Simulations of chemical exchange lineshapes in CP/MAS spectra using Floquet theory and sparse matrix methods

This paper presents a general method for simulating the effect of chemical exchange on MAS NMR spectra of solid samples. The complication in MAS spect ra is that the Hamiltonian itself is time-dependent, due to the spinning of the sample. The approach taken in this work is to use Floquet theory to co nvert the problem into a time-independent form, and then use established me thods (used in liquid NMR simulations) to calculate the lineshape. Floquet theory has been admired for its elegance, but criticized for its computatio nal inefficiencies. This is because it removes the time dependence of the s ystem by expanding the problem in a Fourier-like series. This makes a relat ively small, time-dependent calculation into a much larger time-independent one. Typically, we use twice as many Floquet blocks as there are spinning sidebands, so the increase in size is substantial. The problem that this cr eates stems from the fact that the usual Householder methods far diagonaliz ing a matrix scale as the cube of the size of the matrix. This would make a Floquet calculation prohibitively long. However, the Floquet matrix is inh erently sparse, so sparse matrix methods can produce substantial computatio nal savings. Also, fully diagonalizing a matrix is expensive, but convertin g the matrix to a tridiagonal form (using iterative Lanczos methods) is muc h cheaper. The use of the Lanczos methods makes the Floquet calculations fe asible as a general method for systems of more than one spin. We show how t o set up the full matrix describing chemical exchange in a spinning sample, but the details of how the Lanczos methods work are not included-they are described elsewhere. We then validate the theory by simulating the MAS spec tra of dimethyl sulfone both with natural abundance C-13 and with methyl gr oups labeled with C-13. The latter system has both dipolar and chemical shi elding anisotropy terms contributing to the spectrum. (C) 2000 Academic Pre ss.