Numerical propagation

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... 

In the wave packet dynamics approach to unimolecular predissociation, the quantum dynamics is studied by directly propagating quantum wave packets. After numerically propagating the quantum wave packets, all the detailed information about the reaction dynamics can, in principle, be extracted from the numerical results. 

In the representation of (pj(r,)) (p2( /))> the matrix elements of Vi-i and Vne-i are automatically diagonalized, and the kinetic energy operator in Eq. (398) can also be evaluated straightforwardly. Hence numerically propagating the quantum wave packets reduces to a linear algebra routine. Figure 45 displays the calculated quantum wave packets /(/ , r, f)) at three different times. The initial vibrational state of I2 is taken to be v = 20. 

This decomposition forms the basis for the numerical propagation scheme which will be outlined in the next subsection. 

We have found, by direct numerical propagation under H, that initially Gaussian pseudorotating wave packets often remain fairly well localized over several periods of motion. Even a wave packet excited into the trough of a sym-triazine-like system with k 2 executes more than a... 

NUMERICAL PROPAGATION METHOD 4.1. Propagation in a local interaction picture... 

The equation of motion for Av5 which involves only a commutator with a Hamiltonian, could be solved by expanding the DOp in terms of density amplitudes satisfying a Schrodinger-like equation. More generally, however, the equation for a reduced DOp would contain dissipative rates, and this would make it necessary to solve the equation directly for the DOp. We therefore develop the numerical propagation method for the general case. The computational procedure starts with a basis set of quantum states, arranged as a row matrix l3>) = l< )2,...], taken here to be... 

Thus for each original reaction with oxygen there can be numerous propagation reaction. The peroxides may also cleave to give aldehydes, ketones, acids, or alcohols. Due to this type of molecular cleavage, the product becomes soft with lower average molecular weight. 

As in the single-channel case, the coupled-channel equations are usually solved by numerical propagation. One approach is to start a solution matrix (r) at a distance... 

We may apply the split-operator method (Sec. 3.2.1) to the three matrices lijV) V/), and Vo for a short-time numerical propagation scheme. Ionization is now described by population of the neutral state Xn R,t) transferring to the ionized state partial-wave components Xc,kjix R,t) over time through the interaction represented by the matrix Vo(i ,f)-... 

As reference data for comparison, the full quantum mechanical wave-functions of the coherent-type Gaussian form as in Eq. (6.106) are numerically propagated. The wavefunction is chosen to start at a point xi,X2) = (5.5,0.0) with zero initial momentum on the first diabatic state. The point of conical intersection is located at (xi,X2) = (2.5,0.0), the potential energy of which is exactly the same as the value at (xi, X2) = (5.5,0.0). This symmetric dynamics is performed to purely illustrate how the quantum wave functions are branched behind the conical intersection. The contour plots of the density of nuclear wavefunctions at selected times are given in Fig. 6.12 (solid curves). 

Note that eqn (13.24) is the original form of HEOM, where the ADOs at different tier are of different dimensionality. For numerical propagator techniques, we would also like to scale all ADOs to have a uniform error tolerance. This would facilitate an efficient on-the-fly filtering algorithm that also automatically truncates the level of hierarchy by introducing the following scaled ADOs ... 

In carrying out numerical propagation of eqn (14.41), it is convenient to employ the split operator algorithm again to express the exponential operator of the general potential operator U by using the split operator formula of either... 

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