Stationary points definition

However, in accordance with the definition of a stationary point. Equation (B.1), the first term in Equation (B.3) for AV vanishes irrespective of the value or size of h so that... 

Steepest descent can terminate at any type of stationary point, that is, at any point where the elements of the gradient of /(x) are zero. Thus you must ascertain if the presumed minimum is indeed a local minimum (i.e., a solution) or a saddle point. If it is a saddle point, it is necessary to employ a nongradient method to move away from the point, after which the minimization may continue as before. The stationary point may be tested by examining the Hessian matrix of the objective function as described in Chapter 4. If the Hessian matrix is not positive-definite, the stationary point is a saddle point. Perturbation from the stationary point followed by optimization should lead to a local minimum x. ... 

If no active constraints occur (so x is an unconstrained stationary point), then (8.32a) must hold for all vectors y, and the multipliers A and u are zero, so V L = V /. Hence (8.32a) and (8.32b) reduce to the condition discussed in Section 4.5 that if the Hessian matrix of the objective function, evaluated at x, is positive-definite and x is a stationary point, then x is a local unconstrained minimum of/. 

A sequence of Newton-Raphson iterations is obtained by solving equation (4 4) redefining the zero point, p0, as the new set of parameters recalculating g and H and returning to equation (4 4). Such a procedure converges quadratically, that is, the error vector in iteration n is a quadratic function of the error vector in iteration n-1. This does not necessarily mean that the NR procedure will converge fast, or even at all. However, close to the stationary point we can expect a quadratic behaviour. We shall return later to a more precise definition of what close means in this respect. 

To use this formula one can employ experimental or calculated adiabatic (or vertical, if the species from removal or addition of an electron are not stationary points) values of I and A. This same formula (Eq. 7.32) for / was elegantly derived by Mulliken (1934) [149] using only the definitions of I and A. Consider the reactions... 

This degree of freedom is the reaction coordinate (note that this definition coincides with the definition in Chapter 3). In Appendix E, we show that a multidimensional system close to a stationary point can be described as a set of uncoupled harmonic oscillators, expressed in terms of so-called normal-mode coordinates. The oscillator associated with the reaction coordinate has an imaginary frequency, which implies that the motion in the reaction coordinate is unbound. 

No additional physical concepts are necessary for this static definition of the mechanism. The strategy is well defined and relatively simple to apply to reactions with a simple PES form, i.e. surfaces with a single TS. Actually the topological structure of the surface may be more complex, with several TSs defining accessory stationary points, some of which correspond to intermediates along the RP, others defining alternative routes. 

The stationary point of the system (5.2) is by a definition stable one, if all the roots of its characteristic equation (5.11) have the negative real parts. The Routh-Hurwitz criteria presented in Ref. [206] permits escaping the calculations of these roots to establish the simple relations between the coefficients ock, which allow to point out simple stability conditions. For instance, in the case of terpolymerization the positivity of both coefficients oq and oc2 is regarded to be a criteria of such stability, and as for four-component copolymerization the following non-equality a3 < oq < ot2 has also to be hold. At arbitrary number m of the components the positivity of all ak is regarded to be necessary (but not sufficient) stability condition. For the stability of the boundary SP of m-component system located inside the certain boundary 1-subsimplex of monomers Mk, M2,. .., M, the stability of the above SP in such subsimplex and negativity of all values of X, Xl+1,..., vm x (5.13) are needed. 

A point x is called a stationary point of fti g(x ) = 0 but H(x ) is not necessarily positive-definite. Thus, local and global minima are stationary points, but there are more general stationary points, such as saddle points, which are neither local nor global minima. Special techniques are needed for detection of saddle points, which are often related to structural transitions in molecular applications. 

As we noted above, the kinetic energy is positive definite. Furthermore, it is quadratic in the momenta. As a consequence, we can reduce the search for points of stationary flow in phase space to one of finding the stationary points of the potential energy surface. To see how this comes about, consider the Hamilton s equations for the three velocities... 

Then the stationary points obtained are minima if arbitrary increments to the coordinates Q cause a positive-definite increment of the energy... 

It is very important to remember that this definition of a PES is based on the assumption that the atomic positions can be exactly specified, which is the ultimate condition for the structure or shape of a molecule. This means adoption of the Born-Oppenheimer (B.O.) approximation, in which the nuclei are viewed as stationary point charges, whereas the electrons are described quantum mechanically [5]. This approximation is justified by the fact that the electrons are much lighter than the nuclei and hence are moving faster. The classical nature of the atomic nuclei is usually a valid approximation, but the zero-point vibrational energy of molecules or the tunneling effect, for example, make it evident that it does not always hold. 

The stationary point is a minimum if the Hessian matrix (Eq. (29)) is positive definite. Furthermore, the Cl coefficients must satisfy the eigenvalue equation... 

Equivalent potential functions will be defined as the functions having identical sets of critical points. Such a definition of equivalence of potential functions implies that, as follows from the state equation (1.8) and equations (1.9), (1.10), potential functions of the same form describe physical systems having the same sets of stationary points. 

Since the definition of a stationary point requires that all gradient elements are zero, the closest stationary point can be found by minimizing the sum of squares of elements of the gradient vector. 

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