Paul Hodgkinson1, Dimitris Sakellariou2 and Lyndon Emsley2
1Department of Chemistry, University of Durham,
South Road, Durham DH1 3LE, United Kingdom
2Laboratoire de Stéréochimie et des Interactions
UMR-117 CNRS/ENS, Ecole Normale Supérieure de Lyon, 69364 Lyon, France
Chemical Physics Letters, volume 326, page 515 (2000)
Simulations are very important to help understand the dynamics of systems of multiple (nuclear) spins coupled together by interactions such as the dipolar coupling. Unfortunately exact simulations are limited to relatively small numbers of spins (typically 10 or less) since the Hamiltonian matrices, which grow as 2N where N is the number of I=1/2 spins, rapidly become too large to handle.
Fortunately any symmetry of the system can be used to reduce the difficulty of exact simulation, and has been widely used to interpret the spectra of molecular systems. Here we propose using the translational symmetry of the Hamiltonian (and the density matrix), i.e. a space group symmetry, to "factorise" the simulation problem in crystalline systems. The resulting simulations are orders of magnitude faster for a fixed number of spins and also permit the simulation of a greater number of spins (~17 with 1 spin per unit cell). The number of spins is still restricted, so we consider 1-dimensional systems in which periodic boundary conditions are used to ensure that system is properly periodic.
Using spin diffusion as an example, we show such simulations shed light on the complex dynamics of coupled multi-spin systems. The calculated spectra are seen to converge as the size of the system increases (see figure), suggesting that the restricted system is sufficient to model the behaviour of an infinite lattice (in one dimension at least).
Example zero-quantum spectra for: (Left) 2-spin unit cell as a function of number of cells, illustrating convergence towards infinite system, and (right) 3-spin unit cell as a function of the "order" of couplings, showing that couplings between remote unit cells can readily be neglected.
This approach can be straightforwardly extended to spinning samples and is being used to study the effect of magic-angle spinning on lineshapes - something that has previously only been treatable by approximate analytical methods. Application to two (or more) dimensions is possible, although still restricted by the limited size of spin system.
1D periodic system with RF Other components: calcsignal.m, Fzel.m Functions to calculate spin operators: S, SS, F, singlespinop