Time correlation functions classical bath

There are also situations when one is not in the classical limit, and so Equation (13) would not seem applicable, and instead one would like to approximate one of the quantum mechanical expressions for Ti by relating the relevant quantum time-correlation function to its classical analog. For the sake of definiteness, let us consider the case where the oscillator is harmonic and the oscillator-bath coupling is linear in q, as discussed above. In this case k 0 can be written as... 

In this case, because the bath includes intramolecular coordinates, the evaluation of this formula, which involves products of translational and vibrational time-correlation functions, is quite complicated (12). For a polyatomic solute with a large enough number of vibrational modes, we argue that the time-correlation functions for translations decay on the time scale of the inverse of the characteristic frequencies of the translational bath, which is much slower than the decay of time-correlation functions for the intramolecular vibrations. Therefore we can replace the translational time-correlation functions by their initial values, which we evaluate classically. The upshot is that the rate constant can be written as (12)... 

These normal modes evolve independently of each other. Their classical equations of motion are Uk = —co Uk, whose general solution is given by Eqs (6.81). This bath is assumed to remain in thermal equilibrimn at all times, implying the phase space probability distribution (6.77), the thermal averages (6.78), and equilibrium time correlation functions such as (6.82). The quantum analogs of these relationships were discussed in Section 6.5.3. 

The third alternative is to use the classical correlation functions to define an equivalent quantum mechanical harmonic bath. This approach was pioneered by Warshel as the dispersed polaron method [67, 68]. More recently, this idea has been used in studies of electron transfer systems in solution [64] and in the photosynthetic reaction center [65,69] (see also Ref. 70). This approach is based on the realization that the spectral density describing a linearly coupled harmonic bath [Eq. (29)] can be obtained by cosine transformation of the classical time-correlation function of the bath operator [Eq. (28)]. Comparing the classical correlation function for the linearly coupled harmonic bath [Eqs. (25) and (26)],... 

For a particle evolving in a thermal bath, we focused our interest on the particle displacement, a dynamic variable which does not equilibrate with the bath, even at large times. As far as this variable is concerned, the equilibrium FDT does not hold. We showed how one can instead write a modified FDT relating the displacement response and correlation functions, provided that one introduces an effective temperature, associated with this dynamical variable. Except in the classical limit, the effective temperature is not simply proportional to the bath temperature, so that the FDT violation cannot be reduced to a simple rescaling of the latter. In the classical limit and at large times, the fluctuation-dissipation ratio T/Teff, which is equal to 1 /2 for standard Brownian motion, is a self-similar function of the ratio of the observation time to the waiting time when the diffusion is anomalous. 

A second recent development has been the application 46 of the initial value representation 47 to semiclassically calculate A3,8,13 (and/or the equivalent time integral of the flux-flux correlation function). While this approach has to date only been applied to problems with simplified harmonic baths, it shows considerable promise for applications to realistic systems, particularly those in which the real solvent bath may be adequately treated by a further classical or quasiclassical approximation. 

The treatment up to this point has been fully quantum mechanical Vjj is an operator in the bath degrees of freedom. For many calculations on liquids, however, one wants to treat these degrees of freedom (rotations and translations) classically the question then arises of what is the best way to replace a quantum correlation function with a classical one. A classical autocorrelation function is an even function of the time, a property shared by the anticommutator in (2.11) but not by the one sided correlation function of (2.10). It thus appears that the best place to make a classical approximation is in (2.11) in addition, doing so gives... 

The integration of Equations 9.49 and 9.51 is carried out using the second-order Heun s algorithm, with a very small time step of 0.001. These equations differ from the corresponding classical equations in two ways First, the noise correlation of c-number spin-bath variables r t) are quantum mechanical in nature, as evident from the correlation function in Equation 9.42, which is numerically fitted by the superposition of exponential functions with D, and X . Second, the knowledge of Q requires the quantum correction equations that yield quantum dispersion around the quantum mechanical mean values q and p for the system. Statistical averaging over noise is... 

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