Saddle points Newton-Raphson methods

In practice, of course, the surface is only quadratic to a first approximation and so a number of steps will be required, at each of which the Hessian matrix must be calculated and inverted. The Hessian matrix of second derivatives must be positive definite in a Newton-Raphson minimisation. A positive definite matrix is one for which all the eigenvalues are positive. When the Hessian matrix is not positive definite then the Newton-Raphson method moves to points (e.g. saddle points) where the energy increases. In addition, far from a mimmum the harmonic approximation is not appropriate and the minimisation can become unstable. One solution to this problem is to use a more robust method to get near to the minimum (i.e. where the Hessian is positive definite) before applying the Newton-Raphson method. 

Of course, the surface is not quadratic and the Hessian is not constant from step to step. However, near a critical point, the Newton-Raphson method (Eq. (2)) will converge rapidly. The main difficulty is that the convergence of the Newton-Raphson method is local. Thus the method will converge to the nearest critical point to the starting point. Consequently, one must start the optimization with a Hessian that contains no or one negative eigenvalue according to whether a minimum or a saddle point structure is required. For a minimum, the steepest descent direction... 

Moreover, the second-generation MCSCF parametrizes the wave function in a way that enables the simultaneous optimization of spinors and Cl coefficients, in this context then called orbital or spinor rotation parameters and state transfer parameters, respectively. Then, a Newton-Raphson optimization method is employed which also requires the second derivatives of the MCSCF electronic energy with respect to the molecular spinor coefficients (more precisely, to the orbital rotation parameters) and to the Cl coefficients. As we have seen, in Hartree-Fock theory the second derivatives are usually not calculated to confirm that a solution of the SCF procedure has indeed reached a minimum with respect to the large component and not a saddle point. Now, these general MCSCF methods could, in principle, provide such information, although it is often not needed in practice. 

Several effective methods for direct localization of the transition state points have been evolved which do not require a calculation of the whole PES [35-37]. The Newton-Raphson scheme [38] is a standard method for determining any critical points, however, it converges toward the saddle point only in a region sufficiently close to it. One of the best known methods is the Mclver and Komornicki [39] minimization of the norm of the gradient Sg. 

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