Potential-energy surfaces saddle-point structure

In the phase-space treatment the situation is very similar. However, rather than study the morphology of the potential energy surface, we must focus on the total energy surface. The geometry of this surface, which is defined on phase space instead of coordinate space, can also be characterized by its stationary points and their stability. In this treatment, the rank-one saddles play a fundamental result. They are, in essence, the traffic barriers in phase space. For example, if two states approach such a point and one passes on one side and the other passes on the other side, then one will be reactive and the other nonreactive. Once the stationary points are identified, then the boundaries between the reactive and nonreactive states can be constructed and the dynamical structure of phase space has been determined. As in the case of potential energy surfaces, saddles with rank greater than one occur, especially in systems with high symmetry between outcomes, as in the dissociation of ozone. 

Quantum mechanical calculations (by ab initio method) "" performed on diprotonated methane showed that the symmetrical structure (18) with two stabilizing orthogonal 3c-2e interactions is the only minimum on the potential energy surface. The other structures (19-22), in turn, were calculated to be saddle points on the potential energy surface. These were found to be... 

As mentioned earlier, a potential energy surface may contain saddle points , that is, stationary points where there are one or more directions in which the energy is at a maximum. Asaddle point with one negative eigenvalue corresponds to a transition structure for a chemical reaction of changing isomeric form. Transition structures also exist for reactions involving separated species, for example, in a bimolecular reaction... 

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. 

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. 

There are numerous algorithms of different kinds and quality in routine use for the fast and reliable localization of minima and saddle points on potential energy surfaces (see 47) and refs, therein). Theoretical data about structure and properties of transition states are most interesting due to a lack of experimental facts about activated complexes, whereas there is an abundance of information about educts and products of a reaction. 

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 planar form of phosphole is a first-order saddle point on the potential energy surface, 16—24 kcal/ mol above the minimum (at different levels of the theory). ° (The calculated barriers are the highest at the HF level, which underestimates aromatic stabilization of the planar saddle point, while the MP2 results are at the low end.) It has been demonstrated by calculation of the NMR properties, structural parameters, ° and geometric aromaticity indices as the Bird index ° and the BDSHRT, ° as well as the stabilization energies (with planarized phosphorus in the reference structures) ° and NIGS values ° that the planar form of phosphole has an even larger aromaticity than pyrrole or thiophene. 

Not shown to the quasi-diabatic potential energy surfaces in Fig. (3) there is a adiabatic potential energy surface. This is distinguished by the maximum at the crossing point nil. The system has a saddle-point structure. In the regions about the cis and trans attractors there is no difference between them. Between 2%1 i... 

The strategies for saddle point optimizations are different for electronic wave functions and for potential energy surfaces. First, in electronic structure calculations we are interested in saddle points of any order (although the first-order saddle points are the most important) whereas in surface studies we are interested in first-order saddle points only since these represent transition states. Second, the number of variables in electronic structure calculations is usually very large so that it is impossible to diagonalize the Hessian explicitly. In contrast, in surface studies the number of variables is usually quite small and we may easily trans-... 

Traditional Sequence of Differently Bonded Intermediates. Organic chemists have traditionally considered a reaction mechanism, in its most primitive form, to consist of a sequence of differently bonded intermediates on the path between starting materials and products. In these terms, a mechanism may be considered understood once these chemically distinct species have been correctly identified. For purposes of understanding reaction rates and stereochemistry, it is necessary to expand this set of metastable reaction intermediates to include transition structures at the saddle points between intermediates on a potential energy surface. For photochemistry one must also consider transitions between potential energy surfaces. 

As mentioned in Section 10.2, saddle points on the potential energy surface frequently correspond to the transition states that constitute bottlenecks to reaction. Finding these saddle points can provide a remarkable level of information about the mechanism. Such information about TS structure is not readily available in direct form from experiment. Calculation is then highly complementary with experiment and can be used to confirm a predicted mechanism, cast insight into observed substituent effects, and so on. 

---