A short note on the fast evaluation of dihedral angle potentials and their derivatives

Dihedral angle potentials, which are used in many force fields for molecular dynamics simulations, model the energy of twisting a bond as a function of the angle φijkl between the surfaces spanned by the particles pi, pj and pk, and pj , pk and pl respectively. The potential function may have several forms. In the amber [4], cff [6], charmm [7], dreiding [8], gromos [9] and shapes [1] force fields, the potential is defined as v1(φ) = K [1 + cos(nφ− δ)] , (1) where K is the energy of the potential, n its multiplicity and δ its phase shift. The multiplicity n is an integer and δ is usually an integer multiple of π. In the Unified Force Field (uff) [11], the potential function is generalized as