Vibrational Wavepackets

The following discussion of vibrational wavepackets is unconventional in its emphasis on vibrational eigenstates and on the simple ideas used by frequency domain spectroscopists to understand their spectra. It is written with the hope of beginning to de-mystify the apparently disparate concepts and tools used by the frequency and time domain communities. 

In most vibrational wavepacket experiments, the time-evolving S (t) generally contains more than two vibrational eigenstates. The vibrational quantum numbers and t = to amplitudes of these vibrational eigenstates in k(t) are determined by the nature of the pluck that creates l (t) at f0. If the excitation-pulse is a simple, smooth, and short-time Gaussian, the result is a Franck-Condon pluck 

When the vibrational wavepacket is launched at to from a / = 0 initial eigenstate on the electronic ground state potential surface, 

Provided that the detuning, 5, of the center of the pump-pulse, (wL), from the vertical transition energy 

The situation is much more complicated for a vibrational wavepacket launched from a r 0 initial eigenstate (see Fig. 9.7). However, a simple picture based on the classical Franck-Condon (stationary phase) principle (see Section 5.1.1) captures the essential details of the wavepacket produced at to on the electronically excited potential surface. First, there is the limiting case of an excitation pulse sufficiently short that an exact replica of the electronic ground state vibrational eigenstate, (R v / 0), is created at to on the excited potential surface, 

Transient gratings also can be examined on a femtosecond time scale as a function of the time between the two pulses that create the grating [117]. As our discussion in Sect. 11.3 suggests, the two radiatitm fields do not actually need to be present in the sample simultaneously the second field can interfere constructively or destructively with coherence generated by the first. This makes femtosecond transient-grating experiments potentially useful for exploring relaxations that destroy such coherence. However, photon-echo experiments provide a more thoroughly developed path to this end. 

Constructive and destructive interference between these oscillations give rise to oscillatory features in the fluorescence. 

An excited ensemble with vibrational coherence can be described by a linear combination of vibrational wavefunctions  

Here u represents a dimensionless nuclear coordinate andxk(e) u) again denotes the spatial part of basis function k. Such a combinatiOTi of wavefunctions is called a wavepacket. The coefficients Ck represent averages over the ensemble. Making the Bom-Oppenheimer approximation and neglecting relaxations of the excited state and nonlinear effects in the excitation, they are given by 

11 Pump-Probe Spectroscopy, Photon Echoes and Vibrational Wavepackets 

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Much of the previous section dealt with two-level systems. Real molecules, however, are not two-level systems for many purposes there are only two electronic states that participate, but each of these electronic states has many states corresponding to different quantum levels for vibration and rotation. A coherent femtosecond pulse has a bandwidth which may span many vibrational levels when the pulse impinges on the molecule it excites a coherent superposition of all tliese vibrational states—a vibrational wavepacket. In this section we deal with excitation by one or two femtosecond optical pulses, as well as continuous wave excitation in section A 1.6.4 we will use the concepts developed here to understand nonlinear molecular electronic spectroscopy. 

Figure C3.5.7. Possible modes of vibrational wavepacket (smootli Gaussian curve) motion for a highly vibrationally excited diatomic molecule produced by photodissociation of a linear triatomic such as Hglj, from [8].

The modification of the electronic potentials due to the interaction with the electric field of the laser pulse has another important aspect pertaining to molecules as the nuclear motion can be significantly altered in light-induced potentials. Experimental examples for modifying the course of reactions of neutral molecules after an initial excitation via altering the potential surfaces can be found in Refs 56, 57, where the amount of initial excitation on the molecular potential can be set via Rabi-type oscillations [58]. Nonresonant interaction with an excited vibrational wavepacket can in addition change the population of the vibrational states [59]. Note that this nonresonant Stark control acts on the timescale of the intensity envelope of an ultrashort laser pulse [60]. 

Importantly, Cl s seems to be involved in many classes of physical, chemical and biological processes, from pericyclic reactions to the complex light harvesting and energy conversion functions of chromophores in proteins (See in this volume) and others amply described in this conference. In contrast, direct experimental information on the passage of the vibrational wavepacket through or near Cl s is less abundant. It mostly concerns, femtosecond pump-probe experiments on isolated organic molecules in the gas phase. 

Figure 18. Transient Na3+ signal for strongly attenuated 80-fs pump-probe laser pulses of 620 nm. The frequencies observed in the Fourier transform are due to vibrational wavepacket motion on the B state potential.

For a three-photon and a two-photon process we have shown that vibrational wavepacket propagation excited by an ultrashort laser pulse can be used to drive a molecule to a nuclear configuration where the desired product formation by a second probe pulse is favored (Tannor-Kosloff-Rice scheme). In both cases the relative fragmentation and ionization yield of Na2 was controlled as a function of pump-probe delay. By varying the delay between pump and probe pulses very slowly and therefore controlling the phase relation between the two pulses, additional interference effects could be detected. 

At the same time a vibrational wavepacket is prepared also a rotational wavepacket is formed in our experiments. However, we have not explored that yet. It is clear what happens based upon your earlier experiments. 

In the experiments described thus far, the pump laser simply populates the intermediate excited state. The consequence is that the experiment becomes a means to study that excited state. Often we are more concerned with learning about the ground state than about excited states. For this purpose, it is useful to prepare a vibrational wavepacket of that ground state. One useful means to do this is to excite the species of interest to an allowed excited state and then to down-pump from that excited state back to the ground state, with a pulse that generates a packet rather than a stationary state. The simplest way to do this currently seems to be to raise the power level of the pulsed pump laser [26]. This process is shown schematically in Fig. 7. 

Figure 9. Principle of NeNePo spectroscopy. The probe pulse detaches the photoelectron from the negative ion, introducing a vibrational wavepacket into the ground state of the neutral particle. Its propagation is interrogated by the probe pulse, which ionizes it to a positive ion.

The peak shift data in Fig. 17 show oscillatory character, as is our first two examples (I2 and LH1). This arises from vibrational wavepacket motion. In addition, the very fast drop in peak shift to about 65% of the initial value in -20 fs results from the interference between the wavepackets created in different intramoleculear modes. This conclusion follows directly from obtaining the frequencies and relative coupling strengths of the intramolecular modes from transient grating studies of IR144, carried out in the same solvents (data not shown). Thus, by visual inspection of Fig. 17, an answer to a long-standing question—What fraction of the spectral width arises from intra- and intermolecular motion —is immediately apparent. 

In this chapter we have surveyed recent experimental progress on the investigation of ultrafast nuclear wavepacket dynamics at surfaces. Nuclear (or vibrational) wavepackets of adsorbates are excited with ultrashort laser pulses, and subsequently their evolutions are probed with surface nonlinear spectroscopy such as 2PPE and SHG. These studies provide rich information on the initial stages of photoinduced... 

Matsumoto, Y., Watanabe, K. and Takagi, N. (2005) Excitation mechanism and ultrafast vibrational wavepacket dynamics of alkali-metal atoms on Pt(lll). Surf. Sci., 593, 110-115. 

During the n/2 pulse a coherent population in J is prepared, associated with both the ground- and the excited-electronic-state vibrational wavepacket. Hence, a coherent ro-vibrational wavepacket is formed. In analogy with the case where non-Condon effects play a significant role in the vibrational portion of the wavefunction, the rotational wavepackets in general are not identical on the two surfaces. Nevertheless, the two rotational wavepackets should be sufficiently similar to be strongly coupled. 

Besides the electronic degrees of freedom several vibrational modes contribute and the essential features of the dynamics can only be understood by a multidimensional model. The multidimensional character causes an irreversible course of the ESIPT even though the transfer itself takes only 50 fs and vibrational dephasing occurs on a picosecond time scale. Nevertheless, most of the energetically accessible vibrations do not play a significant role and a realistic description of the ESIPT has to consider only a restricted number of vibrational degrees of freedom. As discussed in Ref [22] we think that these are general features for many ultrafast molecular processes. The observation of vibrational wavepacket dynamics in a number of systems [6, 75, 76], which exhibit other ultrafast processes, supports this conclusion. 

The initial dynamics of hydrogen transfer on this potential energy surface are determined by the propagation of the vibrational wavepacket created upon electronic excitation. 

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