Hessian gradient

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... 

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... 

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. 

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

At each iteration k, the new positions are obtained from the current positions x, the gradient gj. and the current approximation to the inverse Hessian matrix... 

The Hessian and gradient ean also be used to traee out streambeds eonneeting loeal minima to transition states. In doing so, one utilizes a loeal harmonie deseription of the potential energy surfaee... 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

Pragmatically, the procedure considers only one atom at a time, computing the 3x3 Hessian matrix associated with that atom and the 3 components of the gradient for that atom and then inverts the 3x3 matrix and obtains new coordinates for the atom according to the Newton-Raphson formula above. It then goes on to the next atom and moves it in the same way, using first and second derivatives for the second atom that include any previous motion of atoms. 

The procedure uses second derivative information and can be quite efficient compared to conjugate gradient methods. However, th e neglect of couplin g in th e Hessian m atrix can lead to situation s where oscillation is possible. Conjugate gradient methods. 

The column vector with components dUjdxi is the gradient of U it gives the negative of the force, when evaluated at a nuclear position. The p x p matrix with components d Ufdxidxj is called the Hessian. The Hessian is also called... 

A more sophisticated version of the sequential univariate search, the Fletcher-Powell, is actually a derivative method where elements of the gradient vector g and the Hessian matrix H are estimated numerically. 

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