The polaron transformation

In this transformed Hamiltonian Hq again describes uncoupled system and bath the new element being a shift in the state energies resulting from the system-bath interactions. In addition, the interstate coupling operator is transformed to 

To see the significance of this result consider a typical matrix element of this 

The absolute square of these term, which depend on v, y, and the set of shifts Aa , are known as Franck Condon factors. 

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Now let us consider the polaron transformation (146)-(147) applied to the tunneling Hamiltonian... 

The polaron transformation, executed on the Hamiltonian (12.8)-( 12.10) was seen to yield a new Hamiltonian, Eq. (12.15), in which the interstate coupling is renormalized or dressed by an operator that shifts the position coordinates associated with the boson field. This transformation is well known in the solid-state physics literature, however in much of the chemical literature a similar end is achieved via a different route based on the Bom-Oppenheimer (BO) theory of molecular vibronic stmcture (Section 2.5). In the BO approximation, molecular vibronic states are of the form (/) (r,R)x ,v(R) where r and R denote electronic and nuclear coordinates, respectively, R) are eigenfunctions of the electronic Hamiltonian (with corresponding eigenvalues E r ) ) obtained at fixed nuclear coordinates R and... 

Let us now return to the two model Hamiltonians introduced in Section 12.2, and drop from now on the subscripts S and SB from the coupling operators. Using the polaron transformation we can describe both models (12.1), (12.2) and (12.4)-(12.6) in a similar language, where the difference enters in the form ofthe coupling to the boson bath... 

We have already seen that this form of electron-phonon coupling expresses shifts in the vibrational modes equilibrium positions upon electronic transition, a standard model in molecular spectroscopy. Applying the polaron transformation to get a Hamiltonian equivalent to (12.27) and (12.29), then using Eq. (12.34) with 2 = Eg = Eg + flM and 1 = E leads to the electronic absorption lineshape in the form 2-abs( 2<5( g + dui+ vib(v)-Fviblv ))... 

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... 

This Hamiltonian is diagonalized by the canonical transformation (called Lang-Firsov or polaron ) [95-97]... 

To conclude, after the canonical transformation we have two equivalent models (1) the initial model (145) with the eigenstates (160) and (2) the fictional free-particle model (154) with the eigenstates (158). We shall call this second model polaron representation. The relation between the models is established by (155)-(157). It is also clear from the Hamiltonian (148), that the operators < , d. ad. and a describe the initial electrons and vibrons in the fictional model. 

Fourth, Jaime and Salamon (1999) have pointed out that a(T) increases more sharply than exponentially on cooling to rmax in fig. 30 and that the additional entropy transported increases on crossing the () -() phase boundary at tc. This observation is consistent with a progressive transformation from Zener to small polarons in the hole-poor phase as the hole concentration x = 0.30 in this phase is diluted by the trapping of Zener polarons in the hole-rich phase. Such a transformation would double the number of sites available to a polaron and would therefore increase the a of eq. (26) by reducing c = (1 — r)2x toward c = (1 — r)x, where r is the ratio of trapped to free polarons. In the O phase, most of the polarons appear to be small polarons at 7 N. 

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]. 

It is clear that the polaron shifts, fj, directly reduce the linear TLS-bath couplings, so that F would vanish entirely for fj = gf, this is the traditional (nonvariational) small-polaron transformation used in solid-state physics [102]. This choice is not generally optimal, however, because it leads to larger F+ and F couplings. In calculations presented below, the fj are always variationally optimized. 

Results. We can evaluate the effectiveness of the small-polaron transformation by considering a system of just two modes, for which exact free energies can also be obtained using traditional basis-set methods. The results are shown in Fig. 5 as a function of the bath frequency and the TLS coupling, J. Here the fj) in Eqs. (83)-(87) were varied to minimize the free energy Ag of the zero-order Hamiltonian, after the... 

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