Hessian eigenvalues

As a result, the 9x9 mass-weighted Hessian eigenvalue problem ean be sub divided into two 3x3 matrix problems ( of ai and b2 symmetry), one 2x2 matrix of bi symmetry... 

The normal modes of vibration u are obtained as solutions of the Hessian eigenvalue problem,... 

For example, a first-order saddle point has one and only one direction of negative curvature. Clearly, to characterize a stationary point we must be able to calculate the gradient and the Hessian eigenvalues of fix). The figures below illustrate a minimum and a saddle point in two dimensions. 

To see the solutions to Eq. (3.13) more clearly we have plotted below its left-and right-hand sides as functions of p for a function in three variables and with Hessian eigenvalues -1, 2, and 3. The step length function has poles at the eigenvalues as can be seen from Eq. (3.13) in the diagonal representation... 

When solving Eq. (5.8) the level shift parameter p must be chosen such that s(p) is the global minimizer on the boundary. From the discussion in Sec. Ill we know that p must be smaller than the lowest Hessian eigenvalue. Also, p must be negative since otherwise the step becomes longer than the Newton step. The exact value of p may be found by bisection or interpolation. 

If p. is different from all Hessian eigenvalues we obtain the Newton step... 

TABLE II H90 - ALL QUANTITIES ARE IN ATOMIC UNITS. XZARE THE HESSIAN EIGENVALUES. 

TABLE VII EXACT HESSIAN OPTIMIZATION OF H C0 - HESSIAN EIGENVALUES (a.u.). 

Tables VII and VIII examine the nature of the Hessian matrix for CO and NH as the geometry varies during the optimization. In both cases the starting geometry is far from the quadratic region about the minima. After the first geometry update in HjCO (cycle 2) the Hessian eigenvalues are within 10% of their final...

The Hessian matrix is useful in others ways, too. The square root of the mass-weighted Hessian eigenvalue is proportional to the vibrational frequency < ,. 

The main problem with the steepest descent method is the partial nndointr of the direction, as it should be. If, however, one of the Hessian eigenvalues is negative, the... 

---