Internal modes collective coordinates

On the basis of the above physical situation presupposed, the entire state space is now represented in three independent modes (1) a mode of the reaction coordinate R, which is eventually led to translational or dissociative state (2) an internal state, that is, a vibrational mode under focus (denoted by A), which is to be observed as a linear surprisal and (3) all other remaining internal modes (collectively denoted as B). A critical region (denoted by SR) is supposed to exist somewhere on the R coordinate beyond the transition state. At SR the modes A and B are practically separated and uniquely identified. For instance, the path Hamiltonian at... 

The model offers insights on several key points regarding the role of the solvent. But in the final analysis it cannot substitute for realistic dynamical simulations. In particular, the model overlooks any role played the internal modes of the solvent molecules or the internal modes of the solute. What the model provides is insight into the motion along the solvent-modified reaction coordinate. Specifically, the diffusion forward and backward across the barrier, that in the Kramers modef occurs along the reaction coordinate of the isolated solute, is shown in the model to be equivalent to a single crossing of the transition state provided that we use a collective reaction coordinate, one that involves both solute and solvent motion. 

Figure 20. NEXAFS spectra of N in sediments collected from different locations and their isolated humic substances (a), and humic substances isolated by the International Humic Substance Society (b). The spectra are collected in electron yield mode and are sensitive to surface coordination (especially for the sediment samples). [Used with permission from Vairavamurthy and Wang 2002, American Chemical Society],...

The decomposition of coupled harmonic oscillators into a collection of independent oscillators is known as a normal mode expansion and the independent oscillators are called normal modes. Normal modes are defined as modes of vibration where the respective atomic motions of the atoms are in harmony , i.e., they all reach their maximum and minimum displacements at the same time. These normal modes can be expressed in terms of bond stretches and angle deformation (termed internal coordinates) and can be calculated by using a procedure called normal coordinate analysis. 

It is remarked that the use of internal coordinate models is also of interest for normal mode calculations since a reduced number of variables simplifies the eigenvalue problem and also eliminates the high frequency movements associated with bond stretching, which are only weakly coupled to the low frequency collective modes. It has also been demonstrated that the harmonicity of the energy hypersurface can be assumed over a wider range in dihedral angle space than in Cartesian space. This approach, however, requires a special treatment to exclude any motion of the center of mass of the system. ... 

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