Wave packet distribution

Time-Independent Wave-Packet-Distributed Approximating Functional Approach to Quantum Dynamics... 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

This decomposition suggests the strategy for parallel computing. In the simplest implementation, we distribute the job over processors, where Nk is the number of K values in our problem. Each processor is labeled by an index, k = 0,1,..., Nk — 1, associated with a particular value of K. We distribute the wave packet vector, v, over the processors such that processor k contains v( , k) and is responsible for everything related to its propagation, in particular, for computing the action of on v( , fe). Except for the work associated with the Coriolis terms, all of the computation is done locally on each processor, proceeds independently, and involves no communication. 

In general, the topology of interprocessor communication reflects both the structure of the mathematical algorithms being employed and the way that the wave packet is distributed. For example, our very first implementation of parallel algorithms in a study of planar OH - - CO [47] used fast Fourier transforms (FFTs) to compute the action of 7, which also required all-to-all communication but in a topology that is very different from the simple ring-like structure shown in Fig. 5. 

There have been several successful approaches [41, 42] to improving the scaling of parallel algorithms that employ DVRs or FFTs. These rely on more sophisticated methods of distributing the wave packet so as to reduce the amount of communication required. One should consult Refs. [41] and [42] for details. 

To use MPI to distribute the wave packet across nodes rather than processors. 

Figure 8. Schematic of distribution of the wave packet over the nodes and the communication pattern for the hybrid OpenMP/MPI method. The wave packet is distributed over four nodes, each of which contains four processors. All of the communication is handled by the shaded processors the other processors are simultaneously performing computations. Note that there is only nearest-neighbor communication between nodes.

Gray and Wozny [101, 102] later disclosed the role of quantum interference in the vibrational predissociation of He Cl2(B, v, n = 0) and Ne Cl2(B, v, = 0) using three-dimensional wave packet calculations. Their results revealed that the high / tail for the VP product distribution of Ne Cl2(B, v ) was consistent with the final-state interactions during predissociation of the complex, while the node at in the He Cl2(B, v )Av = — 1 rotational distribution could only be accounted for through interference effects. They also implemented this model in calculations of the VP from the T-shaped He I C1(B, v = 3, n = 0) intermolecular level forming He+ I C1(B, v = 2) products [101]. The calculated I C1(B, v = 2,/) product state distribution remarkably resembles the distribution obtained by our group, open circles in Fig. 12(b). 

Thus, the distance /2a may be regarded as a measure of the width of the distribution A k) and is called the half width. The half width may be defined using 1/2 or some other fraction instead of 1/e. The reason for using 1/e is that the value of k at that point is easily obtained without consulting a table of numerical values. These various possible definitions give different numerical values for the half width, but all these values are of the same order of magnitude. Since the value of I (x, r) falls from its maximum value of (2jr) to 1/e of that value when x — v t equals v/lja, the distance flja may be considered the half width of the wave packet. 

If we regard the uncertainty /S.k in the value of k as the half width of the distribution A(k) and the uncertainty Ax in the position of the wave packet as its half width, then the product of these two uncertainties is... 

Figure 1.5 The real part of a wave packet for a gaussian wave number distribution.

Figure 10.8 Examples of controlling rotational wave packets with an optical centrifuge. Shifting the center of a narrow wave packet is shown for centrifuged oxygen on (a), whereas varying the width of the wave packet is demonstrated for nitrogen on (b). Black dots represent the thermal distribution at room temperature.

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