S Global Newton Methods

The results from step 1 form the initial values x of the global Newton s method. Set i = 0. 

Solve H(x,f) = 0 using the global Newton s method (Sec, 4.2.9). The if-values and enthalpies used are determined by the current value... 

To solve Eq. (4.86) we employ the Jacobi-Newton iteration technique, which proceeds iteratively in an alternating sequence of local and global minimization steps. Let p be the local density at lattice site i in the A th local and the /th global minimization step. A local estimate for the corresponding minimum value of fi is obtained via Newton s method (see Eq. (D.6)] that... 

For the optimization of Hartree-Fock wave functions, it is usually sufficient to apply the SCF scheme described in Sec. 3.1. By contrast, the optimization of MCSCF wave functions requires more advanced methods (e.g., the quasi-Newton method or some globally convergent modification of Newton s method, which involves, directly or indirectly, the calculation of the electronic Hessian as well as the electronic gradient at each iteration) [45]. 

If g is convex, then any local minimizer is also a global minimizer. If x is a local minimizer of g(x), that is, Vg(x ) = 0, and if V g(x ) has full rank, then Newton s method will converge to x if started sufficiently close to x. ... 

Newton s method and the Hessian update schemes are in general very robust in the local region of the optimization. In the global part of the search, however, the second-order model does not accurately represent the true surface in the region of the optimizer. Indeed, sufficiently far away from the true minimizer, the Newton step may lead us away from the minimizer. This behavior is easily understood by considering a simple function such as the Gaussian distribution... 

In the present chapter, two different but related methods for the optimization of the MCSCF wave function are treated. The first method, which we discuss in this section, is a straightforward application of Newton s method for the optimization of nonlinear functions, modified so as to ensure global convergence. The second method, discussed in Section 12.4, is based on the solution of an eigenvalue equation which gives steps similar to those in the present section and which reduces to the standard eigenvalue problem of Cl theory when no orbital optimization is included. 

Newton s method is based on a local quadratic model of the energy surface. The orbital rotations generated by this method therefore behave incorrectly for large rotations. In particular, the Newton equations are not periodic in the orbital-rotation parameters as we would expect fix)m a consideration of the global behaviour of the energy function (10.1.22). The one-electron approximation of the SCF method, by contrast, is correct only to zero order in the rotations, but - provided the effective Hamiltonian (i.e. the Fock operator) is a reasonable one - it exhibits the correct global behaviour. In particular, it is periodic in the orbital rotations. 

Any of the global Newton methods can be converted to a relaxation form in Ketchum s method by making both the temperatures and the liquid compositions time dependent and by having the time step increase as the solution is approached. The relaxation technique should be applied to difflcult-to-solve systems and the method of Naphtali and Sandholm (42) is best-suited for nonideal mixtures since both the liquid and vapor compositions are included in the independent variables. Drew and Franks (65) presented a Naphtali-Sandholm method for the dynamic simulation of a reactive distillation column but also stated that this method could be used for finding a steady-state solution. 

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