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The anharmonic features of the short-time dynamics of fluids: The time evolution and mixing of instantaneous normal modes
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15.The connections between these inherent-structure ideas and instantaneous normal modes (and, in particular, with imaginary instantaneous normal modes) were first explored by R. A. LaViolette and F. H. Stillinger, J. Chem. Phys. 83, 4079 (1985).
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19.T. S. Kalbfleisch, L. D. Ziegler, and T. Keyes, J. Chem. Phys. 105, 7034 (1996);
19.S. A. Egorov, M. D. Stephens, and J. L. Skinner, J. Chem. Phys. 107, 10485 (1997).
20.The first quantitative attempts to use imaginary modes to understand diffusion were those of the Keyes group: B. Madan, T. Keyes, and G. Seeley, J. Chem. Phys. 92, 7565 (1990);
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20.G. Seeley, T. Keyes, and B. Madan, J. Chem. Phys. 95, 3847 (1991).
20.Some of their more recent work on the applications and implications of imaginary modes includes P. Moore and T. Keyes, J. Chem. Phys. 100, 6709 (1994);
20.T. Keyes, J. Chem. Phys. 101, 5081 (1994);
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23.W.-X. Li, T. Keyes, and F. Sciortino, J. Chem. Phys. 108, 252 (1998).
24.There have also been a number of less heuristic attempts to see how imaginary modes might enter the INM expressions for spectroscopically observable dynamics. See, for example, J. T. Kindt and C. A. Schmuttenmaer, J. Chem. Phys. 106, 4389 (1997).
25.R. E. Larsen, E. F. David, G. Goodyear, and R. M. Stratt, J. Chem. Phys. 107, 524 (1997).
26.R. E. Larsen and R. M. Stratt (in preparation).
27.So far, the only quantitative binary-mode theories that have been developed have been for dynamics in atomic liquids (Refs. 19, 25, and 26). However, there is evidence that aspects of solvation and vibrational population relaxation in molecular liquids are analogously governed by the motion of the solute and a single solvent. See Ref. 8.
28.As we shall discuss further, papers have already begun to appear discussing the time evolution of instantaneous normal modes. See, in particular, Refs. 29 and 30.
29.M. Buchner and T. Dorfmüller, J. Mol. Liquid 65, 157 (1995).
30.P. Moore and B. Space, J. Chem. Phys. 107, 5635 (1997).
31.We have formulated the INM theory here as it would apply to neat atomic fluids, the subject of this paper. The generalizations to molecular fluids and to both dilute-solute/solvent systems and to other kinds of mixtures have been discussed previously by ourselves and others. See, for example, Refs. 5678, 16, and 18.
32.J. Tully, in Dynamics of Molecular Collisions, edited by W. H. Miller (Plenum, New York, 1976), Part B.
33.E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970) pp. 428–429;
33.G. Fischer, Vibronic Coupling (Academic, London, 1984), pp. 22–24.
34.This result was first observed numerically by M. Buchner (unpublished).
35.As we shall see presently, even with the full, nonperturbative, functions, adiabatic lifetimes can be gross underestimates. In particular, if the modes in the eigenvalue correlation functions are indexed by adiabatic rather than diabatic labels (that is if the labeling, α, is by numerical eigenvalue ordering rather by microscopic identity), correlation functions of the form will decay simply because the true (microscopic) modes are constantly changing their ordering. For the modes we consider, adiabatic lifetimes computed from correlation functions of this sort, such as those in Ref. 30, will turn out to be far smaller than the true mode-mixing times.
36.M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, New York, 1987), Chap. 3.
37.W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge University Press, Cambridge, 1988), Chap. 2.
38.Note that this decomposition is slightly simpler, conceptually, than that given in Ref. 25. Both here and in the previous paper, the only rows and columns containing nonzero elements of the bath matrix are those corresponding to atoms not involved in mnn pairs. However, here, the sum over atoms in the diagonal elements of the bath matrix is also restricted to atoms not involved in mnn pairs. The remaining diagonal-element terms, which arise from sums over atoms which are involved in mnn pairs, are consigned to the residual matrix ΔD. In Eqs. (5.4)–(5.6) of Ref. 25, though, both sets of diagonal terms were lumped into We note that this distinction has no effect on any of the calculations performed in Ref. 25.
39.J. E. Adams and R. M. Stratt, J. Chem. Phys. 93, 1632 (1990).
40.G. Goodyear and R. M. Stratt (unpublished).
41.That is, the atom projections of the modes differ from the ideal, isolated binary-mode values of 0.5 by about as much after the event as they did before the event.
42.In the spirit (though not in the letter) of Brillouin–Wigner perturbation theory, we renormalize by using the exact, molecular-dynamics derived, eigenvalues in Eq. (4.17), rather than the “bare” 0th-order eigenvalues of Eq. (4.5). See, for example, S. Wilson, Electron Correlation in Molecules (Clarendon, Oxford, 1984), Chap. 4. We have two main reasons for taking this step. For one thing, there is a shift in the time evolution of the bare eigenvalues from that the exact dynamics (altering the timing of the scattering event). More importantly, since the bare eigenvalues actually cross one another, using them in Eq. (4.17) would lead to an unphysical divergence. Note that in order to remove the divergence one has to renormalize by using 2nd or higher order formulas for the eigenvalues.
43.P. Moore, A. Tokmakoff, T. Keyes, and M. D. Fayer, J. Chem. Phys. 103, 3325 (1995).
44.Of course, it is this intramode anharmonicity that makes the eigenvalues change in time and eventually become resonant with other modes—which is an important source of anharmonicity.
45.See, for example, S. Palese, S. Mukamel, R. J. Dwayne Miller, and W. T. Lotshaw, J. Phys. Chem. 100, 10380 (1996).
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