Gradient Extremal

We call this line a gradient extremal (GE).16 Unless the function increases indefinitely the gradient approach zero. It is therefore reasonable to expect that by following a GE we sooner or later hit a new stationary point... 

The term gradient extremal was introduced by D. K. Hoffmann, R. S. Nord, and K. Ruedenberg, Theor. Chim. Acta. 69, 265 (1986). Gradient extremals have also been discussed by J. Pancir, Collect. Czech. Chem. Commun. 40, 1112 (1975) and by M. V. Basi-levsky and A. G. Shamov, Chem. Phys. 60, 347 (1981). 

To develop a practical method for tracing gradient extremals we return to the restricted second-order model Eq. (3.10).17 In this model the Hessian is constant... 

Gradient Norm Minimizations 14.5.9 Netvton-Raphson Methods 14.5.10 Gradient Extremal Methods Constrained Optimization Problems Locating the Global Minimum and Conformational Sampling 333 333 338 338 339 Appendix C First and Second Quantization Reference 411 411 412... 

Gradient Extremal (GE), 338 Gradient norm minimization, 333 Gradient of a function, 238 Greens function, 257 GROMOS force field, 40 Gross atomic charge, 218... 

The gradient extremal is a mathematical nature of a hypersurface [32-34]. The most rigid discussion of the gradient extremal is due to Hoffman et al. [32] in 1986. Assuming an M-dimensional surface,... 

On the contour subspace V(x) = Vo, the gradient extremal path (GEP) passes through the point where the square norm of the gradient is extremal. Thus, introducing the Lagrange multiplier X, the following equation must be satisfied on the GEP ... 

P. Jorgensen, H. J. A. Jensen, and T. Helgaker, Theor. Chim. Acta, 73,55 (1988). A Gradient Extremal Walking Algorithm. 

Reaction Path Following Higher-Order Implicit Algorithms. H. B. Schlegel, Theor. Chim. Acta, 83, 15 (1992). Following Gradient Extremal Paths. 

Figure C.4 Graphic representation of the gradient extremal leading from a minimum to a saddle point following Eq. (C.34). The direction qi has followed the lowest vibrational frequency at the minimum.

Figure 2. Steepest descent and gradient extremals (GE) using the Miiller-Brown potential [23], steepest descent path — GE.

Because (i) those parts of the free system PES that are actually explored by either classical trajectories or quantal wave packets describing a chemical reaction, are preferentially the lowest ones and (ii) modern quantum-chemical methods for calculating PES include geometry optimizations, generally gradient optimizations (for gradient extremal methods, see Refs.[64-66]), it is natural to retain for 3", at given 3, the values which minimize V(3). Mathematically, this is expressed as ... 

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