Coulomb potential between electrons

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

If V is the total Coulombic potential between all the nuclei and electrons in the system, then, in the absence of any spin-dependent terms, the electronic Hamiltonian is given by... 

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

Here Zg is the number of tt electrons provided by atom is essentially an ionization potential for an electron extracted from in the presence of the part of the framework associated with atom r alone (a somewhat hypothetical quantity), is a framework resonance integral, and is the coulomb interaction between electrons in orbitals < >, and <(>,. The essential parameters, in the semi-empirical form of the theory, are cug, and and from their definition these quantities are expected to be characteristic of atom r or bond r—s, not of the particular molecule in which they occur (for a discussion see McWeeny, 1964). In the SCF calculation, solution of (95) leads to MO s from which charges and bond orders are calculated using (97) these are used in setting up a revised Hamiltonian according to (98) and (99) and this is put back into (95) which is solved again to get new MO s, the process being continued until self-consistency is achieved. It is now clear that prediction of the variation of the self-consistent E with respect to the parameters is a matter of considerable difficulty. 

In the Lamb shift, the Coulomb potential between proton and electron contributes to the commutator in the hydrogen atom, and the commutator with the free Hamiltonian becomes (h2e2/2)V2(l/r), which gives a delta function that is evaluated in the matrix element when written out by completeness as an integral over space ... 

The excited n-electron may tunnel through a potential barrier in the free state of the neighbouring molecule preserving the energy. The probability for tunnel transition is as a rule, more than the probability of the returning to the initial state. Apparently the energy of the potential barrier may be considered equal to the molecule ionization potential. The barrier form depends on the coulomb potential between the electron and positive ion and affinity of the neutral molecule. 

All the quantities entering the last three equations have been already defined above. The Coulomb repulsion between electrons can be explicitly taken into account by the two-electron operator represented by eq. (21). Vo (r,-) represents the potential from the other (frozen) electrons expressed as... 

Hartree s original idea of the self-consistent field involved only the direct Coulomb interaction between electrons. This is not inconsistent with variational theory [163], but requires an essential modification in order to correspond to the true physics of electrons. In neglecting electronic exchange, the pure Coulombic Hartree mean field inherently allowed an electron to interact with itself, one of the most unsatisfactory aspects of pre-quantum theories. Hartree simply removed the self-interaction by fiat, at the cost of making the mean field different for each electron. Orbital orthogonality, necessary to the concept of independent electrons, could only be imposed by an artificial variational constraint. The need for an ad hoc self-interaction correction (SIC) persists in recent theories based on approximate local exchange potentials. 

The analytic form of the first two terms in the Kohn-Sham effective potential (Vrff [p](r)) is known. They represent the external potential (vext which is the nuclear attraction potential in most cases) and Coulomb repulsion between electrons. The second term is an explicit functional of electron density. The last term, however, represents the quantum many-body effects and has a traditional name of exchange-correlation potential. vxc is the functional derivative of the component of the total energy functional called conventionally exchange-correlation energy (Exc[p]) ... 

Here K(r) is the one-particle potential in which the electron moves and VA — r ) is the bare Coulomb potential between the electrons. Fe(rt) is the external space-and time-dependent potential which acts as a source coupled to the electron density operator, a is the spin index, and is such that... 

Furthermore, the relative permittivity (or dielectric constant) ofthe liquid is an important parameter. The probability P of an electron (ef) which is thermalised at a distance r metres from its geminate positive ion (M +) escaping recombination with it is exp(-r/r) where is the distance at which the Coulomb potential between e and M is equal to thermal energy kT and is given by the Onsager expression where e is the elementary... 

As noted in Section 5.2.1, most thermodynamic approaches to lithium intercalation have been focused on the analysis of the chemical potential of lithium ions, under the assumption that the chemical potential of electrons is constant in the metallic intercalation compounds. When the electrons generated by the intercalation reaction are localized, however, the compound remains semi-conducting as intercalation proceeds. In this case, it is necessary also to take into account the coulombic interaction between electrons and ions. 

It contains the one-electron Dirac Hamiltonian hp plus the nuclear potential, V, and the operator Vy = 1/ry for the instantaneous Coulomb interaction between electrons... 

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