Efficient computation of potential energy first and second derivatives for molecular dynamics, normal coordinate analysis, and molecular mechanics calculations

By using two- and three-body internal coordinates and their derivatives as intermediates, it is possible to enormously simplify formulas for three- and four-body internal coordinates and their derivatives. Using this approach, simple formulas are presented for stretch (two-body), two types of bend (three-body), and wag and two types of torsion (four-body) internal coordinates and their first and second derivatives. The formulas are eminently suitable for economizing molecular dynamics and molecular mechanics calculations and normal coordinate analysis of complicated potential energy surfaces. Efficient methods for computing derivatives of entire potential energy terms, and in particular cross terms or terms with switching functions, are presented.