Saddle point analysis

CPR can be used to find continuous paths for complex transitions that might have hundreds of saddle points and need to be described by thousands of path points. Examples of such transitions include the quaternary transition between the R and T states of hemoglobin [57] and the reorganization of the retinoic acid receptor upon substrate entry [58]. Because CPR yields the exact saddle points as part of the path, it can also be used in conjunction with nonnal mode analysis to estimate the vibrational entropy of activation... 

In realistic systems, the separation of the modes according to their frequencies and subsequent reduction to one dimension is often impossible with the above-described methods. In this case an accurate multidimensional analysis is needed. Another case in which a multidimensional study is required and which obviously cannot be accounted for within the dissipative tunneling model is that of complex PES with several saddle points and therefore with several MEPs and tunneling paths. 

Saddle point. 170 Salt effects. 206-214 Scavenging (see Reactions, trapping) Second-order kinetics. 18-22, 24 in one component, 18-19 in two components (mixed), 19-22 Selectivity. 112 Sensitivity analysis. 118 Sensitivity factor, 239-240 Sequential reactions (see Consecutive reactions)... 

At R > 400 pm the orientation of the reactants looses its importance and the energy level of the educts is calculated (ethene + nonclassical ethyl cation). For smaller values of R and a the potential energy increases rapidly. At R = 278 pm and a = 68° one finds a saddle point of the potential energy surface lying on the central barrier, which can be connected with the activated complex of the reaction (21). This connection can be derived from a vibration analysis which has already been discussed in part 2.3.3. With the assistance of the above, the movement of atoms during so-called imaginary vibrations can be calculated. It has been attempted in Fig. 14 to clarify the movement of the atoms during this vibration (the size of the components of the movement vector... 

Here, pb is the bond critical point (saddle point in three dimensions, a minimum on the path of the maximum electron density). In Eq. (44), and A.2 are the principal curvatures perpendicular to the bond path. The parameters A and B in Eq. (45) determined using various basis sets are given in Bader et al. [83JA(105)5061]. Convenient parameters in the quantitative analysis of a conjugation effect are the relative 7r-character tj (in %) of the CC formal double or single bonds determined with reference to the bond of ethylene (90MI2) ... 

Another interesting limit is the quasistatic limit r 0. Based on the numerical solution of the saddle point equations (160)-(162), it was suggested in Ref. 117 that T q) converged to a constant value over a finite range of work values. Figure 15a shows the results obtained for the heat distributions, whereas the path temperature is shown in Fig. 15b. A more detailed analysis [134] has shown that a plateau is never fully reached for a finite interval of heat values when r 0. The presence of a plateau has been interpreted as the occurrence of a first-order phase transition in the path entropy s q) [134]. 

By performing a normal mode analysis in the initial state and in the saddle point, it is then possible to obtain the harmonic expansion of the potential in the reactant region ... 

A. H. Zewail With regard to Prof. Marcus s comment, we have observed the coherence-in-products first in the IHgl system where the wavepacket is launched near the saddle point. The persistence of coherence in products is fundamentally due to (1) the initial coherent preparation (no random trajectories) and (2) the nature of the potential transverse to the reaction coordinate (no dispersion). The issue of vibrational adiabaticity in the course of the reaction, as you pointed out, must await complete final-state analysis for well-defined initial energy. However, we do know that for a given energy of the initial wavepacket a broad distribution of vibrational coherence (in the diatom) is observed. 

Analysis of the electron density distribution p (r) of numerous molecules has revealed that there exists a one-to-one relation between MED paths, saddle points p and interatomic surfaces on the one side and chemical bonds on the other27,81,82. However, low-density MED paths can also be found in the case of non-bonding interactions between two molecules in a van der Waals complex82. To distinguish covalent bonding fron non-bonded or van der Waals interactions, Cremer and Kraka have given two conditions for the existence of a covalent bond between two atoms A and B8. 

In realistic systems the separation of the modes by their frequencies and subsequent reduction to one dimension with the methods described above is often not possible. In this case an accurate multidimensional analysis is needed. Another case in which a multidimensional study is required and which obviously cannot be accounted for within the dissipative tunneling model is that of a complex PES with several saddle points and therefore several MEPs and tunneling paths. Whereas the goal of the previous models is to carry out analytical calculations and gain insight into the physical picture, the multidimensional calculations are expected to give a quantitative description of concrete chemical systems. However, at present we are just at the beginning of this process, and only a few examples of numerical multidimensional computations, mostly on rather idealized PES s, have been performed so far. Nonetheless, these... 

The potential (6.37) corresponds with the previously discussed projection of the three-dimensional PES V(p,p2,p3) onto the proton coordinate plane (pi,p3), shown in Figure 6.20b. As pointed out by Miller [1983], the bifurcation of reaction path and resulting existence of more than one transition state is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of the PES (6.37) was carried out by Benderskii et al. [1991b], The existence of the onedimensional optimum trajectory with q = 0, corresponding to the concerted transfer, is evident. On the other hand, it is clear that in the classical regime, T > Tcl (Tc] is the crossover temperature for stepwise transfer), the transition should be stepwise and occur through one of the saddle points. Therefore, there may exist another characteristic temperature, Tc2, above which there exists two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points at T = Tcl. The existence of the second crossover temperature Tc2 for two-proton transfer was noted by Dakhnovskii and Semenov [1989]. 

A fourth type of approach relies on analyzing the overall electron density, rather than the density contributed by individual orbitals. A particularly popular analysis of this type is the atoms in molecules analysis (AIM).6 Variation of the density through space can be shown to map onto the bonding pattern. The presence of a bond between two atoms is revealed by the presence of a saddle point, or bond critical point, in the density near the bond midpoint. 

The reaction coordinate is found by a normal-mode analysis at the saddle point and is therefore separable from the other degrees of freedom in the activated complex the motion in this coordinate is treated as that of a free particle. Then, according to Eq. (A. 13), at sufficiently high temperatures, we have... 

First, as before, the potential is expanded to second order in the atomic displacements around the saddle point. From a normal-mode analysis, it follows that, in the vicinity of the saddle point, motion in the reaction coordinate is decoupled from the other degrees of freedom of the activated complex. Furthermore, it is assumed that the motion in the reaction coordinate (r.c.), in this region of the potential energy surface, can be described as classical free (translational) motion. Thus, the Hamiltonian takes the form... 

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