Redfield method

Fig. 3 Absorption spectra of one sample of B850 calculated with different methods for an Ohmic spectral density with / = 3.37 and usc = 0.027 eV. Left Methods with Markov approximation, i.e. Redfield theory with and without secular approximation and TL method. Right TL with and without Markov approximation, TNL and modified Redfield method. (Reproduced from Ref. [37]. Copyright 2006, American Institute of Physics.)...

As we saw in Section 3.4, quadrature phase detection discriminates between frequencies higher and lower than the pulse frequency, but it does not prevent foldover from frequencies higher than the Nyquist frequency. For a desired spectral width FT, there are two common methods for carrying out quadrature phase detection, as was indicated in Section 3.4. One method uses two detectors and samples each detector at FT points per second, thus acquiring 2 FT data in the form of FT complex numbers. The other (commonly called the Redfield method ) requires only a single detector and samples at 2 FT points per second while incrementing the phase of the receiver by 90° after each measurement. (In two-dimensional NMR studies, a variant of this method is usually called the rime-proportional phase incrementation, or TPPI, method.) Because these methods result in quite different treatment of folded resonances, we now consider these approaches in more detail. 

The Redfield method of quadrature phase detection operates as shown in Fig. 3.7a. A spectral width from W/2 to — W/7 is defined by 2 IT7 measurements per second with a single detector, the phase of which is augmented by 90° after each measurement. Each signal magnitude, as illustrated in Fig. 3.7[Pg.67]

Recording and processing the two signals of Eq. 10.14, which are in quadrature, is the analog in the h dimension of the use of two phase-sensitive detectors in the t2 dimension. As we showed in Section 3.7, there is an alternative to use of two detectors. The Redfield method uses one detector but increments the receiver phase by 90°, as illustrated in Fig. 3.8. An analogous technique is available to treat tj data—time-proportional phase incrementation,TPPI.38 108... 

The problem that we noted above with clusters appears also in chemical enumeration when we consider compounds formed by attaching radicals which may be chiral or achiral to a frame which is achiral. In this case, too, Polya s Theorem cannot be used, but the problem can be solved by the appropriate use of Burnside s Lemma. It is also amenable to the methods of Redfield, as shown in [DavRSl] and [LloE85]. 

HarF67c Harary, F., Palmer, E. M The enumeration methods of Redfield. Amer. J. Math. 89 (1967) 373-384. 

A TBma,) explicit functions of the available concentrations of the other nutrients. This approach allows for a pronounced interdependence between the fluxes of the different nutrients but it does not ensure that the Redfield ratios are maintained. In the second approach the contents of the nutrients in the biota reservoir are forced to remain close to the Redfield ratios. This method was used by Mackenzie et al. (1993) in their study of the global cycles of C, N, P, and S and their interactions. They were able to demonstrate how a human perturbation in one of these element cycles could influence the cycles of the other elements. 

There are a number of other possibilities for the explanation of the Doolittle event (Mooers and Redfield, 1996). One of these could be due to the analytical method used by Doolittle, as he assumed a relatively constant rate of amino acid substitution. This assumption may not be justified and should be checked. 

The modified Florence program is well-suited for fitting the experimental NMRD profiles for slowly-rotating complexes of gadolinium(HI), an S = 7/2 ion characterized by relatively low ZFS, whose electron spin relaxation can be considered to be in the Redfield limit. An example of fitting an NMRD profile for aqueous protons, using different methods, for a protein adduct of a Gd(HI) chelate capable of accommodating one water molecule in the first coordination sphere, is displayed in Fig. 11. Other examples will be provided in Chapter 3. 

The problem of a strong coupling between the electron spin system and the classical degrees of freedom (rotation in the first place), as expressed in Eq. (29), can also be treated in another way. This class of methods to circumvent the limitations of the Redfield regime is the subject of this section. 

S. Mukamel In order to represent situations in which nuclear and electronic dynamics take place on the same time scale, one needs to incorporate nuclear degrees of freedom into the description. A frequency-dependent Redfield superoperator can capture some effects, but in general is very limited and may even yield negative probabilities. A method for decomposing a given spectral density into a few collective coordinates and identifying these coordinates was presented in Ref. 1. 

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 all cases where complete elemental analyses have been found, samples were isolated by SPE using XAD resins. As discussed earlier, this method isolates mainly hydrophobic organic compounds from seawater and is strongly biased against N-containing compounds. Polar ionic solutes have little or no affinity for XAD resins. There are perhaps more published results than have been found in this review, but the data in Table 11.4 are hopefully representative of elemental analyses for isolated samples of marine DOM. For reference, two calculated estimates of the bulk chemical composition of marine phytoplankton are included in Table 11.4. All elemental compositions in Table 11.4 are expressed as molar quantities in Redfield format, using an empirical formula that contains 106 moles of C. 

Fast dissipation is treated numerically within the Markoff approximation, which leads to differential equations in time, and dissipative rates most commonly written in the Redfield [9,10] or Lindblad [11,12] forms. Several numerical procedures have been introduced for dissipative dynamics within the Markoff approximation. The differential equations have been solved using a pseudospectral method [13], expansions of the Liouville propagator in terms of polynomials, [14-16] and continued fractions. [17]... 

Redfield, C. 2004. NMR studies of partially folded molten-globule states. Methods Mol Biol 278 233-254. 

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 preceding techniques are applicable only for the measurement of the rate of hydrolysis of peptides and proteins, and the methods employed in the sequence analysis of polypeptides are required for identification of the residues which form the susceptible bonds. These methods have been reviewed in detail elsewhere (Moore and Stein, 1956 Anfinsen and Redfield, 1956 Greenstein and Winitz, 1961 Canfield and Anfinsen, 1963) and do not require comment here. 

Examination of the action of proteolytic enzymes on native proteins (or biologically active peptides) can yield two important types of information. First, determination of the susceptibility of particular bonds in a protein substrate offers a means for evaluation of certain features of the conformation of the protein (Linderstr0m-Lang, 1952 Mihalyi and Harrington, 1959). Second, proteolysis can serve as an important method for modification of the covalent structure of biologically active proteins (Anfinsen and Redfield, 1956). 

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