech in 1993, w h e r e s o m e o f the ideas o f this w o r k w e r e developed. Part of the calculation was p e r f o r m e d on the Cray Y M P at the O h i o S u p e r c o m puter Center.
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307
[8] D. Thirumalai, E.J. Bruskin and B.J. Berne, J. Chem. Phys. 83 (1985) 230. [9] R. Alimi, R.B. Gerber, A.D. Hammerich, R. Kosloff and M.A. Ratner, J. Chem. Phys. 93 (1990) 6484. [10] A. Garcia-Vela, R.B. Gerber and D.G. Imre, J. Chem. Phys. 97 (1992) 7242. [11] Y. Guan, J.T. Muckerman and T. Uzer, J. Chem. Phys. 93 (1990) 4400. [12] K. Haug and H. Metiu, J. Chem. Phys. 99 (1993) 6253. [13] A.E. Roitberg, R.B. Gerber and M.A. Ratner, J. Chem. Phys. 100 (1994) 4355. [14] A.L. Harris, J.K. Brown and C.B. Harris, Ann. Rev. Phys. Chem. 39 (1988) 341. [15] J. Franck and E. Robinovitch, Trans. Faraday Soc. 30 (1934) 120. [16] Q. Liu, J.-K. Wang and A.H. Zewail, Nature 364 (1993) 427. [17] R. Zadoyan, Z. Li, P. Ashjian, C.C. Martens and V.A. Apkarian, Chem. Phys. Letters 218 (1994) 504. [18] R. Zadoyan, Z. Li, C.C. Martens and V.A. Apkarian, J. Chem. Phys. 101 (1994) 6648. [19] Y. Yan, R.M. Whitnell, K.R. Wilson and A.H. Zewail, Chem. Phys. Letters 193 (1992) 402. [20] W. Wang and K.A. Nelson, J. Chem. Phys. 101 (1994) 9663. [21] G. Roberts and A.H. Zewail, J. Phys. Chem. 95 (1991) 7973. [22] A.J. Stace and J.N. Murrell, Mol. Phys. 33 (1977) 1. [23] J.N. Murrell, A.J. Stace and R. Dammel, J. Chem. Soc. Faraday Trans. 74 (1978) 1532. [24] E.D. Potter, Q. Liu and Z.H. Zewail, Chem. Phys. Letters 200 (1992) 605. [25] M. Messina and R.D. Coalson, J. Chem. Phys. 90 (1989) 4015. [26] J. Tellinghuisen, Can. J. Phys. 62 (1984) 1933. [27] P.A.M. Dirac, Proc. Cambridge Phil. Soc. 26 (1930) 376. [28] R.B. Gerber, V. Buch and M.A. Ratner, J. Chem. Phys. 77 (1982) 3022. [29] S.O. Williams and D.G. Imre, J. Phys. Chem. 92 (1988) 6648. [30] H. Guo and A.H. Zewail, Can. J. Chem. 72 (1994) 947. [31] V. Engel and H. Metiu, J. Chem. Phys. 91 (1989) 1596. [32] T.J. Park and J.C. Light, J. Chem. Phys. 85 (1986) 5870. [33] M. Macler and M.C. Heaven, Chem. Phys. 151 (1991) 219. [34] A. Borrmann and C.C. Martens, J. Chem. Phys. 102 (1995) 1905. [35] J.P. Perrot, B. Femela, J.L. Subtil, M. Broyer and J. Chevaleyre, Mol. Phys. 61 (1987) 85. [36] A.H. Zewail, M. Dantus, R.M. Bowman and A. Mokhtari, J. Photochem. Photobiol. A 62 (1992) 301. [37] R.M. Bowman, M. Dantus and A.H. Zewail, Chem. Phys. Letters 161 (1989) 297.
7~
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 237 (1995) 299-307
Quantum/classical hybrid dynamics of I2(A) photodissociation and recombination in matrix Ar, linear chain model Li Liu, Hua Guo Department of Chemistry, University of Toledo, Toledo, OH 43606, USA Received 9 February 1995
Abstract
A quantum/classical hybrid model is developed for the photodissociation and recombination dynamics of an 12 molecule in low temperature matrix Ar. This simplified model consists of an 12 molecule embedded in a linear chain of Ar atoms. The dynamics of 12 on its A state is treated quantum mechanically while the Ar matrix is described classically. The quantum system is self-consistently coupled with the classical bath. Our calculations show substantial energy transfer and retention of coherence in the early stages of the recombination process. The time-delayed pump-probe transients are calculated and compared with experiment and other theoretical simulations.
I. Introduction
With the development of ultrashort laser techniques, more and more chemical reactions have been investigated experimentally with femtosecond resolution [1]. However, exact quantum mechanical treatments for systems with more than three degrees of freedom are still formidable, despite recent advances in time-dependent quantum theory of molecular dynamics [2]. It is even more difficult, if not impossible, to provide a complete quantum mechanical description for chemical processes in condensed phases which involve a much larger number of modes. Alternative methodology in treating dynamical processes in these situations is thus highly desirable. Much effort has been devoted in this direction in the last few decades. It has long been realized that the motion of solvent is usually much slower than that of solute. Hence, it is advantageous to partition a
system into two interacting parts: the reactive (sub)system and the surrounding bath. For example, reduced density matrix approaches to condensed phase problems are almost uniformly based on the assumption that the system and bath modes are separable [3]. By assuming thermal equilibrium for the bath, the dynamics of the system can be characterized by the reduced density matrix which is governed by the Liouville-von Neumann equation. Similar ideas have been implemented without the density matrix formalism. In collision processes, for example, it can be shown [4] that the slow translational motion can be described by trajectories in the classical limit while vibrational modes are treated quantum mechanically. Such hybrid quantum/classical approaches have found wide applications in many dynamical calculations [4-13]. In this work, a quantum/classical hybrid model for the photodissociation and recombination of an 12
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 0 0 9 - 2614(95 ) 0 0 3 1 8 - 5
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L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
in matrix Ar is presented. The I2/liquid system has been extensively investigated theoretically and experimentally [14] and it serves as a prototypical model for studying the so-called 'cage effect' in condensed phases [15]. Here, we will focus on the A state dynamics of 12 in low temperature Ar. Femtosecond experiments indicated prompt recombination (caging) following photodissociation of 12 in both clusters [16] and matrices [17,18]. With a standard pump-probe scheme, these experiments revealed retention of coherence despite substantial energy transfer in the early stages of the dissociation. Classical trajectory calculations [16-20] have been reported for the dissociation and recombination processes and the results are generally consistent with the experimental observations. Although the iodine system is rather heavy and presumably behaves more or less classically, a quantum description is still desirable because some important dynamical features can only be treated quantum mechanically. For instance, the excited state iodine wave packet created by an ultrashort laser pulse is a coherent superposition of several vibrational states. The coherence of the wave packet and its destruction (dephase and relaxation) is a quantum mechanical phenomenon, which has no classical analog. Although many studies have shown that the center of a wave packet can sometimes be described by a classical trajectory [21], it is difficult to describe the complete wave packet dynamics classically. This is particularly conspicuous at turning points where the wave packet may exhibit significant interferences between the incident and reflected waves. Another important quantum feature in the recombination process is non-adiabatic transitions between the excited manifold and the ground electronic state (not treated here). The quantum/classical hybrid method allows one to incorporate an accurate quantum treatment of the system into a classical description of the environment and it may provide a different and more accurate viewpoint of the dynamics.
Ar I
Ar2
Arn. 1
Ar n
[
2. Linear chain (LC) model In this work, we impose a rather oversimplified approximation in our calculations. We assume that the dynamics of photodissociation and recombination of I 2 o c c u r in one dimension. Particularly, our model consists of an 12 molecule embedded in a linear chain of Ar atoms. The iodine molecule is coupled with an Ar atom at each end with a Lennard-Jones 6 - 1 2 potential. Each Ar atom in the bath is connected to its nearest neighbors with the same type of potentials. The last Ar atoms in the chain are connected to stationary walls. The LC model is schematically depicted in Fig. 1. Since the interaction between adjacent atoms is not harmonic, this one-dimensional model represents a non-trivial prototype for impurity system in condensed phases. Because the symmetry of the LC model and the heavy mass of the iodine, the center of mass of the molecule is removed from the model. The A r - I and A r - A r interaction parameters are adapted from previous work [17,22-24], and the intramolecular potentials (for both ground and excited states) of the iodine molecule are those that Potter et al. [24] and Zadoyan et al. [17] used in their molecular dynamics simulations. This LC model is similar to that of Messina and Coalson [25] in studying the emission spectrum of I 2 in Ar, and to that of Roitberg et al. [13] in calculating vibrational relaxation of an impurity in clusters. Admittedly, the LC model is a rudimental one which may not be able to convey a complete picture of dynamics. However, it is our believe that such a simple model should shed light on some general features of the dissociation/recombination as well as energy transfer processes. Since the LC model allows no dissociation, our simulations are only relevant to the recombination process. However, it is noted that no cage escape has been observed in low temperature rare gas matrices [17,18]. In addition, the solute would have little time to rotate during the
I
Am+ 1
Arn+2
Ar2n. 1
Ar2n
Fig. 1. Geometry of the linear chain (LC) model for I2Ar2n. The diatomic bond of 12 is described by the strong Morse potential; the interactions of the I - A t and A r - A r pairs are weak Lennard-Jones potentials.
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L. Liu, 1t. Guo / Chemical Physics Letters 237 (1995) 299-307
first few picoseconds and the LC model should be a reasonable approximation to the early dynamics. More importantly, the LC model presents an idealized and tractable approach to reveal the fundamental physics. It should be pointed out that the extension of the quantum/classical method from the LC model to a full three-dimensional simulation is straightforward.
Table 1 Potential parameters for the 12/mr system Lennard-Jones parameters
Ar-Ar Ar-I
O" (m)
84 130.24
3.40 3.617
Morse parameters
X A
3. Quantum/classical hybrid method
• (cm- 1)
D (cm- l )
Qe
/3 (,~- l)
V0 (cm- 1)
12547.2 1840
2.656 3.10
1.875 2.147
0 7605
Within the LC model described above, the total Hamiltonian can be written as follows:
H(Q, q) =H~ + H
b +Hsb 2n p2
p2
= --
2.
+ V~(Q) + Y'~ i=1
+ Vb(q)
Tmm
+ V~b(Q, q),
(1)
where m and /x are the mass of Ar and the reduced mass of 12, respectively. The quantum degree of freedom Q is the 12 vibrational coordinate while the collective bath coordinate q denotes the classical displacements of the 2n Ar atoms in the matrix. There are three electronic states of 12 involved in the model: the ground state, the A state and the ion-pair state. The ion-pair D state, in its Rittner form, is adapted from Tellinghuisen's work [26]. The X and A states are represented by Morse functions of the form Vs(Q) = O{exp[ - 2/3(Q - Q e ) ]
-2exp[-fl(Q-Qe)]}+V o.
(2)
The bath potential Vb is the sum of the Lennard-Jones type potentials between two adjacent Ar atoms, [[ O-~12
VLj(r)=4,[[r)
6]
--(~)
.
(3)
A similar form of potential is used for the systembath interaction V~b between an iodine atom and an adjacent Ar. All the potential parameters are defined in Table 1. The quantum/classical hybrid method is formally based on the time-dependent Hartree (TDH) approximation [27,28]. The TDH assumes that the total wave function can be factorized into the product of
the system part and the bath part. Such factorization allows one to treat the system and bath separately, with either a full quantum or a quantum/classical description. Because of the time-dependence of the wave function, energy transfer between the system and bath is allowed in the TDH scheme. The wave functions of the system and the bath are subjected to the following effective Hamiltonians: p2 H ~ " = - - + Vs(a ) + (V~b(O, q))q, (4a) 2/x 2n p2 n~ ff= ~ ~ + Vb(q) + (Vsb(Q ,
q))Q.
(4b)
i=1
The coupling terms between the system and bath are introduced by the mean-field potentials. The above approach is based on two considerations, namely, the interaction between the system and the classical environment is usually weak and the time-scale for the primary dynamics is much faster than the solvent motion. The quantum/classical hybrid method goes one step beyond the quantum TDH approximation in that the bath modes are treated classically. Under such circumstances, the effective potential for the quantum mode is modified by the time-dependent coupling which is a function of the solvent coordinate q. On the other hand, the dynamics of the system exerts forces on to the bath atoms through the averaged coupling potential. Specifically, the average over the system coordinate is made by substituting Q with its expectation value ( Q ) . The dynamics of the two subsystems is thus self-consistently coupled. For the quantum degree of freedom, the equation
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
302
of motion can be written as follows (atomic units are used) [29,30]:
/ Ha
0 i - - ~02 = Ot ~3
0 /
H21
H2
H23
q~2 ,
0
H32
H3
¢~3
(5)
where ~t)i (i = 1, 2, 3) are the nuclear wave functions on three electronic surfaces corresponding to the X, A and D states, respectively. The diagonal terms in the Hamiltonian matrix should be understood as effective Hamiltonians for each of the electronic states (Eq. (4a)). Note that the potentials are functions of the solvent coordinates and the dynamics of the system is influenced by the time-dependent fluctuations of the solvent. The off-diagonal terms in the Hamiltonian matrix represent radiative couplings between the electronic states, HI2 =/.~12Al exp('yl t2) exp(itolt),
(6a)
H23 =/XE3A 2 e x p [ - y z ( t -
(6b)
r ) 2] exp(ito2t),
where /~12 and /./,23 are the transition dipole moments for the X ~ A and the A ~ D transitions, respectively; to, A and y are the central frequency, laser intensity and the width of the pulses. The probe pulse is delayed relative to the pump pulse by r. Experimentally, the transient signal is collected by measuring the fluorescence from the probe state (D) after pump and delayed probe pulses. The laser-induced fluorescence (LIF) yield for a given time delay is thus [29,31] P(7-) = lim (q~3(t, r ) [ tP3(t , 7-)). t ---~oo
(7)
The above integral approaches a constant as the probe pulse vanishes. In our calculations, the probe process is treated perturbatively. That is, we assume A 2 is so small that no population trickles down from the D state to the A state during probe pulse duration. This weak-field approximation is identical to assuming H23 = 0 (but H32 ~ 0) in Eq. (5). Computationally, the probe and pump processes are thus separable and the CPU time is reduced by a factor of four. The equations of motion for the classical bath are given below in the Hamilton's form,
d qi dt
0H~ff Opi '
d Pi
0H~ff
dt
Oqi
(8)
The right-hand side of the second equation represents the forces exerted on this atom. Since only the nearest neighbor interactions are included, there are only two force terms for each solvent atom. For the two Ar atoms directly connected with the iodine, the forces are from both the I atom and from an adjacent Ar. For the rest of the Ar atoms, the action from the quantum system is indirect and all the forces are due to the nearby solvent atoms. The time-dependent Schr6dinger equation (Eq. (5)) is solved numerically on a grid by using an efficient fast Fourier transform (FFT) method [2]. Basically, the potential and kinetic energy operators are evaluated in their own diagonal representations and transformation of the wave packet between these representations is accomplished via FF£. We use a short iterative Lanczos method [2,32] to propagate the wave packet. The spatial grid has 512 uniformly distributed points from 2.35 to 7.6 .~ for 12. The widths of the pump and probe pulses are 100 fs (fwhm). For the classical motion of Ar, the Hamilton's equation (Eq. (8)) is solved by using the second-order differencing method. The propagation of the trajectory and the wave packet must be carried out in parallel. The time step for propagation is 0.2 fs. 100 Ar atoms are found to give converged results for dynamics up to 4 ps.
4. Results
4.1. Wave packet dynamics The potential curves of the I2/Ar2n system are given in Fig. 2. The potential energy curves for an isolated 12 are plotted in solid lines while the matrix modified effective potentials are given in either dash-dotted or dashed lines. The potential curve of probe state V3 is vertically downshifted and its equilibrium position is stretched outwards by = 0.2 A from its gas phase counterpart, because the solvation of the dipole stabilizes this state [17,33]. It can be readily seen that there is a repulsive wall at large I - I distances due to the presence of the solvent. This solvent wall is responsible for the recombination of the dissociated fragments (cage effect). Also note that the wall is pushed outwards as the dissociation proceeds towards larger I - I distances.
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
The snap shots of the wave packet are displayed in Fig. 3. The photodissociation of 12 near 734 nm is initiated through a vertical transition from the ground state to the A state at an initial excess energy of ,~ 1 0 0 0 c m - 1 above the dissociation limit. The high initial energy ( = 1500 cm -1 at t < 0 ) is due to transient species that will eventually return to the ground state when the pump pulse finishes. Since the wave packet experiences a strong repulsive force near the Franck-Condon region, it moves rapidly downhill and 12 dissociates. The elongation of the bond is then slowed down by the solvent wall and the I fragments start to recombine geminately. The average I - I distances for the wave packet at two pump wavelengths are plotted against time in Figs. 4b and 4c. It starts with the ground state equilibrium distance and reaches at = 4.2 A at the first encounter with the solvent cage. During the collision, most of the kinetic energy of the wave packet is transferred to the solvent. The energy transfer is extremely efficient since the energy density of the wave packet is very high (nearly in a continuum) and coincides well with the Ar phonon spectrum. The total energy of the 12 system is shown in Figs. 4a and 4d to decrease rapidly: more than 1000 cm-1 of
303
t=0fs
t =80fs
t= 160fs
t = 240 fs
t = 320 fs
t = 480 fs
820 fs
~
t= 1120fs i t
t = 128o fs
] ~ ~ , . ~ . 3
4
5
1 6
7
I-I Distance (/~) Fig. 3. Snap shots of wave packet motion on the A state of 12 at the pump wavelength of 734 nm.
4i A.............. D
.....
3
i
t.
o -1
X ,
,
i
3
. . . .
i
4
. . . .
i
5
. . . .
i
6
. . . .
J
,
7
I-I Distance (/~) Fig. 2. Potential energy curves for an isolated 12 (solid lines) and its effective potential energy curves in matrix Ar. The dashed-dot line ( . . . . . ) corresponds to the A state potential in a frozen Ar matrix before the photo-excitation. The short dashed line ( . . . . . ) and the dotted line ( . . . . . . ) represent the time-dependent potential at 800 and 1900 fs, respectively. The pump-probe scheme is depicted in the figure and the two probe windows are indicated by the two arrows.
energy is lost in the first = 500 fs. The significant energy transfer in the first few hundred femtoseconds is also noted in recent classical simulations in three dimensions [17,18]. Concurrent with the 12 energy loss, the energy of the Ar matrix increases (not shown here) and the total energy of the system is conserved. Before the photoabsorption of the molecule, the solvent is at equilibrium with the ground state 12 and its temperature is assumed to be 0 K for the sake of simplicity. Once the 12 is excited, the solvent atoms start to gradually move outwards due to the expanding 12. At t = 240 fs, the dissociating 12 molecule feels strongly the presence of the solvent cage since the solvent is lagging behind the rapid dissociation of 12 . At this point, part of the wave packet is reflected back by the solvent wall while the remaining part continuously pushes the solvent. As a result,
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
304
the wave packet exhibits a strong interference structure (see Fig. 3, t = 240 fs). At the same time, the solvent itself starts to yield to the highly energetic wave packet and acquire energy from the dissociating 12. The kinetic energy is propagated outwards in the solvent, i.e. sequential excitation of Ar, as shown in Fig. 5a. The Ar atoms are sequentially pushed from their initial equilibrium positions like a shock wave (see Fig. 5). Similar phenomenon has be observed in molecular dynamics simulations of photodissociation of impurity in matrices [34]. The position of the solvent wall at -~ 800 fs is more than 6 A (see Fig. 2, the long-dashed line). Because of the light mass compared with iodine, the Ar atoms, particularly the ones close to the I2, show certain extent of coherent motion. However, the motion of Ar is delayed somewhat from the 12 oscillation and has a slightly lower frequency. As results, the sol-
.........< V > of 12
2 f ;
6
vent wall expands and contracts with the 12 vibration. Subsequent to the initial collision with the solvent wall and the energy loss, the wave packet oscillates in the bound well of the A state. The oscillation has a much smaller amplitude and higher frequency than the first oscillation, as shown by the average I - I distance in Figs. 4b and 4e. The oscillation, which indicates the vibrational coherence, is clearly seen at the 734 nm pump wavelength, but not so well defined at short wavelengths, e.g. 705 nm. As indicated by the total energy of the 12 system in Figs. 4a and 4d, the energy loss after the initial collision is very slow, presumably due to the mismatch between the energy gap among the I - I vibration and the solvent phonon spectrum. This is in contrast to the classical simulation of Zadoyan et al. [17,18] who assigned a 12 ps decay time for the relaxation. It is noted that
~
........ < v > of 12
. . . . . < K > of 12
a
"; 1
. . . . . < K > of 12
- -
\~
- -
Total Energy of 12
d
Total Energy of 12
t
">~2
o -1 5 ,
,
,
e
4.5
3.5
, ,,
, ,
: ,
~
3 2.5
c
"i
f
°
0
1000
2000
3000
Time Delay (Is)
4000
1000
2000
3000
4000
Time Delay (Is)
Fig. 4. Upper panels are the averaged total energy (solid line), kinetic energy (dot-dashed line) and potential energy (dotted line) of 12 . The middle panels are the averaged I - I distance as a function of time. The simulated LIF signals are given in the lower panels. The results at 7 3 4 / 3 6 7 nm are plotted in (a)-(c) while the results at 7 0 5 / 3 5 2 nm in (d)-(f). The dotted lines in (c) and (f) are the simulated LIF signals with the outer probe window switched off.
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307 350
'~
200
,
,,~ !
- . . . . . . Ar~
If~ ;~ . . . . . . . . . .
0
'.~
:
o.6
Ar
,
il
i o., /i!i
.~
0.2
/
b
0 0
1000
2000
3000
4000
Time (fs) Fig. 5. The kinetic energies (a) and displacements (b) of the A r a t o m s w h i c h are nearest to the I a t o m at the right-hand side (see Fig. 1, n = 50) at the 734 nm excitation.
the phonon spectrum of the LC model is much more sparse than a three-dimensional simulation. Thus, the long time dynamics in our 1D model tends to be less reliable. 4.2. L I F transients
The experimental detection of the A state dynamics of 12 is made by using a frequency-doubled probe pulse and collecting the resulting fluorescence from the ion-pair states [17,18]. As indicated by the two probe arrows in Fig. 2, there are two windows for the probe pulse, which correspond to the resonant match between the A and D state potentials in the electromagnetic field. At the pump/probe wavelength of 734/367 nm, the inner and outer windows are approximately at 3.5 and 4.5 .~. Thus, the LIF signal is the probe of the dynamics occurring on the A state near these windows. The transition dipole moment for the A ~ D transition is that of Perrot et al. for the gas phase molecule [35]. The coordinatedependent dipole moment is fitted to a functional form similar to that proposed by Tellinghuisen [26]. The dipole at large I - I distance (Q > 4.2 ,~) is unknown and is extrapolated based on the data in
305
smaller distances. As noted earlier, the D state potential of 12 is adjusted in the calculations, following Zadoyan et al. [17]. However, no dynamic solvent modifications for this state have been considered. The calculated LIF transients are presented in Figs. 4c and 4f for the two wavelengths. Below, we focus our discussion on the dynamics at 734/367 nm. Similar conclusions can be drawn for the 705/352 nm results. As indicated by the first vertical bar in Figs. 4b and 4c, the first peak in the LIF transient can be unequivocally assigned to the first passage of the inner window (Q --- 3.5 ,~). When the wave packet moves to larger I-I distances, the LIF signal decreases. Immediately after the first peak, a small peak emerges at approximately 300 fs. This peak is due to the wave packet entering the outer probe window. At this point, most kinetic energy of the fragments has been lost and wave packet spends a long time at the turning point (see Fig. 3). Since the transition dipole at the outer window is much smaller than that at the inner window [26,35], this peak is not as strong as the first peak. A careful inspection of the LIF signal near 500 fs reveals a third peak (more conspicuous for the 705/352 nm excitation in Fig. 50, which was not seen in earlier classical simulations [17]. This peak can be identified as the inner window passage of the wave packet reflected from the solvent wall after the initial elastic collision. To verify this argument, we carried out simulations which switch off the outer window by carefully damping the transition dipole. The resulting LIF transients are given in the same figure in dashed lines. Since all the peaks here are due to the inner window, the origin of the peak at 500 fs is thus apparent. The reflection of the wave packet by the solvent wall is clearly seen in Fig. 3 near 500 fs. This peak raises some questions regarding the assignment of the experimental data near this region. First of all, our calculations indicate that the recurrence of the LIF signal due to the inner window is within 500 fs, instead of more than 1 ps in the classical study [17,18]. This early reappearance of the population at the inner window is apparently due to the partial reflection of a relatively broad wave packet by the solvent wall. The simultaneous dissociation and recombination are possible because of the
306
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
finite width of the pump pulse which creates a wave train on the excited state. The trailing part of the packet experiences a much farther solvent wall as a result of the energy transfer from the leading part of the packet to the solvent. In a classical simulation, the wave packet is approximated by a swarm of trajectories and each trajectory interacts individually with the solvent. This may be responsible for the absence of this peak in the classical simulations by Zadoyan et al. [17,18]. In addition, experimental data have shown that the peak-to-peak time elapse between the first two LIF peaks at 734/367 nm is approximately 400 fs [17,18]. It is thus difficult to interpret this as the sole consequence of the outer window probing because both the classical and our quantum/classical simulations indicate that such a peak occurs much earlier (~< 300 fs). Here, we propose to interpret the experimental peak at = 500 fs as the combined dynamics at both the inner window (reflected wave packet) and the outer window (outgoing wave packet). Although the exact value of the transition dipole moment at the outer window is unknown, it is generally agreed that it is much smaller than that near the inner window [26,35] (note that the same transition dipole is used for both windows in the classical simulation of I 2 / A r [17]). It is conceivable that the transition dipole at the outer window is so small that the inner window dynamics dominates the LIF transients (dashed lines in Figs. 5c and 5f). The concept of single window probing of the 12 photodissociation/recombination dynamics has been advanced by a number of groups [16,19,24]. This argument is partially supported by the high resolution experimental transients for the direct dissociation on the B state of an isolated I 2 at the pump wavelength of A1 < 505 nm (and a probe wavelength of 310 nm) [16,36,37], which show only a single peak. If the outer probe window were optically active, there would have been two peaks in the LIF signal. Moreover, the higher resolution LIF signal of photodissociation of I e in matrix Kr fails to show double peaks (or fork) near t = 400 fs [18]. Such a double-peak is a signature of double passage (dissociation and reflection) of the wave packet at the outer window. Another weakness of all the simulations is the lack of dynamic solvent modification of the D state potential. It is expected that the potential of the probe state is strongly influenced by the
solvent due to the large charge separation of the ion pair. Of course, an unequivocal assignment of this peak requires more reliable information on the position of the probe windows and the transition dipoles near these windows in a matrix or cluster environment. Following the peaks in the first picosecond, there is a sequence of peaks. Comparison between the single and dual window LIF transients (solid and dashed lines in Figs. 4c and 4f) indicates that the signal from the outer probe window is usually insignificant. This is because the wave packet hardly reaches the outer window after losing all its excess energy. These peaks can be grouped together and the time interval between two nearby given groups is roughly 900 fs (dashed vertical bars) for the pump/probe wavelengths at 734/367 nm. This frequency is clearly indicative of the coherent wave packet passing through the inner probe window. Each group has two strong peaks, corresponding to wave packets moving outwards and inwards. These peaks are usually more intense than the first peak because the kinetic energy of the recombined wave packet is much smaller than that of the initial dissociating wave packet. The former thus spends more time near the probe window, resulting in a larger peak intensity. Our calculated oscillation frequency for the coherent I - I motion is lower than the classical simulations in three dimensions. Again, it can be attributed to the inefficient relaxation of the wave packet after the initial energy loss. In addition, the classical transients exhibit much less structure. It is expected that the removal of the 1D restriction in the simulation should 'wash out' most of the sharp peaks seen here. Because of limitations of the LC model as discussed above, it is not possible for us to extract the long time behavior of the system, which has been shown to be closely related to the relaxation of the recombined 12 [17,18]. Since the energy loss in our 1D quantum/classical simulation is much slower than the experimental rate, a direct comparison has to be deferred to a full three-dimensional treatment. 5. Conclusion
We have shown in this work that the photodissociation and recombination dynamics of molecular
L. Liu, H. Guo / Chemical Physics Letters 237 (1995) 299-307
iodine can be treated w i t h a q u a n t u m / c l a s s i c a l hybrid method. The d y n a m i c s o f system (12) is described with a t i m e - d e p e n d e n t q u a n t u m w a v e packet and the bath (Ar) is treated classically. This m o d e l allows a proper treatment o f the w a v e packet m o t i o n on radiatively coupled electronic states, with the influence of a classical bath. Our m o d e l has b e e n able to reproduce the large energy transfer and the retention of c o h e r e n c e in the early photodissociat i o n / r e c o m b i n a t i o n d y n a m i c s o f 12 in matrix At. It p r o v i d e s a m u c h m o r e solid basis for the detailed analysis o f the q u a n t u m w a v e packet m o t i o n in c o n d e n s e d phases. This m e t h o d appears to be efficient and accurate, and it has the potential to be e x t e n d e d to three d i m e n s i o n s and to d y n a m i c s inv o l v i n g m o r e than one electronic state.
Acknowledgements This w o r k was supported by the donors o f P e t r o l e u m R e s e a r c h Fund, administered by A m e r i c a n C h e m i c a l Society; and by the National S c i e n c e F o u n d a t i o n ( C H E - 9 4 1 1 9 3 4 ) . H G w o u l d like to thank A.H. Z e w a i l for the w a r m hospitality during his visit to Caltech in 1993, w h e r e s o m e o f the ideas o f this w o r k w e r e developed. Part of the calculation was p e r f o r m e d on the Cray Y M P at the O h i o S u p e r c o m puter Center.
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