Hessian method approximate analytic

If the reaction path is not obvious, then the most general techniques require information about the second derivatives. There exist, however, several often successful techniques that do not require this. The MOPAC and AMPAC series of programs utilize, for example, the saddlepoint technique, which attempts to approach the transition state from the reactant and product geometry simultaneously. The ZINDO set of models can utilize a combination of augmented Hessian and analytic geometry techniques. This is a very effective method, but unfortunately the augmented Hessian method does require approximate second derivatives and is somewhat time consuming. 

J. D. Head and M. C. Zerner, Chem. Phys. Lett., 131,359 (1986). An Approximate Hessian for Molecular Geometry Optimization. (Introduced is an approximate analytical Hessian that decreases the amount of work required by a factor of N in ZDO methods, where N is the size of the basis.)... 

There are a few points with respect to this procedure that merit discussion. First, there is the Hessian matrix. With elements, where n is the number of coordinates in the molecular geometry vector, it can grow somewhat expensive to construct this matrix at every step even for functions, like those used in most force fields, that have fairly simple analytical expressions for their second derivatives. Moreover, the matrix must be inverted at every step, and matrix inversion formally scales as where n is the dimensionality of the matrix. Thus, for purposes of efficiency (or in cases where analytic second derivatives are simply not available) approximate Hessian matrices are often used in the optimization process - after aU, the truncation of the Taylor expansion renders the Newton-Raphson method intrinsically approximate. As an optimization progresses, second derivatives can be estimated reasonably well from finite differences in the analytic first derivatives over the last few steps. For the first step, however, this is not an option, and one typically either accepts the cost of computing an initial Hessian analytically for the level of theory in use, or one employs a Hessian obtained at a less expensive level of theory, when such levels are available (which is typically not the case for force fields). To speed up slowly convergent optimizations, it is often helpful to compute an analytic Hessian every few steps and replace the approximate one in use up to that point. For really tricky cases (e.g., where the PES is fairly flat in many directions) one is occasionally forced to compute an analytic Hessian for every step. 

Newton s method and quasi-Newton techniques make use of second-order derivative information. Newton s method is computationally expensive because it requires analytical first-and second-order derivative information, as well as matrix inversion. Quasi-Newton methods rely on approximate second-order derivative information (Hessian) or an approximate Hessian inverse. There are a number of variants of these techniques from various researchers most quasi-Newton techniques attempt to find a Hessian matrix that is positive definite and well-conditioned at each iteration. Quasi-Newton methods are recognized as the most powerful unconstrained optimization methods currently available. 

To discuss the form and cost of analytic gradient and Hessian evaluations, we consider the simple case of Hartree-Fock (HF) calculations. In nearly all chemical applications of HF theory, the molecular orbitals (MOs) are represented by a linear combination of atomic orbitals (LCAO). In the context of most electronic structure methods, the LCAO approximation employs a more convenient set of basis functions such as contracted Gaussians, rather than using actual atomic orbitals. Taken together, the collection of basis functions used to represent the atomic orbitals comprises a basis set. 

The modified Newton methods evaluate the Hessian either analytically or by a numerical approximation in x and solve the linear system with a direct method that exploits matrix symmetry. 

These methods require either the analytical calculation of the Hessian or its numerical approximation. 

The most efficient methods that use gradients, either numerical or analytic, are based upon quasi-Newton update procedures, such as those described below. They are used to approximate the Hessian matrix H, or its inverse G. Equation (C.4) is then used to determine the step direction q to the nearest minimum. The inverse Hessian matrix determines how far to move along a given gradient component of f, and how the various coordinates are coupled. The success of methods that use approximate Hessians rests upon the observation that when f = 0, an extreme point is reached regardless of the accuracy of H, or its inverse, provided that they are reasonable. 

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