Hamiltonian classical path

Tully has discussed how the classical-path method, used originally for gas-phase collisions, can be applied to the study of atom-surface collisions. It is assumed that the motion of the atomic nucleus is associated with an effective potential energy surface and can be treated classically, thus leading to a classical trajectory R(t). The total Hamiltonian for the system can then be reduced to one for electronic motion only, associated with an electronic Hamiltonian Jf(R) = Jf t) which, as indicated, depends parametrically on the nuclear position and through that on time. Therefore, the problem becomes one of solving a time-dependent Schrodinger equation ... 

As long as no approximation is introduced, it is clear that the equations of motion are equivalent in the diabatic and adiabatic representations. This is no longer true, however, once the classical-path approximation is employed the resulting classical-path equations of motion in the adiabatic representation are not equivalent to the diabatic equations of motion. Depending on whether the approximation is employed in the diabatic or in the adiabatic representation, the resulting classical-path Hamiltonian contains identical first-order nonadiabatic couphngs but different... 

In the case of scattering, without electromagnetic fields, the dependence with time of the Hamiltonian comes from the classical path followed by M point-like nuclei. In such case, the external potential t) can be written ... 

The definition of a reduced dimensionality reaction path starts with the full Cartesian coordinate representation of the classical A-particle molecular Hamiltonian,... 

In a study of the rate of isomerization of HCN to CNH, Rice and co-workers [19] suggested exploiting a reaction path Hamiltonian as a device to permit extension of classical statistical reaction rate theory from few-dimensional to many-dimensional systems. In that approach the dynamics of the reacting molecule is reduced to that of a system with a complicated but one-dimensional reactive DOF coupled with other effective DOFs. Although their calculations based on this approach yield an accurate description of the isomerization rate as... 

Additionally, non-Hamiltonian d5mamics can be used in applications/ methodologies such as Path-integral MD, replica-exchange methods, variable transformation techniques, free energy dynamics methods, and other new applications. Generating these alternative statistical ensembles from simulation requires the use of extended systems or extended phase space [9]. In these systems, the simulations do not only include the N coordinate and momentum vectors that are needed to describe a classical Ai-particle system, but they also include a set of additional control or extended variables that are used to drive the fluctuations required by the ensemble of interest. 

Clearly, there are many ways in which the ideas we have proposed must be extended. Among the more important extensions we cite variational optimization of the shape, duration, and separation of the pulses used to generate the selectivity of reactivity, and analysis of the changes induced by the inclusion of all degrees of freedom of the molecule (say in the sense of a reaction path Hamiltonian, or a dynamical path Hamiltonian). For studies involving more degrees of freedom, a swarm of classical trajectories should be a very useful tool. 

One must distinguish between this situation and the statistical theories described under the general heading of statistical mechanics in classical statistical mechanics, the laws which govern the evolution of the system are extremely well known, but the path is so complex that it cannot be followed in detail. Here, the number of states is large, but finite, and the randomness arises from an ensemble of possible Hamiltonians. 

In a pivotal development. Miller, Handy and Adams [12] derived the classical Hamiltonian for a simple potential based on the MEP. The idea of the reaction path Hamiltonian is, conceptually, to consider the potential as a trough or as a stream bed along with 3N-7 harmonic walls that are free to close in or widen out as one proceeds along the trough. The potential energy surface is approximated as the potential energy of the MEP Vo(s) plus a quadratic approximation to the energy in directions perpendicular to the MEP,... 

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