On-site repulsion

Figure 12 Generation of an antiferromagnetic insulator in the Slater approach, (a) The description of the insulator in terms of occupation of spin up and spin down bands and (b) how the up and down spins on adjacent atoms change as the on-site repulsion is gradually increased. When the on-site repulsion is infinite two electrons are not located on the same atom. Compare this diagram with the bottom left-hand picture of Figure 11.

We have interacting 7r-electrons in the PPP model and on-site repulsion in Hub-... 

So far our discussion has been limited to those electronic states that originate from a normal metallic state, and is therefore appropriate when the on-site repulsion is small compared with the bandwidth (i.e. W > U). When U W, electrons are locahzed on lattice sites. Consider the ID lattice (118) that has one orbital and one electron per site, a typical half-filled band system. When U W, favorable electronic states are those in which each site has one electron with either up or down spin. All these states are insulating in nature, because electron hopping from one site to another leads to a situation in which two electrons reside on a single site, thereby causing on-site repulsion (see 119). Such insulating states resulting from partially... 

Regardless of the type of formulation applied to explain HTS, the strong on-site repulsion set by U in (2) must be included. More puzzling could be the fact that such a term can by itself have the ability of pairing particles and ultimately organizing them into the SC state. In this section we give an account of some of the most influential models in the literature that have successfully surmounted these seemingly contradictory conceptions. 

The Hubbard model can be derived from eqn.(2) by (i) making a zero differential overlap (ZDO) approximation and (ii) by assuming the range of Coulomb interactions to be truncated to just on-site repulsion by virtue of strong shielding of the interactions by the conduction electrons in metals. The ZDO approximation [25] restricts the nonzero electron repulsion integrals to those of the form... 

If the on-site repulsion is large (U>4t), then two electrons cannot be accommodated on the same site, and the band fills up to 7c/b (Fig. 5c). The magnetic lattice period is twice the normal period, and one has a Mott-Hubbard semiconductor, in which Bragg or Umklapp scattering occurs at 4kp. 

Table 5. Charge transfer Z and effective on-site repulsion energy Ueff...

Here Eref = haa + hbb + Jab is the reference energy and has been subtracted from all diagonal matrix elements. Kab is the direct exchange, U is the on-site repulsion parameter and tab is the hopping integral and gives a measure of the probability for the electron hopping from site aiob and vice-versa. 

The observed changes suffered by the parameters upon dressing them with the effects that go beyond the valence space can at least partially be rationalized by looking at the interaction of the model space determinants with those in the external space. The interaction of the spin-conserving Ih-lp excitations with the neutral determinants is (nearly) zero due to Brillouin s theorem. On the contrary, the interaction with the ionic determinants is strong (see the right part of Fig. 5.10). Hence, this class of external determinants largely decreases the on-site repulsion U as previously seen in Exercise 6.7 and confirmed here in the example. 

Fig. 5.10 Effect of the spin-conserving (right) and non spin-conserving (lefi) h- p determinants on the on-site repulsion U and the direct exchange parameter The interactions in a valence-only...

In summary, biquadratic exchange interactions are only expected in complexes with sizeable magnetic interaction, small on-site repulsion U, not too large on-site exchange interaction K and different inter-site interactions for the pairs of electrons on the different magnetic centers K i K2a, fi3 f24-... 

Exercise 5.7 Substituting = —2218 cm and AEst = —362 cm in = Afff I AEst gives a value of 13590 cm" (6.74 eV) for the effective on-site repulsion, a lowering of 19eV with respect to the bare valence-only value. 

The simplest possibility to switch on the mutual interaction between rc-electrons is given by the Hubbard scheme. This model takes into account only the most dominant effect of electron-electron interaction the so-called on-site repulsion. This means that if there is an electron on the site (2p orbital) m, putting an additional electron on this site involves a significant increase in energy as a consequence of the Coulomb repulsion. The on-site repulsion can be incorporated into the 7i-electron Hamiltonian by adding a two-electron term corresponding to each site ... 

The one-electron part of the Hubbard Hamiltonian is mostly simplified in the same manner as in the Hiickel model, cf Eqs. (10.5 — 10.7). Having a homogenous system, the on-site repulsion parameters can also be taken equal. This uniform value is often denoted by U, and within this condition the Hubbard Hamiltonian takes the form ... 

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