Multi-dimensional approaches

In the early stage of development of NMR spectroscopy, the applicability of multi-dimensional approach was tested for condensed matter. Today, because of the excellent progress on this field, the "library" of different ND techniques used in SS NMR is pretty rich. On the basic level, these methods can be assigned to two groups homo- (HOMCOR) and heteronuclear correlations (HETCORs). Both correlations employ indirect (through-bond) or direct (through-space) coupling to correlate different spins. 

Retention in SFC is a complex function of temperature, pressure, density and solvent modifier concentration and a more complete understanding of these phenomena should directly benefit the development of SFC. The dynamic processes of stationary phase solvation and solute-solvent interactions in the stationary and mobile phases, respectively, impact on solute retention in SFC. The study of the retention process necessitates a multi-dimensional approach to understand the basic physicochemical processes underlying solute retention in SFC. The discussion in this chapter outlines three interrelated areas of study, probing specific areas of solute... 

This subchapter is devoted to the evaluation of one-dimensional solubility parameters. The methods of the evaluation of components of solubility parameters in multi-dimensional approaches are given in the Subchapter 4.1. 

This approach is, at the same time, the great advantage and a major limitation of membrane potential models. As they are rather compact, models of this type were the first to be used in investigations of the spread of excitation in multi-dimensional tissue representations consisting of relatively large numbers of interconnected excitable elements their role in assessing biophysical behaviour like cardiac impulse propagation is undiminished. 

For k(r) we shall assume at first, as in (19), that the reaction is adiabatic at the distance of closest approach, r = a, and that it is joined there to the nonadiabatic solution which varies as exp(-ar). The adiabatic and nonadiabatic solutions can be joined smoothly. For example, one could try to generalize to the present multi-dimensional potential energy surfaces, a Landau-Zener type treatment (41). For simplicity, however, we will join the adiabatic and nonadiabatic expressions at r = a. We subsequently consider another approximation in which the reaction is treated as being nonadiabatic even at r = a. 

Jamrog, D. C., Zhang, Y., and PhiUips, G. N., Jr. (2003) SOMoRe a multi-dimensional search and optimization approach to molecular replacement. Acta Crystallogr. D 59,304-314. 

Multi-dimensional Chromatography. Multi-dimensional chromatography is the term used to describe a variety of methods where fractions from one chromatographic system are each transferred to another for further separation. Combinations of SEC with thin-layer chrcmatography have been shown to enable separation of copolymers by composition in a "cross-fractionation". OC utilizes a combination of two SECs in a cross-fractionation approach. 

A common approach for the study of activated barrier crossing reactions is the transition state theory (TST), in which the transfer rate over the activation barrier V is given by (0)R/2jt)e where 0)r (the oscillation frequency of the reaction coordinate at the reactant well) is an attempt frequency to overcome the activation barrier. For reactions in solution a multi-dimensional version of TST is used, in which the transfer rate is given by... 

In the last few years we have witnessed the successful development of several methods for the numerical solution of multi-dimensional quantum Hamiltonians Monte Carlo methods centroid methods,mixed quantum-classical methods, and recently a revival of semiclassical methods. We have developed another approach to this problem, the exponential resummation of the evolution operator. - The rest of this Section will explain briefly this method. 

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