Recurrence relations for the direct calculation of spherical multipole interaction tensors and Coulomb-type interaction energies

Recurrence relations for orientation-dependent multipole interaction tensors are derived in the spherical double-tensor formalism. These relations make it possible to calculate all interaction double-tensor components through order L in an expansion in 1/R with computational expenditures that scale as L4. In contrast to Cartesian tensors, the orientation-dependent spherical double-tensors make it possible to evaluate electrostatic and induction energies with algorithms scaling as L4. By introducing an intermediate transformation to a special coordinate system the matrix of spherical interaction tensor elements can be factorized into a product of three sparse matrices, each of which can be calculated within L3 steps. We thus devise algorithms for the electrostatic, induction and dispersion energies that scale through order L in the multipole expansion as L3, L4 and L5, respectively.