J. Phys. Chem. A 2010, 114, 9743–9748
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Dissipative Wave Packet Dynamics of Hydrophobic f Hydrophilic Site Switching in Phenol-Ar Clusters† Ch. Walter,,‡ R. Kritzer,,‡ A. Schubert,,‡ C. Meier,§ O. Dopfer,| and V. Engel*,‡ Institut fu¨r Physikalische und Theoretische Chemie, UniVersita¨t Wu¨rzburg, Am Hubland, D-97074 Wu¨rzburg, Germany, Laboratoire de Collisions, Agre´gats et ReactiVite´, IRSAMC, UniVersite´ Paul Sabatier, 31062 Toulouse, France, and Institut fu¨r Optik und Atomare Physik, Technische UniVersita¨t Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany ReceiVed: March 4, 2010; ReVised Manuscript ReceiVed: April 20, 2010
We analyze the results of recent pump-probe experiments on the site-switching dynamics of Ar within cationic phenol-Ar2 clusters. A reaction-path model is employed for the wave packet dynamics. It is shown that the mechanism of energy dissipation is to be included to understand the features of the transient signals. Therefore, a simple recipe to include energy relaxation to the quantum dynamics is introduced. It is then possible to reproduce the measured pump-probe signals by adjustment of only two parameters. 1. Introduction Intermolecular interactions of aromatic molecules are vital for chemical and biological recognition.1,2 A detailed understanding of these interactions at the molecular level requires accurate knowledge of the intermolecular potential energy surface and the dynamics evolving on this surface. Clusters of phenol with neutral ligands are attractive model systems to investigate the competition of two different fundamental types of intermolecular forces, namely, hydrogen bonding to the acidic OH group (H-bond, hydrophilic interaction) and van der Waals bonding to the highly polarizable π electron system of the aromatic ring (π-bond, stacking, hydrophobic interaction). These different binding sites are illustrated in Figure 1. The relative interaction strengths of both binding motifs strongly depend on the type of ligand and the charge state of phenol.2 The present work deals with the dynamics on the potential energy surface of the phenol-Ar2 cation initiated by ionization of the neutral cluster.3,4 A detailed account of the present theoretical and spectroscopic knowledge on neutral and ionic phenol-Arn clusters has been reported recently.5 In brief, a variety of spectroscopic techniques have demonstrated that neutral phenol-Ar and phenol-Ar2 clusters produced in a molecular beam have π-bonded (1/0) and (1/1) structures, respectively,inlinewithhigh-levelquantumchemicalcalculations.5,6 In these structures, the Ar atoms are located on opposite sides of the aromatic ring, and the attraction is dominated by dispersion interactions of Ar with the polarizable aromatic π-electron system. Only π-bonded complexes of neutral phenol-Ar and phenol-Ar2 were detected so far,6 and it is unclear in the present stage whether the H-bonded isomer of phenol-Ar is a local minimum on the potential of the neutral cluster. Spectroscopic experiments clearly show that the H-bonded isomer is the global minimum on the potential of the phenol-Ar cation, whereas the π-bonded isomer is only a less stable local minimum.7–10 Again, this result was confirmed by quantum chemical calculations.7,11 The †
Part of the “Reinhard Schinke Festschrift”. * Corresponding author. Tel: +49-931-31-86376. Fax: +49-931-3186362. E-mail:
[email protected]. ‡ Universita¨t Wu¨rzburg Am Hubland. § Universite´ Paul Sabatier. | Technische Universita¨t Berlin.
Figure 1. Illustration of π- versus H-binding in phenol-Ar.
attraction in the phenol-Ar cation is dominated by induction forces, which favor H-bonding over π-bonding. The ionizationinduced π f H switch in the preferred recognition motif has been first identified for phenol-Ar2–4,7–10 and has meanwhile been recognized as a general phenomenon for acidic aromatic molecules interacting with nonpolar solvents.2,12–18 In recent experiments, the dynamics of the hydrophobic-tohydrophilic π f H switching motion induced by ionization has been monitored for the first time in real time by picosecond time-resolved UV-UV-IR pump-probe spectroscopy of the phenol-Ar2 trimer.3,4 The results of these experiments can be summarized as follows: (a) the π f H switching dynamics takes
10.1021/jp101964e 2010 American Chemical Society Published on Web 04/30/2010
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Walter et al. excitation of the OH stretch in the H-bonding geometry takes place. Then, the transients reflect the arrival of the Ar atom at the second geometry; that is, it monitors the π f H siteswitching dynamics. To find typical values for the time constants τrot and τd, we calculate pump-probe signals and compare them with experiment. The model and the calculation of the pump-probe signals are described in Section 2. The result section starts with an investigation of the unperturbed wave packet dynamics following the pump excitation. The role of dissipation is discussed afterward; finally, pump-probe signals are presented. Section 4 concludes the article. 2. Theory and Model
Figure 2. Model potential energy curve for the motion of an Ar atom within the phenol+-Ar(VOH ) 0) complex along a reaction coordinate q. The pump-probe excitation scheme is also sketched. Within the pump transition, a wave packet is prepared in the cation ground electronic state, where the OH stretch is in its vibrational ground state (VOH ) 0). Excitation of one vibrational quantum of the OH stretch mode (VOH ) 1) by the probe field is resonant either at the π-binding site (3547 cm-1) or at the H-binding site (3467 cm-1).
place on a time scale of about τ ) 7 ps; (b) the latter time constant is independent of the available internal vibrational energy; and (c) there is no periodic back-and-forth switching, indicating that energy is dissipated in modes different than the reaction coordinate. The aim of the present work is to set up a simple quantum mechanical model that can account, at least approximately, for these experimental results, thereby characterizing time scales of quantum mechanical motion on one hand and energy dissipation on the other. This, of course, is a difficult task. The switching involves the motion of an argon atom on a multidimensional potential energy surface, which is largely unknown. Even if the potential is known with high accuracy, the dynamics to be treated is of high dimensionality, and sophisticated methods to solve the quantum mechanical equations of motion will have to be applied.19 The essence of our approach, which is guided by the experimental results, is summarized in Figure 2, showing a potential curve as a function of an angular coordinate, q. It corresponds to the OH stretch vibrational ground state, VOH ) 0. For a discussion on such vibrationally adiabatic potential energy curves, the reader is referred to the book by Schinke.19 A pump pulse prepares a wave packet ψ0(q, t) in the cationic state. This transition originates from the π-bonding site. If no energy dispersion in degrees of freedom different from the reaction coordinate is present, then the wave packet performs a periodic motion with a period τrot, as long as no substantial spreading induced by the anharmonicity of the potential curve occurs. However, a loss of energy in this coordinate results in a trapping of the wave packet in the H-bonded global minimum of the potential. This trapping is characterized by a time scale τ d. Figure 2 also illustrates the probe transition, which is induced by a probe field of different photon energies. In the case of an energy of εprobe ) 3537 cm-1, one vibrational quantum of OH stretch is excited at the π-bonding site,4 so that the pump-probe signal reflects the motion away from this site. If the probe photon energy is red-shifted to εprobe ) 3467 cm-1,4 then resonant
To describe the experimentally observed dynamics in phenol-Ar2, we employ a simplified 1D model where, within the phenol-Ar complex, the motion of an Ar atom takes place along a reaction coordinate, q. This motion might be imagined as a hindered internal rotation with an effective rotational constant, Beff. The pump ionization originates from a geometry of the neutral cluster with wave function ψg(q) of eigenenergy εg, which is set to a value of εg ) 0. The initial wave function is taken as a Gaussian
ψg(q) )
( 2βπ )
1/4
e-β(q-qe)
2
(1)
with parameters qe ) -2.6 rad and a value of β which amounts to a width (full width at half-maximum, fwhm) of 0.2 rad, as illustrated in Figure 2. The equilibrium angle is chosen such that, within the pump transition, ionic states with energies above the reaction barrier are populated. The potential curve V0(q) corresponds to the OH stretch coordinate being in its vibrational ground state (VOH ) 0) within the complex. The Hamiltonian reads
H0 ) -Beff
d2 + V0(q) dq2
(2)
The potential is parametrized as 4
V0(q) )
∑ an cos[(n - 1)q]
(3)
n)1
with constants a1 ) 311 cm-1, a2 ) -160 cm-1, a3 ) -40 cm-1, and a4 ) -112 cm-1, respectively. In constructing this curve, several experimental findings are taken into account. The binding energy of the π-bound isomer is 535 ( 3 cm-1,20 and the dissociation energy of the H-bonded isomer is ∼300 cm-1 larger.7,9,11 The barrier for π f H isomerization is calculated to be on the order of 100 cm-1.7,11 The parametrization of our potential leads to values for the minima as V0(-2) ) 296.3 (πsite), V0(0) ) 0 (H-site), and a barrier with V0((1.26) ) 384.7 cm-1, which is in accord with the numbers given above. We note, however, that the reaction path connecting the configurations, as displayed in Figure 1, is not known exactly, and our model is only a reasonable guess to approximate this path. During the probe transition, the OH stretch mode is excited by one vibrational quantum (υOH ) 1). It is well known that upon site switching, the vibrational frequency of the bound OH stretch mode is red-shifted from 3537 to 3467 cm-1.4 Therefore,
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the adiabatic potential curve for the motion in the vibrationally excited υOH ) 1 state is parametrized as (in cm-1) -5q2
V1(q) ) V0(q) + 3537 - 70 e
(4)
The last term in the latter equation modifies the form of the V0 potential such that at the global minimum the potential difference (V1 - V0) amounts exactly to 3467 cm-1. The Hamiltonian in the excited state is the one given in eq 2 with the potential curve V0(q) replaced by V1(q). As is discussed in the Introduction, the pump-probe measurements suggest that after ∼7 ps the Ar atom is trapped in the global potential minimum of V0(q) corresponding to the H-site (q ) 0). This can only happen if energy is taken out of the motion along the reaction coordinate, q, and dissipates into other degrees of freedom. Classically, this means that a trajectory starting in the Franck-Condon region of the pump transition ends up at the bottom of the potential well (Figure 2). This, for example, could be described by a classical Langevin equation where the energy loss is caused by a frictional force. Quantum mechanically, however, the picture is more complicated. A correct quantum description involves the solution of the equation of motion for the reduced density operator (acting in the reaction coordinate) coupled to the bath, which here consists of all degrees of freedom other than the reaction coordinate, q.21 Alternatively, one could employ a stochastic Schro¨dinger equation to simulate dissipative effects.22 One strategy is to set up a quantum mechanical equation of motion resembling the classical Langevin equation by including a dissipation and fluctuation term into the Hamiltonian. We include only the dissipative part and add the following term to the Hamiltonians Hn (n ) 0, 1)
γ Hd,n ) -i Vn(q) 2
(5)
where γ is a parameter that determines the time scale for the relaxation process. It is shown in Section 3 that a wave packet moving in the potential V0(q) will, after some time and upon action of Hd,0, be confined to the region of the global potential minimum. The time propagation of the wave packets ψn(t) over one time step ∆t is carried out as:
ψn(t + ∆t) ) e-i/p(Hn+Hd,n)∆tψn(t)
(6)
For the numerical propagation, we employ the split-operator method.23 Because the time evolution operator is no longer unitary (not even to first order in ∆t because the fluctuating term in the Hamiltonian is ignored), the wave functions are renormalized at each time step as
ψn(t + ∆t) f
(
〈ψn(t)|ψn(t)〉 〈ψn(t + ∆t)|ψn(t + ∆t)〉
)
1/2
ψn(t + ∆t)
(7)
The laser-cluster interaction is treated within time-dependent perturbation theory. The wave functions to first- (pump excitation) and second-order (probe excitation) are calculated iteratively as24,25
ψ0(E, q, t) ) -i
∫-∞t dt1 e-i/p(H +E+H 0
d,0)(t-t1)
×
Wpump(t1 - T1)ψg(q) (8)
ψ1(E, q, t, τ) ) -i
∫-∞t dt2 e-i/p(H +H 1
d,1)(t-t2)
×
Wprobe(t2 - T2)ψ0(E, q, t2) (9)
Here the wave functions ψn(E, q, t) correspond to the ejection of a photoelectron with kinetic energy, E.26 The electric dipole interactions are of the form
Wpump(t) ) -µ0g e-β1(t
- T 1 )2
e-iω1(t-T1)
(10)
Wprobe(t) ) -µ00 e-β2(t
- T2)2
e-iω2(t-T2)
(11)
The electric fields of the pump (n ) 1) and probe (n ) 2) pulses are characterized by their frequency, ωn, and Gaussian envelope functions centered at times, Tn, having widths that are determined by the parameters βn. The delay time is denoted as τ ) T2 - T1. Because we calculate only relative numbers and use perturbation theory, the field strengths (and the factor of 0.5, which stems from the application of the rotating wave approximation19) are not explicitly noted in eqs 10 and 11. The transition dipole moment entering into the pump interaction is denoted as µ0g, and the matrix element of the permanent dipole moment in the cationic state is µ00. Both functions are assumed to be independent of the reaction coordinate q and are set to a value of 1. One straightforward approach to treat the ionization step is to discretize the ionization continuum and calculate the nuclear wave functions for a discrete set of electron energies, Em.27,28 Here we ignore the summation over the different exit channels and present results for the case Em ) 0, corresponding to the case of zero kinetic energy photoelectrons. We have numerically checked that the summation results only in a slight broadening of the pump-probe signals to be compared with experiment. The signals thus are calculated as
S(τ) ) lim tf∞
∫ dq |ψ1(E ) 0, q, t, τ)|2
(12)
The population S(τ) settles to a constant when both fields have decayed to zero. 3. Results 3.1. Wave Packet Dynamics. We first regard the rotational motion following the pump-pulse-induced ionization step. Therefore, a numerical value has to be assigned to the constant Beff. Here we fix it to a value of Beff ) 0.02 cm-1 (see below). The value of the parameter β1, which characterizes the Gaussian envelope of the pump pulse, is chosen such that the latter has a full width at half-maximum (fwhm) of 3 ps.4 The pulse center is fixed at T1 ) 3 ps. Figure 3a illustrates the wave packet dynamics in the cationic state obtained for a relative pump excitation energy of 430 cm-1, which exceeds the potential barrier for the π f H switching. In what follows, the energy is referred to the potential minimum, as is indicated in Figure 2. After the field is switched off, a hindered rotational motion takes place, where the packet is focused at times when the potential barriers (at q ) ( 1.26 rad) and the classical turning points (at q ≈ (2.5 rad) are reached. The dynamics is, within the shown time window,
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Figure 3. Wave-packet dynamics after the pump excitation. Shown are probability amplitudes (|ψ1(E ) 0, q, t)|) obtained for three excitation energies corresponding to (a) above-barrier (430 cm-1), (b) barrier (385 cm-1), and (c) below-barrier excitation (350 cm-1). Here the unperturbed dynamics is shown, where no dissipation is present.
periodic with a period of τrot ) 20 ps, and the dynamics of the site-switching takes about 7 ps. The possibility that the barrier along the reaction coordinate is smaller or is not present cannot be ruled out from experiment. To investigate the dynamics in these situations, we do not reparameterize the potential. Instead, calculations are performed for relative excitation energies (pω1), which are close to the top of the and smaller than the potential barrier. Results for photon energies of 385 and 350 cm-1 are shown in Figures 3b,c, respectively. In the former case, the excitation energy equals approximately the barrier height. This has the consequence that the wave packet bifurcates upon hitting the barrier. There is one part that remains (at least for the times regarded) localized in the smaller local minimum, whereas a second part moves outward. There, the second potential barrier is encountered, and another bifurcation occurs giving rise to interference patterns. The third case of below-barrier excitation shows that the main part of the prepared wave packet is trapped in the local potential minimum and tunneling, for the present choice of the rotational constant Beff and the displayed time window, can be ignored. We note that the dynamical features discussed are not sensitive to changes in the width of the initial wave function. If it is placed in the vicinity of the π-site potential minimum, then a trapping of the excited state wave packet occurs and no dynamics is seen. If longer excitation pulses (e.g., 4 ps fwhm) are employed, then the wave-packets are smeared out in the q coordinate but still exhibit a similar dynamics. Next, the effect of dissipation is discussed. For this purpose, an excitation with a 3 ps pump pulse at 430 cm-1 is considered. In Figure 4, the wave packet dynamics is shown for three different values of the parameter γ. For a value of γ ) 0.025 (panel (a)), the motion at early times resembles the case of γ ) 0, as can be seen by comparison with Figure 3a. The damping is effective only after 20 ps, where most of the wave function
Walter et al.
Figure 4. Wave-packet dynamics after the pump-excitation including the effect of dissipation. The case of above-barrier excitation (430 cm-1) is regarded. Values of γ ) (a) 0.025, (b) 0.05, and (c) 0.075 are employed in the calculation.
is trapped in the potential well. For stronger damping (γ ) 0.05), the wave packet is able to pass the potential well only once, but upon its way back toward negative values of the reaction coordinate, it looses sufficient energy so that escaping from the potential well is no longer possible. Finally, for an even larger value of the dissipation parameter (γ ) 0.075, panel (c)), the trapping occurs upon the first passage of the wave packet over the potential well. 3.2. Pump-Probe Signals. Our model employs two parameters that influence the wave packet dynamics and, as a consequence, the appearance of the pump-probe signals. In what follows, we present calculated signals for various values of the rotational constant, Beff, and the dissipation parameter, γ. We determine signals for pulses of 3 ps width (fwhm) fixing first the rotational constant to a value of Beff ) 0.02 cm-1. Figure 5 contains signals for different values of γ, as indicated, and probe photon energies of 3537 (π-site, left-hand panels) and 3467 cm-1 (H-site, right-hand panels), respectively. As was mentioned above, in the first case, the signal monitors the generation of a wave packet at the π-site and its subsequent separation from the Franck-Condon region for the ionization step. If no dissipation is present (γ ) 0.0), then the signal exhibits a periodic variation because the prepared wave packet moves away from its initial position, which is accompanied by a decrease in intensity. When the wave packet reaches its classical turning point at the outer potential wall, resonant vibrational excitation becomes possible because of the symmetry of the potential, and the signal increases again. This happens periodically, and the period seen in the pump-probe signal is twice the period τrot. If dissipation is switched on, then only three peaks of the pump-probe signal are obtained in the case of γ ) 0.025. This means, that the wave packet returns only once to its initial position and afterward is trapped in the potential well, as can be seen upon inspection of the respective wave packet dynamics displayed in Figure 4a. With increasing
Hydrophobic f Hydrophilic Site Switching
Figure 5. Pump-probe signals for resonant excitation at the π-site (left-hand panels) and on the H-site (right-hand panels). Curves are shown for a rotational constant of Beff ) 0.02 cm-1 and different values of the dissipation constant, γ, as indicated.
Figure 6. Same as Figure 5 but for a rotational constant of Beff ) 0.2 cm-1 and three values of the dissipation constant, γ, as indicated.
value of the dissipation constant, the signal becomes more and more restricted to times when pump and probe pulse overlap. A comparison of the curves for no (γ ) 0.0) and the strongest dissipation (γ ) 0.1) shows that indeed only the separation from the π-site is reflected in the signal. This observation is consistent with the appearance of the signal obtained for probing the H-site at 3467 cm-1. Again, the periodic variation of the signal reflects the unperturbed wave packet motion. For resonant excitation to be possible, the prepared wave packet has to move into the region of the global potential minimum. Therefore, the signal obtained at the H-site is phase-shifted with respect to the π-site signal. Switching on the dissipation results in a disappearance of the periodicity. With increasing value of γ, the dissipation leads to a trapping at the H-site at earlier times. This is consistent with the features of the wave packet dynamics discussed in the last subsection. In a next step, the influence of the rotational constant is examined. Figure 6 shows pump-probe signals calculated for a value of Beff ) 0.2 cm-1, which is a factor of 10 larger than the value regarded before (Figure 5). Naturally, the dynamics proceeds on a faster time scale. This is evident from the signals for γ ) 0 for both probe photon energies. Here we find a period of τrot ) 6 ps, which is twice the period seen in the pump-probe
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Figure 7. Comparison between measured (exp.4) and calculated (theo.) pump-probe signal for resonant excitation at the π-site (upper panel) and at the H-site (lower panel), respectively. In the calculation, the values of γ ) 0.1 and Beff ) 0.1 cm-1 were employed.
signal. Because the temporal separation of two adjacent peaks in the signal is comparable to the pulse length, the periodic variations of the signal occur on a background signal. Comparison with the curves displayed in Figure 5 shows that with increasing value of the dissipation constant, γ, the influence of the rotational constant becomes minor. In particular, for a value of γ ) 0.1, the π-site signal exhibits only one major peak which is phase-shifted as compared with the peak obtained for the smaller rotational constant. The reason is that in the latter case, excitation proceeds at the inner potential wall, whereas for the faster motion, the excitation takes place at the outer classical turning point. The H-site signals, however, are very similar for this choice of the dissipation parameter. 3.3. Comparison with Experiment. We now turn to the comparison between theory and experiment. Figure 7 displays the transient signals measured for phenol+-Ar2 for delay times up to 30 ps. It is seen that the signals for the H-site excitation (3467 cm-1) start rising at early delay times and level off at ∼20 ps. This suggests that the trapping in the potential well of the H-bound site sets in rather early. From Figures 5 and 6 it is then obvious that the dissipation constant should assume values around γ ) 0.1. For this choice, a rotational constant of Beff ) 0.02 cm-1 yields a π-site spectrum, which is centered around zero delay time and decays too fast (Figure 5, lower left panel). The spectrum for Beff ) 0.2 cm-1 starts somehow too late but shows an overall behavior that is consistent with experiment. At this point, we take advantage of the fact that the effective internal rotational constant, Beff, of phenol-Ar rotating within phenol-Ar2 can be estimated to be between 0.03 and 0.1 cm-1 from the measured rotational constants of phenol-Ar5 and plausible paths for the internal rotational motion. Taking Beff ) 0.1 cm-1 and fixing the dissipation constant to γ ) 0.1 leads to the calculated signals included in Figure 7. We find an overall excellent agreement taking the fact into account that the model employed here is rather elementary. The calculated π-site signal decreases too fast when compared with experiment. We note that the 1D description does not account for wave packet recurrences into the reaction degrees-of-freedom from other modes. Therefore, the theoretical description is far from being complete.
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4. Summary We have set up a simple model to describe the pump-probe spectroscopy of π- to H-site switching in phenol-Ar cation clusters. The ionization-induced dynamics occurs along a 1D reaction path potential, which is constructed according to information extracted from spectroscopic data. The theoretical description includes two parameters, which are associated with different time scales. The first is an effective rotational constant, which determines the period of the unperturbed bound-state motion. Because the transient measurements point at a dissipation mechanism, which results in a trapping of the wave packet at the H-site, the model is extended to include a dissipative part involving the second parameter, which controls the time scale for relaxation. Pump-probe signals are calculated and compared with experiment. In adjusting the model parameters, we are able to find an overall excellent agreement with the transient spectra, taking the simplicity of the model into account. The conclusion from the model is that the dissipation of energy into modes different from the reaction coordinate is fast. This means that the role of the rotational constant is minor; that is, before a periodic motion sets in, not much energy is left in the reaction coordinate and trapping has occurred. The important question remains into which modes the energy dissipates. This cannot be answered by the approach followed here, but these are almost certainly the low-frequency intermolecular bath modes of the phenol-Ar2 trimer. Acknowledgment. We acknowledge financial support by the DFG within the GRK 1221 and by the DAAD within the PROCOPE program. O.D. acknowledges support from DFG project DO 729/4 and stimulating discussions with M. Fujii on the internal rotation and damping processes in phenol-Arn cation clusters. References and Notes (1) Meyer, E. A.; Castellano, R. K.; Diederich, F. Angew. Chem., Int. Ed. 2003, 42, 1210. (2) Dopfer, O. Z. Phys. Chem. 2005, 219, 125.
Walter et al. (3) Ishiuchi, S.; Sakai, M.; Tsuchida, Y.; Takeda, A.; Kawashima, Y.; Fujii, M.; Dopfer, O.; Mu¨ller-Dethlefs, K. Angew. Chem., Int. Ed. 2005, 44, 6149. (4) Ishiuchi, S. I.; Sakai, M.; Tsuchida, Y.; Takeda, A.; Kawashima, Y.; Dopfer, O.; Mu¨ller-Dethlefs, K.; Fujii, M. J. Chem. Phys. 2007, 127, 114307. (5) Kalkman, I.; Brand, C.; Vu, C.; Meerts, W. L.; Svartsov, Y. N.; Dopfer, O.; Mu¨ller-Dethlefs, K.; Grimme, S.; Schmitt, M. J. Chem. Phys. 2009, 130, 224303. (6) Ishiuchi, S. I.; Tsuchida, Y.; Dopfer, O.; Mu¨ller-Dethlefs, K.; Fujii, M. J. Phys. Chem. A 2007, 111, 7569. (7) Solca, N.; Dopfer, O. J. Mol. Struct. 2001, 563/564, 241. (8) Solca, N.; Dopfer, O. Chem. Phys. Lett. 2000, 325, 354. (9) Solca, N.; Dopfer, O. J. Phys. Chem. A 2001, 105, 5637. (10) Solca, N.; Dopfer, O. Chem. Phys. Lett. 2003, 369, 68. (11) Cerny, J.; Tong, X.; Hobza, P.; Mu¨ller-Dethlefs, K. Phys. Chem. Chem. Phys. 2008, 10, 2780. (12) Solca, N.; Dopfer, O. Euro. Phys. J. D 2002, 20, 469. (13) Solca, N.; Dopfer, O. Phys. Chem. Chem. Phys. 2004, 6, 2732. (14) Andrei, H. S.; Solca, N.; Dopfer, O. Phys. Chem. Chem. Phys. 2004, 6, 3801. (15) Andrei, H. S.; Solca, N.; Dopfer, O. J. Phys. Chem. A 2005, 109, 3598. (16) Takeda, A.; Andrei, H.; Miyazaki, M.; Ishiuchi, S. I.; Sakai, M.; Fujii, M.; Dopfer, O. Chem. Phys. Lett. 2007, 443, 227. (17) Patzer, A.; Knorke, H.; Langer, J.; Dopfer, O. Chem. Phys. Lett. 2008, 457, 298. (18) Patzer, A.; Langer, J.; Knorke, H.; Neitsch, H.; Dopfer, O.; Miyazaki, M.; Hattori, K.; Takeda, A.; Ishiuchi, S. I.; Fujii, M. Chem. Phys. Lett. 2009, 474, 7. (19) Schinke, R. Photodissociation Dynamics; Cambridge University Press: Cambridge, U.K., 1993. (20) Dessent, C. E. H.; Haines, S. R.; Mu¨ller-Dethlefs, K. Chem. Phys. Lett. 1999, 315, 103. (21) May, V.; Ku¨hn, O. Charge and Energy Transfer Dynamics in Molecular Systems; Wiley-VCH: Berlin, 2000. (22) van Kampen, N. G. Stochastic Processes in Physics and Chemistry; Elsevier: Amsterdam, 1992. (23) Feit, M. D.; Fleck, J. A.; Steiger, A. J. Comput. Phys. 1982, 47, 412. (24) Engel, V. Comput. Phys. Commun. 1991, 63, 228. (25) Renziehausen, K.; Marquetand, P.; Engel, V. J. Phys. B: At. Mol. Opt. Phys. 2009, 42, 195402. (26) Wollenhaupt, M.; Engel, V.; Baumert, T. Annu. ReV. Phys. Chem. 2005, 56, 25. (27) Seel, M.; Domcke, W. Chem. Phys. 1991, 151, 59. (28) Engel, V.; Baumert, T.; Meier, C.; Gerber, G. Z. Phys. D: At., Mol. Clusters 1993, 28, 37.
JP101964E
