A Multi-Center Quadrature Scheme for the Molecular Continuum

Computing electronic integrals for polyatomic molecules is a numerical challenge. Becke's partitioning scheme is a possible solution that works well for integrands that fall off rapidly at large distances. When applied to states in the electronic continuum, however, Becke's scheme converges slowly and may be expensive. We present a modified version of Becke's scheme that is applicable to functions in the electronic continuum, such as those involved in molecular photoionization and electron–molecule scattering, which ensure convergence and efficiency. In the new scheme, Becke's atomic weights are smoothly switched off within a range of few bond lengths from their respective nuclei, and complemented by an asymptotically unitary weight. The integrals are evaluated on small spherical grids, centered on each atom, with size commensurate to the support of the corresponding atomic weight. The integral of the interstitial and long-range region is evaluated with a central grid. We demonstrate that the method provides high accuracy for a wide range of integrals and has good scaling properties for the evaluation of the hybrid bi-electronic integrals needed in photoionization and electron-scattering studies.