Saddle methods

Illustration of the saddle method. Energy minima on the hyperspheres are denoted... 

The Sphere optimization technique is related to the saddle method described in... 

The idea in the line then plane (LTP) algorithm is similar to the saddle method, but the constrained optimization is done in the hypeiplane perpendicular to the inteipolation line between the two end-points, rather than on a hypersphere. LTP may also be considered as a variation of LUP where only two intermediate points are used, and the end-points are discarded after each minimization. This is iliustrated in Figure 7. 

The sphere optimization technique is related to the saddle method described in Section 2.2, and involves a. sequence of constrained optimizations on hyperspheres with increasingly larger radii, using the reactant (or product) geometry as a constant expansion point (Figure 9). The lowest-energy point on each successive hypersphere thus traces out a low-energy path on the PES. 

Ionova I V and Carter E A 1993 Ridge method for finding saddle points on potential energy surfaces J. Chem. Phys. 98 6377... 

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

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... 

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. 

HyperChem can calculate transition structures with either semi-empirical quantum mechanics methods or the ab initio quantum mechanics method. A transition state search finds the maximum energy along a reaction coordinate on a potential energy surface. It locates the first-order saddle point that is, the structure with only one imaginary frequency, having one negative eigenvalue. 

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

In realistic systems, the separation of the modes according to their frequencies and subsequent reduction to one dimension is often impossible with the above-described methods. In this case an accurate multidimensional analysis is needed. Another case in which a multidimensional study is required and which obviously cannot be accounted for within the dissipative tunneling model is that of complex PES with several saddle points and therefore with several MEPs and tunneling paths. 

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. 

At first sight, the easiest approach is to fit a set of points near the saddle point to some analytical expression. Derivatives of the fitted function can then be used to locate the saddle point. This method has been well used for small molecules (see Sana, 1981). An accurate fit to a large portion of the potential energy surface is also needed for the study of reaction dynamics by classical or semi-classical trajectory methods. 

A more balanced description requires MCSCF based methods where the orbitals are optimized for each particular state, or optimized for a suitable average of the desired states (state averaged MCSCF). It should be noted that such excited state MCSCF solutions correspond to saddle points in the parameter space for the wave function, and second-order optimization techniques are therefore almost mandatory. In order to obtain accurate excitation energies it is normally necessarily to also include dynamical Correlation, for example by using the CASPT2 method. 

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