Simulations of Chemical Exchange Lineshapes in CP/MAS Spectra Using Floquet Theory and Sparse Matrix Methods P. Hazendonk, Alex D. Bain, 1 H. Grondey,* P. H. M. Harrison, and R. S. Dumont Department of Chemistry, McMaster University, 1280 Main St. W, Hamilton, Ontario L8S 4M1, Canada; and *Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada Received October 11, 1999; revised April 28, 2000 This paper presents a general method for simulating the effect of chemical exchange on MAS NMR spectra of solid samples. The complication in MAS spectra 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 convert the problem into a time-independent form, and then use established methods (used in liquid NMR simulations) to calculate the lineshape. Flo- quet theory has been admired for its elegance, but criticized for its computational inefficiencies. This is because it removes the time dependence of the system by expanding the problem in a Fourier- like series. This makes a relatively small, time-dependent calcu- lation 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 creates stems from the fact that the usual Householder methods for diagonalizing a matrix scale as the cube of the size of the matrix. This would make a Floquet calculation prohibitively long. How- ever, the Floquet matrix is inherently sparse, so sparse matrix methods can produce substantial computational savings. Also, fully diagonalizing a matrix is expensive, but converting the ma- trix to a tridiagonal form (using iterative Lanczos methods) is much cheaper. The use of the Lanczos methods makes the Floquet calculations feasible as a general method forsystems of more than one spin. We show how to 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 else- where. We then validate the theory by simulating the MAS spectra of dimethyl sulfone both with natural abundance 13 C and with methyl groups labeled with 13 C. The latter system has both dipolar and chemical shielding anisotropy terms contributing to the spectrum. © 2000 Academic Press Key Words: chemical exchange; CP/MAS; Floquet theory; spin- ning sidebands. INTRODUCTION The use of cross-polarization (CP) and magic-angle spinning (MAS) in obtaining the NMR spectra of solid samples is well known (1). The lineshape changes in the NMR spectra of liquid samples, due to chemical exchange, are also very familiar (2, 3). However, there are relatively few rigorous studies of the effects of chemical exchange on the patterns of spinning side- bands observed in CP/MAS spectra of complex spin systems. This is because the Hamiltonian is time-dependent, and the spectrum is relatively complicated to calculate (compared to liquids). For simple solid systems (4), exchange effects are often treated in the same way as liquid spectra, using the standard Kubo and Tomita (5) or Gutowsky and Holm formal- ism (6). To provide a full and general description, it is neces- sary to combine the rigorous treatment of liquids with the complexities of the MAS experiment. In this work we use Floquet theory (7–17) to convert the time-dependent MAS problem into a time-independent (but much larger) description. This is not a new approach (18), but it has been hampered by the size of the matrices generated. However, the Floquet matrices are inherently sparse. As such, modern sparse matrix methods reduce the numerical difficul- ties dramatically. We use the dual Lanczos method (19 –24) in Wassam’s formulation (25, 26) to tridiagonalize the resulting matrices. For large matrices, this is much more efficient than the usual Householder method (27). The general formalism is developed and applied to the simulation of the MAS 13 C NMR spectra of doubly 13 C-labeled dimethyl sulfone. Floquet theory was introduced to spectroscopic problems by Shirley (28), who applied this theory to compute the propaga- tor corresponding to a time-dependent non-self-commuting Hamiltonian. This is achieved by changing to a Fourier-spin space, where the Hamiltonian is time-independent, but infinite in dimension. Using similar methods, Vega described multiple- quantum effects in double-frequency pulsed NMR experiments on spin-1/2 and 1 systems (29 –31). Later Zax and Vega used Floquet methods to design broadband pulses (32, 33). Vega also applied Floquet theory to compute sideband patterns of rotating solids. This led to expressions for sideband intensities that were similar to those by Herzfeld and Berger (34) and Maricq and Waugh (35). For multispin systems, numerical methods are required. Based on a perturbation method pro- posed by Maricq (36), spectra of coupled spin pairs were computed to model rotational resonance (13–15, 17, 37) and REDOR dephasing curves (12, 38). Schmidt and Vega applied Floquet theory to uncoupled exchange in rotating solids using the Bloch–McConnell approach (18, 39, 40). In this case, only 1 To whom correspondence should be addressed. Journal of Magnetic Resonance 146, 33– 42 (2000) doi:10.1006/jmre.2000.2111, available online at http://www.idealibrary.com on 33 1090-7807/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved.