Individual density matrix trajectories

Key Words Dynamic NMR, Kinetic Monte Carlo, Chemical exchange, Spin system, Spin set, Individual density matrix, Trajectory, Eigencoherence, Vector model, Mutual exchange, Nonmutual exchange. 

By knowing the trajectory of a spin set, its individual density matrix can be calculated at any time points. The key to the simulation is the determination of the propagating matrix (see Section 3.5). The FID and spectrum of a spin set upon the individual trajectory (one scan) can be determined from the actual values of the time-dependent density matrix. 

In the Monte Carlo simulation, a few hundreds (100-2000) of scans are calculated, and the Fourier transform of their sum gives the simulated spectrum. The trajectories of spin sets are individual which makes their scans different providing the variety of the samples necessary for the simulation, similar to the case of the single spin interpretation. The calculation of the scans remains independent of each other thus the calculation can be parallelised in the case of coupled spin systems as well.101 The density matrix introduced because of the coupling and the increased amount of calculations on the matrix elements emphasise the use of modern architectures in parallel computation.104... 

Notice that while the trajectories jump between the electronic states at a given time, all density matrix elements are propagated continuously over the entire time according to Eqs. 17.14—17.15, or alternatively either Eq. 17.17 or 17.18. Although the individual trajectory is allowed to jump, the fraction of trajectories in a given state, which represents pu as an ensemble average, is also a continuous function of time. The phase of the electronic wavefunction is preserved and our procedure gives... 

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