Augmented Hessian

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 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. 

Step control (augmented Hessian, choice of shift parameter(s)). 

In this formula, p. stands for gTd This is certainly an eigensystem equation. However, we must add a small correction to H, and moreover, this correction is not known in advance. It turns out that this correction can be left out, unless it is important that the linear equation is exactly solved. This is not necessary if the object is to find a good step for a macroiteration. Moreover, it turns out that, in such a context, the discrepancy introduced between this method and the exact NR-steps has the same asymptotic dependence as the error. Therefore, the method is still a second-order method with this modification, and there is no way to say a priori that this method is better or worse than the exact NR-iterations. This method is called the augmented Hessian (AH-)method. It is seen to be equivalent to a Newton-Raphson using a shifted hessian. This can be very advantageous, since this shift tends to keep the step down, and to keep the shifted hessian positive definite, when one is far from a solution. The size of... 

One method, which avoids the problem with undesired negative eigenvalues of the Hessian, and which introduces an automatic damping of the rotations, is the augmented Hessian method (AM). To describe the properties of this method, let us again consider the Newton-Raphson equation (4 4) ... 

We shall now study the secular equation in some detail in order to make a comparison between super-CI and the NR method in the augmented Hessian form. The first thing to note is the non-orthogonality between the SX states ... 

Equation (4 59) can be compared to the unfolded two-step version of the augmented Hessian method, which results in the secular equation ... 

Illustrate the bracketing theorem mentioned in connection with the augmented Hessian method, by solving equation (4 31) for the energy E in the form E = f(E). Plot both the functions E and f(E) and show that the crossing points (the eigenvalues Ej) satisfies the betweenness condition. 

The eigenvalues of Eq. (3.20) coincide with the eigenvalues of the augmented Hessian only when S equals unity. 

Close to a stationary point gc vanishes. One of the eigenvalues of the augmented Hessian Eq. (3.22) then goes to zero and the rest approach those of Gc. The zero-eigenvalue step becomes the Newton step and the remaining n steps become infinite and parallel to the Hessian eigenvectors. 

To summarize, in the RF approach we make the quadratic model bounded by adding higher-order terms. This introduces n+1 stationary points, which are obtained by diagonalizing the augmented Hessian Eq. (3.22). The figure below shows three RF models with S equal to unity, using the same function and expansion points as for the linear and quadratic models above. Each RF model has one maximum and one minimum in contrast to the SO models that have one stationary point only. The minima lie in the direction of the true minimizer. 

Global strategies for minimization are needed whenever the current estimate of the minimizer is so far from x that the local model is not a good approximation to fix) in the neighborhood of x. Three methods are considered in this section the quadratic model with line search, trust region (restricted second-order) minimization and rational function (augmented Hessian) minimization. 

One advantage of the KF minimization over trust region RSO minimization is that we need only calculate the lowest eigenvalue and eigenvector of the augmented Hessian. In the trust region method we must first calculate the lowest eigenvalue of the Hessian and then solve a set of Unear equations to obtain the step. 

In the RF model the (k+l) th eigenvector gives a step in the right direction. To obtain the step we must therefore calculate the k+1 lowest eigenvalues of the augmented Hessian. For example, when optimizing the first excited state we calculate two eigenvalues but do not solve a set linear... 

The starting point in the augmented Hessian approach is the eigenvalue equation... 

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. 

In addition to examining Newton steps, we have also examined methods of the restricted step or augmented Hessian type. 

A comparison of the present algorithm with every one of the various MCSCF procedures used at present in literature is a task beyond the scope of this paper. But some kind of test must be done at least with one of the best procedures described so far. According to Werner /14/ the Augmented Hessian (AH) procedure of Lengsfield /62/ constitutes one of the best... 

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