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 ff. 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 ff with respect to the internal coordinates requires O(N4)O(N4) steps, where NN is the number of atoms involved. We provide analytical expressions for the first and second derivatives of ff, with respect to the Cartesian coordinates, and prove that the gradient and Hessian can be evaluated in O(N2)O(N2), which is the same cost as is required to evaluate ff.