Self-interaction problem

It may not be obvious that the self interaction problem is present in TDDFT, but it can be made evident by comparing CIS and Tamm-Dancoff TDDFT for an excitation describable by just a pair of orbitals (xai = 1) ... 

In the LSDA and GGA approaches, the Coulomb integrals are calculated on the same footing as in HF theory, but using the Kohn-Sham orbitals. However, the exchange contribuhon is taken from a functional of the density and so the interplay of the Coulomb and exchange energies, which is clear under the HF approach, is not so well balanced in DFT. In particular, using Vxc based on the density in Equahon 8.12 implies that each electron contributes to its own potential, the so called self interaction problem. 

A similar correlation functional shown in eq. (6.48) was proposed somewhat later by A. D. Becke (B95) and is one of the few functionals that does not have the self-interaction problem. ... 

The self-interaction problem is responsible for some of the failures of the LDA and the GGA, namely (i) the too small ionization potentials when calculated from ehomo (ii) the non-existence of Rydberg series (iii) the incapacity to bind extra electrons, thus rendering almost impossible the calculation of electron-affinities (EA). 

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. 

A problem with DFT that is not restricted to intermolecular complexes is what might be called overdelocalization . In part because of problems in correcting for the classical self-interaction energy, many functionals overstabilize systems having more highly delocalized densities over more localized alternatives. Such an imbalance can lead to erroneous predictions of higher symmetry structures being preferred over lower symmetry ones, as has been observed, for instance, for phosphoranyl radical structures (Lim et al. 1996), transition-state structures for cationic [4-1-3] cycloadditions (Cramer and Barrows 1998), and in the comparison of cumulenes to poly-ynes (Woodcock, Schaefer, and Schreiner 2002). It can... 

Clearly this problem is a microstmcture effect resulting from the continuous chain limit, which allows for self-interactions of arbitrarily small pieces of the chain. To avoid it, we introduce a cut off. We impose the rule that segment variables Sj are not allowed to approach each other along the chain on distances smellier than a > 0. In some sense this again introduces a finite segment size l a-1/2. Imposing the same cut off at the chain ends we thus write... 

---