Transformation of the kinetic energy

Some examples are given in Figs E.1.1 and E.1.2, for triatomic molecules. A linear triatomic molecule has four (3x3 — 5) vibrational modes two bond-stretching modes and two (degenerate) bending modes. Fig. E.1.2 shows one of the four normal modes 

As shown above, the potential energy can be expressed as a sum of harmonic potentials. We now consider the expression for the kinetic energy in classical as well as quantum mechanical form. In classical mechanics, 

from Eqs (E.12) and (E.10) we see that the classical dynamics of the normal modes is just the dynamics of n uncoupled harmonic oscillators. 

In quantum mechanics the kinetic energy is represented by the operator 

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Impact and Erosion. Impact involves the rapid appHcation of a substantial load to a relatively small area. Most of the kinetic energy from the impacting object is transformed into strain energy for crack propagation. Impact can produce immediate failure if there is complete penetration of the impacted body or if the impact induces a macrostress in the piece, causing it to deflect and then crack catastrophically. Failure can also occur if erosion reduces the cross section and load-bearing capacity of the component, causes a loss of dimensional tolerance, or causes the loss of a protective coating. Detailed information on impact and erosion is available (49). 

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

For droplets of high surface tension, the droplet flattening process may be governed by the transformation of impact kinetic energy to surface energy. In case that this mechanism dominates, the flattening ratio becomes only dependent on the Weber number, as derived by Madej ski by fitting the numerical results of the full analytical model ... 

The way in which local-scaling transformations have been used for the minimization of the kinetic energy functional is as follows [108-111], An arbitrary Slater determinant is selected to be the orbit-generating... 

At the moment of explosion the chemical energy is transformed into thermal and elastic energy, then the thermal and elastic energy are transformed into the kinetic energy of dispersion of the explosion products (EP). Only in the next stage do the EP slow down, giving up their energy to the gas. 

The two delta terms which have been placed side by side encapsulate the main problem with DFT the sum of the kinetic energy deviation from the reference system and the electron-electron repulsion energy deviation from the classical system, called the exchange-correlation energy. In each term an unknown functional transforms electron density into an energy, kinetic and potential respectively. This exchange-correlation energy is a functional of the electron density function ... 

There is still a very important area of contact between LS-DFT and other approaches having to do with the direct evaluation of the Kohn-Sham potentials [6,62-66] from known densities. In this respect, the work, of Zhao, Morrison and Parr [66] is of particular interest as it provides an alternative to local-scaling transformations for a fixed-density variation of the kinetic energy functional. 

Using this result after some transformations the derivative of the kinetic energy integral becomes ... 

Normally the TDSE cannot be solved analytically and must be obtained numerically. In the numerical approach we need a method to render the wave function. In time-dependent quantum molecular reaction dynamics, the wave function is often represented using a discrete variable representation (DVR) [88-91] or Fourier Grid Hamiltonian (FGH) [92,93] method. A Fast Fourier Transform (FFT) can be used to evaluate the action of the kinetic energy operator on the wave function. Assuming the Hamiltonian is time independent, the solution of the TDSE may be written... 

If the recoil is in the direction towards R, part of the kinetic energy E may be transformed by inelastic collision into excitation energy of the recoiling atom 1 and the rest R ... 

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