Newton-Raphson minimization

The implementation of this type of transformation in the multiconfigurational problem has been very successful when associated with energy direct minimization. Newton-Raphson or related techniques have been used for this task /44/. The problem in these implementations is the... 

For a = 0, minimization of this expression yields the Newton-Raphson fomuila for Aq. For large values of a,... 

For nonquadratic but monotonic surfaces, the Newton-Raphson minimization method can be applied near a minimum in an iterative way [24]. 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

FIGURE 4.3. Illustrating the effectiveness of different minimization schemes. The steepest-deicent method requires many steps to reach the minimum, while the Newton-Raphson method locates the minimum in a few steps (at the expense, however, of evaluating the second derivative matrix). 

Exercise 4.3. Use the Newton-Raphson method to minimize the system in Problem 4.1. 

Schaumann T, Braun W, Wilthrich K. The program FANTOM for energy refinement of polypeptides and proteins using a Newton-Raphson minimizer in torsion angle space. Biopolymers 1990 29 679-694. 

To solve for the chemical system at t, we use Newton-Raphson iteration to minimize a set of residual functions, as discussed in Chapter 4. For a kinetic... 

The final step in the MM analysis is based on the assumption that, with all force constants and potential functions correctly specified in terms of the electronic configuration of the molecule, the nuclear arrangement that minimizes the steric strain corresponds to the observable gas-phase molecular structure. The objective therefore is to minimize the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The most popular procedure is by computerized Newton-Raphson minimization. It works on the basis that the vector V/ with elements dVt/dxn the first partial derivatives with respect to cartesian coordinates, vanishes at a minimum point, i.e. = 0. This condition implies zero net force on each atom... 

The steepest-descent method does converge towards the expected solution but convergence is slow in the vicinity of the minimum. In order to scale variations, we can use a second-order method. The most straightforward method consists in applying the Newton-Raphson scheme to the gradient vector of the function/to be minimized. Since the gradient is zero at the minimum we can use the updating scheme... 

When the function to be fitted to data does not depend linearly on the parameters, recursive methods must be used. A slightly modified version of the Newton-Raphson method (Chapter 3) will be used (Hamilton, 1964). Let jc be the vector of the n unknowns Xj and y — f(x) the m-vector of observable functions y, = /(jc). The analytical form of the functions f(x) may be the same or not. Let the vector / represent the m observations / of these functions. A vector jc is sought which minimizes the scalar c2 such that... 

A new feature in MM3 is the full Newton-Raphson minimization algorithm. This allows for the location and verification of transition states and for the calculation of vibrational spectra. Indeed, many of the new potential functions in MM3 were included to provide a better description of the potential energy surface which is required for an accurate calculation of vibrational spectra. 

As stated, the most commonly used procedure for temperature and composition calculations is the versatile computer program of Gordon and McBride [4], who use the minimization of the Gibbs free energy technique and a descent Newton-Raphson method to solve the equations iteratively. A similar method for solving the equations when equilibrium constants are used is shown in Ref. [7],... 

MMI/MMPI incorporates a modified Newton-Raphson energy minimization algorithm that moves atoms one by one and is quite efficient. The force field is parameterized not only for saturated hydrocarbons including cyclopropane, but also for nonconjugated olefins (17c),... 

Several modifications of the Engler-Schleyer force field have appeared. White (27) added olefin parameters and used an efficient two-stage Newton-Raphson minimization modification. The accuracy of the White force field regarding heat of formation calculations is... 

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