Spherical integration grids
This page contains a (slowly increasing) database of common one, two and threeangle 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 ( oneangle sets), spherical harmonics ( twoangle sets) and Wigner D functions ( threeangle 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 ( oneangle sets), sphere ( twoangle sets) or hypersphere ( threeangle sets) and minimizing the Coulomb energy of the resulting arrangement of points with respect to their positions.
3. Icosahedral grids. These are twoangle sets obtained by sequential triangle subdivision, starting from an equilateral icosahedron.
4. Polytope grids. These are threeangle sets corresponding to the vertices of 4dimensional 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 FokkerPlanck module of Spinach makes it possible to avoid spherical grids altogether.
Relevant literature
[coming...]
