Saddle index

Another example is the system shown in Fig. 8.2.1 containing a homoclinic loop r to a saddle-focus. If the saddle index... 

The structure of the phase space in a small neighborhood of F depends on the sign of the saddle index p. We will see that the behavior of the trajectories in a neighborhood of T differs essentially in the two cases p > 1 (simple dynamics) and p < 1 (complex dynamics). 

Here p is the saddle index at the saddle-focus Oi, and z/ is the saddle index at the saddle O2 and P = Re A2/7I, where 7 denotes the positive characteristic exponents of O2, and A2 is the non-leading characteristic exponent of O2 nearest to the imaginary axis (Ai is the leading exponent sou = Xi/ and I < u < u recall also that p > 1 by assumption — the saddle values are negative). 

Another codimension-two homoclinic bifurcation in the Shimizu-Morioka model occurs at (a 0.605,6 0.549) on the curve H2 corresponding to the double homoclinic loops. At this point, the separatrix value A vanishes and the loops become twisted, i.e. we run into inclination-flip bifurcation [see Figs. 13.4.8 and C.7.11]. The geometry of the local two-dimensional Poincare map is shown in Fig. 13.4.5 and 13.4.6. To find out what our case corresponds to in terms of the classification in Sec. 13.6, we need also to determine the saddle index 1/ at this point. Again, as in the case of a homoclinic loop to the saddle-focus, it is very crucial to determine whether 1/ < 1/2 or 1/ > 1/2. Simple calculation shows that z/ > 1/2 for the given parameter values. Therefore,... 

TABLE 7. Calculated relative energies at the Cl level (in kcalmol )21S (ATM = absolute true minima, TM = true minima, SP = saddle point, CP2 = critical point of index 2)... 

Let us take a simple example, namely a generic Sn2 reaction mechanism and construct the state functions for the active precursor and successor complexes. To accomplish this task, it is useful to introduce a coordinate set where an interconversion coordinate (%-) can again be defined. This is sketched in Figure 2. The reactant and product channels are labelled as Hc(i) and Hc(j), and the chemical interconversion step can usually be related to a stationary Hamiltonian Hc(ij) whose characterization, at the adiabatic level, corresponds to a saddle point of index one [89, 175]. The stationarity required for the interconversion Hamiltonian Hc(ij) defines a point (geometry) on the configurational space. We assume that the quantum states of the active precursor and successor complexes that have non zero transition matrix elements, if they exist, will be found in the neighborhood of this point. 

The relationship between the geometry of the saddle point of index one (SPi-1) and the accessibility to the quantum transition states cannot be proved, but it can be postulated [43,172], To some extent, invariance of the geometry associated with the SPi-1 would entail an invariance of the quantum states responsible for the interconversion. Thus, if a chemical process follows the same mechanism in different solvents, the invariance of the geometry of the SPi-1 to solvent effects would ensure the mechanistic invariance. This idea has been proposed by us based on computational evidence during the study of some enzyme catalyzed reactions [94, 96, 97, 100-102, 173, 174, 181-184],... 

The existence of critical solvation numbers for a given process to happen is an important concept. Quantum chemical calculations using ancillary solvent molecules usually produce drastic changes on the electronic nature of saddle points of index one (SPi-1) when comparisons are made with those that have been determined in absence of such solvent molecules. Such results can not be used to show the lack of invariance of a given quantum transition structure without further ado. Solvent cluster calculations must be carefully matched with experimental information on such species, they cannot be used to represent solvation effects in condensed phases. 

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. 

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. 

Levine IN (2000) Quantum chemistry, 5th edn. Prentice Flail, Upper Saddle River, NJ cited in index... 

The minimization of the quadratic performance index in Eq. (8-64), subject to the constraints in Eqs. (8-67) to (8-69) and the step response model in Eq. (8-61), can be formulated as a standard QP (quadratic programming) problem. Consequently, efficient QP solution techniques can be employed. When the inequality constraints in Eqs. (8-67) to (8-69) are omitted, the optimization problem has an analytical solution (Camacho and Bordons, Model Predictive Control, 2d ed., Springer-Verlag, New York, 2004 Maciejowski, Predictive Control with Constraints, Prentice-Hall, Upper Saddle River, N.J., 2002). If the quadratic terms in Eq. (8-64) are replaced by linear terms, an LP (linear programming) problem results that can also be solved by using standard methods. This MPC formulation for SISO control problems can easily be extended to MIMO problems. 

Here the index s runs over the relevant saddle points, those that are visited by an appropriate deformation of the real integration contour, which is the real five-dimensional (t,t, k) space, to complex values, and Sp(t,t, k) s denotes the five-dimensional matrix of the second derivatives of the action (4.5) with respect to t,t and fc, evaluated at the saddle points. The time dependence of the form factors (4.6) and (4.7) is considered as slow, unless stated otherwise (see Sect. 4.5 and [27]). 

When P > 0, using Table 5.5 one can immediately point out the type of the inner azeotrope, and when p < 0, only the node (N) and saddle (S) in this table are replaced with the focus (F) and saddle-focus (SF), respectively. The sign of Ind (X ) of this azeotrope is opposite to the sign of a3 and is determined by indexes of the boundary SPs (see Table 5.6) in accordance with the rule of azeotropy (5.18) which in the case of tetrapolymerization yields ... 

Let N denote the number of the degrees of freedom of a system. We also use the term the index of the saddle to indicate the number of negative eigenvalues of the Hessian matrix of the potential function at the saddle. 

Suppose we have a saddle with index 1. Then, a NHIM of 2N — 2 dimension exists above it in the phase space, with two directions that are normal to it. Along these normal directions, with negative and positive Lyapunov exponents, 2N — 1)-dimensional stable and unstable manifolds exist, respectively. The normal directions of the saddle correspond to the degree of freedom that is the reaction coordinate near the saddle, and they describe how the reaction proceeds locally near the NHIM. 

Thus, based on NHIMs with saddles with index 1, we can construct a theory that is a rigorous reformulation of the conventional Transition State Theory [9,10]. Moreover, the use of the Lie perturbation brings the system locally into the Birkhoff normal form with one inverse harmonic potential [2]. This form is nothing but the Fenichel normal form. 

Moreover, the NHIM with a saddle with index 1 can be connected with NHIMs with saddles with indexes larger than 1. To see this possibility, let us count the dimension of the intersections. Suppose we have a saddle with index L. Then, the NHIM of 2N — 2L dimension exists with (2N — L)-dimensional stable and unstable manifolds. In the equi-energy surface, the dimension of the NHIM is 2N — 2L—1, and that of its stable and unstable manifolds is 2N — L — 1. Thus, the dimension of the intersection, if any, between its stable manifold and the unstable manifold of the NHIM with a saddle with index 1 is 2N — L — 2.lf its value is larger than 0, a path exists which connects these two NHIMs. Therefore, the allowed values of L for systems of 3 degrees of freedom (for example) are 1 and 2, when we also take into account the condition that 2N — 2L—1 (i.e., the dimension of the NHIM with a saddle with index L in the equi-energy surface) should not be negative. 

The basic ideas that are necessary for the first program stage are explained in Sections II, III, and IV. In Section II, we formulate the problem of how to analyze a system that has a gap in characteristic time scales. Our method is to use perturbation theory with respect to a parameter that is the ratio between a long time scale and a short time scale, which is a version of singular perturbation theory. The reason will be explained in Section II. In Section III, the concept of NHIMs is introduced in the context of singular perturbation theory. We will give an intuitive description of NHIMs and explain how the description is implemented, leaving the precise formulation of the NHIM concept to the literature in mathematics. In Section IV, we will show how Lie perturbation theory can be used to transform the system into the Fenichel normal form locally near a NHIM with a saddle with index 1. Our explanation is brief, since a detailed exposition has already been published [2]. 

Until now, we have discussed NHIMs in general dynamical systems. In this section, we limit our argument to Hamiltonian systems and show how singular perturbation theory works. In particular, we discuss NHIMs in the context of reaction dynamics. First, we explain how NHIMs appear in conventional reaction theory. Then, we will show that Lie permrbation theory applied to the Hamiltonian near a saddle with index 1 acmally transforms the equation of motion near the saddle to the Fenichel normal form. This normal form can be considered as an extension of the Birkhoff normal form from stable fixed points to saddles with index 1 [2]. Finally, we discuss the transformation near saddles with index larger than 1. 

In the conventional theory, a saddle with index 1 corresponds to a transition state. Near a saddle, the NHIM Mq exists above it in the phase space. The NHIM Mo consists of those orbits with q = 0 and p = 0—that is, the vibrational motions involving qn,Pn) for n = 2,...,N above the saddle. Thus, its... 

Thus, the Lie transformation brings the Hamiltonian locally near a saddle with index 1 into the Fenichel normal form. In addition, we find that, on the NHIM, tori with sufficiently nonresonant frequencies survive. 

---