Trajectory “swarms

In what is called BO MD, the nuclear wavepacket is simulated by a swarm of trajectories. We emphasize here that this does not necessarily mean that the nuclei are being treated classically. The difference is in the chosen initial conditions. A fully classical treatment takes the initial positions and momenta from a classical ensemble. The use of quantum mechanical distributions instead leads to a seraiclassical simulation. The important topic of choosing initial conditions is the subject of Section II.C. 

This picture is often refeired to as swarms of trajectories, and details are given in Appendix B. The nuclear problem is thus reduced to solving Newton s equations of motion for a number of different starting conditions. To connect... 

The big advantage of the Gaussian wavepacket method over the swarm of trajectory approach is that a wave function is being used, which can be easily manipulated to obtain quantum mechanical information such as the spechum, or reaction cross-sections. The initial Gaussian wave packet is chosen so that it... 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

Figure 1.146, Stre.ss trajectory map.s of southern Northeast Honshu in the late Cenozoic period, after Tsunakawa and Takeuchi (1986) with a slight addition. oh, . trajectory is drawn by smoothing the inferred stress orientations from the selected dike-swarms with K-Ar dates. Selected major faults with age estimation are also shown for indicating types of stress fields. T Extensional stress field, where ay > a 2>cth , and normal or gravity faulting is preferable. P Compre.ssional, oh > ay, reverse or thrust faulting...

However, the total dissociation wavefunction is useful in order to visualize the overall dissociation path in the upper electronic state as illustrated in Figure 2.3(a) for the two-dimensional model system. The variation of the center of the wavefunction with r intriguingly illustrates the substantial vibrational excitation of the product in this case. As we will demonstrate in Chapter 5, I tot closely resembles a swarm of classical trajectories launched in the vicinity of the ground-state equilibrium. Furthermore, we will prove in Chapter 4 that the total dissociation function is the Fourier transform of the evolving wavepacket in the time-dependent formulation of photodissociation. The evolving wavepacket, the swarm of classical trajectories, and the total dissociation wavefunction all lead to the same general picture of the dissociation process. 

Absorption and photodissociation cross sections are calculated within the classical approach by running swarms of individual trajectories on the excited-state PES. Each trajectory contributes to the cross section with a particular weight PM (to) which represents the distribution of all coordinates and all momenta before the vertical transition from the ground to the excited electronic state. P (to) should be a state-specific, quantum mechanical distribution function which reflects, as closely as possible, the initial quantum state (indicated by the superscript i) of the parent molecule before the electronic excitation. The theory pursued in this chapter is actually a hybrid of quantum and classical mechanics the parent molecule in the electronic ground state is treated quantum mechanically while the dynamics in the dissociative state is described by classical mechanics. 

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).

Knowing that the quantum mechanical wavepacket follows — at least for short times — the route of a swarm of classical trajectories, it is plausible to surmise that the periodic orbits also influence the motion of the wavepacket ... 

Figure 15 A swarm of classical trajectories incident on a model PES [49]. The site labelled bridge is initially attractive, but ultimately there is a barrier to dissociation at this site. At die atop site dissociation is activationless (downhill), but molecules can fail to take this path because they are initially steered to die bridge site. Some molecules can trap because die momentum normal to die surface is converted into parallel motion and rotations. After making several bounces, the trapped molecules dissociate or return to die gas-phase. For H2/Pd(l 1 1) die trapping channel contributes a large fraction of the scattered molecules [50].

The third step in the experiment is detection. On the simulation side this is equal to visualizing the results. A time evolution of the ealeulatcd wavefnnetion (or a. swarm of classical trajectories) has to be translated into a physically mean-ingfnl pictorial representation. Finally, at this stage we should link our simulations with the measurements. This means to find measnrablc (luantities which can be compared with the experiment. We can calculate quantities both in the time tmd... 

Wc have taken the initial coordinates and momenta for the swarm of trajectories from the Wigner distribution. It can be easily proven that for a separable wavefunction the Wigner function can be written in a product form. As a basis for photodissociation calculations we have used a wavefunction in the form 3, therefore, the corresponding Wigner distribution is given by an expression... 

Once the initial coordinates, momenta, and weights are generated, a swarm of classical trajectories is proj)agated by riurnerically solving the Newton e[Pg.483]

The result of a molecular dynamics simulation is a time dependent wavefunction (quantum dynamics) or a swarm of trajectories in a phase space (classical dynamics). To analyze what are the processes taking the place during photodissoeiation one can directly look at these. This analysis is, however, impractical for systems with a high dimensionality. We can calculate either (juantities in the time domain or in the energy domain, fn the time domain survival probabilities can be measured by pump-probe experiments [44], in the energy domain the distribution of the hydrogen kinetic energy can be experimentally obtained [8]. 

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