Computation of internal coordinates, derivatives, and gradient expressions: Torsion and improper torsion

Laplacians and gradient dot products are required for the recently develope d internal coordinate quantum Monte Carlo method. New formulas are presente d for these quantities for torsion and improper torsion angles. The Laplaci ans 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 impro per torsion, are presented In addition, calculations of cos tau and sin tau , some suitable for energy subroutines and others for force subroutines, ar e shown. Finally, in a related development, several sets of conditions for three atom Linearity or four atom planarity involving internal coordinate d erivatives are reported. (C) 2000 John Wiley & Sons, Inc.