Saddle point geometry potential energy surfaces

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. 

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 main advantage of MP2/6-31G optimizations over HF/3-21 ( > or HF/ 6-31G ones is not that the geometries are much better, but rather that for a stationary point, MP2 optimizations followed by frequency calculations are more likely to give the correct curvature of the potential energy surface (Chapter 2) for the species than are HF optimizations/frequencies. In other words, the correlated calculation tells us more reliably whether the species is a relative minimum or merely a transition state (or even a higher-order saddle point see Chapter 2). Thus fluorodiazomethane [91] and several oxirenes [53] are (apparently correctly) predicted by MP2 optimizations to be merely transition states, while HF optimizations... 

This transition-state-like point is called a bond critical point. All points at which the first derivatives are zero (caveat above) are critical points, so the nuclei are also critical points. Analogously to the energy/geometry Hessian of a potential energy surface, an electron density function critical point (a relative maximum or minimum or saddle point) can be characterized in terms of its second derivatives by diagonalizing the p/q Hessian([Pg.356]

Previous theoretical calculations [22,35,48-55] have shown that the potential energy surface (PES) of N6 isomers is highly dependent on the level of theory and basis sets. For instance, cyclic N6 with D6h symmetry, isoelectronic with benzene, was calculated to be a minimum at the HF level [49], but to become a higher-order saddle point at the MP2 level on the PES [35,49]. Higher-level calculations indicate that the lowest-energy form of cyclic N6 is not the benzene-like D6h structure but the twisted-boat geometry with D6h symmetry [22,35,50]. This D6h structure is probably not stable at room temperature because the dissociation frequency mode is only 73.6 cm-i at the CCSD(T) level [22],... 

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