Hubbard interaction potential

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... 

Here gives the fraction of occupied Ca + sites in the membrane just adjacent to the cleft. We can now compute the correction to the interaction potential caused by entry of Ca + ions into the presynaptic membrane, and hence the lowering in the free energy of activation for the MEPP frequencies. (At 305°K a decrease in AC of only 0.02 eV will double the frequency v.) We might say that our theory is based on a model of nonspecific calcium-ion sites. The theory of Hubbard, Jones, and Landau is based on a model of... 

In the presence of interactions between the connected segments of a single chain, aforementioned simple diffusion or random walks get affected and the walks are no more random. However, the intricate coupling of the different components such as monomers, solvent, or small ions in the case of polyelectrolytes via the interaction potentials complicates the theoretical analysis. In order to decouple different components, the conformations of the chain can be envisioned as the walks in the presence of fields, which arise solely due to the fact that there are interactions present in the system. This physical argument is the basis of the use ofcertain field theoretical transformations such as Hubbard-Stratonovich [60] transformation, which is well known in the field theory. So, the conformational characteristics of a polymer chain in the presence of different kinds of intrachain interactions can be described once the fields are known. In general, an exact computation of these fields is almost an impossible task. That is the reason theoretical developments resort to certain approximations for computing these fields, which work well for most of the practical purposes. Once these fields are known, the physical properties can be described in terms of these fields. It was shown by Edwards [50] that the similar analysis can be carried out for systems with many chains, where interchain interactions also affect the properties in addition to intrachain interactions. 

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... 

There is one localized,unpaired spin per TCNQ molecule. This presumably follows from 1. if the disorder is sufficiently great as to give complete localization of the one-electron states to a single site or if one has a Mott-Hubbard metal to insulator transition and is in the strong-coup ling limit. However, as we shall see, one does not necessarily have one unpaired spin per site when the disorder potential and interaction are comparable. 

In this section we derive an effective Hamiltonian that describes the high energy physics associated with particle-hole (or ionic) excitations across the charge gap. The Hamiltonian will describe a hole in the lower Hubbard band and a particle in the upper Hubbard band, interacting with an attractive potential. This attractive potential leads to bound, excitonic states. In the next chapter we derive an effective-particle model for these excitons. A real-space representation of an ionic state is illustrated in Fig. 5.5(b). 

An interesting possibility offered by the realization of the self-assembled crystals discussed above is to utilize them as floating mesoscopic lattice potentials to trap other particles, which can be atoms or polar molecules of a different species. We show below that within an experimentally accessible regime of parameters extended Hubbard models with tunable long-range phonon-mediated interactions describe the effective dynamics of the extra-particles dressed by the lattice phonons. 

The LDA-I-U orbital-dependent potential (7.74) gives the energy separation between the upper valence and lower conduction bands equal to the Coulomb parameter U, thus reproducing qualitatively the correct physics for Mott-Hubbard insulators. To construct a calculation in the LDA-I-U scheme one needs to define an orbital basis set and to take into account properly the direct and exchange Coulomb interactions inside a partially filled d- f-) electron subsystem [439]. To realize the LDA-I-U method one needs the identification of regions in a space where the atomic characteristics of the electronic states have largely survived ( atomic spheres ). The most straightforward would be to use an atomic-orbital-type basis set such as LMTO [448]. 

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