Update methods Hessian

A more general update method, widely used in the Gaussian suite of programs [19], is due to Schlegel [13], In this method, the Hessian in the n-dimensional subspace spaimed by taking differences between the current q... 

Some of the most important variations are the so-called Quasi-Newton Methods, which update the Hessian progressively and therefore economize compute requirements considerably. The most successful scheme for that purpose is the so-called BFGS update. For a detailed overview of the mathematical concepts, see [78, 79] an excellent account of optimization methods in chemistry can be found in [80]. 

If the exact Hessian is unavailable or computationally expensive, we may use an approximation. Approximate Hessians are usually obtained by one of several Hessian update methods. The update techniques are designed to determine an approximate Hessian B+ at... 

The augmented Hessian method requires an exact Hessian, or an update method on the Hessian itself. The update formula for the Hessian analysis to the inverse Hessian appear in the Appendix. 

Bakken, V. Millam, J. M. Schlegel, H. B. Ab initio classical trajectories on the Born-Oppenheimer surface Updating methods for Hessian-based integrators, J. Chem. Phys. 1999, 111, 8773-8777. 

Quasi-Newton methods update the Hessian by exploiting the information obtained during the search and update either the factorization of the Hessian or its inverse matrix. 

The gradient approaches zero in the neighborhood of the solution. If it is inaccurate, numerical problems may arise either in the evaluation of the second-order derivatives (modified Newton methods) or in updating the Hessian (quasi-Newton methods). 

The function BuildGradientAndUpdatePositiveHessian is used when we need the Hessian positive definite. It updates the Hessian with the aforementioned technique to preserve its sparsity and, analogously to BuildGradien-tAndPositiv6H6ssian, it makes the Hessian positive definite by adequately increasing diagonal elements similarly to Gill-Murray s method (see Section 3.6.1). 

If the Hessian can be kept in memory, the quasi-Newton methods provide better convergence to the minimum. Some of the more frequently used schemes for updating the Hessian are as follows. 

Note that rtk = 0 yields the DFP method and = 1 the BFGS method. The optimally conditioned (OC) method chooses to minimize the condition number of the Hessian (the condition number is the ratio of the largest to the smallest eigenvalue), thereby improving the behavior of the optimization. The CG, MS and DFP methods are also special cases of the Huang family of algorithms. Equations (7)-(10) can also be used to update the Hessian, B, rather than its inverse, H, provided that Ax and Ag are interchanged when H is replaced by B. ... 

A second important issue is the calculation of the Hessian matrix which can be a computationally expansive task. A method to avoid such calculation is to start with an approximate Hessian, e.g. empirically determined or calculated at a lower level of theory, and to update the Hessian during the optimization using only energies and... 

The efficiency of a quasi-Newton type geometry optimization depends on six factors (a) the initial geometry, (b) the coordinate system, (c) the initial guess for the Hessian, (d) the line search, (e) the Hessian updating method, and (f) step size control. These factors are discussed in the following paragraphs. 

Classical Trajectories on the Born-Oppenheimer Surface Updating Methods for Hessian-Based Integrators. 

In these methods, also known as quasi-Newton methods, the approximate Hessian is improved (updated) based on the results in previous steps. For the exact Hessian and a quadratic surface, the quasi-Newton equation and its analogue = Aq must hold (where - g " and... 

Bofill J M 1994 Updated Hessian matrix and the restricted step method for locating transition structures J. Comput. Chem. 15 1... 

All of these methods use just the new and current points to update the inverse Hessian. The default algorithm used in the Gaussian series of molecular orbital programs [Schlegel 1982] makes use of more of the previous points to construct the Hessian (and thence the inverse Hessian), giving better convergence properties. Another feature of this method is its use... 

The most frequently used methods fall between the Newton method and the steepest descents method. These methods avoid direct calculation of the Hessian (the matrix of second derivatives) instead they start with an approximate Hessian and update it at every iteration. 

Several attempts have been made to devise simpler optimization methods than the lull second order Newton-Raphson approach. Some are approximations of the full method, like the unfolded two-step procedure, mentioned in the preceding section. Others avoid the construction of the Hessian in every iteration by means of update procedures. An entirely different strategy is used in the so called Super - Cl method. Here the approach is to reach the optimal MCSCF wave function by annihilating the singly excited configurations (the Brillouin states) in an iterative procedure. This method will be described below and its relation to the Newton-Raphson method will be illuminated. The method will first be described in the unfolded two-step form. The extension to a folded one-step procedure will be indicated, but not carried out in detail. We therefore assume that every MCSCF iteration starts by solving the secular problem (4 39) with the consequence that the MC reference state does not... 

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