Saddle point, quadratic

Figure 10. Divisions of the Em map for the quadratic Hamiltonian in Eq. (39). Va, Vb, and Vx are the energies of the two isomers and that of the saddle point between them. The solid lines are relative equilibria, and the cusp points, a , indicate points at which the secondary minimum and the saddle point in Fig. 9 merge at a point of inflection.

Alternatively one can make use of No Barrier Theory (NBT), which allows calculation of the free energy of activation for such reactions with no need for an empirical intrinsic barrier. This approach treats a real chemical reaction as a result of several simple processes for each of which the energy would be a quadratic function of a suitable reaction coordinate. This allows interpolation of the reaction hypersurface a search for the lowest saddle point gives the free energy of activation. This method has been applied to enolate formation, ketene hydration, carbonyl hydration, decarboxylation, and the addition of water to carbocations. ... 

Geometry of a quadratic objective function of two independent variables—saddle point. 

As an effect of the linear and quadratic vibronic integrals the adiabatic potential surface stays no longer paraboloid-shaped. It exhibits an additional warping with several local minima and saddle points out of the reference high-symmetry configuration Q0. 

First, we present the dynamics of the initial wavepacket a. Initially the system stands at the equilibrium position of the electronic ground X. The temporal evolution of the wavepacket Pe generated in the electronic excited state is shown in the left-hand column of Fig. 5.9. Apparently, tp originates in the Frank-Condon (FC) region, which is located at the steep inner wall of the electronically excited A state. The repulsive force of the potential l 0 the drives e(t) downhill toward the saddle point and then up the potential ridge, where Pe(t) bifurcates into two asymptotic valleys, with Ye = 0.495 in channel f. The excitation achieved using this simple quadratically chirped pulse is not naturally bond-selective because of the symmetry of the system. The role played by our quadratically chirped pulse is similar to that of the ordinary photodissociation process, except that it can cause near-complete excitation (see Table 5.1 for the efficiency). This is not very exciting, however, because we would like to break the bond selectively. 

In the coordinate-space treatment of TST, certain assumptions must be made concerning the nature of the Hamiltonian of the system. First, it must be assumed that it can be partitioned into the sum of two terms, the kinetic and the potential energy. Furthermore, one must also assume that the kinetic energy is positive definite and is quadratic in the momenta. With these assumptions, then the point of stationary flow in phase space and the saddle point of the potential energy... 

For the parabolic barrier arising from the quadratic expansion about the saddle point, the barrier penetration integral is given by... 

The saddle point in the More O Ferrall-Jencks diagram has a shape which approximates a quadratic equation with a minimum in a direction perpendicular to the reaction coordinate (Figure A3b) and a quadratic equation with a maximum along the reaction coordinate (Figure A3c). In the illustrations the potential energies for movement in parallel and perpendicular directions are given by parallel = -OJx +(/ /10 + 3)x... 

We have introduced the notation qs to signify the value of the coordinate q at the saddle point, and will similarly find it convenient to introduce q to denote the coordinate for the well under consideration. As yet, our statements are general and involve no approximations other than those present in transition state theory itself A standard approximation at this point is to note that in the vicinity of the well it is appropriate to represent the energy surface as a quadratic function in the variable q, and in addition it is asserted that one may make the transcription... 

The most common assumption is one of a reaction path in hyperspace (Miller et al. 1980). A saddle point on the PES is found and the steepest descent path (in mass-weighted coordinates) from this saddle point to reactants and products is defined as the reaction path. The information needed, except for the path and the energies along it, is the local quadratic PES for motion perpendicular to the path. The reaction-path Hamiltonian is only a weakly local method since it can be viewed as an approximation to the full PES and since it is possible to use any of the previously defined global-dynamical methods with this potential. However, it is local because the approximate PES restricts motion to lie around the reaction path. The utility of a reaction-path formalism involves convenient approximations to the dynamics which can be made with the formalism as a starting point. 

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