Stable stationary point

We now assume that the expansion of the potential is around a stationary point (stable or unstable, depending on the sign of the second-order derivatives), that is, all the first-order derivatives vanish. The energy is measured relative to the value at equilibrium, and we obtain... 

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. 

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]. 

A theoretical study at a HF/3-21G level of stationary structures in view of modeling the kinetic and thermodynamic controls by solvent effects was carried out by Andres and coworkers [294], The reaction mechanism for the addition of azide anion to methyl 2,3-dideaoxy-2,3-epimino-oeL-eiythrofuranoside, methyl 2,3-anhydro-a-L-ciythrofuranoside and methyl 2,3-anhydro-P-L-eiythrofuranoside were investigated. The reaction mechanism presents alternative pathways (with two saddle points of index 1) which act in a kinetically competitive way. The results indicate that the inclusion of solvent effects changes the order of stability of products and saddle points. From the structural point of view, the solvent affects the energy of the saddles but not their geometric parameters. Other stationary points geometries are also stable. 

Cameiro et al. have shown (31) that the C-ethonium ion is the most stable isomer among various stationary-point conformations found on the MP2/6-31G potential energy surface for C2H7+. Further studies added ten other stationary points, by using the same level of theory (24). 

Good agreement between a measured enthalpy of activation and that computed at a particular level of theory, fimiishes evidence that calculations at this level of theory are accurate enough to provide reliable information about the enthalpy differences between the reactant, the TS, and other stationary points on the PES for an Rl. As already noted, differences between the heats of formation of an RI and other energy minima on a PES (i.e., stable molecules and other RIs, which are either formed from an RI or from which an RI is formed) are usually harder to measure experimentally than the activation enthalpy for appearance or disappearance of an Rl. Therefore, being able to compute accurately the enthalpy differences between an Rl and other energy minima on a PES can provide very valuable information that is usually not easy to obtain experimentally. 

Thus, the shape of the band energy difference curves in Fig. 6.16(a) can be understood in terms of the relative behaviour of the densities of states in the middle panel. In particular, from eqn (6.111), the stationary points in the upper curve correspond to band occupancies for which A F vanishes in panel (c). Moreover, whether the stationary point is a local maximum or minimum depends on the relative values of the density of states at the Fermi level through eqn (6.113). Thus, the bcc-fcc energy difference curve has a minimum around N = 1.6 where the bcc density of states is lowest, whereas the hcp-fcc curve has a minimum around N = 1.9, where the hep density of states is lowest. The fee structure is most stable around N = 1, where A F fts 0, and the fee density of states is lowest. 

For the hexacoordinate starting complex (1) derived from the crystal structure of the binary complex, the most stable conformer (la, Fig. 6) was found to feature the imidazole ring in His 290 rotated 180° compared to that of the crystal structure, and this conformation of His290 was found to be preferred for all stationary points leading up to the FeIV=0 intermediate. Unless otherwise noted, all energies in this study are given relative to la. 

Also as already noted above, taking advantage of molecular symmetry can provide very large savings in time. However, structures optimized under the constraints of symmetry should always be checked by computation of force constants to verify tlierr nature as stationary points on die full PES. Additionally, it is typically worthwhile to verify that open-shell wave functions obtained for symmetric molecules are stable with respect to orbital changes that would generate other electronic states. 

Even for the present simple case, for which the inflow does not contain the autocatalyst, we have seen a variety of combinations of stable and unstable stationary states with or without stable and unstable limit cycles. Stable limit cycles offer the possibility of sustained oscillatory behaviour (and because we are in an open system, these can be sustained indefinitely). A useful way of cataloguing the different possible combinations is to represent the different possible qualitative forms for the phase plane . The phase plane for this model is a two-dimensional surface of a plotted against j8. As these concentrations vary in time, they also vary with respect to each other. The projection of this motion onto the a-/ plane then draws out a trajectory . Stationary states are represented as points, to which or from which the trajectories tend. If the system has only one stationary state for a given combination of k2 and Tres, there is only one such stationary point. (For the present model the only unique state is the no conversion solution this would have the coordinates a,s = 1, Pss = 0.) If the values of k2 and tres are such that the system is lying at some point along an isola, there will be three stationary states on the phase... 

Fig. 13.1. (a) The phase line for a one-variable system showing three stationary points—two stable (s) and one unstable (u) (b) representation of the potential associated with points along the phase line showing the stationary points as extrema (c), (d) the disallowed motions on the phase line or potential curve which would correspond to oscillatory behaviour, but also to... 

The solution curves in the (, i )-plane are sketched in fig. 29. Evidently all solutions tend to the stationary point, so that they are globally stable. 

Obviously, if as t —> oo the stationary solution dnj( )/d = 0 exists, indeed the asymptotic solution rij(oo) of (2.1.1) is one of the solutions n(- of the set (2.1.14). Here we have an example of a simple but very important case of a stable stationary solution. Other stationary points cannot be ascribed to the asymptotic solutions, i.e., n nj(oo), but they are also important for the qualitative treatment of the set of equations. Note that striving of the solutions for stationary values is not the only way of chemical system behaviour as —> oo another example is concentration oscillations [4, 7, 16]. Their appearance in a set (2.1.2) depends essentially on a nature of... 

If the real parts of all eigenvalues e, Ree, <0 are negative, according to the Lyapunov theorem [14, 15] the stationary point is asymptotically stable... 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

Fig. 2. Types of stationary points on the plane, (a), (c), (e) Stable nodes (b), (d), (f) unstable nodes (g) saddle point (h) stable focus (i) unstable focus, (k) whirl.

Chemistry is essentially the study of the stationary points on potential energy surfaces in studying more or less stable molecules we focus on minima, and in investigating chemical reactions we study the passage of a molecule from a... 

A saddle point is a stationary point on the multidimensional potential energy surface. It is a stable point in all dimensions except one, where the second-order derivative of the potential is negative (see Appendix E). The classical energy threshold Eci or barrier height of the reaction corresponds to the electronic energy at the saddle point relative to the electronic energy of the reactants. 

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