Redfield formalism

If the system under consideration possesses non-adiabatic electronic couplings within the excited-state vibronic manifold, the latter approach no longer is applicable. Recently, we have developed a simple model which allows for the explicit calculation of RF s for electronically nonadiabatic systems coupled to a heat bath [2]. The model is based on a phenomenological dissipation ansatz which describes the major bath-induced relaxation processes excited-state population decay, optical dephasing, and vibrational relaxation. The model has been applied for the calculation of the time and frequency gated spontaneous emission spectra for model nonadiabatic electron-transfer systems. The predictions of the model have been tested against more accurate calculations performed within the Redfield formalism [2]. It is natural, therefore, to extend this... 

For simplicity, in equation (3c) S oc is assumed to be mainly along the director axis n, and n is aligned parallel tol. For Btoc Bo, the total 7, is equal to 7,2, whereas in the opposite case B,oc Bq one has 7, = 7,o- Evidently, the finite local field contribution complicates the control of the angle adjustment without an exact knowledge of, oc, and because of equation (3c) the inclinations of 90° become impossible by external field switches. Furthermore, the spectral densities for 7,0 are not discussed in the literature to the same extent as for 7,2, nor does there exist a critical experimental examination of the validity of the basic expression equation (3a). Approximate predictions about with the Redfield formalism give, for the completely isolated, i.e., uncoupled proton spin-pair (/ = 1) and high spin-temperature approach ... 

A good introductory treatment of the density operator formalism and two-dimensional NMR spectroscopy, nice presentation of Redfield relaxation theory. 

Nuclear spin relaxation is considered here using a semi-classical approach, i.e., the relaxing spin system is treated quantum mechanically, while the thermal bath or lattice is treated classically. Relaxation is a process by which a spin system is restored to its equilibrium state, and the return to equilibrium can be monitored by its relaxation rates, which determine how the NMR signals detected from the spin system evolve as a function of time. The Redfield relaxation theory36 based on a density matrix formalism can provide... 

The time-local approach is based on the Hashitsume-Shibata-Takahashi identity and is also denoted as time-convolutionless formalism [43], partial time ordering prescription (POP) [40-42], or Tokuyama-Mori approach [46]. This can be derived formally from a second-order cumulant expansion of the time-ordered exponential function and yields a resummation of the COP expression [40,42]. Sometimes the approach is also called the time-dependent Redfield theory [47]. As was shown by Gzyl [48] the time-convolutionless formulation of Shibata et al. [10,11] is equivalent to the antecedent version by Fulinski and Kramarczyk [49, 50]. Using the Hashitsume-Shibata-Takahashi identity whose derivation is reviewed in the appendix, one yields in second-order in the system-bath coupling [51]... 

Several formalisms have been applied to relaxation in exchanging radicals. Principal among these are modifications of the classical Bloch equations (8, l ) and the more rigorous quantum mechanical theory of Redfield al. (8 - IJ ). When applied in their simplest form, as in the present case for K3, both approaches lead to the same result. Since the theory has been elegantly described by many authors (8 - 12 ), only those details which pertain to the particular example of K3 will be presented here. Secular terms contribute to the ESR linewidth (r) and transverse relaxation time (T ) by an amount... 

The expressions in the Redfield theory [82] is derived under the assumptions of weak coupling between the spin states and that the correlation of the perturbation, i.e. of Gg-g(r), has decayed to zero for the times of interest. This is formally expressed as... 

Second, we should keep in mind that between the two extreme limits discussed above there exists a regime of intermediate behavior, where dephasing/decoherence and molecular response occur on comparable timescales. In this case the scattering process may exhibit partial coherence. Detailed description of such situations requires treatment of optical response within a formalism that explicitly includes thermal interactions between the system and its environment. In Section 18.5 we will address these issues using the Bloch-Redfield theory of Section 10.5.2. 

This enumeration work has been extended and formalized in a more general sense by utilization of the work of PolyaThe Polya counting polynomial, based on earlier work of Redfield and Lunn and Senior was the basis for useful approaches to enumeration of many chemical classes, including cyclic structures, as demonstrated by Trinajstic. Some detailed examples are given by Balaban and coworkers as well as an extensive set of references. 

Despite the fact that the exact j or TZ can be formally expressed in terms of an infinite series expansion, its evaluation, however, amounts to solve the total composite system of infinite degrees of freedom. In practice, one often has to exploit weak system-bath interaction approximations and the resulting COP [Eq. (1.2)] and POP [Eq. (1.3)] of QDT become nonequivalent due to the different approximation schemes to the partial consideration of higher order contributions. It is further noticed that in many conventional used QDT, such as the generalized quantum master equation, Bloch-Redfield theory and Fokker-Planck equations, there involve not only... 

On less formal grounds a Markovian type of equation was also obtained by Redfield [167]. Redheld used perturbation theory up to second order and derived a Master equation for the evolution of the density matrix Oaa y where a, a denote quantum states of the molecule with energy defined by ha. The simple equation for the relaxation of the density matrix was obtained by neglecting the ensemble-averaged first-order terms. Thus we have [167]... 

The dynamics of the system is irreversible in time, both in the Redfield and the Lindblad formalism. This irreversibility arises from the loss of phase information when averaging over the bath system. 

The classical Liouville equation does have an equivalent in quantum mechanics, which is needed for a consistent description of quantum statistical mechanics the quantum Liouville equation. Equilibrium quantum statistical mechanics requires the introduction of the density operator on an appropriate Hilbert space, and the quantum liouvUle equation for the density operator is a logical and necessary extension of the Schrodinger equation. The quantum Liouville equation can even be written, formally at least, in a form that resembles its classical counterpart. It allows for some weak and almost internally consistent form of dissipative dynamics, known as the Redfield theory, which finds its main use in relating NMR relaxation times to spectral densities arising from solvent fluctuations, although in recent... 

One of the first fields where the quantum nature of the underlying system carmot be ignored, simply because there is no classical limit, is that of NMR. This is also a field where decay is directly observed in spin relaxation back to equilibrium. By giving a radio frequency pulse to an equilibrium system of spins, these are brought out of equilibrium, and, after the pulse, decay back to the equilibrium state. This free induction decay was first modeled phenomenologically by the Bloch equations, which pointed to the existence of two relaxation times, commonly called Tj and T2, but at a later stage Redfield used the density operator formalism to... 

Several remarks are in order here. First, the fact that the correlation functions need to be evaluated at the transition frequency reminds us of the earlier classical results for barrier transitions. A friction coefficient is also related to solvent fluctuations in fact, it can be written as a velocity-velocity correlation function, and as such the Redfleld theory is not much different from classical theories. This is to be expected because these can be derived from a very similar formalism based on the classical Liouville equation, albeit without the problems encountered here. In the Kramers case, we needed to evaluate the friction at zero frequency, which sometimes severely overestimates its role. It would be better to evaluate it at the barrier frequency. In the theory developed by Grote and Hynes, the friction needs to be evaluated not at the original barrier frequency but at the frequency the barrier transition actually takes place as Eq. (9.13) shows. The reason is simple in the latter case the back reaction of the system on the solvent is taken into account, something that was not allowed in the Redfield theory. 

A.G. Redfield, Relaxation Theory, Density Matrix Formalism. Encyclopedia of Nuclear Magnetic Resonance (Wiley, Chichester, 1996)... 

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