Wavepacket initial condition

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. 

In the reactant channel leading up to the transition region, motion along represents the FI atom approaching the molecule, while motion along / is the vibrational motion of the atom. The initial wavepacket is chosen to represent the desired initial conditions. In Figure 2, the FI2 molecule is initially in the ground... 

One way to overcome this problem is to start by setting up the ensemble of trajectories (or wavepacket) at the transition state. If these bajectories are then run back in time into the reactants region, they can be used to set up the distribution of initial conditions that reach the barrier. These can then be run forward to completion, that is, into the products, and by using transition state theory a reaction rate obtained [145]. These ideas have also been recently extended to non-adiabatic systems [146]. 

One expects the timescale of the nonadiabatic transition to broaden for a stationary initial state, where the nuclear wavepacket will be less localized. To mimic the case of a stationary initial state, we have averaged the results of 25 nonstationary initial conditions and the resulting ground-state population is shown as the dashed line in Fig. 8. The expected broadening is seen, but the nonadiabatic events are still close to the impulsive limit. Additional averaging of the results would further smooth the dashed line. 

Gaussian wavepacket propagation, 377-381 initial condition selection, 373-377 nuclear Schrodinger equation, 363-373 Adiabatic-to-diabatic transformation (ADT). 

The evolution of a wavepacket representing the H + H2 scattering reaction for a particular set of initial conditions is plotted on Figure 2 as a series of snapshots. To display the three-dimensional (3D) wavepacket on a two-dimensional (2D) plot, the reduced density... 

The initial conditions of the source can contribute to the character of the emitted electromagnetic radiation. Thus emission from excited atoms occurs in the form of individual photon wavepackets and can give rise to particle-like photon beams under appropriate conditions. Electromagnetic radiation that is excited by the current system of a macroscopic antenna is, on the other hand, expected to produce nearly plane waves at large distances from the source. 

In Fig. 2 we compare results using e = 0.4 for the two mixed quantum-classical methods outlined in this chapter with exact results obtained from MCTDH wavepacket dynamics calculations. To make a reliable comparison the approximate finite temperature calculations were performed at very low temperatures (/ = 25), though a product of ground state wave functions for the independent harmonic oscillator modes could have been used to make the initial conditions identical to those used in the MCTDH calculations. 

In practice one does not proceed as we did in the above derivation. Instead of calculating first all stationary wavefunctions and then constructing the wavepacket according to (4.3), one solves the time-dependent Schrodinger equation (4.1) with the initial condition (4.4) directly. Numerical propagation schemes will be discussed in the next section. Since 4 /(0) is real the autocorrelation function fulfills the symmetry relation... 

Propagation of the wavepacket /(i) in the upper electronic state with initial condition 4>/(0) = pj)... 

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from... 

Evolution of the wavepacket in the excited state with initial condition... 

Knowing the eigenfunctions and eigenvalues implies that any solution to this equation can be written in the form (2.6), with the coefficients determined from the initial condition according to c (Zo) = (Vimportant property of properly chosen wavepackets of harmonic oscillator wavefunctions. 

Let us briefly comment on the relation between the quantum mechanical field-assisted scattering process and its treatment within the classical limit. Therefore, a classical trajectory (R(t)) is determined in the presence of the LCT field-derived quantum mechanically, where the initial condition is defined by the average position and momentum of the initial wavepacket (Eq. (25)). The time evolution of this trajectory is compared to the coordinate expectation value in the lower panel of Fig. 11. It is seen that the trajectory is trapped by the field interaction, leading to a classical vibration at a smaller total energy (0.102 eV as compared to E g = 0.113 eV). Deviations in the two curves are to be expected and arise from the spatial extent of the wavepacket. Here, we encounter a first example for the qualitative relation between quantum and classical dynamics in the case of local control. 

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