all twelve permutations of (denoted as a2). This selection of grid points gives rise to the scheme where A1, A2, and A3 are the weight functions that still need to be determined. Three further types of points can be employed as shown in the table. Each of these types of classes can contribute more than one set of points to the grid. In complete generality, the Lebedev scheme is where the total number of points, N, is The determination of the grid weights is achieved by enforcing the scheme to integrate exactly all polynomials up to a given order. On the unit sphere, this is equivalent to integrating all spherical harmonics up to the same order. This problem is simplified by a theorem of Sergei Lvovich Sobolev implying that this condition need be imposed only on those polynomials which are invariant under the octahedral rotation group with inversion. Enforcing these conditions leads to a set of nonlinear equations which have been solved and tabulated up to order 131 in the polynomial. References External links Fortran code for evaluating Lebedev grid points and weights Python codes: quadpy and CasperBeentjes Downloadable grid points Numerical integration (quadrature)