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

Christof Hättig
Department of Chemistry, Århus University, DK-8000 Århus C, Denmark

Chem. Phys. Lett. 260, 341-351 (1997)
(Received 11 July 1996; in final form 12 August 1996)

Recurrence relations for the 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 the cartesian interaction tensors the orientation-dependent spherical double-tensors make it possible without any further manipulation to evaluate multipole expanded electrostatic and induction energies with algorithms that also scale 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. Employing this factorization, we devise algorithms for the calculation of electrostatic, induction and dispersion energies that scale through order $L$ in the multipole expansion as L3, L4 and L5, respectively.

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