Polynomially scaling spin dynamics simulation algorithms and their application in NMR and Spin Chemistry.

Magnetic resonance, which includes Nuclear Magnetic Resonance (NMR) and Electron Paramagnetic Resonance (EPR), comprises an enormously powerful and versatile range of spectroscopic techniques for exploring the structures, motions and reactivity of molecules. In many of these applications, the spectra cannot satisfactorily be interpreted without performing computer simulations of the response of the spin system to the sequence of radiofrequency and/or microwave pulses that are required to obtain the data. If the system of interest contains fewer than about 10 spins, this is usually reasonably straightforward and many efficient algorithms exist. However, difficulties arise for larger spin systems because the time and storage required scales exponentially with the number of spins. For systems with over 20 spins, the Liouvillian matrix required to evaluate the time-dependence of the density operator is so large that no computer is currently able to store it, let alone diagonalise it. Our preliminary work indicates that a highly efficient algorithm can be obtained that scales polynomially in the number of spins, allowing accurate simulations to be performed for many more than 20 strongly coupled spins. This is achieved by reducing the dimension of the Liouville matrix by intelligently excluding unimportant and unpopulated states, for example high orders of multiple quantum coherence or entanglements of spins that are remote from one another in the coupling network.We wish to extend and improve this method, which is still in its infancy, and to develop applications to two specific simulation problems in magnetic resonance. The algorithm will be adapted for the direct fitting of protein structures to experimental NMR data / something that has not so far been possible. In doing so, we hope to establish a new paradigm for NMR structure determination, wherein the atomic coordinates are directly related to the experimental spectra by a chain of well-defined ab initio simulation algorithms. The second major application will be in the field of Spin Chemistry (broadly defined as the magnetic effects of nuclear and electron spins on the chemistry of paramagnetic molecules). Quantitative interpretation of the data produced by such experiments routinely requires simulation of the coherent evolution of short-lived radical pairs comprising many coupled electron and nuclear spins subject to weak static and/or radiofrequency magnetic fields. Applications will include the elucidation of the biophysical origin of the magnetic compass of migratory birds and determination of the diffusive trajectories of radicals responsible for the magnetic field-sensitivity of the rates and yields of chemical reactions in solution.