Augmented Hessian method

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. 

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. 

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) ... 

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 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. 

The ability of augmented Hessian methods for generating a search toward a first-order ... 

In order to minimize the second-order energy approximation (T) for fixed Cl coefficients a step-restricted augmented Hessian method as outlined in Section II.B (Eqs (30)-(33)) is used. While in other MCSCF methods this technique is employed to minimize the exact energy, it is used here to minimize an approximate energy functional. The parameter vector x is made up of the... 

Augmented Hessian method with step-length control results from Ref 70. Start with orbitals of smaller MCSCF. Final energy - 39.0278826738 hartree. 

The later procedure of Knowles and Werner uses instead an augmented Hessian method to define the orbital corrections. An approximate Hamiltonian operator is constructed that accounts for the simultaneous change of the entire vector of orbital rotations. This procedure may be summarized by the following steps ... 

For the current c, construct the density matrices D and d. From the current exact H, D and d, construct B and w and solve for k using an augmented orbital Hessian method. This orbital correction vector k defines the transformation matrix U. 

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. 

The augmented-Hessian method may therefore be regarded as a generalization of the Cl eigenvalue method to the MCSCF wave function. 

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