Self-interactions

Before leaving this chapter, we introduce the concept of interaction by pointing out an interpretation of the second-order character in the model we have been using. Equation 8.34 can be rewritten as 

Focusing our attention for a moment on the term (P, + P,ac,)jc, it can be seen that over a small region of factor space, the first-order effect of the factor Xy is given by the slope , pj + PnJCi,- But this slope depends on the region of factor space one is observing Another way of stating this is that the effect of the factor depends on the level of the factor. 

In a single-factor system, how many possible designs are there for four experiments (e.g., four experiments at a single factor level, three at one factor level and one at another factor level, etc.) For five experiments  

Show by direct calculation that the numerical value in Equation 8.6 is correct. 

From discussions in this and previous chapters, formulate an algebraic equation that gives the number of degrees of freedom associated with lack of fit. 

There is one more problem which is typical for approximate exchange-correlation functionals. Consider the simple case of a one electron system, such as the hydrogen atom. Clearly, the energy will only depend on the kinetic energy and the external potential due to the nucleus. With only one single electron there is absolutely no electron-electron interaction in such a system. This sounds so trivial that the reader might ask what the point is. But 

This term does not exactly vanish for a one electron system since it contains the spurious interaction of the density with itself. Hence, for equation (6-32) to be correct, we must demand that J[p] exactly equals minus Exc[p] such that the wrong self-interaction is cancelled 

We see that the self-interaction error, J[p] + Exc[p], is in all cases in the order of 10 3 Eh or a few hundredths of an eV In addition, the data in Table 6-2 reiterate some of the facts that we noted before. B3LYP, BP86 and BPW91 yield total energies below the exact result of -0.5 Eh, in an apparent contradiction to the variational principle (see discussion in Sec- 

Of course, this self-correction error is not limited to one electron systems, where it can be identified most easily, but apphes to all systems. Perdew and Zunger, 1981, suggested a self-interaction corrected (SIC) form of approximate functionals in which they explicitly enforced equation (6-34) by substiacting out the unphysical self-interaction terms. Without going into any detail, we just note that the resulting one-electron equations for the SIC orbitals are problematic. Unlike the regular Kohn-Sham scheme, the SIC-KS equations do not share the same potential for all orbitals. Rather, the potential is orbital dependent which introduces a lot of practical complications. As a consequence, there are hardly ary implementations of the Perdew-Zunger scheme for self-interaction correction. 

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Perdew J P and Zunger A 1981 Self-interaction correction to density-functional approximations for many-electron systems Phys. Rev. B 23 5048... 

Svane A and Gunnarsson Q 1990 Transition-metal oxides in the self-interaction-corrected density-functional formalism Phys. Rev. Lett. 65 1148... 

Szotek Z, Temmerman W M and Winter H 1993 Application of the self-interaction correction to transition-metal oxides Phys. Rev. B 47 4029... 

Stampfl C, van de Walle C G, Vogel D, Kruger P and Pollmann J 2000 Native defects and impurities in InN First-principles studies using the local-density approximation and self-interaction and relaxation-corrected pseudopotentials Phys. Rev. B 61 R7846-9... 

Perdew J P and A Zunger 1981. Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems. Physical Review B23 5048-5079. 

Secondly, as we shall see shortly, self-interaction processes occur which alter the properties of even a single-particle system with the result that even heuristically Eqs. (11-72) and (11-73) cannot be correct if are defined by Eqs. (11-56) and (11-57). However, before turning... 

This is precisely the form that would be expected if the self interaction gave rise to a change of the mass of the particle from m to AtniS> + m, in which case the change in the energy of the particle would be... 

For gas-liquid solutions which are only moderately dilute, the equation of Krichevsky and Ilinskaya provides a significant improvement over the equation of Krichevsky and Kasarnovsky. It has been used for the reduction of high-pressure equilibrium data by various investigators, notably by Orentlicher (03), and in slightly modified form by Conolly (C6). For any binary system, its three parameters depend only on temperature. The parameter H (Henry s constant) is by far the most important, and in data reduction, care must be taken to obtain H as accurately as possible, even at the expense of lower accuracy for the remaining parameters. While H must be positive, A and vf may be positive or negative A is called the self-interaction parameter because it takes into account the deviations from infinite-dilution behavior that are caused by the interaction between solute molecules in the solvent matrix. 

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). 

There are several things known about the exact behavior of Vxc(r) and it should be noted that the presently used functionals violate many, if not most, of these conditions. Two of the most dramatic failures are (a) in HF theory, the exchange terms exactly cancel the self-interaction of electrons contained in the Coulomb term. In exact DFT, this must also be so, but in approximate DFT, there is a sizeable self-repulsion error (b) the correct KS potential must decay as 1/r for long distances but in approximate DFT it does not, and it decays much too quickly. As a consequence, weak interactions are not well described by DFT and orbital energies are much too high (5-6 eV) compared to the exact values. 

Encl[p] is the non-classical contribution to the electron-electron interaction containing all the effects of self-interaction correction, exchange and Coulomb correlation described previously. It will come as no surprise that finding explicit expressions for the yet unknown functionals, i. e. T[p] and Encl[p], represents the major challenge in density functional theory and a large fraction of this book will be devoted to that problem. 

Table 13-8. Self-interaction error components for Coulomb and exchange energies (Ej + Ex) as well as for the correlation energy (Ec), and the resulting sum for the H atom, the H2 molecule, and the H3 transition structure [kcal/mol]. Data taken from Csonka and Johnson, 1998.

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