Hessians

If there is no approximate Hessian available, then the unit matrix is frequently used, i.e., a step is made along the gradient. This is the steepest descent method. The unit matrix is arbitrary and has no invariance properties, and thus the... 

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods. 

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... 

For transition state searches, none of the above updates is particularly appropriate as a positive definite Hessian is not desired. A more usefiil update in this case is the Powell update [16] ... 

For a very large number of variables, the question of storing the approximate Hessian or inverse Hessian F becomes important. Wavefunction optimization problems can have a very large number of variables, a million or more. Geometry optimization at the force field level can also have thousands of degrees of freedom. In these cases, the initial inverse Hessian is always taken to be diagonal or sparse, and it is best to store the... 

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... 

An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One... 

Aq becomes asymptotically a g/ g, i.e., the steepest descent fomuila with a step length 1/a. The augmented Hessian method is closely related to eigenvector (mode) following, discussed in section B3.5.5.2. The main difference between rational fiinction and tmst radius optimizations is that, in the latter, the level shift is applied only if the calculated step exceeds a threshold, while in the fonuer it is imposed smoothly and is automatically reduced to zero as convergence is approached. 

The basic self-consistent field (SCF) procedure, i.e., repeated diagonalization of the Fock matrix [26], can be viewed, if sufficiently converged, as local optimization with a fixed, approximate Hessian, i.e., as simple relaxation. To show this, let us consider the closed-shell case and restrict ourselves to real orbitals. The SCF orbital coefficients are not the... 

Due to the large number of variables in wavefiinction optimization problems, it may appear that fiill second-order methods are impractical. For example, the storage of the Hessian for a modest closed-shell wavefiinction with 500... 

Started with a Hessian diagonal in the space of primitive internals usmg the recipe of Schlegel [39]. 

The second tenn in equation B3.5.11 deserves connnent. This tenn shows that Hessian (second-derivative) matrices... 

It is usually not efficient to use the methods described above to refine the transition state to full accuracy. Starting from a qualitatively correct region on the potential surface, in particular one where the Hessian has the right signature, efficient gradient optimization teclmiques, with minor modifications, are usually able to zero in on the transition state quickly. 

The EF algoritlnn [ ] is based on the work of Cerjan and Miller [ ] and, in particular, Simons and coworkers [70,1Y. It is closely related to the augmented Hessian (rational fiinction) approach[25]. We have seen in section B3.5.2.5 that this is equivalent to addmg a constant level shift (damping factor) to the diagonal elements of the approximate Hessian H. An appropriate level shift effectively makes the Hessian positive definite, suitable for minimization. 

Although it was originally developed for locating transition states, the EF algoritlnn is also efficient for minimization and usually perfonns as well as or better than the standard quasi-Newton algorithm. In this case, a single shift parameter is used, and the method is essentially identical to the augmented Hessian method. 

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

Schlegel H B 1984 Estimating the Hessian for gradient-type geometry optimizations Theor. Chim. Acta 66 333... 

LIndh R, Bernhardsson A, Karlstrdm G and Malmqvist P-A 1995 On the use of a Hessian model function In molecular geometry optimizations Chem. Phys. Lett. 241 423... 

The matrix M contains atomic masses on its diagonal, and the Hessian matrix F contains the second derivatives of the potential energy evaluated at Xq. 

The standard analytic procedure involves calculating the orthogonal transformation matrix T that diagonalizes the mass weighted Hessian approximation H = M 2HM 2, namely... 

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