Particle kinetic energy

Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

The electrostatic repulsive forces are a function of particle kinetic energy (/ T), ionic strength, zeta potential, and separation distance. The van der Waals attractive forces are a function of the Hamaker constant and separation distance. 

The third term on the right hand side of this expression is the single-particle kinetic energy of the noninteracting electrons whereas the functional Exc[p contains the additional contribution to the energy that is needed to make Eq. (8) equal to Eq. (6). 

The same conclusion can be reached from energy considerations, noting that the work done by the string tension as r is reduced is equal to the gain in particle kinetic energy, i.e.,... 

The particle theory of matter does not discuss the kinetic energy of particles. Kinetic energy is important, however, when describing the unique properties of gases. 

The Euler equation of the variation problem now involves the functional derivatives of the single-particle kinetic energy density tr corresponding to the exact many-body density p(r), and the exchange and correlation energy density exc. Such a minimization of equation (145) yields immediately... 

As remarked above, considerable progress has resulted from use of the one-body potential of the density description in a one-electron Schrodinger equation approach. In the language of the density description, this is tantamount to treating the single-particle kinetic energy density exactly, as suggested by Kohn... 

If we add and subtract 2/3 of the single-particle kinetic energy from the right-hand-side of this equation, we find... 

As remarked in the main text, since Ts= tr dr is the single-particle kinetic energy, we cannot relate it directly to the total energy E at equilibrium. Rather,... 

See Fig. 3.4 for a pictorial representation of the potential distribution theorem. The inner-averaging weight is a gaussian - corresponding to the free-particle kinetic energy and specified by the path, with no coupling to the solvent particles ... 

Expressing the single-particle kinetic energy Ts as an orbital functional (1.37) prevents direct minimization of the energy functional (1.38) with respect to n. Instead, one commonly employs a scheme suggested by Kohn and Sham [274], which starts by writing... 

One consequence associated with observing elastic collisions in the CM coordinates is that the individual particle kinetic energies are unchanged by the... 

Liquids and solids are called condensed phases (or condensed states) because their particles are extremely close together. Electrostatic forces among the particles, called interparticle forces or, more commonly, intermolecular forces, combine with the particles kinetic energy to create the properties of each phase as well as phase changes, the changes from one phase to another. 

There are several things we need to ej lain. fri the case of particles, kinetic energy is lost by the particle and is transferred to the host lattice via excited phonon states, l.e.- lattice states. This energy is then transferred to the activator center and subsequently appears as excitation and then emission energy. This phenomenon has been extensively studied because of its importance to the TV industry. 

M2T /im + M2) = part of the incident particle kinetic energy available as excitation energy of the compound nucleus Only a fraction of the kinetic energy is available as excitation energy. 

Semiempirical formulas have been developed that give the range as a function of particle kinetic energy. For alpha particles, the range in air at normal temperature and pressure is given by... 

The optimum orbitals have an interesting physical interpretation. Obviously, by construction, they are the components of the best possible single determinant and, as such, they are the solution of a single-particle Schrddinger-like equation, so that an examination of the terms in the Hartree Fock Hamiltonian will tell us a lot about their interpretation. What we might call the parent Hamiltonian of the HF Hamiltonian — the one used in the single-determinant variational method — induces the appearance of the one-particle kinetic energy and nuclear attraction terms in the HF Hamiltonian the difference between the parent and the HF lies in the way in which electron repulsion is represented. 

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