Collective bath coordinate

Figure 13. The whole structure of the phase space in a nutshell. P is the equilibrium point, E is the energy, (q) are the collective bath coordinates, and ( ) is the collective transition coordinates. The cental manifold of P is Cp, and the stable and unstable manifolds are indicated by S/U.

Second, canonical transformation methods may be employed to diagonalize the system-bath Hamiltonian partially by a transformation to new ( dressed ) coordinates. Such methods have been in wide use in solid-state physics for some time, and a large repertoire of transformations for different situations has been developed [101]. In the case of a linearly coupled harmonic bath, the natural transformation is to adopt coordinates in which the oscillators are displaced adiabatically as a function of the system coordinates. This approach, known in solid-state physics as the small-polaron transformation [102], has been used widely and successfully in many contexts. In particular, Harris and Silbey demonstrated that many important features of the spin-boson system can be captured analytically using a variationally optimized small-polaron transformation [45-47]. As we show below, the effectiveness of this technique can be broadened considerably when a collective bath coordinate is first included in the system directly. 

The success of the effective adiabatic approach suggests that including a collective bath coordinate in the system may be a practical way to make... 

The Zusman equation (ZE)/ due mainly to its physically insightful picture on solvation dynamics, is (at least used to be) one of the most commonly used approaches in the study of quantum transfer processes. In this approach, the electronic system degrees of freedom are coupled to a collective bath coordinate that is assumed to be diffusive. The only approximation involved is the classical high temperature treatment of bath. To account for the dynamic Stokes shift, the standard ZE includes also the imaginary part of bath correlation function. This part does not depend on temperature and is therefore exact in the diffusion regime. 

The SES and ESP approximations include the dynamics of solute degrees of freedom as fully as they would be treated in a gas-phase reaction, but these approximations do not address the full complexity of condensed-phase reactions because they do not allow the solvent to participate in the reaction coordinate. Methods that allow this are said to include nonequilibrium solvation. A variety of ways to include nonequilibrium solvation within the context of an implicit or reduced-degree-of-freedom bath are reviewed elsewhere [69]. Here we simply discuss one very general such NES method [76-78] based on collective solvent coordinates [71, 79]. In this method one replaces the solvent with one or more collective solvent coordinates, whose parameters are fit to bulk solvent properties or molecular dynamics simulations. Then one carries out calculations just as in the gas phase but with these extra one or more degrees of freedom. The advantage of this approach is its simplicity (although there are a few subtle technical details). 

The Hamiltonian of Eq. (9.2) couples the reaction coordinate to the environmental oscillator degrees of freedom by terms linear in both reaction coordinate and bath degree of freedom. This is derived in Zwanzig s original approach by an expansion of the full potential in bath coordinates to second order. This innocuous approximation in fact conceals a fair amount of missing physics. We have shown [16a] that this collection of bilinearly coupled oscillators is in fact a microscopic version... 

The parabolic barrier case demonstrates that the effect of the medium is to replace the original reaction coordinate q by a collective mode p along which the dynamics is trivial. It is useful to define a collective bath mode o orthogonal to the unstable mode p as... 

In the following the coordinate p will denote the generalized reaction coordinate (the analog of the coordinate / in the previous section) and the coordinate ct the collective bath mode. 

Sz(i) being the total angular momentum of the i spins. The linear combination 5 (l)/a1 + Sz(2)jx2 is now coupled to the dipole-dipole heat bath, whereas the difference Sz(l)/x1 — Sz(2)jx2 remains constant. It may be shown that the chemical equilibrium condition (31) is then equivalent to an equalization of the temperatures of the collective coordinate Sz(l)jx1 + Sz(2)/x2 and of the dipole-dipole heat bath. 

The second step of the evolution towards equilibrium is the Zeeman dipole-dipole relaxation. Hartmann and Anderson estimated this time using the hypothesis that p at any time is of the form (22). As a consequence of the shortness of the dipole-dipole relaxation time we may assume that the dipole-dipole system always remains in equilibrium we are thus led to treat the evolution of the Zeeman system as the Brownian motion of a collective coordinate in the dipole-dipole heat bath. We assume that the diagonal elements of p have the form... 

We wanted to extend this approach to include dynamical effects on line shapes. As discussed earlier, for this approach one needs a trajectory co t) for the transition frequency for a single chromophore. One could extract a water cluster around the HOD molecule at every time step in an MD simulation and then perform an ab initio calculation, but this would entail millions of such calculations, which is not feasible. Within the Born Oppenheimer approximation the OH stretch potential is a functional of the nuclear coordinates of all the bath atoms, as is the OH transition frequency. Of course we do not know the functional. Suppose that the transition frequency is (approximately) a function of a one or more collective coordinates of these nuclear positions. A priori we do not know which collective coordinates to choose, or what the function is. We explored several such possibilities, and one collective coordinate that worked reasonably well was simply the electric field from all the bath atoms (assuming the point charges as assigned in the simulation potential) on the H atom of the HOD molecule, in the direction of the OH bond. 

In Marcus theory [91,94], the bath is represented by an overdamped harmonic coordinate, coupled linearly to the donor and acceptor states and representing the collective polarization of the environment around the donor-acceptor pair. The system-bath coupling is characterized by the bath reorganization energy. In the work described here [40] the donor and acceptor are treated essentially as single quantum levels and the bath appears only indirectly through its effects (stochastic fluctuations) on the... 

The Golden Rule formula (9.5) for the mean rate constant assumes the Unear response regime of solvent polarization and is completely equivalent in this sense to the result predicted by the spin-boson model, where a two-state electronic system is coupled to a thermal bath of harmonic oscillators with the spectral density of relaxation J(o)) [38,71]. One should keep in mind that the actual coordinates of the solvent are not necessarily harmonic, but if the collective solvent polarization foUows the Unear response, the system can be effectively represented by a set of harmonic oscillators with the spectral density derived from the linear response function [39,182]. Another important point we would like to mention is that the Golden Rule expression is in fact equivalent [183] to the so-called noninteracting blip approximation [71] often used in the context of the spin-boson model. The perturbation theory can be readily applied to... 

The LVC model employed for the bath subspace allows one to introduce coordinate transformations by which a set of effective, or collective modes are extracted that act as generalized reaction coordinates for the dynamics. As shown in Refs. [25,26], Aeff = N N + l)jl such coordinates can be defined for an-state system. Thus, three effective modes are introduced for an electronic two-level system, six effective modes for a three-level system etc., for an arbitrary number of phonon modes that couple to the subsystem according to the LVC model. The subset of effeetive modes entirely determine the short-time dynamics, if the initial excitation is localized in the system subspace [26]. In order to capture the dynamics on longer time scales, chains of such effective modes can be introduced [38,43]. These transformations, which are summarized below, will be shown to yield a unique perspective on highdimensional dynamics in extended systems. 

From the interaction Hamiltonian equation (15.3), we note that the Vb bath modes produce cumulative effects by their coupling to the discretized subsystem. This suggests that the interaction Hamiltonian can be formally re-written in terms of a set of collective coordinates such that... 

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