Rapid and accurate calculation of small-angle scattering profiles using the golden ratio

Calculating the scattering intensity of an N-atom system is a numerically exhausting O(N²) task. A simple approximation technique that scales linearly with the number of atoms is presented. Using an exact expression for the scattering intensity I(q) at a given wavevector q, the rotationally averaged intensity I(q) is computed by evaluating I(q) in several scattering directions. The orientations of the q vectors are taken from a quasi-uniform spherical grid generated by the golden ratio. Using various biomolecules as examples, this technique is compared with an established multipole expansion method. For a given level of speed, the technique is more accurate than the multipole expansion for anisotropically shaped molecules, while comparable in accuracy for globular shapes. The processing time scales sub-linearly in N when the atoms are identical and lie on a lattice. The procedure is easily implemented and should accelerate the analysis of small-angle scattering data.