Divergence coordinates

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

The divergence of a vector field u, written div u, is given in Cartesian coordinates by... 

Several books have recently been published on the subject of CVD, each dealing with a specific aspect of the technology, such as CVD for microelectronics or metallo-organic CVD. With many of his colleagues, the author has felt the need for a general, systematic, objective, and balanced review solely devoted to CVD, which would cover all its scientific, engineering, and applications aspects, coordinate the divergent trends found today in the CVD business, promote interaction and sharpen the focus of research and development. 

Examination of Table 5 immediately reveals that the two anharmonic progressions diverge from the 3.3 p.m origin. One is shifted towards larger wavelengths (CH sym. stretch.) while the other is shifted towards smaller ones (CH asym. stretch.). This behavior can be understood from the shape of the potential as a function of the CH normal coordinates for the two vibrations (Figure 1). As illustrated, the two anharmonic surfaces deviate from the harmonic potentials in different ways. 

The PES is not an analytic function of the atomic coordinates, because it diverges to infinity when any two atoms are at the same position. Using the Znj to describe the PES means that these singularities are banished to infinity, Zn —> oo, resulting in a much better behaved description of the... 

The classical problem is no less daunting The initial state is defined in terms of the 3N coordinates and 3N momenta of the N particles, which together define a 6iV-dimensional phase space. Each point in this phase space defines a specific microstate. Evolution of the system is described by the trajectory of the starting point through phase space. Not only is it beyond the capabilities of the most powerful computer to keep track of these fluctuations, but should only one of the starting parameters be defined incorrectly, the model fatally diverges very rapidly from the actual situation. 

Let M V X tij/) and N = (V x M)/< with p given by Eq. (53), where r is the radius vector of the spherical coordinate system. Then both M and N satisfy the vector wave equation and both have zero divergence. In addition, M and N satisfy... 

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