Frequency space, multidimensional

Forth language, 175 forward-transformation, 94 Fourier transformation, multidimensional, 195 fragment, 71, 72, 75 code, 71,73 fraktur font, 5 frequency space, 90 full-curve... 

We introduce a simple model to investigate and calculate a diffusion coefficient as a basic quantity describing transport in Section II, and then we visualize resonances to detect the structure of the Arnold web and overlapped resonances in Section III. With the aid of this representation, to clarify the relevance of Arnold diffusion and diffusion induced by resonance overlap to global transport in the phase space, we compute transition diagrams in the frequency space in Section IV. In Section V, we extend the resonance overlap criterion to multidimensional systems to identify the pathway for fast transport, and in Section VI we revisit the diffusion coefficient to ensure fast transport affecting the global diffusion. A brief summary is given in Section VII. 

So-called multidimensional NMR techniques can provide important information about macromolecular conformation. In these cases, the sequence of a protein is aheady known, and establishing covalent connectivity between atoms is not the goal. Rather, one seeks through-space information that can reveal the solution conformation of a protein or other macromolecule. Two-or three-dimensional techniques use pulses of radiation at different nuclear frequencies, and the response of the spin system is then recorded as a free-induction decay (FID). Techniques like COSY and NOESY allow one to deduce the structure of proteins with molecular weights less than 20,000-25,000. 

R is called the relaxation superoperator. Expanding the density operator in a suitable basis (e.g., product operators [7]), the a above acquires the meaning of a vector in a multidimensional space, and eq. (2.1) is thereby converted into a system of linear differential equations. R in this formulation is a matrix, sometimes called the relaxation supermatrix. The elements of R are given as linear combinations of the spectral density functions (a ), taken at frequencies corresponding to the energy level differences in the spin system. 

The assignments are made through a battery of techniques to identify the through-bond and through-space interactions. Multidimensional NMR spectra are taken in which the sample is irradiated by two radio frequency fields in... 

A question that arises in consideration of the annihilation pathways is why the reactions between radical ions lead preferentially to the formation of excited state species rather than directly forming products in the ground state. The phenomenon can be explained in the context of electron transfer theory [34-38], Since electron transfer occurs on the Franck-Condon time scale, the reactants have to achieve a structural configuration that is along the path to product formation. The transition state of the electron transfer corresponds to the area of intersection of the reactant and product potential energy surfaces in a multidimensional configuration space. Electron transfer rates are then proportional to the nuclear frequency and probability that a pair of reactants reaches the energy in which they have a common conformation with the products and electron transfer can occur. The electron transfer rate constant can then be expressed as... 

Multidimensional NMR spectra are not restricted to cases where the separate frequency axes encode signals from different nuclear types. Indeed, much of the early work on the development of 2D NMR was performed on cases where both axes involved chemipal shifts. The main value in such spectra comes from the information content in cross peaks between pairs of protons. In COSY-type spectra (COSY = Correlation SpectroscopY) cross peaks occur only between protons that are scalar coupled (i.e., within 2 or 3 bonds) to each other, whereas in NOESY (NOE Spectroscopy) spectra cross peaks occur for protons that are physically close in space (<5 A apart). A combination of these two types of 2D spectra may be used to assign the NMR signals of small proteins and provides sufficient information on internuclear distances to calculate three-dimensional structures. Figure 12.3 includes a panel showing the COSY spectrum of cyclosporin and highlights the relationships between ID H-NMR spectra and corresponding 2D homonuclear (COSY) and heteronuclear (HSQC) spectra. 

Here I is the unit matrix and is the frequency square matrix in the space of nonreactive modes Equation (6.1) is a generalized Langevin equation of the form used in treating the one-dimensional case in Section V, and leads to the result of Eq. (5.25) (with m = 1) for the steady-state probability distribution of the reactive mode near the barrier. In the present multidimensional treatment it is convenient to redefine the distribution according to... 

The interpretation of multidimensional spectra can be quite involved, so only a simple overview will be presented here. Two-dimensional spectra are usually represented as contour plots with two frequency axes forming the base plane (e.g., 8h-5h, 5c-5c, or 6c-Jch). and intensity depicted by curves spaced according to the steepness of the ascent these plots resemble geological survey maps. The stacked plot tilts the frequency-frequency plane, showing the peaks as vertical intensity they look like perspective drawings of mountainous terrain. Three-dimensional spectra may be portrayed as false-color (to show intensity) spots in a cube formed by three frequency axes. 

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