Bosonic spectral functions

The results from the general theory for the vibrational spectrum of a localized harmonic oscillator, linearly coupled with a noninteracting boson continuum (phonons, photons, electron-hole pairs), can be used to estimate the contribution of different relaxation processes at smfaces. The spectral function of the oscillator obtained by normal-mode analysis at zero temperature is [25]... 

Even though there appear many variables and parameters in the equations above, ultimately the spin-boson model, as advertised in [13], is characterized completely by a well defined average property of the system, the spectral function J oj)... 

The key new aspect of our investigation is two-fold first, we base all model parameters on molecular dynamics simulations second, the spin-boson model allows one to account for a very large number of vibrations quantum mechanically. We have demonstrated that the spin-boson model is well suited to describe the coupling between protein motion and electron transfer in biological redox systems. The model, through the spectral function can be matched to correlation functions of the redox energy... 

The main result regarding the electron transfer rates evaluated is that for a spectral function consistent with molecular dynamics simulations the spin-boson model at physiological temperatures predicts transfer rates in close agreement with those predicted by the Marcus theory. However, at low temperatures deviations from the Marcus theory arise. The resulting low temperature rates are in qualitative agreement with observations. The spin-boson model explains, in particular, in a very simple and natural way the slow rise of transfer rates with decreasing temperature, as well as the asymmetric dependence of the redox energy. 

To describe the effect of the environment one usually needs to determine the bath correlation function C(t). Let us start discussing this function for a bosonic bath where the subscript Ph indicates a bath of phonons. Using the numerical decomposition of the spectral density Eq. (2) together with the theorem of residues one obtains the complex bath correlation functions... 

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... 

Abstract Photoinduced processes in extended molecular systems are often ultrafast and involve strong electron-vibration (vibronic) coupling effects which necessitate a non-perturbative treatment. In the approach presented here, high-dimensional vibrational subspaces are expressed in terms of effective modes, and hierarchical chains of such modes which sequentially resolve the dynamics as a function of time. This permits introducing systematic reduction procedures, both for discretized vibrational distributions and for continuous distributions characterized by spectral densities. In the latter case, a sequence of spectral densities is obtained from a Mori/Rubin-type continued fraction representation. The approach is suitable to describe nonadiabatic processes at conical intersections, excitation energy transfer in molecular aggregates, and related transport phenomena that can be described by generalized spin-boson models. 

In the above equations coq is the frequency of a noninteraction oscillator, X(o)) is the eoupling function between the oscillator and the external field, and RHg (ro) is the field density of states. With Eqs (67)-(74) it will now be possible to predict the spectral behavior for localized harmonic oscillators linearly coupled with different model boson continua. 

---