Effective bath coordinates

With the definition of the effective bath coordinate, the spin-boson Hamiltonian can be rewritten as... 

We can now reexamine the two-mode system of Fig. 5, defining an effective bath coordinate as described above the final bath therefore consists of just a single oscillator. Two different treatments are possible at this level, depending on how the polaron transformation is done. In the simpler case the two variational parameters are which defines the effective bath coordinate, and/j, which treats the bath-TLS coupling f>2> which treats the EBC-bath coupling, is held at zero. The results of this treatment are shown as the upper curves in Fig. 6. It is clear that by... 

Figure 6, Normalized solvation free energy for the system of Fig. 5 as a function of the bath cutoff frequency, oi, and the two-level system coupling, J. Here an effective bath coordinate (EBC) was first included in the system and a variational polaron transformation applied to the resulting TLS-bath and EBC-bath couplings. The dashed lines indicate the minimum free energies obtained when only the TLS-bath coupling was treated by the second transformation the dotted line shows results when both were treated. The exact free energies are plotted with solid lines.

The formal structure of (5.77) suggests that the reaction coordinate Q can be combined with the bath coordinates to form a new fictitious bath , so that the Hamiltonian takes the standard form of dissipative TLS (5.55). Suppose that the original spectrum of the bath is ohmic, with friction coefficient q. Then diagonalization of the total system (Q, qj ) gives the new effective spectral density [Garg et al. 1985]... 

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

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. 

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]. 

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

In this section, a general model Hamiltonian is introduced which is of LVC type for a subset of bath coordinates (Sect. 15.2.1). This Hamiltonian is subsequently transformed to the effective-mode representation mentioned above (Sects. 15.2.2 and 15.2.3). The development is presented in a system-bath theory setting, since this facilitates the transition to Sect. 15.3 where we focus upon a description of the vibrational distribution in terms of bath spectral densities. 

The introduction of a set of effective modes g, is the first step in defining an overall orthogonal transformation which leaves the subsystem coordinates xg unaffected while transforming the bath coordinates,... 

If all the PES coordinates are split off in this way, the original multidimensional problem reduces to that of one-dimensional tunneling in the effective barrier (1.10) of a particle which is coupled to the heat bath. This problem is known as the dissipative tunneling problem, which has been intensively studied for the past 15 years, primarily in connection with tunneling phenomena in solid state physics [Caldeira and Leggett 1983]. Interaction with the heat bath leads to the friction force that acts on the particle moving in the one-dimensional potential (1.10), and, as a consequence, a> is replaced by the Kramers frequency [Kramers 1940] defined by... 

More pertinent to the present topic is the indirect dissipation mechanism, when the reaction coordinate is coupled to one or several active modes which characterize the reaction complex and, in turn, are damped because of coupling to a continuous bath. The total effect of the active oscillators and bath may be represented by the effective spectral density For instance, in the... 

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. 

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