Time correlation functions quantum bath

The basic theoretical framework for understanding the rates of these processes is Fermi s golden rule. The solute-solvent Hamiltonian is partitioned into three terms one for selected vibrational modes of the solute, including the vibrational mode that is initially excited, one for all other degrees of freedom (the bath), and one for the interaction between these two sets of variables. One then calculates rate constants for transitions between eigenstates of the first term, taking the interaction term to lowest order in perturbation theory. The rate constants are related to Fourier transforms of quantum time-correlation functions of bath variables. The most common... 

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... 

System-Bath Representation of Quantum Time Correlation Functions... 

To implement the linearized path integral formulation for time correlation functions the initial density operator must be Wigner transformed in the bath variables while it remains an operator in the quantum subsystem space. In the calculations presented below we assume that the system and bath do not interact initially. Consequently total probability density at t = 0 is of the form... 

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)],... 

Lindenberg and West conclude, after analysis of Eq.(59) at low temperatures where kTccIly, that the correlation function decays on a time scale li/ kT rather than 1/y. Thus, the bath can dissipate excitations whose energies lie in the range (0/fi.y), while the spontaneous fluctuations occur only in the range (0,kT) if kTcorrelation time of the fluctuations is therefore the longer of fi/ kT and 1/y. The idea advanced by these authors is that fluctuations and dissipation can have quite distinct time scales [133], This is important if the two quantum states of the system of interest correspond to chemical interconverting states [139, 144, 145],... 

In many applications it is often reasonable to suppose that the bath subsystem dynamics causes slow mixing of the quantum subsystem states. If the relevant experimental measurements involve time scales shorter than the quantum subsystem mixing time, one can proceed as if the bath dynamics occurs in a single quantum subsystem state. This is the adiabatic approximation and in this limit (43) can be simplified by making the following substitutions a = j3 (the forward path begins and ends in the same quantum subsystem state), and similarly for the backward path we have a = / . Thus the adiabatic approximation to the correlation function is obtained as... 

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... 

Recently, a QUAPI procedure was developed suitable for evaluating the full flux correlation function in the case of a one-dimensional quantum system coupled to a dissipative harmonic bath and applied to obtain accurate quantum mechanical reaction rates for a symmetric double well potential coupled to a generic environment. These calculations confirmed the ability of analytical approximations to provide a nearly quantitative picture of such processes in the activated regime, where the reaction rate displays a Kramers turnover as a function of solvent friction and quantum corrections are small or moderate, They also emphasized the significance of dynamical effects not captured in quantum transition state models, in particular under small dissipation conditions where imaginary time calculations can overestimate or even underestimate the reaction rate. These behaviors are summarized in Figure 7. 

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