Coulomb repulsion of electrons

The integrals describing the Coulomb repulsion of electrons in two HOs centered at the same atom appear only in the form of the reduced repulsion integrals for pairs of... 

Coulomb attraction of nuclei (depends on Rf ) J = Coulomb repulsion of electrons and Exc = XC functional. 

The mutual avoidance of electrons in the helium atom or in the hydrogen molecule is caused by Coulombic repulsion of electrons (described in the previous subsection). As we have shown in this chapter, in the Haitree-Fock method the Coulomb hole is absent, whereas methods that account for electron correlation generate sueh a hole. However, electrons avoid each other also for reasons other than their charge. The Pauli principle is another reason this occurs. One of the consequences is the fact that electrons with the same spin coordinate cannot reside in the same place see p. 34. The continuity of the wave function implies that the probability density of them staying in the vicinity of each other is small i.e.. 

We can see that/or some reason, electron 3 has chosen to be in the vicinity of nucleus b. What scared it so much when we placed one electron on each nucleus Electron 3 ran to be as far as possible from electron 1 residing on o. It hates electron 1 so much that it has just ignored the Coulomb repulsion, of electron 2 sitting on b, and jumped on it What has happened ... 

In 1957, the American physicists J. Bardeen, L. N. Cooper, and J. R. Schrieffer (1957) developed a microscopic theory of superconductivity (the BCS theory), and were subsequently awarded the Nobel Prize in physics in 1972. The BCS theory is applicable to metals and alloys (conventional low-Tc superconductors). The Coulombic repulsion of electrons is thought to be overcome by a phonon-mediated mechanism, whereby two electrons with their spins aligned in opposite directions strongly attract each other (Cooper pairing). Cooper pairs condense to a remarkably stable macroscopic quantum state, described by identical wave functions (see Appendix E). 

The terms in square brackets are to do with the nuclear motion the first two of these represent the kinetic energy of the nuclei labelled A and B (each of mass M), and the third term in the square brackets is the Coulomb repulsion between the two nuclei. The fourth and fifth terms give the kinetic energy of the two electrons. The next four negative terms give the mutual Coulomb attraction between the two nuclei A, B and the two eleetrons labelled 1, 2. The final term is the Coulomb repulsion between electrons 1 and 2, with rn the distance between them. As in Chapter 3, I have used the subscript tot to mean nuclear plus electron. 

Here the indices a and b stand for the valence orbitals on the two atoms as before, n is a number operator, c+ and c are creation and annihilation operators, and cr is the spin index. The third and fourth terms in the parentheses effect electron exchange and are responsible for the bonding between the two atoms, while the last two terms stand for the Coulomb repulsion between electrons of opposite spin on the same orbital. As is common in tight binding theory, we assume that the two orbitals a and b are orthogonal we shall correct for this neglect of overlap later. The coupling Vab can be taken as real we set Vab = P < 0. 

Model calculations for the Cs suboxides in comparison with elemental Cs have shown that the decrease in the work function that corresponds to an increase in the Fermi level with respect to the vacuum level can be explained semi-quantitatively with the assumption of a void metal [65], The Coulomb repulsion of the conduction electrons by the cluster centers results in an electronic confinement and a raising of the Fermi energy due to a quantum size effect. 

Unfortunately, if a single configuration is used to approximate the many-electron wave function, electrons of opposite spin remain uncorrelated. The tacit assumption that electrons of opposite spin move independently of each other is, of course, physically incorrect, because, in order to minimize their mutual Coulombic repulsion energy, electrons of opposite spin do certainly tend to avoid each other. Therefore, a wave function, T, that consists of only one configuration will overestimate the Coulombic repulsion energy between electrons of opposite spin. 

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