Transition state theory manifolds

Periodic boundary conditions, Monte Carlo heat flow simulation, nonequilibrium molecular dynamics, 79—81 Periodic-orbit dividing surface (PODS) geometric transition state theory, 196-201 transition state trajectory, 202-213 Perturbation theory, transition state trajectory, deterministically moving manifolds, 224-228... 

As illustrated in the previous section, the kinetics associated with an ET process may be complex when diffusion or relaxation processes create dynamic bottlenecks. In limiting cases, however, a simple model based on transition state theory (TST) suffices. According to TST, the system maintains thermal equilibrium between different positions along the reaction coordinate [87]. We consider the TST rate constant for electron transfer after some preliminary comments about state manifolds and energetics. 

Many but not all of the quantized transition states observed in the densities of state-selected reaction probability are observed as peaks in the total density of reactive states. Some highly bend excited states (e.g., [0 12°], and [0 14°]) are observed as peaks only in the state-selected dynamics. If the closely spaced features in the stretch-excited manifolds for p(i are indicative of supernumerary transition states more closely spaced in energy than the variational transition states (which adiabatic transition state theory also suggests), then only some of the supernumerary transition states, in particular S[20°], 5[22°], 5[24°], 5[30°], 5[32°], 5[34°], and 5[36°], are observed in the total density, i.e., only some are of the first kind. The other supernumerary transition states identified in the state-selected dynamics are of the second kind. 

Scalar equations, transition state trajectory deterministically moving manifolds, 224—228 stochastically moving manifolds, 214—222 Scattering theory, two-pathway excitation,... 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

This very simple Hamiltonian is at the basis of the whole TS approach. It generalizes easily into many dimension (Section IV), is a good basis for perturbation theory [4], and is also the basis for numerical schemes, classical and semiclassical. The inclusion of angular momentum implies that some ingredients must be added (see Section V). Let us thus describe how this very simple, linear Hamiltonian supports normally hyperbolic invariant manifolds (NHIMs see Section IV for a proper discussion) separatrices and a transition state. 

In statistical reaction rate theory, the concept of transition state plays a key role. Transition states are supposed to be the boundaries between reactants and products. However, the precise formulation of the transition state as a dividing surface is only possible when we consider transition states in phase space. This is the place where the concepts of normally hyperbolic invariant manifolds (NHIMs) and their stable and unstable manifolds come into play. 

As we have discussed in Section 2 in classical mechanics the transition state is represented by a lower dimensional invariant subsystem, the center manifold. In the quantum world, due to Heisenberg s uncertainty principle, we cannot localize quantum states entirely on the center manifold, so fhere cannot be any invariant quantum subsystem representing the transition states. Instead we expect a finite lifetime for fhe fransition state. The lifetime of fhe fransifion sfate is determined by the Gamov-Siegert resonances, whose importance in the theory of reacfion rafes has been emphasized in fhe liferafure [61, 62]. 

Oped a so-called reactive island theory the reactive islands are the phase-space areas surrounded by the periodic orbits in the transition state, and reactions are interpreted as occurring along cylindrical invariant manifolds through the islands. Fair et al. [29] also found in their two- and three-dof models of the dissociation reaction of hydrazoic acid that a similar cylinderlike structure emerges in the phase space as it leaves the transition state. However, these are crucially based on the findings and the existence of (pure) periodic orbits for all the dof, at least in the transition states. Hence, some questions remain unresolved, for example, How can one extract these periodic orbits from many-body dof phase space and How can the periodic orbits persist at high energies above the saddle point, where chaos may wipe out any of them ... 

In addition to the transition state, the theory on the dynamics in the saddle region has elucidated other important objects in the phase space. An invariant manifold is a set of points in the phase space such that any point in that set will remain in that set perpetually during the time evolution. As a consequence, if an invariant manifold divides the whole phase space into two disjoint regions, the system can never cross the invariant manifold from one region to the other. 

The theoretical method employed is based on and largely similar to the theory of electron-transfer reactions in solution [123,124,125]. Thus the intramolecular spin conversion may be described as a transition between an initial manifold of states [f((r, qc)ZK,( c)[Pg.94]

The presence of the electron acceptor site adjacent to the donor site creates an electronic perturbation. Application of time dependent perturbation theory to the system in Figure 1 gives a general result for the transition rate between the states D,A and D+,A. The rate constant is the product of three terms 1) 27rv2/fi where V is the electronic resonance energy arising from the perturbation. 2) The vibrational overlap term. 3) The density of states in the product vibrational energy manifold. 

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