Self-interaction region

The microtubule-associated proteins MAP2 and tau both have two separate functional regions (Lewis et al., 1989). One is the microtubule-binding site, which nucleates microtubule assembly and controls the rate of elongation (by slowing the rate of assembly). The second functional domain shared by MAP2 and tau is a short C-terminal a-helical sequence that can cross-link microtubules into bundles by self-interaction. This domain has some of the properties of a leucine zipper. Likely it is responsible for the organization of microtubules into dense stable parallel arrays in axons and dendrites (Lewis et al., 1989). 

For the example of Figure 8.7, in the region where jc, is small, the effect of Xy is to increase the response yj as Xy is increased in the region where x, is large, the effect of Xy is to decrease the response y, as Xy is increased in the region where a , is approximately equal to 5 or 6, there is very little effect of at, on y,. This dependence of a factor effect on the level of the factor will be called self interaction . 

While the LSD exchange-correlation hole is accurate for small interelec-tronic separations (Sect. 2.3), it is less satisfactory at large separations, as discussed in Sect. 2.5. For example, consider the hole for an electron which has wandered out into the classically-forbidden tail region around an atom (or molecule). The exact hole remains localized around the nucleus, and in Sect. 2.5 we give explicit results for its limiting form as the electron moves far away [19]. The LSD hole, however, becomes more and more diffuse as the density at the electron s position gets smaller, and so is quite incorrect. The weighted density approximation (WDA) and the self-interaction correction (SIC) both yield more accurate (but not exact) descriptions of this phenomenon. 

Actually, the classical charge-cloud repulsion is somewhat inappropriate for electrons in that smearing an electron (a particle) out into a cloud forces it to repel itself, as any two regions of the cloud interact repulsively. One way to compensate for this physically incorrect electron self-interaction is with a good exchange-correlation functional (below). 

The linear photoresponse of metal clusters was successfully calculated for spherical [158-160, 163] as well as for spheroidal clusters [164] within the jellium model [188] using the LDA. The results are improved considerably by the use of self-interaction corrected functionals. In the context of response calculations, self-interaction effects occur at three different levels First of all, the static KS orbitals, which enter the response function, have a self-interaction error if calculated within LDA. This is because the LDA xc potential of finite systems shows an exponential rather than the correct — 1/r behaviour in the asymptotic region. As a consequence, the valence electrons of finite systems are too weakly bound and the effective (ground-state) potential does not support high-lying unoccupied states. Apart from the response function Xs, the xc kernel /xc[ o] no matter which approximation is used for it, also has a self-interaction error. This is because /ic[no] is evaluated at the unperturbed ground-state density no(r), and this density exhibits self-interaction errors if the KS orbitals were calculated in LDA. Finally the ALDA form of /,c itself carries another self-interaction error. 

From Table 1 we see the deficiencies of the LDA method, since it underestimates both total and exchange energies. Such discrepancies are increased for small confinement radii. As we discussed in the introduction, the main difference between LDA and LDA-SIC is the self-interaction error, thus this spurious contribution is more important when the electrons are localized within small regions, as it has been pointed out before [21]. 

It can be noticed how the electron-electron interaction in region TZi contributes to terms of smaller order (ZAf than those retained in this equation. This means that some of the approximations performed (exclusion of exchange in this region and therefore inclusion of nonphysical self interaction) do not play any role in the final results. 

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