Introduction to Traditional Geometry Optimization Methods (supporting Documentation for Our Poster at Watoc '99 Entitled " Efficient Methods for Geometry Optimization of Large Molecules " )

Optimization of the structures of large molecules by quantum chemical methods requires stable, efficient, and therefore, elaborate optimization algorithms, usually in the framework of an over-complete, redundant internal coordinate system. Since the derivatives of the energy are calculated with respect to the Cartesian coordinates of the nuclei, they must be transformed into the internal coordinate system to carry out the optimization process. Quasi-Newton algorithms are very efficient for finding minima, particularly when inexpensive gradients (first derivatives) are available. These methods employ a quadratic model of the potential energy surface to take a series of steps that converges to a minimum. The optimization is generally started with an approximate Hessian (second derivative matrix) which is updated at each step using the computed gradients. The stability and rate of convergence of quasi-Newton methods can be improved by controlling the step size, using methods such as rational function optimization (RFO) or the trust radius model (TRM). In this chapter we briefly review various aspects of current optimization methods: a) the quadratic approximation to the potential energy (hyper)surface (PES), b) the use of a complete (and usually redundant) set of internal coordinates, c) BFGS update for the approximate second derivative (Hessian) matrix, d) rational function optimization (RFO) and trust radius model (TRM), e) geometry optimization using direct inversion in the iterative subspace (GDIIS)