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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 *L*^{4}.
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 *L*^{4}.
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 *L*^{3} 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 *L*^{3}, *L*^{4}
and *L*^{5}, respectively.

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