Swarm of trajectories

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. 

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... 

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]. 

Often the integration steps have to be very small because it is not possible to evaluate the second derivative (Hessian) matrix of such systems. In such methods, the real nuclear (quantum) wavepacket must be emulated by a swarm of trajectories. Such trajectories are generated by sampling, which should be extensive enough (i.e., the swarm contains a sufficient variety of trajectories) to ensure that all relevant geometries involved in the chemical event have been explored. 

In the full-quantum dynamics method, the distribution of nuclear positions is accounted for in nuclear wavepacket form, that is, by a function that defines the distribution of momenta of each atom and the distribution of the position in the space of each atom. In classical and semi-classical or quasi-classical dynamics methods, the wavepacket distribution is emulated by a swarm of trajectories. We now briefly discuss how sampling can generate this swarm. 

For each member of the swarm of trajectories, some initial conditions are specified by sampling as shown in Scheme 2.8. 

The main practical problem in the implementation of the mixed quantum-classical dynamics method described in Section 4.2.4 is the nonlocal nature of the force in the equation of motion for the stationary-phase trajectories (Equation 4.29). Surface hopping methods provide an approximate, intuitive, stochastic alternative approach that uses the average dynamics of swarm of trajectories over the coupled surfaces to approximate the behavior of the nonlocal stationary-phase trajectory. The siu--face hopping method of Tully and Preston and Tully describes nonadiabatic dynamics even for systems with many particles. Commonly, the nuclei are treated classically, but it is important to consider a large niunber of trajectories in order to sample the quantum probability distribution in the phase space and, if necessary, a statistical distribution over states. In each of the many independent trajectories, the system evolves from the initial configuration for the time necessary for the description of the event of interest. The integration of a trajec-... 

The term standard studies refers to those based upon tracking the time evolution of swarms of trajectories, so that correlation functions and spectra can be computed. Instead, it is recommended that the analysis be based upon stability parameters of trajectories. Stability analysis of the CH periodic orbit, as a function of stretch-wag coupling, was the subject of an earlier study by Garcia-Ayllon et al. (145). 

Either of the methods (ii) or (iii) discussed in section IV can be used to predict the vibrational structure. Both methods use classical trajectory input, both contain information about the spectroscopic initial state cj), and both make clear the role of nearly periodic classical trajectories as the main reason for a structured absorption or emission spectrum. Neither the wavepacket nor the moving phase space density computed with a swarm of trajectories will show intermediate-resolution structure in the time domain (and thus structure in the frequency domain) unless the trajectories make return visits at short or intermediate times. The spectrum obtained from method (ii) or (iii) simulates a true molecular absorption spectrum. 

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