Newton eigenvector method

Since the Newton eigenvector method is equivalent to the level-shifted Newton method of Section 12.3, it yields the same set of solutions. In our discu.s.sion of the level-shifted Newton method, we noted that, for excited states, it is not always possible to obtain a solution consistent with the first-order condition on the energy and the condition on the length of the step. The same situation may arise in the NEO method as well for excited states, it may not always be possible to generate a step (12.4.22) to the boundary of the trust region. In such situations, we may instead adjust a so as to yield the shortest possible step and then scale the resulting step down to the desired length. 

I he eigenvector-following (or Hessian mode) method implemented in HyperChem is based on an effieien t quasi-Newton like algorithm for loca tin g tran sitiori states, wh ieh can locate tran si-tion states for alternative rearran gern eri t/dissoeiation reactions, even when startin g from th e wron g regio n on th e poten tial en ergy surface. 

The Eigenvector Following method is in some ways similar to the Newton-Raph son method. Instead of explicitly calculating the second derivatives, it uses a diagonalized Hessian matrix to implicitly give the second derivatives of energy with respect to atomic displacements. The initial guess is computed empirically. 

In HyperChem, two different methods for the location of transition structures are available. Both arethecombinationsofseparate algorithms for the maximum energy search and quasi-Newton methods. The first method is the eigenvector-following method, and the second is the synchronous transit method. 

Hessian or " Eigenvector Following (EF), depending on how A is chosen. We wili p pseudo-Newton-Raphson or variable metric methods. It is clear that they do not... 

When only the Davidson part of the quasi-Newton step is used, we observe nearly identical convergence if all vectors are retained in the eigenvector subspace. With truncation of the subspace, the convergence of the Davidson method degrades somewhat relative to the quasi-Newton method. Thus, at the equilibrium geometry, the Davidson method converges to Eh in 15 iterations with two vectors in the subspace, compared with the 12 iterations for the quasi-Newton method. 

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