Franck-Condon wave packet

Fig. 1. Schematic of vertical (a) and non-vertical (b) electronic transitions with the electronic energies represented by displaced harmonic potentials. The initial state is the vibrational ground state on Vi. The wave packet on V2 in (a) is the Franck-Condon wave packet and the dashed arrows mark the positions of the turning points for the oscillation this wave packet will undergo under field-free conditions.

With this setup, the IR field affects the nuclear dynamics in the lower electronic state (injects energy) in the time interval from f0 to T, i.e., prior to excitation to the upper electronic state. In the limit where this time interval is small, the IR field simply boosts the momentum of the wave packet. This implies that the UV delta pulse instead of creating a Franck-Condon wave packet at rest on a down-hill slope on V2 creates a Franck-Condon wave packet with a positive (mean) velocity. 

For a general form of E(t), the excited-state wave function can be thought of as a coherent superposition of Franck-Condon wave packets promoted to the upper state at times t with different weighting factors and phases. At time t each of these wave packets has evolved for a time t—t. ... 

In the control scheme [13,17] that we have focused on, the time evolution of the interference terms plays an important role. We have already discussed more explicit forms of Eq. (7.75). One example is the Franck-Condon wave packet considered in Section 7.2.2 another example, which we considered above, is the oscillating Gaussian wave packet created in a harmonic oscillator by an (intense) IR-pulse. Note that the interference term in Eq. (7.76) becomes independent of time when the two states are degenerate, that is, AE = 0. The magnitude of the interference term still depends, however, on the phase S. This observation is used in another important scheme for coherent control [14]. 

Fig. 12.2 Left The ground (X, solid line), excited (6, dashed line) and dissociative [a1g(3II), dotted line] electronic state potentials of the iodine molecule. The arrow indicates the electronic excitation. The initial excited wave packet is located in the Franck-Condon region near to the inner classical turning point of the B state. The transition from the B to the a state is forbidden by symmetry in the isolated molecule but becomes allowed when the molecule is placed in a solvent.

Excitation of the coupled A2, Bi states results in the decay rate designated X3 which appears to be nearly independent of cluster size. A small increase in the value of x3 appears to occur for (S02)m clusters from the monomer (0.6 ps) to the dimer (0.9 ps), but remains constant at about 1 ps for larger cluster sizes. A likely interpretation of the observed decay process can be found in a detailed computational study [6] which reports that following the initial vertical excitation of the 1 B state, the excited state wave packet travels from the Bi state into the double wells that result from the crossing of the 1A2 and Bi states. The transition of the excited state population into the double wells of the A2 and B states is believed to lead to the decay observed in the pump-probe experiment because the potential energy well minima of both of these states are outside of Franck-Condon region for the absorption of the probe laser pulse. Therefore, ion signal is not observed once the transition has occurred. The primary discrepancy between the computational results of Ref. [6] and the... 

Here we report our exploration of the possibility of inducing an ultrafast non-Franck-Condon transition, which we defined to be the creation of a wave packet at the other turning point of the above-mentioned oscillation, see Fig. 1(b), faster than the time it takes the Franck-Condon packet to reach that turning point due to the natural (field-free) dynamics. We have explored two possible routes for inducing non-Franck-Condon transitions, namely phase-tailoring of a weak-field ultraviolet (UV) pulse [6] tmd a two-pulse scheme combining a transform limited weak-field UV pulse with a strong infrared (IR) field [7]. 

The Franck-Condon point in the transition from the ground state to the excited state potential is indicated in Fig. 27. The velocity distributions are calculated by transferring the wave packet after... 

The TDSE requires an initial condition, i.e. one must specify the wave packet at f = 0. Based on the assumption that the nuclei do not move during an electronic transition but only the electrons, the photo-excitation can be described as a Franck-Condon transition where the wave packet is excited vertically to the excited electronic state. Here fixj is the electronic transition dipole moment of the transition between the ground state X and the th electronically excited state, whereas Xx is a wave function of the electronic groimd state, typically the lowest vibrational state. Assuming a vertical electronic transition the initial wave packet on the excited state PES is given by... 

This initial wave packet tliat is not eigenstate of He, starts to move under its action. The advantage of this approach is that the motion of the wave packet, the center of which remains close to a classical trajectory, can be followed in real time. The motion of the wave packet from the Franck Condon region to the exit channel is described by the autocorrelation function... 

In the example, we choose x = 0 and p 0 and consider only the broadening of the momentum distribution V (p) by varying its width a. The results can be seen in Fig. 2 The transition probability for a = 0.1 is shown in Fig. 2 (a). The distribution is centered around n - 5 and has the Poisson-like shape of a Franck-Condon type of interaction, as is expected for a quasiconstant interaction with small a. With increasing a, the maximum is shifted slightly towards higher values of n. But the most obvious effect is the broadening of the probability distribution. For a — 00, an extremely narrow position wave packet with its center at x = 0 is excited on the upper potential. In this limit, the probability distribution is... 

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