Quasi classical orbits

When they observed final states of m = 0, with both lasers polarized along the field direction, they observed the familiar quasi-Landau resonances spaced at 3a J2. In contrast, when they excited the m = 1 final state via the intermediate 2p m = 1 state these resonances were absent. Only resonances spaced by 0.64 wc were observed. In the final m = 0 states the oscillations at 3coJ2 are evident, but in the final m = 1 states the oscillations spaced by 0.64 yc are less so, although they are clearly evident if the data are smoothed over 2 cm-1. The periodicity is even more apparent in the Fourier transform of the spectrum, shown in Fig. 9.7. In Fig. 9.7(b) the peak corresponding to the resonance spacing of 0.64 yc is readily apparent. 

The origin of this resonance was identified by extending Reinhardt s21 wave packet notion. Realizing that the wave packet evolves along the classical trajectories of the electron, Holle et al. searched for classical trajectories in which the electron leaving the origin returned in a time of 9.5 ps, 1/2tt (0.64 yc). The orbit they discovered which has this return, or recurrence, time does not lie in the x,y plane, and thus cannot be predicted by the Edmonds-Starace approach.12,13 

It is clear that it is only possible to observe these resonances if their recurrence times are shorter than the coherence time of the exciting laser. With this point in mind, Main et al.23 made a five fold improvement in their spectral resolution and were able to see new resonances in the Fourier transform spectrum with longer recurrence times, as shown in Fig. 9.8. Tc is the recurrence time for a cyclotron orbit. The spectrum vs tuning energy is not shown since it is composed of 

An interesting hybrid way of calculating the spectrum is one used by Du and Delos.24 They start with a quantum mechanical wave packet at the origin and let it propagate to r = 50ao, where they use the normals of phase fronts of the wave packets to define classical electron trajectories. These classical trajectories are then followed. Some of the trajectories are reflected back to the origin, and when 

The fact that the resonances in the Fourier transform spectrum of Fig. 9.8 can be predicted by finding classical orbits which return to the origin suggests a better 

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To perform a PO analysis of nonadiabatic quantum dynamics, we employ a quasi-classical approximation that expresses time-dependent quantities of a vibronically coupled system in terms of the vibronic POs of the system [123]. Considering the quasi-classical expression (16) for the time-dependent expectation value of an observable A, this approximation assumes that the integrable islands in phase space represent the most significant contributions to the dynamics of the observables considered [236]. As a consequence, the short-time dynamics of the system is determined by its shortest POs and can be approximated by a time average over these orbits. Denoting the A th PO with period 7 by qk t),Pk t) we obtain [123]... 

Figure 36, (A) Classical time evolution of the reaction coordinate (p as obtained for a representative trajectory describing nonadiabatic photoisomerization. (B) The vibronic motion of the system can be characterized by various quasi-periodic orbits, which are schematically drawn here as lines in the adiabatic potentials Wo and Wi. (C) As an example of a mixed orbit, the time evolution between t — 2.7 ps and t2 =3.9 ps is shown here in more detail.

As we discussed in Section II in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasi-classical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits periods, we have to assume that die level density is well approximated as a sum over periodic orbits whose periods Tp = 3eSp and amplitudes vary slowly over the energy window [ - e, E + e]. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain... 

Table 9 shows that the latter prediction is supported by the EDA results as far as the contributions of the orbitals with different symmetry to the AEo term is concerned. 90.4% of the latter come from the ti orbitals. The d orbitals of Xe do not play a role for the Xe F bonds. The contributions of the eg and t2g orbital interactions are neghgible. However, the EDA results show also that the attractive orbital interactions are compensated by the repulsive Pauli term. Xenon hexafluoride is stable because there is strong quasi-classical Coulomb attraction between Xe and Fe. The sum of the quantum theoretical expressions (AEorb and AEpauu) is destabilizing. 

The introduction of / -parameter should be considered as further development of quasi-classical concepts with quantum-mechanical data on atom structure to obtain the criteria of phase-formation energy conditions. For the systems of similarly charged (e.g., orbitals in the given atom) homogeneous systems the principle of algebraic addition of such parameters is preserved ... 

Open problems in writing the basic organic chonistry textbook include the selection of concepts for the representation of the material, but also the level of the explanation of the complex phenomena such as reaction mechanisms or the electron structure. Here I propose the compromises. First compromise is related to the mode of the systematization of the contents, which can traditionally be based either on the classes of compounds, or on the classes of reactions. Here, the main chapter titles contain the reaction types, but the subtitles involve the compound classes. The electronic effects as well as the nature of the chemical bond is described by using the quasi-classical approach starting with the wave nature of the electron, and building the molecular orbitals from the linear combination of the atomic orbitals on the principle of the qualitative MO model. Hybridization is avoided because all the phenomena on this level can be simply explained by non-hybridized molecular orbitals. 

The behaviour of Q is quasi-classical. On the one hand we must have small electrostatic contributions from the active orbitals of the form... 

The GMS wave function [1,2] combines the advantages of the MO and VB models, preserving the classical chemical structures, but dealing with self-consistently optimized orbitals. From a formal point of view, it is able to reproduce all VB or MO based variational electronic wave functions in its framework. Besides that, it can deal in a straightforward way with the nonadiabatic effects of degenerate or quasi-degenerate states, calculating their interaction and properties. 

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