Numerical simulations of vibrational relaxation

Numerical simulations have become a central tool in studies of condensed phase processes. The science, technique, and art of this tool are subjects of several excellent texts. Here we assume that important problems such as choosing a 

For a detailed treatment of this model see R. B. Gerber and M. Berkowitz, Phys. Rev. Lett. 39, 1000 (1977). 

for example, M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1987) D. C. Rapaport, The Art of Molecular Dynamics Simulation, 2nd edn (Cambridge University Press, Cambridge, 2004) Daan Frenkel and B. Smit, Understanding Molecular Simulation, 2nd edn (Academic Press, San Diego, CA, 2002) J. M. Haile, Molecular Dynamics Simulation Elementary Methods ( Wiley, New York, 1997). 

It should be noted that a similar problem exists also in systems with vastly different lengthscales. The smallest characteristic lengthscale in atomic-level condensed phase simulations is the interatomic distance which is of the same order as the atomic size. To simulate phenomena that involve much larger lengthscales we 

Coming back to the timescale issue, it is clear that direct observation of signals such as shown in Fig. 13.2 cannot be achieved with numerical simulations. Fortunately an alternative approach is suggested by Eq. (13.26), which provides a way to compute the vibrational relaxation rate directly. This calculation involves the autocorrelation function of the force exerted by the solvent atoms on the frozen oscillator coordinate. Because such correlation functions decay to zero relatively fast (on timescales in the range of pico to nano seconds depending on temperature), its numerical evaluation requires much shorter simulations. Several points should be noted  

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Molecular dynamic simulations are very useful for solvation dynamic studies. In contrast to the difficulties described in applying numerical methods to the problems of vibrational relaxation (Section 13.6) and barrier crossing (Section 14.7), solvation dynamics is a short-time downhill process that takes place (in pure simple solvents) on timescales easily accessible to numerical work. 

The procedure described here is an example for combining theory (that relates rates and currents to time correlation functions) with numerical simulations to provide a practical tool for rate evaluation. Note that this calculation assumes that the process under study is indeed a simple rate process characterized by a single rate. For example, this level of the theory cannot account for the nonexponential relaxation of the v = 10 vibrational level of O2 in Argon matrix as observed in Fig. 13.2. 

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