Electron-spin spectral density functions

The spectral density function, f(a>), reflects the electronic spin motion and depends sensitively on the dimensionality of the process. For ID diffusion,... 

Fig. 5. The optical absorption spectrum and the electronic structure of Vs" ", (a) Experimental data, where a photodissociation action spectrum of a rare-gas complex, Vs+Ar, was measured by observing a photofragment, Vs+. (b) Density-functional calculation of the spectrum for the most stable isomer illustrated in the inset. The bars show oscillator strengths the solid line a spectral profile, (c) Density-of-states profiles of the majority or and the minority-spin electrons obtained by the density-functional calculation. The shadows indicate occupied electronic levels. The vanadium pentamer ion. Vs" ", was shown to be in the spin triplet state with a trigonal bipyramid structure, where the average bond length was 2.4...

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... 

One dimensional (ID) diffusion of the electron spin along the chain The spectral density (Pqid(co) as a function of frequency, m, can be obtained by a solution of the ID diffusion equation ... 

For an external magnetic field Bo along the z direction, the electron spins are oriented parallel or antiparallel to the z direction. Modulation of the components of the local field in the xy plane due to a stochastic process then induces stochastic electron spin transitions (spin flips) that contribute to longitudinal relaxation with time constant T. For historical reasons longitudinal relaxation is often termed spin-lattice relaxation. The relaxation rate T is proportional to the spectral density /(co) of the stochastic process at the resonance frequency Mo of the transition under consideration. This spectral density is maximum for a correlation time Tc of the stochastic process that fulfils the condition wqTc = 1. As correlation times usually are a monotonic function of temperature, there is a temperature for which the relaxation rate attains a maximum and T attains a minimum. Measurements of 7] as a function of temperature can thus be used to infer the correlation time of a dynamic process. By varying the external field Bo and thus mq, the time scale can be shifted to which EPR experiments are most sensitive. 

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. 

The usual way of solving eqn (7) requires its transformation into the interaction representation (Dirac picture) that is often called rotating frame for a particular case, when static part of the spin Hamiltonian is restricted to the electron Zeeman interaction. In the Dirac picture only the stochastic dipolar interaction is left in the spin Hamiltonian, its matrix elements get additional oscillatory factors due to the static Hamiltonian transitions. The integral on each matrix element of the double commutator in eqn (7) thus evolves into the Fourier transform /(co ) of the correlation function for the corresponding stochastic process. This Fourier transform is often called spectral density of the stochastic process and it is to be taken at a frequency co of a particular transition of the static Hamiltonian operator, driven by a single transition operator ki ... 

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