Wavepacket dynamics short-time

How well do these quantum-semiclassical methods work in describing the dynamics of non-adiabatic systems There are two sources of errors, one due to the approximations in the methods themselves, and the other due to errors in their application, for example, lack of convergence. For example, an obvious source of error in surface hopping and Ehrenfest dynamics is that coherence effects due to the phases of the nuclear wavepackets on the different surfaces are not included. This information is important for the description of short-time (few femtoseconds) quantum mechanical effects. For longer timescales, however, this loss of information should be less of a problem as dephasing washes out this information. Note that surface hopping should be run in an adiabatic representation, whereas the other methods show no preference for diabatic or adiabatic. 

In order to show the main features of the reaction dynamics after EP, it is relevant to follow the wavepacket evolution in time, and in Eigs.5 and 6 two different snapshots are shown for t= 7.75 and 15.25 fs. The dynamics leads rapidly to LiE products at short times because of the node present in the r coordinate. At longer times, however, there is a relative small proportion of the wavepacket that remains... 

Messina et al. [25] test the time-dependent Hartree reduced representation with a simple two-degree-of-freedom model consisting of the h vibration coupled to a one-harmonic-oscillator bath. The objective function is a minimum-uncertainty wavepacket on the B state potential curve of I2. Figure 12, which displays a typical result, shows that this approximate representation gives a rather good account of the short-time dynamics of the system. 

The dynamics of the wavepacket on the upper potential surface determines both the absorption spectrum and the Raman spectrum. The emission spectrum is determined by the dynamics on the ground state potential surface with the same displacements as those which determine the absorption and Raman. In the short time limit, the intensities in the Raman spectrum are related to the displacements by eq. 7. In the short time limit, the absorption spectrum becomes... 

Figure 5 shows a plot of the magnitude of the overlap for / = 0, K0/ (f)> sl. versus time. The magnitude of the overlap decreases as the wavepacket spreads out. There is no recurrence. The steeper the inverted potential (i.e., the higher coj), the faster the wavepacket spreads out and the faster the overlap decreases. Because the inverted harmonic potential surface can model only a small area around the Frank-Condon region, this model can only be applied to short time dynamics. 

The r dependence of the relative resonance Raman intensities of two modes is much more sensitive to their difference in frequency than to their difference in displacement. The reason for this dependence is found in the short time dynamics of the wavepackets which is much more sensitive to the frequency at a given displacement than to the displacement for a given frequency. This simple physical picture provides an explanation for why very low frequency metal-ligand modes in a big molecule often do not appear in the resonance Raman spectrum even though the modes have appreciable displacements. 

Recently, in this laboratory, we have applied time-dependent quantum mechanics-wavepacket dynamics to several bona fide time-domain spectroscopies. Specifically, we have formulated time-dependent theories of coherent-pulse-seque nee-induced control of photochemical reaction, picosecond CARS spectroscopy, and photon echoes. These processes all involve multiple pulse sequences in which the pulses are short or comparable in time scale to the... 

In both cases, la and lb, the total photodissociation cross section is completely determined by the short-time dynamics in the Franck-Condon region. In contrast, the partial cross sections, which determine the vibrational, rotational, and electronic-state distributions of the products, involves longer time dynamics. To obtain all of the relevant information about the reaction, the wavepacket evolution must be followed out into the product region of the potential energy surface and projected onto the various different vibrational and rotational states of the fragments. The partial cross section for scattering... 

PES for a long time, i.e. which does not dissociate either on the upper or on the lower PES. As discussed by Weide et the short-time dynamics of the wavepacket which does not quickly dissociate is governed by a periodic classical orbit that basically performs bending motion in the deep potential well. For a more detailed discussion of diffuse structures and periodic classical orbits, see Ref. 1. Thus, in contrast to H2S and the excitation of ozone in the Chappuis band, the diffuse structures are due to bending excitation rather than due to excitation of the symmetric stretch mode. Because the equilibrium angles in the two electronic states are so drastically different, the progression is long. 

The control is exercised by means of a few-cycle control pulse field with a cycle duration comparable to the time-scale of the nuclear dynamics and a short pulse width such that the control can be switched on/off at the right moment with respect to the position of the evolving wavepacket. First we survey the effects the moving crossing has on the nuclear wavepacket dynamics by looking at the results of a simplified single-cycle control pulse of the form... 

Assume we have an electron wavepacket associated natural orbitals electron dynamics for a short time... 

An exponential signal rise is expected if the dynamics can be described as a rate-governed population transfer between two states. This seems to be an inadequate model for the ESIPT. The step function, on the other hand, points to an almost classical ballistic motion along the PES [31]. The wavepacket produced by the optical excitation seems to move completely to the product state without pronounced spreading or splitting. The population appears delayed but within a very short time interval in the product state. A ballistic wavepacket motion is incompatible with a tunneling process of the proton from the enol to the keto site. 

A wavepacket initially at rest experiences displacement due to the slope of the potential. For short times, the displacement in momentum is proportional to the time that in position is proportional to time squared. Thus, a quick way to approximate the short-time dynamics of (t) is to write... 

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

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