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Spherical integration grids

This page contains a (slowly increasing) database of common one-, two- and three-angle spherical grids that have been suggested, at various points in the past, for the calculation of powder average NMR and EPR spectra. Data format is either ASCII text files or Matlab *.mat files. Most grids come from the publications by Mattias Eden, Malcolm Levitt and Baltzar Stevensson.
1. Lebedev type grids. These are mathematically elegant because they integrate spherical Legendre polynomials (one-angle sets), spherical harmonics (two-angle sets) and Wigner D functions (three-angle sets) up to the specified rank to machine precision. The catch is that, for any spherical rank exceeding the specified one, the integration result is a random number.
2. Repulsion type grids. These are obtained by assigning notional charges to points on a circle (one-angle sets), sphere (two-angle sets) or hypersphere (three-angle sets) and minimizing the Coulomb energy of the resulting arrangement of points with respect to their positions.
3. Icosahedral grids. These are two-angle sets obtained by sequential triangle subdivision, starting from an equilateral icosahedron.
4. Polytope grids. These are three-angle sets corresponding to the vertices of 4-dimensional regular polytopes.
Spinach library provides spherical grid evaluation and manipulation functionality, including spherical rank accuracy profiling, SHREWD function for point weight assignment and grid direct product function that tiles one grid using the angles found in the other. Note also that the Fokker-Planck module of Spinach makes it possible to avoid spherical grids altogether.

Relevant literature


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