Gradient matrix, stationary point

The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

It should be noted that the force constant matrix can be calculated at any geometry, but the transformation to nonnal coordinates is only valid at a stationary point, i.e. where the first derivative is zero. At a non-stationary geometry, a set of 3A—7 generalized frequencies may be defined by removing the gradient direction from the force constant matrix (for example by projection techniques, eq. (13.17)) before transformation to normal coordinates. 

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. ... 

This equation provides a prescription for the location of stationary points. In principle, starting from an arbitrary structure having coordinates q , one would compute its gradient vector g and its Hessian matrix H, and then select a new geometry q( +0 according to Eq. (2.41). Equation (2.40) shows that the gradient vector for this new structure will be the 0 vector, so we will have a stationary point. 

A point Q0 = QJ0 where all gradient matrix elements are equal to zero is referred to as a stationary point ... 

There are two problems here. One is easily disposed of The gradient is zero at r = 0 (at the saddle point), so the scheme in Eq. (3.10) does not progress away from the saddle point. One can show, however [40], that a MEP must approach a stationary point along the direction of the eigenvector of the force constant matrix, F, with lowest eigenvalue. At the saddle point, there is one negative eigenvalue of F, so we can simply replace the first step in Eq. (3.10) by... 

Here gg, Fg, and Gg are, respectively, the first (gradient), second (force constants), and third energy derivatives evaluated at Xg. The square brackets indicate that the three-dimensional array of third derivatives is contracted with the vector of coordinate changes to yield a square matrix. If Xg is a stationary point, i.e., a minimum, then the usual theory of small vibrations applies the gradient term vanishes, and truncation after the second-order term leads to separable, harmonic normal modes of vibration. However, on the MEP, the gradient term generally is not zero. The second relevant expansion is the Taylor series representation of the path (of the solution to Eq. [8]) in the arc length parameter, s, about the same point, Xg ... 

Here g is a transposed vector (gradient) containing all the partial first derivatives, and H is the (Hessian) matrix containing the partial second derivatives. In many cases, the expansion point Xo is a stationary point for the real function, making the first derivative disappear, and the zeroth-order term can be removed by a shift of the origin. 

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