Fundamentals: general discussion Stuart C. Althorpe, Vijay Beniwal, Peter G. Bolhuis, Jo ˜ ao Brand ˜ ao, David C. Clary, John Ellis, Wei Fang, David R. Glowacki, Timothy J. H. Hele, Hannes J ´ onsson, Johannes K ¨ astner, Nancy Makri, David E. Manolopoulos, Laura K. McKemmish, Georg Menzl, Thomas F. Miller III, William H. Miller, Eli Pollak, Sergio Rampino, Jeremy O. Richardson, Martin Richter, Priyadarshi Roy Chowdhury, Dmitry Shalashilin, Jonathan Tennyson and Ralph Welsch DOI: 10.1039/c6fd90077a Laura McKemmish opened discussion of the introductory lecture by William Miller: Can you please elaborate on the relationship between the classical and quantum Hamiltonian, and between the classical and quantum methods of solving Hamiltonians. William Miller responded: The quantum Hamiltonian (operator), where the coordinates and momentum variables are the usual quantum coordinate and momentum operators, is exact; i.e., if it were used in the Schr¨ odinger equation it would generate the exact quantum dynamics. The classical Meyer–Miller (MM) Hamiltonian is this same quantity with the coordinate and momentum operators now classical coordinate and momentum variables, whose time evolution is determined by integrating Hamilton's equations (that are generated from the classical Hamiltonian in the usual way). Jeremy Richardson asked: Your results for the time dependence of the elec- tronic states are excellent. However, can such good information be obtained about the nuclear dynamics using your method? In particular there might be difficulties associated with inverted potentials, caused when the state population becomes less than 0. Will this cause problems when you apply the method to more realistic systems? William Miller answered: In the published version of my introductory lecture (DOI: 10.1039/c6fd00181e) the issue you mention is discussed, i.e., when some of the nuclear potential energy surfaces (PESs) have very harsh repulsive walls (an innitely hard wall would be the most dramatic case) and the state populations become less than 0. We saw this in earlier work 1 when the MM Hamiltonian was implemented semiclassically (using the initial value representation). Here some classical trajectories diverged (‘ran away') because of the situation you describe, but in that case this was ~10% or fewer of them, so they were simply discarded This journal is © The Royal Society of Chemistry 2016 Faraday Discuss. , 2016, 195, 139–169 | 139 Faraday Discussions Cite this: Faraday Discuss. , 2016, 195, 139 DISCUSSIONS Published on 12 December 2016. Downloaded by University College London on 23/02/2017 11:33:27. View Article Online View Journal | View Issue