Repulsive Coulomb interaction

The standard quantum chemical model for the molecular hamiltonian Hm contains, besides purely electronic terms, the Coulomb repulsion among the nuclei Vnn and the kinetic energy operator K]. The electronic terms are the electron kinetic energy operator Ke and the electron-electron Coulomb repulsion interaction Vee and interactions of electrons with the nuclei, these latter acting as sources of external (to the electrons) potential designated as Ve]q. The electronic hamiltonian He includes and is defined as... 

The next stage in the defect formation process involves the transfer of energy from the electronic excitation to the lattice. Although the exact details of the necessary excited states which induce the instability are the subject of some controversy, it is known that the basic cause of the transformation is the coulombic repulsive interaction between the electron and the X2 molecule. 

The distance between two electrons at a given site is given as ri2. The electron wave function for one of the electrons is given as (p(ri) and the wave function for the second electron, with antiparallel spin, is Hubbard intra-atomic energy and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. Finally, it should also be noted that the Coulomb repulsion interaction had been introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson, 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov, 1995). Hence, Eq. 7.3 is sometimes referred to as the Anderson-Hubbard repulsion term. 

A simple model for the Verwey transition has been proposed (Honig, Spalek Gopalan, 1990) octahedral sites in magnetite were represented by a site pair, with a ground energy state (an electron trapped), a first excited state (the electron resonating between the two components of the site pair) and a second excited state (two electrons in the site pair). An important characteristic of this model was that the Verwey transition was driven by the coulomb repulsive interaction between electrons in the site pair. 

From our own data, together with other data found in the literature, it is obvious that non electrostatic cohesive interactions are predominant at low degree of ionization and are responsible for the maintenance of very compact structures. In such ionization ranges, the coulombic repulsive interactions are therefore more or less completely screened, depending on the strength of cohesive forces as clearly indicated by the respective behaviour of PM A, partially hydrolyzed N,N-disubstituted polyacrylamide (COPy), PLL and related polyelectrolytes. 

The saturated, short-ranged nature of the attractive nucleon-nucleon interaction creates an approximately uniform mean field inside the nucleus, giving rise to a nearly flat behavior of the nuclear potential. Near the nuclear periphery, the long-range Coulomb repulsive interaction overpowers the short-range nuclear attraction, giving rise to the Coulomb barrier and... 

On the other hand, the two-electron operator represents the coulomb repulsion interaction between the electrons occupying the impurity states... 

Many-electron atoms present a more complicated picture because of the electron-electron interaction terms that enter the Hamiltonians for the Schrodinger equations. These interaction terms are the Coulombic repulsive interaction of like charged particles. This couples the motions of the different electrons, and that precludes separation of variables. Nonetheless, important understanding of the features of the wavefunctions of many-electron atoms can be recognized on the basis of the one-electron atoms we have considered so far. 

A solid is an assembly of charged particles, electrons and atomic nuclei. Because of the difference in mass, atomic nuclei are veiy slow when compared to electrons and their positions can be considered merely as parameters and not as unknown functions of time, solutions of the Schrodinger equatioiL Furthermore, Coulomb repulsive interactions of one electron with others are averaged over the electronic population. These are the two basic assumptions of the band theory, assumptions which are often not entirely justified in the case of ceramic materials (this will be discussed later on). It is therefore sufficient to consider the motion of a single electron placed in an effective potential and to determine its energy levels and states... 

Themiodynamic stability requires a repulsive core m the interatomic potential of atoms and molecules, which is a manifestation of the Pauli exclusion principle operating at short distances. This means that the Coulomb and dipole interaction potentials between charged and uncharged real atoms or molecules must be supplemented by a hard core or other repulsive interactions. Examples are as follows. 

DFT methods compute electron correlation via general functionals of the electron density (see Appendix A for details). DFT functionals partition the electronic energy into several components which are computed separately the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electron-electron interaction (which is itself... 

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. 

Electrons do of course interact with each other through their mutual Coulomb electrostatic potential, so an alternative step to greater sophistication might be to allow electron repulsion into the free-electron model. We therefore start again from the free-electron model but allow for the Coulomb repulsion between the electrons. We don t worry about the fermion nature of electrons at this point. 

Monte Carlo simulations [17, 18], the valence bond approach [19, 20], and g-ology [21-24] indicate that the Peierls instability in half-filled chains survives the presence of electron-electron interactions (at least, for some range of interaction parameters). This holds for a variety of different models, such as the Peierls-Hubbard model with the onsite Coulomb repulsion, or the Pariser-Parr-Pople model, where also long-range Coulomb interactions are taken into account ]2]. As the dimerization persists in the presence of electron-electron interactions, also the soliton concept survives. An important difference with the SSH model is that neu-... 

The second in these polypyridinocrowns [26] again is distinctly lower due to Coulomb repulsion between the positive charges (see p. 67). The larger the ring, the smaller is this interaction pK 2 (dipyridinocrown-12 [26b]) <3, pATa2 (tripyridinocrown-18 [26a]) 3.7, pK 2 (tetrapyridinocrown-24 [26c]) >3. 

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