Scaled Hamiltonians separate scalings

To discuss the separation of time scales, we begin with the argument that a system that reaches the continuum via a narrow bottleneck can exhibit more than one time scale [45a,b,f, 51]. Particular attention will be given to the question of when this will be the case. The argument begins by considering the time evolution in the bound subspace. As is well known [52,53, 54], one can confine attention to the bound levels by the introduction of an effective Hamiltonian H in which the coupling to the continuum is accounted foT by a rate operator T ... 

In the previous section, we have shown that switching the picture from the nearly integrable Hamiltonian to the Hamiltonian with internal structures may make it possible to solve several controversial issues listed in Section IV. In this section we shall examine the validity of an alternative scenario by reconsidering the analyses done in MD simulations of liquid water. As mentioned in Section III, since a water molecule is modeled by a rigid rotor, and has both translational and rotational degrees of freedom. So, the equation of motion involves the Euler equation for the rigid body, coupled with ordinary Hamiltonian equations describing the translational motions. The precise Hamiltonian is therefore different from that of the Hamiltonian in Eq. (1), but they are common in that the systems have internal structures, and the separation of the time scale between subsystems appears if system parameters are appropriately set. 

The conformers give spectra as if they were separate molecules. Molecular motions that are fast on the NMR time scale but slow relative to molecular tumbling are considered to be between conformers. The observed NMR spectra correspond to a spin Hamiltonian that is a weighted mean of those of the individual conformers. In general, each such con-former will be characterized by its own structure, molecule-fixed coordinate system, and motional constants. 

This separation-of-time-scales assumption allows the electrons to be described by electronic wavefunctions that smoothly ride the molecule s atomic framework. These electronic functions are found by solving a Schrodinger equation whose Hamiltonian contains the kinetic energy of the electrons, the Coulomb... 

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