The number and location of the grid points together with a corresponding set of integration weights are determined by enforcing the exact integration of polynomials (or equivalently, spherical harmonics) up to a given order, leading to a sequence of increasingly dense grids analogous to the one-dimensional Gauss-Legendre scheme.
The use of a single sum, rather than two one dimensional schemes from discretizing the θ and φ integrals individually, leads to more efficient procedure: fewer total grid points are required to obtain similar accuracy.
The Lebedev grid points are constructed so as to lie on the surface of the three-dimensional unit sphere and to be invariant under the octahedral rotation group with inversion.
The smallest such set of points is constructed from all six permutations of (±1, 0, 0) (collectively denoted as a1), leading to an integration scheme where the grid weight is A1.
Geometrically these points correspond to the vertices of a regular octahedron when aligned with the Cartesian axes.
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.
[5] Enforcing these conditions leads to a set of nonlinear equations which have been solved and tabulated up to order 131 in the polynomial.