J . Phys. Chem. 1993,97,
Reply to “Comment on ‘Monte Carlo Evaluation of Real-Time Feynman Path Integrals for Quantal Many-Body Dynamics: Distributed Approximating Functions and Gaussian Sampling’
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Received: April 20, I993 The graphical and numerical comparison between the Distributed Approximating Function (DAF) propagator and Makri’s Truncated Plane Wave (TPW) propagator presented in our work was designed to show explicitly that the two methods-and resulting propagators-are fundamentally different. Although the two approaches have some similarity, Makri starts with an exact propagator on a subspace of the Hilbert space of interest. In contrast, the DAF supplies an exact propagator for the DAF class, which is nor a subspace of this Hilbert space. This feature of the DAF is of fundamental importance, and consequently the differences between the two methods are well worth discussing. However, the explicit comparison of the two approaches appears in our paper solely because it was requested by the referee. Our example clearly demonstrates that the two methods are, in fact, quite different, and in that sense we consider the comparison “objective”. To address each of Makri’s three points: 1. She is correct that the parameters chosen for the graphical comparison were not based on considerations of equal accuracy with respect to propagation by matrix multiplication. Rather, as stated in our paper, they were chosen to compare and contrast the functional forms of the two propagators, and they were designed strictly to do the real-time Monte Carlo path integration for exactly the time interval reported in Makri’s original papers.
We show that if the two propagators are close in the region of small Ix - X I (which is expected to provide the dominant contribution in numerical path integral calculations), then the DAF propagator will decay exponentiallyon a much shorter length scale. Thus, the two propagators are fundamentally different in functional form. We assumed-and indeed, the nature of this whole approach demands-that the respective short-time propagators were sufficiently accurate for the calculation originally reported by Makri. The validity of this assumption is demonstrated in our Table 11, where, as we explicitly note, all of the short time propagators give sufficiently accurate results for the relevant time period when used in matrix multiplication methods. 2. We chose our DAF to yield accurate results for the Monte Carlo calculation as reported in Makri’s original papers (which was roughly one-half of a cycle). We see no reason why we should be expected to use a DAF that extends to longer times. 3. The extra restriction of cutting off sampling for Ix - X I greater than some value will indeed alter the sampling in such a way as to invalidate our statement concerning divergenceof the effective integrand in the Uwings”of the Gaussian. However, this extra restriction, and the accompanying parameter it necessitates, was neither discussed in Makri’s original paper nor conveyed to us when she provided the other parameters used in her calculations. The TPW propagator is not zero in the 1x1 > xmpxregion (and, in fact, when divided by the Gaussian sampling density the magnitude of the resulting function is quite large). In any event, this only reinforces our basic proposition,that the DAF and TPW approaches are fundamentally different, for no such modified sampling is used or needed for our Monte Carlo calculations. Certainly, enormous progress has been made in the field of real time path integration in recent years. In addition to our work and Makri’s, there have been contributions to stationary phase Monte Carlo as well as significantresults utilizing a coherent state formalism, and it would be of interest to compare all of these methods (with all of the necessary parameters given) on a common problem. However, we take the view that the true test of any method is its ability to perform accurate and speedy calculations on real world problems. We fully expect the DAF Monte Carlo approach to pass this test and hope to report such results in the near future.
The Ames Laboratory is operated for the Department of Energy by Iowa State University under Contract No. 2-7405-ENG82.
Acknowledgment. We thank Dr. T. L. Marchioro I1 for helpful discussions.
Donald J. Kouri,’ Wei Zhu, Xin Ma, and B. Montgomery Pettitt
Department of Chemistry, University of Houston, Houston, Texas 77204-5641 David K. Hoffman
Department of Chemistry and Ames Laboratory,f Iowa State University, Ames, Iowa 5001 I
0022-3654/93/2097-8107$04.00/0 0 1993 American Chemical Society
