Adiabatic time evolution

Next we employ chirped pulses and investigate the aspects of adiabatic time evolution in the strong-field excitation of the five-state system of Figure 6.9. Similar scenarios to invert multistate systems using chirped pulses have been reported by... 

T. T. Nguyen-Dang,/. Chem. Phys., 90, 2657 (1989).Adiabatic Time-Evolution of Atoms and Molecules in Intense Radiation Fields. 

Care must be exercised to distinguish the concept of adiabatic Floquet dynamics introduced here, which refers to an adiabatic time-evolution, or to the slow variations of the Floquet basis with time, from the concept of adiabatic representation defined in the previous section, which refers to the slow variations of the electronic Hamiltonian (Floquet or not) with respect to nuclear motions (i.e., the noncommutativity of the electronic Hamiltonian Hei and the nuclear KE operator Tjv). Where confusion is possible and to be avoided, we shall refer to this concept of adiabaticity related to the BO approximation as the R-adiabaticity, while adiabaticity in actual time evolution will be termed t-adiabaticity. Non-adiabatic effects in time evolution are due to a fast variation of the (Floquet) Hamiltonian with time, causing Floquet states to change rapidly in time, to the extent that in going from one time slice to another, a resonance may be projected onto many new resonances as well as diffusion (continuum) states [40], and the Floquet analysis breaks down completely. We will see in Section 5 how one can take advantage of such effects to image nuclear motions by an ultrafast pump-probe process. 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

Fig. 11.5 Schematic comparison of (a) sudden and (b) adiabatic inversion of the z-component of the polarization vector. In the sudden case a n-pulse is applied while in the adiabatic case a frequency sweep is shown. The time evolution of the z-polarization as a function of the pulse duration...

Due to the large-level density of the lower-lying adiabatic electronic state, the chances of a back transfer of the adiabatic population are quite small for a multidimensional molecular system. To a good approximation, one may therefore assume that subsequent to an electronic transition a random walker will stay on the lower adiabatic potential-energy surface [175]. This observation suggests a physically appealing computational scheme to calculate the time evolution of the system for longer times. First, the initial decay of the adiabatic population is calculated within the QCL approach up to a time to, when the... 

Although the phase space of the nonadiabatic photoisomerization system is largely irregular, Fig. 36A demonstrates that the time evolution of a long trajectory can be characterized by a sequence of a few types of quasi-periodic orbits. The term quasi-periodic refers here to orbits that are close to an unstable periodic orbit and are, over a certain timescale, exactly periodic in the slow torsional mode and approximately periodic in the high-frequency vibrational and electronic degrees of freedom. In Fig. 36B, these orbits are schematically drawn as lines in the adiabatic potential-energy curves Wo and Wi. The first class of quasi-periodic orbits we wish to consider are orbits that predominantly... 

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.

Further evaluation of Eq. (2.35) requires an expression connecting 0(g)) (assumed to be nondegenerate) with Ox 5 ) (also assumed to be nondegenerate). This link is established via the interaction-picture time-evolution operator i.e. by an adiabatic switching oiHi ... 

Even for purely adiabatic reactions, the inadequacies of classical MD simulations are well known. The inability to keep zero-point energy in all of the oscillators of a molecule leads to unphysical behavior of classical trajectories after more than about a picosecond of their time evolution." It also means that some important physical organic phenomena, such as isotope effects, which are easily explained in a TST model, cannot be reproduced with classical molecular dynamics. So it is clear that there is much room for improvement of both the computational and experimental methods currently employed by those of us interested in reaction dynamics of organic molecules. Perhaps some of the readers of this book will be provide some of the solutions to these problems. 

Consider the case that the columns of U t) are the eigenvectors of H(t). Then U(t)H t)U t) is diagonal and the adiabatic approximation asserts that if U t)H t)U t) the time evolution of an initial superposition of eigenstates of H t), denoted k)(, remains unchanged aside from the phases... 

Adiabatic passage schemes are particularly suited to control population transfer between states, since the adiabatic following condition assesses the stability of the dynamics to fluctuations in the pulse duration and intensity [3]. The time evolution of the wave function does not depend on the dynamical phase, and is therefore slow in comparison with the vibrational time scale. This fact guarantees that the time variation of the observables during the controlled dynamics will be slow. Adiabatic methods can therefore be of great utility to control dynamic observables that do not commute with the Hamiltonian. We are interested in the control of the bond length of a diatomic molecule [4]. 

Time Resolved Spectra. We have studied the time evolution of the fluorescence of BA in acetone as shown in Figure 29. The results are in qualitative agreement with the simulated time-dependent spectra using the simulated p(z,t) and Eq. (38) (see Figure 30). This strongly supports the validity of the adiabatic GLE model for the charge transfer of S, BA. 

All types of time evolution are present in dynamical solvation effects. It is difficult, and perhaps not convenient, to define a general formulation of the Hamiltonian which can be used to treat all the possible cases. It is better to treat separately more homogeneous families of phenomena. The usual classification into three main types adiabatic, impulsive and oscillatory, may be of some help. The time dependence of the phenomenon may remain in the solute, and this will be the main case in our survey, but also in the solvent in both cases the coupling will oblige us to consider the dynamic behaviour of the whole system. We shall limit ourselves here to a selection of phenomena which will be considered in the following contributions for which extensions of the basic equilibrium QM approach are used, mainly phenomena related to spectroscopy. Other phenomena will be considered in the next section. 

Fig. 9.3 Computed time evolution of Rabi frequencies, adiabatic eigenvalues, and popula- tion of states 1) and 2). (After Fig. 3, Ref. [245].), ,j ...

Fig. 10 The time evolution of the adiabatic wavepacket in the space of coordinates Q and Q2 for the 2D three-state model of the relaxation of Cr(CO)3 after formation by photodissociation superimposed on the lower adiabatic potential energy surface. The D,k conical intersection is at (0,0). Adapted from [17]...

Figure 6 Adiabatic ab initio potentials with snapshots of tbe time evolution of tbe laser-induced wave packets (at f = 0 fs, original ground state t = 50 fs, dasbed envelope t — 135 fs, solid envelope), (a) Simulation of tbe pinnp/probe experiment with competing pathways leading to CpMn(CO)3+ and CpMn(CO)2 (R1 and R2 primary processes) (b) simulation of the control pathway leading exclusively to the CpMn(CO)3+ ions by quenching the internal conversion from the c A and b A" absorbing states to the low-lying dissociative states b A and a A"...

Time evolution of the ground state hole as well as fluorescence spectra initiated by a short pulse laser irradiation in solution has been conventionally explained in terms of the two-dimensional configuration coordinate model by Kinoshita . According to his theory, two adiabatic potential curves corresponding to the ground and excited states are assumed to have the same curvature but have the different potential minimum in the configuration coordinate. 

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