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. (c) 2007 Elsevier B.V. All rights reserved.2272135141