Wavepackets classical energy

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

Classical behavior is expected to occur on time scales that are shorter than the Heisenberg time (3.4). As the energy of the initial wavepacket increases, the quasiclassical and semiclassical regimes will extend over longer time intervals if the level density increases, as is usually the case in bounded... 

Fig. 3. The average time At expressed in units of la/v [spent by the wavepacket in a square well with (2mV0a2/tiY/2 = 315], Note that at high energies the classical value of unity is approached (figure reproduced from Merzbacher21).

The wavepacket /(t), on the other hand, is constructed in a completely different way. In view of (4.4), the initial state multiplied by the transition dipole function is instantaneously promoted to the excited electronic state. It can be regarded as the state created by an infinitely short light pulse. This picture is essentially classical (Franck principle) the electronic excitation induced by the external field does not change the coordinate and the momentum distributions of the parent molecule. As a consequence of the instantaneous excitation process, the wavepacket /(t) contains the stationary wavefunctions for all energies Ef, weighted by the amplitudes t(Ef,n) [see Equations (4.3) and (4.5)]. When the wavepacket attains the excited state, it immediately begins to move under the influence of the intramolecular forces. The time dependence of the excitation of the molecule due to the external perturbation and the evolution of the nuclear wavepacket /(t) on the excited-state PES must not be confused (Rama Krishna and Coalson 1988 Williams and Imre 1988a,b)... 

Figure 3.2 shows the same trajectory as in Figure 5.1 in a different representation together with the time-independent total dissociation wave-function StotiR,7)- The energy for the trajectory is the same as for the wavefunction. It comes as no surprise that, on the average, the wave-function closely follows the classical trajectory. Another example of this remarkable coincidence between quantum mechanical wavefunctions on one hand and classical trajectories on the other hand is shown in Figure 4.2. There, the time-dependent wavepacket (f) follows essentially the trajectory which starts at the equilibrium geometry of the ground electronic state. This is not unexpected in view of Equations (4.29) which state that the parameters of the center of the wavepacket obey the same equations of motion as the classical trajectory provided anharmonic effects are small. Figures 3.2 and 4.2 elucidate the correspondence between classical and quantum mechanics, at least for short times and fast evolving systems ...

Figure 8.2 depicts a typical potential energy surface (PES) for a symmetric molecule ABA with intramolecular bond distances R and R2] the ABA bond angle is assumed to be 180° (collinear configuration). The PES is symmetric with respect to the line defined by Ri = R2 it has a saddle point at short distances and decreases monotonically from the saddle point out into the two identical product channels A + BA and AB + A (see also Figure 7.18). The shaded area indicates the Franck-Condon (FC) region accessed via photon absorption and the two arrows illustrate the main dissociation paths for the quantum mechanical wavepacket or, equivalently, a swarm of classical trajectories. Because no barrier obstructs dissociation, the majority of trajectories immediately evanesce in either one of the two product channels without ever returning to the vicinity of the FC point.

Fig. 8.2. Typical potential energy surface for a symmetric triatomic molecule ABA. The potential energy surface of H2O in the first excited electronic state for a fixed bending angle has a similar overall shape. The two thin arrows illustrate the symmetric and the anti-symmetric stretch coordinates usually employed to characterize the bound motion in the electronic ground state. The two heavy arrows indicate the dissociation path of the major part of the wavepacket or a swarm of classical trajectories originating in the FC region which is represented by the shaded circle. Reproduced from Schinke, Weide, Heumann, and Engel (1991).

According to Section 4.1.1 the wavepacket is a superposition of stationary wavefunctions corresponding to a relatively wide range of energies. This and the superposition of three apparently different types of internal vibrations additionally obscures details of the underlying molecular motion that causes the recurrences. A particularly clear picture emerges, however, if we analyze the fragmentation dynamics in terms of classical trajectories. 

Fig. 8.11. (cont.) the unstable periodic orbit, represented by the solid line, influences the dissociation dynamics all direct trajectories, which fragment immediately without any recurrence, are discarded. The times range from 0 fs in (a) to 50.8 fs in (h). The arrows schematically indicate the evolution of the classical wavepacket and the heavy dot marks the equilibrium of the R-state potential energy surface. Adapted from Weide, Kiihl, and Schinke (1989). 

This is inherently impossible in the time-independent approach because the wavefunction contains the entire history of the wavepacket. The real understanding, however, is provided by classical mechanics. Plotting individual trajectories easily shows the type of internal motion leading to the recurrences which subsequently cause the diffuse structures in the energy domain. The next obvious step, finding the underlying periodic orbits, is rather straightforward. 

Fig. 5.2. Electronic excitation of LiH wavepacket from the outer classical turning point ( 6 a0) of the ground X1L7+ state. The B 1J7 X 1S+ transition is considered and the initial wavepacket is a 3ao shifted ground vibrational eigenfunction of LiH. The potential energy curves and the transition dipole moment of LiH are taken from the ab initio work of Partridge and Langhoff [55]...

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