Rotational Wavepackets

The rotational selection rule for electronic transitions between states at the same Hund s case (see Section 3.2.1) limit is A J = AN = 1,0. If either electronic state is not at a Hund s limiting case or if the limiting cases are not identical in the upper and lower electronic states, less restrictive rotational selection rules apply. However, the number of rotational eigenstate J or N components present in a (to) created by a typical spectroscopically realizable pluck (i.e., a single, non-chirped, not-saturating excitation pulse) is small, typically 2 or 3. The possibility that each of these rotational components is split by spin fine structure (see Section 3.4) is neglected in the following discussion. 

The tf(to) prepared by a Z-polarized pump pulse from each thermally-populated J (or N ) initial state has the form 

N is the total population density, and [kT/hcB ] is the rotational partition function. This complete p(t) has terms that oscillate as e l27rcBv[2m](t-to) where 

Rotational recurrences may be detected in polarization selected spontaneous fluorescence (provided the photodetector has a sufficiently fast response) or by a variety of sub-nanosecond pump/probe schemes (Felker and Zewail, 1987 Felker, 1992 Hartland, et al. 1992 Joireman, et al.. 1992 Smith, et al., 2003a,b). 

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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. 

Figure 11. Time-resolved PADs from ionization of DABCO for linearly polarized pump and probe pulses. Here, the optically bright S E state internally converts to the dark 5i state on picosecond time scales, (a) PADs at 200 fs time delay for pump and probe polarization vector both parallel to the spectrometer axis. The difference in electronic symmetry between S2 and Si leads to significant changes in the form of the PAD. (b) The PADs at 200 fs time delay for pump polarization parallel and probe polarization perpendicular to the spectrometer axis, showing the effects of lab frame molecular alignment, (c) and (d) The PADs evolve as a function of time due to molecular axis rotational wavepacket dynamics. Taken with permission from C. C. Hayden, unpublished.

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. 

The two essential components of a nuclear wavepacket experiment in a diatomic molecule are an excitation pulse that is of sufficiently short duration that it creates, from a single v", J" initial eigenstate, a state f (t) that contains amplitudes in at least two J = J" + 1 and J = J" — 1 rotational eigenstates (a rotational wavepacket with motion in the rotational coordinates, 0, , which specify the orientation of the molecule fixed coordinate system relative to the... 

It is seen that the bifurcations are nearly removed and that now the motion is much more in favor of a counterclockwise rotation. In fact, one finds that 83% of the rotational wavepackets move in this direction. The classical trajectories exhibit a dynamics similar to that of the quantum densities. This is true for the ones moving freely but also for the trapped trajectories which do not have enough energy to escape the potential wells for a discussion of similar features and their relation to femtosecond pump-probe signals, see Ref. 214. 

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. 

Photodissociation has been referred to as a half-collision. The molecule starts in a well-defined initial state and ends up in a final scattering state. The intial bound-state vibrational-rotational wavefunction provides a natural initial wavepacket in this case. It is in connection with this type of spectroscopic process that Heller [1-3] introduced and popularized the use of wavepackets. 

Figure 14. (a) Potential-energy surfaces, with a trajectory showing the coherent vibrational motion as the diatom separates from the I atom. Two snapshots of the wavepacket motion (quantum molecular dynamics calculations) are shown for the same reaction at / = 0 and t = 600 fs. (b) Femtosecond dynamics of barrier reactions, IHgl system. Experimental observations of the vibrational (femtosecond) and rotational (picosecond) motions for the barrier (saddle-point transition state) descent, [IHgl] - Hgl(vib, rot) + I, are shown. The vibrational coherence in the reaction trajectories (oscillations) is observed in both polarizations of FTS. The rotational orientation can be seen in the decay of FTS spectra (parallel) and buildup of FTS (perpendicular) as the Hgl rotates during bond breakage (bottom). 

What about rotationally selected wavepackets in Na3 as reported in the new scheme shown by Leone s group for Li2 ... 

S. A. Rice Prof. Woste, your data indicate that rotational dephasing of the coherent wavepacket is unimportant for the time regime you have studied. Unless your beam has an unusually low rotational temperature, it is to be expected that a heavy molecule such as K2 will have many rotational states excited. Because the different isotopic species you have studied, one homonuclear and the other heteronu-clear, would then have different numbers of rotational states in the initial wavepacket, one should expect to observe different rotational dephasing times for the two species. What is the effective rotational temperature of your beam Is it likely that only a very few rotational states are present in the initial wavepacket ... 

We show how one can image the amplitude and phase of bound, quasibound and continuum wavefunctions, using time-resolved and frequency-resolved fluorescence. The case of unpolarized rotating molecules is considered. Explicit formulae for the extraction of the angular and radial dependence of the excited-state wavepackets are developed. The procedure is demonstrated in Na2 for excited-state wavepackets created by ultra-short pulse excitations. 

Consider the fluorescence from a molecular wavepacket excited from the ground electronic state by a short pulse of light. We assume that the initial eneigy of the molecule is EVgjg, where v, j denote, respectively, vibrational and rotational quantum numbers, with well defined magnetic quantum m,... 

In order to check our imaging procedure we have to first stimulate the fluorescence emitted by excited polarized (and unpolarized) Na2 wavepackets. In these simulation we assume that the molecule, which exists initially in a (Xvg,jg) Na2 (X1 5 ) vib-rotational state, is excited by a pulse to a superposition of (xs) vib-rotational states belonging to the Na2(B IIu) electronic-states. 

We can say that such a static device is a U( ) unipolar, set rotational axis, sampling device and the fast polarization (and rotation) modulated beam is a multipolar, multirotation axis, SU(2) beam. The reader may ask how many situations are there in which a sampling device, at set unvarying polarization, samples at a slower rate than the modulation rate of a radiated beam The answer is that there is an infinite number, because from the point of the view of the writer, nature is set up to be that way [26], For example, the period of modulation can be faster than the electronic or vibrational or dipole relaxation times of any atom or molecule. In other words, pulses or wavepackets (which, in temporal length, constitute the sampling of a continuous wave, continuously polarization and rotation modulated, but sampled only over a temporal length between arrival and departure time at the instantaneous polarization of the sampler of set polarization and rotation—in this case an electronic or vibrational state or dipole) have an internal modulation at a rate greater than that of the relaxation or absorption time of the electronic or vibrational state. 

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from... 

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