Stationary-point structures

Fig. 14 Stationary-point structures for the epoxidation of ethene with hydrogen peroxide in the absence and in the presence of one molecule of HFIP, optimized at RB3LYP/6-31+ G(d,p) (selected bond lengths in A RB3LYP/6-311++G(d,p) results in parentheses)...

In the second family of approaches, explicit solvent molecules are placed around the gas phase stationary point structures. In some cases, the gas phase geometries are held constant and only the geometries and/or positions of the surrounding solvent molecules are optimized, and in other cases, the structure of the whole system (often called a supermolecule 32) is optimized. The supermolecule approach generally only involves explicit solvent molecules from the first (and occasionally second) solvation shell of the solute. 

In the analysis of autonomous non-gradient systems the methods of gradient system theory have proved useful. The notions such as a stationary (critical) point, degenerate stationary point, structural stability (instability), morsification, phase portrait can be directly transferred to autonomous systems. A qualitative description of dynamical autonomous systems is constructed analogously with the description of gradient systems. 

Fig. 4.7 Calculated geometries of the three NFnHs-n molecules that were found to be stationary points ( structurally stable ) at the MP2/6-31++G level [26]...

Example I hc reaction coordinate for rotation about the central carbon-carbon bond in rt-bulane has several stationary points.. A, C, H, and G are m in im a and H, D, an d F arc tn axirn a. Only the structures at the m in im a represen t stable species an d of these, the art/[ con form ation is more stable th an ihc nauchc. 

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... 

The optimization facility can be used to locate transition structures as well as ground states structures since both correspond to stationary points on the potential energy-surface. However, finding a desired transition structure directly by specifying u reasonable guess for its geometry can be chaUenging in many cases. 

Because of the nature of the computations involved, firequency calculations are valid only at stationary points on the potential energy surface. Thus, frequency calculations must be performed on optimized structures. For this reason, it is necessary to run a geometry optimization prior to doing a frequency calculation. The most convenient way of ensuring this is to include both Opt and Freq in the route section of the job, which requests a geometry optimization followed immediately by a firequency calculation. Alternatively, you can give an optimized geometry as the molecule specification section for a stand-alone frequency job. 

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. 

We have already considered two reactions on the H2CO potential energy surface. In doing so, we studied five stationary points three minima—formaldehyde, trans hydroxycarbene, and carbon monoxide plus hydrogen molecule—and the two transition structures connecting formaldehyde with the two sets of products. One obvious remaining step is to find a path between the two sets of products. 

Verifying that the stationary point is a transition structure, and computing its zero-point energy. 

Chapter 3, Geometry Optimizations, describes how to locate equilibrium structures of molecules, or, more technically, stationary points on the potential energy surface. It includes an overview of the various commonly used optimization techniques and a consideration of optimizing transition strucmres as well as minimizations. 

Iti Chapter 1, we dealt at length with molecular mechanics. MM is a classical model where atoms are treated as composite but interacting particles. In the MM model, we assume a simple mutual potential energy for the particles making up a molecular system, and then look for stationary points on the potential energy surface. Minima correspond to equilibrium structures. 

There have a number of computational studies of hypothetical RMMR species [10-13, 40, 411. The simplest compounds are the hydrides HMMH. Some calculated structural parameters and energies of the linear and trans-bent metal-metal bonded forms of the hydrides are given in Table 1. It can be seen that in each case the frans-bent structure is lower in energy than the linear configuration. However, these structures represent stationary points on the potential energy surface, and are not the most stable forms. There also exist mono-bridged, vinylidene or doubly bridged isomers as shown in Fig. 2... 

As expected, the hexagonal chair form of Se with 03a symmetry, occurring in the solid hexasulfur, is the most stable form of hexasulfur, due to its minimal strain. The boat conformer of C2V symmetry is 50 kj mol less stable than the chair form [54]. The Dsa—>C2v interconversion requires to overcome a barrier of ca. 125 kJ mol A structure of C2 symmetry, which is a local minimum at the HF/3-21G level [49, 50], is not a stationary point at higher levels of theory [54, 55]. 

UB3LYP theory predicts four minima of Sg which possess Cj, Cjhy O2 d C2 symmetries. At the UMP2 level of theory, no stationary point corresponding to the C2 minimum can be located and two new local minima with and D2 symmetries appear. The Sg conformers are found to be very prone to pseudorotation and are predicted to interconvert readily. For this reason, Cioslowski et al. refer to Sg as a fluxional species [93]. Interestingly, they found that the structures corresponding to local minima are not directly interconvertible. 

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