Newton-Raphson minimization algorithm

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. 

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. 

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),... 

The system of N(M + 1) equations is then solved by using standard nonlinear solvers such as the Newton-Raphson method or the conjugate-gradient minimization algorithm, both of which are described in Press et al. (1992). 

To compute the concentrations of the species in equilibrium, a Newton-Raphson algorithm is applied to the direct minimization of Gibbs Free Energy. Lagrange multipliers are used to incorporate the restrictions of the problem. Its implementation is adapted to problems with high dynamic minerals appearance/disappearance as in the case presented in next section. 

Algorithms for the solution of quadratic programs, such as the Wolfe (1959) algorithm, are very reliable and readily available. Hence, these have been used in preference to the implementation of the Newton-Raphson method. For each iteration, the quadratic objective function is minimized subject to linearized equality and inequality constraints. Clearly, the most computationally expensive step in carrying out an iteration is in the evaluation of the Lapla-cian of the Lagrangian, V xL x , X which is also the Hessian matrix of the La-grangian that is, the matrix of second derivatives with respect to X . 

As a rule, independent data points are required to solve a harmonic function with N variables numerically. Because a gradient is a vector N long, the best one can hope for in a gradient-based minimizer is to converge in N steps. However, if one can exploit second-derivative information, an optimization could converge in one step, because each second derivative is an V x V matrix. This is the principle behind the variable metric optimization algorithms such as Newton-Raphson method. 

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