Saddle points local methods

A technical difference from other Gaussian wavepacket based methods is that the local hamionic approximation has not been used to evaluate any integrals, but instead Maiti nez et al. use what they term a saddle-point approximation. This uses the localization of the functions to evaluate the integrals by... 

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. ... 

LSKMC) method [81], the search for saddle points in the vicinity of a given local minimum by the dimer method was used [82], For the found new local minima, the rate constants of transitions are calculated based on the TST... 

Fukui s reaction path, corresponding to a vibrationless and rotationless trajectory, passes gradually into the normal decomposition mode of the reactants (or products) or into the transition vector of the activated complex. Strictly speaking, Fukui s concept requires the knowledge of an accurately localized saddle point. However, it is also possible to exploit Fukui s procedure for approximately localized saddle points these points are usually obtained by an independent method. The respective resulting path is then an approximation to the real intrinsic reaction path. 

With suitable definitions of search functions, EA methods can also be used to locate more features on the PES than just low-energy local and global minima. Chaudhury et al. [144,145] have implemented methods for finding first-order saddle points and reaction paths, applying them to LJ clusters up to n=30. It remains to be tested, however, if these method can be competitive with deterministic exhaustive searches for critical points for small systems [146], on the one hand, and with the large arsenal of methods for finding saddles and reaction paths between two known minima for larger systems [63], on the other hand. 

Some crucial aspects of studying the GHF wave functions are connected with the relationship between the GHF and UHF methods. First of all, it is evident that the RHF and UHF wave functions are particular solutions also to the GHF problem In this case the components p " and p" of the Fock-Dlrac density matrix are zero, and the GHF equations separate into two sets of equations for the orbitals of spins a and p, respectively. The system of equations obtained in this way is identical to that of the ordinary UHF scheme. We note that the two sets of equations are still coupled through the components p++ and p". The situation is in some way analogous to the case of the UHF equations for a closed-shell system, for which the RHF functions always provide a particular solution. Similarly to the RHF versus UHF case, the UHF (or RHF) solution can, in principle, represent either a true (local) minimum or a saddle point for the GHF problem. 

The purpose of the later study of Tsai and Jordan8 was different. They used an eigenmode method of Section 2.4 in identifying as many local total-energy minima and saddle points as possible for 7 < N < 13. Some of their results were presented above in Figure 2 and Table 1. 

In their discussion of the eigenmode method, Tsai and Jordan8 illustrated the approach through two simple systems. One of those was Lennard-Jones clusters, and the other was small clusters of water molecules. For both they tried to identify as many local total-energy minima and saddle points as possible. The numbers for the Lennard-Jones clusters are reproduced in Table 1 and it is remarkable to see that the number of transition states exceeds by far the number of local total-energy minima. This was also the case for the clusters of water molecules. 

In the last work we shall discuss here, Maeda and Ohno138 used their scaled hypersphere search method, related to approximating the total-energy surface with some analytical form (see Section 5.8) for a couple of smaller systems. Their results on formaldehyde, summarized in Figure 36, represent the best way of closing the presentation of the problem of determining the structure in electronic-structure calculations. This molecule has only four atoms but nevertheless, the total-energy surface is complicated and full of local minima and saddle points ... 

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