Saddle computation

Palais J C 1992 Fiber Optic Communication 3rd edn (Englewood Cliffs, NJ Prentice Hall) Tanenbaum AS 1996 Computer Networks 3rd edn (Upper Saddle River, NJ Prentice Hall)... 

A saddle point approximation to the above integral provides the definition for optimal trajectories. The computations of most probable trajectories were discussed at length [1]. We consider the optimization of a discrete version of the action. 

Orbital-based methods can be used to compute transition structures. When a negative frequency is computed, it indicates that the geometry of the molecule corresponds to a maximum of potential energy with respect to the positions of the nuclei. The transition state of a reaction is characterized by having one negative frequency. Structures with two negative frequencies are called second-order saddle points. These structures have little relevance to chemistry since it is extremely unlikely that the molecule will be found with that structure. 

The primary means of verifying a transition structure is to compute the vibrational frequencies. A saddle point should have one negative frequency. The vibrational motion associated with this negative frequency is the motion going... 

Hanna, O. T, and O. C. SandaU. Computational Methods in Chemical Engineering, Prentice Hall, Upper Saddle River, NJ (1994). 

First, we perform an optimization of the transition structure for the reaction, yielding the planar structure at the left. A frequency calculation on the optimized structure confirms that it is a first-order saddle point and hence a transition structure, having a zero-point corrected energy of -113.67941 hartrees. The frequency calculation also prepares for the IRC computation to follow. 

Many problems in computational chemistry can be formulated as an optimization of a multidimensional function/ Optimization is a general term for finding stationary points of a function, i.e. points where tlie first derivative is zero. In the majority of cases the desired stationary point is a minimum, i.e. all the second derivatives should be positive. In some cases the desired point is a first-order saddle point, i.e. the second derivative is negative in one, and positive in all other, directions. Some examples ... 

The partial differential equations used to model the dynamic behavior of physicochemical processes often exhibit complicated, non-recurrent dynamic behavior. Simple simulation is often not capable of correlating and interpreting such results. We present two illustrative cases in which the computation of unstable, saddle-type solutions and their stable and unstable manifolds is critical to the understanding of the system dynamics. Implementation characteristics of algorithms that perform such computations are also discussed. 

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

Location of the saddle-type objects whose stable and/or unstable manifolds we want to compute by establish numerical methods. 

H. Waalkens, A. Burbanks, and S. Wiggins, A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill s problem, J. Phys. A 37, L257 (2004). 

As explained above, the QM/MM-FE method requires the calculation of the MEP. The MEP for a potential energy surface is the steepest descent path that connects a first order saddle point (transition state) with two minima (reactant and product). Several methods have been recently adapted by our lab to calculate MEPs in enzymes. These methods include coordinate driving (CD) [13,19], nudged elastic band (NEB) [20-25], a second order parallel path optimizer method [25, 26], a procedure that combines these last two methods in order to improve computational efficiency [27],... 

Table III summarizes the reported relative energies appropriate to the BC, Si—AB, and B—AB sites for the H—B pair in silicon. In many calculations, the B—AB site has been predicted from calculations to be not only higher in energy than the BC site but also to be a saddle point for reorientation between equivalent BC sites. Some calculations predict that the Si—AB site may be metastable, consistent with evidence from channeling and PAC studies. The computed total-energy differences, however, between the stable (BC) and metastable (Si—AB) configurations might be too large to explain the simultaneous observation of both species at the reported temperatures.

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