Computation of Orientational Averages in Solid-State NMR by Gaussian Spherical Quadrature

We investigate Gaussian spherical quadrature as a method for calculating orientational averages in solid-state NMR. For the case of magic-angle-spinning sideband amplitudes of isolated spins-1/2, we demonstrate the superiority of Gaussian spherical quadrature over other orientational averaging methods. Depending on the shift anisotropy parameters and the desired accuracy, the computation speed is enhanced by a large factor (between two and many hundreds). In addition, a method for improving any present sampling scheme is devised. Such schemes are called SHREWD (Spherical Harmonic Reduction or Elimination by a Weighted Distribution). The role of orientational symmetry in solid-state NMR is explored. We also discuss the limitations of the Gaussian spherical quadrature methods.