Minima and Saddle Point on the PES

Both the minima and the saddle points are stationary points on the PES the derivative of the energy with respect to each of the geometric coordinates, and thus the force on each of the atoms, is zero. 

The set of coordinates q can be any sort of internal or Cartesian coordinates. The positions of the stationary points are independent of the choice of coordinates. However, away from the stationary points the shape of the potential energy surface no longer is independent of the choice of coordinates it depends among other things on the choice of internal or Cartesian coordinates, on whether the coordinates are mass-weighted, and on whether angles are expressed in degrees or radians. 

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With respect to a more quantitative characterization of the PE function along the reaction coordinate, in terms of local minima and barriers between them, the minimum energy reaction (MER) path concept is more promising. The MER path approach, also called the intrinsic reaction path approach [39], is defined as the steepest descent path from the transition state (a saddle point on the PE surface) down to the local minima that are equilibrium geometries of reactant and product. It has been shown [40] how to express the Hamiltonian of an N atom molecular system in terms of the reaction coordinate, and this approach has been used successfully to describe a variety of processes in polyatomic reaction dynamics. It is also well known that for many reactions (PT process among them) the MER path is very sharply curved, so that the relevant dynamical motion deviates far from it. This is not a particularly important point for our present purpose of a qualitative characterization of the PE surfaces relevant for the PT reaction, as far as we do not consider the dynamics of... 

Saddle points, just like passes in a mountain area, provide the easiest way to pass from one valley to another. On the PES. saddle points therefore represent transition states providing the lowest energy barriers for conformational interconversions. The goal of conformational analysis (see Conformational Analysis 2 Conformational Analysis 3 and Conformational Anafysis Homotopy) is to find the set of interrelated energy minima and saddle points revealing the network of conformational interconversions (NCI). The NCI describes the dynamics of concerted atomic motion characteristic of the conformational behavior of a molecule. Of course, the interesting conformations are those of sufficiently low free energy to be populated at reasonable temperatures. 

The conformations of a molecule comprise the full set of low-energy minima on the PES derived from an appropriate force field applicable to the molecule at hand. Conformational searching and analysis includes the search for the low-energy minima as well as the low-energy saddle points on the PES, and to identify the conformational interconversions involving low-energy conformational barriers. 

Eig. 6 Two-dimensional PES depicted with lines of equal potential energy. Here, the stable states A and B correspond to minima in the PES and the transition state TS that needs to be crossed during a transition between the stable states is a saddle point on the PES. The thick arrow indicates the direction of the unstable mode and the dashed line is the planar dividing surface orthogonal to this direction. The reactive trajectory shown in red recrosses the dividing surface twice... 

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... 

Extremum points on a JT PES for an orbital doublet will coincide with epikernel configurations. If the distortion space conserves only one type of epikernel, minima and saddle points will be found on opposite sides of the same epikernel distortion. If the distortion space conserves two types of epikernels, minima and saddle points will be characterized by different epikernel symmetries. 

In Figure 1 we depict the PES for rotation of the tp, ip torsion angles of N-acetylalanine N-methylamide. The isoenergy contours shown on a plot of vs. (similar to isobars on a weather map) are depicted in the same color and indicate quite clearly where the minima and saddle points are located. 

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