Locating Stationary Points

A good place to start our study- is with a function of a single variable f(x). Consider the function 

A simple way of finding the roots of an equation, other than by divine inspiration, symmetry or guesswork is afforded by the Newton method. We start at some point denoted x l along the x-axis, and calculate the tangent to the curve at 

Newton s method can be easily re-written for the problem of finding stationary points, where (d//dx) = 0 rather than j = 0. The formula 14.9 becomes 

Finally, there is the question of availablity of analytical derivatives. Minima, maxima and saddle points can be characterized by their first and second derivatives. Over the last 25 years, there has been a rapid development in this area, and analytical gradient formulae are now known for most of the common techniques discussed in this volume. The great advantage is that those methods that use analytical gradients tend to out-perform in speed of execution those methods where gradients have to be estimated numerically. 

Such algorithms can be found in any standard book on numerical analysis. These methods have the widest range of applicability, but they tend to be slower than those methods that make use of analytical formulae for the gradient. Two examples will give the gist. 

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Analytical gradient energy expressions have been reported for many of the standard models discussed in this book. Analytical second derivatives are also widely available. The main use of analytical gradient methods is to locate stationary points on potential energy surfaces. So, for example, in order to find an expression for the gradient of a closed-shell HF-LCAO wavefunction we might start with the electronic energy expression from Chapter 6,... 

Chapter 2, Michael L. McKee and Michael Page address an important issue for bench chemists how to go from reactant to product. They describe how to compute reaction pathways. The chapter begins with an introduction of how to locate stationary points on a potential energy surface. Then they describe methods of computing minimum energy reactions pathways and explain the reaction path Hamiltonian and variational transition state theory. 

The optimization methods described in Sections 12.2-12.4 concentrate on locating stationary points on an energy surface. The important points for discussing chemical reactions are minima, corresponding to reactant(s) and product(s), and saddle points. 

Locating stationary points on a reaction coordinate is essential for characterizing a reaaion. Such points are, however, as their name implies, stationary no information regarding the speed with which the path is traversed... 

It should be noted that in quantum chemical geometry optimization carried out to locate stationary points of the PES, the three... 

All the reviewed programs carry out fundamental tasks of a computational chemist or a computational molecular physicist calculation of energy for various hamUtonians evaluation of gradients of energy (needed to locate stationary points on the potential energy surface) evaluation of the energy hessian (required to analyze the character of the located stationary point, identify local minima and saddle points, and perform vibrational frequency calculation) and evaluation of basic properties (population analysis, dipole moments). The components of the programs include basis set libraries and pseudopotentials. 

Analytic gradients and hessians for RHF, ROHF, UHF, GVB, and MCSCF wavefunctions that are used to locate stationary points on the potential energy surface and identify their character (local minimum or transition state). 

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