Hubbard interaction

To the above, we need to add the interactions between electrons. Here we invoke the so-called Hubbard interaction potential U that specifies the potential energy when two electron with reversed spins reside on the same site with probability rj. Again, we neglect the somewhat weaker (but in many cases, not negligible) interactions between electrons on adjacent or more distant sites. The total energy for the simple model under the consideration reads... 

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-... 

Relations (2.46) and (2.47) are equivalent formulations of the fact that, in a dense medium, increase in frequency of collisions retards molecular reorientation. As this fact was established by Hubbard within Langevin phenomenology [30] it is compatible with any sort of molecule-neighbourhood interaction (binary or collective) that results in diffusion of angular momentum. In the gas phase it is related to weak collisions only. On the other hand, the perturbation theory derivation of the Hubbard relation shows that it is valid for dense media but only for collisions of arbitrary strength. Hence the Hubbard relation has a more general and universal character than that originally accredited to it. 

This brief analysis explains why it is very important to know whether the Hubbard relation is reproduced in the liquid cage model. The existence of the Hubbard limit means that orientational relaxation is insensitive to the precise details of the interaction. Below, it is shown that this is the case. 

Hubbard P. S. Theory of nuclear magnetic relaxation by spin-rotational interactions in liquids. Phys. Rev. 131, 1155-65 (1963). 

Orientational disorder and packing irregularities in terms of a modified Anderson-Hubbard Hamiltonian [63,64] will lead to a distribution of the on-site Coulomb interaction as well as of the interaction of electrons on different (at least neighboring) sites as it was explicitly pointed out by Cuevas et al. [65]. Compared to the Coulomb-gap model of Efros and Sklovskii [66], they took into account three different states of charge of the mesoscopic particles, i.e. neutral, positively and negatively charged. The VRH behavior, which dominates the electrical properties at low temperatures, can conclusively be explained with this model. 

First we consider the origin of band gaps and characters of the valence and conduction electron states in 3d transition-metal compounds [104]. Band structure calculations using effective one-particle potentials predict often either metallic behavior or gaps which are much too small. This is due to the fact that the electron-electron interactions are underestimated. In the Mott-Hubbard theory excited states which are essentially MMCT states are taken into account dfd -y The subscripts i and] label the transition-metal sites. These... 

Electron spin resonance, nuclear magnetic resonance, and neutron diffraction methods allow a quantitative determination of the degree of covalence. The reasonance methods utilize the hyperfine interaction between the spin of the transferred electrons and the nuclear spin of the ligands (Stevens, 1953), whereas the neutron diffraction methods use the reduction of spin of the metallic ion as well as the expansion of the form factor [Hubbard and Marshall, 1965). The Mossbauer isomer shift which depends on the total electron density of the nucleus (Walker et al., 1961 Danon, 1966) can be used in the case of Fe. It will be particularly influenced by transfer to the empty 4 s orbitals, but transfer to 3 d orbitals will indirectly influence the 1 s, 2 s, and 3 s electron density at the nucleus. 

This discussion is also applicable to the %-d interacting system with slight modification. The Hubbard Hamiltonian for a pair of re-electron donor and (i-electron magnetic anion is expressed as... 

Liechtenstein AI, Anisimov VI, Zaanen J (1995) Density-functional theory and strong interactions orbital ordering in Mott-Hubbard insulators. Phys Rev B 52(8) R5467... 

Functional Theory and Strong Interactions Orbital Ordering in Mott-Hubbard Insulators. 

In order to perform the functional integrations over the quark fields q and q we use the formalism of bosonisation which is based on the Hubbard-Stratono-vich transformation of the four-fermion interaction. The resulting transformed partition function in terms of bosonic variables will be considered in the mean-field approximation... 

Kuschert, G.S., F. Coulin, C.A. Power, A.E. Proudfoot, R.E. Hubbard, A.J. Hoogewerf, and T.N. Wells. 1999. Glycosaminoglycans interact selectively with chemokines and modulate receptor binding and cellular responses. Biochemistry 38 12959-12968. 

Many electron systems such as molecules and quantum dots show the complex phenomena of electron correlation caused by Coulomb interactions. These phenomena can be described to some extent by the Hubbard model [76]. This is a simple model that captures the main physics of the problem and admits an exact solution in some special cases [77]. To calculate the entanglement for electrons described by this model, we will use Zanardi s measure, which is given in Fock space as the von Neumann entropy [78]. 

In order to take into account these intra-atomic terms, and in a way similar to the Stoner s model, Hubbard ), see also adds to the Hamiltonian (11) an exchange interaction term ... 

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