Coupled spin systems exchange processes

Contents 1. Introduction 176 2. Static NMR Spectra and the Description of Dynamic Exchange Processes 178 2.1. Simulation of static NMR spectra 178 2.2. Simulation of DNMR spectra with average density matrix method 180 3. Calculation of DNMR Spectra with the Kinetic Monte Carlo Method 182 3.1. Kinetic description of the exchange processes 183 3.2. Kinetic Monte Carlo simulation of DNMR spectra for uncoupled spin systems 188 3.3. Kinetic Monte Carlo simulation of coupled spin systems 196 3.4. The individual density matrix 198 3.5. Calculating the FID of a coupled spin system 200 3.6. Vector model and density matrix in case of dynamic processes 205 4. Summary 211 Acknowledgements 212 References 212... 

It can be seen that, in the average density matrix formalism which is based on spin system model, the scalar couplings and the exchange processes are handled simultaneously. Thus they cannot be separated and a larger atomic basis (spin system) is required for their description. Meanwhile, the Monte Carlo method based on spin sets separates the two interactions, and thus spin systems can be reduced to smaller spin sets. 

The calculation method presented here also provides a possible extension of the single spin vector model. This extension is performed in two steps first to weakly coupled spin systems, then to strongly coupled ones. In the first case, the introduction of the well-known product basis functions and their coherences is sufficient while in the latter one the solution is not so trivial. The crucial point is the interpretation of the linear transformation between the basis functions and the eigenfunctions (or coherences) during the detection and exchange processes. These two processes can be described by the population changes of single quantum... 

Calculation of DNMR spectra is based on the construction of operators of interactions of the system and solving their eigenvalue problem. The most complicated interactions are the perturbation effects of the chemical shielding and the scalar coupling on the resonance frequency. The descriptions for these interactions are the same in both dynamic and static (free from chemical exchange processes) systems. Let n be the number of nuclei in a spin system, fi and /( the indices of the nuclei in it, v/( the frequency (in Hz) corresponding to the chemical shift of fi nuclei and the coupling... 

On the contrary to spin sets, the spin system contains all the nuclei of all conformers. Therefore, the spin system is a closed set with respect to exchange processes as well, not only to J coupling, and thus it is usually larger than a spin set. 

The term spin diffusion has been coined by Bloembergen [1] to characterize the polarization-exchange process in a strongly dipolar-coupled many-spin system. As pointed out by Bloembergen, this process leads to a spatial spread of polarization originating on a given spin that mimics, under certain conditions, a diffusion process. In a true diffusion process, the entropy increases monotonically. In the exact quantum description of the spin-diffusion process, however, the entropy is conserved and the process is, in principle, fully reversible. 

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