Wavepacket rotational

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 ... 

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. 

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... 

Excitation of ClNO(Ti) in any one of the three vibrational bands yields exclusively NO products in vibrational state n — n (Qian et al. 1990). The left-hand side of Figure 9.12 depicts the results of a three-dimensional wavepacket calculation including all three degrees of freedom and using an ab initio PES (Solter et al. 1992). This calculation reproduces the absorption spectrum and the final vibrational and rotational distributions of NO in good agreement with experiment. 

The set of coupled equations (15.11) represents an example of time-dependent close-coupling as described in Section 4.2.3. It is formally equivalent to (4.25), for example, and can be solved by exactly the same numerical recipes. The dependence on the two stretching coordinates R and r is treated by discretizing the two nuclear wavepackets on a two-dimensional grid and the Fourier-expansion method is employed to evaluate the second-order derivatives in R and r. If we additionally include the rotational degree of freedom, we may expand each wavepacket in terms of... 

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.

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