System-bath coupling separation

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

To evaluate the correlation functions in Eqs. (12) and (13), it is usual to complete the separation of the system and bath by decomposing the system-bath coupling into a sum of products of pure system and bath operators. This allows the correlation functions of the system-bath coupling to be replaced, without loss of generality, by correlation functions of bath operators alone, evolving under the uncoupled bath Hamiltonian. Moreover, as we have previously pointed out [39,40], this decomposition of the system-bath coupling make it possible to write the Redfield equation in a highly compact form, without explicit reference to the Redfield tensor at all. 

When the system-bath coupling is linear in the bath coordinates, as in the spin-boson Hamiltonian, the physical interpretation is that the minimum position of each bath oscillator is shifted proportionately to the value of the system variable to which it is coupled. The small-polaron transformation redefines the Hamiltonian in terms of oscillators shifted adiabatically as a function of the system coordinate here the system coordinate is tr, so that the oscillators will be implicitly displaced equally but in opposite directions for each quantum state. Note that in the limit that the TLS coupling J vanishes, this transformation completely separates the system and bath. This makes it an effective transformation for cases of small coupling, and it has in fact been long and widely used in many types of physical problems, although typically in a nonvariational form [102]. Harris and Silbey showed that while simple enough to handle analytically, a variational small-polaron transformation contained the flexibility to treat the spin-boson problem effectively in most parameter regimes (see below) [45-47]. 

Thus, an exchange between two-qubit entanglement and system-bath entanglement may take place and be dynamically controlled by modulations. If the systems couple to two baths that have common modes, one can observe the transfer of coherence and buildup of entanglement between the two systems via these modes. If, on the other hand, the baths were completely separate, the coherence transfer between each system and its bath can modulate the amount of system-system entanglement, first lost and then regained. 

The basic features of ET energetics are summarized here for the case of an ET system (solute) linearly coupled to a bath (nuclear modes of the solute and medium) [11,30]. We further assume that the individual modes of the bath (whether localized or extended collective modes) are separable, harmonic, and classical (i.e., hv < kBT for each mode, where v is the harmonic frequency and kB is the Boltzmann constant). Consistent with the overall linear model, the frequencies are taken as the same for initial and final ET states. According to the FC control discussed above, the nuclear modes are frozen on the timescale of the actual ET event, while the medium electrons respond instantaneously (further aspects of this response are dealt with in Section 3.5.4, Reaction Field Hamiltonian). The energetics introduced below correspond to free energies. Solvation free energies may have entropic contributions, as discussed elsewhere [19], Before turning to the DC representation of solvent energetics, we first display the somewhat more transparent expressions for a discrete set of modes. 

Equation (10.155) was obtained under three approximations. The first two are the neglect of initial correlations and the assumption of weak coupling that was used to approximate Eq. (10.110) by Eq. (10.112). The third is the assumption of timescale separation between the (fast) bath and the (slow) system used to get the final Markovian form. 

The temperature was monitored by separately coupling the water, CO2 and solutes subsystems to a thermal bath at the reference temperature (350 K) with a relaxation time of 0.2 ps for the solvents and 0.5 ps for the solutes. Non-bonded interactions were calculated with a residue-based twin cutoff of 12/15 A for all systems, excepted for D and E for which we used a 13 A cutoff with a reaction field correction for the electrostatic interactions. 

In atomic scale simulations, there is often a clear separation of timescales. The rate of rare events, e.g., chemical reactions, in a system coupled to a heat bath can be estimated by evaluating the free energy barriers for the transitions. Transition State Theory (TST) [9] is the foundation for this approach. Due to the large difference in time scale between atomic vibrations and typical thermally induced processes such as chemical reactions or diffusion, this would require immense computational power to directly simulate dynamical trajectories for a sufficient period of time to include these rare events. Identification of transition states is often the critical step in assessing rates of chemical reactions and path techniques like the nudged elastic band method is often used to identify these states [10-12,109]. 

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