Kinetic energy of nuclei

Here the adiahatic approximation is employed (to assume that the kinetic energy of nuclei is zero), and it is assumed that 47reo = 1 for simplicity. R is the internuclear distance, rj2 is the distance hetween the two electrons, and Ri is the distance hetween the ith electron and the ath nucleus. A is the vector potential of the magnetic field that obeys the relation... 

We have not explicitly included the nuclear motions (molecular vibrations) so far, the fact being frequently misunderstood. This is the second stage the lowest sheet of the adiabatic potential surface, Etp, adopts the role of potential energy in solving the electron-nuclear Schrodinger equation containing explicitly the kinetic energy of nuclei... 

A large number of elementary molecular collision processes proceeding via (or in) excited electronic states are known at present. A prominent feature of all these is that as a rule they can not be interpreted (even at a very low kinetic energy of nuclei) in terms of the motion of a representative point over a multidimensional potential-energy surface. The breakdown of the Born-Oppenheimer approximation, which manifests itself in the so-called nonadiabatic coupling of electronic and nuclear motion, induces transitions between electronic states that remain still well defined at infinitely large intermolecular distances. 

This section discusses the energetics of a, p, and y decay, demonstrating how the kinetic energies of the products of the decay can be calculated from the masses of the particles involved. In all cases, it will be assumed that the original unstable nucleus is at rest—i.e., it has zero kinetic energy and linear momentum. This assumption is very realistic because the actual kinetic energies of nuclei due to thermal motion are of the order of kT (of the order of eV), where k is the Boltzmann constant and T the temperature (Kelvin), while the energy released in most decays is of the order of MeV. 

O = Kinetic energy of nuclei 0 = Nuclear-nuclear repulsions = Kinetic energy of electrons O = Nuclear-electron attraction 0 = Electron-electron repulsion... 

It basically unfolds the sum over the expected values of kinetic energies, of nuclei-electrons, of electrons-electrons and of nuclei-nuclei interactions. 

The TB-QC method is constructed using a single- Slater-type basis set for each type of atom. The total energy of TB-QC is the sum of the kinetic energy of nuclei, the occupied molecular orbital energies, the coulomb energy, and a repulsion term ... 

In the adiabatic limit the values of can be evaluated simply through the coupled perturbed Hartree-Fock method and the kinetic energy Ekm is identical with the kinetic energy of nuclei 7. Now the crucial step is coming At the case when the... 

So far we have discussed surface properties for the case where the atoms which constitute a crystal occupy their equilibrium positions. Taking into account of the kinetic energy of nuclei leads to the equation for their vibrational... 

To obtain the Hamiltonian at zeroth-order of approximation, it is necessary not only to exclude the kinetic energy of the nuclei, but also to assume that the nuclear internal coordinates are frozen at R = Ro, where Ro is a certain reference nucleai configuration, for example, the absolute minimum or the conical intersection. Thus, as an initial basis, the states t / (r,s) = t / (r,s Ro) are the eigenfunctions of the Hamiltonian s, R ). Accordingly, instead of Eq. (3), one has... 

Here, the first term is the kinetic energy of the electrons only. The second term is the attraction of electrons to nuclei. The third term is the repulsion between electrons. The repulsion between nuclei is added onto the energy at the end of the calculation. The motion of nuclei can be described by considering this entire formulation to be a potential energy surface on which nuclei move. 

The first two terms represent the kinetic energy of the nuclei A and B (each of mass M), whilst the fourth term represents the kinetic energy of the electron (of mass m). The fifth and sixth (negative) terms give the Coulomb attraction between the nuclei and the electron. The third term is the Coulomb repulsion between the nuclei. 1 have used the subscript tot to mean nuclei plus electron, and used a capital I. ... 

The Fock operator is an effective one-electron energy operator, describing the kinetic energy of an electron, the attraction to all the nuclei and the repulsion to all the other electrons (via the J and K operators). Note that the Fock operator is associated with the variation of the total energy, not the energy itself. The Hamilton operator (3.23) is not a sum of Fock operators. 

The energy from nuclear fission is released mainly as kinetic energy of the new, smaller nuclei and neutrons that are produced. This kinetic energy is essentially heat, which is used to boil water to generate steam that turns turbines to drive electrical generators. In a nuclear power plant, the electrical generation area is essentially the same as in a plant that burns fossil fuels to boil the water. 

If subatomic particles moving at speeds close to the speed of light collide with nuclei and electrons, new phenomena take place that do not occur in collisions of these particles at slow speeds. For example, in a collision some of the kinetic energy of the moving particles can create new particles that are not contained in ordinaiy matter. Some of these created particles, such as antiparticles of the proton and elec-... 

---