Gauss-Legendre Quadrature

While the Lebedev grids are very efficient they cannot be used for arbitrary precision. For this reason it is sometimes desirable to perform the $\theta$ and $\phi$ integration separately, even though it is likely to be less efficient.

The $\phi$ integration is performed with equally spaced points, while the $\theta$ integration uses Gauss-Legendre quadrature, which is designed to exactly integrate all polynomials up to degree 2NP-1, where NP is the number of points. To exactly integrate all spherical harmonics up to degree L, the $\theta$ quadrature requires (L+1)/2 points and the $\phi$ quadrature L+1 points, thus
$$\begin{align}\mathrm{NP}(L) = \frac{(L+1)^{2}}{2} \end{align}$$
which is 2/3 the efficiency of the Lebedev scheme.