Analytic evaluation of the gradient and Hessian of molecular potential energy functions

The three-dimensional structure of a protein is believed to be associated with the global minimum of its potential energy function f . We can describe potential energy functions using Cartesian coordinates or internal coordinates (bond lengths, bond angles, and torsion angles). Analytic evaluation of the gradient of f with respect to the internal coordinates requires O ( N 4 ) steps, where N is the number of atoms involved. We provide analytical expressions for the first and second derivatives of f , with respect to the Cartesian coordinates, and prove that the gradient and Hessian can be evaluated in O ( N 2 ) , which is the same cost as is required to evaluate f .