Abstract
The numerical simulation of the internal motions of a molecule undergoing a unimolecular reaction on an assumed potential energy surface requires the step-by-step solution of a set of simultaneous differential equations. After several thousand time steps, due to differences in the handling of rounding errors in different computing systems, the situation often arises that no two computing machines will give the same result for a given trajectory, even when running the identical algorithm. Such effects are demonstrated for a simple unimolecular isomerisation reaction. In general, it is only when reliance is placed on the integration of a single trajectory, rather than on an ensemble of similar trajectories, that conclusions may be unreliable. Moreover, under certain conditions, small molecules may show signs of chaotic internal motions; conversely, but for a different reason, large molecules may exhibit non-statistical characteristics rather than RRKM behaviour. The rounding error problem, in a slightly different guise, has come to be dubbed the “butterfly effect” in popular culture, and the original proposition is re-examined using 16- and 32-decimal precision arithmetic. 
[1] D. Shen, H.O. Pritchard, J. Chem. Soc., Faraday Trans. 92, 4357 (1996) http://dx.doi.org/10.1039/ft996920435710.1039/ft9969204357Search in Google Scholar
[2] D. Shen, H.O. Pritchard, Int. J. Chem. Kinetics 26, 729 (1994) http://dx.doi.org/10.1002/kin.55026070610.1002/kin.550260706Search in Google Scholar
[3] W-T. Chan, Ph. D. thesis, York University (York University, Toronto, Canada, 1995) Search in Google Scholar
[4] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1985) Search in Google Scholar
[5] W.L. Hase, D.G. Buckowski, J. Comp. Chem. 3, 335 (1982) http://dx.doi.org/10.1002/jcc.54003030810.1002/jcc.540030308Search in Google Scholar
[6] M. Winger, D. Trzesniak, R. Barron, W.F. van Gunsteren, Phys. Chem. Chem. Phys. 11, 1934 (2009) Search in Google Scholar
[7] H.O. Pritchard, J. Phys. Chem. A 109, 1400 (2005) http://dx.doi.org/10.1021/jp045262s10.1021/jp045262sSearch in Google Scholar PubMed
[8] E.N. Lorenz, Trans. New York Acad. Sci., Ser. II 25, 409 (1963) 10.1111/j.2164-0947.1963.tb01464.xSearch in Google Scholar
[9] E.N. Lorenz, Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas? (AAAS, Boston, 1972) http://eapsweb.mit.edu/research/Lorenz/publications.htm Search in Google Scholar
[10] J. Gleik, Chaos, Making a New Science (Penguin, New York, 1987) Search in Google Scholar
[11] R. Redford, Havana (Universal Studios, Hollywood, CA, 1990) Search in Google Scholar
[12] D. Shen, Ph. D. thesis, York University (York University, Toronto, Canada, 1991) Search in Google Scholar
[13] Pilot and Air Traffic Controller Guide to Wake Turbulence, Training Manual 04SEC2 (Federal Aviation Administration, Washington, DC, 2004) Search in Google Scholar
[14] Aircraft Wake Turbulence, Circular AC90-23F (Federal Aviation Administration, Washington, DC, 2002) Search in Google Scholar
[15] W-T. Chan, D. Shen, H.O. Pritchard, J. Chem. Soc., Faraday Trans. 91, 1717 (1995) http://dx.doi.org/10.1039/ft995910171710.1039/ft9959101717Search in Google Scholar
[16] D. Shen, H.O. Pritchard, J. Chem. Soc., Faraday Trans. 86, 3171 (1990) http://dx.doi.org/10.1039/ft990860317110.1039/ft9908603171Search in Google Scholar
[17] D. Shen, H.O. Pritchard, J. Chem. Soc., Faraday Trans. 87, 3595 (1991) http://dx.doi.org/10.1039/ft991870359510.1039/ft9918703595Search in Google Scholar
[18] H.O. Pritchard, Quantum theory of unimolecular reactions (Cambridge University Press, Cambridge, 1984) Section 7.1 http://dx.doi.org/10.1017/CBO978051173583710.1017/CBO9780511735837Search in Google Scholar
[19] D. Shen, H.O. Pritchard, Mol. Phys. 80, 1135 (1993) http://dx.doi.org/10.1080/0026897930010293110.1080/00268979300102931Search in Google Scholar
[20] W.D. Price, P.D. Schnier, R.A. Jockusch, E.F. Strittmatter, E.R. Williams, J. Am. Chem. Soc. 118, 10640 (1996) http://dx.doi.org/10.1021/ja961812r10.1021/ja961812rSearch in Google Scholar PubMed PubMed Central
[21] R.L. Hayes, E. Fattal, N. Govind, E.A. Carter, J. Am. Chem. Soc. 123, 641 (2001) http://dx.doi.org/10.1021/ja000907x10.1021/ja000907xSearch in Google Scholar PubMed
[22] L. Sun, W.L. Hase, K. Song, J. Am. Chem. Soc. 123, 5753 (2001) http://dx.doi.org/10.1021/ja004077z10.1021/ja004077zSearch in Google Scholar PubMed
[23] R. Schork, H. Köppel, J. Chem. Phys. 115, 7907 (2001) http://dx.doi.org/10.1063/1.140512010.1063/1.1405120Search in Google Scholar
[24] D.J. Mann, W.L. Hase, J. Am. Chem. Soc. 124, 3208 (2002) http://dx.doi.org/10.1021/ja017343x10.1021/ja017343xSearch in Google Scholar PubMed
[25] T.R. Quinn, S. Tremaine, M. Duncan, Astron. J. 101, 2287 (1991) http://dx.doi.org/10.1086/11585010.1086/115850Search in Google Scholar
[26] D. Shen, H.O. Pritchard, J. Chem. Soc., Faraday Trans. 92, 1297 (1996) http://dx.doi.org/10.1039/ft996920129710.1039/ft9969201297Search in Google Scholar
[27] H.O. Pritchard, S.R. Vatsya, D. Shen, J. Chem. Phys. 110, 9384 (1999) http://dx.doi.org/10.1063/1.47890310.1063/1.478903Search in Google Scholar
[28] B.G. Sumpter, D.L. Thompson, J. Chem. Phys. 87, 5809 (1987) http://dx.doi.org/10.1063/1.45350510.1063/1.453505Search in Google Scholar
[29] P.J. Stimac, J.R. Barker, J. Phys. Chem. A 112, 2553 (2008) http://dx.doi.org/10.1021/jp710016n10.1021/jp710016nSearch in Google Scholar PubMed
[30] N.B. Slater, Theory of unimolecular reactions (Methuen, London, 1959) Section 8.3 Search in Google Scholar
[31] P.J. Robinson, K.A. Holbrook, Unimolecular reactions (Wiley, London, 1972) Section 7.2 Search in Google Scholar
[32] D. Shen, H.O. Pritchard, J. Chem. Soc., Faraday Trans. 88, 2985 (1992) http://dx.doi.org/10.1039/ft992880298510.1039/ft9928802985Search in Google Scholar
[33] E.N. Lorenz, Tellus 16, 1 (1964) http://dx.doi.org/10.1111/j.2153-3490.1964.tb00136.x10.1111/j.2153-3490.1964.tb00136.xSearch in Google Scholar
[34] E.N. Lorenz, Physica D 35, 299 (1989) http://dx.doi.org/10.1016/0167-2789(89)90072-910.1016/0167-2789(89)90072-9Search in Google Scholar
[35] R.H. Dalling, M.E. Goggin, Am. J. Phys. 62, 563 (1994) http://dx.doi.org/10.1119/1.1751910.1119/1.17519Search in Google Scholar
[36] D. Auerbach, Am. J. Phys. 63, 276 (1995) http://dx.doi.org/10.1119/1.1794110.1119/1.17941Search in Google Scholar
[37] M.E. Goggin, R.H. Dalling, Am. J. Phys. 63, 277 (1995) http://dx.doi.org/10.1119/1.1794210.1119/1.17942Search in Google Scholar
[38] P. Brumer, Adv. Chem. Phys. 47, 201 (1981) http://dx.doi.org/10.1002/9780470142677.ch310.1002/9780470142677.ch3Search in Google Scholar
[39] P. Brumer, S. Shapiro, Adv. Chem. Phys. 70, 365 (1988) http://dx.doi.org/10.1002/9780470141199.ch910.1002/9780470141199.ch9Search in Google Scholar
[40] D. Butler, Nature 466, 804 (2010) http://dx.doi.org/10.1038/466804a10.1038/466804aSearch in Google Scholar PubMed
[41] Description of the NCAR Community Atmospheric Model, NCAR Technical Note TN-464 (National Center for Atmospheric Research, Boulder, CO, 2004) Search in Google Scholar
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