Redfield matrix

Relaxation or chemical exchange can be easily added in Liouville space, by including a Redfield matrix, R, for relaxation, or a kinetic matrix, K, to describe exchange. The equation of motion for a general spin system becomes equation (B2.4.28). 

The Redfield matrix elements are defined in full analogy with the case where the conventional electron spin relaxation processes in a non-equilibrium ensemble are considered, but the rates are in general different from... 

The matrix obtained after the F Fourier transformation and rearrangement of the data set contains a number of spectra. If we look down the columns of these spectra parallel to h, we can see the variation of signal intensities with different evolution periods. Subdivision of the data matrix parallel to gives columns of data containing both the real and the imaginary parts of each spectrum. An equal number of zeros is now added and the data sets subjected to Fourier transformation along I,. This Fourier transformation may be either a Redfield transform, if the h data are acquired alternately (as on the Bruker instruments), or a complex Fourier transform, if the <2 data are collected as simultaneous A and B quadrature pairs (as on the Varian instruments). Window multiplication for may be with the same function as that employed for (e.g., in COSY), or it may be with a different function (e.g., in 2D /-resolved or heteronuclear-shift-correlation experiments). 

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... 

A more general approach is required to interpret the current experiments, Jean and co-workers have developed multilevel Redfield theory into a versatile tool for describing ultrafast spectroscopic experiments [22-25], In this approach, terms neglected at the Bloch level play an important role for example, coherence transfer terms that transform a coherence between levels i and j into a coherence between levels j and k ( /t - = 2) or between levels k and l ( f - j - 2, k-j = 2) and couplings between populations and coherences. Coherence transfer processes can often compete effectively with vibrational relaxation and dephasing processes, as shown in Fig. 4 for a single harmonic well, initially prepared in a superposition of levels 6 and 7. The lower panel shows the population of levels 6 and 7 as a function of time, whereas the upper panels display off-diagonal density matrix ele-... 

Product operators can thus be used to predict the behavior of an NMR experiment. The calculations are relatively simple to perform. Computer programs are available that also take into account the effects of phase-cycling to select the desired terms and reject unwanted ones. A drawback of the product operators approach is that, in its simplest version, it does not take into account the effect of relaxation. This is a must when dealing with paramagnetic substances. Exponential decay terms can be introduced to multiply each term and take relaxation into account. The method then becomes more cumbersome, and the effect of relaxation is introduced in a phenomenological way. A more detailed approach is that of using the concept of Redfield density matrix [1,2]. 

In the coherent (Hamiltonian) approach to the four-state spin system, the state populations are just the diagonal elements of the corresponding density matrix p(r, f), which obeys the Bloch or Redfield equation [211] ... 

The conventional approach to the theory of electron spin relaxation is to use a density matrix approach developed by Redfield. (32) However, this method is only valid when x Xg2- Thus, cases of very fast electronic relaxation leading to sharp NMR lines, which are in general of particular interest, are strictly excluded from this theoretical approach. Doddrell et al. (33) have developed a more general theory for... 

The Redfield equation describes the time evolution of the reduced density matrix of a system coupled to an equilibrium bath. The effect of the bath enters via the average coupling V = and the relaxation operator, the last sum on the right of Eq. (10.155). The physical implications of this term will be discussed below. 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

Redfield A G 1996 Relaxation theory density matrix formulation Encyclopedia of Nuclear Magnetic Resonance ed D M Grant and R K Harris (Chichester Wiley) pp 4085-92... 

Following Cohen-Tannoudji and using the Markoffian approximation we have derived the equations of motion for the off-diagonal elements of the reduced density matrix, which determine the dephasing constant (7 ) and the optical lineshape. The following Redfield-like equations were obtained ... 

This result is the Redfield-Liouville-von Neumann equation of motion or, simply, the Redfield equation [29,30,49-53]. Here the influence of the bath is contained entirely in the Redfield relaxation tensor, 3i, which is added to the Liouville operator for the isolated subsystem to give the dissipative Redfield-Liouville superoperator (tensor) that propagates (T. Expanded in the eigenstates of the subsystem Hamiltonian, H, Eq. (9) yields a set of coupled linear differential equations for the matrix... 

This result demonstrates that the Redfield tensor can be applied to any operator without the explicit construction of the full tensor. The G and G are ordinary operators, so numerical evaluation of the right side of Eq. (20) in an W-dimensional basis involves only the storage and multiplication oi N y N matrices. The full Redfield tensor, in contrast, is an X operator in Liouville space, whose application to an x density matrix requires 0 N ) scalar multiplications, as opposed to the 0 N ) operations required to multiply NxN matrices. The ability to apply the Redfield tensor to the density matrix using Eqs. (20) or (23) therefore allows a significant savings in both computer time and memory, particularly as N becomes large. 

The final form of the Redfield equation [Eq. (20)] is superficially similar to the equation of motion that arises in the axiomatic semigroup theory of Lindblad, Gorini et al. [48,54-57]. They showed that the most general equation of motion that preserves the positivity of the density matrix must have the general form... 

Here the W are operators of the subsystem and the superscript dagger denote the Hermitian conjugate. The Redfield equation can be written in this form only when an additional symmetrization of the bath correlation functions is performed [48]. Note that this alternative equation also expresses the dissipative evolution of the density matrix in terms of N x N... 

There are three important issues to consider in the numerical solution of the Redfield equation. The first is the evaluation of the Redfield tensor matrix elements I ,To obtain these matrix elements, it is necessary to have a representation of the system-bath coupling operator and of the bath Hamiltonian. Two fundamental types of models are used. First, the system-bath coupling can be described using stochastic fluctuation operators, without reference to a microscopic model. In this case, the correlation functions appearing in the formulas for parame-... 

An important feature of the reduced-density-matrix approach is that it allows the bath to be treated at different levels of approximation. In the Redfield equation, the bath enters only through the correlation functions of the coupled bath variables in Eq. (18). This means that a substantial part of the complexity of a realistic condensed-phase environment is... 

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