Computation of Internal Coordinates, Derivatives, and Gradient Expressions: Torsion and Improper Torsion

Laplacians and gradient dot products are required for the recently developed internal coordinate quantum Monte Carlo method. New formulas are presented for these quantities for torsion and improper torsion angles. The Laplacians can also be used to economize calculation of sets of second derivatives used in molecular mechanics and other methods. Formulas for torsion angle gradient dot products and Laplacians, and completely new formulas for improper torsion, are presented. In addition, calculations of cos τ and sin τ, some suitable for energy subroutines and others for force subroutines, are shown. Finally, in a related development, several sets of conditions for three atom linearity or four atom planarity involving internal coordinate derivatives are reported.