Saddle-point geometry, potential energy

Both minima and saddle points are of interest. In the case of wave functions, the ground state is a minimum and the excited states are saddle points of the electronic energy function.6 On potential surfaces minima and first-order saddle points correspond to equilibrium geometries and transition states. Higher-order saddle points on potential energy surfaces are of no interest. 

Saddle points in potential energy surfaces of HlHj. Reproduced from Das and Balasubramanian (1991d). Geometries of the saddle points were estimated at the CASSCF level of calculation. 

Figure 2.3 Contour plot of the LSTH potential energy surface at fixed internuclear angle 90 , cutoff at 2.0 eV. Also shown are the contours of the difference potential Ylsth — Ydmbe) 3.t 90 where dotdash = 0.1, dash = 0.3, and dot = 0.5 kcalfmol. The two surfaces differ by 0.51 kcal/mol (LSTH is higher) at this fixed angle saddle point geometry.

Once one has the gradient of the energy with respect to the nuclear coordinates, one can use it to efficiently determine various characteristics of a potential energy surface, such as equilibrium and saddle point geometries and force constants. 

Using the coordinates of special geometries, minima, and saddle points, together with the nearby values of potential energy, you can calculate spectroscopic properties and macroscopic therm ody-riatriic and kinetic parameters, sncfi as enthalpies, entropies, and thermal rate constants. HyperChem can provide the geometries and energy values for many of these ealeulatiori s. 

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. 

Another use of frequency calculations is to determine the nature of a stationary point found by a geometry optimization. As we ve noted, geometry optimizations converge to a structure on the potential energy surface where the forces on the system are essentially zero. The final structure may correspond to a minimum on the potential energy surface, or it may represent a saddle point, which is a minimum with respect to some directions on the surface and a maximum in one or more others. First order saddle points—which are a maximum in exactly one direction and a minimum in all other orthogonal directions—correspond to transition state structures linking two minima. 

An IRC calculation examines the reaction path leading down from a transition structure on a potential energy surface. Such a calculation starts at the saddle point and follows the path in both directions from the transition state, optimizing the geometry of the molecular system at each point along the path. In this way, an IRC calculation definitively connects two minima on the potential energy surface by a path which passes through the transition state between them. 

Secondly, it is usual to calculate only a few points which are assumed to be characteristic with full optimization of geometry instead of the complete potential energy surface 48). For a pure thermodynamical view it is enough to know the minima of the educts and products, but kinetic assertions require the knowledge of the educts and the activated complex as a saddle point at the potential energy surface (see also part 3.1). 

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

The point of minimal potential energy in the trajectory of reactants to products in a chemical reaction. A reaction s saddle point (or coF) indicates the geometry and energy of reactants as they approach and pass the transition state of a reaction. 

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