Saddle points quadratic region

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. 

For an arbitrary potential function V(x,Q), a more general dependence E (Q) can be derived by approximating V(x,Q) in the vicinity of its maximum by a parabolic function, which is usually possible in the saddle-point region of a potential energy surface. For this purpose we conveniently write this quadratic function in the form /37a,o/... 

Our objective in this chapter is to survey the theory and practice of the computation of transition state structure. We take the word structure in this context to encompass the geometry of the saddle point on the surface, a local quadratic approximation to the surface at that point (the second derivative matrix), and the nature of the reaction path in the region of the... 

We now turn to methods for first-order saddle points. As already noted, saddle points present no problems in the local region provided the exact Hessian is calculated at each step. The problem with saddle point optimizations is that in the global region of the search, there are no simple criteria that allow us to select the step unambiguously. Thus, whereas for minimization methods it is often possible to give a proof of convergence with no significant restrictions on the function to be minimized, no such proofs are known for saddle-point methods, except, of course, for quadratic surfaces. Nevertheless, over the years several useful techniques have been developed for the determination of saddle points. We here discuss some of these techniques with no pretence at completeness. 

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