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On the zero point energy in classical trajectory computations
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19.In a recent paper (Ref. 15) where a good classical-quantal correspondence is reported, an unphysically low value of Planck’s constant (0.04 or 0.06ℏ) is used instead of lowering the energy by the zpe. The effect is the same as that of lowering the energy because as will be illustrated below, the dimensionless mass parameter is, as in one dimension proportional to [H. Harrison and R. B. Bernstein, J. Chem. Phys. 38, 2135 (1963)].
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21.Specifically, for a Hamiltonian with self anharmonicity terms one cannot use a harmonic approximation. We emphasized this in our original paper (Ref. 10) and reiterate this point below. Guo et al. (Ref. 15) have not followed our procedure and so the results they report do not constitute a test of our method.
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28.Within our approximation that is sufficiently slowly varying that its derivative is constant, Eq. (11), this pair of variables is, in fact, canonical where the curly brackets are the Poisson brackets. Of course, this is not exactly consistent with Eq. (13). The factorization method (Ref. 27) has identified all the potentials for which the factorization (7)–(9) is exact.
29.Nor is it reasonable to use x for as was apparently done in Ref. 15. As noted therein, the harmonic approximation can introduce spurious resonances which are not evident in the exact dynamics and this is the reason for the unphysical oscillations reported there. The (self) anharmonicities need to be realistically handled.
30.The question of “partial remedy” has also been discussed by McCormack and Lim (Ref. 14) in their reply to Schlier (Ref. 13).
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35.One can argue that quantum mechanically dissociation is still possible by tunneling. That is in principle correct but the required time scale is not what one would estimate by allowing the molecule to move classically using its zpe to overcome the barrier, except if there are partial barriers to the energy transfer [M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986)
35.and S. K. Gray, M. J. Davis, and S. A. Rice, J. Phys. Chem. 90, 3476 (1986)]. Take as an example a simple RRK estimate using a mean frequency of Then in the ground state the mean energy content is where s is the number of vibrational modes. Say the barrier to dissociation is Then the RRK reaction rate is This is faster than for all (Benzene has For a barrier to isomerization of the rate is faster than for all For very small values of s where the zpe is low, the RRK rate is indeed quite slow.
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