Selective Probing of Vibrational Hot States in Bromine Using Time

Article pubs.acs.org/JPCA

Selective Probing of Vibrational Hot States in Bromine Using TimeResolved Coherent Anti-Stokes Raman Scattering Mahesh Namboodiri, Jörg Liebers, Ulrich Kleinekathöfer, and Arnulf Materny* Center for Functional Materials and Nanomolecular Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany ABSTRACT: In previous work (Scaria, A.; et al. Chem. Phys. Lett. 2009, 470, 39− 43) it was shown that the excitation of the electronic B state in bromine can be characterized by transitions starting from vibrational hot states of the electronic ground X state. This contribution is strongly depending on the specific Franck− Condon factors for the chosen wavelength (in that work 540 nm) used for excitation. For the investigation of the resulting excited state dynamics, a pumpdegenerate four-wave mixing (pump-DFWM) experiment was applied. To increase the vibrational selectivity, in the present work we have performed temperaturedependent time-resolved coherent anti-Stokes Raman scattering (CARS) spectroscopy to probe the B state dynamics of bromine. Also here, the wavelength of the excitation (in this case, the pump laser of the CARS process) was set to 540 nm for all measurements. The hot state contribution is small, even at high temperatures. It can be probed by tuning the Stokes wavelength to resonance. The time delay between the probe pulse and the time-coincident pump/Stokes pulse pair of the CARS process is scanned, giving access to the wave packet dynamics in the excited B state. The experimental observations are supported by quantum dynamical calculations.

■

INTRODUCTION The advent of femtosecond laser sources made the observation of ultrafast molecular dynamics feasible. A great number of studies have been reported on different samples ranging from simple diatomic molecules to complex biological systems. Standard pump−probe experiments as well as more advanced spectroscopy techniques like four-wave mixing experiments have been employed to access the molecular dynamics of molecules involving different electronic and vibrational energy states, some of them not accessible by linear spectroscopic methods.1−3 The coherent motion of vibrational wave packets on ground and excited state potential energy surfaces have been probed by detecting laser induced fluorescence (LIF), degenerate four-wave mixing (DFWM), coherent anti-Stokes Raman scattering (CARS), transient grating (TG) spectroscopy, etc. signals.1,2,4,5 Leonhardt et al.,6 first employed timeresolved CARS to monitor the vibrational dephasing and quantum beats in condensed phase materials. Hayden and Chandler7 applied this technique to monitor wave packet evolution in gas phase benzene. In the present paper we will introduce results obtained on the investigation of wave packet dynamics excited in gas phase bromine molecules extending earlier work. Schmitt et al.8 studied ground and excited state wave packet evolution of gas phase iodine and bromine using femtosecond time-resolved CARS. The ground state dynamics of bromine were not observed due to the narrow bandwidth of the pulses used; the electronic ground state of bromine has a relatively large vibrational energy level spacing of approximately 325 cm−1). To overcome this problem, Lausten et al.9 performed CARS experiments using sub 25 fs pulses. These authors monitored the wave packet evolution in molecular bromine and iodine. © 2012 American Chemical Society

Due to the broad spectrum of the short pulses also ground state wave packet dynamics in bromine could be observed. Knopp et al.10 introduced two-dimensional time-dependent CARS (TD2 CARS) to probe highly excited vibrational wave packets on the ground electronic state of molecular iodine. In most timeresolved CARS experiments the time delay between the timecoincident pair of pump and Stokes laser pulses and the probe laser pulse is scanned giving access to both electronic ground and excited state wave packet dynamics. The TD2 CARS introduced an additional delay between pump and Stokes pulses. During this time the excited state wave packet propagated toward a Franck−Condon (FC) window, which optimized the Stokes dump process for creating vibrational wave packets in vibrational overtone states of the electronic ground state. Faeder et al.11 discussed this pump−dump−pump mechanism in detail. To better access the dynamics in different excited electronic states of gas phase molecules, Scaria et al.12,13 performed pumpDFWM experiments where an initial pump laser prepared a wave packet in the excited electronic state and a four-wave mixing interaction was used to then probe dynamics of this state as well as even higher lying electronic states in resonance with the nonlinear optical probe transitions. This technique was applied to bromine and helped to probe the ion pair state dynamics. These authors have shown that an initial pump excitation with a wavelength of 540 nm has also appreciable FC Special Issue: Jörn Manz Festschrift Received: June 7, 2012 Revised: July 3, 2012 Published: July 3, 2012 11341

dx.doi.org/10.1021/jp305579t | J. Phys. Chem. A 2012, 116, 11341−11346

The Journal of Physical Chemistry A

Article

factors for transitions from vibrational hot states of the ground electronic X state to the excited electronic B state (Figure 1).

Figure 1. Potential energy surface diagram of Br2, showing the electronic X ground and B excited PESs and some vibrational levels. The pump at 540 nm populates B state vibrational levels from both ground and hot vibrational levels in the X state.

Therefore, they could selectively probe different vibrational wave packets by varying the DFWM wavelength, which is not the case for iodine. The quantum dynamical calculations by Liebers et al.14 support these experimental observations. In our present work we intended to verify our earlier results by making use of the state selectivity of a femtosecond timeresolved CARS process (here, using a degenerate pump−probe pulse pair). To this end, two resonance conditions can be used. First, the pump and probe laser wavelengths can be chosen to be resonant with transitions between different electronic states. Second, the pump−Stokes energy difference can be tuned to any vibrational energy. In the present contribution femtosecond time-resolved CARS experiments are reported, which probe the vibrational dynamics of the bromine B state. Similar to the pump-DFWM experiments mentioned above, in which the initial pump with a wavelength of 540 nm resulted in the interesting wave packet dynamics, here the pump and probe wavelengths were chosen to have the same wavelength for enhancing the hot state contribution. In addition, the hot state population was enhanced by heating the bromine sample. Such the conclusions made earlier, which were based on theoretical simulations, could now be checked and verified.

Figure 2. Sketch of the experimental setup used for the femtosecond time-resolved CARS experiments (BS = beam splitter, BC = Berek compensator for polarization rotation, RM = retro-reflecting mirrors).

arrangement, which fulfils the phase matching condition and allows us to filter the anti-Stokes signal spatially. With a 100 mm lens the pulses were subsequently focused into the sample cell. By monitoring the cross correlation and autocorrelation signals produced in a BBO crystal, we determined the temporal overlap between the three pulses. The such generated CARS signal was collimated and monitored by a monochromator (Triax 180/190, Horiba Jobin Yvon; 1200 lines/mm grating), which helped to filter out background radiation and to disperse the spectrally broad anti-Stokes signal. For detection a CCD detector was used. The signal was recorded as a function of the time delay between the time coincident pump/Stokes pair and the probe pulse. As previously reported,8,9 for probing the excited electronic state wave packet dynamics a negative time delay had to be chosen; i.e., the probe laser pulse preceded the time-coincident pump−Stokes pulse pair. The bromine gas was contained in a vacuum-sealed 2 cm cylindrical cuvette, and the sample holder was equipped with an additional heating unit for temperature-dependent measurements.

■

EXPERIMENTAL SECTION The scheme of the experimental setup is given in Figure 2. The 150 fs (775 nm, 1 kHz repetition rate, 1 mJ Energy/pulse) pulses from the regeneratively amplified Ti:sapphire laser (CPA 2010, Clark MXR) are used to pump two optical parametric amplifiers (TOPAS, Light Conversion). One of the OPAs serves as source for the pump and probe pulses and the other one yields the Stokes pulse. Pump, probe, and Stokes pulses are compressed to ≃100 fs using prism pair set-ups. The output of the first OPA is split into two equal parts to obtain independent pump and probe pulses used in the degenerate pump−probe CARS experiment. The timing of the individual pulses was controlled using computer-controlled delay stages in a Michelson interferometer like setup. Moreover, the pulses were aligned into a three-dimensional folded BoxCARS

■

THEORY The present simulations were performed using perturbation theory in the field-matter interaction.15 To this end, the model employed for the Br2 molecule takes into account the electronic ground state X and the first excited state B. These electronic states are coupled by the electric field of the three CARS beams. The system Hamiltonian is given by 11342



Article

Ĥ (t ) = Ĥ mol + Ĥ int(t )

in the different electronic states involved can be easily tracked during the excitation processes in the simulation. Using only those spectroscopic pathways, which contribute to the polarization in the direction of the CARS signal results in20

(1)

In this equation, Ĥ mol denotes the Hamiltonian of the unperturbed system, which can be given in terms of the vibrational eigenstates of the electronic potential energy surfaces Ĥ mol =

(0) (3) P(3) = ⟨Ψ(0)|μ|̂ Ψ(3) k1− k 2 + k3⟩ + ⟨Ψ |μ |̂ Ψ k3− k 2 + k1⟩ (1) (2) (1) + ⟨Ψ(2) k 2 − k3|μ |̂ Ψ k1 ⟩ + ⟨Ψ k 2 − k1|μ |̂ Ψ k3 ⟩ + c.c.

∑ |av⟩⟨av|ℏωav (2)

a, v

Here, the first term corresponds to the process, in which a wave packet is created by the excitation with pulse k3 in the B state and subsequently freely evolving in time until it is probed by a two-photon process with the pulses k2 and k1. The second term is similar but now the wave packet in the B state is created by k1 and subsequently interrogated by pulses k2 and k3. Moreover, the third term emerges from the overlap between the wave packet prepared by k1 and the second-order wave packet created by absorption of k2 and stimulated emission of k3. The last term is again similar to the third one with the order of pulses exchanged. For negative time delays, Δt < 0, there are only two spectroscopic pathways, which mainly contribute to the signal15

The letter “a” denotes the electronic state (either ground state g or excited state e) and v is the quantum number of the vibrational eigenstate. To model the system accurately, we used Rydberg−Rees−Klein (RKR) potential energy surfaces (PES) based on Dunham coefficients from ref 16 to calculate the eigenstates. These PESs are based on high-resolution spectroscopic measurements though assumptions are being made during the semiclassical RKR procedure, which might introduce small errors into the PES. The coupling of the molecules to the electric field of the laser pulses E(t) through the dipole operator μ̂ was treated in dipole approximation ̂ t) Ĥ int = −μE(

(3)

(2) (1) P(3) = ⟨Ψ(0)|μ|̂ Ψ(3) k3− k 2 + k1⟩ + ⟨Ψ k 2 − k1|μ |̂ Ψ k3 ⟩ + c.c.

Furthermore, the laser pulses are assumed to be of the form E(t ) =

∑ En(t ) = ∑ ε(t − Tn)e−iω (t− T ) + ik x + c.c. n

n

n

n

(4)

with Gaussian shaped envelopes ε(t) centered at T n, frequencies ωn, and wave vectors kn. To determine the CARS signal, several different techniques exist. In previous studies12,13 on the same or similar systems, we employed nonperturbative18,19 as well as perturbative schemes in the laser-matter interaction to obtain the polarization P(t) or the third-order polarization P(3)(t), respectively. No signifcant differences in the CARS signals for the contrasting approaches were obtained, and therefore, we used the perturbative scheme here for simplicity. This scheme allows us to select only those spectroscopic pathways, which contribute to the polarization in the direction of the CARS signal

■

RESULTS AND DISCUSSION The major observations in the pump-DFWM experiments12,14 can be summarized as follows. As a first point, the 540 nm initial pump wavelength has a high FC factor for the transition from v′ = 1 (first overtone) in the X state to v″ = 26 in the B state. The wavenumber of the initial pump (λ = 540 nm) of 18 519 cm−1 corresponds to the transition between the vibrational ground state (v′ = 0) of the electronic ground X state to the vibrational overtone state around v″ = 21 in the excited electronic B state. Second, a DFWM probe process at a wavelength of 310 nm probes the vibrational wave packets around v″ = 21 in the B state. This corresponds to a transient oscillating with a period of 395 fs. A vibrational energy spacing of approximately 83 cm−1 is observed in the Fourier transform spectrum of the transient. As a third and last point, changing the wavelength of the DFWM interaction to 300 nm results in probing vibrational wave packets at around v″ = 26 in the excited electronic B state. This corresponds to a transient signal oscillation with a period of 510 fs. The respective vibrational energy spacing is roughly 67 cm−1 as observed in the Fourier transform of the transient. The CARS signal oscillates as a function of the time delay Δt between the probe pulse and the time-coincident pump/Stokes pulse pair. For Δt < 0 the probe comes before the timecoincident pump/Stokes pulse pair and creates a wave packet on the excited electronic potential energy curve. The time evolution of this wave packet is probed by the interaction of the time-coincident pump/Stokes pulse pair, after a time delay Δt,

(5)

Within perturbation theory the third-order polarization induced in the system is given by P(3) = ⟨Ψ(0)|μ|̂ Ψ(3)⟩ + ⟨Ψ(2)|μ|̂ Ψ(1)⟩ + c.c.

(6)

where c.c. denotes the complex conjugate. The wave functions in the different orders can be calculated iteratively17 iℏ

d |Ψ(n)(t )⟩ = Ĥ mol|Ψ(n)(t )⟩ + Ĥ int(t )|Ψ(n − 1)(t )⟩ dt

(9)

In case the CARS process starts with the preparation of an excited state wave packet, which is subsequently probed after a delay time Δt, only the wave packet dynamics on the excited state determines the transient behavior. In the present calculations the polarization was determined over a time range of 40 ps and afterward Fourier transformed to obtain the respective spectra corresponding to those spectra measured in the experiment using the CCD camera. To this end, the signals were recorded for the same detection wavelengths as in experiment.

n

k s = k1 − k 2 + k3

(8)

(7)

For the present system, these equations were solved in energy representation. In this representation it is possible to reduce the system to those vibrational states, which are actually populated during the excitation processes. This helped to reduce the time for the numerical calculations drastically compared to a gridbased scheme. Because the wave packet simulations do not explicitly contain a temperature, the contributions of the different vibrational states of the electronic ground state were evaluated separately. Subsequently, the total signal was determined as the sum of these contributions weighted by the Boltzmann distribution for the respective temperature. Furthermore, the characteristics of the vibrational populations 11343



Article

giving rise to the CARS signal. This signal oscillates at beat frequencies corresponding to the vibrational as well as rotational energy spacings in the excited electronic B state. For Δt > 0, the coherent interaction of the pump/Stokes pair creates a wave packet in the electronic ground state, which is subsequently probed by the time delayed probe pulse, giving rise to the CARS signal. In this case the signal reveals the ground state dynamics. At Δt = 0, a sharp peak is observed due to the coherent interaction of all three pulses, which is called the “coherent artifact”. This is due to nonresonant contributions to the CARS polarization possessing very short lifetimes and therefore reflecting the cross correlation of the involved pulses.5,8 The Fourier transform (FT) spectra of the transient show the vibrational as well as rotational energy spacings seen as beatings in the CARS transients. The peaks in the FT calculated from the CARS transients obtained from bromine show coherences between nearest neighbor vibrational levels (v and v + 1), which are centered at the average vibrational frequency of the wave packet. Coherences between next nearest neighbors (v and v + 2) can be seen at the second harmonic frequency. Three sets of experiments have been performed to see the contributions of the hot states in the electronic ground X state leading to the excitation of the vibrational wave packet in the excited B state. The main parameter is the wavelength of the laser pulses involved in the excitation and probe processes. In all cases, the maximum population has to be expected in the ground state. Therefore, to be able to observe hot-state contribution, the probe laser wavelength of the CARS process (which serves as “pump laser” in the arrangement with Δt < 0) is responsible for the wave packet excitation in the B state and thus has to be chosen such that transitions from hot states of the X state are possible; i.e., it has to have a wavelength fitting to a FC window favoring these transitions. As mentioned above, we have chosen 540 nm for this purpose. The “probing” in this temporal sequence of pulses is performed by the timecoincident pump−Stokes pulse pair. Here, the “dump transition” induced by the Stokes laser can be selected such that the FC to the hot state is greater than that to the ground vibrational state. In addition to this, the temperature of the sample can be used to influence the population density in the hot vibrational states. The first experiment was performed at room temperature (≈300 K), resulting in a small population of the hot state. The CARS transient (upper panel) and FT spectra (experimental, theory) of the transient recorded at room temperature are shown in Figure 3. The pump wavelength was selected to be 540 nm and the Stokes wavelength to be 560 nm. Moreover, the signal was detected at the anti-Stokes wavelength of 521.5 nm. The bromine B state dynamics could be observed for negative delay times (Δt < 0). For positive delay times (Δt > 0) we could not observe any ground state dynamics because of the narrow bandwidth of the laser pulses used in accordance to earlier observations.8 For negative delay times, however, the transient oscillates with a period of 395 fs. The FT spectrum shows major peaks around ≈83 cm−1 and its second harmonic (≈ 162 cm−1). This reveals that mainly vibrational levels around v″ = 21 contribute to the observed wave packet dynamics. The resolution of the experimental FT is ≈2 cm−1. In addition, the calculated FT spectrum agrees well with the experimental observations. Rotational contributions were not included in the calculations. Therefore, the intensities vary and some peaks are absent in the calculated spectrum compared to

Figure 3. CARS transient and its Fourier transform spectrum recorded at room temperature (≈300 K) with pump and probe wavelengths set to 540 nm and Stokes wavelength at 560 nm. The transient oscillates with a period of 395 fs. The calculated and experimental FT spectra show major peaks around 83 cm−1 and its second harmonic (162 cm−1). This corresponds to a beating between the vibrational levels around v″ = 21.

the experimental one. Even though the transitions from hot states are favored, the 540 nm pump prepares wave packets around both v″ = 21 (from v′ = 0 in the X) and v″ = 26 (from v′ = 1 in the X) in the B state due to the high population density of the vibrational groundstate at room temperature. The time-coincident pump/Stokes pulse pair probes both wave packets. Nevertheless, practically no hot state contribution is detected in the experimental transient and the corresponding FT spectrum. The calculations show that the Stokes (560 nm) FC factor favors wave packet contributions around v″ = 21 over those at v″ = 26 for the Stokes−dump process. This, together with the fact that the population in the hot state is small, answers the question why the hot state contributions are nearly absent in room temperature measurements. To increase the contribution of the transitions starting from the vibrational hot state of the electronic ground X state, the sample was heated. The Boltzmann distribution calculations showed that an increase of temperature by 100 K decreases the population in the vibrational ground state v′ = 0 (X state) by 14%. This change is of course directly reflected by an increase in the hot state populations. Figure 4 shows the resulting CARS transient and FT spectrum recorded at higher temperature (≈400 K). Again the pump wavelength was set to 540 nm. Moreover, the Stokes laser wavelength was left unchanged at 560 nm. As before, the anti-Stokes signal was detected at the anti-Stokes wavelength of 521.5 nm. In the FT spectrum, the energy bands around 65 cm−1 show a small hot state contribution. The calculated FT spectrum agrees with the experimentally observed trend. The major contribution is still from vibrational levels around v″ = 21. The increase of hot state population is clearly visible in both experimental and calculated 11344



Article

Figure 5. CARS transient and its Fourier transform spectrum recorded at room temperature (≈300 K) by changing the Stokes wavelength to 545 nm. Probe and pump wavelengths were kept at 540 nm. The transient oscillates have a period of ≈510 fs. The calculated and experimental FT spectra show major peaks around 65 cm −1 corresponding to the beating between the vibrational levels around v″ = 26. Thus a clearly increased contribution by the vibrational hot state can be detected.

Figure 4. CARS transient and its Fourier transform spectrum recorded at increased temperature (≈400 K) with pump and probe wavelengths set to 540 nm and Stokes wavelength at 560 nm. The calculated and experimental FT spectra shows major peaks around 83 cm−1. The small peaks around 65 cm−1 indicate a small hot state contribution.

spectra. The calculated B state vibrational energy spacings are given in Table 1.

■

CONCLUSIONS In previous pump-DFWM experiments a wavelength change of the degenerate four-wave mixing (DFWM) process resulted in different beating frequencies observed in the transients. In this kind of experiment one probes the wave packet dynamics induced in the electronic excited B state of bromine by the initial pump laser. These findings were explained by an excitation of the wave packets starting from both vibrational ground and hot states in the electronic ground X state.12 To confirm this assumption, we have now performed femtosecond time-resolved coherent anti-Stokes Raman scattering (CARS) experiments on bromine vapor. We found that also in these experiments the variation of the probe wavelength results in transients with different beating contributions. This can be assigned to wave packets originating from excitations from vibrational ground and hot states. To obtain further evidence for this model assuming hot state contributions, the temperature of the sample was varied. The relative change of ground and hot state population was directly reflected by the experimental data. Moreover, quantum dynamics calculations of the CARS process in bromine confirmed the experimental data and showed that the Franck−Condon factors for both excitation and probing can be chosen accordingly by making use of the degrees of freedom the nonlinear spectroscopy offers in terms of laser pulse wavelengths.

Table 1. B State Vibrational Energy Spacings vibrational level (v″)

energy (cm−1)

E(v″+1) − Ev″

E(v″+2) − Ev″

19 20 21 22 23 24 25 26 27 28 29 30 31 32

18457.79 18549.53 18636.81 18720.48 18799.87 18875.58 18947.35 19015.00 19079.22 19139.49 19196.04 19249.17 19298.75 19344.87

91.73 87.28 83.67 79.38 75.71 71.77 67.65 64.21 60.27 56.54 53.13 49.57 46.12 42.95

179.01 170.95 163.05 155.09 147.48 139.42 131.86 124.48 116.81 109.68 102.71 95.70 89.07 82.75

The calculations have indicated that a Stokes wavelength of 545 nm results in an optimized FC factor for dump transitions favoring vibrational levels around v″ = 26. Considering the wavelengths and bandwidth of the pump (540 nm) and the Stokes pulse (545 nm), this experiment is close to a DFWM process. The CARS transient recorded at room temperature for the changed Stokes wavelength at 545 nm is shown in Figure 5. The transient oscillates with a period of ≈510 fs. In this case, the hot state contributions are selectively probed and the experimental and theoretical FT spectra show two major peaks at around 65 cm−1. These correspond to the energy spacing between vibrational levels around v″ = 26 and v″ = 25.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 11345



■

Article

ACKNOWLEDGMENTS All authors have been inspired by the work of Prof. Jörn Manz. Especially, U.K. and AM. are very grateful for many discussions on various topics of “femtochemistry” they had with him in the past. We are thankful for the financial support by the Center for Functional Materials and Nanomolecular Science, Jacobs University Bremen. Dr. Torsten Balster’s help with measurement software related issues is highly acknowledged.

■

REFERENCES

(1) Zewail, A. H. Femtochemistry. Ultrafast dynamics of the chemical bond, Vol. I + II; World Scientific: Singapore, 1994. (2) Manz, J.; Wöste, L. Femtosecond chemistry; VCH: Weinheim, 1995. (3) Manz, J. In Femtochemistry and Femtobiology: Ultrafast Reaction Dynamics at Atomic-Scale Resolution; Sundström, V., Ed.; Nobel Symposium; Imperial College Press: London, 1996; Vol. 101, pp 80−318. (4) Motzkus, M.; Pedersen, S.; Zewail, A. J. Phys. Chem. 1996, 100, 5620−5633. (5) Materny, A.; Chen, T.; Schmitt, M.; Siebert, T.; Vierheilig, A.; Engel, V.; Kiefer, W. Appl. Phys. B: Laser Opt. 2000, 71, 299−317. (6) Leonhardt, R.; Holzapfel, W.; Zinth, W.; Kaiser, W. Chem. Phys. Lett. 1987, 133, 373−377. (7) Hayden, C. C.; Chandler, D. W. J. Chem. Phys. 1995, 103, 10465−10472. (8) Schmitt, M.; Knopp, G.; Materny, A.; Kiefer, W. J. Phys. Chem. A 1998, 102, 4059−4065. (9) Lausten, R.; Smirnova, O.; Sussman, B. J.; Graefe, S.; Mouritzen, A. S.; Stolow, A. J. Chem. Phys. 2008, 128, 244310/1−13. (10) Knopp, G.; Pinkas, I.; Prior, Y. J. Raman Spectrosc. 2000, 31, 51− 58. (11) Faeder, J.; Pinkas, I.; Knopp, G.; Prior, Y.; Tannor, D. J. Chem. Phys. 2001, 115, 8440−8454. (12) Scaria, A.; Liebers, J.; Kleinekathöfer, U.; Materny, A. Chem. Phys. Lett. 2009, 470, 39−43. (13) Liebers, J.; Scaria, A.; Materny, A.; Kleinekathöfer, U. J. Raman Spectrosc. 2009, 40, 822−827. (14) Liebers, J.; Scaria, A.; Materny, A.; Kleinekathöfer, U. Phys. Chem. Chem. Phys. 2010, 12, 1351−1356. (15) Meyer, S.; Schmitt, M.; Materny, A.; Kiefer, W.; Engel, V. Chem. Phys. Lett. 1997, 287, 753−754. (16) Gerstenkorn, S.; Luc, P. J. Phys. (Paris) 1989, 50, 1417−1432. (17) Engel, V. Comput. Phys. Commun. 1991, 63, 228−242. (18) Meyer, S.; Engel, V. Appl. Phys. B: Laser Opt. 2000, 71, 293− 297. (19) Seidner, L.; Stock, G.; Domcke, W. J. Chem. Phys. 1995, 103, 3998−4011. (20) Tannor, D. J. Introduction to Quantum Mechanics: A TimeDependent Perspective; University Science Books: Sausalito, CA, 2007.

11346


Selective Probing of Vibrational Hot States in Bromine Using Time

Recommend Documents