Coordinant models, fixed

We divide the airshed models discussed here into two basic categories, moving cell models and fixed coordinate models. In the moving cell approach a hypothetical column of air, which may or may not be well mixed vertically, is followed through the airshed as it is advected by the wind. Pollutants are injected into the column at its base, and chemical reactions may take place within the column. In the fixed coordinate approach the airshed is divided into a three-dimensional grid. 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

When the IRP is traced, successive points are obtained following the energy gradient. Because there is no external force or torque, the path is irrotational and leaves the center of mass fixed. Sets of points coming from separate geometry optimizations (as in the case of the DC model) introduce the additional problem of their relative orientation. In fact, the distance in MW coordinates between adjacent points is altered by the rotation or translation of their respeetive referenee axes. The problem of translation has the trivial solution of centering the referenee axes at the eenter of mass of the system. On the other hand for non planar systems, the problem of rotations does not have an analytical solution and must be solved by numeiieal minimization of the distanee between sueeessive points as a funetion of the Euler angles of the system [16,24]. 

Below we will use Eq. (16), which, in certain models in the Born-Oppenheimer approximation, enables us to take into account both the dependence of the proton tunneling between fixed vibrational states on the coordinates of other nuclei and the contribution to the transition probability arising from the excited vibrational states of the proton. Taking into account that the proton is the easiest nucleus and that proton transfer reactions occur often between heavy donor and acceptor molecules we will not consider here the effects of the inertia, nonadiabaticity, and mixing of the normal coordinates. These effects will be considered in Section V in the discussion of the processes of the transfer of heavier atoms. 

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