Wave packet representation

Sawada S and Metiu H 1986 A multiple trajectory theory for curve crossing problems obtained by using a Gaussian wave packet representation of the nuclear motion J. Chem. Phys. 84 227-38... 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

It is possible to propagate just the real part of a wave packet if the representation of H is real-valued, as it often is in chemical reaction dynamics. 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

The theory behind body-fixed representations and the associated angular momentum function expansions of the wavefunction (or wave packet) in terms of bases parameterized by the relevant constants of the motion and approximate constants of the motion is highly technical. Some pertinent results will simply be stated. The two good constants of the motion are total angular momentum, J, and parity, p = +1 or 1. An approximate constant of the motion is K, the body-fixed projection of total angular momentum on the body-fixed axis. For simplicity, we will restrict attention to the helicity-decoupled or centrifugal sudden (CS) approximation in which K can be assumed to be a constant of the motion. In terms of aU its components, and the iteration number k, the real wave packet is taken to be [21]... 

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

A typical initial condition in ordinary wave packet dynamics is an incoming Gaussian wave packet consistent with particular diatomic vibrational and rotational quantum numbers. In the present case, of course, one has two diatomics and with the rotational basis representation of Eq. (30) one would have, for the full complex wave packet. 

In this section we represent wave packets as vectors v and refer to the individual components of v in the computer-friendly form of multidimensional arrays with indices, for example, v i,j,k), where the indices i,j, and k refer to the various degrees of freedom. These vectors may be real, such as the real vector V of the previous sections, or they may be complex representations of the... 

Again, this relation arises from the representation of a particle by a wave packet and is a property of Fourier transforms. 

In Eq. (109), y denotes an n-dimensional diagonal matrix, with element y being the width parameter for the coherent state of the y th dimension. The coordinate space representation of an n-dimensional coherent state is the product of n onedimensional minimum uncertainty wave packets... 

It is instructive to consider the momentum-space representation of the Gaussian wave packet. In this representation, the states are projected onto the eigenstates of the momentum operator, i.e., P p) = p p), which in the coordinate representation takes the form... 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

FIGURE 15 Wigner representation of the optimal electric field for l2 wave packet control. [Reproduced with permission from Krause, Whitnell, J. L., Wilson, R. M., K. R., and Yan, Y. (1993). J. Chem. Phys. 99, 6562. Copyright American Institute of Physics.]... 

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. 

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