On the calculation of derivatives for Coulomb-type interaction energies and general anisotropic pair potentials

We derive efficient algorithms for the derivatives, in particular the forces, torques and the Hessian, for Coulomb-type interactions and for general anisotropic pair potentials. We use spherical tensor techniques and an intermediate transformation to a quasi-internal coordinate system, proposed recently to obtain a factorization of the multipole interaction tensors. By these means we obtain a unified approach to the derivatives at small extra computational cost to the energy calculation. For the special case of an electrostatic interaction the computational cost scales for a system of N molecules through rank L in the multiple expansion as N2 x O(L3) for both the energy and its derivatives.