Motion, kinetic energy and

The total energy in an Molecular Orbital calculation is the net result of electronic kinetic energies and the interactions between all electrons and atomic cores in the system. This is the potential energy for nuclear motion in the Born-Oppenheimer approximation (see page 32). 

Initializing the initial kinetic energy and temperature of the system it is necessary to start the motion at some level, eg, assume a Boltzmann (random) distribution of atomic velocities, at 300 K. 

Imagine a model hydrogen molecule with non-interacting electrons, such that their Coulomb repulsion is zero. Each electron in our model still has kinetic energy and is still attracted to both nuclei, but the electron motions are completely independent of each other because the electron-electron interaction term is zero. We would, therefore, expect that the electronic wavefunction for the pair of electrons would be a product of the wavefunctions for two independent electrons in H2+ (Figure 4.1), which I will write X(rO and F(r2). Thus X(ri) and T(r2) are molecular orbitals which describe independently the two electrons in our non-interacting electron model. 

Figure 7-3 shows an experimental study of the collision of hard spheres. The experimenter imparts energy of motion to the white ball (see Figure 7-3A, 7-3B). He does so by doing work by striking the ball with the end of a cylindrical stick (a cue). The amount of energy of motion (kinetic energy) received by the ball is fixed by the amount of work done. If the ball is struck softly (little work being done), it moves slowly. If the ball is struck hard (much work being done), it moves rapidly. The kinetic energy of the white ball appears because work was done—the amount of work, IV, determines and equals the amount of kinetic energy, (KE),. In symbols,...

By introducing the simplest semi-classical approximation to the propagators, in which the nuclear motion kinetic energy is assumed to commute with the anion and neutral potential energy functions and with the non BO coupling operators, one obtains... 

The motion on the PES corresponds to an interchange of potential and kinetic energies and asymptotically rearrangement of particles in a reactive collision. For a given / , we can plot the time variation of the coordinates on the top of the PE contours as shown in Fig. 9.25. These are called the trajectories and their behavior would tell as about molecular collisions. 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

The Schrodinger equation for nuclear motion contains a Hamiltonian operator Hop,nuc consisting of the nuclear kinetic energy and a potential energy term which is Eeiec(S) of Equation 2.7. Thus... 

The smaller drops follow. Both liquids accelerate in the holes, because the sum of the cross section of all the holes is less than half the column cross section. However, this motion is retarded within a short distance, whereby a zone of drop compaction results above the trays. These phenomena are modeled based on a balance of maximum and minimum kinetic energy and the cohesive energy of the droplets [1]. After that, the resulting equation for the maximum stable drop diameter in the field of pulsing is ... 

The motion of polydispersed particulate phase is modeled making use of a stochastic approach. A group of representative model particles is distinguished. Motion of these particles is simulated directly taking into account the influence of the mean stream of gas and pulsations of parameters in gas phase. Properties of the gas flow — the mean kinetic energy and the rate of pulsations decay — make it possible to simulate the stochastic motion of the particles under the assumption of the Poisson flow of events. 

Kinetic energy is the energy of motion. Gas particles have a lot of kinetic energy and constantly zip about, colliding with one another or with other objects. The picture is complicated, but scientists simplified things by making several assumptions about the behavior of gas pcirticles. These assumptions are called the postulates of the kinetic molecular theory. They apply to a theoretical ideal gas ... 

Basic anatomy of an electric generator. Electricity is generated in a looped wire as the wire rotates through a magnetic field. This motion causes electrons in the wire to slosh back and forth. Because the electrons are moving, they possess kinetic energy and so have the capacity to do work. 

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